Intellectual Mathematics: Recent Episodes
Intellectual Mathematics
Copernicus’s planetary models contain elements also found in the works of late medieval Islamic astronomers associated with the Maragha School, including the Tusi couple and Ibn al-Shatir’s models for the Moon and Mercury. On this basis many historians have concluded that Copernicus must have gotten his hands on these Maragha ideas somehow or other, even though no direct evidence for such transmission has been found. Let us consider the evidence as to whether Copernicus plagiarized these Arabic sources or not.
See PDF slides for figures and references.
Einstein’s theory of special relativity defines time and space operationally, that is to say, in terms of the actions performed to measure them. This is analogous to the constructivist spirit of classical geometry. Transcript Oh no, we are chained to a wall! Aaah! This is going to mess up our geometry big time. Remember what … Continue reading Operational Einstein: constructivist principles of special relativity
Reviel Netz’s New History of Greek Mathematics contains a number of factual errors, both mathematical and historical. Netz is dismissive of traditional scholarship in the field, but in some ways represents a step backwards with respect to that tradition. I argue against Netz’s dismissal of many anecdotal historical testimonies as fabrications, and his “ludic proof” … Continue reading Review of Netz’s New History of Greek Mathematics
A strangely petty review by Victor Pambuccian of my article on Greek operationalism has appeared in zbMATH. Its supposed critiques are as follows. “Although the reader is told that the concept [of operationalism] will be defined in 2.2.1, one doesn’t find a definition there (except for a quotation from P. W. Bridgman [The logic of … Continue reading Reply to Pambuccian on Greek constructivism
In 2014 and 2018 I challenged the common view that Copernicus stole ideas from Maragha astronomers. My arguments were discussed by Nikfahm-Khubravan & Ragep (2019), who are believers in the “influence thesis” that I was challenging. Some people have asked me what I think of their take. Nikfahm-Khubravan & Ragep do not try to refute … Continue reading Reply to Nikfahm-Khubravan & Ragep on Copernicus
Geometry might be innate in the same way as language. There are many languages, each of which is an equally coherent and viable paradigm of thought, and the same can be said for Euclidean and non-Euclidean geometries. As our native language is shaped by experience, so might our “native geometry” be. Yet substantive innate conceptions … Continue reading The “universal grammar” of space: what geometry is innate?
The discovery of non-Euclidean geometry in the 19th century radically undermined traditional conceptions of the relation between mathematics and the world. Instead of assuming that physical space was the subject matter of geometry, mathematicians elaborated numerous alternative geometries abstractly and formally, distancing themselves from reality and intuition.
Transcript
The mathematician has only one nightmare: to claim to have proved something that later turns out to be false. There are thousands of theorems in Greek geometry, and every last one of them is correct. That’s what mathematical proofs are supposed to do: eliminate any risk of being wrong. If this armor is compromised, if proofs are fallible, then what’s left of mathematics? It would ruin everything.
But the nightmare came true in the 19th century. What had been thought to have been proofs were exposed as fallacies. Top mathematicians had made mistakes. Mistakes! Like some commoner. It’s going to be hell to pay for this, as you can imagine.
I’m referring to Euclid’s fifth postulate, the parallel postulate. Euclid’s postulate had rubbed a lot of people the wrong way even since antiquity. It sounds more like a theorem.
The earlier postulates were very straightforward: there’s a line between any two points, stuff like that. Very primitive truths. It makes sense that the bedrock axioms of geometry should be the simplest possible things, such as the existence of a line between any two points.
The parallel postulate, by contrast, is not very simple at all. It’s not a primordial intuition like the other postulates. It states that two lines will cross if a rather elaborate condition is met. That’s the kind of thing theorems say. This particular type of configuration has such-and-such a particular property. That’s how theorems go in Euclid. Very different from the other postulates, which are things like: any line segment can be spun around to make a circle.
So, people tried to prove the parallel postulate as a theorem. Many felt that it should not be necessary to assume the parallel postulate as Euclid had done. The parallel postulate should be a consequence of the core notions of geometry, not a separate assumption.
Many people tried to “improve” on Euclid in this way. From antiquity all the way to the 19th century. Around 1800, people like Lagrange and Legendre proposed so-called proofs of the parallel postulate. Those are big-name mathematicians. Their names are engraved in gold on the Eiffel Tower. Lagrange was even buried in the Panthéon in Paris. Elite establishment stuff.
But even these bigwigs were wrong. Their proofs contain hidden mistakes. It’s astonishing that this was more than 2000 years after Euclid. People tried to improve on Euclid for millennia. And not a few claimed to have succeeded. But the fact is that Euclid was right all along. The parallel postulate really does need to be a separate assumptions, just as Euclid had made it. It cannot be proved from the other axioms, as so many mathematicians during those millennia had mistakenly believed.
The Greeks, you know, they were really something else. It’s so easy to make subtle mistakes in the theory of parallels. History shows that there are a hundred ways to make tiny invisible mistakes that fool even the best mathematicians. Top mathematicians who were never wrong about anything else stumbled on this one issue.
Somehow Euclid got it exactly right. He didn’t make any of those hundred mistakes that later mathematicians did. That’s not luck, in my opinion. Arguably, the Greeks were more sophisticated foundational geometers than even the Paris elite in 1800. Unbelievable but true. Euclid’s Elements is not a classic for nothing. Euclid is not a symbol of exact reasoning because of some lazy Eurocentric birth right. Euclid’s Elements really is that good.
When Euclid made the parallel postulate an axiom, he seems to be suggesting that it cannot be proved from the other axioms. And he was right. But, as I said, many people had a hunch that he was wrong about this. They thought it would be impossible for the other axioms to be true and the parallel postulate not true.
So many mathematicians figured they could prove this by contradiction: Suppose the parallel postulate is false. If we could show that that assumption would contradict other geometrical truths, then the assumption must be false. So this way we could prove that the parallel postulate must be true, by showing that it would be incoherent or impossible for it to be false.
Indeed, it was found that negating the parallel postulate had various strange consequences. For example, if the parallel postulate is false then squares do not exist. Suppose you try to make a square. So you have a base segment, and you raise two perpendiculars of equal length from the two endpoints of the segment. Then you connect the two top points of these two perpendiculars. That ought to make a square. In Euclid’s world it does.
But proving that this really makes a square requires the parallel postulate. If the parallel postulate is false, one can instead prove that this construction does not make a square but rather a weirdly disfigured quadrilateral. Because the last side of the “square” doesn’t make right angles with the other sides. So even though you made sure you had right angles at the base of the quadrilateral, and that the perpendicular sides were equal, the fourth and final side still somehow manages to “miss the mark” so to speak. It makes non-right angles.
It’s as if the sides are sort of bent. It’s as if you had four perfectly equal sticks of wood, but then you stored them carelessly and they were exposed to humidity and so on and they were warped. So now they’re kind of mismatched in terms of length and straightness, and when you try to piece them together to make a square they don’t fit right. They make some wobbly not-quite-square shape.
Doing geometry without Euclid’s parallel postulate feels a bit like that. It’s sort of bent out of shape and nothing fits the way it should anymore.
One person who investigated this was Saccheri. He wrote a big book discussing this misshaped square and other things like that, in 1733. Saccheri felt that he had justified Euclid’s parallel postulate by examples such as theses. The square that’s not a square and other such deformities, Saccheri declared to be “repugnant to the nature of the straight line.”
But one might say that he used this emotional language to compensate or cover up a shortcoming in the mathematical argument. He had indeed showed that if the parallel postulate is false then geometry is weird. Then you have squares that don’t fit, and other things that feel like doing carpentry with crooked wood.
But weird is not the same as self-contradictory. Despite their best efforts, mathematicians could not find a clear-cut proof that negating the parallel postulate led to directly contradictory conclusions. This is why Saccheri had to say “repugnant” rather than contradictory. You only get “repugnantly” deformed squares, not direct contradictions such as 2=1 or a part being greater than the whole. Those things would be logical contradictions and you wouldn’t need emotions like repugnance.
In fact, a hundred years after Saccheri, mathematicians came to accept that this strange non-Euclidean world of the warped wood is not contradictory. It is coherent and consistent. It is merely another kind of geometry. An alternative to Euclid.
People used to shout and scream that all kinds of things were repugnant, such as homosexuality, for instance. That doesn’t really prove anything except the narrow-mindedness of those accusers. Mathematicians had been equally narrow-minded. They had tried to justify the status quo for thousands of years. They had tried to prove that their way of doing things–their geometry–was the only right way. Only in the 19th century did they finally realize that it was much more productive to embrace diversity, to accept all the geometries of the rainbow.
For so many years mathematicians could not get away from the idea that the “straight” squares of Euclid were the only “normal” ones, and that the “repugnant” alternative squares of non-Euclidean geometry were birth defects. But they were wrong. Non-Euclidean geometry is as legitimate as any other. It was a creative watershed shift in perspective in mathematics to finally accept this instead of trying to prove the opposite.
Here’s how Gauss, the greatest mathematician at the time, put it in the early 19th century. Negating Euclid’s parallel postulate “leads to a geometry quite different from Euclid’s, logically coherent, and one that I am entirely satisfied with. The theorems [of this non-Euclidean geometry] are paradoxical but not self-contradictory or illogical.” “The necessity of our [Euclidean] geometry cannot be proved. Geometry must stand, not with arithmetic which is pure a priori, but with mechanics.”
Geometry has become like mechanics in the sense that it is empirically testable. The theorems of geometry are not absolute truths but hypotheses like the hypotheses of physics that have to be checked in a lab and perhaps corrected if they don’t agree with measurements.
For example, Euclid proves that the angle sum of a triangle is 180 degrees. But this theorem depends on the parallel postulate, just as Euclid’s proof reveals it to do. In non-Euclidean geometries, angle sums of triangles will be different. So that’s something testable. Measure some triangles to see which geometry is right, just as you drop some weights or whatever in a physics lab to see which law of gravity is right.
Let me quote Lobachevsky, one of the other discoverers of non-Euclidean geometry. Here’s how he makes this point in his book of 1855: “[Non-Euclidean geometry] proves that the assumption that the value of the sum of the three angles of any rectilinear triangle is constant, an assumption which is explicitly or implicitly adopted in ordinary geometry, is not a consequence of our notions of space. Only experience can confirm the truth of this assumption, for instance, by effectively measuring the sum of three angles of a rectilinear triangle. One must give preference to triangles whose edges are very large, since according to [Non-Euclidean geometry], the difference between two right angles and the three angles of a rectilinear triangle increases as the edges increase.” So you need big triangles to tell the difference, just as the earth is round but looks flat from where we’re standing because we only see a small part of it. In the same way we need big triangles to detect the nature of space. Therefore Lobachevsky recommends that we should use astronomical measurements for this: “The distances between the celestial bodies provide us with a means for observing the angles of triangles whose edges are very large.”
Let’s think about the logical structure involved in the realization that non-Euclidean geometry is possible. It used to be thought that Euclid’s parallel postulate was a necessary consequence of the other axioms. Although Euclid seems to have been wise enough to realize that it was not, others erroneously believed that this was a mistake rather than an insight on Euclid’s part.
So the question is: Does the parallel postulate follow from the other axioms? If the answer is yes, then the way to settle matter is to provide a proof, a deduction, starting from the other axioms and ending up with the parallel postulate. So that would be like adding another theorem to Euclid’s Elements.
On the other hand, suppose the answer is no, the parallel postulate does not follow from the other axioms. How then could we prove that? It’s very different in this case. It is no longer about proving a theorem. Rather it is about proving that something cannot be proved. It’s much more “meta” than just proving a particular theorem.
But here’s how you do such a thing. Consider this analogy. Suppose someone believes that all odd numbers are prime numbers. 3 is prime, 5 is prime, 7 is prime, and so on. So someone has become convinced that all odd numbers are prime numbers, and they set out to prove it. The start with what it means to be odd, and from that information they try to prove that that implies that it must be prime as well.
But this is of course wrongheaded. Trying to prove that being prime follows from being odd is just as futile as trying to prove that the parallel postulate follows from the other axioms of Euclid.
How could we set this mathematician straight? How could we prove that what he’s trying to prove is impossible to prove? The way to do this is not by some general proof, but by a specific example.
Look at the number 9. It’s odd, but it’s still not prime. Because it’s 3 times 3, so not a prime number.
The obvious way to interpret this is to say that the guy was wrong with his hypothesis. The claim that being odd implies being prime is false.
But from a logical point of view it is interesting to look at it in slightly different terms. Let’s not think about it in terms of right and wrong. Logic doesn’t care about right and wrong. Logic cares only about what follows from what. When logic looks at a proposition, logic doesn’t ask: is it true or false? Logic asks: does it follow from a particular set of axioms?
Logic is about entailment relations. What follows from what. Logic doesn’t care what assumptions or axioms you use. It only cares about what follows from those axioms.
So in terms of our example with the odd numbers, we shouldn’t focus on the question “are all odd numbers prime numbers?” Instead, from a logical point of view, the better question is: “does being odd entail being prime?” Or “is primeness a logical consequence of oddness?”
We had a counterexample: the number 9. From the logical point of view, we interpret this a bit differently. Not as proving the falsity of the conjecture, because we’re not interested in true or false. Instead, what the example of 9 shows is that it is not possible to derive the property of being prime from the property of being odd.
When we put it this way, we have an answer to that challenging meta question: How can we prove that it’s impossible to prove something? We just did! It’s impossible to prove primeness from oddness. Because if there was a proof that showed that any odd number must be prime, then that proof would apply to 9, since it’s odd, and it would prove that 9 is prime, which it is not. Therefore no such proof could exist.
It was the same in geometry. People thought the parallel postulate was a logical consequence of the other axioms. The way to prove this wrong is to exhibit an example in which the other axioms are true but the parallel postulate is false. Just as in the number theory case we had to find an example where oddness was true but primeness was false.
This is indeed what happened. Mathematicians discovered something that corresponded to the number 9. This proved the logical independence of the parallel postulate, just as the number 9 proves that primeness is not a logical consequence of oddness.
In the geometry case, the role of the number 9 was played by models of hyperbolic geometry. These are visualizations that prove that there are perfectly coherent worlds in which the parallel postulate is false while all the other axioms of Euclid are true.
Once mathematicians started thinking in these kinds of terms, it turned out to be not so difficult to find models like that. Mathematicians really could have done that a lot earlier. Even hundreds of years earlier, or even in Greek times. It’s a bit of an embarrassment that it took so long.
Imagine how embarrassing it would be to sit around for hundreds of years trying to prove that all odd numbers are prime numbers, and ranting about how the very idea of an odd non-prime is “repugnant to the nature of an odd number” only to then discover that, whoops, actually there’s a pretty straightforward counterexample right there, the number 9.
The mistake mathematicians made in geometry was of course not quite so glaring but still in a way it was quite similar. The counterexamples were not that difficult to find. Once mathematicians opened their minds to the possibility of such counterexample, they found them fairly easily.
Mathematicians had missed these rather simple counterexamples for thousands of years because of their closed-minded perspective and preconceived notions. Mathematicians had relied too much on emotions, intuitions, such as repugnance. And they had assumed that there can only be one reasonable geometry, because geometry must correspond to physical space.
Mathematicians could not afford to make those mistakes again. These mistakes are what made the nightmare come true, namely that what mathematicians had thought they had “proved” was actually false.
It was a time for soul searching and repentance. And the lessons from this whole embarrassment were quite obvious. The sources of error were intuitions, such as feelings about how straight lines “should” behave, as well as the notion that geometry means the geometry of the physical space around us.
Those ideas were the losers of the story. The winner was logic. The breakthrough had come by detaching geometry from intuition and reality. By abstracting geometry away to its logical structure only. That was the winning perspective.
To spell out what this means for geometry and its relation to the world, let me quote Einstein’s essay Geometry and Experience. Einstein wrote this is 1921, but he is really just summarizing a standard consensus that had been firmly established decades earlier. But why not use the words of the famous Einstein, they are as good as any to make this point. Here’s what Einstein says:
“How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality? Is human reason, then, without experience, merely by taking thought, able to fathom the properties of real things? In my opinion the answer to this question is, briefly, this: as far as the propositions of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality. It seems to me that complete clarity as to this state of things became common property only through that trend in mathematics, which is known by the name of ‘axiomatics’.” That’s Einstein’s word for what I called the logic perspective. Same thing. Einstein continues:
“Let us for a moment consider from this point of view any axiom of geometry, for instance, the following: through two points in space there always passes one and only one straight line. How is this axiom to be interpreted in the older sense and in the more modern sense?”
“The older interpretation [is]: everyone knows what a straight line is, and what a point is. Whether this knowledge springs from an ability of the human mind or from experience, from some cooperation of the two or from some other source, is not for the mathematician to decide. He leaves the question to the philosopher. Being based upon this knowledge, which precedes all mathematics, the axiom stated above is, like all other axioms, self-evident, that is, it is the expression of a part of this a priori knowledge.”
“The more modern interpretation [is]: geometry treats of objects, which are denoted by the words straight line, point, etc. No knowledge or intuition of these objects is assumed but only the validity of the axioms, such as the one stated above, which are to be taken in a purely formal sense, that is, as void of all content of intuition or experience. These axioms are free creations of the human mind. All other propositions of geometry are logical inferences from the axioms (which are to be taken in the nominalistic sense only). In axiomatic geometry the words ‘point’, ‘straight line’, etc., stand only for empty conceptual schemata. That which gives them content is not relevant to mathematics.”
“‘Practical geometry’ [arises if we] add the proposition: solid bodies are related, with respect to their possible dispositions, as are bodies in Euclidean geometry of three dimensions. Then the propositions of Euclid contain affirmations as to the behavior of practically-rigid bodies. All length-measurements in physics constitute practical geometry in this sense, so, too, do geodetic and astronomical length measurements, if one utilizes the empirical law that light is propagated in a straight line, and indeed in a straight line in the sense of practical geometry. I attach special importance to the view of geometry, which I have just set forth, because without it I should have been unable to formulate the theory of relativity. From the latest results of the theory of relativity it is probable that our three-dimensional space is approximately spherical, that is, that the laws of disposition of rigid bodies in it are not given by Euclidean geometry, but approximately by spherical geometry.” That is to say, the angle sums of triangles are more than 180 degrees.
So all of that I quoted from Einstein. But Einstein speaks for basically the entire mathematical community here. He is describing what was, in his time, the standard view that almost everyone took for granted.
Indeed, these points about mathematics turning to pure axiomatics and so on, apply not only to geometry but to mathematics as a whole. Mathematicians took that lesson to heart and never looked back, basically. So the discovery of non-Euclidean geometry was the birth of modernity, you might say, in mathematics. It led mathematicians to conceive their field exclusively in terms of logic and formalism, and forget everything about intuition or the idea that mathematics is linked to physical reality. And that’s pretty much where we are today, almost two centuries later, with few exceptions.
In the 19th century, you could be forgiven for thinking that this was a case of straightforward progress. Mathematicians had simply discovered the right way to do mathematics, or the best way known so far anyway. The new logic perspective was simply better than the old intuitive or empirical stuff. We shedded the old errors like so many superstitions and became enlightened.
Around 1900, that was a pretty credible narrative. The logic perspective had gone from win to win, and done a clean sweep of mathematics. Everything it touched seemed to become instantly clearer and better. Hilbert was a leading mathematician at this time who may be taken as a symbol of this. He turned from field to field and made everything clear and clean and modern with this logical Midas’ touch.
But the winning streak did not last forever. With one knock-out win after another behind him, Hilbert turned to the foundations of the entire subject of mathematics and tried to do the same trick there. Many people were optimistic. The trick had worked every time before, and now the world’s greatest mathematician was going to use it to definitively settle all the questions of the foundations of mathematics, such as proving that mathematics is consistent.
But the trick broke this time, even though it had worked every time before. Hopes of a quick victory proved as delusional as the equally hubristic delusions of the war planners who were marching into the First World War at the same time.
The world came crashing down around the great Hilbert. He was German, and these were not good times for Germany. First the students and younger generation died in the war. Then the many prominent Jewish faculty were driven out of the country. Hilbert’s once vibrant university was quickly turned into a shadow of its former self. Hilbert himself contracted a rare decease for which the only treatment was eating lots and lots of raw liver every day.
1933 was a year of not one but two disasters. The Nazis took power, but there was an equal blow in the world of mathematics, when Gödel proved that the logician’s dream was impossible. Logical formalism could not prove its own consistency. In other words, the program of detaching mathematics from intuition and experience turned out to be inherently limited. Its utopian dream proved to be unreachable, and demonstrably so in fact.
Kant used a beautiful analogy that is relevant here. It goes like this:
“Deceived by the power of reason, we can perceive no limits to the extension of our knowledge. The light dove cleaving in free flight the thin air, whose resistance it feels, might imagine that her movements would be far more free and rapid in airless space.”
Which is of course not true. The dove may think that air causes nothing but resistance, but if all air is removed, the dove would quickly be taught a different lesson of course. Not only would the dove crash to the ground at once, it would also suffocate in seconds.
A similar fate awaited the movement to purge mathematics of intuition and physical content. People like Hilbert were so keen to remove the old dependence on intuition and the physicality of geometry as if these things were nothing but “air resistance” that prevented the flight of pure logic in a perfectly clean vacuum.
But birds cannot fly without air, and neither could mathematics. Gödel’s theorem of 1933 proved that logical formalism cannot prove it own consistency, which in terms of this analogy is like proving that the dove cannot fly in a vacuum.
This setback within mathematics was perhaps just as unnerving to Hilbert and other mathematicians as all those jarring disasters that were piling on in the outside world. It’s cruel joke of history that it had both these worlds collapse at the same time.
Maybe the parallel extends further. World War One was a horror of horrors, but that didn’t prevent us from doing it all over again soon thereafter. And we still don’t know how to get rid of war.
Mathematics has a similar attachment to formalism and logic. As with war, the romantics among us are not too happy about formalistic mathematics. Its power cannot be denied. Some, or maybe even many, of its victories were for the best. But still it does not feel right in one’s heart to drill young people into an army of formalists. Seeing mathematics as nothing but logical inferences from arbitrary axioms is as heartless as realpolitik. It reigns to this day, despite a now checkered record, because the only alternatives are hippie fantasies with no realistic prospects of ruling. Modern mathematics and modern politics are alike in this regard.
Well, that makes for a bleak ending. Perhaps non-Euclidean geometry does not deserve to be associated with all this misery. It’s not non-Euclidean geometry’s fault that mathematicians had made mistakes about the parallel postulate. Nevertheless the impact of the discovery of non-Euclidean geometry on the mathematical psyche was dramatic and long-lasting. It sent mathematicians on a soul-searching bender, the hangover of which is still felt today.
Kant developed a philosophy of geometry that explained how geometry can be both knowable in pure thought and applicable to physical reality. Namely, because geometry is built into not only our minds but also the way in which we perceive the world. In this way, Kant solved the applicability problem of classical rationalism, albeit at the cost of making our perception of the world around us inextricably subjective. Kant’s theory also showed how rationalism, and philosophy generally, could be reconciled with Newtonian science, with which it had been seen as embarrassingly out of touch. In particular, Kant’s perspective shows how Newton’s notion of absolute space, which had seemed philosophically repugnant, can be accommodated from an epistemological point of view.
Transcript
Rationalism says that geometrical knowledge comes from pure thought. Empiricism says that it comes from sensory experience. Neither is very satisfactory, because geometry is clearly both: it’s too good a fit for the physical world to be only thought, and it relies too much on abstract proofs to be only experience.
So it seems the “right” philosophy of mathematics must be a little bit of both. But how? Rationalism and empiricism mix like oil and water. They are so different, so opposite, that it seems impossible to find any sort of middle ground that has half of each.
Nevertheless there is a golden mean of sorts. The philosophy of Kant. Immanuel Kant, the late 18th-century philosopher. His view of geometry is in a way the best of both worlds: combining the best rationalism with the best of empiricism. Let’s see how he pulled that off.
Kant’s theory is another innateness theory. Geometry is innate; it is hardwired into our minds. That’s what the rationalists said too, of course. Innateness was why the uneducated boy in Plato’s Meno could reach substantial geometrical insights without instruction. The innate intuitions of geometry are reliable because God is not a deceiver, said Descartes. And they apply to the physical world, because the Creator put into our minds the same ideas he had used to design the universe, said Kepler and others.
But Kant’s innateness is different. He lived a century and a half after Kepler and Descartes, and the reliance on God to support the rationalist worldview had become much less fashionable during the intervening years. And indeed Kant does away with it.
By taking geometry to be innate, Kant automatically inherits the best of rationalism. That is to say, he is able to account for the prominent role of pure reason in geometry in a convincing way, just as the earlier rationalists had done before him.
But Kant solves the weakness of rationalism very differently. If geometry is innate and susceptible to purely theoretical elaboration, then why does it always agree so well with experience? Not because “God was a geometer,” as Plato and Kepler said. No, Kant’s solution is: Not only are geometrical principles innate in our thoughts, they are also innate in our perceptions.
Aha, plot twist! The so-called success of mathematics in the real world is no miracle. It’s a rigged game. Our eyes and our senses are biased. They can only see Euclidean geometry.
It’s an illusion to think that we can compare our mathematical deductions with reality. We think we can “test” whether theoretically established theorems are true or not in the physical world; for example, by measuring the sides of right-angle triangles and comparing the results to what Euclid says it should be. But we were naive to think that these two things were really independent. Just as our thoughts are shaped by our innate intuitions, so also our perceptions of physical reality are shaped by the same intuitions.
We don’t have direct access to “raw data” about the physical world. When we think we observe the world, we really observe an interpreted version of the world. The mind can only process information that is interpreted or converted to fit a particular format. In terms of geometry, that means Euclidean geometry.
We see the world through Euclid-colored glasses. Like those pink sunglasses that John Lennon used to wear. Of course if you wear pink glasses, then everything looks pink. But of course John Lennon could not say: aha, I told you, the world is rosy; I argued theoretically that the world should be rosy, and now I have confirmed by observation of the actual world is in fact rosy, because look how everything is pink. John Lennon would be fooling himself if he argued that way.
Euclidean geometry is exactly like that, according to Kant. With experience and observation, we do not discover that Euclidean geometry applies to the real world. What we really observe is our own biases. We discover not facts about the world, but facts about what kind of glasses we are wearing.
John Lennon made the world rosy by putting on his glasses, and in the same way we make the world Euclidean by a hidden lens in the mind’s eye.
Think of a chili pepper. This pepper proves that Kant was right. How so? Think about it. What are the characteristics of a chili pepper?
It’s red. What is red? Is the pepper “really” red? That is to say, is its redness an objective property of reality? Well, yes and no, right? The way science accounts for colors is in terms of wave length of light. What we in our minds regard as red correspond in reality to a particular frequency of light waves.
So there’s something there corresponding to red, but our minds put a major interpretative spin on it. Red is really just a wavelength, “just as number,” so to speak. But our minds turn it into something more qualitative.
And the mind is very selective as well. The remote control of your TV sends its signal using infrared light, which the minds chooses not to see at all. Even though infrared light is exactly the same kind of thing as the color red from the pepper. They are exactly the same kind of waves, only with slightly different frequencies. “Objectively” they are very similar. But subjectively, in our minds, they couldn’t be more different. One is the color red that is one of the fundamental categories in terms of which we navigate reality, and the other is completely invisible.
So perception is very far from direct access to “objective” raw data. The mind interferes very heavily: it selects, transforms, distorts. Maybe when we think Euclidean geometry fits reality it’s only because of this. Only because things that don’t fit a Euclidean mold are as invisible to us as infrared light.
The chili pepper is also very hot to taste. Again, is that an objective property of pepper itself, or is it just a subjective matter of how it interacts with our tongues?
In fact, birds eat chili peppers and to them they are not hot. Chili peppers evolved to be hot because it’s better to be eaten by birds than to be eaten by mammals. The seeds spread better that way. So chili peppers developed this characteristic of being repellently hot to mammals like us, but but perfectly appetizing to birds.
Could it be the same with our geometrical intuitions? “Two lines cannot enclose a space,” Euclid says. The very thought is repellent! Well, maybe that’s just our subjective experience, just as we find it repellent to bite into a hot pepper. Maybe other creatures have completely different geometrical intuitions. Just as birds think pepper are not hot, maybe they also think space has five dimensions or whatever. Who knows.
Different animals can have all kinds of different intuitions and modes of perception that fits the way they navigate the world, the way they find food, and so on. Maybe Euclidean geometry is just something that happened to be convenient to us, just as night vision is convenient to a cat, or a heightened sense of smell is to a dog.
That would make geometrical experience quite subjective. Or at least subjectively contaminated, or entangled with subjective factors.
This is how Kant is able to bridge the gap between rationalism and empiricism. We can sit in a closed room and figure out in our heads geometry that then turns out to be true in the real world. Because, says Kant, what is really happening is that the mind is analyzing itself. What we discover through geometrical meditation is not objective facts, but facts about the geometrical preconceptions hardwired into our minds and the consequences of those conceptions. And these results apply to the world not because they are really out there but because our perception actively processes and converts and interprets any sensory input in Euclidean terms.
As Kant himself says: “Space is not something objective and real; instead, it is subjective and ideal, and originates from the mind’s nature as a scheme, as it were, for coordinating everything sensed externally.” “It is from the human point of view only that we can speak of space, extended objects, etc.”
In the introduction I framed the issue in a way that is flattering to Kant. I said: Everybody knew that you needed some kind of middle ground between rationalism and empiricism, but no one could figure out how to do it. Until Kant, he finally cracked the nut.
But maybe that’s the wrong way to look at it. It often is, this kind of narrative. “Everybody tried to do such-and-such but no one could do it, until finally one guy was smart enough to figure it out.” That’s not usually how history works. If we look into the details we often find that contextual factors explain why key steps happened at certain times.
It’s true that everyone knew that neither rationalism nor empiricism were perfect for hundreds of years. And Kant’s proposal is certainly a very clever way of tackling that problem. So why did it take all the way to the late 18th century before these ideas were proposed?
We already mentioned some relevant factors. Kant makes geometrical knowledge in a sense subjective. That’s a major disappointment, one might say. Most philosophers had certainly hoped to be able to defend a much grander claim. Kant “solves” the rationalism-empiricism problem only by as it were belittling geometrical knowledge, which is a very high price to pay.
The main alternative, as we have seen, was to give God a major role in epistemology. So there’s a trade-off: either you pin geometry to God and you can have it be the most amazing thing, the most perfect knowledge, or else you detach it from God and make it stand on its own legs, but then it’s a lot weaker; it’s a mere subjective human thing and no longer this almighty pinnacle of pure intellect.
The exchange rate, as it were, between these two options fluctuated over time. As God became less popular, the cost of switching to Kantianism went down.
But there’s another reason too why Kant’s theory made more sense in the 18th century than in the 17th. Namely what we said before about how Newton’s science was a blow to rationalism.
We spoke about how that was the case. Rationalism requires knowledge to be generated from within the mind. All knowledge needs to be gradually built up from the most simple intuitions, according to the rationalist point of view. In geometry, that meant ruler and compass and other tools for generating geometrical objects. In physics, it meant contact mechanics; that is to say, seeing complex physical phenomena as an aggregate of lots and lots of little collisions of bodies.
Newton’s physics cannot be reduced to contact mechanics. Or to any other simple intuition. It is in fact counterintuitive. So it cannot be generated from within the mind, through an elaboration in thought of the most undoubtable truths. This is why Newtonian physics is a problem for rationalism.
But the story is a bit more general than that. In fact, Newton’s physics can be seen as a blow to philosophy altogether.
From the rationalist point of view, philosophy comes before science. You start with general philosophical thought. “I think therefore I am”: That’s a very general philosophical truth, and you start there because it’s the most knowable. You start by asking yourself what kinds of things are knowable. From that starting point you arrive at the idea that in physics one of the most primitive knowable things is the contact mechanics of bodies.
From this point of view, philosophy is the boss of science. Philosophy is telling science what to do. Before even starting on science, you have already determined through introspection and meditation what the primitive intuitions of physics are. Any science that follows needs to conform to these predetermined rules that philosophy has established beforehand.
From a rationalist point of view, this makes sense. If knowledge fundamentally comes from within the mind, it makes sense to work from the inside out; to start with the most general philosophical core and then build on that to get to things like physics and other stuff that are more connected to the outside world. That’s a core commitment of the rationalist worldview. This is why it requires philosophy to be prior to science, and the boss of science.
Newton does it the other way around. To him, science is the boss of philosophy. This is a natural consequence of his empiricist, “reading backwards” mindset that we have emphasized before. Thought starts not with inward reflection on our basic intuitions, but in the wild jungle of complex phenomena. Science reasons as it were backwards from there to discover the basics principles, such as axioms of geometry and fundamental laws of physics.
If you continue this process one further step you get to philosophy. Just as the laws of physics are whatever is needed to explain the phenomena, so the principles of philosophy are whatever is needed to make that physics possible. So philosophy is subordinated to science. It doesn’t tell science what to do, but the other way around.
To the rationalists, philosophy set the ground rules that science must obey. To the empiricists, to Newton, philosophy merely describes what assumptions are necessary for science after science has already been established. To the rationalists, philosophy is prescriptive: it gives orders, it says how science has to be. To the empiricists, philosophy is descriptive: it’s an observer, a backseat journalist, that merely says how science is, without having any influence over it.
So we see how the basic outlooks of rationalism and empiricism imply these opposite views of the relation between science and philosophy. And Newton’s physics was extremely successful. So its success lent credibility to the empiricist outlook overall, including the demotion of philosophy.
But in fact this is still not the end of it. There is yet another respect in which Newton’s physics dealt an additional death blow to philosophy. Namely on the issue of absolute versus relative space.
Newton clashed with Descartes and Leibniz on this issue as well. It goes like this. What can we know about the spatial properties of a body, such as its position and velocity?
Descartes and Leibniz were relativists about space. Everything we could ever know about positions and velocities of bodies is relative. That is to say, you can only specify the position or speed of a body by comparing it to another body. The chair is so-and-so far from the table. The train is moving away from the station at such-and-such a speed. You cannot speak of the position of the chair or the speed of the train without comparing it to something. You need to relate it to some reference point.
Descartes and Leibniz insisted on this. Here’s how Descartes puts it: “The names ‘place’ or ‘space’ only designate its size, shape and situation among other bodies.” “So when we say that a thing is in a certain place, we understand only that it is in a certain situation in relation to other things.” Leibniz agreed. “Motion is nothing but a change in the positions of bodies with respect to one another, and so, motion is not something absolute, but consists in a relation.”
It takes two to tango, and it takes two bodies to be able to speak of position and velocity. Because you can only describe the position or velocity of the second body by using the first as a reference point.
If there was only one body in the universe, it wouldn’t make any sense to ask whether it was moving or not. Since there’s nothing to use as a reference point, the very concept of motion becomes meaningless is such a situation. According to the relativist conception of space.
This fits very well with our previous emphasis on operations in geometry. Relative positions and relative velocities correspond very well to operations. You can specify what it means for one object to be so-and-so far from another object, or moving with such-and-such a speed with respect to the other object, in terms of concrete measurements. I take a measuring tape, I stretch it from one to the other, that’s how far apart they are.
If there is only one object in the universe, there is no operation we can perform to check whether it is moving or not. So to introduce the idea of every body having some absolute state of motion, independently of any other body, is equivalent to introducing concepts by means other than operations. We know from geometry that this is dangerous, as we saw with the superright triangle and other examples.
Yet Newton does exactly this. Newtonian physics presupposes absolute space. That is to say, it assumes that every body has some definitive position and velocity, completely independently of any other body, and completely independently of what is measurable or knowable to us.
So from the Newtonian, absolutist point of view, if there is only a single object in the universe, then that object still has some definite velocity. It’s either moving or not. Whether it’s moving or not is physically undetectable. There is no way to tell, with a physical experiment, whether it is moving or not. Nevertheless, the question of whether it is “really” moving or not still makes sense and has a definite answer, according to Newton.
This notion–that any body has an “absolute” position and velocity–is necessary for Newton’s physics. Think of the law of inertia. It says: If there is no outside force acting on a body, then the body keeps going in a straight line with the same speed. Forever. Like a metal ball rolling on a marble table, when there is no friction and no obstacles, it keeps going with the same velocity. Without external influence, the state of motion remains the same.
But note that this law talks about the state of motion of a body without reference to other bodies. The law of inertia presupposes that the body has some inherent velocity, a true velocity. That’s the thing that stays the same in absence of interference. Obviously this is not dependent on some particular reference point. The body in and of itself has a state of motion associated with it. The state of motion of the body is an absolute property, not a relative one.
This clash between the absolute and relative space points of view is another clash between science and philosophy. Relative space is clearly the “best” view in terms of philosophy. The philosophical objections to absolute space are very compelling: Absolute space is unknowable. Absolute space introduces concepts that are empirically untestable, unverifiable, unoperationalisable.
The reply from the other side, from Newton’s side, is not to dispute that philosophy is on the side of relative space. Instead it is to belittle the authority of philosophical arguments. Indeed, absolute space makes no sense philosophically. But the conclusion from this is: tough break for philosophy.
Absolute space is a necessary precondition to state the law of inertia, and the law of inertia is an integral part of Newton’s extremely powerful physics, so inertia and hence absolute space must be accepted. Philosophy is just going to have to deal with it.
So this once again reinforces Newton’s point that philosophy is basically a spectator sport. Philosophy can’t tell science what to do. If philosophy clashes with science, as it does regarding absolute space, then philosophy has to give way.
Physicist Stephen Hawking famously declared that “philosophy is dead.” He had in mind 20th-century developments. That’s how many modern scientists think. But philosophy was dead once before. Newton killed philosophy.
If you want to get somewhere in science and mathematics, you can’t get caught up in pointless speculations and debates about “what it all means.” You just have to do the math, get on with it. That was the case in the 18th century, and again in the 20th century.
Another prominent modern physicist, Lee Smolin, put it as follows: “When I learned physics in the 1970s, it was almost as if we were being taught to look down on people who thought about foundational problems. When we asked about the foundational issues in quantum theory, we were told that no one fully understood them but that concern with them was no longer part of science. The job was to take quantum mechanics as given and apply it to new problems. The spirit was pragmatic; ‘Shut up and calculate’ was the mantra. People who couldn’t let go of their misgivings over the meaning of quantum theory were regarded as losers who couldn’t do the work.”
It was exactly the same thing in the 18th century. Then too scientists and mathematicians figured they were better off just ignoring philosophy. And with good reason since Newton’s physics was an obvious winner in terms of mathematics and science, but a complete non-starter philosophically according to many.
The greatest mathematician and physicist of the 18th century, Euler, realized this perfectly well. He knew that absolute space was junk philosophy but essential to science.
He knew that the law of inertia demanded absolute space. As Euler says: “For if space and place were nothing but the relation among co-existing bodies, what would be the same direction? Identity of direction, which is an essential circumstance in the general principles of motion, is not to be explicated by the relation of co-existing bodies.”
Euler also knew that there were powerful philosophical objections to absolute space. The objections of Descartes and Leibniz that I already mentioned. Let me quote here how Ernst Mach later made the same point in the late 19th century. Mach is basically reviving the 17th-century criticism of absolute space. Here’s how Mach puts it:
“Absolute space and absolute motion are pure things of thought, pure mental constructs, that cannot be produced in experience. [They have] therefore neither a practical nor a scientific value; and no one is justified in saying that he knows aught about it. It is an idle metaphysical conception. All our principles of mechanics are experimental knowledge concerning the relative positions and motions of bodies. No one is warranted in extending these principles beyond the boundaries of experience. In fact, such an extension is meaningless, as no one [can] make [any] use of it.”
Euler and others in the 18th century were aware of this problem with the notion of absolute space that is so essential to Newtonian science. They didn’t know how to solve this philosophical problem, except to ignore philosophy altogether. Euler pretty much says so. Listen to this quote:
“I do not want to enter the discussion of the objections that are made against the reality of [absolute] space and place; since having demonstrated that this reality can no longer be drawn into doubt, it follows necessarily that all these objections must be poorly founded; even if we were not in a position to respond to them.”
So Euler admits that he cannot answer the philosophical objections. Instead his solution is: forget philosophy. Philosophy became obsolete with the Newtonian revolution in science. It was out of touch.
Kant is the savior of philosophy. Kant makes philosophy relevant to science again, after a century of being obsolete. Kant’s theory is a way to bring philosophy up to date with science. It is a philosophy that is compatible with Newtonian science, unlike earlier versions of rationalism.
Against this background we can understand why Kant was willing to make mathematical knowledge subjective. That part of his theory was a huge betrayal of a major tenet of classical rationalism. But times had become desperate enough. Philosophy was the laughing stock of scientists. It had to do something, anything.
So Kant decided to bite the bullet on subjectivity in order to at least salvage something of philosophy. Save what can be saved.
Rationalism had once been a mighty kingdom, but it was bleeding territory. Newton’s science was taking the world by storm, and it seemed a real risk that rationalism would not only lose ground but might even be wiped off the map altogether.
Kant’s plan for saving rationalism shows how far it had fallen. In its glory days, rationalism would have scoffed at the notion that geometry is subjective. But now, it was that or death. Like royalty eating peasants’ porridge, rationalism had to adapt or die. Rationalism had to sacrifice the pride of its forefathers–the objective truth of geometry.
But despite this humiliating concession, Kant’s reinvention of rationalism was an astonishing success. Rationalism was back with a vengeance.
Not only was rationalism no longer obsolete or out of touch with science, it was even ahead of the game. Kant had not only stopped the rot but even brought rationalism back on the winning side. Kant’s account not only showed that some parts of classical rationalism could be saved; it also provided the best available account of how the success of Newtonian science could be explained philosophically. Where people like Euler had merely given up on philosophy because of the magnitude of the problems it faced, Kant had shown that philosophy could answer the challenge and more. Philosophy was relevant again. Philosophy was no longer dead.
Rationalism says mathematical knowledge comes from within, from pure thought; empiricism that it comes from without, from experience and observation. Rationalism led Kepler to look for divine design in the universe, and Descartes to reduce all mechanical phenomena to contact mechanics and all curves in geometry to instrumental generation. Empiricism led Newton to ignore the cause of gravity and dismiss the foundational importance of constructions in geometry.
Transcript
Here’s a fundamental problem in the philosophy of mathematics. You can sit in an isolated room, in an arm chair, and prove theorems about triangles, such as the angle sum of a triangle or the Pythagorean theorem. When you do this, you have the feeling that you have established these results with absolute certainty. You feel that they must be true because of how compelling the proof is. And you feel that you have established this by thought alone, by purely intellectual means.
Mathematics is unique in this respect. In other subjects, thinking is a powerful tool, but it is always supplemented by observation and experience. If you spent your whole life isolated in a locked room, you would not be able to say anything about the laws of astronomy or the anatomy of the digestive system, because without observation, with only pure thought, it is impossible to even get started in those field. But you could figure out everything about triangles. If one day you were released from your prison where you had been sitting for decades, you could go out and measure actual triangles and you would find that, indeed, their angle sum is always two right angles, the Pythagorean theorem always holds for right-angle triangles and so on. Just as you had predicted by pure thought.
This is a bit of a mystery. Because it shows that there are two sides of mathematics that are difficult to reconcile. On the one hand, the internal, mental conviction that mathematics establishes absolute truths purely by reasoning. On the other hand, the external, physical fact that mathematics works in the real world.
What is the bridge between these two worlds? It is as if there is a natural harmony between our minds and the outer world. What is the cause of that harmony?
These two poles can be called rationalism and empiricism. Rationalism takes mathematics to be fundamentally a matter of pure thought. This fits well with the sense we have when doing mathematics, when reading Euclid, that we are establishing absolute truths by sheer reasoning. But it doesn’t explain why mathematics works so well in the physical world.
We have encountered some rationalists already: Plato, Descartes. We saw how Descartes solved the problem. Mathematics is pure thought, and it works in the physical world because the Creator put mathematical ideas in our minds. As the Bible says, “God created man in his image.” That is to say, God created the world based on mathematical ideas, and then created humans and sort of pre-programmed their minds with the same kinds of ideas that he had used to create the world.
So no wonder there’s a harmony between the mental and the physical worlds: they both stem from the same source, the Creator, who used the same principles when designing both. Descartes said basically this quite explicitly, as we recall. Plato pretty much hints at the same idea. God is a mathematician. That is a central belief in Platonist thought as well. And it is a necessary thesis for the rationalists to explain why mathematics works so well.
We have already encountered some empiricist as well: Aristotle, Francis Bacon. They think knowledge ultimately comes from the world around us. From that point of view, it is no mystery that mathematics works on physical triangles. It stems from physical experience to begin with, so of course it conforms to physical experience.
The challenge for the empiricists is instead to explain the mental experience of doing mathematics; our feeling that it brings absolute truth by pure thought in a way that no other subject does. From the empiricist point of view, this feeling is a mistake, a delusion. We think we are doing pure thought, but actually mathematical thought is generalized experience. We think we can sit in a closed room, an arm chair, and figure things out about an outside world that we have never even seen. But it only feels that way.
We have seen and touched many lines and triangles and squares our entire life, since the year we were born. We have internalized this experience. It has become second nature to us. Basic truths of geometry, such as Euclid’s axioms, may feel like core intuitions that are much more pure and absolute and undoubtable than things we know from experience. But that feeling is a delusion, according to the empiricists. Our minds, our feelings have imperfect self-awareness. Just as we are not aware through introspection how our digestive system works, so we are not conscious of the psychological origins of our mathematical intuitions.
I think we can agree that rationalism and empiricism both face big challenges. The challenge for rationalism is to explain why mathematics applies to the physical world. Traditional rationalism had an answer that was very compelling at the time: the explanation in terms of God, the Creator. But nowadays we may want an atheistic answer. And then rationalism is back to square one, facing the original problem all over again, without any solution in sight.
Empiricism doesn’t have that problem, but it has other ones. If mathematics comes from experience, how can it seem so absolute and undoubtable? How can an exact science come from inexact sensory impressions? If mathematics is based on experience like everything else, why does it seem to be such a different kind of knowledge in so many respects? Those are challenges for the empiricist to answer.
It matters how you answer these questions. It shapes the kind of science that you do.
Consider for instance Kepler, the 17th-century astronomer. He was another rationalist. As Kepler says: “Nature loves [mathematical] relationships in everything. They are also loved by the intellect of man who is an image of the Creator.” That’s almost word for word how I described the rationalist position just moments ago.
Kepler felt that the world was designed with the intent that we should study the universe mathematically. As he says: “Whenever I consider in my thoughts the beautiful order [of the universe] then it is as though I had read a divine text, written onto the world itself saying: Man, stretch thy reason hither, so that thou mayest comprehend these things.”
In fact, scientific facts support this view, in Kepler’s opinion. For example, as he says, “Sun and moon have the same apparent sizes, so that the eclipses, one of the spectacles arranged by the Creator for instructing observing creatures in the orbital relations of the sun and the moon, can occur.”
That is indeed a striking fact: that the moon is exactly the right size to precisely block out the sun at the moment of a solar eclipse. From the point of view of modern science, this is a remarkable coincidence. It’s pure chance that the moon is exactly the right size.
You can understand why the explanation in terms of purpose was more compelling in Kepler’s time. Witnessing a solar eclipse is a spiritual experience. It all seems so perfect. Much too perfect to chalk it up to chance. It’s very disappointing that modern science offers nothing more than this non-explanation of such an emotionally compelling spectacle.
And not just modern science. Such views were around already in Kepler’s time. Atomism is a classical worldview that is indeed happy to attribute almost everything, eclipses included, to chance and randomness. According to Kepler’s teacher, Melanchthon, such views “wage war against human nature, which was clearly founded to understand divine things.”
So here we have again that double challenge to empiricism. If mathematics is just one type of knowledge among many that we pick up from experience, then, first of all, why does the universe show so many signs of being mathematically designed? Like the thing with the eclipses, but there are also countless other examples one could use to make this point. Empiricism has no answer to this. It thinks that’s all just a bunch of coincidences, and we are just fooling ourselves by looking for purpose and design that isn’t there.
And secondly, if empiricism is right, and mathematics is just experiential knowledge like everything else, then why does mathematical reasoning feel so uniquely compelling and convincing? As Melanchthon says, mathematics is as natural to a human being as “swimming to a fish or singing to a nightingale.” Just as animals are born with these instincts, so our minds are innately predisposed to do mathematics. Empiricism does not explain why that is the case, or why that seems to be the case.
So it’s understandable that Kepler was a convinced rationalist instead. And this conviction shaped his scientific work. Astronomers are “priests of the book of nature,” as Kepler said. So he was always looking for meaning and purpose and design.
For example, the telescope was a new invention in Kepler’s time, and it was a big moment when the moons of Jupiter were discovered. Kepler immediately looked for the purpose behind the existence of these moons. He concluded that Jupiter must be inhabited. Why else would it have moons? As Kepler says: “For whose sake, the question arises, if there are no people on Jupiter to behold this wonderfully varied display with their own eyes? We deduce with the highest degree of probability that Jupiter is inhabited.”
Another of Kepler’s attempts at uncovering divine design was his theory of planetary distances. According to Kepler, the Creator had chosen the number and position of the planets according to a very beautiful and pleasing mathematical design. Namely, a plan based on the five regular polyhedra.
Euclid discusses the regular polyhedra at length in the Elements. There are precisely five of them, as Euclid indeed proves in the very last theorem of the Elements.
Kepler figured God was as fascinated by these shapes as Euclid had been. So when God asked himself how many planets there should be in the solar system, and how far from the sun to put them, God figured that the most mathematically pleasing way would be to choose six planets, and to have the spaces between them chosen in such a way that the five regular polyhedra fit between them like a nesting doll.
Kepler’s theory in fact fit the data very well. You could calculate planetary distances from astronomical measurements, and you could calculate size proportions of the regular polyhedra from Euclid’s Elements. If you put these things side by side in two columns they come out remarkably close to one another.
So again Kepler explained things that modern science doesn’t explain at all. Why are there six planets? Why are they positioned at those particular distances form the sun? Why does the moon fit precisely on top of the sun during an eclipse?
Kepler explained all of these things. If you accept the basic outlook that it makes sense to think of the creator of the universe as a Geometer, then Kepler’s explanations are very good. This is Kepler, the best mathematical astronomer of his age. These are not some whimsical religious musings. It’s very serious science. Very good science, one might argue.
Meanwhile, modern science doesn’t explain any of these things. There is no explanation, there is no why, according to modern science, of course. It’s all just chance. The solar system was formed by a bunch of random rocks getting caught in a gravitational field. Whatever positions they took up is just random.
It’s easy for us to judge Kepler. But shouldn’t science explain more things as it develops? Not fewer things. You would think that science should take things that are not explained and explain them. Instead of taking things that are already explained and attributing them to coincidence instead. And yet that is precisely what happened when Kepler’s theories were abandoned.
In any case, this Kepler stuff is interesting for all kinds of reasons, but for our purposes, what I wanted to show was that it matters whether you are a rationalist or an empiricist. Rationalism, as we saw, almost requires the hypothesis that God was a Geometer, just as Plato and Descartes and Kepler all said. And that assumption has major implications for how you practice mathematical science. It suggests looking for deliberate design put into the world by a mind that is essentially like our mind, as far as mathematics is concerned.
So that’s one way in which the rationalism-empiricism divide strongly shaped scientific practice in the early 17th century. But that was not the end of it. Here’s another example: the contrasting ways in which Descartes and Newton approached cubic curves.
Cubic curves are the next step beyond conic sections. Conic sections are curves of degree 2. They were studied in great depth by the Greeks. Cubic curves are called cubic because that have degree 3. So they are the more complicated cousins of the conic sections. In the 17th century, this was natural direction to take geometry: to understand curves of degree 3 and higher in the same depth that the Greeks had understood conic sections.
For instance, conic sections come in three classes: ellipse, parabola, hyperbola. Can one find an analogous way of classifying cubic curves? There are going to be more classes because cubics are more complicated. But maybe with the right principle of taxonomy one can impose order among their variety in way that is as useful as the division into ellipse, parabola, and hyperbola is in the theory of conics.
Newton did precisely this. He gave a very detailed and advanced technical study in which he classified cubic curves in several different ways. He divided cubic curves into “species” as he says. That’s Newton’s own term, and it’s a vivid one.
Taxonomising curves into “species” makes Newton sound like a pioneering explorer-scientist forging into unknown jungles and studying all the strange creatures. When you find a new exotic insect, you put it under a microscope and study all its properties. How many legs does it have, how many eggs does it lay, and so on. It’s the same when studying curves. How many crossing points, how many inflections points, and so on. It’s the zoology of mathematics.
This metaphor fits very well with the epistemological ideals of empiricism. You learn by studying the great diversity of things out there. Into the jungle! That’s the call of empiricism. That’s how you learn things. By immersing yourself in the unknown.
“The best geologist is one who has seen the most rocks.” That’s another slogan of empiricism. Experience is the source of knowledge, in other words. If you want to understand rocks, you need to look at a whole lot of rocks. And if you want to understand cubic curves, you need to look at a whole lot of cubic curves, first of all. Once you have built up a store of experience, then maybe you will see some patterns starting to emerge and you can begin the process of systematising or taxonomising the “rocks.”
Empiricism is all about diving in at the deep end and figuring it out as you go. This corresponds to reading Euclid backwards. You start with the complicated stuff, the Pythagorean theorem and such things. Those kinds of things are the exotic beasts that you encounter “in the jungle.” Gradually, you seek to bring order into the chaos by finding general principles that account for the phenomena you observe.
That’s empiricism. And it’s completely backwards according to rationalism. That’s not how you learn things. You can’t start with observations, with the phenomena. Perception is unreliable. Aimless exploration unguided by the intellect is bound to be a waste of time leading nowhere.
The way to knowledge is thinking. To “meditate,” as people used to say. You have heard of Descartes’s Meditations. That’s even the title of one of his works. The source of knowledge is meditation. That is to say, deep thought where you basically close yourself off from the world. Sitting in an armchair in a closed room. That’s where you make progress in understanding, not running around in the jungle.
So Newton’s way to study cubic curves was the empiricist way. Get your machete out and start chopping your way through the thick of it. Eventually you become familiar with all these wild things you encounter, and you start to see what kinds of species there are and how they are related.
Descartes was the opposite of this. A rationalist. Descartes studied cubic curves too, but through meditation. His big book is La Géométrie (1637). He doesn’t study cubics specifically, but all algebraic curves. So curves of any degree, not just degree 3.
Already we see a typical rationalist characteristic: rationalism starts from the general; empiricism starts from the specific.
Rationalists withdraw into meditation because they do not trust individual observations. Thought is more reliable. If you sit back in an armchair and introspect about what is knowable, you are bound to come up with very general and abstract truths: I think therefore I am; the whole is greater than the part; two lines cannot enclose a space. Gradually, you have to work your way from there, step by step, to any specific fact you need to explain. Just as Euclid gradually works his way up to more and more complex and detailed material by starting with very general principles that ultimately entail all the rest.
So the rationalist in interested in all-encompassing abstract law or axioms. It is important to the rationalist that all truths can in principle be deduced from these axioms. But it’s less important to actually do this. The rationalist is most interested in the fundamental axioms or laws, because those are the source of the certainty of knowledge. The specifics derived from them merely inherit their certainty from the certainty of these foundational axioms.
So the very first principles of the entire field is where you need to focus your attention if you are a great rationalist philosopher. And that’s exactly what Descartes does in his book, La Géométrie. Even the title fits with this point of view: The Geometry; it’s a very total, definitive account of geometry as a whole, just as the rationalist epistemological ideal demands.
This is further confirmed in the very first sentence of the text: “All the problems of geometry …”––that’s how Descartes opens his book. He starts with extreme generality, just as rationalism suggests one should. He wants to find the principles that can be used to solve “all the problems of geometry,” in principle.
Descartes doesn’t care so much about the details. He is very keen to explain why his principles are sufficient to solve “all the problems of geometry,” but has very little patience for actually solving any of those problems. This is reflected in the very last sentence of his book.
Descartes writes: “I hope that posterity will judge me kindly, not only as to what I have explained, but also as to what I have intentionally omitted so as to leave to others the pleasure of discovery.”
This is a bit dishonest, of course. He did not omit the details merely out of kindness to the reader, obviously. His focus on the general and lack of interest in the specific is a consequence of his rationalist outlook.
Newton is the opposite. He loves the details; he loves getting stuck in with some obscure technical problem. In fact, his long treatise on cubic curves is full of technical details but he gives very little attention to explaining any general conclusions. It’s hard to see the forest for the trees.
That’s good empiricism, of course. Rationalism thinks you can trust specific results because they are derived from reliable general principles. The certainty of knowledge resides in the axioms, the general principles. That’s where you need to focus your attention to secure the rigour and reliability of reasoning. And that’s what Descartes does.
Empiricism looks at it the other way around. It is the details, the little things, that are the most knowable. Knowledge starts from the directly observed phenomena, with all their specificity. That’s the root of reliability and certainty. Abstract principles are trustworthy only insofar as they are inferred from a large body of facts.
It’s the same in physics. To Newton, the empiricist, the starting points are specific facts. The orbital time of Jupiter, the speed of Saturn. Specific observable facts. You have to start there and then infer general laws like the law of gravity by showing that it fits a long list of facts. It is the specific facts that give credibility to the general law.
Not so to Descartes. The introspective, meditative, rationalist way of doing physics is to figure out first what properties of moving bodies are the most undoubtable. What are the things that are like Euclid’s axioms, but for mechanics?
Descartes did physics exactly this way. In his view, the most undoubtable core principles of physics are the laws of collision of two bodies. If one body bumps into another, what happens? Well, if one is twice as heavy but they have the same speed, then so-and-so happens; if one is twice as heavy but the other is twice as fast, then so-and-so happens; etc. Those are the kinds of principles that Descartes thought one could establish through pure thought and meditation.
Descartes saw this as analogous to Euclid’s geometry. Euclid’s axioms are about lines and circles: the basic building blocks of all geometrical figures. More complex figures are built up from there by combinations of lines and circle, or ruler and compass. In the same way, in physics, complex phenomena can be regarded as ultimately generated by the simple root phenomenon of the collision of two bodies.
Indeed, modern science kind of agrees about that part. If you exhale on a cold day, you breath forms a cloud that moves in complex ways. It seems to flow or float, but really it’s just lots and lots of tiny molecules crashing into each other millions of times, and that gives rise to this kind of flowing pattern that you see on a larger scale.
So simple generative principles can be enough to account for all kinds of things behind their immediate reach, though elaborate repeated composition. Just as lines and circles kind of “give birth” to all geometry, including very complicated shapes that aren’t just round or straight.
Actually, lines and circles are not enough to generate all geometry. They can’t generate cubic curves for example. Descartes is very interested in this issue. And indeed, in his book La Géométrie, he supplements the ruler and compass with another basic generative principle for drawing curves. A kind of linkage principle. You can build a sort of machine that consist of multiple rulers and pegs interlinked in certain ways, and as you push one part of the machine the other parts move in specific ways because of how all the parts are interconnected. An ordinary compass is sort two rulers nailed together. In the same way you can make more elaborate devices composed of more rulers. This gives rise to “new compasses,” as Descartes calls them. And these are sufficient to encompass “all the problems of geometry,” according to Descartes.
In a way it might seem contradictory that it was the rationalists, like Descartes and Leibniz, who were so concerned with the making of geometrical figures with concrete devices. Shouldn’t a proper rationalist hate physical instruments, like Plato did?
But there is no contradiction. Descartes cared about geometrical instruments for theoretical reasons. As I just emphasised, constructions in geometry go naturally with the general rationalist idea of the mind generating all knowledge from within itself. It’s a form of self-reliance. It doesn’t need anything from the outside world.
And earlier we have spoken about how constructions are connected to the epistemological foundations of geometry. Maker’s knowledge. Constructions are the most knowable thing, and the most secure form of geometrical knowledge, protected against many threats of paradoxes and contradictions. So that’s another way in which constructions go well with rationalism, which is of course very much concerned with what are the most undoubtably knowable things.
So these instruments like the ruler and compass and the generalisations of them that Descartes conceived are theoretical, not practical. There’s a funny anecdote that sums this up in the Brief Lives by Aubrey—a late 17th-century collection of biographical stories, maybe not super reliable exactly but this story could very well be true. Here’s what this biographer Aubrey says:
“[Descartes] was so learned that all learned men made visits to him, and many of them would desire him to show them his instruments. He would drawe out a little drawer under his table, and show them a paire of Compasses with one of the legges broken: and then, for his ruler, he used a sheet of paper folded double.”
Quite amusing, and it fits with what I said about the constructions being theoretical.
So we see that the idea of drawing curves with instruments in geometry is analogous to the idea of explaining all of physics in terms of collisions of little bodies. They are both simple, intuitable principles that generate the entire world of phenomena.
From a rationalist point of view, you need such principles. You start in the simple and pure world of meditation and you need to reason your way to the complicated and messy outside world. So you need a bridge that goes from the simple to the complex. Contact mechanics is such a bridge in physics, and ruler and compass is such a bridge in geometry.
But this is only necessary if you are a rationalist. If you insist on starting with pure intuition and thought, then you need such a bridge to the phenomena and the outside world.
But if you are an empiricist you take the outside world—the jungle—for granted as given, as a starting point, so you don’t need to explain how it can be generated by repeated composition of simple principles.
Indeed, Newton rejects both contact mechanics and geometrical constructions at the same time, for precisely this reason.
The fact that these two things are intimately related is not lost on Newton. This is why he starts his big masterpiece on physics by talking about the construction of line and circle in geometry. A very weird way to start a physics treatise to modern eyes, but it makes perfect sense if we keep in mind the background of Descartes and rationalism and everything I just outlined.
I’m referring to Newton’s Principia of 1687. Descartes was long dead by then, but his ideas about the foundations of physics were as relevant as ever. Leibniz, who was a contemporary of Newton, was a rationalist like Descartes. Like Descartes, Leibniz attached great importance to contact mechanics in physics and constructions in geometry.
So when Newton’s Principia came out, Leibniz was very upset that Newton had abandoned the principle of contact mechanics, which was so essential to the entire rationalist worldview. Let me quote Leibniz on this point. Here’s what he said: “A body is never moved naturally except by another body that touches and pushes it. Any other kind of operation on bodies in either miraculous or imaginary.”
Newtonian gravity is precisely one such “other operation”; something that cannot be explained in terms of particles bumping into one another. This is why Leibniz condemns very fiercely the notion of gravity as a foundational principle of physics: “I maintain that the attraction of bodies is a miraculous thing, since it cannot be explained by the nature of bodies.”
That is to say, Newton’s law of gravity cannot be explained or arrived at from a rationalist point of view. Newton in fact agreed. If anything, he makes this point in even stronger terms than Leibniz. Here’s what he says: “It is inconceivable that inanimate brute matter should, without the mediation of something else, which is not material, operate upon and affect other matter without mutual contact, as it must be, if gravitation be essential and inherent in it. That gravity should be innate, inherent, and essential to matter, so that one body may act upon another at a distance through a vacuum without the mediation of anything else, is to me so great an absurdity, that I believe no man who has in philosophical matters a competent faculty of thinking can ever fall into it.”
Very strong words there from Newton. And we can understand why. He wants to discard the rationalist outlook entirely. He is not interested in winning broad support for his theory by trying to argue that it sort of fits with rationalism somehow. He could have given that a shot. He clashed with many influential people: Descartes, Huygens, Leibniz. He could have tried to go a diplomatic route and try to come up with reasons for why his way of doing science was compatible with their rationalist commitments. But he chose not to. This is why he comes on so strongly in these quotes about how gravity is rationally inconceivable and so on.
In this way, Newton moves the conflict into the area of rationalism versus empiricism generally, instead of arguing about the interpretation or meaning of gravity specifically. “With the cause of gravity I meddle not,” says Newton, since “I have so little fancy to things of this nature.”
So what Newton wants to justify is not gravity specifically, but a the empiricist way of doing science generally, in which you don’t care about such questions at all. Questions such as how to give a rationalistic account of gravity, or explaining how a meditating mind in an armchair could arrive at the necessity of the law of universal gravitation. Those questions should simply be ignored, says Newton. Which makes sense from an empiricist points of view, but is sheer madness from a rationalist point of view.
So Newton bites the bullet on the cause of gravity. He says: yeah, I know my physics completely clashes with the core beliefs and methodology of rationalism, but rationalism is wrong anyway.
Now, as I said, the role of contact mechanics in physics is analogous to the role of constructions in geometry. Newton knows this, and this is why, to justify his physics, he starts by talking about how to interpret the role of constructions in geometry. Here is what he says right at the beginning of the Principia:
“The description of right lines and circles, upon which Geometry is founded, belongs to Mechanics. Geometry does not teach us to draw these lines, but requires them to be drawn. For it requires that the learner should first be taught to describe these accurately, before he enters upon Geometry; then it shews how by these operations problems may be solved. To describe right lines and circles are problems, but not geometrical problems. The solution of these problems is required from Mechanics; and by geometry the use of them, when so solved, is shewn. And it is the glory of Geometry that from those few principles, fetched from without, it is able to produce so many things. Therefore Geometry is founded in mechanical practice.”
So that’s a clearly empiricist account of geometry. Not only because it obviously grounds geometry in the physical world, in physical practice and experience. But also because it takes away the idea that the axioms need to be justified by being intuitive and undoubtable. That was important to the rationalists, but Newton does away with that.
This is how Newton can justify that he “meddle not with the cause of gravity.” Geometry likewise doesn’t “meddle” with the construction of curves, but merely postulates their description—in fact, geometry postulates these things precisely “because it knows not how to teach the mode of effection,” just as physics does not know how to teach the cause of gravity.
So Newton has twisted Euclid into support for his physics. This is why the preface to the Principia is about constructions in geometry, such as the ruler and compass of Euclid. If geometry doesn’t really know how to generate these curves, but only takes them for granted and goes from there, then physics can do the same with gravity.
So Newton and Leibniz clashed along such lines. And not only them. One could argue that there’s a geographical element to this divide. Empiricism is to some extent a British movement more generally: not only Newton but also Francis Bacon, John Locke, Wallis—just to name some people we have already encountered before. Meanwhile, Leibniz’s rationalistic tendencies in his science and mathematics were shared by his leading colleagues in Continental Europe, such as Descartes and Huygens.
By way of summary, let me read a passage by Newton on his scientific method, and I will insert comments on how what he says fits exactly with what we have discussed. The passage begins:
“As in mathematics, so in natural philosophy, …”
Already very interesting. In other words, Newton is announcing that his scientific method is based on the method of mathematics; the method of Euclid basically. Ok, so what is this this methodological principle that is common to both mathematics and science? The sentence continues:
“… the investigation of difficult things by the method of analysis, ought ever to precede the method of composition.”
Analysis corresponds to reading Euclid backwards. To analyse is to break down into smaller pieces. Composition corresponds to reading Euclid forwards. To compose is to put simpler pieces together to form more complex results. Newton continues:
“Analysis consists in making experiments and observations, and in drawing general conclusions from them by induction. By this way of analysis we may proceed from compounds to ingredients, and from motions to the forces producing them; and in general, from effects to their causes, and from particular causes to more general ones, till the argument end in the most general.”
“General” indeed: a key words here. From observations, that is to say from specific facts, one infers more general underlying principles. Empiricism goes from the specific to the general; rationalism the other way around.
It is also nice that Newton mentions that analysis goes “from compounds to ingredients”: this is precisely the chemistry or cooking metaphor that we used before when discussing how to read Euclid backwards.
Newton continues:
“This is the method of analysis, and the synthesis consists in assuming the causes discovered, and established as principles, and by them explaining the phenomena proceeding from them, and proving the explanations.”
That is to say, reading Euclid forwards is of course also essential. The method of analysis that the empiricist uses does not dispense with this directions of Euclid; it merely reveals that a preliminary stage is necessary to understand its meaning. It is through the preliminary analysis, the backwards direction of reasoning, that one arrives at the principles—not by direct intuition, as the rationalists would have it. Then the forward direction, the composition or synthesis, proves that these principles really work; that is to say, that they are sufficient to prove everything. That part is the same to both rationalists and empiricists. The key difference is how they account for where the principles or axioms came from.
So those are Newton’s own words, corresponding very closely to the story I have told. Of course Newton and Leibniz and all these guys were acutely aware of all of this. In this way they were much more philosophically conscious than most scientists of later ages. And clearly it shaped their science very profoundly, as I have shown by several examples. So that’s all the more reason to keep pursuing these questions. As indeed we will.
Euclid inspired Gothic architecture and taught Renaissance painters how to create depth and perspective. More generally, the success of mathematics went to its head, according to some, and created dogmatic individuals dismissive of other branches of learning. Some thought the uncompromising rigour of Euclid went hand in hand with totalitarianism in political and spiritual domains, while others thought creative mathematics was inherently free and liberal.
Transcript
Gothic architecture is known for its pointed arches. Unlike round arches like a classical Roman aqueduct for example. Those are semi-circular, but Gothic arches are steeper, pointier. Gothic buildings, cathedrals, have these arches everywhere: windows, doorways, and so on.
Gothic arches consist of two circular arcs. You can make it like this. First make a rectangular shape. Like a plain window or door. A boring old rectangle. Now let’s spice it up. Take out your compass, and put it along the top side of the rectangle. Draw two circular arcs going up above the rectangle. Use the top side of the rectangle as the radius, and its two endpoints as the two midpoints of the two arcs you are drawing. The two arcs make a pointed extension of the rectangle. Now you have your Gothic window.
If you have your Euclid in fresh memory you will recognize at once that this is precisely the type of construction involved in Proposition 1 of the Elements. Coincidence? No, I don’t think so. The Gothic style of architecture arose in Europe in the early 12th century, within a decade or two of the first Latin translation of Euclid’s Elements. If that’s not cause and effect, it‘s an incredible coincidence.
There is little direct documentation about this, but, I am quoting now from Otto von Simson’s book The Gothic Cathedral, “at least one literary document survives that explains the use of geometry in Gothic architecture: the minutes of architectural conferences held in 1391 in Milan. The question debated at Milan is not whether the cathedral is to be built according to a geometrical formula, but merely whether the figure to be used is to be the square or the equilateral triangle. The minutes of one particularly stormy session relate an angry dispute between the French expert, Jean Mignot, and the Italians. Overruled by them on a technical issue, Mignot remarks bitterly that his opponents have set aside the rules of geometry by alleging science to be one thing and art another. Art, however, he concludes, is nothing without science, ars sine scientia nihil est. This argument was considered unassailable even by Mignot’s opponents. They hasten to affirm that they are in complete agreement as regards this theoretical point and have nothing but contempt for an architect who presumes to ignore the dictates of geometry.”
So the geometrical ethos was very strong indeed. This hardline view probably softened a bit over time. Renaissance art is more expressive, emotive, more alive, one might say, than this rigid late medieval stuff. That’s if we fast-forward two hundred years from these Gothic conferences about how art is nothing without geometry.
Then you have people like Michelangelo who said: “the painter should have compasses in his eyes, not in his hands.” I suppose it means that art should go a little more by feeling and intuition, and not be completely dictated by mathematics. But you still have “compasses in your eyes,” so there’s still a very significant role for geometry, it seems.
It is also revealing, perhaps, that Michelangelo thought it was important to point this out at all. I guess there were a lot of artists with compasses in their hands running around back then. Why else would Michelangelo feel the need to criticise that practice?
In fact, geometry proved useful to art again, in new ways, in the Renaissance. At this time artists discovered (or perhaps rediscovered) the geometrical principles of perspective.
Accurate representation of depth in a painting follows simple geometrical principles. The key construction is that of a tiled floor. Like a chessboard type of pattern of floor tiles, but seen in perspective, so tiles that are further away appear smaller in the picture. There are a lot of tiled floors in Renaissance art, because they are great for conveying a sense of depth.
You draw it like this. Draw two horizontal lines: one is where the floor starts, and one is the horizon. Divide the floor line into many equal pieces, representing the size of the tiles. Connect all these points to a fixed point on the horizon. This is because all the parallel rows of the floor will appear to converge in one point, just as whenever you are looking at parallel lines that go off into the distance, such as railroad tracks for instance, they appear to meet at the horizon.
Next, draw a second horizontal line representing the other edge of the front row of floor tiles. Now here comes the magic step. Draw the diagonal of the first tile, and extend it. Where this line cuts the other lines you have already drawn, those are all corners of other tiles.
This is because of the geometrical principle that a straight line, no matter how you look at it, from whatever angle, will always still be straight. A lot of stuff looks funny in perspective: big things can look small, parallel lines appear to meet, perfectly round things appear to be oval, and so on. Perspective distorts shapes in all kind of ways. But straight lines remain straight lines. That is an invariant in all of this. This is why the diagonal of the first floor tile is also the diagonal of successive tiles. Since that makes a straight line in the real world, it must make a straight line in the picture as well.
Once you have this diagonal it is easy to complete the rest of the floor. It looks great. It creates a photorealistic sense of depth. No wonder so many artists chose to set their scenes in locales that just happened to have lots of tiled floors.
But the insight is much deeper than this. It’s not really about tiled floors; it’s about the correct perspective representation of depth and size generally. Even if you don’t want a tiled floor in your finished picture, it’s still very useful to draw one in pencil on your canvas as a reference grid. You can use the tiles as a guide to transfer and compare sizes at different depths and distances. Then later you can paint it over with some trees or whatever, so nobody can see the grid anymore; you just used it behind the scenes to get the proportions right.
The Greeks probably knew about this stuff, but basically no paintings from antiquity have been preserved. But we know they were skilled artists. One guy is said to have painted grapes so realistically that birds came and pecked at it.
It seems the Greeks knew the geometrical principles of perspective and used it to creates scenes for the theatre for example. As Vitruvius says, “by this deception a faithful representation of the appearance of buildings might be given in painted scenery, so that, though all is drawn on a vertical flat facade, some parts may seem to be withdrawing into the background, and others to be standing out in front.”
This wasn’t just a party trick to the Greeks. It also had philosophical implications. To Plato they raised profound epistemological conundrums. He was concerned that optical illusion painting has “powers that are little short of magical,” “because they exploit this weakness in our nature,” bypassing “the rational part of the soul.” The solution to this problem, as Plato saw it, was a solid mathematical education. Since “sense perception seems to produce no sound result” with these illusory paintings, “it makes all the difference whether someone is a geometer or not.”
“The power of appearance often makes us wander all over the place in confusion, often changing our minds about the same thing and regretting our actions and choices with respect to things large and small.” “The art of measurement,” by contrast, “would make the appearances lose their power” and “give us peace of mind firmly rooted in the truth.”
Those are all Plato’s words. A rousing case for mathematics! But Plato perhaps drew his conclusions a step too far, rejecting categorically the role of observational data in science: “there’s no knowledge of sensible things, whether by gaping upward or squinting downward.” Science must be based on “the naturally intelligent part of the soul,” not observation. For example, “let’s study astronomy by means of problems, as we do geometry, and leave the things in the sky alone.” With such attitudes, perhaps it is no wonder that the Greeks excelled more in mathematics than in the sciences. But indeed the threat of optical illusions is a legitimate argument in Plato’s defense.
When the principles of perspective were rediscovered in Renaissance Italy, they were again at the heart of scientific developments, but this time on the side of empirical science. Galileo looked at the moon through a telescope and concluded that it had mountains and craters. Of course the image one sees through a telescope is flat. But mountains and craters are revealed by the shadows they cast.
This is not necessarily very easy to see, or not necessarily a very evident conclusion. Some scholars have argued that the artistic tradition, and its extensive study of perspective and shadows, was a necessary training for the eye to be able to correctly interpret the telescope data.
Is it a coincidence that Galileo the telescopic astronomer came from the same land as the great Italian Renaissance painters? Galileo was born and raised in Tuscany, right where so many of these masters had worked. Maybe only someone immersed in this artistic culture had the right eyes to interpret the heavens. A far-fetched theory, in my opinion, but it’s a nice story.
Let’s put aside the art stuff now and look at another theme in how mathematics was received in the early modern world. Namely, the status of mathematics in relation to other fields. Geometry carried a certain authority. This led to many tensions.
Let’s jump right into the action, with an eyewitness report from 1703. “There has been much canvassing and intrigue made use of, as if the fate of the Kingdome depended on it.” “On the eve of Newton’s election as president [of the Royal Society], matters had deteriorated to such an extent that various fellows could be restrained only with difficulty from a public exchange of blows (or, in one case, the drawing of swords).”
Yikes. So what was this conflict on which “the fate of the Kingdome” depended? It was a battle between the mathematical and the non-mathematical sciences within the Royal Society in London.
The “philomats” who identified with Newton thought the non-mathematical sciences were hardly science at all. Botany, geology, stuff like that. They just collect data and write down obvious things. There’s no real thinking involved, no advanced theoretical progress, no genius.
Here’s how they put it, when they made the case that Isaac Newton, the great mathematician, ought to be the new president of the society to ensure its intellectual quality: “That Great Man [Newton] was sensible, that something more than knowing the Name, the Shape and obvious Qualities of an Insect, a Pebble, a Plant, or a Shell, was requisite to form a Philosopher, even of the lowest rank, much more to qualifie one to sit at the Head of so great and learned a Body.”
So science is divided into two camps: mathematical geniuses like Newton, and then people who just know the names of a bunch of insects.
As you can imagine, the other side saw it rather differently. They identified with Francis Bacon, who had complained about “the daintiness and pride of mathematicians, who will needs have this science almost domineer over Physic. For it has come to pass, I know not how, that Mathematics and Logic, which ought to be but the handmaids of Physic, nevertheless presume on the strength of the certainty which they possess to exercise dominion over it.”
So mathematicians have an inflated ego. They are so full of themselves that they think they have the right to tell others how to think.
Here’s how this point was put in 1700: “The World is become most immoderately fond of Mathematical Arguments, looking upon every thing as trivial, that bears no relation to the Compasse, and establishing the most distant parts of Humane Knowledge, all Speculations, whether Physical, Logical, Ethical, Political, or any other upon the particular results of number and Magnitude. In any other commonwealth but that of Learning such attempts towards an absolute monarchy would quickly meet with opposition. It may be a kind of treason, perhaps, to intimate thus much; but who can any longer forbear, when he sees the most noble, and most usefull portions of Philosophy lie fallow and deserted for opportunities of learning how to prove the Whole bigger than the Part.”
So mathematics corrupts mind and soul by fostering delusions of grandeur, and by focusing on obscure technical questions instead of on what is really important.
Roger Ascham made a similar point in 1570: “Some wits, moderate enough by nature, be many times marred by over much study and use of some sciences, namely arithmetic and geometry. These sciences sharpen men’s wits over much. Mark all mathematical heads, which be wholly and only bent to those sciences, how solitary they be themselves, how unapt to serve in the world.”
Meanwhile, the mathematicians, for their part, thought that an exclusive focus on the merely practical is anti-intellectual and beneath a true thinker. Others scientists may use basic mathematics, but the real accomplishment is to understand it.
Mathematician William Oughtred put it like this: “The true way of Art is not by Instruments, but by demonstration. It is a preposterous course of vulgar Teachers, to beginne with Instruments, and not with the Sciences, and so to make their Schollers onely doers of tricks, and as it were jugglers.”
Very relatable for a modern mathematics teacher. Students are so dependent on calculators that they are “onely doers of tricks.” That’s what you get when mathematics is not respected as an end in itself, but only as a tool for what is practically useful.
There’s an interesting twist to this story though. Part of what these opponents of mathematics were attacking was the pedantic focus on theoretical subtleties. Instead of tackling real problems, mathematicians sit around and muse about nuances of definition and postulates that only matter for very subtle foundational debates, not for actual problem solving. A valid critique, you might say, after reading Euclid with all his foundational pedantry.
But here’s the twist: Many mathematicians didn’t like that stuff either. Many mathematicians in the 17th century felt that the Greek geometrical style was much too formal. They recognized the value of the Euclidean style for foundational investigations, but they felt that creative mathematics must be much more free and loose.
Here’s how Clairaut put it in the 18th century:
“[Euclid’s] geometry had to convince stubborn sophists who prided themselves on refusing [to believe] the most evident truths. It was necessary then that geometry have the help of forms of reasoning to shut the idiots up. But times have changed. All reasoning which applied to that which good sense knows in advance is a pure loss and serves only to obscure truth and disgust the reader.”
This fits pretty well with what we have said about the Greek context. Euclid’s special style of geometry arose in a critical philosophical climate. Mathematicians had to anticipate attacks from philosophers who wanted to undermine the entire notion that geometrical reasoning was a rigorous way of finding truth.
Without this external pressure from philosophy, mathematicians may have been happy with a much more informal style, as they were in other cultures and societies. And as indeed they became again in the 17th century.
Almost all mathematicians in the 17th century were very happy to take a freewheeling approach for example when exploring a lot of stuff related to what we call calculus today. For example, John Wallis, a leading mathematician, did work on infinite series that was based on daring, unrigorous extrapolations and generalisations, which he considered “a very good Method of Investigation which doth very often lead us to the early discovery of a General Rule.” In fact, “it need not any further Demonstration,” according to Wallis.
This is very unlike Euclid or anything you find in Greek sources. It’s explorative trial and error, and a readiness to trust the patterns and rules you discover without the minutiae of carefully writing out meticulous proofs of every little thing.
When mathematicians chose this approach, they did not think of themselves as going against the ancient tradition. Instead they imagined—and they were probably right, of course—that Greek mathematics too would have been developed this way, in an informal way.
Euclid’s style of mathematics is very powerful for certain foundational purposes, but of course Euclid’s proofs do not reflect how people initially discovered these things. There must have been an exploratory side to Greek mathematics that is not revealed in surviving sources.
Euclid’s Elements is the end product of a long process of discovery and exploration. That process would not have been conducted in the pedantic and overly polished style of the finished Elements. It is necessary to start with a much freer creative phase. Then its fruits can be systematized and analyzed in the manner of Euclid.
Torricelli, for example, expressed a view typical among 17th-century mathematicians: “For my part I would not dare to assert that this Geometry of Indivisibles is a thoroughly new invention. Rather, I would have believed that the old geometers used this one method in the discovery of the most difficult theorems, although they would have produced another way more acceptable in their demonstrations, either for concealing the secret of the art or lest any opportunity for contradiction be proffered to envious detractors.” Many mathematicians agreed with Torricelli on this point.
The Greek historian Herodotus says about Persian political leaders that they “deliberate while drunk, and decide while sober.” That’s how you have to do mathematics too. First you need to generate ideas. For this you have to be “drunk,” that is to say, try out wild ideas, be uninhibited. Then you have to go over the same material again while “sober”: that is to say, you scrutinize everything critically, discarding and correcting all the mistakes you made while “drunk.”
The documentation we have for Greek mathematics is only the “sober” part. But there must have been a “drunk” part too. The sober part is what gives mathematics its distinctive precision and exactness and reliability. But the sober part alone is sterile. It needs the fertile input of daring ideas from the drunk part. Creative mathematics requires both.
Note that if you want to create new mathematics, then this is essential to realize. So working mathematicians, research mathematicians, will absolutely agree with his.
But many people in the 17th century wanted to use the example of mathematics to support various agendas, without having any interest in discovering new mathematics. From that point of view, it is possible to ignore the drunk phase. If you are merely preserving and admiring past mathematics, and you don’t need creativity, you don’t need new ideas, then you can stick entirely to the sober mode, the Euclidean mode, and maintain that that alone is the essence of mathematics.
This matters if you want to use the authority and status of mathematics to legitimate other, non-mathematical agendas. Indeed, it suited some people very well in the 17th century to emphasise the “soberness” of Euclid. They wanted mathematics to be like that, because they had political or philosophical ideals that fit that image.
Amir Alexander’s book Infinitesimal has some nice examples of this. Let’s look at those. I mentioned Wallis as an example of a creative mathematician who very much embraced the “drunk” style of mathematics. His arch enemy was Hobbes, who, by contrast, appealed to the authority and rigour of Euclidean geometry as a model for reasoning as well as political organisation.
As Amir Alexander says: “Wallis and Hobbes both believed that mathematical order was the foundation of the social and political order, but beyond this common assumption, they could agree on practically nothing else. Hobbes advocated a strict and rigorous deductive mathematical method, which was his model for an absolutist, rigid, and hierarchical state. Wallis advocated a modest, tolerant, and consensus-driven mathematics, which was designed to encourage the same qualities in the body politic as a whole.”
Wallis’s vision of mathematics was very agreeable to the experimental scientists of the Royal Society. “Experimentalism is a humbling pursuit, very different from the brilliance and dash of systematic philosophers such as Descartes and Hobbes. It ‘teaches men humility and acquaints them with their own errors’. And that is precisely what the founders of the Royal Society liked about it. Experimentalism ‘removes all haughtiness of mind and swelling imaginations’, teaching men to work hard, to acknowledge their own failures, and to recognize the contributions of others.”
“Mathematics, [the Royal Society founders] believed, was the ally and the tool of the dogmatic philosopher. It was the model for the elaborate systems of the rationalists, and the pride of the mathematicians was the foundation of the pride of Descartes and Hobbes. And just as the dogmatism of those rationalists would lead to intolerance, confrontation, and even civil war, so it was with mathematics. Mathematical results, after all, left no room for competing opinions, discussions, or compromise of the kind cherished by the Royal Society. Mathematical results were produced in private, not in a public demonstration, by a tiny priesthood of professionals who spoke their own language, used their own methods, and accepted no input from laymen. Once introduced, mathematical results imposed themselves with tyrannical power, demanding perfect assent and no opposition. This, of course, was precisely what Hobbes so admired about mathematics, but it was also what Boyle and his fellows feared: mathematics, by its very nature, they believed, leads to claims of absolute truth, dogmatism, threats of tyranny.”
But note that this image of mathematics as totalitarian and absolutist is linked to its sober phase. By playing up the liberal, drunk way of doing mathematics, one changes its political implications.
So that’s how things played out in England. Conservatives appealed to Euclid’s rigour to justify hardline reactionary politics, while creative mathematicians saw the freedom of creation and discovery in mathematics as suggesting that society as a whole should have a high tolerance for unconventional ideas and novel approaches.
The situation in Italy was quite analogous. The Jesuits were the intellectual leaders of the Catholic world in the 17th century. They ran hundreds of colleges across Europe, notable as much for their “sheer educational quality” as for their doctrinal role “in the fight to defeat Protestantism.”
The Jesuit colleges placed great emphasis on Euclidean mathematics, which to them “represented a deeper ideological commitment. Geometry, being rigorous and hierarchical, was, to the Jesuit, the ideal science. The mathematical sciences that followed—astronomy, geography, perspective, music—were all derived from the truths of geometry. Consequently, [the Jesuit] mathematical curriculum demonstrated how absolute eternal truths shape the world and govern it,” thereby serving as a model for their religious doctrine and worldview. “Euclidean geometry thus came to be associated with a particular form of social and political organization, which the Jesuits strived for: rigid, unchanging, hierarchical, and encompassing all aspects of life.”
For this reason, “the Jesuits reacted with fury to the rise of infinitesimal methods”—which is “drunk” mathematics. “The mathematics of the infinitely small was everything that Euclidean geometry was not. Where geometry began with clear universal principles, the new methods began with a vague and unreliable intuition that objects were made of a multitude of minuscule parts. Most devastatingly, whereas the truths of geometry were incontestable, the results of the method of indivisibles were anything but,” thereby undermining “the Jesuit quest for a single, authortized, and universally accepted truth.”
Thus infinitesimal mathematics was dangerous to the Jesuits not for intrinsic mathematical reasons but because it was associated with diversity of thought unchecked by authority. As one Jesuit leader declared: “Unless mind are contained within certain limits, their excursions into exotic and new doctrines will be infinite, [which would lead to] great confusion and perturbation to the Church.”
One God, one Bible, one Euclid. Set in stone for all eternity. That’s what these guys wanted, and that’s why they liked Euclid. And that’s why, “in a fierce decades-long campaign, the Jesuits worked relentlessly to discredit the doctrine of the infinitely small and deprive its adherents of standing and voice in the mathematical community.” You have to stifle this dangerous new “drunk” mathematics, in which people think for themselves, explore diverse perspectives, and look for new truths (as if there was such a thing!).
So, in summary, mathematics had many possible connotations that could be exploited to various ends. It’s like when someone becomes a celebrity, everyone wants them to endorse their product or sign their petition and so on. A sponsored post on their Instagram is prime real estate. Mathematics had become a celebrity in the 17th century. It had status, for better or for worse. And everyone wanted a piece of it. Coke or Pepsi, PC or Mac—who would get the coveted endorsement of mathematics? Mathematics never sold out or picked a side, but it’s illuminating to see the pitches the PR departments of all these various movements made on its behalf.
Philosophical movements in the 17th century tried to mimic the geometrical method of the ancients. Some saw Euclid—with his ruler and compass in hand—as a “doer,” and thus characterised geometry as a “maker’s knowledge.” Others got into a feud about what to do when Euclid was at odds with Aristotle. Descartes thought Euclid’s axioms should be justified via theology.
Transcript
Everybody has seen the painting The School of Athens, the famous Renaissance fresco by Rafael. It shows all the great thinkers of antiquity engaged in lively intellectual activity. Plato and Aristotle are debating the relative merits of the world of ideas and the world of the senses, both gesticulating to emphasize their point. Others are absorbed in other debates and lectures, somebody’s reading, somebody’s writing.
But here’s something most people don’t notice in this painting. There is one and only one person in this entire pantheon who is actually making something. Everybody is thinking, arguing, reading, writing. Except Euclid. Euclid is drawing with his compass. He is producing the subject matter he is studying. He is active with his hands. He’s practically a craftsman among all these philosophers.
In the ancient world, the mathematician is the maker. Geometry is the most hands-on of all the branches of philosophy and higher learning.
Today the cliche is that a math nerd is almost comically feeble in anything having to do with physical action.
But ancient geometry was in the thick of the action. You had to roll up your sleeves to do geometry. Even in theoretical geometry you would constantly draw, construct, work with instruments. It was a short step to engineering. The greatest ancient mathematician, Archimedes, is almost as famous for his feats in engineering. Such as mechanical devices for lifting and moving heavy objects, and for transporting water. Archimedes and other mathematicians were also at the front lines of war, building catapults and many other warfare machines according to precise calculations. They were architects. The Hagia Sophia in Istanbul for example, was designed by a mathematician, Isidore, who had written an appendix to Euclid’s Elements.
In early modern modern times, like the 17th century, this link between mathematics and concrete action was well understood and appreciated.
Francis Bacon was sick of traditional philosophy because “it can talk, but it cannot generate.” This frustration led him to the radical counterproposal: to know is to do. “What in operation is most useful, that in knowledge is most true.” And on the other hand “to study or feign inactive principles of things is the part of those who would sow talk and nourish disputations.” So we have to condemn much traditional philosophy and turn more to action, to doing.
Perhaps the most important difference between ancient mathematics and ancient philosophy is precisely this. That mathematics is active, while philosophy merely “sows talk and nourishes disputations.” Perhaps that is the explanation for why mathematics proved so fruitful, still thousands of years later, both for intricate theory, such as planetary motions, and for practice, such as engineering, navigation, and so on. Try doing that with Aristotle’s doctrine of causes or Plato’s theory of the soul. Those things are great for “sowing disputations” but if doing is the goal then you can’t get much mileage out of them.
Thomas Hobbes, another famous 17th-century philosopher, very much agreed with this analysis. Hobbes famously declared that “Geometry is the only science that it hath pleased God hitherto to bestow on mankind.” How so? What makes geometry different from all other branches of philosophy and science?
Constructions, of course. Hobbes is very explicit about this. “If the first principles contain not the generation of the subject, there can be nothing demonstrated as it ought to be.” This is what makes mathematics different. Its principles contain the generation of the subject: Euclid’s postulates correspond to ruler and compass, and these are tools that generate the figures that geometry is about.
All philosophical and scientific theories are based on some assumptions or axioms. But they are not generative axioms. They are not a recipe for producing everything the theory talks about from nothing.
In this light we can readily appreciate for instance Hobbes’s otherwise peculiar-sounding claim that political philosophy, rather than physics or astronomy, is the field of knowledge most susceptible to mathematical rigour. Here’s how he puts it:
“Of arts, some are demonstrable, others indemonstrable; and demonstrable are those the construction of the subject whereof is in the power of the artist himself, who, in his demonstration, does no more but deduce the consequences of his own operation. The reason whereof is this, that the science of every subject is derived from a precognition of the causes, generation, and construction of the same; and consequently where the causes are known, there is place for demonstration, but not where the causes are to seek for. Geometry therefore is demonstrable, for the lines and figures from which we reason are drawn and described by ourselves; and civil philosophy is demonstrable, because we make the commonwealth ourselves.”
As bizarre as this may sound to modern ears, it makes perfect sense when we keep in mind the all-important role of constructions in classical geometry.
Indeed there are many things that only the person who made it truly understand. At this time, the 17th century, various mechanical devices were becoming more common. Such as pocket watches and all kinds of other machines based on gears and cogwheels and so on. The person who made it knows what all the parts are for, but an outsider cannot see this very easily at all. Today another example might be computer programs. The person who wrote it knows how it works, what it can do, how it could be changed, what might cause it to fail, and so on. It would be very difficult for someone else to get a similar sense of how it all works, even if they had access to the code, or they could pop the hood and look at the gears so to speak. Only the maker truly knows: “maker’s knowledge” is a slogan often repeated in the 17th century.
Hobbes took this idea and built a general philosophy from it. His general philosophical program can be read as a direct generalisation of the constructivist precept to the domain of general philosophy. Here’s how Hobbes defines philosophy: “Philosophy is such knowledge of effects or appearances as we acquire by true [reasoning] from their causes or generation.” This is basically a direct equivalent in more general terms of the principle that constructions are the source of mathematical knowledge and meaning.
Indeed, Hobbes explicitly draws out this parallel: “How the knowledge of any effect may be gotten from the knowledge of the generation thereof, may easily be understood by the example of a circle: for if there be set before us a plain figure, having, as near as may be, the figure of a circle, we cannot possibly perceive by sense whether it be a true circle or no. [But if] it be known that the figure was made by the circumduction of a body whereof one end remained unmoved” then the properties of a circle become evident. You understand a circle because you make it, in other words.
Another way of putting it is that “The subject of Philosophy, or the matter it treats of, is every body of which we can conceive any generation.” Just as, classically, the domain of geometry is the set of all constructible figures.
Concepts that are not constructively defined can easily be contradictory or meaningless: a common problem outside of geometry. As Hobbes says: “senseless and insignificant language cannot be avoided by those that will teach philosophy without having first attained great knowledge in geometry.”
Again, as we have discussed before, anchoring geometrical entities in physical reality is a warrant of consistency. Hobbes makes this point as well. “Nature itself cannot err”; that is to say, physical experiences “are not subject to absurdity.”
It is notable that Hobbes and other 17th-century thinkers who invoked geometry did not have in mind simple school geometry and some superficial remarks in Plato or Aristotle. Rather, they were referring to the rich picture of the geometrical method that emerges from a thorough study of advanced Greek geometry and technical writers. When they call upon geometry they mean not some simplistic idea of axiomatic-deductive method but a rich methodology only conveyed implicitly in the finest ancient works of advanced geometry.
This is why the constructive aspect shines through so clearly. It’s importance is evident if you study the mathematicians and build your idea of philosophy of mathematics from there. You’re not going to learn anything about that by reading Plato and Aristotle.
Hobbes is very clear about this. As he says, his philosophy of geometry is “to an attentive reader versed in the demonstrations of mathematicians without any offensive novelty.” Indeed, one must be “an attentive reader,” because one must draw out the philosophical implications left implicit in these sources. And one must be “versed in the demonstrations of mathematicians,” meaning the technical Greek authors. As Hobbes calls them, those “very skillful masters in the most distant ages: above all in geometry Euclid, Archimedes, Apollonius, Pappus, and others from ancient Greece.” This is why Hobbes, in one of his works, “thought it fit to admonish the reader that he take into his hands the works of Euclid, Archimedes, Apollonius, and others.”
Many other 17th-century philosophers picked up the same themes. Some took it to the epistemological extreme of saying that anything other than concrete, specific experience is strictly unknowable. Gassendi, for instance, did not hesitate to take this leap: “Things not yet created and having no existence, but being merely possible, have no reality and no truth.” “The moment you pass beyond things that are apparent, or fall under the province of the senses and experience, in order to inquire about deeper matters, both mathematics and all other branches of knowledge become completely shrouded in darkness.” Mathematical objects must be “considered in actual things”; indeed, “as soon as numbers and figures are considered abstractly then they are nothing at all.” Those are all quotes from Gassendi, and his point of view makes sense. He merely spells out the consequence of taking concrete construction to be essential to knowledge, just as the mathematical tradition suggests.
Other philosophers agreed too. Vico put it like this: “We are able to demonstrate geometrical propositions because we create them; were it possible for us to supply demonstrations of propositions of physics, we would be capable of creating them ex nihilo as well.” So once again the link between creation and knowledge is all-important, and geometry is the key example of this.
Paolo Sarpi made much the same point: “We know for certain both the existence and the cause of those things which we understand fully how to make [just as] in mathematics someone who composes [that is to say, demonstrates synthetically, in the manner of Euclid] knows because he makes.”
It’s striking how many of these early modern thinkers who were well versed in the Greek tradition seized upon the constructive element as the essence of the more geometrico, “the manner of the geometers.”
But there were of course other perspectives on mathematics as well. A lot of people read too much Aristotle and not enough Archimedes. Then as now, one might add. Anyway, these Aristotelians didn’t like mathematics much, and they tried to undermine its authority.
Here is their main point of attack: Mathematical proofs, such as those in Euclid, show that the theorem is true, but not why it is true. In other words, mathematics does not demonstrate “from causes,” as a true science should, according to Aristotle.
Here’s one typical expression of this argument, from Aristotelian philosopher Pereyra in the 16th century:
“My opinion is that the mathematical disciplines are not proper sciences. To have science is to acquire knowledge of a thing through the cause on account of which the thing is. However, the most perfect kind of demonstration must depend upon those things which are proper to that which is demonstrated; indeed, those things which are accidental and in common are excluded from perfect demonstrations.”
Euclid’s geometry is not a “science” in this sense, according to this point of view. For example, Pereyra, says, consider the theorem that the angle sum of any triangle is two right angles (Euclid’s Proposition 32). “The geometer proves [this theorem] on account of the fact that the external angle which results from extending the side of that triangle is equal to two angles of the same triangle which are opposed to it. Who does not see that this middle is not the cause of the property which is demonstrated? [The external angle] is related in an altogether accidental way to [the angle sum of the triangle]. Indeed, whether the side is produced and the external angle is formed or not, or rather even if we imagine that the production of the one side and the bringing about of the external angle is impossible, nonetheless that property will belong to the triangle; but, what else is the definition of an accident than what may belong or not belong to the thing without its corruption?”
So in other words, Euclid’s proof of the angle sum theorem does not reveal the actual reason why the theorem is true. Instead it proves the result via a non-essential thing, the external angle sticking out from the triangle. This external part was obviously added by the geometer quite gratuitously; it’s not essential to the very nature of the triangle. So it’s a kind of artificial trick to add this extra angle and base the proof on it. Truly explanatory and causal demonstrations should not be based on artificial tricks but on what is truly essential to the situation.
Schopenhauer later ranted against Euclid along similar lines. That’s in the 19th century. These ideas were more important and influential in the 16th century, when Aristotelianism was a dominant philosophy. But it’s fun to quote Schopenhauer anyway, because he expresses the same ideas in a charming way. Here’s what he says:
“Perception is the primary source of all evidence, and the shortest way to this is always the surest, as every interposition of concepts means exposure to many deceptions. If we turn with this conviction to mathematics, as it was established as a science by Euclid, and has remained as a whole to our own day, we cannot help regarding the method it adopts, as strange and indeed perverted. We ask that every logical proof shall be traced back to an origin in perception; but mathematics, on the contrary, is at great pains deliberately to throw away the evidence of perception which is peculiar to it, and always at hand, that it may substitute for it a logical demonstration. This must seem to us like the action of a man who cuts off his legs in order to go on crutches.”
“Instead of giving a thorough insight into the nature of the triangle, [Euclid] sets up certain disconnected arbitrarily chosen propositions concerning the triangle, and gives a logical ground of knowledge of them, through a laborious logical demonstration, based upon the principle of contradiction. We are very much in the position of a man to whom the different effects of an ingenious machine are shown, but from whom its inner connection and construction are withheld. We are compelled by the principle of contradiction to admit that what Euclid demonstrates is true, but we do not comprehend why it is so. We have therefore almost the same uncomfortable feeling that we experience after a juggling trick, and, in fact, most of Euclid’s demonstrations are remarkably like such feats. The truth almost always enters by the back door, for it manifests itself per accidens through some contingent circumstance. Often a reductio ad absurdum shuts all the doors one after another, until only one is left through which we are therefore compelled to enter. Often, as in the proposition of Pythagoras, lines are drawn, we don’t know why, and it afterwards appears that they were traps which close unexpectedly and take prisoner the assent of the astonished learner. The proposition of Pythagoras teaches us a qualitas occulta of the right-angled triangle. In our eyes this method of Euclid in mathematics can appear only as a very brilliant piece of perversity.”
So Schopenhauer agrees with the 16th-century Aristotelians that Euclid’s proofs are not explanatory. Instead they proceed by some kind of trick. Euclid is constantly setting logical mousetraps that force the reader to accept the conclusion even though nothing has truly been explained.
It’s interesting though that Schopenhauer uses the example of a machine that is shown to someone who doesn’t know how it was made and therefore is baffled by it and cannot understand how it works. The people of the constructivist tradition we discussed earlier of course used the same image to prove the opposite point: namely that in geometry we are the makers of the machines we use and precisely for that reason that we have genuine knowledge and understanding of it. The people who looked at it that way were basing themselves on mathematical sources. Schopenhauer and the 16th-century Aristotelian who hated mathematics so much were also the ones who knew the least about it. They had not studied the technical Greek writers like Archimedes, Apollonius, and Pappus. Some of these technical sources had not even been translated into Latin yet at the time the Aristotelians were writing in the 16th century. And by the time of Schopenhauer they had been forgotten again among philosophers.
But these Aristotelian guys in the 16th-century also had further interesting arguments to support their point. For example, consider Euclid’s area theorems for parallelograms and triangles in Propositions 35 and 37. The theorems say that same base and same height implies the same area. The first theorem says this for parallelograms and the other one for triangles. The proof of the second theorem is based on the first one: a triangle is just half a parallelogram, so since we already have the result for parallelograms it follows almost immediately that it is also true for triangles.
But we could just as well have done it the other way around: we could have proved the theorems first for triangles, and the infer the result for parallelograms by saying that parallelograms are basically just double triangles.
Euclid chose to start with the parallelogram and then do the triangle, but this was essentially an arbitrary choice. It doesn’t reflect any causal relation. The two theorems are equivalent. It’s not that one of them is more fundamental and therefore explains or causes the other. Neither of the two theorems is more of a cause than the other. So Euclid’s procedure doesn’t fit Aristotle’s decree that demonstrations should proceed from causes.
These guys, like I said, didn’t keep in mind the whole construction business. They were not aware of that because they had not read much mathematics. Later, Leibniz, who knew both the mathematical and the philosophical traditions very well, argued that the construction perspective solves the problem that the Aristotelians raised. Here’s what Leibniz says:
“[Geometry] does demonstrate from causes. For it demonstrates figures from motion; from the motion of a point a line arises, from the motion of a line a surface, from the motion of a surface a body. Thus the constructions of figures are motions, and the properties of figures, being demonstrated from their constructions, therefore come from motion, and hence, from a cause.”
So basing geometry on constructions imposes a natural order—a causal hierarchy, as it were—on its theorems whence Aristotle’s ideal of demonstrative understanding can be maintained. According to Leibniz anyway.
Let’s have a look at Descartes as well. He also had interesting ideas about what made mathematics such a special type of knowledge, and how its success could be emulated in other fields.
In his Discourse on Method of 1637, Descartes explained his philosophical program and how he arrived at it. In an autobiographical introduction he explains:
“I was most keen on mathematics, because of its certainty and the incontrovertibility of its proofs. Considering that of all those who had up to now sought truth in the sphere of human knowledge, only mathematicians have been able to discover any proofs, that is, any certain and incontrovertible arguments, I did not doubt that I should begin as they had done.”
Those are the words of Descartes, famous for doubting everything; his very method has been called the method of doubt. Yet as he himself says: “I did not doubt” that I should follow the mathematicians.
You just had to extend the mathematical method to other areas as well, to philosophy in general. As Descartes says:
“Believing as I did that its only application was to the mechanical arts, I was astonished that nothing more exalted had been built on such sure and solid foundations.”
Just imagine the amazing things that could be achieved if other fields were as successful as mathematics. This was a common sentiment. Here’s how Hobbes put the same point:
“The geometricians have very admirably performed their part. For whatsoever assistance doth accrue to the life of man, whether from the observation of the heavens, or from the description of the earth, from the notation of times, or from the remotest experiments of navigation; finally, whatsoever things they are in which this present age doth differ from the rude simpleness of antiquity, we must acknowledge to be a debt which we owe merely to geometry. If the moral philosophers had as happily discharged their duty, I know not what could have been added by humane Industry to the completion of that happiness, which is consistent with humane life.”
So the goal of philosophy is to be as good as mathematics. So let’s see what Descartes considers to be the foundations of mathematics. He formulates a method for how to philosophise in general, and he intends for this to be a generalization of the mathematical method.
So you might say his methodological program is part descriptive and part prescriptive. It is descriptive because it describes how geometry works; it’s an analysis meant to capture what made Euclid so great. And at the same time it is prescriptive in that it gives orders as to how one should philosophise. Namely, whatever Euclid did in geometry, that philosophers should do in every field, such as physics, ethics, theology, and so on.
Here’s what Descartes says about the axioms or starting points of a theory. We discussed before whether the axioms should necessarily be obvious. Descartes comes down very firmly on that issue.
“The first [principle of my method] was never to accept anything as true that I did not incontrovertibly know to be so; that is to say, carefully to avoid both prejudice and premature conclusions; and to include nothing in my judgements other than that which presented itself to my mind so clearly and distinctly, that I would have no occasion to doubt it.”
So we should start only from the most obvious things, in other words. Things that are so clear that they cannot be doubted. Things known by immediate intuition, in other words. That’s supposed to correspond to the axioms of Euclid.
So Descartes has a lot of faith in innate intuition. As Descartes says, there are “basic roots of truth implanted in the human mind by nature, which we extinguish in ourselves daily by reading and hearing many varied errors.” So this inner “natural light” is more reliable than book learning.
So we should, Descartes says, “conduct thoughts in a given order, beginning with the simplest and most easily understood objects, and gradually ascending, as it were step by step, to the knowledge of the most complex.” And for the sake of this stepwise process, it is necessary to “divide all the difficulties under examination into as many parts as possible.”
You can see how philosophy is going to look a lot like Euclid if people follow these rules that Descartes lays down.
It is interesting that Descartes also specifically says that one should “posit an order even on those [things] which do not have a natural order or precedence.” This is a kind of reply to the Aristotelian point we mentioned above.
The Aristotelians were arguing that when two theorems are equivalent—such as the areas theorems for triangles and parallelograms—then it is artificial and unscientific to impose a particular order that makes one logically prior to the other, as Euclid does. Because then you haven’t given a causal explanation, as Aristotle says one should.
Descartes turns the tables on this. Instead of criticising Euclid when his method seems to go against philosophical sense, he makes Euclid the boss of philosophy. Whatever Euclid does, that’s good method. So if Euclid imposes an artificial logical order on equivalent theorems, then that’s what one should do in philosophy, Descartes concludes.
It goes against Aristotle—so what? Those people I quoted from the 16th century, a hundred years before Descartes, they thought Aristotle had more authority than Euclid, so they used Aristotle to criticise Euclid. Now, a hundred years later with Descartes, it is the other way around. Descartes would rather use Euclid to criticise Aristotle.
A lot had happened in those hundred years. A lot of new science: Copernicus, Galileo, Kepler, etc. Science had made terrific progress by using Euclid and ignoring Aristotle.
By the time of Descartes, the Aristotelians were dinosaurs. Descartes didn’t pull any punches when making this point: he condemned the Aristotelians as “less knowledgeable than if they had abstained from study.”
This new hierarchy, where mathematics has greater authority than philosophy was soon widely accepted. John Locke, the famous philosopher, put it like this half a century later: “in an age that produces such masters as the great Huygenius and the incomparable Mr. Newton, it is ambition enough to be employed as an under-labourer in clearing the ground a little, and removing some of the rubbish that lies in the way to knowledge.” So philosophy is just an under-labourer to mathematical science. The real geniuses, the real creative forces are mathematicians such as Huygens and Newton. Philosophers take on a subordinated role. The task of the philosopher is to explain to others how to follow the lead of the mathematical sciences. This is why Locke calls himself a mere under-labourer.
So that was the general methodological influence of mathematics on Descartes. But Descartes was not content with merely adopting the Euclidean method in philosophy. He also wants to justify this method; to explain why it is so reliable. He does this in his Principles of Philosophy of 1644.
In the very first sentence of this book, Descartes says: “whoever is searching for truth must, once in his life, doubt all things.” As we just saw, in his earlier work he had said that he did not doubt the method of the mathematicians. Now he’s going to fix this gap.
Let’s say you did doubt the mathematical method, the method of Euclid. According to Descartes, as we saw, the foundations of the method was intuition. Euclid starts from axioms such as “if equals are added to equals, the results will be equal.” Intuitively, these basic truths feel completely undoubtable. We are so convinced that they must be true, even though we cannot prove these things.
You might argue: there will always be something we cannot prove. In a deductive system, one thing is deduced from another, but you have to start somewhere. If I tried to prove Euclid’s axioms, I would have to deduce them from something. Whatever those somethings are, they will become the new axioms. So then they have to be assumed. We can never escape this cycle. We always have to assume something.
Unless. Unless we find axioms that are somehow logically self-justifying. This is the idea of the consequentia mirabilis that we discussed before. Axioms can be self-justifying if it is incoherent to try to refute them. If asserting that the axioms is false actually implies accepting the axiom, then the axiom is self-justifying. That way we can find an end to the problem of infinite regress; the problem of always having to prove everything from something else in a never-ending cycle.
This is going to be Descartes solution. He will give an axiom of that type, and then derive the Euclidean axioms from it. Then he will have closed the loop: there are no loose ends, nothing unjustified, anymore.
Here’s the axiom: I think therefore I am. This is the undeniable truth which cannot be denied because denying it would be contradictory.
Here’s how Descartes puts it: “We can indeed easily suppose that there is no God, no heaven, no material bodies; and yet even that we ourselves have no hands, or feet, in short, no body; yet we do not on that account suppose that we, who are thinking such things, are nothing: for it is contradictory for us to believe that that which thinks, at the very time when it is thinking, does not exist. And, accordingly, this knowledge, I think, therefore I am, is the first and most certain to be acquired by and present itself to anyone who is philosophizing in correct order.”
Ok, so that’s the axiom that cannot be denied because to deny it would be contradictory. How are you supposed to prove Euclid’s axioms from there? That seems difficult. How am I supposed from prove geometrical statements from “I think therefore I am”? Well, Descartes has an answer.
“The knowledge of remaining things [including geometry] depend on a knowledge of God,” because the next things the mind feels certain of are basic mathematical facts, but it cannot trust these judgments unless it knows that its creator is not deceitful. “The mind discovers [in itself] certain common notions [such as the axioms of Euclid], and forms various proofs from these; and as long as it is concentrating on these proofs it is entirely convinced that they are true. Thus, for example, the mind has in itself the ideas of numbers and figures, and also has among its common notions, that if equals are added to equals, the results will be equal, and other similar ones; from which it is easily proved that the three angles of a triangle are equal to two right angles, etc.”
But the mind “does not yet know whether it was perhaps created of such a nature that it errs even in those things which appear most evident to it.” Therefore “the mind sees that it rightly doubts such things, and cannot have any certain knowledge until it has come to know the author of its origin.”
So mathematics depends on intuition, and intuition is something implanted into the mind. So God made us have these intuitions. So justifying our innate intuitions depends on the nature of God.
Here is Descartes’s proof that “a supremely perfect being exists”: “That which is more perfect is not produced by a cause which is less perfect. There cannot be in us the idea or image of anything, of which there does not exist somewhere, some Original, which truly contains all its perfections. And because we in no way find in ourselves those supreme perfections of which we have the idea; from that fact alone we rightly conclude that they exist, or certainly once existed, in something different from us; that is, in God.”
“It follows from this that all the things which we clearly perceive are true, and that the doubts previously listed are removed,” since “God is not the cause of errors,” owing to his perfection, because “the will to deceive certainly never proceeds from anything other than malice, or fear, or weakness; and, consequently, cannot occur in God.” “Thus, Mathematical truths must no longer be mistrusted by us, since they are most manifest.”
So, in summary: Euclid’s axioms are true because we innately feel them to be true, and this intuition was implanted into us by God. Our intuition is reliable because God is not a deceiver because he is a perfect being. God must be perfect, because we have the idea of perfection, and we could only get that idea from actual perfection. Since we can conceive of perfection, there must be perfection, there must be a perfect being, a perfect God. That God has hardwired truths such as the Euclidean axioms into our minds. And they must be right because God wouldn’t be perfect anymore if he tricked us by implanting false beliefs in our minds.
That’s Descartes’s argument. I think it’s interesting how we can tell this entire story as driven almost completely by the analysis of Euclid. This whole thing about God and so on it almost like an afterthought, or a minor stepping-stone. The real goal is to justify the geometrical method or explain why Euclid’s axioms should be believed. All this philosophy and theology stuff—I think therefore I am, the existence of God—those are just supporting characters or secondary concerns. Or at least that’s one way of reading Descartes.
In any case, in Descartes as in the previous philosophers we have discussed today, we have seen the very profound influence of ancient geometry. Euclid was still setting the course for philosophy, two thousand years after his death. All the more reason to study him further.
The use of diagrams in geometry raise questions about the place of the physical, the sensory, the human in mathematical reasoning. Multiple sources of evidence speak to how these dilemmas were tackled in antiquity: the linguistics of diagram construction, the state of drawings in the oldest extant manuscripts, commentaries of philosophers, and implicit assumptions in mathematical proofs.
Transcript
Diagrams. What are their role in geometry? Some people like to think that the logic of a geometrical proof doesn’t need the diagram. Mathematics is supposed to be pure and absolute. Diagrams seem connected to the visual, the intuitive, that makes it kind of psychological, and perhaps therefore even subjective.
Certain people don’t like that association one bit, so they try to minimise the role of diagrams. Maybe diagrams are just crutches to help those with weaker minds, whereas a perfect logical reader could follow the proof from the text alone. Some people like to think so. It’s a dogma that fits modern tastes.
But, historically, that interpretation is a pretty poor fit. In some ways, classical geometry appears to have embraced visuality rather than tried to replace it with abstract logic.
There are signs of this attitude in the very language of Greek geometry. The word for proving is the same as the word for constructing: grafein, to draw. To prove something is literally to make it graphic. And a theorem, in ancient Greek, is a diagramma, a diagram. Instead of the Pythagorean Theorem the Greeks would say the Pythagorean Diagram.
Indeed there is always one diagram for each theorem in Greek mathematics. That’s a very rigid rule. In modern mathematics we often find it natural to have several pictures for some proofs, and no pictures at all for many other proofs. Just do what comes natural to explain the particular content. But not the Greeks. One theorem, one picture: this rule was extremely firmly ingrained in their conception of geometry.
And not only in geometry, in fact. Euclid follows this rule slavishly even when he writes about number theory. For example, he proves (Elements VII.30) that if a prime number divides a times b, then it divides either a or b. A very important theorem that is still proved in every modern book on number theory. But no modern book would include a picture for this. It’s just not a visual thing at all, so it makes little sense to draw a picture to go with it.
But Euclid does. The numbers that he is talking about he draws as line segments. The bigger the number the longer the line. But this has little to do with his proof. The proof is not visual. It’s just as abstract as the ones in the modern books. So the diagram doesn’t really do anything. And it’s like that theorem after theorem after theorem: Euclid has these useless diagrams that are basically irrelevant to the content. But he insists on the rule “one theorem, one diagram” even where it doesn’t really seem to serve any purpose.
At least it doesn’t serve any purpose in terms of capturing or visualizing the steps of the proof. Maybe it has other purposes. One purpose could be to signal that number theory is subsumed by geometry. The number 5 really just means a line segment of length 5 units, Euclid seems to be saying with these diagrams. So since Euclid has established the foundations of geometry, and number theory so to speak lives within geometry, then it follows that Euclid has established the foundations for number theory as well. Number theory doesn’t need separate foundations since it is subsumed by geometry. Maybe this is what Euclid is trying to emphasize with his pictures of numbers.
Or maybe Euclid needs pictures because he doesn’t have algebra. A modern proof of theorems like these are very dependent on algebraic notation. If p divides ab, then p divides a or b. In the course of the proof you keep referring to relationships between these number all the time. Suppose p divides ab but not a. Etc., etc. It would be hard to get all that across without algebraic symbols.
If you have a picture you don’t need algebra, because you can point. Instead of the letters a, b, p you have line segments of different lengths that you can point to and say: suppose that one dives that one. You don’t need algebraic symbols or letters, because you are pointing to a picture. The mode of presentation is oral; you have your audience in front of you, and you have drawn the diagram in the sand with a stick, and you point to it as you reason your way through the proof.
You might say: But Euclid does have labels, like A, B, C, etc. So he is referring to entities by letter or label designation, not merely by pointing visually. Well, maybe. But one could argue that that’s not really what Euclid’s A, B, Cs mean.
When Euclid calls things alpha, beta, gamma, it is perhaps inaccurate to translate this as A, B, C. Because it would also mean 1, 2, 3, or first, second, third. The Greeks wrote numbers this way, using the letters of the alphabet. Alpha meant 1, beta meant 2, and so on. So perhaps we shouldn’t think of Euclid’s ABC as algebraic designations. Perhaps it simply means “the first point,” “the second point,” and so on.
This makes it seem a lot closer to the pointing hypothesis. Perhaps the standard way for mathematicians to explain their reasoning was to point to a picture and say “this one,” “that one,” and so on. Then to encode this in writing they used alpha, beta, gamma, to mean “the first one I mentioned,” “the second one I mentioned,” and so on.
If this is right, then the letters in the English version of the Elements are a bit deceptive. They seem more algebraic, more modern, than they really are. From that point of view, diagrams in number theory make some sense.
In fact, in early modern geometry, in the 17th century, you sometimes see people labeling points in diagrams 1, 2, 3 instead of A, B, C. Because they thought this was the right way to translate Greek into Latin. Euclid’s alpha is really a 1, and so on. They were more sensitive to Greek culture back then. Nowadays people have forgotten about that stuff.
Here’s another fun linguistic-cultural perspective on diagrams in Greek geometry. The language in which Euclid describes constructions is quite odd. “Let the circle ABC have been described.” The language of Greek mathematics “makes the author and temporality disappear from a proof,” as one historian has put it. Euclid is not saying that he’s drawing the diagram, and he’s not telling the reader to draw the diagram. He’s just sort of commanding the diagram into existence.
You know the book of Genesis in the Bible: “Let there be light,” God said, and there was light. Euclid uses literally the same kind of construction. It’s exactly the same verb form as in the Ancient Greek version of the Bible. Just as God makes heaven and earth by merely pronouncing that they exist, so Euclid makes geometrical objects appear just ordering them to be. It’s not “I draw” or “you draw” but “let it have been done.”
You could read this as supporting a Platonic conception of mathematics. Euclid is distancing himself from actual drawing. The objects of mathematics just are. They are not something you or I have to make.
But here’s a counter argument to this interpretation. Netz argues that actually Euclid’s grammatical construction merely reflects a purely practical circumstance of the Greek tradition. Namely, that Greek mathematicians had to prepare their diagrams in advance due to technical limitations of the visual media available. Here’s what Nets writes:
“Of the media available to the Greeks none had ease of writing and rewriting. [Standard media were papyri and wax tablets, and, for larger audiences, such as Aristotle’s lectures,] the only practical option was wood painted white. None of these [ways of representing figures] is essentially different from a diagram as it appears in a book. The limitations of the media available suggest the preparation of the diagram prior to the communicative act---a consequence of the inability to erase. This, in fact, is the simple explanation for the use of perfect imperatives [such as] ‘let the point A have been taken’. It reflects nothing more than the fact that, by the time one comes to discuss the diagram, it has already been drawn.”
That’s Netz’s interpretation, and if he’s right then Euclid’s grammatical choice reflects only incidental cultural circumstances and says nothing about philosophical commitments.
So “let it have been done” just means “I did it yesterday”. It doesn’t mean that geometry is set apart from concrete action and that doing has no place in mathematics.
It’s fascinating how the same aspect of the text takes on such a different meaning when cultural context is taken into account, compared to a purely philosophical reading. In fact, let me tell you about another striking aspect of Greek manuscripts which is also like that. Namely, the way diagrams are drawn in manuscripts of Greek geometry.
Diagrams in manuscripts of Greek mathematical treatises are very often very poorly drawn. They are oversimplified and crudely schematic. Ellipses, parabolas, and hyperbolas are represented as pieces of circles and so on. Very poor pictorial accuracy.
Also the simplicity and specificity of the diagrams often obscure important mathematical points. For example, the figure for the Pythagorean Theorem is often drawn in manuscripts with the two legs of the triangle being equal, even though the theorem holds for any right-angle triangle. The diagram thereby gives the misleading impression that the theorem is less general than it really is.
So you might think: aha, clearly the Greeks didn’t care about the diagrams. They are poorly executed, poorly thought through. So diagrams couldn’t have been an important part of geometry then.
Well, not so fast. The diagrams are drawn this way in the manuscripts that exist today. But who wrote these manuscripts, and when? In fact, the oldest manuscript of Euclid’s Elements that exists today is closer to us in time that it is to Euclid. It’s from the Middle Ages. A thousand years ago. That might seem ancient enough, but Euclid lived thirteen hundred years before that.
There was no printing press until the 15th century, so for well over a thousand years the book had to be copied by hand. You had to hire a scribe to write the whole thing out.
Manuscripts are fragile. The Greeks wrote on papyrus. It takes a miracle for a roll of papyrus to survive more than two thousand years. Just think of books from the 19th century, maybe some old book from your grandparents. They are already falling apart, and that was only a hundred years ago. Imagine storing that for twenty times as long. It will fall apart on its own, and that’s not even counting the risk of fires, or floods, or insects, or wars, and so on.
So few documents from Greek times survive to this day, and hardly any of those are mathematical. Only the tiniest little scraps of mathematics from antiquity itself are still around. And they are not enough to say anything about how the Greeks dealt with diagrams.
We only have these later copies. Or better put: a copy of a copy of a copy of a copy and so on. Our oldest manuscript may very well be, who knows, maybe twenty or thirty copying steps away from Euclid’s original.
The state of the diagrams in these manuscripts perhaps says more about the copying and the copyists than it does about Greek geometry. The scribes who copied these manuscripts probably often knew little or no mathematics. They probably had some training as scribes; training in Greek, in writing. Perhaps they mostly copied literary texts or whatever.
So they were probably pretty good at copying text, but not at copying diagrams. It’s pretty straightforward to copy text if you know the language. An A is an A. You can’t really misinterpret it.
Diagrams are a lot more subtle. Often you can only understand what aspects of a diagram are essential by studying the text, the logic of the proof that goes with it. But the scribes would not have done this. They were hired copyists, not research students. They didn’t study the content, they just blindly copied it for a paycheck, like a photocopier.
This is enough to explain why the diagrams are so simplistic. It is natural in such a context of copying that the diagrams gradually degenerate and converge to more simplistic versions. This is the predictable outcome of repeated copying by generations of scribes largely ignorant of mathematical content. For a very simple reason: an ignorant copyist can easily misinterpret a subtle diagram in a simplistic way while going the other way around, toward a more subtle and exact diagram, could only be done by someone with a solid understanding of the mathematical content, who would restore the diagram based on what the text suggests.
For example, in the case of the Pythagorean Theorem, a scribe might get a version of the figure where the two legs look approximately similar and then mistakenly assume that exact equality was intended. He then copies it this way, and specificity is introduced. Now others will keep copying this simplified diagram. No one will restore more generality in the diagram, because that would require revising the figures based on mathematical understanding, which was not the task of copying scribes.
There’s a fun paper on this by Christian Carman in a recent volume of Historia Mathematica. Carman tested this hypothesis with his students. He had them go in a circle and copy a mathematical diagram from one another, like the children’s game Chinese whispers or telephone where you whisper something, then they try to pass it on, and so on. By the time the message has made it full circle it has become something else. It’s the same with diagrams.
You can see also how the specificity aspect emerges from this. The original diagram might show two lines meeting at an angle of, say, 75 degrees. Copying is a bit imperfect, so maybe someone copies it more like 82 degrees. Then the next guy thinks: well, this is probably supposed to be 90 degrees, they just drew it a little bit wrong. So they make it 90 exactly. Then from that point on everybody copies it as 90 degrees. Because exactly 90 degrees looks a lot more intentional than 82. This is why the process almost always goes toward more specificity.
So we cannot conclude anything about ancient philosophy of mathematics from the way diagrams are drawn in the manuscripts. This aspect of the manuscript sources is very likely an artefact of transmission that says nothing about ancient geometry.
So we still don’t know what Euclid thought about diagrams. We know what Plato thought. His opinion was reportedly that mathematicians who “descended to the things of sense” were “corrupters and destroyers of the pure excellence of geometry.” That’s how Plutarch describes Plato’s opinion. So basically an anti-diagram agenda.
But there is no evidence that mathematicians shared these sentiments. On the contrary, the combative way in which this view is presented in the sources clearly show that they were far from a consensus opinion. Plato himself openly puts his view in diametrical contrast with that of the geometers. Here’s what he says in the Republic (VII 527):
“No one with even a little experience of geometry will dispute that this science is entirely the opposite of what is said about it in the accounts of its practitioners. They give ridiculous accounts of it, for they speak like practical men, and all their accounts refer to doing things. They talk of squaring, applying, adding, and the like, whereas the entire subject is pursued for the sake of knowledge [and] for the sake of knowing what always is, not what comes into being and passes away.”
Again, Plutarch reports on the same conflict and makes it crystal clear that Plato’s views on geometrical method was diametrically opposed to that of the leading mathematicians of his day. Here’s what Plutarch says: “Plato himself censured Eudoxus and Archytas and Menaechmus for endeavouring to solve the doubling of the cube by instruments and mechanical constructions.”
So not only is there no evidence that any notable Greek mathematician was a Platonist, but the Platonic sources themselves clearly and openly admit that their view is an ideological extreme that was not widely shared, especially not among mathematicians.
So what’s the alternative? If the mathematicians were not Platonists, what were they? Maybe the didn’t care about philosophy at all. Here’s how Netz puts it:
“Undoubtedly, many mathematicians would simply assume that geometry is about spatial, physical objects, the sort of thing a diagram is. The centrality of the diagram meant that the Greek mathematician would not have to speak up for his ontology. The diagram acted, effectively, as a substitute for ontology. One went directly to diagrams, did the dirty work, and, when asked what the ontology behind it was, one mumbled something about the weather and went back to work.”
That’s what Netz thinks, and it seems consistent with Plato’s rants against the geometers that they would have been disinterested in these questions indeed.
But I think Netz is selling the mathematicians short. I do not believe that Greek mathematicians “simply assumed” these things, and could only “mumble something about the weather” if pressed on the issue. I suspect that, on the contrary, Greek mathematicians had a philosophically sophisticated defence of their ontological stance, based on the operationalist ideas that we discussed before.
Let’s see how this plays out in a concrete mathematical example. From a modern point of view, the right way to do geometry is as a formal axiomatic-deductive system. The Greek tradition has often be interpreted as aspiring toward, but falling short of, this ideal. According to this view, Euclid’s Elements was a brave and admirable attempt at a formal treatment of geometry, especially for its time, but that it contains some fundamental flaws stemming from Euclid’s inability to fully avoid implicit reliance on intuitive and visual assumptions.
Operationalism, by contrast, embraces visual reasoning and keeps abstract logic at arm’s length. This arguably fits the Greek geometrical tradition better than modern formalistic conceptions of geometry. Indeed it is well known that Greek geometry sometimes bases inferences on diagrammatic considerations that are not explicitly formalised.
The most famous example is Proposition 1 of the Elements. In this proposition, the existence of a point of intersection of two circles is tacitly assumed but can arguably not be formally justified from Euclid’s definitions and postulates.
The modern mathematician rejects anything not obtained through logical deduction from formal axioms. The operationalist classical geometer rejects anything not obtained through concretely defined operational procedures. We can formulate the difference between the two points of view in terms of what kind of audience the geometer is trying to convince. If we adopt the modernistic point of view, we can picture the audience of a mathematical proof as a veritable logic-parsing machine. The mathematician feeds in statements, in the form of symbolic strings in a suitable formal language, one by one, and the machine tests whether each statement follow from the one before it based on basic logical inference rules or previously established theorems. This point of view fits very uneasily with classical geometry for a range of reasons, including the use of diagram-based reasoning.
The operationalist point of view, on the other hand, envisions the audience of a mathematical proof differently. A Euclidean proof is addressed at a person with a ruler and compass. This person is every bit as critical as the logic machine of the modernists. He is hell-bent on trying to argue against us at every stage. But our strategy for convincing him to nevertheless concede the truth of our theorems is not by appeals to formal logical inferences. Instead we make him draw things. We build our results up from simple operations with ruler and compasses. In this way we put our critic in a difficult position. He is forced to either agree with us, or to deny a very specific, concrete claim about a very specific, concrete figure that he himself has drawn.
For instance, what is the person with the ruler and compass supposed to say regarding the intersection of the circles in Proposition 1 of the Elements? He just drew the two circles himself on a piece of paper. It would be ridiculous for him to claim that there is no justification for the assumption that they intersect. They clearly intersect right there in front of his eyes, and it was he himself who drew it using tools whose validity he had admitted.
Since operationalism gives absolute primacy to the concretely constructed diagram, the sceptic has no other foothold from which to reject the proof. The logic machine of the modernist paradigm would catch the gap in Proposition 1 at once, and shoot down our proof. But operationalist mathematics is not susceptible to that kind of critique. Geometrical proofs are claims about what happens when you carry out concrete constructions. Constructed diagrams is all there is, so the only way to question a geometrical proof is to question what it says about a concretely constructed diagram. The sceptic cannot hide behind sophistical logic and vague generalities, but is forced to either concede the validity of the proof or deny something so obvious that he will look ridiculous.
The conception of a proof as addressing a sceptic fits the Greek context well. It’s just like a Socratic dialogue. You extracting concessions from a determined opponent in incremental steps. Just as Socrates does in the dialogues of Plato. And just as disputants would aim to do in a stage debate of the kind the Greeks loved.
I think one could argue that the diagrammatic inferences Euclid permits are precisely those that such a sceptic, who has drawn the diagram himself, could not reasonably doubt. This fits well with Kenneth Manders’ observation that Euclid permits diagrammatic inferences only of properties of the diagram that are invariant under minor variations or imperfections in the drawing process.
For example, in Proposition 1 of the Elements, the equality of the legs of the triangle can of course not be established merely by visual inspection of the diagram; rather, these equalities have to be derived from postulates and definitions, as do all exact properties of diagrams in Euclid’s geometry. Indeed, a sceptic could very well question whether such properties hold, despite having just constructed the diagram himself. The equality of the legs is not immediate from the diagram in and of itself, but only follows when we remind ourselves that we used the same radius for both circles and so on. You could draw the diagram without keeping such things in mind. You could not, however, draw the diagram without directly experiencing one circle cutting unequivocally right through the other one.
Operationalism relies on diagrammatic reasoning only in this restricted sense. It attributes foundational status to diagrams in certain respects, but of course it does not go so far as to say that the truth of propositions or veracity of solutions to problems can be verified merely by measurements in a diagram. Of course such things have to be established by rigorous demonstration, which is obviously the main preoccupation of Greek mathematical sources.
What Plato says about inferring geometric truths from diagrams remains true also for operationalists. This is a quote from the Republic (VII 529): “If someone experienced in geometry were to come upon [diagrams] very carefully drawn and worked out, he’d consider them to be very finely executed, but he’d think it ridiculous to examine them seriously in order to find the truth in them about the equal, the double, or any other ratio.”
Indeed, exact properties such as ratios cannot be inferred from diagrams, no matter how carefully drawn, just as Plato says. But the operationalist enterprise does not rely on such epistemic overreach. Instead, its use of diagrammatic reasoning is much more restrictive and limited to essentially qualitative or topological or inexact inferences from diagrams.
So operationalism makes sense of Euclidean practice with regard to diagrammatic reasoning. It eliminates the need to attribute to Euclid a big logical blunder in his very first proof, or the need to denigrate the more visual aspects of Euclid’s reasoning as lowly intuition and an imperfect form of mathematics. Instead it articulates a philosophy of mathematics that incorporates this aspect of Euclidean mathematical practice into a coherent and purposeful whole.
So that’s one way to argue that the so-called logical gap in Euclid’s Proposition 1 is not a gap at all. It’s only a gap if you want geometry to be completely reduced to formal logic. From the point of view of operationalism it is not a gap.
There are other ways to try to save Euclid’s proof. More conservative ways. If you know some modern mathematics it’s a fun game to play to try to read all kinds of things into Euclid’s definitions.
For instance, Euclid’s definition of a circle specifies that it is contained by a single curve, and that it has an inside, and by implication also an outside. In Proposition 1, when you draw the second circle, it is evident that the second circle will have some points inside and some points outside the first circle. So you could argue that, topologically, there’s no way a continuous curve could go from the inside of a closed curve to the outside of it without crossing it. Therefore the existence of the intersection point can be regarded as implied by Euclid’s definition rather than a logical gap.
If you are a modern mathematician you might reply: well, that depends on the underlying field! The argument works for the plane of real numbers, but not if the underlying field is that of rational numbers only. Then indeed the intersection does not exist. So Euclid would have to specify the underlying field before the argument based on inside and outside could work.
Interestingly, one could argue that Euclid sort of does this actually. Because he says in Definition 3 that “the extremities of lines are points.” Now if you wear your modern glasses, you can read this as saying that lines contain their limit points. So the Euclidean plane is a complete metric space. So that rules out the argument based on the rational numbers.
Well, if you know modern mathematics it’s fun to think along these lines, but for my part I vote for the operationalisation reading of Euclid as the more historically plausible way of saving Euclid’s proof of Proposition 1.
Here’s an objection though to the operationalist interpretation. The so-called generality problem. Geometrical theorems are about entire classes of objects---infinite sets of them. For instance, the angle sum of all triangles. Yet all geometrical proofs in the classical tradition are always illustrated with, and reason based on, one particular diagram. The standard way to defend geometrical reasoning against this challenge is to say that geometrical proofs concern only properties that hold generally and do not rely on incidental properties that hold only for the particular diagram. This view was expressed already by Proclus.
Operationalism suggests a very different way of dealing with the generality problem: it denies the premiss that there is such a thing as “all triangles” in the first place. Before you have put your pen on the paper, there is no geometry. There are no lines, no circles, no triangles. We do not make the metaphysical assumption, as the modernists do, that there is some preexisting universe of these things “out there” about which geometry looks for universal truths.
From this point of view, the “problem” of generality ceases to exist. The theorem is not: there is an infinitude of triangles and all of those have angle sum 180 degrees. Instead it is: any triangle has angle sum 180 degrees. Which really means: if you put your ruler down and draw a line segment, then another one, then another one, then the angles of that one triangle has angle sum 180 degrees. The theorem has no other meaning than that. And the proof is not a logical schema talking about an infinite class of objects. Rather, it is a set of instructions for the sceptic to carry out that will convince him, regardless of which triangle he started with, that the theorem is true for that triangle. It is precisely the strength of the insistence on constructions to reduce everything from the abstract to the concrete in this way. We only talk about what we can see and draw and put on the table right in front of us. To do otherwise would be to engage in empty metaphysics, according to operationalism.
Greek geometry is remarkably consistent with such a reading. Indeed, as Netz has observed, Greek mathematical texts never explicitly claim generality beyond the concrete proof based on a particular diagram.
From a modern point of view, any reliance on diagrams in mathematics is inherently problematic, since mathematics is in essence independent of diagrams. On this view, diagrams are merely a secondary representation of mathematics, and furthermore one contaminated by intuition and other limitations. How, then, can diagrammatically based reasoning be a legitimate way of doing mathematics? That is, how could we ever be sure that what is true of diagrams is true of the “actual” content of mathematics? Operationalism does not answer the question but rejects it. There is nothing more “actual” than the diagram.
So the generality problem is dissolved since operationalism rejects the Platonist ontology of mathematics on which it is based. Nothing exists except what the geometer has constructed.
This view re-emerged in modern mathematics for reasons independent of classical geometry. Here’s how famous Dutch intuitionist Brouwer puts it it his dissertation:
“Wheresoever in logic the word all or every is used, this word, in order to make sense, tacitly involves the restriction: insofar as belonging to a mathematical structure which is supposed to be constructed beforehand.”
There is no “all triangles.” There is only “all the triangles you have made.”
To be sure, many who are concerned about the generality problem will feel that operationalism “solves” the problem only by introducing further problems of equal or greater magnitude. For one thing, operationalism implies that “the very nature of meaning itself makes it impossible to get away from the human reference point,” as Bridgman puts it, since nothing exists or has meaning in geometry except through human agency. But operationalism denies that this is a problem, as Platonists would have it.
Regarding the generality problem more specifically, a modern mind may feel that the operationalist solution merely shifts the problem one step over. Even the operationalist is committed to a form of generality, in the sense that the proof of, say, the angle sum theorem must always work for any given triangle. Isn’t the operationalist mathematician still obligated to somehow justify that the proof has this form of generality, which is essentially the original generality problem in slightly different guise?
It is of course true that the proof is intended to be general in this sense, but officially the operationalist mathematician does not need to be committed to having proved that it is. The operationalist mathematician can simply say: I assert that such-and-such a construction will always have such-and-such an outcome; if you want to prove me wrong, feel free to try to come up with a counterexample.
Of course, psychologically the mathematician presenting a proof must be convinced that it will always work, for if a counterexample would be forthcoming he would be exposed as a fool. But this can be left to the discretion of the mathematician’s intuition. Internally, operationalist mathematicians are of course concerned with this kind of generality. But externally, as a reply to sceptical and philosophical challenges to the epistemological status of mathematics, there is no need for them to saddle themselves with the burden of claiming that their proofs themselves have inherent characteristics that strictly ensure such generality. Instead they can restrict themselves to presenting the proof as a challenge to any sceptic: apply these construction and inference steps to any one figure that fulfils the conditions stated, and you will find that you cannot credibly doubt the validity of any step, and hence you will become convinced that the proposition holds for that figure. It is possible, for the operationalist, to maintain that this is what a proof is.
One may well feel that this restrictive view of what a proof is sells mathematics short and fails to account for the nature and status of mathematical knowledge. However that may be, the fact remains that operationalism makes it possible to take such a stance. The restrictive view of the nature of proofs fits naturally with the operationalist conception of mathematical content and meaning, while it is incompatible with a Platonist conception of the nature of a mathematical theorem.
The restrictive view is a scorched-earth defensive position that can be useful when under philosophical attack. Saying that this is the only sense of mathematics one is willing to defend against sceptical attack does not preclude one from holding more expansive, Platonist beliefs in private. But it is a powerful way of cutting off lines of philosophical attack without changing the practice of mathematics substantially.
So, in conclusion, I have argued that Greek mathematicians were prepared to base geometry on actual diagrams. Despite their physicality, despite their links to human action and perception. Greek mathematics went against modern tastes in this respect.
One could argue against this by pointing to the crudeness of diagrams in surviving manuscripts, or the strangely passive language that Euclid and others used to describe constructions of diagrams. But we have seen that those things can better be explained as the result of cultural context rather than philosophy of mathematics.
The modern view that geometry should be studied through abstract reasoning not dependent on the visual and the physical also has ancient support in Plato’s philosophy. But Plato was not a mathematician. In the words of Francis Bacon, when “human learning suffered shipwreck [at the end of classical antiquity], the systems of Aristotle and Plato, like planks of lighter and less solid material, floated on the waves of time and were preserved,” while more mathematically advanced works were lost forever.
To understand ancient mathematics we must look beyond the surface. We must look beyond loudmouths like Plato. We must seek instead the assumptions conveyed implicitly in the way the mathematicians wrote their proofs. Based on this kind of evidence, a diagram-based mathematical practice can be plausibly reconstructed.
Euclid spends a lot of time in the Elements constructing figures with his ubiquitous ruler and compass. Why did he think this was important? Why did he think this was better than a geometry that has only theorems and no constructions? In fact, constructions protect geometry from foundational problems to which it would otherwise be susceptible, such as inconsistencies, hidden assumptions, verbal logic fallacies, and diagrammatic fallacies.
Transcript
Ancient Greek geometers were obsessed with constructions. Why?
Euclid’s Elements spends almost as much time showing how to draw geometrical figures as it does proving theorems about them. In fact, it seems Euclid thought drawing was a prerequisite for proving. For instance, the first theorem involving squares is the Pythagorean Theorem. In the proposition right before it, Euclid explains in detail how to construct a square by ruler and compass. The same goes for every other geometrical entity ever used in the Elements: first you construct it, and only then can you say anything about it. Without constructions there can be no geometry, Euclid seems to be saying.
And not only Euclid. All the best Greek geometers had their own signature constructions. Three famous construction problems dominated higher geometry for centuries: doubling the cube, trisecting the angle, squaring the circle. The long list of mathematicians who contributed their own distinctive solutions to these problems is a who’s who of everybody who was anybody in ancient geometry.
What fundamental motivations—what philosophy—drove ancient Greek geometers to this fixation with constructions? Why did Greek mathematicians think it was a good idea to spend hundreds of years trying to make an angle the third of another, or a cube twice the volume of another, in dozens of different ways? Why did they so stubbornly bang their heads against the same wall for century upon century? What sin could be so grave that they imposed on themselves such a Sisyphean task?
Why indeed make things at all? And why do so only sometimes? Why meticulously articulate recipes for transferring line segments by ruler and compass, only to then suddenly move entire triangles like it’s nobody’s business in the very next proposition, as Euclid seemingly does? (When he uses superposition to prove triangle congruence in Proposition 4.)
Euclid knew what he was doing, in my opinion. Constructions were a deliberate strategy to guard against fundamental threats to the reliability and rigour of geometry. If our house is built on rotten pillars it’s only a matter of time before it comes crashing down.
Some ancients critics of geometry indeed identified some ominous cracks in its foundations. Remember, the quarrelsome Greeks, they questioned everything with zeal. Some people tried to take down geometry. They were determined to show that it was pseudo-knowledge that was by no means as certain and exact as the mathematicians claimed.
Geometers had to deal with such external attacks. More so than in other centuries and cultures, mathematicians in Ancient Greece were under constant critical-philosophical attack. They had to formulate a defense. And they did, in my opinion. This is where their obsession with constructions comes from.
So what were these philosophical attacks, to which constructions were the answer? There are a number of them. Here’s one:
False diagram fallacies. If you draw diagrams that are slightly off, you can easily fool yourself when doing geometry. There’s a famous example for instance where one proves that any triangle is isosceles. The conclusion is obviously absurd. But it is “proved” in a way that looks just like any other proof in Euclid.
The false proof is made possible by a subtle error in the diagram. The proof involves bisecting one of the angles of the triangle, and then raising the perpendicular bisector of the opposite side. These two lines meet somewhere. That’s drawn in a plausible-looking way in the diagram. The proof then proceeds based on the diagram, just as Euclid does in his proofs.
But the way the diagram was drawn was erroneous. The two lines were drawn as meeting inside the triangle when in fact their true intersection would be outside the triangle. This is a subtle issue that is easy to miss. So when we reasoned based on the diagram we made some hidden assumptions that we were hardly even aware of.
The rest of the proof was typical Euclid-style stuff. So this example shows that a small and subtle mistake in the way we drew a diagram can destroy the certainty of geometrical reasoning. All the other steps of the proof were very carefully justified, just as Euclid always justifies each of his steps. But that was all for nothing since the subtle error in the diagram poisoned the well and destroyed the whole thing.
The Greeks were evidently well aware of this type of problem. Plato mentions it explicitly: “geometrical diagrams have often a slight and invisible flaw in the first part of the process, and are consistently mistaken in the long deductions which follow.” (Cratylus, 436d) Plato is exactly right. “A slight and invisible flaw” at the outset is enough to ruin “the long deductions which follow.”
We even know for a fact that Euclid himself wrote a (now lost) treatise on fallacies in geometry which is likely to have dealt with these kinds of issues. So the Greeks were clearly well aware of this threat to geometrical certainty. What did they conclude from this?
Today, the issue of diagram fallacies is taken to show how dangerous it is to rely on visual and intuitive assumptions. The solution is to purge geometry of any kind of reasoning based on diagrams. In the late 19th century this view was expressed forcefully by leading geometers, and it has remained the mainstream view ever since. “A theorem is only proved when the proof is completely independent of the diagram,” as Hilbert said for example. Instead of relying on pictures, geometry must be made to proceed through purely logical deduction.
But this is not the only possible diagnosis and treatment of the problem with the erroneous proof. Another point of view is to say: the problem is not that the proof relied too much on diagrammatic reasoning, but that it did so too little. The problem is not that the proof is insufficiently divorced from visual considerations, but that it is too divorced from them. The example doesn’t show that diagrams are dangerous even if they are just schematic accompaniments to otherwise logically solid proofs, but rather that diagrams are dangerous when they are merely treated as such.
The solution is not to place less emphasis on diagrams, but more. That is, to demand diagrams to be not merely schematically sketched but in fact precisely constructed according to the most exacting standards and rigorous proofs that these constructions accomplish the configurations in question. This would indeed prevent errors of this type from occurring. No one adhering to this mode of doing geometry would ever find themselves reasoning about false diagrams like the one in the above example.
This diagnosis of the source of error in the false proof above leads immediately to the conclusion that precise constructions of angle bisectors, bisectors of segments, and perpendicular lines are foundationally very important, and that no proof must ever be formulated without constructive recipes for all entities occurring in it having been established beforehand.
And this is exactly what we find in Euclid’s Elements. Without fail, Euclid always meticulously shows how to construct all entities involved in all of his propositions. And all the constructions needed to ensure that we end up with the correct figure rather than the deceptive one in the above example are carefully spelled out as core propositions right at the heart of the Elements: how to bisect an angle (Proposition 9), how to bisect a line segment (Proposition 10), how to raise a perpendicular from a point on a line (Proposition 11), how to drop a perpendicular from a point to a line (Proposition 12).
In other words, right off the bat of the Elements, Euclid carefully explicates precisely the tools needed to solve the false diagram problem mentioned by Plato. Coincidence? I don’t think so. Euclid knows the problem. Euclid knows how to solve it. That’s why he’s obsessed with constructions.
Or rather, it’s one of the reasons. There are other, equally compelling grounds to base one’s geometry on constructions.
Constructions are related to existence issues. It is impossible to conduct a serious axiomatic study of geometry without paying attention to existence issues. For example, do squares exist? Existence is separate from definition. Euclid defines what a square means in his definitions. But that doesn’t mean there are any. You could also define what “unicorn” means. That doesn’t mean unicorns exist.
You might think it’s obvious: of course there are squares, any child realizes that, you can see it with your own eyes. But it’s more subtle than you might think. In fact the existence of squares implies the parallel postulate. Wallis showed this in the 17th century. You could replace Euclid’s parallel postulate with the assumption that you could make a square on a given line segment. Then you could prove Euclid’s parallel postulate and all his other theorems based on that assumption. So it’s no small matter to assume that there are squares.
Therefore, any investigation that aims to elucidate the fundamental assumptions of geometry cannot treat any object whose existence has not first been either proved or explicitly postulated. To do otherwise would be to render the entire enterprise of axiomatic geometry useless and moot, since it would open a back door through which any number of hidden assumptions can creep in. The point of an existence proof for squares, then, would not so much be to establish that there is such a thing as squares, but to ensure that any foundational assumptions involved in supposing the existence of squares have been systematically accounted for.
Another example of this type occurs in Legendre’s attempt to prove by contradiction, using only the first four postulates of Euclid, that the angle sum of a triangle cannot be less than 180 degrees. His proof implicitly assumes that given two intersecting lines, and a point not on those lines, it is possible to draw a line through that point that intersects the two given lines. This assumption does not hold in hyperbolic geometry. Therefore Legendre’s attempted proof is worthless, since the contradiction did not come from the assumption he intended to refute, but from an innocent-seeming existence assumption introduced along the way in his argument.
This shows once again the danger of letting even the most harmless-looking existence or construction assumptions proliferate without explicit control. Inconsistencies can arise from even the most inconspicuous of assumptions. The moral of the story is that the mathematician must stick to a minimalistic set of stringently controlled construction principles, whose consistency should be as unquestionable as possible.
Issues of this nature were recognised in antiquity. Quite possibly, even the specific issue of Legendre’s assumption may have been investigated in works that are no longer extant, such as the lost treatise On Parallel Lines by Archimedes. At any rate, closely related issues emerge explicitly in the treatments of parallels by Simplicius and Al Jawhari.
On a more conceptual level, Aristotle pinpoints the same type of fallacy in the work of some “who suppose that they are constructing parallel straight lines: for they fail to see that they are assuming facts which it is impossible to demonstrate unless parallels exist. So it turns out that those who reason thus merely say that a particular thing is, if it is.” (Prior Analytics, 65a) Indeed. Aristotle is right. Circular assumptions are easy to make, especially with respect to existence issues and subtle foundational questions in the theory of parallels.
Aristotle draws the obvious conclusion that existence issues must be controlled by either explicit postulates or existence proofs. “What is denoted by the first [terms] is assumed; but, as regards their existence, this must be assumed for the principles but proved for the rest,” says Aristotle (Posterior Analytics, 76a). For example, “what a triangle is, the geometer assumes, but that it exists he proves.” That’s Aristotle again (Posterior Analytics, 92b). He’s quite right.
Constructions are a way to ensure existence. Euclid’s first proposition proves that equilateral triangles exist. His 46th proposition proves that squares exist. And so on.
But there’s much more to constructions than merely establishing existence. Construction also establish consistency. That is, it shows that objects are not self-contradictory.
For example, suppose I add to Euclid’s Elements the definition: “A superright triangle is a triangle each of whose angles is a right angle.” Then its angle sum is three right angles by definition, but also two right angles according to a theorem of Euclid’s. So two right angles equal three right angles—an obvious contradiction.
The definition of a superright triangle is disturbingly similar to that of an equilateral or isosceles triangle, and applying Euclid’s theorem to it sounds just like the kind of thing we do in geometry all the time. So it casts doubt on the entire enterprise of geometry. How do we know that the propositions of the Elements are not one or two steps away from leading to contradictions? The geometers must reply with some definitive criterion that explains why none of their theorems are susceptible to this kind of error.
In a way it is clear what the problem is. There are no superright triangles. Hence one can consider the problem solved by ensuring the existence of the objects one speaks of. One way of accomplishing this would be to say: Only constructive definitions, that imply a recipe for making the object defined, are permitted in mathematics. This is clearly not the path taken by Euclid, however. For instance, Euclid defines a square at the outset but only shows how to produce one much later, in Proposition 46, based on substantial previous results.
Another strategy would be to demand that we cannot make a propositional statement about a particular class of objects unless we have first shown beforehand that the class in question is nonempty. Thus the types of inferences made in the false argument are only warranted if supported by suitable existence proofs, and that is why theorems about triangles cannot be applied to superright triangles, but can be applied to equilateral and isosceles triangles, which Euclid indeed proves exist by means of constructions.
But existence is not the only aspect that should be emphasised here. Another important lesson from the superright triangle example is the danger of defining objects through multiple conditions. A superright triangle is defined as: having three sides; having one right angle; another right angle; ant yet another right angle. The first two conditions were fine. It was taking all of them together that was impossible. The more conditions you add, the greater the risk of ending up with an inconsistency.
Another example of this is to say: Let ABC be a triangle such that: one angle is a right angle; the sides next to the right angle have lengths 4 and 7; the third side has length 9. Actually I have taken this example from a 16th-century geometry textbook. But the book messed up. Some of these conditions would have been fine on their own, but all of them taken together are inconsistent. You cannot make a right triangle with those side lengths. Those numbers contradict the Pythagorean Theorem.
Hence defining or introducing an object through a list of specifications of its properties is unacceptable. Doing so would leave the door wide open for possible inconsistency to enter mathematics, and hence ruin the claim to certainty of mathematical reasoning.
A rigorous mathematical theory needs a systematic guarantee that such errors cannot be committed. Constructions are a way to provide such a guarantee. Instead of introducing objects by a list of properties, construction builds it up step by step. Thus properties can no longer be ascribed to an object merely by decree. Rather they must be introduced by a rigorously controlled stepwise process. Each step in this process involves the application of a construction postulate or a demonstrated construction proposition or theorem, which means that assumptions and conditions of validity are carefully monitored and reduced to a few axiomatic principles.
One could argue that the challenge posed by the superright triangle fallacy is not convincingly solved by the insistence on existence proofs. This solution diagnoses the problem as effectively just another variation on the existence issue discussed before, which Aristotle mentioned and so on. But one can readily see it as pointing to a deeper problem. It arguably casts doubt on the credibility of verbal logic altogether. While it is clear that being more careful about existence issues would eliminate the particular problem of the superright triangle, it is not clear whether this is the only problem with relying on verbal logic.
We know for a fact that logical paradoxes and fallacies figured prominently in Greek thought in the classical era. Some of these are clearly relevant to mathematics, such as Zeno’s paradoxes of motion. But there are others.
The liar paradox arguably shows that natural-language propositional logic is incoherent. It shows that verbal logic allows propositional statements to be formulated that are inherently contradictory. “This statement is false” or “I am lying” are examples of such statements. If the statement is true, it follows that it is false. And if the statement is false, then it follow that it is true. So there is no way of assigning a truth value to such a statement without ending up with a contradiction. This kind of thing clearly poses an issue for a logic-based conception of mathematics, not least in connection with proofs by contradiction.
Another example of a paradox discussed in ancient times was that of the horn: What you have not lost, you have; but you have not lost horns; therefore, you have horns. Here again the blind, mechanical application of logical inferences in a quasi-algebraic manner leads to an absurd conclusion. As with the superright triangle fallacy, it is possible to attribute the problem to some specific cause: in this case not so much an existence issue as a certain misleading ambiguity in the first premiss. Furthermore, the fallacy may be regarded as “obvious.” But trying to defuse the paradox in these ways does not solve the core issue exposed by the paradox, namely that “blind” logic, in and of itself, seems to lead to erroneous conclusions.
This multitude of logical paradoxes arguably validates the suspicion that when we supplemented verbal logic with existence proofs we had perhaps not gotten to the bottom of all its problems yet.
It would not have been out of character for the Greeks to have taken radical steps to protect themselves from logical fallacies and paradoxes. The situation may be somewhat comparable to the discovery that the square root of 2 is irrational. This would have been in the very early days of Greek geometry and we don’t know much about it for certain. But a development more or less along the following lines has often been imagined.
In the beginning, the Greeks seem to have blissfully assumed that arithmetic and geometry would always be in natural harmony. The square root of 2 discovery ruined this by showing that natural geometric entities such as the diagonal of a unit square could not be represented by “numbers” (that is to say, rational numbers). The Greeks had burned their fingers and would not make that mistake twice: their response was an extreme foundational purge that eradicated any foundational status of arithmetic and then some. From that moment on, everything is at bottom geometry. Even where a modern mind wishes to see algebra, Euclid and the other Greek mathematicians insist on geometrical formulations with a pedantry bordering on paranoia.
The historical evidence, or absence thereof, of such a “square root of 2 crisis” is a much-debated issue among historians. But the basic point—that Greek mathematicians may very well have gone to great lengths to protect themselves from foundational objections—is plausible. It was a time when the foundations of any subject was constantly under attack from rival philosophers, and people were ready to go to the ends of the earth to rebut such charges.
Extreme action in response to paradoxes that call the bedrock of mathematics into question is a quite plausible scenario. And arguably one with a strong historical precedence in the form of the square root of 2 case. It is squarely within the realm of historical possibility that such a context may have led to the radical proposal of denying any reliance on abstract logic in mathematics and instead founding all of geometry on concretely constructed figures.
So verbal logic is dangerous. It invites paradoxes, like the liar paradox and the paradox of the horns. It has no guard against reasoning about inconsistent objects such as superright triangles. Making constructions, rather than logic, the foundation of geometry solves these problems.
It is suggestive in this connection that Euclid’s proofs are all “purely quantifier free.” That is to say, they never make assertions of the form “there exists” or “for all.” From the point of view of modern mathematics, which is a logic-oriented mathematics, those phrases are fundamental and are used all the time. These phrases are so commonly used that mathematicians do not even have the patience to spell out these two-syllable expressions every time they use them. So they have made up special symbols to abbreviate them. A backwards E and an upside-down A.
I mentioned Hilbert, a leading pioneer of the modernist movement, around 1900. I mentioned that Hilbert wanted to translate all visual information or any inferences based on diagrams into purely logical form. That leads precisely to formulations with those favorite phrases of the mathematician: “these exists” so-and-so; “for all” objects of such-and-such a class, this and that property holds.
That’s the language of modern mathematics. But not of Euclid. He never uses that manner of speaking which is so natural to logic-oriented mathematics. This fits very well with the hypothesis that Greek mathematicians vehemently rejected the notion that their reasoning was based on syllogistic or propositional logic. Instead they relied on constructions. In fact, they did so in part precisely because logic is so problematic.
Or so I have claimed. As usual we cannot know for certain. Euclid didn’t say why he’s so obsessed with constructions. But I think this is a good way of making sense of it.
Ok, so we have seen a number of specific considerations that point toward the foundational importance of constructions. Let’s bring these ideas together into a single philosophy. In fact, there is such a philosophy. I call it operationalism.
Operationalism is a term most closely associated with a 20th-century movement in philosophy of science that grew out of relativity theory and quantum mechanics. But several of its key ideas are much older and more universal. I propose that this rich tradition in philosophy of science was largely foreshadowed in Greek philosophy of geometry. The key commitments and motivations of modern operationalism and related traditions could very plausibly have been precisely mirrored in Greek geometrical thought.
One of the leading modern defenders of operationalism is the Harvard physicist Percy Bridgman, a Nobel Prize winner. Here’s how he formulates the core principle of operationalism: “we mean by any concept nothing more than a set of operations; the concept is synonymous with the corresponding set of operations.”
Bridgman had physical concepts in mind, but it works equally well for geometry. For example, what does “triangle” mean? The operationalist answer is that “triangle” means: the figure obtained when drawing three intersecting lines with a ruler. This diagram is not a drawing of a triangle, or a physical instantiation of the formal concept of a triangle, or in some other way subordinated to or derived from some purer concept of triangle. No, a diagram resulting from these operations simply is what a triangle is. The act of drawing itself is the root meaning of the concept of “triangle.” The act of drawing is the foundational bedrock on which any claim about triangles ultimately rests. When Euclid says “let ABC be a triangle,” he strictly speaking simply means: draw one line, then another, then another (making three points of intersection).
In the same way, what is a line? Take a piece of string and pull the ends; that’s a straight line. What is a circle? Take a piece of string and hold one end fixed and move the other end while keeping the string taut; that’s a circle. What does it mean for two things to be equal? Put one on top of the other; if they align, and neither sticks out beyond the other, then they are equal. As Euclid says in Common Notion 4. What is a right angle? Cut the space on one side of a line into two equal pieces; that’s a right angle. As Euclid says in Definition 10. And so on.
Every statement Euclid makes in the Elements can be read as a statement about operations or the outcome of operations. Not every geometry book is like that. Far from it. Most geometry treatises of later eras do not allow themselves to be interpreted in operationalist terms.
Consider for example the parallel postulate, Postulate 5. This postulate is very convoluted and hard to read the way Euclid states it. It goes like this: “if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.”
Basically, this says that if two straight lines are heading toward each other then they meet. And the postulate includes a criterion for checking whether two lines are heading toward each other or not. Namely: cut across them with a third line, and check the angles it makes. Less than 180 degrees on one sides means the lines are inclines toward each other, so they will eventually meet on that side, says the postulate.
In fact it is clear already from the Elements itself that Euclid could have used a simpler, equivalent statement in place of this complicated thing. Such as: given any line and any point not on this line, there is no more than one parallel to the line through that point.
Why did Euclid opt for his much more convoluted formulation of the postulate? From the point of view of modern mathematics, his choice is strange, as witnessed by the majority of more modern treatments that much prefer the formulation in terms of existence of parallels. But from an operationalist point of view Euclid’s choice makes perfect sense. Euclid’s version of the postulate is purely about operations: if you draw two lines, and discover by an operational test that they stand in such-and-such a relation, then if you extend them such-and-such a thing will happen. Everything is formulated in terms of actions that the geometer performs. The existence formulation, on the other hand, is incompatible with operationalist principles. It only makes sense in some kind of preformationist framework that assumes that all the objects of geometry are already “out there,” independently of any geometer.
Similarly, Euclid doesn’t say “there are infinitely many prime numbers” but rather: if you have a list of prime numbers, you can make a larger list of prime numbers (Elements, Book IX, Proposition 20). This achieves the same thing but without needlessly entangling itself with the quasi-metaphysical assumption that “the set of all prime numbers” is a preexisting entity whose properties we are proving theorems about. There is no need for mathematics to make assumptions of that type. Doing so would only invite attacks from philosophical sceptics.
Operationalism avoids the dubious ontological assumption that the totality of all objects of geometry are somehow already at our disposal. The modern formulation of the parallel postulate assumes that mathematics can, so to speak, survey the totality of all lines through a particular point and make proclamations about this infinite set. Operationalism doesn’t make such an assumption.
We can also read for example the Pythagorean Theorem this way. Again, this theorem does not say that every element of the infinite set of all right-angle triangles has a particular property. Rather, operationally speaking, it says: if you have drawn a right-angled triangle, and if you then draw the square on each of the sides, then the areas of those particular squares are related in such-and-such a way. Until you have drawn a right-angled triangle, the theorem can be said to have no content.
Operationalism cuts away a huge amount of philosophical baggage, yet still allows us to retain virtually all mathematical content. Insofar as mathematical practice needs to be adapted when operationalism is adopted, this is in the form of explicating constructions for all objects dealt with. But, as we have seen, there are in any case strong internal reasons for mathematicians to adhere to this ideal.
Indeed, operationalist geometry is automatically protected from the fallacies discussed above in straightforward ways. The existence and false diagram issues are resolved because they could never arise in strict operationalist practice. And the verbal logic problems do not arise since verbal logic is not accorded any foundational role in operationalist mathematics. Thus operationalism very conveniently cuts off in one fell swoop numerous lines of attack of philosophical scepticism directed at mathematics, without the need for any sacrifices in mathematical content.
Operationalism is related to positivism. Science can only speak about observable facts, according to positivism. So positivism implies a strong adherence to a scientific worldview as the only source of knowledge, and a rejection of other humanistic or philosophical theories or belief systems. Science, positivists say, prudently restricts itself to what is actually knowable, while other forms of philosophy speculate futilely about the ultimate nature of things and all sorts of other concepts that transcend observable reality. On this view, much grandiose philosophising is wrongheaded and even strictly meaningless.
Here’s how Bridgman puts this point. “It is quite possible, even disquietingly easy, to invent expressions or to ask questions that are meaningless. It constitutes a great advance in our critical attitude toward nature to realize that a great many of the questions that we uncritically ask are without meaning. If a specific question has meaning, it must be possible to find operations by which an answer may be given to it.”
You know how all courses have to have “learning goals” these days? You can’t say: “this course is about quadratic equations.” Instead you have to say: “after completing this course, the student will be able to obtain the solutions to equations of the form blah blah blah.” So you must operationalise what it means to succeed in the course. You must state it in terms of what the student can do. Not in terms of just naming the topics.
Greek geometry was like that too. Everything is formulated in terms of doing. It’s not enough to just give names to things: you must make those names meaningful by explaining what you can actually do with them.
Here’s another quote from Bridgman: “Politics, philosophy and religion are full of purely verbal concepts. Such concepts are outside the field of the physicist. Only in this way can the physicist keep his feet on the ground or achieve a satisfactory degree of precision.”
So positivism and operationalism go hand in hand with an “us versus them”—scientists versus philosophers—type of attitude that is as much about rejecting other perspectives as it is about affirming its own principles.
It is possible that this dynamic was directly paralleled in antiquity. Ancient mathematicians would have felt that their geometry was a lot more grounded in reality than even quasi-science such as the four elements theory, not to mention more abstract philosophy such as, say, Aristotle’s doctrine of causes. Ancient mathematicians would have felt that their results were qualitatively different from philosophy in terms of reliability, objectivity, and many other dimensions. They may even have felt that much philosophy was empty gibberish. Perhaps this would have led them to articulate general methodological principles that would “explain” why their form of reasoning and knowledge was superior to that of the philosophers, as many scientists have been inclined to do ever since.
What methodological dicta might Greek mathematicians have seized upon to set their field apart from philosophy? Certainly not anything like the modern identification of mathematics with logic and axiomatic-deductive reasoning. Logic and deduction were already highly prized among Greek philosophers. If anything, they were too obsessed with deductive logic: Zeno’s argument that there can be no such thing as motion is one example among many of extreme faith in abstract deductive reasoning even when it is in blatant conflict with the most basic common sense. So ancient mathematicians could certainly not hope to stand out by their reliance on abstract deductive reasoning.
Axiomatics too was far from the exclusive purview of mathematicians; indeed it is obvious that basing one’s theories on a list of allegedly evident but ultimately unjustified axioms is very convenient for mathematicians and sophists alike. It may even be reasonable to say that the abundance of deductive philosophical systems that were clearly in conflict with one another would rather have been an incentive for the mathematics to insist that, unlike the philosophers, they did not rely on abstract logic.
Operationalism would have been an alternative readily at hand. Constructions had always been a central part of geometry, from the time of the Egyptian “rope-stretchers” whom the Greek identified as the originators of the field. Later theoretical developments, such as the irrationality of the square root of 2, had spoken in favour of taking geometry as the foundational bedrock of all mathematics. It would have been a short and natural step for the mathematicians to tie the foundations of their subject to their already ubiquitous ruler and compass.
To the mathematicians it would have cost little to embrace all-out radical operationalism. Virtually all of mathematics was readily susceptible to being reframed in such a paradigm. It would have been a way of legitimating existing practice that would have necessitated little or no deviation from what they were already doing. Meanwhile, other branches of philosophy stood no chance of founding their teachings on an operationalist basis. So if the mathematicians were looking for a way to set themselves apart from the philosophers—to explain why their field had cumulative progress, universal agreement, and inviolable truths while philosophy had paradoxes and schools in constant disagreement with one another without any prospect of reconciliation—then operationalism would have been the obvious way to go.
Another virtue of positivism that could be held up as distinctive is that it restricts all knowledge claims to the domain of what is actually knowable in a straightforward empirical sense. Failing to adhere to positivism means making statements that are, by their very nature, empirically unverifiable and hence arguably unknowable almost by definition. Unlike most of philosophy, any statement of geometry is readily equated with a claim regarding certain empirical circumstances. Ancient mathematicians had a golden opportunity to highlight this natural attribute of their field as an epistemic virtue. They could pose to head-in-the-clouds philosophers the very difficult challenge of explaining what good a theory is if it has no “cash value” in the real world, in the form of empirically testable claims. And they could stress that geometry, by contrast, has no need to engage in that kind of theorising.
Related to this is the ideal of falsifiability. When the geometers claim that any triangle has an angle sum of two right angles, they are sticking their necks out. If their claim was false, it should be simple enough to find a counterexample. The operationalist formulation of geometry makes it possible to press this point very strongly. The theorem simply means: if you put a ruler down on a piece of paper and draw three intersecting lines, then cut out the three corners and put them point-to-point, then the three pieces fill precisely the angle on one side of a straight line. The very meaning of the theorem directly contains a concrete recipe for testing and potentially falsifying it.
Karl Popper, the philosopher of science, is the name most prominently associated with the philosophy of falsificationism. This was in the first half of the 20th century. To Popper, falsifiability is what set science apart from non-science.
As examples of non-scientific theories, Popper had in mind things like the theories of Marx and Freud, which were influential at that time. These theories had a sort of quasi-scientific appearance. They postulated fundamental laws and used these to explain many phenomena. But according to Popper it was pseudo-science. Because, no matter what phenomena were observed, they could always tell some story about how that fits with their laws.
So these theories pretended to have laws, but they were vague enough to allow many different possible applications, so that almost anything could be construed as consistent with these laws one way or the other. Just as astrological horoscopes in the newspaper make so-called predictions about the future, but in fact they are so vague that they can often be interpreted as having been correct no matter what happens.
This is why Popper emphasized the importance of falsifiability. For a prediction to be scientific, there must be clearly specified condition under which it would be regarded as having failed. The scientist must say: if such-and-such a thing happens, then I was wrong. Before making an experiment or observation, the scientist has already set down those criteria, that is to say, the conditions under which the theory must be regarded as having been falsified.
Non-scientific theories like those of Freud or Marx are not like that, according to Popper. Advocates of those theories use them to “explain” all kinds of things, but they never say: if such-and-such a thing were to happen, then that would prove me wrong and I would give up the theory.
Formulating geometry in terms of constructive operations is a great way of making it scientific in Popper’s sense. It makes the theorems of geometry directly testable. Euclid’s constructions are like lab instructions for carrying out such a hypothesis test. Do the construction and measure for yourself if it came out the way the theorem said.
Euclid’s parallel postulate is something that can be performed and tested in a very concrete way. It says: here’s what going to happen if you draw this kind of configuration.
Alternatives to the parallel postulate are not like that. Instead of the parallel postulate, you could say: Given a line and a point, there is precisely one parallel to the given line through the given point.
How would you test that? It’s in the form of a metaphysical statement, rather than in the form of a falsifiable scientific hypothesis. There is one and only one parallel. It’s like saying: There’s one and only one God. How can you verify that? How could you even prove it wrong? You can’t. Unlike scientific hypotheses, and unlike operationalist geometry, statements of that form do not come with a concrete set of operations one can perform to see if it works or not.
Operationalising geometry makes it falsifiable. It also makes geometry theory-independent. You do not need to accept the definitions and postulates of the mathematicians in order to perform this empirical test. Sceptics who try to criticise mathematics in general terms can thus be confronted with a concrete challenge: regardless of whether you accept any of our assumptions or modes of reasoning, we offer you hundreds upon hundreds of claims of the form: if you perform such-and-such concrete operations, then the outcome will always be one particular way rather than another. Feel free to prove us wrong, the mathematician can say. It would be impossible to meet the challenge and very difficult to try to dismiss it as illegitimate.
The operationalist formulation of mathematical statements is reducible to straightforward recipes whose neutrality and objectivity is very difficult to deny. This is in stark contrast with many philosophical claims, which must often be bought into or rejected wholesale along with an entire theory because all the parts of the theory are interdependent. Even the very meaning of the concepts the theory uses is inherently bound up with the system as a whole.
Operationalism ensures that geometry is not like that. Operationalist geometry is not an entangled holism.
Here’s an analogy for this. Consider a casino. It has roulette and black jack and so on. You play with casino money. Plastic chips that only have meaning and value inside the casino. Once you leave the casino you can’t buy anything for those worthless poker chips.
Non-scientific thought-systems such as philosophy or religion are like the casino. Internally, they have all kinds of intricate laws and explanations for how everything fits together. And it’s easy to get caught up in the system once you buy into it. But to link it to the real world, you have to ask yourself: what’s the actual cash value of this stuff? That is to say, what could I actually do with any of this in the real world, concretely?
Operationalism is “cash value” geometry. It translates everything into real-world operations that anyone can perform. It’s cash money. You can use it directly and it works. It’s not casino money, which only makes sense if you accept the entire premise of the casino with all its internal rules.
Even someone who doesn’t believe in the postulates of Euclid, or doesn’t believe in geometrical proofs, etc. Even such a person can test these things. They can cut the corners off a triangle and see if they fit together the way Euclid says. Or they can draw squares on the sides of a right-angle triangle and see if the areas are equal the way Pythagoras says. Those are scientifically, concretely, real-world testable claims.
Let’s summarise. Operationalism safeguards mathematics against a multitude of plagues. It prevents us from reasoning about entities and concepts that are inconsistent, incoherent, non-existent, or imaginary.
Mathematicians would have had every reason to articulate such a philosophy. Greek antiquity was an age of sceptical philosophical attacks. Mathematics would have found itself under fire, and its enemies were no fools. The logic and rigour of mathematical proofs were by and large hugely impressive. Yet it had a conspicuous Achilles heel: a veritable self-destruct button that could bring the entire edifice crashing down at the slightest trigger. For if there was any way an inconsistency could slip into mathematical reasoning undetected, then everything that followed would immediately be rendered logically worthless. What guarantee do we have that this will never happen, or indeed that it has not already happened?
This vulnerability pertains especially to the way objects are introduced into mathematical discourse. It is safe to say “let ABC be a right-angled triangle,” but if you say “let ABC be a triangle with two right angles,” then you have introduced an inconsistency and all is lost. Then you can prove that 2 is equal to 1, and the entire credibility of mathematics collapses. So geometry needs to systematically guarantee that it could never commit an error of this type. In other words, it needs a meticulous gatekeeping policy that only allows the most carefully vetted entities to enter mathematical discourse.
Constructions are the answer to this problem. By insisting that geometry only speaks of entities that are constructed, the mathematician immediately knocks the legs out under boogeymen examples of inconsistent objects such as the superright triangle.
Constructions also ground mathematics in reality and gives a straightforward account of what geometry is and what geometrical statements mean. This can be used to set geometry apart from empty philosophy, from metaphysics, religion, astrology, all kinds of empty pseudo-science.
Philosophers of science in the 20th century spent a lot of effort trying to formulate the criteria that distinguished science from non-science. One of their answers was falsifiability: scientists bravely specify what would prove them wrong. They say: try this for yourself, and if it doesn’t come out the way I said I promise I will admit that I was wrong and that my theory should be rejected.
They also found that to follow through on this program it was important to translate abstract theoretical notions into observable real-world terms. Instead of merely speaking abstractly about for instance the concept of the force of gravity, it is necessary to translate the meaning of that theory into something doable, testable, such as: if you hang this led weight from this spring, then the spring will extend by so-and-so many centimeters. Things like that is what the concept of gravity comes down to in practical terms. This concreteness is essential to science, and essential to separate science from fancy games with words.
Euclid’s geometry is a perfect fit for all this stuff. It’s almost as if Euclid had read these 20th-century philosophers of science. Maybe Euclid and his friends had many of the same ideas. Maybe they too wanted to set their theory apart and explain why it was superior to other branches of philosophy. The way they based geometry on constructions is a perfect fit for making those kinds of arguments.
So there you go. These are many reasons to ground geometry in constructions. It is not for nothing that all depictions of Euclid shows him with ruler and compass in hand. These are no mere practitioner’s tools. They are in fact essential even to the theoretical foundations of geometry in numerous respects. That is what I have tried to argue.
The etymology of the term “postulate” suggests that Euclid’s axioms were once questioned. Indeed, the drawing of lines and circles can be regarded as depending on motion, which is supposedly proved impossible by Zeno’s paradoxes. Although whether these postulates correspond to ruler and compass or not is debatable, especially since Euclid seems to restrict himself to a “collapsible” compass in Proposition 2. Furthermore, why did Euclid feel the need to postulate that “all right angles are equal”? Perhaps in order to rule out non-flat surfaces such as cones.
Transcript
Euclid’s Postulate 4 is super weird. It says: “all right angles are equal.” What kind of a postulate is that? 90 degrees equals 90 degrees? A right angle is equal to itself?
Why would you need to state that as an axiom? And if you do need to state it as an axiom, why only right angles? Why wouldn’t you need other axioms starting that various things are equal to themselves? 10 degrees equals 10 degrees, 1+1 equals 2: Why don’t we need axioms like those? What’s so special about right angles? Why do they need to be singled out like that, in their very own postulate?
But Euclid knew exactly what he was doing. His postulate only appears crazy and weird. There’s a way to make sense of it. We must reconstruct the original context and intent of the postulate.
I say: Euclid included this postulate in order to rule out cone points. I will explain what I mean by this. But let’s note first the historical methodology we are using here.
There is nothing in Euclid, and in fact nothing anywhere in any ancient source, that actually says that this was the intent of the postulate. The interpretation that the postulate has to do with cone points is purely a hypothetical reconstruction by historians, formulated thousands of years after Euclid.
Yet the reconstruction is so convincing. It just has to be right. If it’s right, everything fits; everything makes sense. If it’s not right, then we can’t explain the postulate, and we just have to assume that Euclid hadn’t really thought it through all that carefully and just put the postulate down on kind of whim or whatever and it doesn’t mean all that much in the greater scheme of things.
This is the difference between a great text and an average one. Great texts in intellectual history, like Euclid’s, reward reflection. If something seems weird it’s because you haven’t understood it. There’s a reason behind every step of the text. The text is the tip of an iceberg. It’s built on a huge body of supporting thought. This is why the text rewards reflection. The text is not just whatever popped into Euclid’s head. It is the fruit of an intellectual culture where these ideas had been scrutinized and criticized forwards and backwards and inside and out.
This is why you should read great texts like Euclid’s. These are the kinds of texts that, every time you dig into even the weirdest parts you realize that, huh, that’s actually a good point. The more you probe the text, the more compelling it becomes. It’s the mark of a great text that when you scrutinize an apparent weakness, it turns out to be a strength.
Euclid’s right angle postulate is an example of this. It looks silly and weird at first sight, but when we think about it, it opens our eyes to new and unexpected perspectives and insights.
So, “all right angles are equal,” what’s the deal with that? First of all, what does “right angle” mean? Euclid defines it in Definition 10. Draw a line. Consider the space on one side of the line. Cut that space in half with another line. That’s a right angle. A right angle is half the space on one side of a line.
So what does “all right angles are equal” mean? It means: Suppose you have made a bunch of right angles. That is to say, you have drawn various lines and then cut the space on one side in half. So you have a piece of paper full of what looks like a bunch of copies of the letter capital T. There are a bunch of T’s scattered across the paper, at random angles and positions. “All right angles are equal” means: if you cut out one of those T’s and put it on top of one of the other ones, then it fits. All the different T’s align perfectly with each other, as far as angles are concerned.
So the right angle postulate is really a kind of homogeneity postulate. It effectively says that no part of the paper is different than any other. The space on the side of a line is the same anywhere.
A cone is an example of a surface where that is not the case. Like an ice cream cone. The cone is non-homogenous. It has an exceptional point, the cone point, the apex of the cone, which is different from the other points.
Euclid’s postulate is false on the cone. A right angle at the cone point is smaller than a right angle elsewhere.
You can see this if you think about how a cone is made. You can make a cone like this. Start with a circular piece of paper. Then cut out a wedge from it, like a pizza slice. Then grab the two sides of the cut and pull them together. Now you have a cone.
Think about the amount of space around each point. Most points are surrounded by the same amount of space as they were originally, on the paper we started with. 360 degrees’ worth of space, so to speak.
But the cone point is different. It’s surrounded by “less space” or “fewer degrees” than before. The pizza wedge you cut out took away some of the angle sum around this point. Not so for any other point. Even the points along the sides of the cut are still normal. They lost 180 degrees, but then you pasted another 180 degrees right back in. So they are back to normal. But not the cone point. It lost some of its angles and never got them back.
So the right angle postulate is false on the cone because right angles are smaller at the cone point. Since right angles are defined in terms of cutting the space on the side of a line in half, then if there’s less space around some points compared to others, then the right angles there will be smaller too.
I believe this is what Euclid had in mind when he wrote his postulate. We can’t prove that this is what Euclid meant, but it is the most satisfying explanation.
Here’s a little cultural sidelight. The Declaration of Independence of the United States starts in a kind of Euclidean manner. It says: “We hold these truths to be self-evident,” and then it lists a number of “truths” the first of which is “that all men are created equal.”
So the Declaration of Independence has self-evident axioms just like Euclid, and they sound the same too: “all right angles are equal”; “all men are created equal.”
That’s no coincidence. The founding fathers of the United States were obsessed with antiquity. They used the ancient world as a model all the time. As a model for their political system, of course. The senate, for instance, is straight up copied from Rome, and so on with many other things. Euclid was part of that package as well. A very conscious revival of ancient enlightenment.
So the founding fathers of the United States called their axioms “self-evident.” And of course many people have interpreted Euclid that way too. You don’t have to prove the postulates because they are immediately obvious. You can draw a line from any point to any point: yes, of course you can, it’s too simple to even prove, but it’s impossible to doubt. That’s one way to think about Euclid’s postulates. A common way.
But one could argue that it’s a bit more complex. This is suggested even by the word that Euclid uses: postulate. These simple and self-evident starting points are called postulates. But this term doesn’t suggest that these things are self-evident or impossible to doubt. To postulate is more of a demand or a request. So the term doesn’t seem to take assent for granted but rather the opposite: it seems to imply that some people might oppose these statements, no matter how obvious they might seem.
How could anybody deny that you can draw a line from any point to any point? In fact, some people in Ancient Greece did deny this, and they were not crazy; they had some very compelling reasons.
A useful book on this is The Beginnings of Greek Mathematics by Arpad Szabo. I will summarize it for you.
“Mathematics grew out of the more ancient subject of dialectic”—that is to say, philosophical debate. Just as we discussed before, the argumentative Greeks, they loved debating. Two philosophers passionately disagreeing and trying to poke holes in each others’ arguments in a lively disputation before an audience: that was their idea of a good time. Instead of “dinner and a movie” you would go to a philosophy debate.
So that’s “dialectic”—a debate with two warring sides. Terms such as axiom, postulate and many others seem to have originated in this setting. These terms were imported into mathematics from dialectic. Today only their mathematical meaning survives. Therefore to us these terms have rather different connotations than they did for the ancient Greeks. That’s how the idea that axioms or postulates are supposed to be self-evident has become associated with the terms even though that was not the original intent or meaning.
The terms axiom and postulate originally mean something like “concessions which the participants in a discussion have agreed to make.” “We know that the term aitema [=postulate] came from dialectic where it was used to denote a ‘demand’ about which the second partner in a dialogue had reservations.”
“Let us see whether there is any connection between this early meaning of the word and Euclid’s postulates. At first glance, Postulates 1-3 appear to be such simple, self-evident and easily fulfilled ‘demands’ that one is tempted to disregard the literal meaning of their name.” But no.
Euclid’s postulates arguably rely on motion. To draw a straight line from any point to any point: how do you do that? You put a ruler down and trace the line with a pen. The pen is moving: you put it at one point and move it to the second point. Same thing with circles: you draw them with a compass, which is also a moving instrument.
It’s quite possible to deny that such things can be done. In fact, you may have heard about the famous paradoxes of Zeno, which purport to prove that motion is impossible. One of them goes like this.
Suppose I have to walk from A to B. Before I can walk all the way to B, I first have to walk half the way to B. Then, when I’m at the halfway point, before I can get to B I have to walk half of what’s left. And so on. Whatever distance is left, I always first have to go half of it.
But this process never ends. There’s always “another half to go.” So to go from A to B you have to “do an infinite number of things,” so to speak.
You can think of it this way. When I have gone half the way from A to B, I say: one. Then when I have gone half again of what’s left, I say: two. I go half of what’s left: three. And so on. This implies that if in fact I can go all the way from A to B, I will have shouted out all the numbers that exist: one, two, three, four, five, … all of them.
So to say that you can go from A to B is to say that you can count through all the numbers in finite time. But of course you can’t. Nobody has ever counted through all the numbers. So therefore you can’t move either. Motion is impossible. It must be an illusion.
We only think we move. That’s feeble sensory “knowledge,” or so-called knowledge. We discussed before the extreme rationalistic tendency of Greek philosophy: reliance on pure deductive reason at the expense of all other forms of knowledge. Zeno’s paradox is an example of this. The senses say we can move, but deductive “reason” says we cannot.
We discussed before how the stage debate format incentivized philosophers to pick the side of reason in such cases, no matter how extreme and outrageous the conclusion may be. “All is water”, “all is fire”: the crazier the better. Proofs of radically unexpected conclusions is perfect for the stage debate setting.
Zeno’s argument is great way to dazzle an audience and to show how clever you are. Being reasonable and arguing that one can walk from A to B is boring. Who wants to hear that? You won’t become a blockbuster debate star by arguing for the obvious. You gotta have some signature absurdities that you claim to prove.
Zeno also had a second form of his argument that is equally amusing. Here’s how Simplicius describes it:
“The argument is called the Achilles because of the introduction into it of Achilles, who, the argument says, cannot possibly overtake the tortoise he is pursuing. For the overtaker must, before he overtakes the pursued, first come to the point from which the pursued started. But during the time taken by the pursuer to reach this point, the pursued always advances a certain distance; even if this distance is less than that covered by the pursuer, because the pursued is the slower of the two, yet none the less it does advance, for it is not at rest. And again during the time which the pursuer takes to clever this distance which the pursued has advanced, the pursued again covers a certain distance. And so, during every period of time in which the pursuer is covering the distance which the pursued has already advanced, the pursued advances a yet further distance; for even though this distance decreases at each step, yet, since the pursued is also definitely in motion, it does advance some positive distance. And so we arrive at the conclusion that not only will Hector never be overcome by Achilles, but not even the tortoise.”
So that’s another way to prove that motion is impossible. Those who believe in motion believe that Achilles can out-run a tortoise. But that contradicts reason, as we have just seen. Therefore those who believe in motion must be wrong.
Why did Zeno prove the same thing in two ways? Maybe he was just like: Hey guys, I thought of another funny one, it has a tortoise in it, I’m sure you’ll get a kick of it. Or is there more to it than that? Do Zeno’s two forms of the argument differ in substantial respects?
I think they are subtly different. You might say that the Achilles argument assumes the possibility of motion and derives a contradiction. It so to speak plays along with those who believe in motion for a bit, only to then trap them in a paradox.
The other argument—the dichotomy, or half half half argument—doesn’t really need to even presuppose motion at all. It derives the impossibility of motion more from the nature of length. It has more to do with the infinite divisibility of the continuum than with motion as such.
So in that respect the dichotomy argument is more “pure” as it were. Since it doesn’t need to use motion to refute motion.
But on the other hand it is less pure in another respect. It assumes metricity; that is to say, an absolute notion of distance. For the argument to work, it must be possible to talk about the half of something. But half involves quantification. You need to put a number on the full length before you can know what half of it is.
So Zeno’s opponents could say: Your argument doesn’t disprove my beliefs, because although I believe in motion I do not believe in metricity. I do not believe that numerical lengths can be assigned objectively to the paths between points. Therefore the whole business about halfs doesn’t work, and you haven’t really disproved motion after all.
If Zeno’s opponents tried to wiggle out from under the dichotomy argument along those lines, then Zeno could just hit them with the Achilles argument. Because the Achilles story doesn’t involve assigning numerical lengths to anything. It purely about relative positions: the tortoise is in front of Achilles. It doesn’t say by how much. The argument doesn’t need the notion of being in front to be quantifiable. It needs only relative positions. So in that sense the Achilles argument is the purer one.
Well, that’s fun to think about, but let’s get back to our original purpose. I brought up Zeno’s paradoxes because they are related to the issue of whether Euclid’s postulates are obvious or not.
“If we bear [Zeno’s paradoxes] in mind, it is easy to understand why Euclid’s first three postulates had to be laid down. They really are demands (aitemata) and not agreements (homologema); for they postulate motion [such as the motion of a pen that is drawing a circle], and anyone who adhered consistently to [Zeno’s] teaching would not have been able to accept statements of this kind as a basis for further discussion.”
So when Euclid is presenting his postulates, he doesn’t seem to be saying: surely you all agree with these statements; they are clear even without a proof. Instead Euclid seems to mean by postulate: these are assumptions that must be accepted for the sake of argument if we are to do geometry; if you don’t like them, then we just have to agree to disagree.
The same goes for Euclid’s Common Notions. “Our text of Euclid” has a separate heading called common notions, but this was not a well-entrenched term and these principles “obviously bore the name axioma in pre-Euclidean times,” and “the noun axioma, when used as a dialectical term, originally just meant a ‘demand’ or ‘request’.”
Indeed the common notions could be doubted. They “are assertions which are justified by practical experience and, in some cases, directly by sense-perception. [One of them] states that ‘things which coincide with one another are equal to one another’. It can literally be seen that plane figures which coincide are actually equal; hence this axiom is verified by sensory experience.” Therefore the common notions “could not have been accepted by [those] who required that all knowledge be obtained by purely intellectual means and without appealing to the senses.” And there were plenty of people like that. Just as Zeno’s argument implies: an extreme trust in purely intellectual reasoning, even when it goes flatly against even the most basic and immediate experience.
This is why “these principles were originally called demands (axiomata): because the other party in a dialectical debate had reservations about accepting them as a basis for further inquiry or, in other words, because their acceptance could only be demanded.” People would not have agreed that these things were self-evident; that’s why they had to be “demanded”, or postulated.
There’s yet another way to criticize Euclid’s principle that “things which coincide with one another are equal to one another.” Not only does it rely on sense evidence, it is also arguably conceptually incoherent. If “two” things coincide and are equal, doesn’t that mean that they are actually one thing? Does it even make sense “to speak of two things unless they can be distinguished from one another“?
So we see how Euclid’s axioms can be questioned in various ways. The Greeks loved to quarrel. Mathematics was born in this kind of climate. Everybody criticizing everything, trying to poke holes in it.
So that’s why Euclid’s text starts with “demands.” Many later readers were happy to accept them as self-evident, but Ancient Greek geometers could not have expected to get away with that.
So the terminology of “postulates” and “axioms” points to this ancient context. But the meaning of the terms morphed over time. In the very early days, mathematics lived within the dialectical tradition and was a subordinate part of it. But mathematics took on a life of its own and soon outlived dialectic.
Soon “the essentials of [the old] dialectic [context] were no longer very well understood; hence the ancient term axioma acquired a new meaning. Since it had always been used to refer to a group of principles which, from the viewpoint of common sense, were evidently valid, it came now to denote those statements whose truth was ‘accepted as a matter of course’.”
So that American phrase—“we hold these truths to be self-evident”—is perhaps not as Euclidean as Jefferson and those guys thought.
Here’s another interesting aspect of Euclid’s postulates. The first three postulates basically state that lines and circles can be drawn. That is to say, lines and circles can be taken to exist. That’s a primitive assumption of geometry.
Lines and circles are so to speak the Adam and Eve of geometry. In the beginning there are only these two, these male and female generative principles. You couldn’t get very far with just one of them, but together they combine to make rich offsprings. They eventually populate the entire Euclidean universe. Everything that ever happens in Euclid’s world comes from these two parents, the line and the circle.
The line and the circle are also embodied in physical tools: the ruler and the compass. To what extent is that important? Is this physical realizability important to the credibility of these postulates? Or is Euclid merely talking about lines and circles in the abstract, and it’s just a coincidence that they correspond to physical tools?
There is no simple answer. Euclid’s text is ambiguous in this respect. You can read it either way.
Insofar as we can say anything about what Euclid meant in this respect, we must infer it from the technical material later in the text. Euclid never tells us: “here’s my philosophy.” We can only read his proofs and ask ourselves what implicit assumptions appear to be made and what implicit philosophy might have guided the particular choices Euclid makes in technical arguments.
Already Proposition 2 is very interesting in this regard. Euclid shows in Proposition 2 how to transfer a length from one position to another, using only his postulates about line and circle, or ruler and compass.
In other words, somebody has drawn a line segment on a piece of paper, and now you want to draw an equally long line segment somewhere else on the paper.
Euclid accomplishes this by a very elaborate construction. It involves drawing numerous circles and an equilateral triangle. Very elegantly, this leads to exactly what you need: the given segment has been reproduced in the new position, with exact mathematical precision.
That’s all very neat, but it’s also weird, isn’t it? It seems totally out of touch with reality. If a craftsman or engineer or architect would need to transfer a length, surely they would not use Euclid’s absolutely baroque procedure.
First of all you might say: just use a ruler. Measure the given length. It’s so-and-so many centimeters. Then put the ruler wherever you want the length to go, and mark off the same number of centimeters there. Done. No need for drawing a bunch of circles and god knows what else.
Why doesn’t Euclid accept this and save himself some time? Actually it’s not so crazy. In a way you might it’s a mistake to think that length lives in the ruler. Actually, out of the two “parents” ruler and compass, length comes from the DNA of the compass, not the ruler.
We are so used to working with rulers, measuring things with rulers. It’s the prototypical manifestation of length. But think about it. Where do rulers come from? How do you make a ruler? How do you put the centimeter marks on it?
You do it with a compass. You set the compass to a fixed opening, and you mark off the size of that opening repeatedly along the ruler. Can you feel it? You start with a blank ruler, just a straight piece of wood. Now you take your compass and make it so to speak “walk” along the edge of the ruler. Left foot, right foot, left foot, right foot. The places where the compass “stepped” so to speak become the marks of the ruler. So when you use a ruler to measure things, you are really relying on the compass. Length is born from the compass.
This suggests that Euclid is on to something when he involves circles in his proof of Proposition 2. But it still doesn’t explain why it has to be quite so complicated.
A compass can solve the problem directly. Just open the compass to the length you want to move, then lift it and put it back down wherever you want the length to go. The length you wanted in the new position where you wanted it is directly manifested in the form of the distance between the two legs of the compass. Piece of cake. There’s nothing to it. You can move lengths directly with the compass without any hassle.
Euclid acts as if this is not possible. One might say that Euclid behaves as if his compass is “collapsible”: it stays at a fixed opening while drawing a particular circle but as soon as it is lifted from the paper it “collapses,” or closes up, so that the opening to which it was set is lost and cannot be used elsewhere.
Of course there are no collapsible compasses. It’s not a real thing. So you might say: aha! This proves that Euclid is in fact talking about lines and circles abstractly, maybe in the manner of Plato and his world of ideals. From that point of view Euclid’s proof is not problematic. It’s a dazzling intellectual construction. Great stuff. Hopelessly impractical, to be sure, but that’s just all the better of course as far as Plato is concerned.
Meanwhile, if you want to say that Euclid’s postulates correspond to actual rulers and compasses, then you have to bend over backwards and make up stories about “collapsible compasses,” which don’t exist.
So it seems we have a clear winner. Only the abstract, non-physical reading of Euclid makes sense.
But I’m not so sure. Maybe “abstract versus physical” is the wrong lens to use here. We can also make sense of Euclid’s peculiar proof from a different point of view that is independent of this issue of physical versus abstract.
This point of view is: assumption minimalism. Euclid masterfully reveals the minimum assumptions necessary for geometry. Remember: reduce, reduce, reduce. That seems to be Euclid’s mantra. That’s the philosophy of “reading backwards.” If you can avoid an assumption, then you should avoid that assumption.
This kind of minimalism or purism doesn’t depend on whether geometry is physical or abstract. Either way, if something can be proved rather than assumed, then that’s regarded as a win. This kind of reduction is about exploring and clarifying the ultimate foundations of geometry and the bedrock source of geometrical knowledge. It is applicable regardless of whether geometry is physical or abstract.
This perspective of minimalism demands that we do not allow lengths to be merely transferred directly by a compass. Even if we do think physical compasses are somehow important to geometry, we should still pursue this reduction. It is our duty to always reduce.
Just as a chemist reduces molecules to atoms. Of course molecules are great. The best level at which to explain many things is molecules, not atoms. But since they can be reduced, they must be reduced. It is our scientific duty to run the reduction as far as it goes. Of course we still retain the explanatory power of molecules. The reduction to atoms is just a supplement.
Maybe so also in Euclid. Maybe the physical compass should be seen as the operative tool throughout the Elements, just as molecules are the right level of analysis for many chemical phenomena. But even so it makes sense to show up front how it could, in principle, be reduced even further to more basic building blocks. We might say that Euclid does this in Proposition 2.
The fact that one can do away with the assumption that a compass can transfer length is an interesting foundational insight. Since Euclid can prove this, he does. This does not imply that he is opposed to the idea of a non-collapsible compass. One could simply delete Proposition 2 from the Elements and all the rest would still stand verbatim as a treatise about constructions with non-collapsible compasses.
So Proposition 2 can be viewed as an optional exercise in foundational minimalism within a paradigm otherwise fully based on physical compasses. Rather than as evidence of conceptions fundamentally at odds with such a physical point of view.
Analogous situations occur in modern mathematics all the time. For example, open any textbook on abstract algebra and turn to the definition of a group. The definition of a group says that any group has an identity element: anything multiplied by the identity stays the same. As far as this definition is concerned, there could potentially be several identity elements in any given group. However, all textbooks immediately proceed to show that the identity element is in fact unique. Other groups axioms imply that it must be unique.
These textbook authors could have made life easier for themselves by simply making the uniqueness of the identity element part of the definition. Then there would have been no need to prove it a separate theorem. But it is better to keep definitions and axioms as simple and minimalistic as possible, for instance in order to minimise the risk of inconsistency, or because proving properties instead of gratuitously including them in the definition illuminates fundamental relationships.
But note that one cannot infer from this that the uniqueness of the identity is somehow a secondary or less embraced aspect of the group concept. It is proven as a theorem rather than included in the definition solely because of the technical possibility of doing so, not because it was seen as less essential than the definitional group properties. This does not show that the fundamental conception of a group that mathematicians have in mind is ambivalent regarding the uniqueness of the identity. On the contrary, this is arguably a core aspect of the intuitive notion of a group that has, in itself, no less of a claim to being fundamental than the definitional properties. But if one tries to find the smallest set of key properties of a group to take as definitional, then one finds that uniqueness of the identity is a property that can most efficiently be made into a theorem.
In the same way, then, one might argue, Proposition 2 of the Elements does not show that Euclid’s fundamental notion of the circle-drawing constructions and postulates were divorced from a physical compass. It does not prove this any more than a modern textbook proves that the uniqueness of the identity is fundamentally divorced from the group concept.
Just as a modern algebra textbook would have nothing a priori against including uniqueness of the identity in the definition of a group, so Euclid may very well have had nothing a priori against assuming a non-collapsible ruler. Just as the modern algebra textbooks nevertheless arrives at the conclusion that it is better to make the uniqueness of the identity into a theorem because that enables the minimisation of definitional properties overall, so Euclid may very well have decided to assume only a non-collapsible ruler purely for reasons of axiomatic minimalism. If so, it would be a mistake to infer from this proposition that he didn’t care about physical tools like the compass.
Even if you’re not familiar with group theory I’m sure you have encountered a similar aesthetic elsewhere. For example, some people, when they cook pasta, they save a few spoonfuls of the cooking water and toss it into the dish. To make the pasta less dry.
I always thought it’s a little pretentious when TV chefs do this. Obviously you could achieve the same result various other ways. Instead of adding some of the cooking water, you could add other water, oil, make your sauce a bit runnier, etc. I’m sure nobody could tell the difference.
But it’s cool somehow to use the actual cooking water. It makes you feel creative and spontaneous. Almost spiritual: it’s like you’re in synch with the universe like some ancient Indian who lived in harmony with the land. Making use of everything, every part of the pig, even the cooking water. It takes skill and true understanding to use things for something other than their intended purpose. It’s a rock ‘n’ roll move. Anybody can cook the way it says on the tin, but I’m such a creative rebel that I use the very cooking water itself.
Euclid’s Proposition 2 is a bit like that. Of course you could accept the transfer of lengths as a separate assumption, or implied by the compass. But it’s cooler if you can do without it, and instead use what is already at hand in an unexpected new way. Euclid uses the cooking water, so to speak. He uses the assumptions from the postulates that were already necessary anyway. By cleverly combining these, he shows that you don’t need anything else. It’s satisfying in the same way the pasta trick is satisfying.
I think this is enough to explain why Euclid wanted to include Proposition 2. So we don’t need to attribute to Euclid any anti-compass agenda. It’s enough that he thought this was a cool trick.
So the question is still open then whether Euclid meant his postulates to correspond to ruler and compass or not. We will have to keep reading to find out more. Let’s do that.
Euclid’s definitions of point, line, and straightness allow a range of mathematical and philosophical interpretation. Historically, however, these definitions may not have been in the original text of the Elements at all. Regardless, the subtlety of defining fundamental concepts such as straightness is best seen by considering the geometry not only of a flat plane but also of curved surfaces.
Transcript
“A point is that which has no part.” What a bonkers way to start a book. But that’s Euclid for you. Let’s start the whole thing off with a negative, Euclid apparently told himself. He’s like: Let me tell you what a point is. Think of things that have parts. It’s not that. It’s the other stuff. Stuff that doesn’t have a part. Pretty weird that the first thing you introduce is actually defined by exclusion, in terms of what it is not. But anyway, never mind that. There are important interpretative issues at stake here.
The first two lines of Euclid’s Elements are the most misunderstood. They define the concepts of point and line. “A point is that which has no part” and “a line is a length without breadth.” We might interpret this as saying that a line is 1-dimensional, and a point is 0-dimensional.
Here’s how people misunderstand this. They say: Aha, told you! Geometry is not about physical things; it’s about objects in some ideal realm, just like Plato said. Because if you draw a line with a pen for example, it will always have some breadth, no matter how thin it may be. No physical object can ever be a “breathless length.” This proves that Euclid is not talking about physical space.
But that is a terrible argument, which makes no sense. It is demonstrably false. Yet you hear it repeated again and again. Some ancient philosophers made this argument. Aristotle mentions it in the Metaphysics (998a). Still today, many modern scholars walk into this fallacy all the time. But don’t worry, I’m here to save you from this mistake.
There is no inconsistency between Euclid’s definitions and a physicalist view of geometry. On the contrary, these kinds of idealisations are an essential part of any physical theory. Ptolemy, the astronomer, treats the moon as a point for the purposes of many of his demonstrations, for instance. Obviously no one would infer that he is therefore believes the moon is a mathematical point with no extension. The convention of treating the moon as a point is simply a common-sense idealisation that is the only sensible thing to do for many mathematical purposes, regardless of what one’s estimation of the actual body of the moon may be.
It is the same for instance in Archimedes’s work on levers, where the lever arm is a weightless mathematical line and the weights are applied at mathematical points. Since such idealisations are unequivocally used all the time without further ado in applied mathematics, it makes no sense to take them to be inconsistent with a physicalist view of geometry. On the contrary, such idealisations are exactly the standard assumption one would expect in physicalist geometry, just as one invariably finds it any other mathematical theory pertaining to the real world.
So if this argument is right, that Euclid’s definitions prove that his geometry is divorced from reality would, then it is equally true that the Greeks did not intend their astronomy or their statics to apply to the real world either, which is obviously absurd. So it’s madness to infer from Euclid’s definitions that he thinks geometry is non-physical.
It is more plausible to read these definitions as specifications of idealisations made in geometry, rather than as claims about the ultimate nature of geometrical objects. Indeed you can find support for this in ancient sources. Heron, for example, clearly takes such a view. He writes:
“Already in ordinary language use we have the notion of a line as something which has only length, but not at the same time width and thickness. For we say: a road of 50 stades, as we concern ourselves with the length only, but not at the same time its width.”
Here the identification of geometry with everyday physical objects is evident. The allegedly Platonic or ontological aspects of the definitions is merely a common-sense matter of simplifying assumptions and directing attention only to the relevant aspects of the situation.
Proclus makes the same point as Heron. He also uses the example of a road. And he attributes this view to “the followers of Apollonius.” In other words, Proclus puts this view right at the mainstream of Greek geometry at its peak. Apollonius is at the heart of the mathematical establishment. Heron was also a mathematical author. So mathematicians were the ones who thought a road was a good example of a line. Meanwhile, those who tried to use Euclid’s definition to drive a wedge between mathematics and physical reality were philosophers.
It’s typical, of course, that philosophers focus on the first two lines of Euclid and try to dismiss the relevance or status of geometry on that basis. Perhaps they never made it past the first page of the Elements. How convenient that they immediately found an excuse to dismiss geometry based on the first two definitions. How convenient that their objective analysis just happend to justify ignoring all technical mathematics.
Such a motivation is quite transparent in at least one of these philosophical authors, Sextus Empiricus. He gives probably the most extensive articulation of this idea that the first definitions of Euclid undermines the credibility of mathematics. The very title of his work is Against the Mathematicians. “The mathematicians talk idly,” he accuses, “for the straight line shown to us on the board has length and breadth, whereas the straight line conceived by them is ‘length without breadth’.” Gotcha, huh? You can decide for yourself if you think Sextus Empiricus is a razor-sharp philosophical mind who has outsmarted all the mathematicians, or whether he’s a guy who doesn’t like mathematics and wants to rationalize his own ignorance.
Those of us who read Euclid beyond the first page quickly realize that there is a further compelling argument for why one must not make too much of the alleged ontological import of Euclid’s definitions of point and line. Namely, that these definitions are the most extraneous part of the Elements.
The Elements is obviously a very carefully constructed logical theory, where almost every statement is carefully formulated to correspond precisely to the justification of specific inferences in deductive proofs. Obviously postulates and propositions are of this type, and so are many definitions, such as the definition of a circle which is used already in the very first proposition to infer that since two line segments are radii of the same circle, they must be equal.
However, the definitions of point and line are not of this type. These definitions serve no direct role in the deductive structure of the theory. They are effectively ornamental. They are arguably the most inconsequential parts of the entire Elements, since they are never actually used in any proof. Yet these are the very lines always cited as virtually the only textual evidence in mathematical sources of alleged anti-physicalist tendencies in Greek geometry. Madness.
In fact, these definitions may not even have been part of Euclid’s original text of the Elements at all. The version of the Elements we have has been edited, unfortunately. When Euclid wrote it, it was a sophisticated analysis of the foundations of geometry. It’s readers were high-level mathematicians. Later it became a textbook for schools. Editors interfered to make it more accessible. Possibly adding the first couple of definitions for example.
This is especially clear with respect to Definition 4 of the Elements, the definition of a straight line. Here’s what it says: “A straight line is a line which lies evenly with the points on itself.” This definition is meaningless drivel. What does it even mean to “lie evenly with itself”? How can such a masterful work, which is clearly written by a top-quality mathematician, open with such junk?
There’s a compelling answer to this conundrum, proposed by Lucio Russo. It goes as follows. Euclid didn’t define “straight line” at all. His focus was on the overall deductive structure of geometry, and for this purpose the definition of “straight line” is essentially irrelevant. Indeed, the utterly useless Definition 4 is never actually used anywhere in the Elements.
Archimedes agreed with Euclid as far as the Elements were concerned, but in the course of his further researches he found himself needing the assumption that among all lines or curves with the same endpoints the straight line has the minimum length. He therefore stated this property of a straight line as a postulate in the context where it was needed.
The Hellenistic era, which included Euclid and Archimedes, was one of superb intellectual quality. Unfortunately it did not last forever.
Heron of Alexandria lived about 300 years later. This was a much dumber time. Fewer people were capable of appreciating the great accomplishments of the Hellenistic era. But Heron was one of the best of his generation. He could glimpse some of the greatness of the past and tried his best to revive it.
To this end Heron tried to make Euclid’s Elements more accessible to a less sophisticated audience who didn’t have the background knowledge and understanding that Euclid’s original readers would have had. He therefore wrote commentaries on Euclid, trying to explain the meaning of the text.
To these new, more ignorant readers of geometry, it was necessary to explain for example what a straight line is. Heron realised that Archimedes’s postulate about the line as the shortest distance captured well the essence of a straight line. However, it is not itself suitable as a definition, because a line should have the property of being the shortest distance between any two of its points, not be phrased in terms of only two fixed points, as Archimedes. Remember, Archimedes was not trying to define a straight line, only to make explicit an assumption about straight lines that was particularly relevant in a particular work of his.
To adapt Archimedes’s idea into a definition, Heron therefore explained that, and now I quote him: “a straight line is [a line] which, uniformly in respect to [all] its points, lies upright and stretched to the utmost towards the ends, such that, given two points, it is the shortest of the lines having them as ends.”
In this passage, the phrase “uniformly …” obviously refers to the universality of the shortest-distance property. The point of this phrase is to highlight that this property applies to any two points on the line.
This is what later becomes Euclid’s phrase “evenly with the points on itself.” The original purpose of this phrase was to say that the distance-minimization property of the straight line holds for any pair of points on the line: that is to say, the property holds “uniformity” or “evenly” across the entire line. Not only for the endpoints.
The definition in the Elements is a mutilated version of what Heron said. Heron’s point is that no matter which two points on the curve you pick, the straight line is always the shortest path between them. The mutilated version ignores the part about shortest distances, and distorts the part about it applying across all points into the vague phrase about evenness of all points.
How did that happen? To understand this we need to fast-forward another 300 years. Intellectual quality has now plunged deeper still. Geometry is in the hands of rank fools. Euclid’s Elements, which was once written for connoisseurs of mathematical subtlety, is now used by schoolboys who rarely get past Book I and “learn” that only by mindless rote and memorisation. A time-tested way (still in widespread use today) to “teach” advanced material to students who do not have the capacity to actually understand anything is to have them blindly memorise a bunch of definitions of terms.
In this context, therefore, there is a need for an edition of the Elements which includes many definitions of basic terms, which must be short and memorisable, and which don’t need to make mathematical sense.
In this era of third-rate minds, some compiler set out to put together an edition of the Elements that would satisfy these conditions. Heron’s commentary on the Elements is appealing in this context since it affords opportunities to focus on trivial verbiage instead of hard mathematics. But Heron’s description of a straight line is still too complicated. It’s too long to memorise as a “soundbite” and the mathematical point it makes is moderately sophisticated.
The compiler therefore makes the decision to simply cut off Heron’s description after the bit about “uniformly in respect to [all] its points.” This solves all his problems: the definition becomes shorter and easier. The only drawback is that the “definition” becomes utter and complete nonsense. But since the whole purpose of it is nothing but blind memorisation anyway this doesn’t matter anymore.
This is how the ridiculous Definition 4 ended up in “Euclid’s” Elements. It’s a mutilated version of what was once a very good definition. According to Russo’s hypothesis, which is compelling.
As Russo also observes, in the works of other great Greek mathematicians such as Archimedes and Apollonius (who “belong to the same scientific tradition” as Euclid) “there is nothing analogous to the pseudo-definitions of fundamental geometrical entities contained in the Elements. The introduction of terms implicitly defined through postulates is instead frequent.” So this supports the hypothesis that the Elements was corrupted due to its association with introductory teaching. While these more advanced works remained less tampered with.
If we want a definition of a straight line consistent with Greek geometry, I would propose defining it as follows: a straight line is the path of a stretched string. In other words, a straight line is a curve that doesn’t change shape when you pull its endpoints.
This is closely related to the notion of the shortest distance between two points. Related, but not equivalent. To get to the bottom of the notion of straightness it is useful to consider not only the usual plane but also other surfaces. Euclid’s geometry is the geometry of a flat plane, a flat piece of paper so to speak. Other surfaces have other geometries. A cylinder, for instance, like a Pringles can. It has its own geometry. Pringles lines, Pringles triangles.
To appreciate the geometry of a surface we should forget for a moment that it is located in three-dimensional space. We should look at it through the eyes of a little bug who crawls around on it and thinks about its geometry but who cannot leave the surface and is unaware of any other space beyond this surface. Think of for example those little water striders that you see running across the surfaces of ponds. They know the surface of the pond ever so well. They can feel any little movement on it. But they are quite oblivious to the existence of a third dimension outside of their surface world. This makes the water strider an easy prey for a bird or a fish that strikes it without first upsetting the surface of the water.
It is instructive to think about the intrinsic geometry of surfaces in this way. It forces us to realise that many things we take for granted as “obvious” objective truths in geometry are really a lot more specific to our mental constitution and unconscious assumptions than we realise. In some ways we are as ignorant of our own limitations as the water strider.
Let’s transport ourselves into the cylinder world to practice seeing geometry from a different point of view. On a cylinder there are stretched-string curves that are not the shortest path between its two endpoints. Wrap a shoelace around a Pringles can. You can make various spirals that are stretched strings. Or a helix as it’s called, a corkscrew curve. So these are straight lines, according to my definition. But they are not the shortest distances between their endpoints. Even if you have to stay on the surface of the cylinder, you can still get from one endpoint to the other more directly than by a spiral that winds around and around an excessive number of times.
So “stretched string lines” and “shortest distance lines” are not the same thing, as this example shows. It is arguably the stretched string that gets it right. It makes straightness a “local” property.
We can alter the distance characterisation of straightness to be local too. Then we would say: a curve is a locally shortest path if, for any given point on the curve, there is a neighborhood around that point such that the distance along the curve between any two points on the curve in that neighborhood is the shortest possible distance between those points. This picks out the same straight lines as the stretched string definition. Being a stretched string is the same thing as being a locally shortest path: it’s the shortest path between points on the line when you zoom in, but not necessarily between points on the line that are far apart.
Straight lines can also be defined as curves possessing half-turn symmetry about every point: a curve has half-turn symmetry if, for any given point P on the curve, there is a neighbourhood around that point such that when this neighbourhood is rotated about P by half the angle-measure around P then the curve ends up on top of itself. More loosely, a curve is straight if it always “cuts angles in half”; it “leaves the same amount of space on either side.” To test for this kind of straightness on surfaces one can use the “ribbon test”: if a ribbon or band can be laid flatly along the curve without creasing on either side, then the curve is straight.
Try it on your Pringles can. You can use a measuring tape for instance, for instance those free paper ones you can get at hardware stores or furniture stores. That’s your “ribbon.” Try wrapping it around the Pringles can. Some ways of wrapping it makes it lay flat against the surface; those are straight lines. Other ways of wrapping it makes it crease up on one side or the other; those are not straight lines because they don’t leave the same amount of space on either side.
Here’s a fun thing to investigate and think about. We have now defined straight lines on a Pringles can in two different ways: one in terms of a stretched string, like a shoelace, and one in terms of a flat ribbon, like a measuring tape. Are they the same? Are there some lines that are “shoelace-straight” but not “ribbon-straight” or the other way around? I’ll leave that to you to explore.
So we have two notions of straightness, and both of them get at something very fundamental:
The stretched string highlights the idea of straightness as minimization, or as a tight fit. This idea is reflected in many real-world occurrences of straightness. For instance, the path of a cross-Atlantic flight. You know that when you look at the path on a map, in the flight tracker, it looks curved. It looks like you’re flying from Paris up toward the North Pole, and then back down again to get to New York. Why not go “straight across” instead? Of course the path is in fact straight. It looks curved only because the map is an imperfect representation. If you have a globe you can stretch a string between Paris and New York and feel for yourself that the shortest path indeed goes “up” toward the North Pole. But that path is straight, according to the stretched string definition.
But we also have the second idea of straightness: that of straightness meaning “the same amount of stuff on both sides.” This is also reflected in various familiar situations. For instance, when you fold a piece of paper, the edge is straight. Why is that? This doesn’t have to do with stretched strings or least distances. Instead it has to do with the sameness of both sides. To fold something you match up points on one side with points on the other. Folding is only possible if the two halves are precisely equal.
There is also a kind of three-dimensional version of this. Namely the axis of rotation when a solid body is rotated. For example a döner spit at a Middle Eastern restaurant, or a basketball spinning on your finger tip. The axis of rotation is a straight line. Why? This is again because of sameness on all sides. The moving parts have to fit into each others’ space. So they have to be equal on either side.
Here’s an example from engineering. Mirrors are made flat by rubbing two of them against each other face-to-face, with a fine sand or other polishing agent applied between them. This too embodies the idea of flatness or straightness as equivalent to sameness on both sides.
Another example is rowing a boat. You go straight in a rowboat if you apply equal force to each oar. This is again symmetry-straightness, not stretched-string straightness. It’s not built into the very rowing process that this necessarily corresponds to the shortest distance between the endpoints of the journey. But it is built into the very act of rowing this way that you leave equal amounts of space on either side.
Light rays are straight. But this is more like the stretched string again. Light “cares” about minimizing the time of travel, so to speak. Just like the airline. The airline stretched a string across the globe to find out how to fly from Paris to New York. They also tightened their purse strings, so to speak, with the same move, because the shortest path is also the cheapest path. Light is a bit of a penny-pincher too, it would seem; or it is impatient, perhaps. Because it chooses the quickest path. For instance, if it has to go from point A to point B via a flat mirror, then it chooses to bounce off the point on the mirror that makes the total distance as short as possible.
You can reproduce this path with a stretched string. Suppose A and B are two points on a wooden table. Let’s hammer two nails into those points. One of the edges of the table we regard as the mirror. Take a vertical metal bar and put it against the edge of the table. Now wrap a string from A, around the metal bar at the end of the table, and then to B. Now pull the string as tight as you can. The metal bar forces the string to go to the edge of the table and back. But the bar can move along the edge of the table. When we pull the string we force the bar into a particular position, namely the position that minimizes the total distance. The path of the string is the same as the path of light between these points via a mirror at the edge of the table. You can try it out with a laser pointer if you don’t believe me.
So light is like stretched strings. Indeed artists use this sometimes. The pull strings to simulate light rays in order to get vantage points and perspectives just right.
I’m trying to emphasize with these examples how thinking about what straightness means is connected to many aspects of culture and experience. Isn’t it fascinating how the mathematical notion of straightness is a sort of root of all these diverse phenomena? Once you’ve read the Elements you see geometry everywhere. Flight paths, döner spits, spinning basketballs, light and mirrors, rowboats, Pringles cans––henceforth, anytime you encounter these things you will go: ah, of course, that reminds me of Euclid’s Definition 4!
The idea of straightness as corresponding to stretched string also generalizes well to other surfaces that are not homogenous. So far we have mentioned the plane, the cylinder, and the sphere. These surfaces are all homogenous in that every point is alike. If you cut out a piece of the surface, it fits on top of any other part of the surface.
Some surfaces are not like that. For example, the surface of the human face. It has regions of different curvatures, as we say. A flat piece of paper has zero curvature: it’s not curved at all. A ball has positive curvature: it curves the same way in all directions. A saddle has negative curvature: it curves in different ways in different directions. A saddle for riding a horse. It curves “upwards” along the spine of the horse, and “downwards” where your legs go. Opposite directions of curving. This is what makes the curvature negative.
The human face has both negative and positive curvature. Some parts are like a saddle. For instance the side of the nose, or the area just below your mouth. If you put your finger there and run it top-to-bottom, then it curves one way. But if you ruin it side-to-side, it curves the others way. So those are regions of negative curvature. They are like a saddle.
Other parts of the face have positive curvature, like a ball. For instance the chin and the cheeks. There the surface curves the same way no matter which direction you run your finger.
Felix Klein, a 19th-century mathematician, thought this might be the key to a mathematical analysis of the elusive concept of human beauty. Since the face has regions of positive curvature and regions of negative curvature, there’s a diving line running between them. Between the cheek and the nose, between the lips and the chin, and up again on the other side.
So Klein drew this line of zero curvature on a classical sculpture. You can google it, Felix Klein Apollo Belvedere, and you can see photos of this. Klein was hoping that a simple pattern would emerge that would “explain” the beauty of this face. But it didn’t work. No such pattern was discernible.
Still it makes for a good story. It’s also a good piece of “first date mathematics.” You can explain this idea to your date over some glasses of wine. And of course slowly reach out and sensually trace these curves on their face and so on. Great stuff.
But where were we? I wanted to discuss how the notion of straightness extends to these other surfaces. Surfaces with variable curvature. We can still say that straight lines are stretched strings. We often call them geodesics rather than straight lines in such cases. But the stretched-string idea is still the same.
Here are some examples. Think of bandaging an injured limb. The bandage needs to be tightly wrapped. This means that it must follow a geodesic path, a stretched-string path. The bandage is a “straight line” in the sense that it is a stretched string. In other words, it always takes the locally shortest distance. Of course not the shortest distance overall, since it wraps around and around. But the shortest distance between any two nearby points on its path, because otherwise it would create slack which you would never do of course.
Another example: The heart beats through the contraction of muscular threads across its surface. These muscular threads must be geodesics. They must be stretched-string paths. Because the heart beats by contracting these threads. If these muscular threads were not positioned along geodesic paths, then when they contracted they would just slide around on the surface of the heart instead of contracting it. The human heart is carefully designed with this geometry in mind. And if it wasn’t we would all die very quickly. So the stretched-string notion of straightness is truly a matter of life and death.
How should axioms be justified? By appeal to intuition, or sensory perception? Or are axioms legitimated merely indirectly, by their logical consequences? Plato and Aristotle disagreed, and later Newton disagreed even more. Their philosophies can be seen as rival interpretations of Euclid’s Elements.
Transcript
What kinds of axioms do we want in our geometry? How do you tell a good axiom from a bad one? Should an axiom be intuitively obvious? Should it be empirical, physically testable? Should it be logically self-justifying, or are axioms logically arbitrary?
The time has come to take a stand. As we have been reading Euclid backwards, we have seen how the Pythagorean Theorem can be reduced to a theorem on the areas of parallelograms, and how this theorem in turn can be reduced to triangle congruence. So now we have to prove triangle congruence somehow.
If two triangles have the same side-angle-side, then they are the same triangle. How to prove such a thing? We can’t keep playing our game of reducing every theorem to a simpler one, because we’re running out of “simpler.” Maybe this theorem is as simple as it gets? Maybe it’s the rock bottom? How do you decide anyway what’s simpler than what? It’s becoming more philosophy than mathematics to answer these questions.
This theorem—side-angle-side triangle congruence—is Euclid’s Proposition 4. Ok, so it’s a proposition, not an axiom. So apparently he has reduced it to something. But what?
Let’s read the proof. So we have two triangles, and they have certain measurements in common. Two sides of one triangle are equal to two sides of the other, and also the angle between those sides are equal in both triangles.
Euclid says he can prove that the other measurements are equal too. The remaining side, the remaining angles: it’s all equal. They’re the same triangle basically.
And here’s how Euclid says you can prove this. Take one of the triangles and put it on top of the other. We know that they have side-angle-side in common, so those parts line up perfectly. These three attributes are enough to “lock” the entire triangle into one unique shape, in fact.
Because suppose it wasn’t. Suppose the two triangles were different. Since they have side-angle-side in common, they lined up at least on those parts. This “locks” into position two of the sides and all three of the vertices. There is no way one of the triangles can stick out beyond the other in terms of these two sides or in terms of any one vertex. So the only way the triangles could be not equal would be if the third side somehow missed.
This would mean that the endpoints of the third sides were the same for both triangles, but the line joining them would be different. Impossible! You can’t have multiple lines connecting the same two points. Or Euclid puts it: two straight lines cannot enclose a space. You can’t draw a straight line from A to B, and then another straight line from A to B, in such a way that these two lines miss each other and have some space in between them.
Since this is impossible, the third sides of the triangles must line up on top of each other, and therefore the two triangles are identical, or congruent. That’s the proof.
Once again the point of the proof is not to convince us that the theorem is true, but to reveal how its truth can be reduced to more basic truths. Euclid has now taken this as far as he can. We’re all the way down to the axioms: things that cannot be broken down any further.
The proof of the triangle congruence theorem rests most prominently on two axioms. One, as we saw, is that “two lines cannot enclose a space.” Which is equivalent to saying that, for any two points, there is only one straight line between them. This corresponds to Euclid’s Postulate 1, which states as an axiomatic principle that we can “draw a straight line from any point to any point.” It is understood that this line is unique. That is to say, there’s only one way you draw that line. So that’s an axiom. You can’t reduce it any further.
But there was another axiom involved as well in our proof of the triangle congruence theorem. Namely the assumption that we could put one triangle on top of the other. This corresponds to Euclid’s Common Notion 4: “things coinciding with one another are equal to one another.”
This is basically a definition of equality. What does it mean for two things to be equal? Put one on top of the other. If neither sticks out beyond the other, then they are equal. That’s what equal means. In fancier words you could say: equality means alignment under superposition. So that’s another axiom that Euclid states at the beginning of his work, and which he cannot prove from more basic principles.
What should we make of these two axioms? Since we can’t prove them from other things, they must be justified some other way. What way would that be?
Euclid apparently thought these two principles were especially suited to be axioms. He could have done it differently. He could have chosen other axioms. For example, the triangle congruence theorem itself could have been taken as an axiom. That’s what Hilbert later did, in his modern and very authoritative axiomatisation of geometry. So from the point of view of modern mathematics it makes a lot of sense to take the triangle congruence principle as axiomatic. From a logical point of view that’s perhaps the best approach. Modern logicians don’t like Euclid’s proof one bit. Bertrand Russell called it “logically worthless.” If you want mathematics to be logic, then that makes sense.
But what is “good” mathematics? That depends on your philosophy of mathematics. You must first decide what kind of thing mathematical knowledge is. What it should be. Only after you have made that philosophical decision do you have any basis for judging whether Euclid’s approach is better or worse than that of others.
Euclid’s choice of superposition as an axiomatic principle is quite interesting in this regard. It seems almost physical or empirical. In the proof of the triangle congruence theorem, you are literally, physically picking up one of the triangles and placing it on top of the other triangle. This seems to assume that triangles are physical objects, like cardboard cutouts or some such thing. And the idea that equality means alignment under superposition also has a somewhat physical feel. The thing fits on top of the thing. It’s something you could test practically, in the real world.
The modern authors I mentioned do not approve of these connotations. They don’t like it one bit that mathematics is so to speak contaminated by empirical considerations. They want mathematics to be pure reason. They don’t want it to depend on sense perception and physical experience.
But Euclid’s use of superposition suggests that he was less dogmatic about this. It could be interpreted as a sign that he was open to the idea of geometry as ultimately physical.
Of course geometry is still very theoretical. Obviously, to Euclid, you can’t justify things like the Pythagorean Theorem just by measuring things, the way you would verify a physical law by making a bunch of measurements in a lab. Of course geometry is not like that.
But the fact remains that the axioms cannot be justified by the axiomatic-deductive process itself. What axioms are the “right” axioms, or the “best” axioms, is a question that cannot be answered by purely mathematical means. Some philosophical assumptions will necessarily be involved in such judgements.
I wanted to use this as a bridge to discuss some Plato and Aristotle. I’m trying to emphasize how these things go together. Mathematics and philosophy. Reading Euclid leads naturally to philosophical questions. We reduced the Pythagorean Theorem down to superposition and uniqueness of lines. We faced the questions: Why stop there? Why these principles and not others? What kinds of foundations should geometrical knowledge be built upon?
This is the right time to read philosophy, with these burning questions fresh in our minds. Mathematics itself does not answer these questions. As Aristotle says in the Posterior Analytics: “for the principles a geometer as geometer should not supply arguments.”
So there is a kind of division of labor. Justifying the axioms is not the business of the geometer “as geometer.” But of course Aristotle didn’t mean by this that you should have mathematicians over there and philosophers over here and there’s no point for them to talk to each other. A better way to read it, I think, is this: geometers, as geometers, cannot justify their axioms, and therefore any geometer needs to be a philosopher as well.
Aristotle discussed the axiomatic-deductive method at length in this treatise, the Posterior Analytics. Here’s a quote that sums up his view: “Demonstrative understanding must proceed from items which are true and primitive and immediate and more familiar than and prior to and explanatory of the conclusions.”
Quite a list of demands! Axioms, such as those of geometry, should have all of those characteristics, according to Aristotle.
Obviously this means that the axiomatic-deductive method is a whole lot more than merely logical deductions from arbitrary assumptions. Indeed Aristotle says as much: “There can be a deduction even if these conditions are not met, but there cannot be a demonstration––for it will not bring about understanding.”
This places very significant restrictions on what could be a legitimate axiom in geometry. It must be “primitive and immediate and more familiar than and prior to and explanatory of the [theorems].” So axioms need to be self-evident, in other words, it seems. That’s more or less what Aristotle means by “immediate,” I suppose. And axioms must also be irreducible, not in turn derivable from some other principle. That seems to be the meaning of Aristotle’s demand that they be “primitive” and so on.
It gets pretty interesting when Aristotle elaborates further on what he means by some of these terms, because then he commits himself to the perhaps controversial stance that axioms are ultimately grounded in physical experience. Here’s what he says: “I call prior and more familiar in relation to us items which are nearer perception.” So immediate perception must be the ultimate foundations of “demonstrative understanding.” Not pure thought, but sensory perception.
The axioms are generalized or idealized facts of experience. As Aristotle says: “We must get to know the primitives [that is to say, axioms] by induction; for this is the way in which perception instills universals.” For instance, for any two points there is a unique line connecting them. This is fact of experience, but of course generalized––“by induction,” as Aristotle says. That is to say, we have observed this in many examples. For this particular pair of points there’s a unique line, and for that pair, and so on. These are facts of perception. And then “perception instills universals by induction”: that is to say, we generalize from these examples to the general principle that the principle will work for any two points, not just the numerous examples we have witnessed.
So Aristotle thinks the axioms of geometry ultimately come from concrete experience. The credibility of the axioms, the certainty of the axioms, derives from immediate sensory experience.
This fits pretty well with the principles to which Euclid reduced everything. It is known through experience that there is a unique line from any point to any point. For instance by pulling a string between two points you can get a very direct sensory feeling for the existence and uniqueness of that line. And the principle of superposition, of putting one triangle on top of the other, can likewise be seen as an idealized version of a very immediate and basic physical experience.
But not everyone agreed. Plato is the opposite of Aristotle. He has complete contempt for the physical world, and he loves mathematics precisely because it is something purer and higher than physical experience.
Let me quote Proclus expressing this view. Proclus is a follower of Plato. He is keen to argue that mathematics stems from the soul, not sense experience. He addresses the Aristotelian view, and he sums it up like this: “Should we admit that [the objects of mathematics] are derived from sense objects, either by abstraction, as is commonly said, or by collection from particulars to one common definition?” That’s what Aristotle had argued, but Proclus says: No, we should not accept that.
And here’s why. Geometry cannot be based on physical experience, Proclus says, because “The unchangeable, stable, and incontrovertible character of the propositions [of mathematics] shows that it is superior to the kinds of things that move about in matter. And how can we get the exactness of our precise and irrefutable concepts from things that are not precise? We must therefore posit the soul as the generatrix of mathematical forms and ideas,” not physical reality.
Plato was quite obsessed with this idea that pure thought is the highest and most noble thing in human life. In the Timaeus he elaborates on this idea in a rather amusing and poetic way. To philosophise is the purpose of life. Human anatomy is merely an appendix to the soul and the mind. “The entire body” was created “as its vehicle,” Plato says. That is to say, the body exists only to make philosophising possible.
For example, consider the intestines of the human digestive system. They are very long and winding, right? Like you roll up an extension cord when putting it away in a drawer; it looks like that in our insides. Food doesn’t go in a straight line from the mouth and out the other end. Instead the body passes it through the intestines that go back and forth, back and forth, a very long distance.
Plato thinks he knows why. Here’s how he explains it: “The intestines are wound round in coils to prevent the nourishment from passing through so quickly that the body would of necessity require fresh nourishment just as quickly, there by rendering it insatiable. Such gluttony would make our whole race incapable of philosophy and the arts, and incapable of heeding the most divine part within us.”
So the human body is just a means to an end. The only thing worth anything is philosophy. Eating doesn’t have any value in itself. The only purpose of eating is to put off the annoying needs of the body for a while, so as to give us time to think.
Plato has a similar theory regarding eyesight. To Aristotle, the senses were a source of knowledge. The foundations of geometry rested on sensory experience. Of course Plato disagrees. The purpose of eyesight is just like that of the intestines: it’s just a physical crutch whose ultimate goal is to support pure philosophy. Here’s how Plato puts it:
“Our ability to see the periods of day and night, of months and of years, of equinoxes and solstices, has led to the invention of number and has given us the idea of time and opened the path to inquiry into the nature of the universe. These pursuits have given us philosophy, a gift from the gods to the mortal race whose value neither has been nor ever will be surpassed. I’m quite prepared to declare this to be the supreme good our eyesight offers us.”
So eyesight is not a good in itself, but merely a stepping-stone toward philosophy. It’s a kind of necessary evil, like the intestines. It would be better if we didn’t have to eat at all, but given that we live in this feeble physical world, the best we can do is to make the food take a long time to go through us so we have as much time as possible to think in between meals.
In the same way, ideally, we wouldn’t need eyesight. Ideally, we would do pure philosophy, which transcends feeble physical reality. But we are stuck in physical form and with imperfect minds. So we need these support mechanisms to push us toward philosophy. Eyesight leads to astronomy which leads to mathematics and thus philosophy, and then we’re in business.
It would have been better if we could have skipped those preliminary steps and gone straight to philosophy. Then eyesight would have been redundant. Eyesight isn’t actually needed for true philosophy. We only need it because of our imperfections. We need this little push to get us started on philosophy, but once we’re up and running with philosophy we can pretty much poke our eyes out because they’re not needed anymore.
In this passage, Plato was talking about astronomy but he could just as well have said the same thing for geometry. This is how we must think about the role of geometrical diagrams and sensory perceptions in Plato’s philosophy of mathematics. True mathematics is independent of all that physical stuff, according to Plato. Geometry is not based on physical and sensory experiences with moving figures, drawing lines, and so on, as Aristotle claimed. Diagrams and reliance on the senses are only a stepping stone to true geometry. We need this crutch because our minds and bodies are feeble and imperfect. But once we’ve reached the philosophical level of doing geometry, we can kick away this ladder because then it serves no purpose anymore.
Here’s another colorful image Plato has for this. He’s explaining why birds exist. “[Birds] descended from simpleminded men––men who studied the heavenly bodies but in their naiveté believed that the most reliable proofs concerning them could be based upon visual observation.” And conversely, “land animals came from men who had no tincture of philosophy and who made no study of the heavens whatsoever. As a consequence they carried their forelimbs and their heads dragging toward the ground.”
So the philosophising human is the perfect balance between these poles: not focused on worldly gratification like the beasts, but also not making the mistake of trying to understand thing by looking. The birds thought that the best way to understand the stars was to get as close as possible to get a good look. But humans know better. We understand that the best way to understand the stars is by thinking, by philosophising, not looking.
Once again the same can be said for geometry. Too much looking and not enough thinking: that is the cardinal sin that we must avoid not only in astronomy but in geometry as well.
This also fits well with another work by Plato, the Meno. In this work, Plato shows how an ordinary uneducated slave boy can be led to recognize geometric truths, such as a special case of the Pythagorean Theorem. Socrates draws a simple diagram and asks some simple questions, and step by step the boy fills in the reasoning and arrives at the theorem.
Plato interprets this as a sort of awakening. Learning is a form of recollection, he claims. That is to say, the boy did not reach this geometric insight through instruction, or through empirical investigation dependent on the senses. Rather, the boy realized that he knew something that he didn’t know that he knew, so to speak. His inner philosopher was awakened. External input was the trigger for this awakening, but the knowledge had really been there all along. The senses are just a trigger for reawakening this knowledge, not an actual basis for that knowledge. This story sums up the role of the senses in geometry, according to Plato.
So what does this mean for the axioms of Euclid? What kinds of things do the axioms of geometry need to be to conform with Plato’s vision of geometry as this kind of pure philosophy, a work purely of the mind? I think it comes down to a kind of innateness theory of axioms. The axioms of geometry need to be essentially pre-programmed into our minds.
This fits with the idea that learning is recollection, and that mathematics is merely making the mind conscious of things it didn’t know that it knew. There is no external source of this knowledge, according to Plato. The mind just knows it, within itself.
So axioms should be intuitive, instinctive. You should read them and you should go: of course! They should feel like the most natural and undoubtable thing in the world. That’s what Plato’s theory suggests.
Proclus of course agrees. He’s Plato’s mouthpiece, and here’s what he says about axioms: “axioms take for granted things that are immediately evident to our knowledge and easily grasped by our untaught understanding”; “[axioms] must always be superior to their consequences in being simpler, indemonstrable, and evident in themselves.”
That’s almost exactly what Aristotle said. So Plato and Aristotle arrive at the same view of axioms despite their very different outlooks. They disagree on the ultimate origin and foundation of this knowledge: whether it comes from sensory experience and the external world, or whether it comes purely from within our philosophical faculties.
This opposition is famously captured in the the iconic fresco The School of Athens painted by Raphael. Plato is pointing to the sky, Aristotle is pointing straight ahead. They are basically pointing to where they think knowledge comes from. Aristotle thinks the source of knowledge is the world before our noses. Plato thinks knowledge resides in a higher realm, above the physical.
But despite this orthogonal disagreement, Plato and Aristotle agree on the properties that axioms must have. Axioms need to be the simplest and most obvious first truths.
Do you agree with them? No, you don’t. You don’t think axioms need to be obvious and intuitive. Either that, or else you think Newtonian physics is a hoax.
Newtonian physics is an example of an axiomatic theory where the axioms are completely non-intuitive. In fact, they are very strongly counter-intuitive. The basic axiom of Newtonian physics is the law of universal gravitation. Any rock is pulling on any other rock, even if they are separated by thousands of miles of empty space. That’s just sheer witchcraft. In fact, you yourself is in a direct bond with all the universe through this mysterious force. It’s like something straight out of science fiction or new age spirituality. Every last one of the thousand stars in the night sky is actively and directly exerting a force on you at any given moment. That’s crazier than any occult astrology you’ve ever heard. Yet that’s Newtonian physics, the most successful scientific theory of all time.
In fact, this example of Newtonian physics corresponds precisely to a kind of blind spot that we should have seen coming in our discussion of axiomatic philosophy. On the one hand we said axioms should be obvious, simple truths, but on the other hand we said axioms are what you are left with after you start with theorems like the Pythagorean Theorem and reduce and reduce and reduce.
Those are two different ideals. And they are not necessarily compatible. The idea of reducing complex theorems into smaller part does not entail that the axioms you end up with are obvious truths. Axioms are just whatever results when you reduce many theorems to a few core principles. This process could be seen as agnostic as to the nature of the axioms. We just follow the reductive process where it takes us. Just like a chemist cannot decide in advance what kinds of elements he wants the period table to contain, so also the mathematician reducing geometry to its building blocks has to keep and open mind and follow the reductive process where it takes him.
At least that’s how Newton interpreted the geometrical method. He’s very clear about this. He’s very explicit about this reductive process being the same in physics as in geometry. Geometry starts with things like the Pythagorean Theorem; physics starts with things like the speeds of the planets and so on. These are the “phenomena” as Newton calls them. And from the phenomena you reason backwards to the underlying causes or unifying principles. That’s what you do in geometry when you show how many theorems can be reduced to a few key principles, and that’s what you do in physics when you show that lots of astronomical data can be derived from a few laws.
Newton is adamant that these two things are the same. “As in mathematics, so in natural philosophy,” he says. “Natural philosophy” means physics. The two are the same, in terms of methodology. That’s how Newton justifies his radical physics. By saying that it’s nothing but what the geometers had been doing all along.
To make this shoe fit, Newton has to sacrifice the idea that axioms are obvious truths, as Aristotle and Plato had claimed. But his interpretation is not crazy. You could read Euclid that way. You could say: Euclid doesn’t care whether the axioms are obvious or not. He just follows the reductive process where it leads. He’s agnostic or open-minded about what kinds of axioms will be the outcome of this process.
Of course that clashes with what Plato and Aristotle said, but they are philosophers so it doesn’t really matter. The important thing is what the mathematicians thought, and their texts are ambiguous enough to allow for the possibility of Newton’s interpretation.
So Newton interprets Euclid a certain way in order to justify his own methodology. Newton’s interpretation is hardly very likely, but it’s also not provably wrong exactly. He’s a clever guy, Newton. He knows his physics is crazy and occult, so he massages an interpretation of the Euclidean tradition to legitimate it.
I don’t think Newton was right in the way he interpreted Euclid. But his perspective is very illuminating nonetheless. For one thing it’s striking that Euclid’s geometry was so authoritative still 2000 years after it was written that cutting-edge modern science was justified on the grounds that its method was the same as that of Euclid. There was no more solid pillar of respectability than Euclid, to anchor your theory to. Even then, 2000 years after the Elements was written. Euclid’s city, Alexandria, had burned any number of times, and seen several new religions come and go. But the impact of the geometrical method was above such transient circumstances.
But even just for understanding Greek philosophy of geometry in itself the Newtonian example is useful. Greek philosophers seem to have been blissfully unaware of the possibility of such a theory, where the reductive process leads to non-obvious axioms.
In fact, in Aristotle’s Posterior Analytics there is a phrase that pretty much sums this up. Here’s what Aristotle says: ”I call the same things principles and primitives.” Principles are the logical starting points of a deductive system, and primitives are the immediately given truths grounded in perception. Aristotle thinks you might as well regard these as synonyms, apparently. He does not serious consider the possibility of viable scientific theory in which these two concepts would not align.
But Newtonian physics is such a theory. It has principles that are not primitives. That is to say, it has axioms obtained by reducing the phenomena down to their smallest parts, but those axioms are not obvious and not intuitive and not known by direct experience.
So the Greeks could have their cake and eat it too. They could have the idea of “reasoning backwards”–of reducing geometry to a few core principles–and at the same time maintain that these core principles should conform to various predetermined philosophical requirements as well, such as being obvious.
Newtonian physics shows that you can’t always have it both ways. At a certain point, you have to pick sides. So you have to decide which of the two you’d rather sacrifice.
Newton picked the brave side, I think. The path less travelled. He sacrificed the idea that axioms should be intuitive. A huge sacrifice, almost unthinkable. It’s like a military general sacrificing 90% of his troops in an audacious manoeuvre. Few people would have dared to even contemplate such a move. But it worked. Even though it was a huge sacrifice, it got Newton into such a strong position that he won the war anyway.
Many people at the time thought Newton was crazy for making this sacrifice. He got a lot of pushback for this. Reduction to non-obvious axioms?! It’s such a radical idea. It goes against everything Plato and Aristotle said.
But in a way Newton’s idea is already contained in Euclid. It’s the idea of reading Euclid backwards. Newton’s perspective may not have been Euclid’s exactly, but it’s useful to keep the example of Newtonian physics in mind to highlight what’s at stake in this tension between the “backwards” and “forwards” directions of reading Euclid.
Euclid’s Elements, read backwards, reduces complex truths to simpler ones, such as the Pythagorean Theorem to the parallelogram area theorem, and that in turn to triangle congruence. How far can this reductive process be taken, and what should be its ultimate goals? Some have advocated that the axiomatic-deductive program in mathematics is best seen in purely logical terms, but this perspective leaves some fundamental challenges unresolved.
Transcript
Here’s a way to think about one of the key ideas involved in Euclid’s proof of the Pythagorean Theorem. Picture a stack of books sitting on your desk. It has the shape of a rectangle. Let’s say you’re looking at the side with the spines of the books; they make a rectangle. Now, give the stack of books a whack with your hand. So the pile is knocked askew. The shape of the stack is now a parallelogram instead of a rectangle. But the area is the same. I mean the area of the side facing toward you, the side with the spines of the books.
It’s obviously the same area because it’s made up of the same books as before. You just moved the books around. You moved the same amount of area into a new configuration.
Also the height is the same: the height from the desk to the top of the pile. This is still equal to the sum of the thicknesses of each book.
This illustrates the geometrical theorem that the area of a parallelogram is equal to the area of a rectangle with the same base and height. This is Euclid’s Proposition 35.
This is a key ingredient in Euclid’s proof of the Pythagorean Theorem. To prove the Pythagorean Theorem we need to show that the area of the squares on the sides is equal to the area of the square on the hypothenuse. We do this by starting with one of the small squares on the sides and showing that its area can be remolded and made to fit into the big square in such a way the theorem becomes clear.
So the idea of Euclid’s proof is to transform one area into another. Its shape is transformed but the area remains the same. And the transformation he uses is basically this one with the stack of books knocked over into a parallelogram shape.
Euclid starts with a stack of books corresponding to one of the small squares. He knocks it over into a parallelogram shape. He rotates the parallelogram by 90 degrees so it’s now aligned with the big square instead. And he straightens the parallelogram back out again, just like you would straighten out a stack of books. This is how he shows the equality of areas that the Pythagorean Theorem asserts.
The book analogy is not perfect because Euclid so to speak slices his stack of books two different ways. If we want to think of his first step, transforming a square into a parallelogram, in terms of a book stack, then we must visualise the spines to go a particular way. Then when Euclid is straightening the parallelogram back out later, if we want to visualise that in terms of books, we need to picture the spines of the books differently, sitting in another direction. It’s a different stack of books, so to speak. Different but equal. If you have Euclid’s text in front of you, you can draw this into the diagram, how the books need to be oriented for each step to work, and you will see clearly that you have to change perspective halfway through. Euclid is talking about triangles instead of rectangles and parallelograms but that doesn’t matter, the principle is the same.
So we are continuing our adventure of reading Euclid backwards. We reduced the Pythagorean Theorem to a more basic proposition, the book stack proposition, 35. What does that in turn depend on? Remember that we are trying to boil everything down to its molecular components. How does Euclid prove Proposition 35? That is to say, how does he reduce this this proposition to more basic ones?
I should clarify that Euclid doesn’t do anything like this stuff with the books. I explained this theorem with this analogy to a stack of books, but certainly Euclid’s logic doesn’t depend on anything like that. That would be much too informal. The books need to be “infinitely thin” for the argument to work perfectly, and that’s a whole can of worms foundationally that Euclid certainly doesn’t want to go in to. Instead he offers a purely finitistic proof.
Euclid’s proof of Proposition 35 is very clear and satisfying. Euclid proves that one area is equal to another by adding and subtracting pieces in a clever way. So he decomposes it into a couple of puzzle pieces that fit just right with each other. Even though the two areas as wholes have entirely different shape, Euclid shows that there is a clever way of cutting the situation into puzzle pieces that are equally suited to each area.
The two areas are two parallelograms of different shape; they’re like two different languages so to speak. You would have thought that they couldn’t communicate very easily. But these puzzle pieces establish a common understanding; something that is equally natural and understandable in either language. So these puzzle pieces, this universal language, can be used to translate one area into the other.
If we think in terms of reducing the truth of the theorem to more basic facts, this means that, with the puzzle pieces, we have basically reduced the equality of the entire areas to the equality of the each corresponding puzzle piece separately. The puzzle pieces are all triangles, and the fact that corresponding ones are equal comes down to triangle congruence theorems. That is to say: Under what conditions are two triangles the same? For example, they are the same if the have side-angle-side in common.
That turns out to be the next step down if we keep reducing the Pythagorean Theorem. Like a French chef simmers a sauce to make it thicker, so we keep boiling the Pythagorean Theorem, and now we’re down to this. Triangle congruence, and some stuff about parallels as well. We have to keep reading Euclid to find out what happens if you keep cooking it.
But before we keep wilting down the Pythagorean Theorem on the Bunsen burner to see what it’s made of, let’s take a moment to reflect on this theorem about the stack of books, or the areas of parallelograms.
Proclus has an entertaining remark about this theorem in his ancient commentary on the Elements. He points out that it shows that the same area can have many different perimeters. The stack of books, if you make it more askew you will increase the perimeter while keeping the area the same. A very stretched-out parallelogram has a lot of perimeter but not a lot of area.
According to Proclus, military commanders in antiquity did not understand this, with detrimental consequences. Suppose an enemy army is advancing toward your borders. You want to know how many they are. So you send a spy in the cover of darkness at night to scout the situation. The spy sneaks up on the enemy’s night camp and stealthily walks around it, counting the number of step. He then rides back and reports this number.
So the number of steps around the camp is taken to be a measure of its size. For cities as well you could do this: How big is the city? Just walk around the city walls and count the steps. It’s so-and-so many steps big.
Of course this is a mathematical mistake, because it measures the perimeter when you really wanted to know the area. And the stack-of-books theorem shows that they are not at all the same.
Anyway, that’s just a fun story. All the propositions of Euclid have some cultural significance like this. It’s like you see sometimes the period table of chemistry and for each element they’ve added a little example of some familiar real-world thing where this element occurs. “You know kids, lithium isn’t just some weird science thing, you use it every day!” It’s in whatever, toothpaste or something. So you can do that with Euclid’s Elements as well. A little story for each theorem to lighten the mood and make things a bit more culturally relevant. But that’s just for kicks and giggles.
Let’s get back to the more scientific purpose: the systematic reduction of all geometrical knowledge to some sort of ultimate minimum foundation. We are just a few steps in to this process and it’s already starting to raise some philosophical conundrums. It was natural enough to take apart the Pythagorean Theorem into more basic results, like the one about areas of parallelograms. Then that in turn could be reduced to triangle congruence.
But this can’t go on forever. And we’re already down to such basic facts that it’s becoming very difficult to see how there could be anything “more basic” to reduce them to.
This path of reduction, it looked so natural when we set out on it. Starting from the Pythagorean Theorem, this seemed like an obvious way to go. But our clear path through the woods is now becoming darker and thornier. It’s no longer clear where to go from here. Instead of blindly forging ahead in the same direction, we need to take a step back and think about where it is we want to go. What kinds of things should the foundations of geometry be?
There are in fact a number of possible answers to this that are very different and completely incompatible with each other, yet each of them is quite plausible in their own right. Let’s have a look at some of the main ones. I mean philosophical views of the status of axioms, or starting points, in mathematics. Or what pretty much comes to the same thing: philosophical interpretation of the ultimate nature of mathematical reasoning and the source of its credibility.
Do you think mathematics is ultimately empirical, like physics? Is geometry just the science of physical space? If so, that suggests that the axioms of geometry should be the most fundamental and testable things from an empirical point of view. Geometry should start from things you can check in the field or in a lab. Measuring things with rulers, for instance. That should be the starting point of geometry if you think the certainty of geometrical reasoning ultimately derives from sensory experience and data collected from the world around you.
Or do you think mathematics is ultimately pure reason? Then the axioms don’t need to be physically testable but rather mentally fundamental. That suggests that goal of the reductive process is to boil theorems down to the most obvious or intuitively undoubtable starting points.
This divide between empiricism and pure reason is mirrored in Aristotle and Plato, one might argue. We will look into that in more depth another time.
Let’s focus now on yet another point of view: That of logic. There are two ways you can say mathematics is pure reason: One associates reason with the human mind. Intuition, aha-moments. Those are mental experiences, maybe to some extent subjective experiences. Another characterisation of pure reason is logic. This envisions the laws of reason as detached from human considerations, such as the mind and its subjective experiences. Instead it tries to give a purely objective account of reasoning.
Suppose we try to argue that mathematics is basically logic. So it’s not based on anything contaminated by humanness, such as the senses or the mind. Instead mathematical truths are simply necessary truths in some absolute sense. Their truth follow from absolute laws of reason that are some kind of abstract truths more fundamental than human experience or physical reality.
This point of view doesn’t really impose any evident restrictions on what kinds of things the axioms of mathematics should be. The starting points of mathematics do not need to be physically measurable, nor intuitively obvious, and so on. Logic does not imply such prescriptions, like the other views did.
Mathematicians just deduce consequences of definitions and axioms. Mathematics doesn’t care what the axioms are. From this point of view, mathematics doesn’t make any claim to establishing absolute truths. All of mathematics is just “if ... then ...” statements. If these axioms are true, then these theorems follow.
The axioms themselves, then, can be pretty much arbitrary for all the mathematician cares. This is a very modern view. Modern mathematicians pretty much accept this. It’s certainly a very convenient view for the mathematician. It’s a sort of abdication of responsibility.
What is a philosophy of mathematics supposed to do? What is it for? Surely it should explain the obvious facts about mathematical reasoning, such as that it somehow establishes seemingly absolute truths. When we read a proof such as Euclid’s proof of the Pythagorean Theorem or the parallelogram area theorem, the proof is so compelling. It gives us complete conviction that the theorem must be true. It’s unlike anything we ever see in other domains. There are no such absolutely compelling and irrefutable proofs in politics or ethics. Why not? What’s so special about mathematics?
History reinforces the point. Every last one of Euclid’s theorems are as true today as they were when they were written well over two thousand years ago. Every civilisation accepts these universal truths. Why does this happen only in mathematics?
A philosophy of mathematics should answer these questions. But the logic interpretation of mathematics does not. It doesn’t pinpoint any particular characteristic of geometrical reasoning that explains why it should be so unique in these regards. It doesn’t explain why the particular axioms of geometry that Euclid investigated were universally accepted in so many contexts, and turned out to be so uniquely suited to describe the physical world in all kinds of scientific advances that the Greeks had not even dream of yet.
So in this way the logic philosophy of mathematics is perhaps a kind of coward’s philosophy. It’s a non-philosophy, as far as many key questions are concerned. It just doesn’t have any kind of answer to the major questions that other philosophies of mathematics sees it as their duty to address.
There’s a famous essay called “The unreasonable effectiveness of mathematics in the natural sciences.” Famous physicist Eugene Wigner said this in 1960. Everybody cites it all the time.
But ask yourself: Why did no one say this until 1960? Did the effectiveness of mathematics somehow become unreasonable only then? Of course not. The effectiveness of mathematics in the natural sciences had been around forever. Including the effectiveness of ideas that were first developed for purely mathematical reasons but later proved to have hugely important and completely unforeseen scientific applications. For instance, the Greeks studies ellipses in great mathematical detail, and then two thousand years later it turned out, completely unexpectedly, that planetary orbits are ellipses. So this purely geometric topic became hugely important in science, which no one had predicted.
Why didn’t people say then: the effectiveness of mathematics is unreasonable? Why would it take all the way to 1960 before anyone drew this obvious conclusion?
I’ll tell you why. Because the conclusion that the effectiveness of mathematics is unreasonable only follows if one assumes the logic interpretation of mathematics. If mathematics is nothing but logical inferences from arbitrary axioms, then sure enough it’s a complete mystery, it’s completely unreasonable that mathematics can work so well.
But what people used to conclude from this is that it is the logic conception of mathematics that must be unreasonable. It is unreasonable to think that mathematics is nothing but logical deductions. Because that completely fails to explain so much of what we know about mathematics.
In 1960 the logic conception of mathematics had become the modern dogma that it remain to this day. It had become so ingrained in the mathematical psyche that mathematicians could no longer even conceive of rejecting it. Then they had no choice but to declare the effectiveness of mathematics in physics to be unreasonable. That’s why Wigner’s famous phrase is from 1960 and not 450 BC.
It’s not a fact that effectiveness of mathematics is unreasonable. Rather, one of two things is unreasonable: either the effectiveness of mathematics is unreasonable, or the conception of mathematics as nothing but logic is unreasonable.
For thousands of years people preferred to conclude from this that there must be more to mathematics than just logic. Euclid is not just “the axiomatic-deductive method.” This can’t be the whole picture. The axioms must be somehow more than arbitrary. What makes the axioms true? Logic itself doesn’t care and cannot help us with this question. So we need something more than logic in our philosophy of mathematics.
So I claim that only in very modern times did the logic conception become the norm. Maybe in some future episode I will discuss what circumstances made that come about. The important thing for our present purposes, as we read Euclid, is to understand that with the reduction process that we have begun, that consists of breaking down theorems into smaller and smaller pieces, the end pieces, the ultimate rock-bottom pieces, need to have some sort of claim to credibility. They cannot simply be whatever you’re left with when you keep reducing and reducing.
Or can they? I say everyone rejected that view, but I could play devil’s advocate. Listen for example to this fragment from Eudemus’ Physics: “As for the principles they talk about, mathematicians do not attempt to demonstrate them, they even claim that it is not their business to consider them, but, having reached agreement about them, they prove what follows from them.”
This is a bit of a disturbing quote, in my opinion. It seems to almost assert that logic view that I said was regarded as unacceptable at that time. Mathematicians only prove what follows from axioms, and they claim that “it is not their business” to worry about the status or truth of those axioms. Sounds strangely modern, just the view I assigned to the 20th century.
I think that’s not really what the quote says for various reasons. In part what Eudemus is saying is that the justification of the “principles” (that is to say the axioms) shouldn’t be regarded as part of mathematics but rather part of some other field, some more philosophical domain. But whatever, that’s just putting labels on things. That still means that the axioms are to be justified some way. So they are not arbitrary. The justification is “philosophy” rather than “mathematics”—sure, whatever, call it what you want, but it’s in any case very different from not justifying or being concerned with the nature of the axioms at all.
The quote also said, if you noticed, that the mathematicians don’t care about the axioms, “having reached agreement about them.” What does that entail? On what basis did mathematicians reach such an “agreement”? This opens the door for all kinds of considerations of the status and nature of the axioms within mathematics, even according to this quote, the devil’s advocate quote.
So I think it’s safe to say that the logic view by itself was not satisfactory. The starting points, or axioms, of mathematics need to have some kind of justification.
In fact, there is one way in which logic itself can provide such a justification. So the problem we need to solve is this. We started with the Pythagorean Theorem, we reduced it to more basic statements, then those to more basic ones, and so on. Where do we stop this process?
Do we stop when we just don’t see how to go any further? This is what I just criticised as untenable. Because this would mean declaring whatever we’re left with to be axioms, without convincing criteria of justification for which kind of things should be allowed to be axioms and which not. The axioms can’t just be arbitrary because then we can’t explain the successes of mathematics.
One hope of some logicians has been that everything could be reduced to definitions. There are no axioms! Everything is at bottom just definitions. The meaning of words. Mathematics is about drawing out consequences contained in the definitions of concepts, without any assumptions being made.
That would be great for the logician and some people have tried to fit geometry into such a mold. But it doesn’t work. Geometry needs assumptions, genuine axioms. You can’t get away with only definitions. You can’t reduce mathematics to a purely linguistic game. And besides, even if you could, what would be the guarantee that the definition corresponded to anything? That the entities defined actually exist? And that the definitions are not self-contradictory or inconsistent? Definitions alone cannot carry this burden of justification. You need something more.
But there’s one more ace up the logician’s sleeve, and it’s a pretty clever one. There are statements that are logically self-justifying. Statements such that, if you try to deny them, you have actually committed yourself to accepting them.
An example is the famous statement by Descartes: I think, therefore I am. How could you deny such a thing? What would you say if you wanted to deny it? “No, I don’t think that.” Or: “I think that’s wrong.” As you can hear, you walked right into the trap. By trying to deny that you are a thinking being, you made statements that actually presuppose that you are thinking being. The denial is self-defeating. You can’t deny the statement without actually implicitly conceding it.
Such statements are justified by “consequentia mirabilis,” as it’s called.
There’s an argument of this form already in an Aristotelian fragment. Aristotle uses it to prove the proposition: We ought to philosophise. Try to deny it. So you say: No, we should not philosophise. Well, in that case, it would be important to reach the conclusion that we should not philosophise. Reasoning our way to this conclusion would spare us from the mistake of philosophising. Then we could do more important things with our time instead of philosophising.
But now we are caught in a trap again. We wanted to establish that we shouldn’t philosophise, but in trying to argue this we actually committed ourselves to the position that we should philosophise, namely we should philosophise in order to establish the conclusion that we shouldn’t philosophise. So once again the attempted rejection of the proposition actually implies acceptance of the proposition.
Could it be that all the axioms of mathematics could be of this type? That would be a logician’s dream. That would be a great way of justifying ending the chain of reductions of theorems to lower and lower constituent parts. We have to keep reducing until you’re left with nothing but logically self-justifying statements. Consequentia mirabilis axioms only, which must be accepted as true because it is logically incoherent to try to deny them.
This view had its adherents. Clavius was fond of the consequentia mirabilis. Clavius was influential in discussions of Euclid around 1600; he was the editor of the standard Latin version of Euclid that everybody used. Even Saccheri, who did some very sophisticated work on the foundations of geometry in the 18th century, was keen on trying to reduce the foundations of geometry to consequentia mirabilis.
So this idea was clearly seen as very attractive. People really tried to make it work. But ultimately it failed. It was an approach based more on what the logician wanted than on what mathematics is really like and how mathematics wants to be understood.
So altogether, the reduction of mathematics to logic is an idea that has had great appeal to many. Several times in history, a complete reduction of mathematics to logic has seemed within reach, only for the quest to end in bitter disappointment. This is also what happened with Frege and Russell, Hilbert and Gödel, and so on, centuries later.
Bertrand Russell put it in interesting terms. Here’s what he says in his autobiography: “I wanted certainty in the kind of way in which people want religious faith.” He’s talking about his early career, around 1900. At this time he worked on an enormously ambitious project to reduce all of mathematics to logic. It didn’t work. As Russell himself says: “After some twenty years of very arduous toil, I came to the conclusion that there was nothing more that I could do in the way of making mathematical knowledge indubitable.”
Russell’s case is quite typical, one might argue. Others have had the same experience when they have tried to achieve the same goal. It’s a great temptation: one logic to rule them all; “my precious.” Many have been seduced by that idea, and spent twenty years obsessed with it only to fail, as Russell did.
Let’s look at the most famous of the problems Russell ran in to: the so-called Russell’s Paradox. A popularised version of Russell’s Paradox goes like this. A barber shaves everyone who do not shave themselves. Who shaves the barber? There is no coherent answer. The barber cannot shave himself because he only shaves those who do not shave themselves. But he also could not not shave himself. Because if he didn’t shave himself he would by definition be one of the people he does shave, which is everybody who do not shave themselves. So either way leads to a contradiction.
Mathematicians unknowingly allowed this type of paradox to enter their logical systems. This stuff about the barber is just a translation into everyday terms of something that first occurred within mathematics itself.
Russell thought this problem was fixable. But others thought it was a comeuppance for logic that was both deserved and bound to happen. Consider for example the reply by Brouwer, an influential but eccentric mathematician in the early 20th century. Here’s what he says: “Exactly because Russell’s logic is no more than a linguistic system, there is no reason why no contradictions would appear.”
That is to say, since logic is divorced from meaning, divorced from the real world, why wouldn’t it be inconsistent and self-contradictory? History shows that inconsistencies can very easily creep into formal axiomatic systems, against the best efforts of even top mathematicians devoted specifically to building rigorous and coherent foundations. A long list of leading logicians have published systems of logic which turned out to be inconsistent.
According to Brouwer: “The language of Euclidean geometry is reliable only because the mathematical systems and relations, which are symbolized by the words of that language as conventional signs, have been constructed beforehand independently of that language.”
That is to say, it is precisely because it is not merely logic that Euclidean geometry is so reliable. It is anchored in the real world, and the physical world has a much better track record of being consistent than the thought-constructs of logicians.
Emil Post was another rebel at that time who likewise called for “a reversal of the entire axiomatic trend of the late 19th and early 20th centuries, with a return to meaning,” as he put it.
Logic had gone too far. Some formalisation and logic are powerful tools in mathematics. But you can take it so far that mathematical theories lose all bond with reality and meaning. Then there is no grounding anymore to protect you from contradiction and inconsistency.
Logic is “the hygiene which the mathematician practices to keep his ideas healthy and strong,” said Hermann Weyl, another contemporary of these guys. But, like hygiene, you can overdo it. Some hygiene is much better than none, of course, but obsessive hygiene can undermine the natural state of the body and the immune system. Maybe logic is like that. It’s like cleaning everything away with bleach all the time. It’s good to clean, but if you overdo it you eventually clean away the very thing you were trying to protect.
There were big debates about such questions in the early 20th century; the people I quoted were all part of those heated debates about logic. But that’s a story for another day. For our purposes, we are interested specifically in logic-centric attempts at interpreting Euclid, and accounting for the success of Greek geometry. Indeed, such logic-centered interpretations have been sought eagerly. They are very agreeable for some purposes; they have an almost religious appeal, as Russell said. But ultimately there are severe limitations inherent in such views, which have meant that most people from antiquity to early modern times have felt that some additional ingredient, beyond mere logic, is needed for a successful philosophy of mathematics.
And as we read Euclid backwards, the closer we get to the beginning, the more essential it becomes for us to make up our minds about our philosophy of mathematics. Any moment now we have reached all the way down to the axioms and then push comes to shove. We’re going to have to take a stand and say: this is why we stop at these particular axioms and why you should believe them. Let’s keep reading Euclid and see how we can answer this challenge.
The Pythagorean Theorem might have been used in antiquity to build the pyramids, dig tunnels through mountains, and predict eclipse durations, it has been said. But maybe the main interest in the theorem was always more theoretical. Euclid’s proof of the Pythagorean Theorem is perhaps best thought of not as establishing the truth of the theorem but as breaking the truth of the theorem apart into its constituent parts to analyse what makes it tick. Euclid’s Elements as a whole can be read in this way, as a project of epistemological analysis.
Transcript
Let’s read Euclid together. Euclid’s Elements, one of the most important and influential works in human history, who wouldn’t want to read that? “Euclid alone has looked on beauty bare,” as the poets say.
Let’s do some episodes on this where we go through Euclid’s Elements Book I. And here’s the first twist: Let’s read it backwards. Well, not quite. But it’s a good idea to start at the end. Book I of the Elements ends with the Pythagorean Theorem and its converse. It’s not a murder mystery, it won’t spoil the fun to know the ending.
I will explain why I think this is a good idea. This has to do with appreciating the refined goals of the Elements. It’s a very subtle work, in ways that are easy to miss. So I will use this idea of starting at the end as a way of highlighting some things to keep in mind in that regard, so that we approach the text with appreciation of these subtleties.
It might be a bit dry to do only that, so I will also mix it up with some lighter things. Some stories related to the Pythagorean Theorem. Did the Egyptians use the Pythagorean Theorem to build the pyraminds, for example? Is that how they got the angles just right? We will discuss that soon. And I will also play a clip of RoboCop.
I will try to do this for the Elements as a whole: a serious discussion of its finer points, as well as some entertaining tangents exploring the many cultural links of the various parts of the Elements.
So here we go. My first goal is to outline the mindset with which we must approach Euclid’s text.
If you’re a young person, you may look at Euclid’s Elements and say: yeah yeah, triangles and stuff, I saw all of that in high school too; our textbook had proofs just like this thing by Euclid; it’s pretty much the same thing. No, no, no. That’s like listening to Mozart and saying: yeah yeah, big deal, music is music.
Forget it. There’s a world of difference. Euclid is on a whole other level of sophistication than some crappy high school textbook. You wouldn’t know it just by looking at the text though. The text looks the same as any other geometry text. Triangle ABC blah blah blah. It’s the same with musical scores, isn’t it? They all look the same when you just glance at the pages. You can’t tell Mozart from some hack.
We must look deeper to appreciate the subtlety and genius of Euclid. The text itself doesn’t spell that out, just as a Mozart quartet doesn’t have a narrator telling you what’s great about it. But great works reward reflection. The more you study Euclid, the more you interrogate the text, the more you puzzle over its oddities, the more you come to appreciate the mastery that went into crafting everything just right. Euclid knew exactly what he was doing. His work is orders of magnitude more sophisticated than other superficially similar works in the same genre.
The exercise of reading backwards is one angle we can use to start getting a handle on this. If we read Euclid from cover to cover, in the order it’s written, we get a strictly “bottom-up” perspective: we start with the most basic things and gradually get to higher and higher levels of sophistication. That’s how mathematics is typically written down. And with good reason. But the way mathematics comes into being is much more bidirectional. Mathematics grows like a tree: as the branches extend, so do the roots. Starting our Euclid adventure with the Pythagorean Theorem is a way of making us think about this.
Of course when we read Euclid’s proof of the Pythagorean Theorem we find that it is based on earlier results. So you might say: Obviously you have to read those first before you can understand this proof. But that’s a bit simplistic. You could also say: Actually you need to look at the Pythagorean Theorem first because only then can you understand what the purpose is of those earlier propositions. From a purely logical perspective you have to read it linearly from start to finish, but to understand the meaning and purpose of these logical constructions you have to take a step back and interrogate the text from other angles as well. For a dogmatic understanding, it is enough to read it linearly, and parse the logical steps like a machine. But for a critical, independent understanding you want to not only verify the logic but also see how one could arrive at such logical constructions organically.
That goes for any formal mathematics text, still to this day. Or maybe even more so today than ever. The definitions and axioms are the starting points of the way mathematics is written, but often they are almost the end product of the actual creative thought process. Only after you have figured out the hard parts of your theory do you know what the starting points need to be. Or at least there’s an interaction, a back-and-forth negotiation between the top and the bottom of the theory. Each is adapted to the other.
So that’s one reason to read Euclid backwards. It’s a reason that applies to any formal mathematical theory, because they all have this element of bidirectionality.
Actually geometry might be among the more unidirectional formal mathematical theories in how it was conceived, because the results of geometry were known in great detail, long before they were formalised. The tree came before the roots, so to speak.
Here’s another way of visualising it. Think of the Pythagorean Theorem as the apex of a pyramid. The proof reveals which lower, more foundational stones it rests on. Those stones in turn rest on other stones, and so on. Something has to be the bedrock that is considered solid enough not to need any further support beneath it. Euclid’s Elements can be read in two directions: as a way of building up a more and more elaborate structure on top of solid foundations, or as a way of reducing advanced results to their basic components. So when we read the proof of the Pythagorean Theorem, one of the perspectives we should use is to think of it as “boiling down” this somewhat advanced result to more basic ones. This will help us appreciate the purpose and achievement of the more fundamental parts of the Elements when we get to those.
Indeed, by the time Euclid wrote the Elements, the theorems themselves—such as the Pythagorean Theorem—had been known for hundreds or even thousands of years. Even proving the theorem wasn’t all that new. There were plenty of proofs. I bet Euclid knew two dozen proofs of the Pythagorean Theorem.
We shouldn’t think of Euclid as saying: Hey guys, I discovered some things about triangles and stuff; check out this book where I explain how I came up with these theorems.
No, no, no. That’s not at all what Euclid is doing. We must understand, when we read the Elements, that we’re way beyond that.
If you just wanted to convince a random person that the Pythagorean Theorem is true, then there are much better proofs than Euclid’s. Simpler ones. More intuitive, based on simple diagrams. If all you want is a psychologically compelling argument that the Pythagorean Theorem is true then there are better options than Euclid.
Euclid knew all of that, and he chose his proof very deliberately. Because it’s the best proof for his purposes. Namely the purpose of carefully analysing how the truth of the Pythagorean Theorem can be broken down into smaller truths. And more generally to do the same thing for all the truths of geometry in a comprehensive and systematic manner.
So the proof of the Pythagorean Theorem isn’t so much about showing that the theorem is true. It’s more about showing what its ultimate foundations are.
Here’s another metaphor for this. Think of a mathematical theorem as a dish that you cook. The Pythagorean Theorem is like a soup, let’s say. You can whip it up very quickly with store-bought ingredients like stock cubes or just microwaving something from a can. But Euclid doesn’t do store-bought. He’s going to do everything from scratch. And I mean really from scratch. If there’s going to be carrots in there, then Euclid is going to grow his own carrots.
In fact you might say that Euclid is not so interested in cooking at all, even though a proof is like a recipe. Euclid is like a cookbook author who doesn’t like cooking and has no interest in feeding anyone.
Instead he’s more like a chemist who is analyzing the molecular composition of foods. His recipes are not meant as a practical cooking guide but as an analysis of what the core ingredients of the dish are if you deconstruct the recipe as far as you possibly can.
Here we have the idea of reading backwards again: Euclid isn’t really interested in making Pythagorean Theorem soup, but in starting with Pythagorean Theorem soup and taking it apart in the lab. Put it on the Bunsen burner. Different ingredients have different boiling points and so on, so you can carefully separate them out again.
There was already plenty of geometry before Euclid. If theorems are food, everyone was already well fed, so to speak. Everyone already had their favourite dishes and neither they nor Euclid were looking to replace the traditional menus. What Euclid is bringing to the table is not new food but a refined theoretical perspective that stands apart from actual cooking.
The idea of reading Euclid backwards is also related to a famous anecdote recorded about Thomas Hobbes, the 17th-century philosopher. Here’s what it says about Hobbes:
“He was 40 years old before he looked on geometry; which happened accidentally. Being in a gentleman’s library, Euclid’s Elements lay open, and ‘twas the [47th Proposition of Elements Book I, the Pythagorean Theorem]. He read the proposition. By God, sayd he, this is impossible! So he reads the Demonstration of it, which referred him back to such a Proposition; which proposition he read. That referred him back to another, which he also read. [And so on], that at last he was demonstratively convinced of that trueth. This made him in love with geometry.”
It is interesting that Hobbes ended up reading Euclid backwards by accident like this. Precisely what I recommended as a deliberate strategy. But he doesn’t seem to have appreciated the point of doing so the way I have described it. Maybe he could just as well have read the book forwards and had the same experience, as far as this anecdote goes.
Hobbes fell “in love with geometry” by reading it backwards, but others had the same experience reading it forwards. Bertrand Russell, another famous philosopher, read Euclid the conventional way, starting at the beginning, and he still found it, as he later said, “as dazzling as first love”: “I had not imagined there was anything so delicious in the world.” Bertrand Russell was eleven at the time, while Hobbes was 40 when he stumbled upon Euclid. They lived almost three centuries apart. So these anecdotes speak to the universality of Euclid’s text: young or old, forwards or backwards, conservative or socialist, in a society of cars or one of horses—the one thing they have in common is the love that Euclid stirred up in them.
That’s all very nice, but it kind of misses the point in terms of what I have tried to argue was the goal of Euclid’s Elements. What Hobbes and Russell fell in love with was the idea of geometrical proof, it seems. Historically, those epiphanies are better associated with a pre-Euclidean period. We discussed Thales before, and there were plenty of others in the centuries between him and Euclid.
So when you read Euclid, by all means, do fall in love. Be seduced like so many others have been. But also keep in mind that these charms are only part of the greatness of Euclid. Euclid’s Elements can be as good a vehicle as any to have that epiphany of the beauty of mathematics. But to Euclid and many of his readers that was old news.
Euclid wanted to do more than that. He didn’t want to just show how cool it is to prove stuff, although that is lovely. More than that, he wanted to explore the very essence of geometrical knowledge. What are its preconditions, and the source of its certainty? Just as a chemist seeks to decompose any substance into the elements of the periodic table, so Euclid sought to find the “periodic table” of geometry, so to speak: he wanted to uncover the ultimate building blocks of this entire branch of knowledge.
Ok, so that’s my lesson one in how to read Euclid. Start at the back and keep in mind this theme of distillation into ultimate foundations.
So I urge you to go read Euclid that way. I’m not going to go through the proof here; you’ll have to follow along in your own copy of the Elements. I recommend my own edition, for which I added illustrations for each step of the proofs. It’s a joy to read, in my opinion. But it’s too visual to translate into this medium, so I’ll leave that to you to pursue.
Now I wanted to take this opportunity to think about the origin of the Pythagorean Theorem. Part of the appeal of reading Euclid’s Elements is how embedded it is many aspects of human culture and history. So in parallel with our reading of Euclid I wanted to bring up such themes as well.
The Pythagorean Theorem has little to do with Pythagoras. It was discovered independently in several cultures, some of them long before Pythagoras. But never mind the name. The more interesting question is: Why were people interested in this theorem? Why would anybody want to calculate a bunch of hypothenuses?
If you look in a modern geometry textbook, you won’t find any good answers. The book will give you the formula and ask you to apply it in all kinds supposedly real-world cases, but they are all fake and transparently ridiculous. How to calculate the diagonal of a field when you know the lengths of the sides: When would you ever use this? Why wouldn’t you just measure the diagonal then if that’s what you want to know?
Ladder problems is another one of those fake classics. The foot of the ladder is so-and-so far from the wall, and the ladder is so-and-so long, will it reach to such-and-such a height, maybe for instance the ladder of a fire truck to save someone from a burning building? Not a very realistic scenario. Wouldn’t you just try it and see if it worked? Wouldn’t that be just as easy as sitting around making calculations? And why would the distance from the wall to the foot of the ladder be some exact given number? And so on.
It doesn’t make sense that people discovered the Pythagorean Theorem because they were wrestling with practical problems like those. They would not have needed mathematics for that. If they wanted to solve those problems they would have used trial and error and direct measurements.
Unfortunately, ancient textbooks are as ridiculous as modern ones in this regard. Here’s an example from a Chinese text from about the time of Euclid. A 10-feet-high stem of bamboo broke in the wind. It broke into two straight prices. One part remains upright, perpendicular to the ground. But the other part, that broke off but is still attached, tipped over and is now touching the ground, 3 feet away from the base of the stem. How high up the stem did the break occur?
You can calculate this with the Pythagorean Theorem, sure enough, but of course there is no way anyone would ever do something so absurd in the real world. Just measure it, if you want to know. You apparently already measure the distance along the ground and the full height somehow, so why couldn’t you just as well measure this thing? Doesn’t make any sense.
Here’s another scenario some have claimed involves the Pythagorean Theorem. On the Greek island of Samos, there’s an ancient tunnel, which was dug in fact right in the lifetime of Pythagoras.
This tunnel is a marvelous thing, a tribute to the engineering skills of the Greeks. It’s still there today. The tunnel is over one kilometer in length through a big mountain. It was dug to supply the capital with fresh water.
Digging the tunnel was certainly a geometrical project. In fact, the walls still have letters on them, like the lettering of a geometrical diagram. Evidently there was a plan of the tunnel in the form of a drawn diagram, with points makes by letters, and then as it was dug these letters were inscribed on the wall to keep track of how the actual tunnel corresponded to the geometrical plan.
This was all the more essential since the tunnel had to be dug from both ends, in order to complete it in half the time. So the diggers had to be coordinated to ensure they met in the middle. A highly non-trivial problem, which the Greek geometers solved flawlessly.
In fact, at some point the plan even had to change because the rock was becoming to porous. So there was a risk that tunnel would collapse. Therefore it was necessary to make a bend in the tunnel that took it more toward the core of the mountain, which had harder rock. The geometers dealt with this flawlessly as well. They added a shallow isosceles triangle to the diagram. So each digging team had started out along straight lines that would have met in the middle, but halfway through both teams were instructed to make a slight turn which was specified with geometrical precision. So the whole tunnel has a kind of V-shaped bend in the middle. But it still worked. The two digging teams met just as the geometers had calculated.
That’s great stuff, but is it the Pythagorean Theorem? Let me play to you a clip from the History Channel documentary series Engineering an Empire, which claims that it is.
“Eupalinos dug tunnels from each side of the mountain, until they met in the middle. To succeed, Eupalinos had to make sure that each tunnel started at the same vertical height., on opposite sides of the mountain. The tunnels also had to match up on a horizontal plane. Otherwise, they would pass each other like ships in the night.”
“By forging a path from the spring to the city, in short perpendicular lines, Eupalinos could measure each small length in order to calculate two sides of a right triangle. With two known sides of the triangle, the hypothenuse became the path of the tunnel through the mountain.”
So according to the History Channel, the plan for the tunnel was based on the Pythagorean Theorem. The History Channel are not even taking into account the alterations of the plans midway through, by the way. They just discuss the problem of making a straight tunnel.
The presenter of this documentary is Peter Weller, who is also the actor who played RoboCop in the 1987 movie. Turns out he’s also a historian.
I must say though that I disagree with RoboCop’s analysis. The tunnel of Samos was great geometry but it wasn’t the Pythagorean Theorem. The way RoboCop puts it in the documentary, it sounds as if the point was to calculate the length of the tunnel. That’s the hypothenuse that RoboCop is talking about in that clip. But of course the real problem is the coordination of the two digging teams, so they won’t miss each other “like ships in the night,” as RoboCop himself said. How is the length of the hypothenuse supposed to be useful for this? Knowing how long the tunnel is supposed to be doesn’t help you determine the direction of digging.
So I don’t think this tunnel stuff is a great example of real-world motivation for the Pythagorean Theorem. We have to keep looking for where ancient man could have had reason to discover or apply this theorem.
Here’s another such scenario. Did the Egyptians use the Pythagorean Theorem to build the pyramids? I’ll play another clip from another documentary series that claims: yes. This is from The Story of Maths, a BBC documentary presented by Marcus du Sautoy.
“The most impressing thing about the pyramids in the mathematical brilliance that went into making them. Including the first inkling of one of the great theorems of the ancient world: Pythagoras’s Theorem. In order to get perfect right-angled corners on their buildings and pyramids, the Egyptians would have used a rope with knots tied in it. At some point, the Egyptians realised that if they took a triangle with sides marked with 3 knots, 4 knots, and 5 knots, it guaranteed them a perfect right angle.”
The theorem involved here is not the Pythagorean Theorem itself, but the converse of it, which is Proposition 48 in Euclid.
In terms of historical evidence, we really don’t know if the Egyptians did this or not. It’s plausible that they knew this but there’s very little documentary evidence from way back then.
Obviously you can’t believe anything just because Marcus du Sautoy said it in a BBC documentary. Marcus du Sautoy is not a historian, he’s just clowning around. But let’s see, if we’re serious about it, does it make any sense?
I used to be skeptical about this, but I have come to think maybe it’s not so bad. I think the standard formulation about a rope with 3+4+5 equally spaced knots on it is a bit silly. Seems very complicated to get the knots just right.
But you don’t really need one triangular rope. Instead you can just use three separate ropes, of lengths 3, 4, and 5. That’s easy to make. Then when you need to make a right triangle you stretch the 3 and 4 ropes along the intended sides, and you check if the 5 rope fits between their endpoints. Then you have the guy holding the end of the 4 rope move a bit this way or that until it lines up perfectly.
I have to admit, if I was building a pyramid I would probably go with this method. Especially because of the scale of the project. The base of the pyramid is enormous. You would use ropes with lengths 3, 4, 5, but not in feet or meters but some bigger unit. Maybe 30 meters, 40 meters, 50 meters. The ratio is all that matters of course. The longer the ropes, the less significant the measurement error becomes. So it’s a pretty good method I think.
Let me read you a quote here from the book Euclid’s Window by Leonard Mlodinow. I thought it was quite funny.
“Picture a windswept, desolate desert, the date, 2580 B.C. The architect had laid out a papyrus with the plans for your structure. His job was easy—square base, triangular faces—and, oh yeah, it has to be 480 feet high and made of solid stone blocks weighing over 2 tons each. You were charged with overseeing completion of structure. Sorry, no laser sight, no fancy surveyor’s instruments at your disposal, just some wood and rope. As many homeowners know, marking the foundation of a building or the perimeter of even a simple patio using only a carpenter’s square and measuring tape is a difficult task. In building this pyramid, just a degree off from true, and thousands of tons of rocks, thousands of person-years later, hundreds of feet in the air, the triangular faces of your pyramid miss, forming not an apex but a sloppy four-pointed spike. The Pharaohs, worshipped as gods, with armies who cut the phalluses off enemy dead just to help them keep count, were not the kind of all-powerful deities you would want to present with a crooked pyramid. Applied Egyptian geometry became a well-developed subject.”
So that’s a quite comical way of putting it, but the point is well taken, I think. Indeed it does make some sense, this whole thing. The historical and societal context, the mathematics available at that time, the need to make exact right angles, the method for doing so using strings and a Pythagorean triple: that is all quite plausible, I would say.
It’s hardly plausible that they would have discovered the Pythagorean Theorem this way, by starting with the problem of making right angles. But it is plausible that may have used knowledge of the 3-4-5 special case of the converse of the Pythagorean Theorem to make right angles.
Here’s another proposal for the possible origins of the Pythagorean Theorem. This proposal is from van der Waerden’s book Geometry and Algebra in Ancient Civilizations. He proposes that the original motivation for the discovery of the Pythagorean Theorem might have been related to eclipses. Namely, calculating the duration of a lunar eclipse.
Indeed, astronomy was important to many ancient peoples. You know the Stonehenge, Maya temples aligned with solstices and so on. People cared a lot about the sky back then.
Eclipses were a big deal. Probably they were often seen as having some kind of theological significance, some sort of omen, and so on. They were also scientifically important, for instance for exact calendar keeping.
So what do eclipses have to do with the Pythagorean Theorem? Mathematically, this is a neat example. Fun to use in a geometry class.
A lunar eclipse occurs when the moon passes through the shadow cast by the earth. The earth’s shadow is about twice the size of the moon, at that distance. So the moon is approaching this dark spot, it enters it, and keeps moving through it, and comes out at the other side. The whole thing takes maybe an hour or two, it differs.
We can predict in advance how long a particular lunar eclipse is going to last. The determining factor is whether the path of the moon goes right through the middle of the earth’s shadow, or cuts across it off center. The moon’s orbit is complicated and it’s different each time. Sometimes it’s coming in a bit high and sometimes a bit low. We can see this by comparing its position to the stars.
So this means that the problem of calculating the duration of an eclipse comes down to calculating the length of a line cutting through a circle, not necessarily through the middle. We assume that the moon’s speed is constant throughout the eclipse. So the duration of the eclipse is determined by how big of a segment of the moon’s path is in the circular shadow cast by the earth.
This indeed becomes a Pythagorean Theorem problem. You can picture it like this. Draw a circle. That’s the shadow cast by the earth. Now draw a line cutting through the circle, but not through the midpoint. That’s the path the moon is moving along. We want to know the length of the segment inside the circle. This is what determines the duration of the eclipse.
Find the midpoint of this segment. Connect it to the center of the circle. This is a known length, because it corresponds to how far off-center the moon was in its approach, which we can determine by comparing its position to the stars. So the distance from the midpoint of the segment to the center of the circle was known before the eclipse began.
Let’s add one more line to the diagram: the line from the center of the circle to the point where the moon’s path entered the circle. That’s of course a radius of the circle, which is known because the size of the earth’s shadow is known.
So now you see why it’s a Pythagorean Theorem problem. The two knowns are two sides of a right-angle triangle, and the sought length is the remaining side.
Could this be how ancient man discovered the Pythagorean Theorem? This hypothesis has one thing going for it, namely that the sought quantity cannot be measured directly in advance of the eclipse. You genuinely need the Pythagorean Theorem to do this. It’s not one of those fake ones where you could just as easily have measured the side you are looking for, instead of measuring the sides you don’t want and then calculating the you do want, as in those fake textbook problems.
Mathematically, that’s all very satisfying. Unfortunately this hypothesis is not very plausible historically. In the Babylonian tradition, mathematics came long before mathematical astronomy. Serious mathematical astronomy such as this, with detailed eclipse calculations and so on, was a preoccupation of the second flowering of ancient Babylonian mathematics. That’s about a thousand years after the first golden age of Babylonian mathematics.
Already the older period had excellent mathematics, including something like the Pythagorean Theorem. One of the most famous old Babylonian clay tablets states the ratio between the side and the diagonal of a square. So it’s essentially a numerical approximation of the square root of 2, in other words. The numerical value the tablet states is very nearly accurate to six decimal places. That’s very accurate indeed. Suppose you used it to compute the diagonal of a square field with a side of a hundred meters. So a football field, basically. Then the Babylonian approximation is off from the exact answer by less than one millimeter.
That’s more than a thousand years before Babylonian priests became obsessed with eclipses for the sake of ensuring the calendric accuracy of their rituals. So the mathematically pleasing hypothesis about the Pythagorean Theorem being discovered to calculate eclipse durations doesn’t really fit the historical record unfortunately.
So what can we conclude from all this? I think it’s safe to say that practical need was never the main driver of mathematics that goes even a bit beyond the basics. The Pythagorean Theorem was discovered because people were fascinated by mathematics for its own sake, not because they needed to calculate stuff. The Chinese didn’t need to know the breaking points of bamboos, the Babylonians didn’t need to know the diagonal of a football field with millimeter accuracy. They were fascinated by the power of mathematical reasoning to discover hidden relationships, and that’s why they explored these things.
This is also how we should read Euclid. The proof of the Pythagorean Theorem is not so much about proving that the theorem is true. It’s more about exploring the basis for this knowledge. Mathematics was always explored for this reason.
Discovering mathematics was like discovering magic. It impresses us as a powerful force that can do incredible things. We want to understand it: How is this possible? What makes this magic tick? It is so unlike anything else we are familiar with, it’s like a portal to a divine realm. We feel a spiritual imperative to understand it.
Already ancient civilisations started along this path, and Euclid does the same. If mathematics is magic, Euclid’s Elements is not a book of spells, but a scientific investigation of how there can be such a thing as magic at all.
Or to use another metaphor, we have to dissect mathematics like an alien corpse to discover the secrets of it mysterious inner workings. The Pythagorean Theorem is the alien: a weird thing that seems to have superhuman powers. Euclid’s proof is not a recipe to give you alien abilities; rather, it is the result of his through dissection of an alien he found in the wild.
So let’s read Euclid this way, as an exploration into the inner mechanisms—the heartbeat—of these strange entities, these superhuman theorems, that have impressed mankind with their seemingly magical and divine aura for many thousands of years.
Greek geometry is written in a style adapted to oral teaching. Mathematicians memorised theorems the way bards memorised poems. Several oddities about how Euclid’s Elements is written can be explained this way.
Transcript
Greek geometry is oral geometry. Mathematicians memorised theorems the way bards memorised poems. Euclid’s Elements was almost like a song book or the script of a play: it was something the connoisseur was meant to memorise and internalise word for word. Actually we can see this most clearly in purely technical texts, believe it or not. It is the mathematical details of Euclid's proofs that testify to this cultural practice. That sounds almost paradoxical, but I’m sure I will convince you.
The surviving documentation about ancient Greek geometry consists almost entirely of formal treatises. Very stilted and dry texts. Definition, theorem, proof. Pedantically written. Highly standardised, formalised. Completely void of any kind of personality. Where is the flesh and blood, the hopes and dreams, the lived experience of the ancient geometer? It’s as if they were determined to erase any traces of all of those things, and leave only a logical skeleton.
But it’s not as hopeless as it seems. At first glance it looks as if these texts have been scrubbed of all humanity. But, in fact, if we read between the lines we can extract quite a bit of information. There are implicit clues in these texts that reveal more than the authors intended.
That’s our topic for today: How these seemingly purely logical texts actually say quite a lot about the social context in which they were produced.
One thing we learn this way is that we should think of the Greek geometrical tradition as spoken geometry, not written geometry. Today we think of written texts as the primary manifestation of mathematics. When mathematicians disseminate their ideas, the published article is the official, definitive, primary expression of those ideas. The mathematician crafts a written document with the expectation that reading the text on paper is going to be the primary way in which people will access this material.
Not so in antiquity. Oral transmission was considered the primary mode of explaining mathematics. Written documents were a last resort when personal contact was not possible. And the written document was not meant to be a primary exposition in its own right. Writing was merely the oral explanation put down on paper (or papyrus, rather).
At least it must have been like that in the early days. Many conventions of Greek mathematical writing only make sense from this point of view. They must have been formed in an oral mathematical culture. Probably in later antiquity the situation was not so clear cut. Writing probably gradually became more of a thing in its own right, rather than merely a record of oral exposition. But even then, the conventions of written mathematics remained largely fixed. Greek mathematics never liberated itself from these conventions that had been set in an oral culture. They lived on. Perhaps in part due to tradition and conservatism, but probably also because the oral element remained a significant part of mathematical culture, perhaps especially in teaching.
Here’s an example of this, which I have taken from Reviel Netz’s book The Shaping of Deduction in Greek Mathematics. Consider the equation A+B=C+D. Here’s how the Greeks expressed this in writing: THEAANDTHEBTAKENTOGETHERAREEQUALTOTHECANDTHED. This is written as one single string of all-caps letters. No punctuation, no spacing, no indication of where one word stops and the next one begins.
A Greek text is basically a tape recording. It records the sounds being spoken. There is a letter of the alphabet for each sound one makes when speaking. The scribe just stenographically puts them down one after the other. From this point of view there is no distinction between upper or lower case letters: a letter just stands for a sound and that’s it. And there is no punctuation or separation of words, because those are not spoken sounds. And of course no mathematical symbols such as plus or equal signs, because that also does not exist in spoken discourse.
The only way to understand a text like that is to read it out loud. You have to read it like a child who is just learning to read: you sound it out letter by letter, and then interpret the sounds, rather than interpret the writing directly.
So the Greeks had a very limited conception of writing. They thought of writing only as a way of recording speech. They completely missed the opportunities that writing provides when embraced as a primary medium in its own right. Writing is a better way of representing equations, for example, than speech. But the Greeks completely missed that opportunity because they were stuck with the limited notion of writing as merely recorded sounds.
I like to compare this with early movies. Think of those classic movies from, say, the 1950s or so. They are basically recorded stage plays. There are limitations inherent in the medium of theatre. The actors have to speak quite loudly, articulately, to be heard by the audience in the back of the theatre. And the scenery on stage cannot easily be changed or moved. In a play you better stick to one or two sets, such as the interior of a room. That you can set up carefully with furniture and all kind of stuff on the walls and so on. But because you can’t change it easily, you have to have to have large parts of the play take place in that single setting.
These technical limitations constrain the artistic freedom of the playwright. You have to come up with a story where all the various characters have some reason or other to come and go into a single room, and once there to have loud conversations that drive the plot. All emotional depth and so on must be conveyed in this particular form.
These things became second nature to writers. So when film came around they kept doing the same thing even though that was no longer necessary. Many treated film as simply a way of recording plays. So in early movies you still have a lot of these static scenes with a fixed camera at one end of a room, and characters coming and going, having loud conversations.
Film affords new artistic possibilities. You are no longer limited to a static camera showing a fixed set, the way the audience of a theatre would be looking through the “fourth wall” of a room. You have many more options to convey things visually, instead of being limited to strongly articulated stage dialogs as the only driver of the plot.
But many early movies didn’t take advantage of that. They just kept doing what they had always been doing at the theatre and just recorded that. They saw the new medium of film merely as a way of “bottling” existing practice. It’s just a storage medium. They didn’t consider that the new medium was in some ways better than the old one and enabled you to do completely new things.
It was the same with writing in antiquity. Writing was merely for storing speech. They failed to take advantage of the ways in which writing could not only preserve existing cognitive practice but in fact transform it and improve it. Such as working with equations symbolically.
Here is another consequence of this: the absence of cross-referencing. If a mathematical text is like a tape recording, you can’t easily access a particular place in the tape. The only way to make sense of the text is to “hit play,” so to speak, and translate it back into sounds. Only then can it be understood. You can “fast forward” and “rewind”—that is to say, start reading at any point in the manuscript. But you can’t turn to a particular place, such as Theorem 8.
Modern editions of Euclid’s Elements are full of cross-references. Each step of a proof is justified by a parenthetical reference to a previous theorem or definition or postulate. But that’s inserted by later editors.
There is no such thing in the original text. Because it’s a tape recording of a spoken explanation. Referring back to “Theorem 8” is only useful if the audience has a written document in front of them. If they are merely listening to a long lecture, or a tape recording of a lecture, then there is no use referring back to “Theorem 8”, because the audience has no way of going back specifically to that particular place in the exposition.
For this reason, oral mathematics involves committing a lot of material to memory. In the arts, people memorise poems and song lyrics. Actors memorise the dialogues of plays. Ancient mathematics was like that as well. You would learn to recite theorems the same way you learn to sing along to your favourite song.
This aspect of the oral culture thoroughly shaped the way ancient mathematical texts are written. Euclid’s Elements and many other texts follow a certain stylistic template that at first sight seems quite irrational, but which starts to make sense once we consider the oral context.
Consider for example Proposition 4 of Euclid’s Elements. This is the side-angle-side triangle congruence theorem. It’s completely typical, I’m just picking a theorem at random. Let’s look at the text of this proposition. First we have the statement of the theorem in purely verbal terms. It goes like this:
“If two triangles have two sides equal to two sides, respectively, and have the angle enclosed by the equal straight lines equal, then they will also have the base equal to the base, and the triangle will be equal to the triangle, and the remaining angles subtended by the equal sides will be equal to the corresponding remaining angles.”
Ok, so: two triangles have side-angle-side equal, the it follows that they also have all the other things equal. Namely the remaining side, the remaining angles, and the area. “The triangle will be equal to the triangle,” says Euclid: this is his way of saying that they have equal area.
After Euclid has stated this, he goes on to re-state the same thing, but now in terms the diagram. “Let ABC and DEF be two triangles having the two sides AB and AC equal to the two sides DE and DF, respectively. AB to DE, and AC to DF. And the angle BAC equal to the angle EDF. I say that the base BC is also equal to the base EF, and triangle ABC will be equal to triangle DEF, and the remaining angles subtended by the equal sides will be equal to the corresponding remaining angles. ABC to DEF, and ACB to DFE.”
This is exactly the same thing that he just said in words. But now he’s saying it with reference to the diagram. He always does this. He always has these two version of every proposition: the purely verbal one, and the one full of letters referring to the diagram.
For simple propositions you can understand the value of both formulations. But quite soon, when the material gets more technical, it often happens that the verbal version becomes so abstract that it’s quite impossible to follow. This happens quite soon already in Euclid. Ken Saito has a recent paper on this, “traces of oral teaching in Euclid’s Elements.” He takes as an example Proposition 37 from Book 3 of the Elements. I’ll read it to you just to convince you how convoluted and unnatural it is to state theorems in this purely verbal form. Here it is, Euclid’s statement of this proposition:
“If a point be taken outside a circle and from the point two straight lines fall on the circle, and if one of them cut the circle, and the other fall on it, and if further the rectangle contained by the whole of the straight line which cuts the circle and the straight line intercepted on it outside between the point and the convex circumference be equal to the square on the straight line which falls on the circle, the straight line which falls on it will touch the circle.”
That’s very difficult to follow. Of course, as always, Euclid immediately goes on to state the same thing, but in terms of the diagram. That part is much easier to follow, and it turns out to be a pretty straightforward claim. The theorem is a kind of formula for the length of a tangent; how far it is to the point of tangency from a given point outside the circle. But you would hardly know that by reading the verbal statement only.
For some reason the Greeks insisted that the verbal formulation should be one single, rambling sentence. No matter how complicated your theorem is, you have to cram all the conditions and all the consequences, everything you want to say, into one single sentence.
This is taken to absurd lengths in Apollonius for example. Let me read to you an example from the Conics of Apollonius. This is Proposition 15: one of the earliest. It only gets worse from there, but this is bad enough, I’m sure you will agree when I read it to you. The proposition is a kind of change-of-variables theorem for ellipses: it tells you the equation for an ellipse in a new coordinate system conjugate to the first. So it has to specify what the equation of the ellipse was in the first coordinate system and what the assumptions for that was, then how the change of coordinates is defined, and then what the equation of the ellipse is in the new coordinate system. And it has to do all of that purely verbally, and in one single sentence, one big “if ... then ...” statement. So you get this crazy monstrosity of a sentence, it goes like this:
“If in an ellipse a straight line, drawn ordinatewise from the midpoint of the diameter, is produced both ways to the section, and if it is contrived that as the produced straight line is to the diameter so is the diameter to some straight line, then any straight line which is drawn parallel to the diameter from the section to the produced straight line will equal in square the area which is applied to this third proportional and which has as breadth the produced straight line from the section to where the straight line drawn parallel to the diameter cuts it off, but such that this area is deficient by a figure similar to the rectangle contained by the produced straight line to which the straight lines are drawn and by the parameter.”
What’s going on with this crazy stuff? Were the Greeks some kind of aliens with brains that could understand that type of thing? No. When encountering a theorem like this, they surely did not try to parse a sentence like that in the abstract. Instead they would turn to the diagram explication for help. Just as Euclid always does, so also Apollonius always goes on to restate the theorem in terms of labelled point in a diagram. And this explanation is not one big crazy sentence, but nicely broken into small steps. Much easier to follow.
At a certain point you may ask yourself: Why even include the purely verbal formulation at all? It’s so abstract, so difficult to follow. Surely any reader or listener will be lost before you have even gotten halfway through a sentence like that. And since you’re going to restate the theorem immediately anyway, why bother? You might as well only do the diagram version of the theorem. That’s the one you are going to use for the proof anyway.
That’s something of a puzzle in itself, but here’s the real kicker though. Not only does Euclid insist on including the abstruse verbal formulation of every theorem, he actually includes it twice! This is because, at the end of the proof, his last sentence is always “therefore ...” and then he literally repeats the entire verbal statement of the theorem. It is literally the exact same statement, word for word, repeated verbatim. You say the exact same thing when you state the proposition and then again when you conclude the proof. Copy-paste. The exact same text just a few paragraphs apart.
Astonishing. What a waste of papyrus and scribal effort. This was an enormous cost back then. There were no printing presses. You had to copy all of this by hand. Writing materials were expensive, copying was expensive, preservation was expensive. They had every incentive to cut and keep things minimal, yet they included this massive redundancy of repeating the rambling verbal statement of every proposition twice in short succession.
You may recall that an important treatise by Archimedes was scrubbed off its parchment because the parchment itself was so valuable even when recycled. And medieval scribes were big on minimising writing. Think of “etc.”, “e.g.”, “i.e.”: we still use those shortened versions of Latin expressions. They were invented back when people were writing and copying manuscripts by hand. Very understandable.
Yet despite all of that, for some strange reason, including the entire verbal statement of the proposition twice was somehow found valuable enough to warrant the enormous cost.
In the case of the side-angle-side theorem for example, the verbal statement of the theorem takes up about 15% of the total text of the proposition and proof. And then another 15% for the redundant recapitulation. So that’s 30% of the total text that could simply be cut. The remaining 70% of the text would still contain the full statement of the theorem in its diagram form, and the complete proof.
You’d think the temptation would be great to cut at least those last 15% of pure recapitulation. Even the standard English edition of the Elements by Heath simply writes “therefore etc.” at the end of the proofs, instead of repeating the full statement like the original did.
So what was the value of this very expensive business of repeating the statement of the proposition? The oral tradition explains it. The verbal statement of the proposition is like the chorus of a song. It’s the key part, the key message, the most important part to memorise. It is repeated for the same reason the chorus of a song is repeated. It’s the sing-along part.
In a written culture you can refer back to propositions and expect the reader to have the text in front of them. Not so in an oral culture. You need to evoke the memory of the proposition to an audience who do not have a text in front of them but who have learned the propositions by heart, word by word, exactly as it was stated, the way you memorise a poem or song.
This is why, anytime Euclid uses a particular theorem at a particular point in a proof, he doesn’t says “this follows by Theorem 8” or anything like that. He doesn’t refer to earlier theorems by number or name. Instead he evokes the earlier theorem by mimicking its exact wording. Just as you just have to hear a few words of your favourite chorus and you can immediately fill in the rest. So also the reader, or listener, of a Euclidean proof would immediately recognise certain phrasings as corresponding word for word to particular earlier propositions. They would have memorised the earlier propositions not only in terms of content but in terms of the exact verbal phrasing, almost melodically, rhythmically. Just hearing the first few words of such a formula repeated would trigger the full memory to flow out naturally and unstoppably, like singing along to the chorus of a song you love.
You can see an example of this already in Euclid’s Proposition 5. We already discussed his Proposition 4, the side-angle-side triangle congruence theorem. Euclid applies this result twice in the course of the proof of Proposition 5. However, he really only needs part of the theorem. Remember that Proposition 4 concluded several things: that the remaining sides were equal, that the remaining angles were equal, and that the areas of the two triangles are equal. Areas are completely irrelevant to Proposition 5, which is a statement purely about angles. Yet each time Euclid applies the side-angle-side theorem he spells out the full conclusion. Including the needless remark that the areas are equal.
In one case it is even irrelevant that the remain sides are equal as well, but Euclid still needlessly remarks on this pointless information in the course of the proof of Proposition 5 even though it has no logical bearing on the proof. Go look up Euclid’s proof if you want to see this nonsense for yourself. Ask yourself why Euclid points out that “the base BC is common” to both triangles the second time he applies the side-angle-side theorem in the proof of Proposition 5. It’s completely redundant and worthless. He could have just omitted that remark, and it wouldn’t have affected the logic of the proof at all.
But from the oral point of view it makes sense. Applying a theorem is a kind of package deal. You get the whole thing whether you need it or not. Once you’ve triggered the memory of the previous theorem with the appropriate key phrases, then the whole conclusion comes blurting out. Once you’ve committed to singing the chorus there’s no going back. You can’t sing only the part of the chorus you need. The whole thing goes together. You have memorised it in one flow. Once you hit play on that memory you automatically run through the whole thing.
This is why Euclid is needlessly talking about areas in the proof of Proposition 5, even though that serves no logical function whatsoever. He is mimicking word for word the phrasing of the previous proposition, filling in the specifics of the case at hand as he goes along. You sing the “chorus” of the side-angle-side theorem and you “fill in the blanks” as it were. The purely verbal statement of the side-angle-side theorem spoke of sides and angles and so on in the abstract. To apply the theorem is to repeat that exact same phrasing, but inserting AB, BCF, and so on, into that formula to specify what the sides and angles are in the particular case at hand.
It’s like singing “happy birthday”: it has a fill-in-the-blank part. Just as you would go: “Happy birthday dear Euclid”, so also you would go: “If the side AB equals CD, then the angle is ...” and so on, something like that.
Here’s maybe another consequence of this: Euclid’s odd formulation of the side-side-side triangle congruence theorem. This is Euclid’s Proposition 8. As we saw, in the side-angle-side case, Euclid drew all the possible conclusion: about sides, about angles, about area. So the theorem became a mouthful, and led to the introduction of superfluous remarks any time the theorem is applied, because you have to repeat all the conclusions whether you need them or not.
To avoid this problem it might be tempting to state theorems in less general form. And this is exactly what Euclid does with the side-side-side theorem. He introduces an asymmetry in the statement of the theorem. Instead of three sides, he speaks of two sides and a base. And his statement of the conclusion is that one particular angle (the angle between the two “sides”) is equal in both triangles. Of course it is completely arbitrary which side you designate as the “base.” And of course you could just as well have concluded that the other angles too correspond to each other in these congruent triangles. Yet Euclid choses to arbitrarily limit the generality of his theorem, and introduce arbitrary specificity and asymmetry. You’d think that would be anathema to a mathematician.
But if we think of the downsides of the way he formulated the side-angle-side case, we can understand why he went with this non-general formulation in the side-side-side case. Any time you are going to apply the side-side-side theorem, you probably want to conclude something about a specific angle, not all three angles of a triangle. So if you formulated the theorem generally, then every time you applied it you couldn’t stop yourself of course from reciting the entire chorus and hence you would end up with one conclusion that you actually needed, about one angle, and then needless spelling out two other conclusions about the other two angles that you don’t want at all. So this way you will only clutter your proofs with needless and irrelevant remarks. So the strangely specific, non-general formulation of the side-side-side theorem is actually well chosen given this constraint that you have to repeat the full theorem verbatim any time you apply it.
It’s pretty fascinating, I think, how textual aspects that appear to be purely technical and mathematical, such as a few barely noticeable superfluous bits of information in the proof of Proposition 5, can open a window like this into an entire cultural practice. The oral tradition must have been there, and the best proof of this is hiding in the ABCDs of Euclid’s formal text. It’s the beauty of history that historical texts can be read on so many levels. They carry so much hidden information about the culture that produced them. You would think Euclid’s ultra-formalised proofs would be the last place to find such clues, but here they are. We’re just a few proposition into the Elements and from the smallest technical quirks we have already recreated a rich picture of the ancient singing geometers and the strange culture in which they worked.
Proof-oriented geometry began with Thales. The theorems attributed to him encapsulate two modes of doing mathematics, suggesting that the idea of proof could have come from either of two sources: attention to patterns and relations that emerge from explorative construction and play, or the realisation that “obvious” things can be demonstrated using formal definitions and proof by contradiction.
Transcript
How did proofs begin? It’s like a chicken-or-the-egg conundrum. Why would anyone sit down and say to themselves “I’m gonna prove some theorems today” when nobody had ever done such a thing before? How could that idea enter someone’s mind out of the blue like that?
In fact, we kind of know the answer. The Greek tradition tells us who had this lightbulb moment: Thales. Around the year -600 or so. Hundreds of years before we have any direct historical sources for Greek geometry. But we still sort of know what Thales proved, more or less. Later sources tell us about Thales. History is perhaps mixed with legend in those kinds of accounts, but key aspects are likely to be quite reliable. More fact than fiction. Let’s analyse that question, the credibility question, in a bit more depth later, but first let’s take the stories at face value and see how we can relive the creation of deductive geometry as it is conveyed in these Greek histories.
So, here we go: What was the first theorem ever proved? What was the spark that started the wildfire of axiomatic-deductive mathematics? The best guess, based on historical evidence, goes like this. That love-at-first-sight moment, that theorem that opened our eyes to the power of mathematical proof, was: That a diameter cuts a circle in half.
Pretty disappointing, isn’t it? What a lame theorem. It’s barely even a theorem at all. How can you fall in love with geometry by proving something so trivial and obvious?
But don’t despair. It is nice, actually. It’s not about the theorem, it’s about the proof.
Here’s how you prove it. Suppose not. This is going to be a proof by contradiction. Suppose the diameter does not divide the circle into two equal halves. Very well, so we have a line going through the midpoint of a circle, and it’s cut into two pieces. And we suppose that those two pieces are not the same. Take one of the pieces and flip it onto the other. Like you fold an omelet or a crepe. The pieces were not equal, we assumed, so when you flip one on top of the other they don’t match up. So there must be some place where one of the two pieces is sticking out beyond the other. Now, draw a radius in that direction, from the midpoint of the circle to the place on the perimeter where the two halves don’t match up. Then one radius is longer than the other. But this means that the thing wasn’t a circle to start with. A circle is a figure that’s equally far away from the midpoint in all directions. That’s what being a circle means.
So we have proved that two things are incompatible with one another: You can’t be both a circle, and have mis-matched halves. Because if you have mis-matched halves you also have “unequal radii” and that means you’re not a circle.
So a circle must have equal halves. Bam. Theorem. It’s a boring result but a gorgeous proof. Or a suggestive proof. It’s a proof that hints at a new world.
Thales must have felt like a wizard who just discovered he had superpowers. “Woah, you can do that?!” By pure reasoning, by drawing out consequences of a definition, one can prove beyond any shadow of a doubt that certain statements could not possibly be wrong? That’s a thing? That’s something one can do? Wow. Let’s do that to everything! Right?
So that’s how Thales discovered proof. As best as we can guess.
A few other theorems are attributed to Thales as well. I want to bring up one in particular that I think is also a kind of archetype of what mathematics is all about.
The theorem we just saw, about the diameter bisecting the circle, perfectly embodies one prototypical mode of mathematical reasoning. The pure mathematics paradigm, you might call it. Logical consequences of definitions, proofs by contradiction. That kind of thing. Thales’s proof really hits the nail on the head with that whole aesthetic. We’ve been doing the same thing over and over ever since. A modern course in, say, group theory, for example, is just Thales’s proof idea applied five hundred times over, basically.
Now I want to take another one of the results attributed to Thales, and I want to argue that it is emblematic of another mode of mathematical thought. It’s a second road to proof. This second way is based more on play, exploration, discovery, rather than logic and definitions.
The example I want to use to make this point is what is indeed often called simply “Thales’s Theorem.” Which states that any triangle raised on the diameter of a circle has a right angle. So, in other words, picture a circle. Cut it in half with a diameter. Now raise a triangle, using this diameter as one of its sides, and the third vertex of the triangle is on the circle somewhere. So it looks like a kind of tent, sticking up from the diameter. And it could be an asymmetrical tent that is pointed more to one side or the other. No matter how you pitch this tent, as long as the tip of it is any point on the circle, then the angle between the two walls of the tent at that point, at the tip, is going to be a right angle, 90 degrees. That’s Thales’s Theorem.
How might Thales have proved this theorem? We don’t really know that based on historical evidence unfortunately. But let’s consider one hypothesis that makes sense contextually.
We must imagine that Thales would have stumbled upon the proof somehow. We are not trying to explain how someone might think of a proof of this theorem per se. That’s the wrong perspective because it takes for granted that in mathematics one tries to prove things. What we need to explain is where this vision to prove everything in geometry came from in the first place. How could someone have struck upon Thales’s Theorem unintentionally, as it were, and through that accident become aware of the idea of deductive geometry?
Indeed Thales’s Theorem is not terribly interesting or important in itself. If you had this vision of subjecting all of geometry to systematic proofs, why would you start with this theorem, or make this theorem such a center piece, as Thales supposedly did? You wouldn’t.
The interesting thing about Thales’s Theorem is not that is was one of the first results to which mathematicians applied deductive proof. Rather, the interesting thing about it is that it was the occasion for mathematicians to stumble upon the very idea of proof itself, unintentionally.
There’s a story about Thales falling into a well because he got so caught up in astronomical reasoning that he forgot his surroundings. It’s recorded in Plato: “While he was studying the stars and looking upwards, he fell into a pit. Because he was so eager to know the things in the sky, he could not see what was before him at his very feet.”
A legend maybe, but the discovery of Thales’s Theorem must have been a little bit like that too. Discovering mathematical proof must have been like falling into a pit. You are looking in one direction, and boom, suddenly you find yourself having accidentally smashed face first into this completely unrelated new thing that you didn’t know existed.
How could Thales’s Theorem be like that? Among all the world’s theorems, what makes Thales’s Theorem particularly conducive to this kind of fortuitous discovery of proof?
Here’s my hypothesis. In this age of innocence, before anyone knew anything about proof, people still liked shapes. The had ruler and compass. They used these tools for measuring fields and whatnot, but they also liked the aesthetic of it.
They were playing around with ruler and compass. Playing with shapes. After five minutes of playing with a compass you discover how to draw a regular hexagon. Remember? You probably did this as a kid. Draw a circle, and then, without changing the compass opening, run the compass along the circumference. It fits exactly six times. A very pleasing shape.
We know for a fact that people did this before Thales. There are hexagonal tiling patterns in Mesopotamian mosaics from as early as about -700.
Dodecahedra are another one of those things. The dodecahedron is like those twelve-side dice that you use in Dungeons and Dragons and stuff like that. Do-deca-hedron, it’s literally: two-ten-sided. So twelve-sided, in other words. Twelve faces, each of which is a regular pentagon. These things are in the archeological record. People made them of stone and bronze. A couple of dozen of dodecahedra from antiquity have been found, the oldest ones even predating Thales. They were used perhaps for oracular purposes, like tarot cards or something. Or maybe for board games, who knows?
In any case, my point is that people were interested in geometrical designs for various purposes: artistic, cultural, and so on. Not just measuring fields for tax purposes. And they were clearly working with instruments such as ruler and compass to make these things.
It’s easy to arrive at Thales’s Theorem by just playing around with ruler and compass, trying to draw pretty things. Start with a rectangle. Draw its diagonals. Put the needle of a compass where they cross, right in the midpoint of the rectangle. Set the pen of the compass to one of the corners of the rectangle. Now spin it. You get a circle that fits perfectly, snugly, around the rectangle.
But look what emerged. A diagonal of the rectangle becomes a diameter of the circle. And the rectangle pieces sticking out from it are precisely those kind of “tent” triangles that Thales’s Theorem is talking about. This suddenly makes the theorem obvious.
Why is Thales’s Theorem true? Why does any of those “tents” raised on the diameter of a circle have a right angle? It’s because it comes from a rectangle. Any such tent is half a rectangle. This is a powerful shift of perspective. By looking at the triangle this way we reveal hidden relationships, a hidden order in the nature of things. Certain angles must always be right angles by a sort of metaphysical necessity, as it were. Our eyes have been opened, maybe for the first time, to the existence of these kinds of necessities, these kinds of hidden relationships that are out there for the thinking person to uncover.
So the key is this shift of perspective that the triangle is “really” half a rectangle. Suppose instead that we had been stuck in the point of view is that we are staring at a triangle inscribed in a circle. Then the kinds of associations and ideas that suggest themselves to us are not so useful for proving this theorem. From that point of view, if you were looking for a proof, what would you do? Maybe you would for example connect the midpoint of the circle to the tip of the triangle. So now you have two smaller triangles. What are you going to do with those? Something with angle sums and so on? Or maybe you would be tempted to drop the perpendicular instead from the tip of the triangle, and then you can use the Pythagorean Theorem of the two small triangles you get.
These kinds of things are not what we want. Those kinds of approaches quickly become too technical. This was supposed to be the beginnings of geometry, remember. You are not supposed to use a bunch previous results for the proof. It should be a proof from first principles. A proof before all other proofs.
The idea that the triangle is “really” half a rectangle is different. It transforms how we look at the diagram. It changes the emphasis. It changes what we think of as primary. Now the rectangle comes first, and the triangle second, and the circle last. The theorem actually isn’t so much about circles at all, so to speak, from this point of view. The circle is just a kind of secondary artefact.
With this proof we are like artists. We take a step back from the canvas and tilt our heads and have this epiphany. And the epiphany was made possible by the way we had played with these ideas previously. We were just playing around with ruler and compass, we explored triangles and rectangles and circles with an open-minded affection. Epiphanies like Thales’s Theorem emerge from this play. Inspiration comes naturally in that context.
Unlike those other boring proofs I alluded to, that were based on cutting the triangle up and throwing the book at it: angle sums, Pythagorean Theorem, everything we can think of. That’s an uninspired approach, a brute force approach. It lacks that aesthetic inspiration, that epiphany of revealing the true nature of the triangle, and its other half that it was destined to be reunited with.
Geometry could not have started with these kinds of by-the-book proofs, because they only make sense after there is a geometry book to begin with. But geometry could have started with the epiphany type of proof. So that’s a way in which someone like Thales might have arrived at the idea of proof through playing around with ruler and compass.
Perhaps you are familiar with “Lockhart’s Lament”: a great essay on what is wrong with mathematics education. Go read it, it’s available online. It is interesting that Lockhart uses this very example to make his point. He describes how his students discovered Thales’s Theorem basically the way I’m saying that Thales might have done so. He also eloquently captures how this is so much more satisfying than a dry by-the-book proof.
It’s not for nothing that history and education go together on this point. Proof must have started with a compelling aesthetic experience or wow moment. There was no other way at the time. There was no one to force Thales to memorise facts for an exam. Discovery compelled him to value mathematics. If we want to foster intrinsic motivation in our students, it’s a good idea to consider what made people fall in love with these ideas in the first place. First love is always the purest and most innocent. Modern textbooks are like arranged marriages forced upon the students. But history always has the true love story.
Nevertheless, for all this, you might still think that Thales’s Theorem is a bit boring. Something something is always a right angle. So what? Who cares?
As I tried to argue, it was probably not the theorem per se that was impressive to Thales and his contemporaries, but rather the idea that there is such a thing as theorems and proofs at all. There are hidden truths out there that can be uncovered through reasoning. Remarkable.
But in fact even the theorem itself is quite interesting. Let me show you something cool you can do with Thales’s Theorem.
There’s an ancient legend about Queen Dido. Daughter of the king of Tyre, a major city in antiquity. You can still see the ruins of this ancient city in present-day Lebanon. At a certain point Dido had to flee, because of court intrigues. Murders and betrayals and so on. So she grabs a couple of diadems off her nightstand, maybe a chest of gold she put aside for a rainy day, and hastily sails off into the night. With hardly a friend left in the world.
She has to go all the way to present-day Tunisia, thousands of kilometers away, and try to start over somehow, in a manner befitting a royal. Using her treasure chest, she strikes a bargain to buy some land. As much land as she can enclose with the skin of an ox, the story goes. So she cuts the ox hide into thin strips and ties them together, and now what? So now she has this long string, which she can use as a kind of fence to seal off the land she wants.
But what shape to make it? A square, a rectangle, a triangle? No. Dido knows better. Perhaps her royal education included mathematics. Make it round. That’s the best way. The circle has the maximal area among all figures with a given perimeter. Or in this case, since she was by the ocean: a semi-circle, with the shoreline as a natural boundary on the other side.
Let’s prove this. That the semi-circle is the best choice. I’m going to prove this by contradiction: Suppose somebody has fenced in an area that is not semi-circular; then I can show how to make it better: how to move the fence so that the area becomes even bigger, without adding any more fence.
Ok, so you have the shoreline, that’s a straight line. And from one point on the shore, going inland you have this fence which then comes back down and meets the shore again in some other point. So together with the shoreline it closes off a certain area.
Suppose this shape is not a semi-circle. If it was a semi-circle, Thales’s Theorem would apply. And it would tell you that this angle, what I called the tent angle, at any point along the fence would be a right angle. So if the shape is not a semi-circle, there must be some point along the fence where this angle is not a right angle.
I say that making this angle a right angle improves the amount of area covered. You can picture it like this. So you have this shape enclosed by the fence: imagine that you have that cut out of cardboard. And on the perimeter you have some point marked where the tent angle is not a right angel. So on your cardboard you have that triangle drawn: a triangle consisting of the straight shoreline on one side, and the two lines from its endpoints going up to meet at the tent point on the perimeter.
Let’s cut that triangle out of the cardboard. So you’re left with two pieces: whatever bits that were sticking out from the triangle sides. Now move those two pieces so that you make the tent angle a right angle. This means moving the endpoints along the shoreline. As you move the two points on the shoreline, you change the angle at which the two cardboard pieces meet. The two cardboard pieces meet in a single point, the tent point, and that’s like a hinge that can open or close to a bigger or smaller angle. So you slide these things around until that hinge angle becomes 90 degrees.
Note that you didn’t change the perimeter this way. You just moved the same amount of fence around.
But you did increase the area enclosed, in fact. Because if you have two sticks of fixed length, and you want to make the biggest triangle you can with those sticks, the best way is to make the angle between them a right angle. That’s quite clear intuitively. You know that the area of a triangle is base times height over two. So if one of your sticks is the base, then to maximise the area you want to maximise the height, that is to say the perpendicular height going up from the base, which is obviously done by pointing the other stick straight up at right angles.
So what this proves is that, for any fence enclosure that is not a semi-circle, you can make a better one. You can move the fence around and make the area bigger. So the semi-circle is the best solution, and all other ones are less good.
I don’t know if you could visualise all of that. But maybe try reconstructing this argument for yourself later. It really is very intuitive and beautiful.
So what’s the moral of the story then? Mathematically, it is an answer to the “so what?” question regarding Thales’s Theorem. It may have seemed like a boring enough theorem, but here we see it in action in a beautiful and unexpected way, as a key ingredient in this proof about how to enclose land. Who would have seen that coming?
This suggests that mathematics has a kind of snowballing or self-fertilising aspect to it. Thales’s Theorem, what’s the big deal? Just some boring observation about a triangle in a circle. May not seem like much. But one thing leads to another. Once Thales’s Theorem is a thing to you, you start seeing it in other places, unexpected places. Like this problem about area. You wouldn’t think it was related, but the more mathematics you do, the more connections you find.
Pick any theorem, no matter how boring, like Thales’s Theorem, and you can find these amazing things where the boring theorem is actually a key insight that opens entirely new ways of thinking about seemingly unrelated problems. That’s mathematics for you. No wonder it caught on like a bug among the Greeks, once they got the ball rolling. One moment you stumble upon some random result like Thales’s Theorem, and the next thing you know you’re seeing mathematics everywhere.
So that’s the mathematical moral of the story. Now we must go back and say something about the historical side of all this. What do we really know about Thales and his theorems and Queen Dido and all that? How much is history and how much is legend?
If we start with Dido, that story comes to use primarily through Virgil. The Aeneid, the famous epic poem. That was written in Roman times, around the year -20. But it is referring to historical, or supposedly historical, events that took place even before Thales, maybe two centuries before Thales, so -800-ish. We have Virgil’s version, that’s what has come down to us, but he is just stealing an older story. These things would have been around for centuries in Greek culture, in various literary and historical retellings that are now lost.
It is perfectly plausible that there really was such a historical queen, who really did flee her royal home in Tyre, and really did land on the north shores of Africa where she founded this new settlement, which was to become the great city of Carthage. Maybe indeed she even made the city walls semi-circular, who knows? It is perfectly conceivable that she might have wanted to minimise the perimeter for whatever reason, and that she might have known that a semi-circular shape was optimal for this purpose.
But at that time there would not have been any mathematical proofs of this, like the one I sketched above. The proof I outlined is from Jakob Steiner, in the early 19th century. From Greek times we have a different proof of this result. So they were certainly very much aware of the result, that the semi-circle is optimal, if perhaps not the particular proof I suggested.
If the story of Queen Dido says anything about the history of mathematics, it probably illuminates most neither the time when the events took place, around -800, nor the time when the sources we have were written, around year 0. But maybe it says something about the centuries in between, where the story would have been passed on and reworked.
The story was marinated, as it were, in Greek culture. Maybe they were the ones who gave it a mathematical flavour. The shoe fits: The Greeks valued wise, aristocratic, well-educated rulers, who design rational policy for the common good informed by reason and mathematics. Maybe they let these ideals colour the way they retold the story of Queen Dido and her round city.
From this point of view we could also speculate that by the time Virgil comes around and writes the Roman version of the story, this appreciation of mathematics is no longer what it once was. Indeed Virgil doesn’t really spell out the mathematical optimisation aspect of the story. Dido is just a side character altogether. His epic is about Aeneas, who is on a quest that will eventually lead to the founding Rome.
Aeneas is shipwrecked and blown ashore at Carthage, Dido’s round city. Dido falls in love with him, but he does not return her love. He sails away and Dido kills herself because of her broken heart. Morris Kline concludes the story: “And so an ungrateful and unreceptive man with a rigid mind caused the loss of a potential mathematician. This was the first blow to mathematics which the Romans dealt.” Sure enough there’s plenty more where that came from.
One can view this story as symbolic of this transition from the wise philosopher kings (or queens in this case) of the Greek world, who cherished mathematics and used it to improve the world. The transition from that to the heartless Roman, who only think of themselves and couldn’t care less about Thales’s Theorem. In the Greek world math nerds were considered attractive, but somehow these ignorant Romans didn’t think a geometer queen was girlfriend material at all evidently.
Ok, so the story about Dido and the round city and the optimisation proof and all that, it is very interesting in terms of the broader mathematical and cultural points its connects to, but in and of itself its is not directly history per se.
It’s different with Thales. That’s more fact than legend. As best as we can determine, Thales really did prove that diameter bisects a circle, most likely with the proof discussed above.
The sources that we have for this are far from perfect. Primarily Proclus, who was writing in year 450 or so, basically one thousand years after Thales lived. These kinds of late sources are hit and miss. They have no authority in and of themselves. Proclus was nobody. His own understanding of history and mathematics is very poor. A mediocre thinker, a mediocre scholar, living in a mediocre age.
Those are the kinds of sources that we have. Basically as authoritative as a factoid you read on the back of a cereal box or something.
But there is hope. Back in its glory days, Greece was just an outstanding intellectual culture. And some of the stuff about for example Thales can be traced back to then, which makes it highly credible. Aristotle’s student, Eudemus, wrote a history of geometry. It’s no longer with us alas. Ignorant ages neglected it and now it’s gone. But what a work that would have been.
These people knew what they were doing. Later people like Proclus are like some online rando posting half-baked ideas on blogspot or poorly informed comments on Facebook. That’s how credible they are.
But people like Eudemus is a very different story. That is more like a first-rate scholar at a research institution with all the infrastructure one could dream of: libraries, extremely knowledgable and intelligent colleagues with a range of expertises, broad financial and cultural support from the public and from politicians, and so on. Eudemus’s History of Geometry would be a proper “University Press” book, peer-reviewed to the teeth and with a nice dust-jacket blurb by Aristotle.
People like Eudemus were not in the business of passing on random gossip and unchecked factoids because they sound cool. They were proper scholars and intellectuals.
And indeed a lot of the stuff about Thales can be traced back to this lost source. When Proclus says that Thales was the first to prove that a circle is bisected by its diameter, the source of this is Eudemus. Hence it is very credible. This Thales stuff really happened. Actually that part about the diameter bisecting the circle is more certain than the part about Thales’s Theorem. Was Thales’s Theorem really Thales’s? Maybe. But we cannot trace that part specifically back to the best sources. Unlike the diameter bisection one and some other details. But contextually it makes sense.
The stories of Thales and the origin of geometry were evidently well known not only to specialised scholars but to the general Athenian public. Aristophanes the playwright uses the name of Thales as a symbol of geometry a few times in his plays. Just as today one might use the name of Einstein for instance to evoke the image of a scientist. Aristophanes has one speakers in a dialogue say: “The man is a Thales.” Meaning that the person is a geometer. Evidently the theatre-going public in classical Athens could be expected to understand this reference. Every educated person would know about Thales and the origins of geometry.
In fact, public respect for geometry and its history was apparently so great that Aristophanes even has one of his characters lament it as excessive, saying: “Why do we go on admiring old Thales?” What a time to be alive that would have been. When playwrights had to tackle issues such as there being too much respect and interest in mathematics among the general public. “Hey guys, maybe we need to cool it with how much we love geometry.” What a luxury problem. Hardly one that Hollywood blockbusters today have to grapple with.
Anyway, we should maybe not read too much into those isolated quotes. But the general intellectual credibility of this age is important. These very intelligent and serious people recorded in scholarly histories the accounts about Thales founding deductive geometry and proving that a circle is bisected by its diameter. That’s only some two or three hundred years after Thales, and in a direct lineage from him, probably with entire works by Thales still around in libraries and so on.
So there you go. The origins of proof and deductive geometry. We really do know quite a bit about it, and it’s a story worth knowing if you ask me.
How did Leibniz view the foundations of infinitesimals? A new paper by Rabouin & Arthur gives one answer. Among other things, they address some critiques I raised. My reflections on their new paper are as follows.
First some context. There were a range of options available for justifying infinitesimal methods in the 17th century. One could argue that:
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Infinitesimals can be completely avoided in the manner of the method of exhaustion of the Greeks, which is functionally equivalent to infinitesimal reasoning yet is impeccably rigorous since it is based entirely on finitistic reasoning. Anytime one uses infinitesimal language, this is to be understood as a mere shorthand that could always in principle be translated into a proof “in the manner of the Ancients” if needed.
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Infinitesimals can be thought of as very small numbers (such as dx=0.00001), in which case infinitesimal results are strictly speaking only approximations, but approximations that can be made arbitrarily accurate (that is to say, the error can be made less than any assignable magnitude) by making the dx smaller and smaller.
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Infinitesimals can be thought of as new entities that enlarge the universe of real numbers, analogous to complex numbers or points at infinity in projective geometry.
(4. More exotically: Nature is inherently infinitesimal. The calculus does not approximate curves by polygons; rather, all curves genuinely are actually straight on the micro level.)
(5. Perhaps not an independent strategy but a shorthand manner of speaking that ultimately reduces to one of the above: Infinitesimals are “useful fictions” — heuristic devices that work regardless of their existential or foundational status.)
Leibniz often alluded to these ways of justifying infinitesimals. He seems to have been quite happy to take a pluralistic approach: not only is each of these ways of justifying infinitesimals quite convincing separately, but the sheer multitude of plausible approaches in itself adds further credibility.
Indeed, it is not necessary for the working mathematician to insist on one of these approaches and exclude the others; on the contrary, the flexibility afforded by the multiplicity of lenses is creatively useful. It is “the business of the metaphysician,” as Leibniz says, to worry about which is the “real” or “true” foundation. Since this is an issue of little consequence to actual mathematical practice, it is wiser for the mathematician to remain neutral and agnostic and not waste time with it.
Many things Leibniz says fits this picture well. But Rabouin & Arthur argue against it. The say that, on the contrary, Leibniz had “very definite views” on what the right foundations of infinitesimals were, “from which he never wavered.” According to this interpretation, the pluralistic tendencies of Leibniz’s public statements on the foundations of infinitesimals are to be seen more as argumentative strategies adopted to the context and audience than genuine expressions of Leibniz’s own views.
For my part, I think it makes a lot of sense to take Leibniz’s pluralistic statements at face value. Moreover, I don’t think the difference between this and the view of Rabouin & Arthur amounts to all that much, for many purposes. The fact remains, any way you slice it, that Leibniz argued skilfully for the pluralistic approach in many writings, especially in direct reply to criticism of the foundations of the calculus. This effectively makes him a pluralist for most intents and purposes, I would say. I find this more interesting and important for the development of mathematics than esoteric debates about what Leibniz allegedly “really” thought privately, based on unpublished manuscripts.
In any case, that is the general background. Now let’s look at the specific points of contention between my paper and that of Rabouin & Arthur.
Some scholars (including Rabouin and Arthur in previous publications) have claimed that Leibniz had a very sophisticated understanding of the foundations of infinitesimal methods that was far ahead of his time. But he didn’t actually publish these ideas. On the contrary, in public communication he was often laid back about the foundations of the calculus. He seemed uninterested in addressing the issue, and only made some rather laconic remarks about it when pressed by others. Nevertheless, some scholars maintain that the unpublished De quadratura arithmetica (DQA), written in his youth, a decade before his first calculus publication, contains these brilliant foundational insights.
I disagree. In my view, as far as the foundations of infinitesimals is concerned, the DQA is quite unremarkable and basically just rehashes ideas that were commonplace among leading mathematicians at the time. I also believe that this was Leibniz’s own view. That’s what I argued in my paper mentioned above.
Against this, Rabouin & Arthur claim that:
there are in fact many documents in which Leibniz refers to the DQA, most of the time very explicitly, as the place to go to find a justification for the use of infinitesimals.
I say: No, that’s not what those documents show. Leibniz references the DQA but hardly any of these references even concern the justification of infinitesimals at all, let alone explicitly say that the DQA is “the [!] place to go” for such justifications.
Let’s go through them all one at a time.
Gerhardt published a Compendium of the DQA, which Leibniz prepared for publication.
In this Compendium, Leibniz explicates at length the specific geometrical results of the treatise, and gives very short shrift to the allegedly so insightful parts on the foundations of infinitesimal methods.
Moreover, Leibniz certainly sought to publish the treatise itself. In an exchange of letters in 1682, he discusses the project of publishing the DQA.
1682 is before any calculus publication, and hence irrelevant for our purposes.
More interestingly, though, ten years later Leibniz reopened the possibility of publishing the DQA. In a letter …, he wrote: “… the distractions that I then had did not permit me to lay it out in full, and I contented myself by giving certain abstracts in the Actes of Leipzig. … One could add a preamble containing some curious particulars on what Mons. Descartes invented or took from elsewhere.”
Again it seems that Leibniz does not have in mind the foundational parts, but the particular geometrical results of the treatise — “curious particulars” indeed, rather than unprecedented foundational material. The allegedly innovative foundational parts are not partially published and abstracted in the Actes but the geometrical results are, such as the series for pi/4. Nor do the foundational parts have any evident connection to Descartes. So foundations of infinitesimals does not appear to be the aspect of the DQA that Leibniz has in mind.
Bodenhausen signalled to Leibniz that it would be very useful to have at one’s disposal a gentle introduction to shut the mouths of the Euclidean “Pied Pipers” (Rattenfänger), who were hostile to the new method [i.e., the calculus]. Leibniz responded positively to the demand and sent … a presentation of the calculus for those who were trained in the “manner of the Ancients”. And what did he provide on this occasion? A presentation of Prop. 6 of the DQA accompanied by a translation into the differential calculus corresponding to Prop. 8. To be sure, all of these results are at the time superseded by the many researches in which Leibniz had been engaged since then, and he does warn his correspondent that the results from the DQA are almost immediate with the new calculus. But precisely, it is all the more striking that when coming to a translation of this calculus into the language of the Ancients, the only example he has to provide in 1690 is still prop. 6 of the DQA.
None of this contradicts my interpretation. This is all consistent with Leibniz regarding the DQA as a tedious explication of standard material as far as foundations is concerned. This explains why he never did such things again, and why he would only use it to give to those who were out of touch with the mathematics of the time.
When Leibniz is pressed to explain his method of quadrature to someone having difficulty with it in 1695, the “most elegant” way he can conceive of demonstrating it is not in terms of the more powerful methods he has developed since his youth, but by reference to the very presentation in the DQA which those methods had, according to Jesseph and Blåsjö, rendered inadequate and obsolete.
I think Rabouin & Arthur are quite deceptive here. They could have made it a lot clearer that Leibniz’s “method of quadrature” here means his method of quadrature of the circle, i.e., his series for pi/4. That is to say, what is at stake is one particular result, not the general method of quadrature employed in the calculus. The fact that certain very specific results can be proved “elegantly” by DQA methods obviously says absolutely nothing about the alleged significance of the DQA as foundational for infinitesimal methods overall.
Moreover, contrary to Blåsjö’s claim that he never quoted any of its results in foundational discussions, we have … also what Leibniz wrote to Johann Bernoulli in 1698.
Here indeed Leibniz mentions that a specific and not particularly important point Bernoulli made is similar to one he made in an obscure part of DQA. This obviously has nothing to do with Leibniz claiming that the DQA was anything like “the place to go to find a justification for the use of infinitesimals” in any way, shape, or form. On the contrary, Rabouin & Arthur themselves quote Leibniz as immediately saying in the same letter that “it is always the case that what is concluded by means of the infinite and infinitely small can be evinced by a reductio ad absurdum by my method of incomparables (the Lemmas for which I gave in the Acta)” — in other words, Leibniz is referring to his published works for the foundations of his calculus, with no indication whatsoever that the earlier DQA that he just mentioned has a more profound treatment of those very issues.
In sum, I do not see any compelling evidence that Leibniz thought of the DQA as foundationally important.
In my paper I also argued the same point internalistically: the mathematical details of the DQA are nothing special as far as foundations are concerned, and certainly not as remarkably rigorous and general as has been claimed.
Rabouin & Arthur claim to address this: in their introduction they promise to “show how Leibniz’s method in the DQA builds on and improves upon the extant method of indivisibles.” But in fact they say hardly anything about this. The only explicit claims about how Leibniz allegedly goes beyond his predecessors in this way that I could find are a few sentences around notes 30-32, 39-41. I very much doubt that the rather incidental things pointed to there were either new or significant. In any case, one could certainly not conclude that they were from these vague allusions.
A high burden of proof falls on those who make extravagant claims about the allegedly profound and proto-modern insights in the unpublished DQA. The new paper by Rabouin & Arthur is interesting and useful in many respects, but it certainly falls short of meeting that burden.
In ancient Mesopotamia and Egypt, mathematics meant law and order. Specialised mathematical technocrats were deployed to settle conflicts regarding taxes, trade contracts, and inheritance. Mathematics enabled states to develop civil branches of government instead of relying on force and violence. Mathematics enabled complex economies in which people could count on technically competent administration and an objective justice system.
Transcript
How did geometry start? Who was doing it, and why, in early civilisations? The Greeks invented theorem and proof, but long before them there was geometry in Egypt and Mesopotamia. So that’s practical geometry, applied geometry.
Or is it? Actually even the oldest sources have lots of pseudo-applications in them. Such as: Find the sides of a rectangular field if you know the perimeter and the diagonal. Or: I have two fields, and I know how much grain each field produces per unit area, and I know the total grain produced by both of them, and I know the difference between their areas, now tell me how to find the area of each field.
Not the kind of situations you find yourself in every day exactly. You can judge for yourself if that deserves to be called applied mathematics. Given obscure and convoluted information, find something that should have been much easier to measure directly than this artificial data you somehow had access to.
In any case, geometry like that, whatever you want to call it, was highly developed almost four thousand years ago. Why? What made people do this? Let’s try to find out.
Early mathematics emerged where there was fertile soil. Rivers that made this possible. Agricultural abundance meant resources enough to expend some people specialising in mathematics instead of having all hands on the ploughs.
Look at a modern population density map of Egypt. You will find that virtually the entire population is concentrated along the Nile; all the rest is pretty much desert. That’s still the case today. Even with the assistance of modern technologies the river area is by far the most liveable. Even more so back then when geometry started, thousands of years ago.
It was the same in Mesopotamia, present-day Iraq. Also a river civilisation with very good agricultural conditions. They had legendary gardens that were praised in ancient sources. Google it: The Hanging Gardens of Babylon. You will see some nice pictures of what these luxurious gardens might have looked like. That’s a nice visual for this idea that it was agricultural abundance that made a specialised pursuit like mathematics possible in those societies.
So that explains why they had the resources to support mathematics. But why would they want to? What did they stand to gain from geometry?
Basically, mathematics was for a long time about commerce and taxes; bureaucratic management of workers and produce; inheritance law. Those kinds of things.
Eleanor Robson’s book is very illuminating about this. “Mathematics in Ancient Iraq: A Social History”, the book is called. She emphasises especially that mathematics was very strongly associated with justice. A society without a functioning justice system is hampered by constant disputes about land, taxes, inheritance. Everybody is fighting with everybody. Like the old American West, you board yourself up and mind your own business and if there’s a disagreement, well, that’s what guns are for, isn’t it?
Mathematics is the way out of this primitive state. Mathematics is objective. It can settle these disputes in a fair way. If everybody is wasting a huge amount of effort and resources on petty disputes in a lawless no-man’s land, who you gonna call? The mathematicians, that’s who. That’s how it went in ancient Iraq.
A specialised, highly trained mathematician would come in and delineate all the plots of land, compute all the taxes owed, and distribute every inheritance. All according to exact calculations. This stuff used to be ruled by emotions, personal animosity, and the law of the jungle. But now, thanks to mathematics, that is replaced by objective rules. Who can argue with a calculation? Mathematics takes the worst sides of human nature out of the equation.
When society is run by fair, universal rules, people no longer have to constantly look over their shoulder and fear that some lawless eruption of force could destroy everything they have at any moment. A functioning justice system enables people to work for the collective good and to plan for the long term.
It is the authority of mathematics that makes this possible. These skilled mathematical technocrats had great credibility because people recognised that they were above the subjective and the emotional. They were bound by dispassionate calculation. Mathematics compelled them to be fair and rational.
Indeed they explicitly said so themselves. As one mathematical scribe put it: “When I go to divide a plot, I can divide it; So that when wronged men have a quarrel I soothe their hearts. Brother will be at peace with brother.” That’s a quote by one of those mathematical technocrats, explaining what geometry accomplishes. Note that it has both of those elements I emphasised. Mathematics is the opposite of emotional disputes. It soothes heated hearts, it creates peace between warring brothers. And the quote also highlights that this happens because of the expertise of the mathematician: I know how to do this kind of thing, the technocrat is saying. It takes special training.
The quote is from Eleanor Robson’s book. Here’s another thing she points out that is yet more evidence of the importance of mathematics in this context. The Sumerian word for justice literally means straightness, equality, squareness. Also in Akkadian: justice is the “means of making straight.”
Again, another major indicator of this: “the royal regalia of justice were the measuring rod and rope.” Think of those Lady Justice statues that you see sometimes. She’s blindfolded because that shows that she’s unbiased, and she has these scales, showing that she’s considering both sides and weighing them carefully and fairly. That’s the symbol of justice in our society. But, in ancient Babylon, the symbols of justice were not a blindfold and a set of scales. Instead, Lady Justice was a geometer. She held her land-measuring tools. Those were the instruments of justice in ancient society.
Maybe it’s pretty much the same today, four thousand years later. Back then, the trustworthiness of mathematics was a cornerstone of society. If people didn’t trust mathematics, there could be no law and order, no state bureaucracy, no complex economy, no civilisation. Today, that link is perhaps less evident. But perhaps no less crucial. We have added many layers of complexity to our society, but perhaps looking back at historical societies is the same thing as looking into the inner essence of our own. Maybe without faith in mathematics the entire fabric of our society would unravel. Maybe without mathematicians mediating their disputes, “brother would be at war with brother” as that ancient scribe feared.
It is interesting also that this role of mathematics that I have outlined is really as much psychological as it is scientific. What makes this whole system work is not only that mathematics can give useful answers to certain technical problems. The psychological side is equally essential: mathematics has a kind of aura of objectivity, of trustworthiness, of professional expertise. That goes well beyond merely calculating the taxation rate of some field, or how many goats you can buy for a silver shekel. The system rests on a more nebulous trust in the mathematician class by the population at large. The idea of mathematics, the image of mathematics, is more important than the sum of its actual applications.
That’s an important conclusion because it explains that striking feature of ancient mathematics: namely that many of the problems the ancients texts solve are super fake. They are pseudo-applications.
For instance: Find the two sides of a rectangle, given that the sum of the length and the width is 24, and that the area plus the length minus the width = 120. So in other words, you basically have two equations in x and y, and if you solve for y in one and plug it into the other you have a quadratic equation in x. Lots and lots and lots of problems like that in Babylonian mathematics.
Obviously nobody would ever face a problem like that in any real-word situation. It’s very often like this: you are looking for something simple, like the sides of a rectangle x and y, and you are given something super weird, like some convoluted combination of x and y is three eights of some other convoluted combination of x and y.
Here’s another actual one: The width of a rectangle is a quarter less than the length. The diagonal is 40. What are the length and the width?
In what real-world scenario can you realistically end up knowing the diagonal of a rectangle, and the difference between the sides, but not the sides themselves? And why couldn’t you just measure the sides? Someone did measure the diagonal, apparently, so why not the sides?
Sometimes these texts hardly even try to hide how fake they are. One problem goes: I found a stone, but did not weigh it. I cut away one-seventh and then one-thirteenth, and then it weighed so-and-so much. What was the original weight of the stone?
Who among us has not “found” whatever random stone, then chipped away an extremely exact ratio of it, and then suffered some kind of stone-cutter’s remorse I guess, and tried to reconstruct the original weight of the stone for some reason.
Very relatable, isn’t it? Actually it kind of is. Not because we are sitting around cutting one-thirteenth out of random stones, or because we are running around measuring the diagonals of various fields and then later wish we had measured the sides instead. That never happens to any sane person in the real world. But it does happen in math books. Still today, we torture our students with such questions, one more artificial and unrealistic than the other.
Some people think that kind of thing is modern pedagogy run amok. They see these kinds of problems in modern textbooks and they think: How silly modern pedagogy has become! These naive educators are bending over backwards to make math “relevant” to kids, but they just end up with silly fake problems.
History offers a different perspective. The problems may be silly, but the cause is not a misguided obsession with real-world relevance among modern educators. Fake problems are as old as written mathematics itself. For as long as there has been mathematics education, students have been forced to go through page after page after page of pseudo-problems that only superficially, or linguistically, appear to be talking about real-world things, while actually corresponding to absurd scenarios that would never happen.
In a way one might argue that history vindicates these problems. They are not so silly after all, if we consider them in the light of the role of mathematics in ancient Babylonian society. Mathematics doesn’t support the economy merely by keeping the account books. It’s more than that. Mathematics is what instills confidence in monetary law and order, without which any kind of complex economy would be impossible in the first place.
For this system to work, there needs to be a specialised class of number-crunching technocrats. These people need to embody logic and reason and objectivity. They need to be math machines, detached from politics and emotion. A long schooling in artificial pseudo-problems makes some sense as a means of creating this class.
From this point of view, it is even a strength that these problems are artificially divorced from real-world problems, because the mathematical technocrat is supposed to be detached from such concerns anyway. Mathematicians are valuable to society precisely because they are so disinterested in the needs of people of flesh and blood. It is this disinterestedness that makes people willing to trust the mathematicians to be the arbiters of disputes.
The sheer volume of training in pointless problems also has its point. It is not enough that people at large know some mathematics: they could use mathematics as a tool for evil, as just one more incidental weapon in a society still ruled by greed and conflict. For a complex economy to take off, there needs to be faith that the law and the state administrative bureaucracy are fair and consistent. This faith comes from the credibility of mathematics. The mathematical technocrats need to be proper experts to justify the confidence placed in them. They need to embody mathematics; they need to single-mindedly look at any situation or conflict and see only the mathematics in it.
Society needs the mathematicians to not only get the right answer, but to have great authority as proper experts. And it needs them to be “nerds,” so to speak, who are so one-sidedly developed that they can only see mathematics anywhere they look, and not let emotions or politics influence their work. A long and rigorous training in fake applied problems is not a bad recipe for bringing this about. Arguably, we pretty much still use the same recipe to the same end today, thousands of years later.
So that’s the Babylonian tradition. We know quite a bit about it because they wrote on clay which is pretty durable. In Egypt, mathematics was recorded on papyrus, which isn’t going to survive for thousands of years normally. So we only have two or three or maybe four papyri that beat the odds and were conserved. But it seems the Egyptian situation may very well have been quite similar to the Mesopotamian one in terms of the role of the mathematicians.
“Geometry” means “earth-measurement.” That’s from the Greek: geo metria. The ancient Egyptians had the same idea but their word for it was more concrete: a geometer was literally a “rope-stretcher.” A land surveyor stretches ropes to measure distances and delineate fields.
A rope is pretty much equivalent to a ruler and compass. Pull the ends of the rope and you have a straight line. Hold one end fixed and move the other one while keeping the rope stretched: now you have a circle.
Euclid explains how to make a square with ruler and compass. That’s Proposition 46 of the Elements. The Egyptians would have done that long before with their stretched ropes. Try it for yourself, it’s fun: go out into a field with a friend and try to make a perfect square using nothing but a piece of string. You will see why geometers were called rope-stretchers.
Do you think you could make a square? Do you think anyone could? Back in the day, this skill could have given you a leg up in life. Suppose you make one square field, and then a rectangular field with the same perimeter. The square field will have greater area. But you could trick those less knowledgeable in mathematics. You could say: you get that field and I get this one, fair and square. Just try it for yourself, you would say, let’s walk around the fields and count the number of steps. 400 steps around my field, 400 steps around yours: aha, our fields are the same size. That’s what you tell the other guy, who isn’t such a math person. But you know that of course 100100 is way more than 50150. So later you get a much greater harvest. But of course you would pretend that that’s because you worked so hard while the other guy was lazy. Maybe that’s another way in which ancient society is like ours: privileged people use their privilege to rig the game in their favour, and then pretend it was all due to merit.
According to Proclus, this kind of mathematical deceit did indeed happen: “The participants in a division of land have sometimes misled their partners. Having acquired a lot with a longer periphery, they later exchanged it for lands with a shorter boundary and so, while getting more than their fellow colonists, have gained a reputation for superior honesty.”
Here’s how Thomas Heath paraphrases this in his History of Greek Mathematics: “Proclus mentions certain members of communistic societies who cheated their fellow members by giving them land of greater perimeter but less area than the plots which they took themselves, so that, while they got a reputation for greater honesty, they in fact took more than their share of the produce.”
A dubious paraphrase, in my opinion. Can you spot the suspicious part of it? Good old Heath put something in there that was not in the original source. Hint: turn to the title page of Heath’s book. There are some clues there. The book was published by Oxford University Press in 1921. Heath’s name comes with some bells and whistles: it’s Sir Thomas Heath, in fact, and then K.C.B, K.C.V.O. That’s Knight Commander of the Royal Victorian Order etc.
Titles upon titles. It’s an establishment guy, this Sir Thomas. A gentleman scholar, who was a civil servant as his day job at the Treasury.
What part of Sir Thomas’s paraphrase of the ancient mathematical land deceit reflects his own social context more than that of the ancients he is trying to describe? I’m thinking of his phrase that these were “communistic societies.” The original source says nothing at all about this having anything to do with communism. But you can understand how Sir Thomas would have been concerned about communism at this time. The Russian Revolution started in 1917, Heath’s book is published in 1921. While writing the book, Heath was a secretary at the British Treasury. He would have read all about Lenin and Bolsheviks in The Times while having his afternoon tea. And those worries would have been at the top of his mind when he sat down in his study to do his scholarly work in the evening. It didn’t take much provocation, one imagines, for him to have a swing at how “communistic societies” were dreadful and corrupt.
We must always read historical sources this way. Context matters.
Now, the “original” in this case was Proclus. But that’s not much of an “original” to speak of. Proclus is nobody. He’s not particularly trustworthy. He was writing in the year 450 or so, thousands of years after the historical events he is talking about. So it’s anybody’s guess how much truth there is in what he is saying. And in any case, like so many other mediocre writers, both ancient and modern, Proclus is just copying what others had said.
Let’s illustrate this point. Let’s see what we can learn by looking at Proclus’s account of the origins of geometry in Egypt. Here’s what Proclus says:
“Geometry was first discovered by the Egyptians and originated in the remeasuring of their lands. This was necessary for them because the Nile overflows and obliterates the boundary lines between their properties. It is not surprising that the discovery of this and the other sciences had its origin in necessity, since everything in the world of generation proceeds from imperfection to perfection. Thus they would naturally pass from sense-perception to calculation and from calculation to reason. Just as among the Phoenicians the necessities of trade gave the impetus to the accurate study of number, so also among the Egyptians the invention of geometry came about from the cause mentioned.”
Ok, sounds pretty plausible. But it’s worth running Proclus through a plagiarism checker, just as we do with modern student essays these days. Cutting-and-pasting from Wikipedia is nothing new. Proclus had many Wikipedia equivalents available to him. Perhaps he stole the whole thing for example from the Geography of Strabo, which was written more than 400 years before. Here’s what Strabo says:
“An exact and minute division of the country was required by the frequent confusion of boundaries occasioned at the time of the rise of the Nile, which takes away, adds, and alters the various shapes of the bounds, and obliterates other marks by which the property of one person is distinguished from that of another. It was consequently necessary to measure the land repeatedly. Hence it is said geometry originated here, as the art of keeping accounts and arithmetic originated with the Phoenicians, in consequence of their commerce.”
Basically a dead ringer for the Proclus passage. Plagiarism detected, SafeAssign would say.
Actually Proclus has added something that is not in Strabo, namely the claim that this historical episode illustrates how human though passes from the world of the senses to the higher realm of reason. This is card-carrying Platonism. Proclus is a sycophantic follower of Plato. He sees everything through Plato-coloured glasses. Which is not helpful if we want to use him as a source of historical information. As Heath had his anti-communism, so Proclus has his Platonic axe to grind and it infects everything he says.
Actually we can go back even earlier than Strabo. Let’s take an equal jump back in time again: another 450 years still. From Roman Strabo to classical Greek Herodotus. He too speaks of the origins of geometry in Egypt. Let’s listen to his account:
“This king [Sesostris] also (they said) divided the country among all the Egyptians by giving each an equal parcel of land, and made this his source of revenue, assessing the payment of a yearly tax. And any man who was robbed by the river of part of his land could come to Sesostris and declare what had happened; then the king would send men to look into it and calculate the part by which the land was diminished, so that thereafter it should pay in proportion to the tax originally imposed. From this, in my opinion, the Greeks learned the art of measuring land.”
Ok, I have to admit that this makes Thomas Heath look a bit better. “The king gave to each an equal parcel of land”: That is a bit more like communism. Heath said he was paraphrasing Proclus where there is no such phrase about equality. But Herodotus, the better source, kind of vindicates him a bit. You could imagine, in the scenario that Herodotus describes, that certain administrators in charge of implementing the king’s decree might secure a nice big square plot for themselves and trick the mathematically illiterate into a smaller plot with the perimeter trick. Perhaps not entirely unlike how corrupt middle-managers in the Soviet bureaucracy might manipulate the system for personal gain. But be that as it may.
I think there’s another interesting thing about Herodotus’s description compared to Strabo’s. Strabo and Proclus give a cleaner and simpler account: the flooding of the Nile obliterates everything and you have to start afresh each year with the drawing of boundaries. Herodotus’s account is much less dramatic: some parts of properties might become damaged by the floods, and the task of the mathematician is not to redraw the entire agricultural map each year but rather to calculate what proportion of area has been lost in each case for taxation purposes.
One can easily imagine how a desire to simplify and tell a clear and dramatic story might have led authors like Strabo and Proclus to prefer their version. The older source is a bit more “boring” but perhaps that makes it more credible.
Indeed, Herodotus’s account fits better with what we said about the role of mathematics in Mesopotamian society. In Herodotus’s version, the mathematician’s task is more technical, more specialised, more bureaucratic. Note his phrase: “the king would send men” to do the calculations. You have to send mathematicians. They are a small, specialised class of technocratic experts that are dispatched to solve disputes with authority and objectivity. That’s precisely the main point I have made today, so let us end there.
The Greek islands were geographically predisposed to democracy. The ritualised, antagonistic debates of parliaments and law courts were then generalised to all philosophical domains, creating a unique intellectual climate that put a premium on adversarialism and pure reason. This style of thought proved ideal for mathematics.
Transcript
Why the Greeks, of all people? Why did mathematics start there, on a few scrawny little islands in the Mediterranean?
The very idea that mathematics is about systematically proving things is an exclusively Greek invention. Axiomatic-deductive mathematics has been discovered only once in human history. No other culture independently developed anything like it.
The lettered diagram is another uniquely Greek invention. Triangle ABC, the line AB, stuff like that. Geometrical diagrams with the points denoted by letters. Only in Greece did they feel the need to do geometry this way. If you find it elsewhere, it’s because they copied it from the Greeks.
Not that the lettered diagram is a big deal in itself, of course. But it’s a symbol; it’s emblematic of how so many aspects of mathematics that we now consider so essential and indispensable were in fact discovered once and only once in human history, at a particular time and place.
So what was it about that time and that place that made it explode with intellectual progress?
You can make a pretty good case for geographical determinism. The seeds of excellence was not in the blood or the genes of these people, but it was in the land and the sea.
Islands. That is the key. Greece is a country of a thousand islands. In fact, you can hear this in the very names of the great mathematicians of that time.
Consider Pythagoras, for example. More fully you often see his name given as Pythagoras of Samos, his place of birth. Which is an island. One of those typical picturesque Greek islands.
The same goes for other great Greek mathematicians. Hippocrates of Chios, Aristarchus of Samos, Archimedes of Syracuse, Hipparchus of Rhodes: island, island, island, island. Everybody is an islander in Greek mathematics. There’s also Eudoxus of Cnidus, and Diocles of Carystus: those are technically peninsulas, but pretty nearly islands basically.
What’s with all these islands? Let’s see where this geographical argument leads us.
First of all, islands are excellent for trade. Back then, it was a thousand times easier to transport goods by water than by land. Even the Romans, centuries later, used to import huge amounts of grain from Egypt for example. And that’s the Romans, who are famous for their excellent roads. Even to them it was much more of a hassle to get grain from mainland Europe than to swish it across the sea with some efficient ships.
So the Greeks became tradespeople. Because they had so much access to the water.
And what did they have to trade? Think of the typical landscape of a Greek island. It’s hilly and full of slopes and kind of dry, rocky soil. Not the typical agriculture landscape you would have on the irrigated flats of mainland Europe or America. That kind of stuff would slide right off the Greek hills. In Greece you need tougher plants with roots that really dig in and hang on for their life as a rain shower threatens to wash the whole thing down with it down the hill.
Hence: olives and grapes. These plants love a good slope. They thrive there.
And what luck for the Greeks! These plants are perfect for trade. Think about it. You use them to make olive oil and wine: expensive, non-perishable luxury products.
Vegetable and fruit is highly perishable: by the time you get to your destination to sell it, half of it is rotten or eaten by worms. And it’s also very bulky: a big barrel of cabbage isn’t going to fetch you a whole lot of cash. It doesn’t have many calories. So it’s a lot of work to transport for so little payoff. The cabbage business isn’t very lucrative.
But olive oil and wine is perfect. Olive oil is a calorie bomb: a little goes a long way, so it’s easy to transport a fortune’s worth of it. And these products don’t mind being stored. Just stick them in an urn with a good cork on it and you’re set. Wine can even get better by sitting around. Unlike a sack of cucumbers that will spoil before you put your sandals on.
Olive oil and wine are also highly processed. A lot of work goes into the production. What are you gonna do with a bag of cucumbers? They are what they are, you just eat them. But the grapes and olives are processed by expert artisans. Lots of added value. The labor theory of value, you know, that Marx talked about and so on.
So the Greek islands are a recipe for wealth. Perfect products for trade, and perfect access to the sea for trading. This creates wealth, which creates a large middle class with lots of leisure time. That is certainly a precondition for intellectual culture.
Maybe also trade is itself a recipe for a certain open-mindedness and diversity of thought. There was no Internet back then. Travel was a good way to get exposed to other ideas, other ways of doing things. And therefore to start thinking more critically about the idiosyncrasies of your own habits and worldview.
Plus, a merchant needs to trade with whoever is paying. That may be people of different religions and so on. So you get used to dealing with people different than yourself. You develop and kind of tolerance for differences of opinion, and strategies for reasoning with people you disagree with.
All that from trade. But there is a second big consequence of the islands: independent city states. Islands are naturally isolated units. It will be much harder for a single despot to impose a unified rule on a bunch of scattered islands than on a solid land mass.
This is the geography of democracy. And democracy means debate. You don’t have “do this because I’m the king and I’ll chop your head off.” Instead you have one guy presenting reasons for this, the other guy presenting reasons for that, and people are weighing the arguments and making up their own minds.
This is going to be the setting that gives birth to mathematics and philosophy. Geography created this rich, democratic, cosmopolitan people who fell in love with clashes of ideas and took that concept to the extreme.
Geoffrey Lloyd the Cambridge professor has written good stuff about this. I’m going to quote extensively from his works.
“The level of technology and economic development” in ancient Greece was high indeed. In fact, it was “far in advance of many modern non-industrialised societies” today. And “Aristotle [explicitly] associated the development of speculative thought with the leisure produced by wealth.” And not for nothing.
However, “Egypt and Babylonia were, economically, incomparably more powerful than any of the Greek city-states.” So the explanation for the “additional distinctively Greek factor” of “generalised scepticism” and “critical inquiry directed at fundamental issues” must be something other than wealth alone.
The answer may lie in “a particular social and political situation in ancient Greece, especially the experience of radical political debate and confrontation in small-scale, face-to-face societies. The institutions of the city-state put a premium on skill in speaking and produced a public who appreciated and the exercise of that skill. Claims to particular wisdom and knowledge in other fields besides the political were similarly liable to scrutiny, and in the competition between many and varied new claimants to such knowledge those who deployed evidence and argument were at an advantage compared with those who did not.”
The Greeks were so fond of debates and clashes of ideas that they developed a refined social machinery for it. They ritualised and institutionalised the concept of a philosophical debate. “Public debates between contending speakers in front of a lay audience” was a prominent part of ancient Greek culture. Science and philosophy were born on this stage. Many otherwise peculiar characteristics of Greek thought are explained by this format.
For example, the stage debate requires the speakers to proclaim bold and provocative theses, and to strive to avoid reconciliation with other viewpoints at all costs. This is why early Greek thought is rife with crackpot claims such as that motion is impossible or “that man is all air, or fire, or water, or earth.” Indeed, the format demands a multiplicity of such viewpoints in competition with one another, whence “the remarkable proliferation of theories dealing with the same central issues” that “may well be considered one of the great strengths of Presocratic natural philosophy.”
Indeed, this used to always puzzle me. How can anyone in their right mind genuinely believe themselves to have discovered that “all is fire” or “all is water”? What were these people smoking, right? And that’s just a couple of generations before peak Greek philosophy and its many very refined insights in mathematics and science. How can they have been such crackpots and then gone from 0 to 100 in the blink of an eye?
But in fact it makes sense in the stage debate setting. “All is fire” is perfect for that. It’s like a dangerous stunt. Jumping across a ravine with a motorcycle, or juggling with three chainsaws. To go on stage and say “all is fire,” now try to prove me wrong, I will answer any counterargument. If somebody pulls that off, credit to them. The crazier and the more implausible their initial thesis is, the more impressive it is if they manage to parry objections and defend their thesis with clever arguments.
Nobody ever actually believed that “all is fire,” but they admired the guts of someone who was prepared to argue as if they did believe it. They glorified the ability to argue unconventional ideas well. This was a great move for stimulating philosophy.
The stage debate setting also explains why these kinds of crazy theses were always defended by abstract deductive reasoning, not empirical investigation. “Given an interested but inexpert audience, technical detail, and even careful marshalling of data, might well be quite inappropriate, and would, in any event, be likely to be less telling than the well-chosen plausible—or would-be demonstrative—argument.” Hence we understand why “with the Eleatics logos—reasoned argument—comes to be recognised explicitly as the method of philosophical inquiry.” This “notion of the supremacy of pure reason may be said to have promoted some of the triumphs of Greek science.”
However, these triumphs of reason “were sometimes bought at the price of a certain impoverishment of the empirical content of the inquiry.” In early Greek science, “observations are cited to illustrate and support particular doctrines, almost, we might say, as one of the dialectical devices available to the advocates of the thesis in question.” Also, “observations and tests could be deployed destructively [to disprove an opponent’s thesis], as they were by Aristotle especially, with great effect.”
These uses of observation fit well within the stage debate format. However, “theories were not put at risk by being checked against further observations carried out open-endedly and without prejudice as regards the outcome.” We can understand why since “The speaker’s role was to advocate his own cause, to present his own thesis in as favourable a light as possible. It was not his responsibility to scrutinise the weaknesses of his own case with the same keenness with which he probed those of his opponent.”
Of course, everyone was well aware of the deceptive potential of sly rhetoric for “making the worse argument appear the better.” So much so, in fact, that “early on it became a commonplace to insist on your own lack of skill in speaking.” But the Greeks did not see this problem, the rhetoric problem, as a reason to abandon the stage disputation format altogether. Instead they focussed on explicating “the correct rules of procedure for conducting a dialectical inquiry,” to ensure the intellectual integrity of the debates.
What I just described is basically a summary of Geoffrey Lloyd’s book “Magic, Reason and Experience,” about the origin of Greek scientific thought. Also very illuminating is Lloyd’s later book contrasting the Greek contrarian climate of thought with its opposite paradigm: reverential, conservative thought, typified for instance by the ancient Chinese tradition. The book title hints at this division: “Adversaries and Authorities,” it is called. Here is the argument.
“Any acquaintance with early Greek natural philosophy immediately brings to light a very large number of instances of philosophers criticising other thinkers.” Being a philosopher means being “subjected to blistering attack.” That could pretty much be considered the definition of philosophy in Greek antiquity. “From the list of occasions when philosophers are attacked by name, one could pretty well reconstruct the main lines of the development of Hellenistic philosophy itself.” Nor is this limited specifically to philosophy only. On the contrary, “hard-hitting polemic” is the name of the game in mathematics, medicine, and art as well. There is a “lack of great authority figures”; even Homer “is attacked more often than revered.”
This Greek style of philosophy is connected to its social context. “Greek pupils could and did pick and choose between teachers. Direct criticism of teachers is possible, and even quite common. Argument and debate are one of the means of attracting and holding students, and secondly they serve to mark the boundaries of [schools of thought].” “The Greek schools were there not just, and not even primarily, to hand on a body of learned texts, but to attract pupils and to win arguments with their rivals. They may even be said to have needed their rivals, the better to define their own positions by contrast with theirs.” “Dialectical debate, on which the reputations of philosophers and scientists alike so often depended, stimulated, when it did not dictate, confrontation. The recurrent confrontations between rival masters of truth left little room for the development of a consensus, let alone an orthodoxy; [and] little sense of the need or desirability of a common intellectual programme.”
“It was the rivalry between competing claimants to intellectual leadership and prestige in Greece, that stimulated the analysis of proving and of proof.” “Many have assume that the internal dynamic of the development of mathematics itself would, somehow inevitably, eventually lead to a demand for strict axiomatic-deductive demonstration, and that there is accordingly no need to pustulate any external stimulus such as [this.] Yet the difficulty for that view is [that] other, non-Greek, ancient mathematical traditions — Babylonian, Egyptian, Hindu, Chinese — all got along perfectly well without any notion corresponding to axioms and the particular notion of strict demonstration that went with it.”
The underlying cause is perhaps captured by the dichotomy between “adversarial Greeks and irenic, authority-bound, Chinese.”
The different philosophical styles of ancient Greece and China reflect differences in their political systems. “Extensive political and legal debates, in the assemblies, councils and law-courts, were a prominent feature of the life of Greek citizens.” Democracy primes people for debate, for listening to and assessing different points of view and conflicting claims.
“Greek philosophical and medical schools used, as the chief means for the expression of their own ideas and theories, both lectures and open, often public debates, sometimes modelled directly on the adversarial exchanges so familiar in Greek law-courts and political assemblies.” They imported democratic practices and put them to work in the sciences.
It was very different in China. “Many Greeks seem to have positively delighted in litigation; [they developed] taste for confrontational argument in that context and became quite expert in [evaluating such arguments]. [The Chinese, by contrast,] avoided any brush with the law as far as they could. Disputes that could not be resolved by arbitration were felt to be a breakdown of due order and as such reflect unfavourably on both parties, whoever was in the right.”
“The typical target audience envisaged in Greek rhetoric is some group of fellow citizens,” just as “in Greek law-courts the decisions rested with [peers] chosen by lot [who] combined the roles of both judge and jury.” “In China, the [intended] audience for much philosophical and scientific work was very different: the ruler or emperor himself.” “The Chinese were never in any doubt that the wise and benevolent rule of a monarch is the ideal.”
“We often find Greek philosophers adopting a stance of fierce independence vis-a-vis rulers. With this independence came a disadvantage. Compared with their Chinese counterparts, Greek philosophers and scientists had appreciably less chance of having their ideas put into practice. Autocrats — as in China — could and did move swiftly from theoretical approval to practical implementation.” Not so in Greece. Greek philosophers had little hope of real power, and perhaps that’s why they liked to pretend that they didn’t want any anyway. “The superiority of theory to practice is a theme repeatedly taken up by scientists as well as philosophers in Greece: but that was sometimes to make a virtue out of necessity.”
“Unlike in classical Greece, the bid to consolidate a comprehensive unified world-view was largely successful in China.” “The prime duty of members of a Chinese Jia was the preservation and transmission of a received body of texts. In that context, pupils did not criticise teachers, and any given Jia did not see it as a primary task to take on and defeat other Jia in argument.”
While the Greeks “adopted a stance of aggressive egotism in debate, the tactics of Chinese advisers was rather to build on what could be taken as common ground, [and] certainly on what could be represented as sanctioned by tradition.” “The emphasis is not on points at which [earlier philosophers] disagreed, but rather on what each of them had positively to contribute, how each succeeded, at least in part, in grasping some part of the Dao,” the true or right way.
So there you have it. The source of Greek exceptionalism in intellectual history comes down to this: to glorifying extreme adversarialism; to waking up in the morning and going “today I’m gonna point out errors in other people’s arguments.” The Greeks lived for that stuff. And it was this that made them mathematicians, eventually. But that was not a planned child. Geography led to democracy, which led to this combative philosophical climate.
When some fragments of mathematics from Egypt and the orient were dropped into this petri dish, the reaction was explosive. These two were made for each other. Mathematics and argumentative debate was a match made in heaven. The Greek philosophical context triggered an avalanche of mathematical progress that took geometry from a set of obscure calculation rules to mankind’s best exemplar of perfect knowledge.
Galileo is the most overrated figure in the history of science. That is the thesis of Season 1 of my podcast, which consists of the following 18 episodes.
Galileo bad, Archimedes good
Galileo’s bumbling attempts at determining the area of the cycloid suggests a radical new interpretation of his scientific opus. Archimedes’s work on floating bodies is an example of excellent Greek science that has not been sufficiently appreciated.
Mathematics versus philosophy, then and now
Divergent interpretations of Galileo’s alleged greatness cut across disciplinary divides: mathematics versus philosophy, science versus humanities. Understanding Galileo means dealing with these fundamental tensions.
Galilean science in antiquity?
Ancient Greek scientists studied the dynamics of falling bodies. Were “Galileo’s” discoveries anticipated in these treatises that have since been lost? This question leads to a bigger one regarding relativism versus universalism in the history of thought.
The case against Galileo on the law of fall
Galileo is praised for his work on falling bodies, but his arguments were dishonest and his trifling discoveries were not new.
Galileo’s errors on projectile motion and inertia
Galileo gets credit he does not deserve for the parabolic nature of projectile motion, the law of inertia, and the “Galilean” principle of relativity. In reality, his treatments of all of these matters were riddled with errors and fundamental misunderstandings.
Why Galileo is like Nostradamus
Galileo committed scores of errors in his physics. These are bad in themselves and also undermine Galileo’s claim to credit for the things he did get right.
Galileo’s theory of tides
Galileo dismissed the notion that the moon influences the tides as “childish” and “occult.” Instead he argued that tides are a kind of sloshing due to the motion of the earth. This very poor theory is inconsistent with several of his own scientific principles.
Heliocentrism in antiquity
Two thousand years before Galileo, Greek astronomers argued that the heavenly bodies revolve around the sun. Their reasoning involved sophisticated mathematics and sound physical considerations.
Heliocentrism before the telescope
Galileo is credited with defeating Ptolemaic earth-centered astronomy, but most mathematical astronomers had already abandoned this theory long before Galileo.
The telescope
The telescope offered a shortcut to stardom for Galileo. We offer some fun cynical twists on the standard story.
Blemished sun
Galileo thought sunspots were one of the three best arguments for heliocentrism. He was wrong.
Phases of Venus
Telescopic observations of Venus provided evidence for the Copernican view of the solar system. But was Galileo the first to see this, as he claims? Or did he steal the idea from a colleague and lie about having made the observations months before?
Galileo’s theory of comets is hot air
Galileo thought comets were an atmospheric phenomenon, not physical bodies in outer space. How could he be so wrong when all his colleagues got it right? Perhaps because his theory was a convenient excuse for not doing any mathematical astronomy of comets. We also discuss his unsavoury ways of dealing with data in the case of double stars and the rings of Saturn.
Galileo and the Church
Galileo’s sentencing by the Inquisition was avoidable. The Church had no interest in prosecuting mathematical astronomers, but since Galileo had so little to contribute in that domain he foolishly got himself involved with Biblical interpretation. His scriptural interpretations not only got him into hot water: they are also scientifically unsound and blatantly inconsistent with his own science.
Galileo was the first to … what exactly?
Was Galileo “the father of modern science” because he was the first to unite mathematics and physics? Or the first to base science on data and experiments? No. Galileo was not the first to do any of these things, despite often being erroneously credited with these innovations.
More things Galileo didn’t do first
What was Galileo’s great innovation in science? To give practical experience more authority than philosophical systems? To insist on mechanical as opposed to teleological or supernatural explanations of natural phenomena? To take mathematical physics as our best window into the fundamental nature of reality as opposed to just a computational tool for a small set of technical problems? No, none of the above. All of these things had been old hat for thousands of years.
Historiography of Galileo’s relation to antiquity and middle ages
Our picture of Greek antiquity is distorted. Only a fraction of the masterpieces of antiquity have survived. Decisions on what to preserve were made by in ages of vastly inferior intellectual levels. Aristotelian philosophy is more accessible for mediocre minds than advanced mathematics and science. Hence this simpler part of Greek intellectual achievement was eagerly pursued, while technical works were neglected and perished. The alleged predominance of an Aristotelian worldview in antiquity is an illusion created by this distortion of sources. The “continuity thesis” that paints 17th-century science as building on medieval thought is doubly mistaken, as it misconstrues both ancient science and Galileo’s role in the scientific revolution.
The mathematicians’ view of Galileo
What did 17th-century mathematicians such as Newton and Huygens think of Galileo? Not very highly, it turns out. I summarise my case against Galileo using their perspectives and a mathematical lens more generally.
What did 17th-century mathematicians such as Newton and Huygens think of Galileo? Not very highly, it turns out. I summarise my case against Galileo using their perspectives and a mathematical lens more generally.
Transcript
I’m going to conclude my case against Galileo with this final episode on this subject. Here’s a little anecdote I found that can be used to frame the overall point that I have made. Galileo was sentenced by the church in 1633. And to go along with this there was a bit of a crackdown on Galileo sympathisers. Somebody in Florence was going to publish a book that made reference to the “most distinguished Galileo.” But the Inquisition intervened and demanded that this phrase should be changed. Instead of “most distinguished Galileo,” the phrase should be changed to: “Galileo, man of noted name.” I am not generally on the side of the Inquisition, but I have come to the conclusion that this particular decree is sound. Instead of “Galileo, father of modern science,” we would be better off saying “Galileo, man of noted name.”
That’s what I have argued before. Today I will offer a bit of a roundup with some new perspectives on these issues. And that will be the end of my 18-episode rant against Galileo.
My main claim has been that Galileo was a poor mathematician. Historians are still blind to this fact. People still speak of “Galileo’s mathematical genius.” That persistent myth must certainly die. John Heilbron, the UC Berkeley historian of science, published an authoritative biography of Galileo in 2010. There Galileo is called “the greatest mathematician in Italy, and perhaps the world” in his time.
Galileo was no such thing. In reality, tell-tale signs of mathematical mediocrity permeate all his works. Many pages of Galileo would not be out of place somewhere in the middle of the piles of slipshod student homework that some of us grade for a living.
A number of Galileo’s numerous mathematical errors even concern some of his core achievements. I have discussed all of his notable scientific contributions and found much to object in every single case. For instance, Galileo uses “his” law of fall erroneously on a number of occasions: when he tries to explain the orbital speeds of the planets, when he tries to calculate how long it would take for the moon to fall to the earth, when erroneous claims to have proved that centrifugal whirling could never throw objects off the earth regardless of speed, and when he erroneously describes the path of a falling object in a reference frame not rotating with the earth. Galileo is praised for having discovered the law of fall, but the fact is that he derived as many false conclusions from it as correct ones.
He also not infrequently presents arguments that are demonstrably inconsistent with his core beliefs, such as his tidal theory contradicting his own principle of relativity, his Joshua argument contradicting his own principle of inertia, and his objection to the geocentric explanation of sunspots being inconsistent with his own heliocentrism.
I should say that, of course, other people made mistakes too. It was the early days of science after all. Suppose I concede that everyone has an equal comedy of errors to their name. Even so, this would still prove my point that Galileo was a dime a dozen scientist and not at all a singular “father of modern science.” But I do not in fact need to concede this much. Galileo’s sum of errors are not just par for the course. They are exceptionally poor, and in matters of mathematics especially they are astonishing.
We have seen time and time again that virtually all of “Galileo’s” achievements were either anticipated or at least made independently by others. To name just the most striking case:
“Let us hypothetically assume that a scholar contemporary to Galileo pursued experiments with falling bodies and discovered the law of fall as well as the parabolic shape of the projectile trajectory, that he found the law of the inclined plane, directed the newly invented telescope to the heavens and discovered the mountains on the moon, observed the moons of the planet Jupiter and the sunspots, that he calculated the orbits of heavenly bodies using methods and data of Kepler with whom he corresponded, and that he composed extensive notes dealing with all these issues. In short, let us assume that this man made essentially the same discoveries as Galileo and did his research in precisely the same way with only one qualification: he never in his life published a single line of it. As a matter of fact, the above description refers to a real person, Thomas Harriot.”
Actually these discoveries are not identical with those of Galileo but rather go beyond them, because Galileo never “calculated the orbits of heavenly bodies using methods and data of Kepler,” as Harriot did, who was a better mathematician.
So the history of science would have been much the same without Galileo, because people like Harriot and others were doing all of that stuff independently anyway.
It’s instructive to compare Galileo to Kepler in these kinds of terms. We can find independent contemporary discoveries for almost everything Galileo did, but not so for Kepler’s achievements, even though many of them are still central in modern science. Harriot was a “second Galileo” and you could go on to a third or a fourth stand-in without much loss. It would be much harder to find a “second Kepler.”
In my view it is not hard to see why: Kepler was an excellent mathematician who worked on difficult things, while Galileo didn’t know much mathematics and therefore focussed on much easier tasks. The standard story has it that Galileo’s insights were more “conceptual,” yet at least as deep as technical mathematics. On this account it is imagined that basic conceptions of science that we consider commonsensical today were once far from obvious: we greatly underestimate the magnitude of the conceptual breakthroughs required for these developments because we are biased our modern education and anachronistic perspective.
But if this is true, how come that Galileo’s ideas—for all their alleged “conceptual” avant-gardism—spontaneously sprung up like mushrooms all over Europe? And how come all of those ideas can easily be explained to any high school student today, if they are supposedly so profound and advanced? The same cannot be said for Kepler’s ideas. They were neither simultaneously developed by dozens of scientists, nor can they be taught to a modern student without years of specialised training. Perhaps this contrast between Galileo and Kepler says something about what genuine depth in the mathematical sciences looks like.
In my opinion, mathematicians at the time realised this perfectly well. I have already spoken before about the very harsh words that Descartes and sometimes Kepler had for Galileo. “He is eloquent to refute Aristotle but that is not hard,” as Descartes said. There are a number of quotes like that from mathematicians. And of course they spotted numerous mathematical blunders in Galileo, which they condemned.
Let’s look at what some other competent mathematicians thought of Galileo.
Christiaan Huygens was perhaps the greatest physicist and mathematician of the generation between Galileo and Newton. He is often portrayed as continuing the scientific program of Galileo. Huygens’s collected works is 22 thick volumes. Go ahead and try to find any strong praise of Galileo in there, let alone anything remotely like calling him a “father of science.” Somehow Huygens never got around to saying any such thing, in these tens of thousands of pages on physics and mathematics and astronomy that we wrote. Hmm, what a mystery.
The closest Huygens ever gets to mentioning Galileo favourably is in the context of a critique of Cartesianism. In the late 17th century, the teachings of Descartes had attracted a strong following. In the eyes of many mathematicians, the way Cartesianism had become an entrenched belief system was uncomfortably similar to how Aristotelianism had been an all too dominant dogma a century before. Huygens makes this parallel explicit:
“Descartes had a great desire to be regarded as the author of a new philosophy [and] it appears that he wished to have it taught in the academies in place of Aristotle. [Descartes] should have proposed his system of physics as an essay on what can be said with probability. That would have been admirable. But in wishing to be thought to have found the truth, he has done something which is a great detriment to the progress of philosophy. For those who believe him and who have become his disciples imagine themselves to possess an understanding of the causes of everything that it is possible to know; in this way, they often lose time in supporting the teaching of their master and not studying enough to fathom the true reasons of this great number of phenomena of which Descartes has only spread idle fancies.”
It is in direct contrast with this that Huygens slips in a few kind words for Galileo: “[Galileo] had neither the audacity nor the vanity to wish to be the head of a sect. He was modest and loved the truth too much.” Historians have observed that Huygens in all likelihood quite consciously intended this passage to apply to himself as much as to Galileo. Perhaps this is why Huygens is surely too generous in praising Galileo’s alleged “modesty.” Galileo was anything but modest, of course.
In any case, it is very interesting to see what Huygens says about Galileo’s actual science in this passage. Let us read it, and keep in mind that this is as close as Huygens ever gets to praising Galileo, and that the context of the passage—a scathing condemnation of Cartesianism—gives Huygens a notable incentive to put Galileo’s scientific achievements in the most positive terms for the sake of contrast.
In light of this, Huygens’s ostensible praise for Galileo is most remarkable, I think, for how qualified and restrained it is. Huygens’s praise begins like this:
“Galileo had, in spirit and awareness of mathematics, all that is needed to make progress in physics …”
Interesting phrasing. Huygens seems to be saying: Galileo said all the right things about about mathematics and scientific method, but he didn’t actually carry through on it. Given Galileo’s rhetoric, he ought to have been able to do it, but be didn’t.
Interestingly, Huygens does not say that Galileo had great mathematical ability or demonstrable achievements, only that he was “aware” that mathematics is necessary for physics. In this respect, Galileo “had all that is needed to make progress in physics,” Huygens says. Why not simply say that Galileo made great progress in physics, instead of this convoluted and qualified “he had what was needed to do so”? So really Huygens’s ostensible praise for Galileo is actually quite backhanded. At least that’s how it seems to me.
Let’s continue reading because the Huygens quote goes on. Here is the rest of the sentence:
“… and one has to admit that he was the first to make very beautiful discoveries concerning the nature of motion …”
Galileo wasn’t the first, as we now know. Huygens didn’t know about the unpublished work of Harriot etc., so he is overly generous in that regard. But never mind that. Huygens’s formulation is still very restrained in an interesting way: did you notice that strange phrase “one has to admit”? “One has to admit” that Galileo was the first to make certain discoveries. Who speaks of their greatest hero in such terms? One “has to admit” that he made some discoveries? That seems more like the kind of phrasing you use to describe the work of someone who is overrated, not someone you esteem as the founder of science.
Huygens wrote in French. The phrase is “il faut avouer.” I’m not a linguist but I think the translation I gave is the most natural one. “Il faut avouer”: “one has to admit”; it suggests a reluctance to concede the point. I’m not sure if it’s possible to argue that taken in context it could also be construed as “even a Cartesian would have to admit” or something like that. If you’re an expert of 17th-century French I would like to hear your opinion about this.
Let’s see, the Huygens quote continues even further and here’s how it concludes. After this remark about Galileo having made discoveries concerning motion, Huygens adds:
“ … although he left very considerable things to be done.”
Well, yes. That’s my point exactly. What is most striking and remarkable about the work of Galileo is not the few discoveries he “admittedly” made, but how very little he actually accomplished despite all his posturing about mathematics and scientific method. It seems to me that Huygens and I agree on this. Even in his most pro-Galilean sentence in all his works, Huygens is undermining Galileo as much as he is praising him.
What about Isaac Newton? What did he think of Galileo?
Newton famously said that “if I have seen further it is by standing on the shoulders of giants.” Many have erroneously assumed that Galileo was one of these “giants.” One scholar even proposes to explain that “when Newton credits Galileo with being one of the giants on whose shoulders he stood, he means …” blah blah blah. We do not need to listen to what this philosopher thinks Newton meant, because the first part of the sentence is false already. The assumption that Galileo was one of the scientific giants in question has no basis in fact.
The closest Newton gets to praising Galileo is in the Principia, his most important work. After introducing his laws of motion, Newton adds some notes on their history.
“The principles I have set forth are accepted by mathematicians and confirmed by experiments of many kinds. By means of the first two laws and the first two corollaries Galileo found that the descent of heavy bodies is in the squared ratio of the time and that the motion of projectiles occurs in a parabola.”
The laws and corollaries in question are: the law of inertia, which Galileo did not know, as we have seen; then Newton’s second law, the force law F=ma, which Galileo also did not know; and the composition of forces and motions, which was established in antiquity.
Note that Newton doesn’t say that Galileo was the discoverer of these laws. All Newton says is that Galileo used these laws to find the path of projectiles. Indeed, as one historian has pointed out, “Newton’s Latin contains some ambiguity” for it “can have two very different meanings: that the two laws were completely accepted by Galileo before he found that projectiles follow a parabolic path, or that these two laws were already generally accepted by scientists at the time that Galileo made his discovery of the parabolic path.”
Either way, Newton is wrong. Of course, once you are looking at the world though Newtonian mechanics it is natural to think that surely Galileo must have had these laws, because that is so obviously the right way to think about parabolic motion. Therefore, as Dijksterhuis says, “[according to] the myth in which he appears as the founder of classical dynamics, [Galileo] must surely have known the proportionality of force and acceleration. But to those who have become acquainted with Galileo through his own works, not at second hand, there can be no doubt that he never possessed this insight.”
Quite so, and indeed Newton was not acquainted with Galileo’s work directly. As I.B. Cohen says, “Newton almost certainly did not read [Galileo’s] Discorsi until some considerable time after he had published the Principia,” if ever. On the other hand “early in his scientific career, [Newton] had read [Galileo’s] Dialogo”—but that is of course his work on Copernicanism, not his work on mechanics and the laws of motion, which is what Newton is referencing in the Principia.
“Hence Newton (rather too generously, for once!) allowed to Galileo the discovery of the first two laws of motion.” And the reason for Newton’s excessive charity is not hard to divine. To quote I.B. Cohen again, Newton’s Principia is marked by an obvious and vehement “anti-Cartesian bias.” “Because of his strongly anti-Cartesian position, Newton might have preferred to think of Galileo rather than Descartes as the originator of the First Law.” Whereas, “in point of fact, the Prima Lex [that is, the law of inertia] of Newton’s Principia was derived directly from the Prima Lex of Descartes’s Principia”—that is, the correct law of inertia. Descartes stated the correct law of inertia with crystal clarity in this published key work, while Galileo never stated it anywhere, nor believed it.
Clearly, then, Newton’s attribution of these laws to Galileo means next to nothing. Galileo demonstrably did not know these laws; Newton hadn’t read Galileo anyway; and Newton had an obvious bias and incentive to overstate Galileo’s importance in order to belittle the influence of Descartes which he did not want to admit.
Newton’s words aren’t high praise in any case. In fact, that becomes ever clearer if we read on in Newton’s text. For when Newton continues his historical discussion he says on the very same page:
“Sir Christopher Wren, Dr. John Wallis, and Mr. Christiaan Huygens, easily the foremost geometers of the previous generation, independently found the rules of the collisions and reflections of hard bodies.”
So evidently Newton was in the mood when writing this to point out who “the foremost geometers” of the past were. Yet on the very same page he had no such words for Galileo. A telling omission. Altogether there is no evidence that Newton regarded Galileo particularly highly, let alone considered him anywhere near a “father of modern science.”
The time has come to wrap up my Galileo story. Perhaps I can sum it up like this.
Say you go to the library and find the shelves with philosophical texts ordered chronologically. You pick the books up one by one and see what they have to say about science. Century after century you find the same thing. Aristotle, Aristotle, Aristotle. Then commentaries on Aristotle. Then commentaries on commentaries on Aristotle. Then people who ostensibly try to think more independently, yet cling desperately to Aristotelian concepts and terminology as if their life depended on it, even when they try to challenge isolated claims of Aristotelian dogma. Then, suddenly: Galileo. What a breath of fresh air this is. The Aristotelian shackles are emphatically discarded, and all the nowadays familiar principles of modern science are articulated in lucid and entertaining prose. At once after him everyone is a scientist. The Aristotelianism that ran rampant for centuries had suddenly stopped dead in its tracks. How can one not admire this singular father of the scientific worldview, this pivotal hero who divides the entire history of thought into two disjoint worlds separated by such an abyss?
Alas, you made one mistake. You went to the philosophy shelves. You should have gone to subbasement 3, where the mathematics books are kept. This may not have been an obvious choice. Perhaps you were educated in the humanities and therefore naturally drawn to the sprawling and well-attended shelves in your part of the library. The out-of-the-way mathematics section never caught anyone’s eye. Isn’t it just for nerds in training who need to double-check their formulas? Apparently there are a few books there from Greek times, but blink and you miss them between thick modern textbooks on algebraic topology and partial differential equations. And if you do open one of those old math books, it’s full of technical diagrams and equations anyway. Who would ever think to look there for man’s view of the world? Surely that’s what philosophy is for?
In reality, we sent Galileo to your shelves because he wasn’t good enough for ours. Galileo wasn’t the first to do anything except explain what mathematicians had always known in such basic terms that even philosophers could understand. Galileo once wrote to a fellow philosopher:
“If philosophy is that which is contained in Aristotle’s books, you would be the best philosopher in the world. But the book of [natural] philosophy [or science] is that which is perpetually open to our eyes. But being written in characters different from those of our alphabet, it cannot be read by everyone; the characters of this book are triangles, squares, circles, spheres, cones, pyramids and other mathematical figures, the most suited for this sort of reading.”
That is Galileo’s advice to the philosophers of his day. I say much the same thing to modern scholars. If the history of science is that which is contained in philosophical books, you would be the best historians in the world. But the real truth is perpetually open to our eyes, if only we take the trouble to read mathematics.
Our picture of Greek antiquity is distorted. Only a fraction of the masterpieces of antiquity have survived. Decisions on what to preserve were made by in ages of vastly inferior intellectual levels. Aristotelian philosophy is more accessible for mediocre minds than advanced mathematics and science. Hence this simpler part of Greek intellectual achievement was eagerly pursued, while technical works were neglected and perished. The alleged predominance of an Aristotelian worldview in antiquity is an illusion created by this distortion of sources. The “continuity thesis” that paints 17th-century science as building on medieval thought is doubly mistaken, as it misconstrues both ancient science and Galileo’s role in the scientific revolution.
Transcript
To praise Galileo is to criticise the Greeks. The contrast class of “Aristotelian” science is constantly invoked to explain Galileo’s alleged greatness, both in Galileo’s own works and in modern scholarship. But this narrative gets it all wrong, in my opinion. It is based on a caricature of Greek science that effectively ignores the Greek mathematical tradition.
Francis Bacon put it well: when “human learning suffered shipwreck” with the death of the classical world, “the systems of Aristotle and Plato, like planks of lighter and less solid material, floated on the waves of time and were preserved,” while treasure troves of much more mathematically advanced works were lost forever.
Aristotelian science is not the pinnacle of Greek scientific thought. Far from it. It is not the best part of Greek science, but the part of Greek science that was most accessible and appealing to the generations of mathematically ignorant people who populated the universities in medieval Europe for hundreds of years. And perhaps some generations who still do.
Mathematicians have always felt differently. “So many great findings of the Ancients lie with the roaches and worms,” said Fermat. They are lost, in other words, these mathematical masterpieces that once existed. That’s how Fermat put it, and all his mathematical colleagues agreed. And they were right.
In the 20th century a few such masterpieces were recovered. So these 17th-century mathematicians were proven right in their intuition that great works were forgotten and hidden away among “roaches and worms” indeed.
In 1906, a work of Archimedes that had been lost since antiquity was rediscovered in a dusty Constantinople library. The valuable parchment on which it was written had been scrubbed and reused for some religious text. But the original could still just about be made out underneath it. As one historian put it: “Our admiration of the genius of the greatest mathematician of antiquity must surely be increased, if that were possible,” by this “astounding” work, which draws creative inspiration from the mechanical law of the lever to solve advanced geometrical problems. If even this brilliant work by antiquity’s greatest geometer only survived by the skin of its teeth and dumb luck, just imagine how many more works are lost forever.
Also in the 20th century, divers chanced upon an ancient shipwreck, which turned out to contain a complex machine (the so-called Antikythera mechanism). Again historians were astonished: “From all we know of science and technology in the Hellenistic age we should have felt that such a device could not exist.” “This singular artifact is now identified as an astronomical or calendrical calculating device involving a very sophisticated arrangement of more than thirty gear-wheels. It transcends all that we had previously known from textual and literary sources and may involve a completely new appraisal of the scientific technology of the Hellenistic period.”
Another example. The Greeks appear to have been much further ahead than conventional sources would lead one to believe in a number of mathematical fields. One example is combinatorics. Of this entire mathematical field little more survives than one stray remark mentioned parenthetically in a non-mathematical work by Plutarch:
“Chrysippus said that the number of intertwinings obtainable from ten simple statements is over one million. Hipparchus contradicted him, showing that affirmatively there are 103,049 intertwinings.”
“This passage stumped commentators until 1994,” when a mathematician realised that it corresponds to the correct solution of a complex combinatorial problem worked out in modern Europe in 1870, thereby forcing “a reevaluation of our notions of what was known about combinatorics in Antiquity.” It is undeniable from this evidence that this entire field of mathematics must have reached an advanced stage, yet not one single treatise on it survives.
These are just a few striking examples illustrating an indisputable point: the Hellenistic age was extremely sophisticated mathematically and scientifically, and we don’t even know the half of it.
Scores of key treatises are lost, and we are forced to rely on later commentators and compilers for accounts of the works of Hellenistic authors. It’s like trying to understand modern science and mathematics from popularisations in the Sunday newspaper. It’s vastly oversimplified and dumbed-down. It reduces complex science to one or two simplistic ideas while conveying nothing whatsoever of the often massive technical groundwork that it is based on. That’s the state of our sources for much Greek science: all that has come down to use are some clickbait headlines and blurbs by people who are themselves not scientists and wouldn’t understand the first thing about the technical details of the works they are trying to summarise.
Actually this is a misleading analogy. The situation is even worse than this. Here is how one historian puts it:
“Nearly all that we know on observations and experiments among the Greeks comes from compilations and manuals composed centuries later, by men who were not themselves interested in science, and for readers who were even less so. Even worse, these works were to a great extent inspired by the desire to discredit science by emphasizing the way in which men of science contradicted each other, and the paradoxical character of the conclusions at which they arrived. This being the object, it was obviously useless, and even out of place, to say much about the methods employed in arriving at the conclusions. It suited Epicurean and Sceptic, as also Christian, writers to represent them as arbitrary dogmas. We can get a slight idea of the situation by imagining, some centuries hence, contemporary science as represented by elementary manuals, second- and third-hand compilations, drawn up in a spirit hostile to science and scientific methods. Such being the nature of the evidence with which we have to deal, it is obvious that all the actual examples of the use of sound scientific methods that we can discover will carry much more weight than would otherwise be the case. If we can point to indubitable examples of the use of experiment and observation, we are justified in supposing that there were others of which we know nothing because they did not happen to interest the compilers on whom we are dependent. As a matter of fact, there are a fair number of such examples.”
In previous episodes we have discussed the many ways in which Greek sources already showed full awareness of many things often attributed to Galileo. Taking this context of filtering and lost sources into account means that we should give all the more weight to those arguments.
Sadly, however, the lack of appreciation for science among these ignorant commentators continues among scholars today. I collected some quotes on this by some very respectable classicists of today.
“The state of editions and translations of ancient scientific works as a whole remains scandalous by comparison with the torrent of modern works on anything unscientific — about 100 papers per year on Homer, for example. An embarrassingly large number of classicists are ignorant of Greek scientific works.”
“Classicists include many who have chosen Latin and Greek precisely to escape from science at the very early stage of specialisation that our schools’ curricula permit: and often a very successful escape it is, to judge from the depth of ignorance of science ancient and modern that it often secures.”
It is remarkable how strongly these authors make this point. The first quote is from Lloyd, the Cambridge professor. It takes a lot for people like that to almost condemn their colleagues to their face. They wouldn’t do this if it wasn’t serious.
Little wonder then that Greek science is systematically misunderstood and undervalued, and that simplistic ideas of philosophical authors and commentators are substituted for the real thing.
Galileo’s relation to the preceding philosophical tradition has been systematically misunderstood because of this.
How did modern science grow out of mathematical and philosophical tradition? The humanistic perspective is that science needed both: it was born through the unification of the technical but insular know-how of the mathematicians with the conceptual depth and holistic vision of the philosophers. The mathematical perspective is that science is what the mathematicians were doing all along. Science did not need philosophy to be its eye-opener and better half; it merely needed the philosophers to step out of the way and let the mathematicians do their thing. So which is it?
Many historians have tried to stress commonalities between Galileo and the Aristotelian philosophers who preceded him. That is to say, they argue for the “continuity thesis” which says that the so-called “Scientific Revolution” was not a radical or revolutionary break with previous thought. Here is what they say:
“Galileo essentially pursued a progressive Aristotelianism [during the first half of his life—the period of] positive growth that laid the foundation for the new sciences.”
“A particular school of Renaissance Aristotelians, located at the University of Padua, constructed a very sophisticated methodology for experimental science; … Galileo knew this school of thought and built upon its results; this goes a long way toward explaining the birth of early modern science.”
“The mechanical and physical science of which the present day is so proud comes to us through an uninterrupted sequence of almost imperceptible refinements from the doctrines professed within the Schools of the Middle Ages.”
“Galileo was clearly the heir of the medieval kinematicists.”
I agree with these authors that “those great truths for which Galileo received credit” are not his. But the notion that they were first conceived in Aristotelian schools of philosophy is wrongheaded.
The argument of these historians is based on a simple logic. First they show that various concepts of “Galilean” science are prefigured in earlier sources. Then they want to infer from this that these sources marked the true beginning of the scientific revolution. But in order to draw this inference they need two assumptions: first, that Galileo was the father of modern science; and second, that the Greeks were nowhere near the same accomplishments. These two assumptions are simply taken for granted by these authors, as a matter of common knowledge. But in reality both assumptions are dead wrong, and therefore the inference to the significance of the Aristotelian sources is unwarranted.
It is interesting that the continuity thesis on the one hand devalues the contributions of Galileo, yet at the same time desperately needs to reassert the traditional view that “Galileo has a clear and undisputed title as the ‘father of modern science’,” as one of these historians puts it. They need to say this because this is what gives them the one point of connection they are able to establish between medieval and modern science. The entire argument stands and falls with this false premiss. Therefore, if one proves, as I have done before, that Galileo was a mediocre scientists of negligible importance to the mathematically competent people who actually achieved the scientific revolution, then the continuity thesis collapses like a house of cards.
The defenders of the continuity thesis are equally ineffectual in establishing the second false premiss of their argument, namely the alleged absence of these “new” ideas in Greek thought. In fact, even continuity thesis advocates make no secret of the fact that the medieval tradition was built on “remnants of Alexandrian science.” For example, “although we are left with few monuments from the profound research of the Ancients into the laws of equilibrium, those few are worthy of eternal admiration.” Obviously, “masterpieces of Greek science [such as the works of] Pappus, and especially Archimedes, are proof that the deductive method can be applied with as much rigor to the field of mechanics as to the demonstrations of geometry.” All of that are quotes form Pierre Duhem, a passionate advocate of the continuity thesis.
How can people like Duhem acknowledge these “masterpieces” “worthy of eternal admiration” from antiquity, yet at the same time attribute the scientific revolution to medieval or renaissance philosophers? Here’s how. By writing off those ancient works as minor technical footnotes to an otherwise thoroughly Aristotelian paradigm. Only if this picture is accepted can any kind of greatness be ascribed to the pre-Galileans, as is evident from passages such as these:
“Some philosophers in medieval universities were teaching ideas about motion and mechanics that were totally non-Aristotelian [and] were consciously based on criticisms of Aristotle’s own pronouncements.”
“Admittedly, most of these significant medieval mechanical doctrines were formed within the Aristotelian framework of mechanics. But these medieval doctrines contained within them the seeds of a critical refutation of that mechanics.”
“The medieval mechanics occupied an important middle position between Aristotelian and Newtonian mechanics. [Hence it was] an important link in man’s efforts to represent the laws that concern bodies at rest and in movement.”
“The impressive set of departures from Aristotelianism achieved by medieval science nevertheless failed to produce genuine efforts to reconstruct, or replace, the Aristotelian world picture.”
If Aristotle is taken as the baseline, this looks quite impressive indeed. But why should Aristotle be accepted as the default opinion? Aristotle was one particular philosopher who was a nobody in mathematics and lived well before the golden age of Greek science. Medieval and renaissance thinkers indeed mustered up the courage to challenge isolated claims of his teachings almost two thousand years later, while mostly retaining his overall outlook. This does not constitute great open-mindedness and progress. Rather it is a sign of small-mindedness that these people paid so much attention to Aristotle at all in the first place. In my view, it is not so much impressive that they deviated a bit from Aristotle as it is deplorable that they framed so much of what they did relative to Aristotle, even when they disagreed with him. This is very different from post-Aristotelian thought in Greek times, where there is no evidence that any mathematician paid any attention to Aristotle’s mechanics.
In any case, “extravagant claims for the modernity of medieval concepts” suffer from “serious defects.” One historian has summarised it well:
“There was no such thing as a fourteenth-century science of mechanics in the sense of a general theory of local motion applicable throughout nature, and based on a few unified principles. By searching the literature of late medieval physics for just those ideas and those pieces of quantitative analysis that turned out, three centuries later, to be important in seventeenth-century mechanics, one can find them; and one can construct a “medieval science of mechanics” that appears to form a coherent whole and to be built on new foundations replacing those of Aristotle’s physics. But this is an illusion, and an anachronistic fiction, which we are able to construct only because Galileo and Newton gave us the pattern by which to select the right pieces and put them together.”
The main piece of such precursorism is the so-called “mean speed theorem.” This is a completely trivial result. You can visualise it in terms of a graph with time on the x-axis and velocity on the y-axis. Suppose you plot the graph of a uniformly accelerated motion, such as a freely falling object. It makes a straight line going from the bottom left to the to right. It starts from no velocity and goes to a certain final velocity. How far did the thing travel? Distance travelled is the area under the graph. So it’s the area of a triangle. Base times height over 2. That is to say, the time of fall, times half the final velocity. Or another way of putting it is that half the final velocity is the same thing as the average velocity. The triangle has the same area as a rectangle with the same base and half the height. The “mean speed theorem” is just this. In terms of distance covered, a uniformly accelerated motion is equivalent to a constant-speed motion with the same average speed. A very simple thing to see.
Some people praise this as an “impressive” achievement of the middle ages—”probably the most outstanding single medieval contribution to the history of physics,” derived by “admirable and ingenious” reasoning, according to one historian. Even though these medieval authors did absolutely nothing with this trivial theorem and only deduced it to illustrate the notion of uniform change abstractly within Aristotelian philosophy. Later the theorem became central in “Galilean” mechanics since free fall is uniformly accelerated. But it “was, in fact, never applied to motion in fall from rest during the 14th, or even in the 15th century” (only in the mid-16th century there is a passing remark to this effect within the Aristotelian tradition, “without any accompanying evidence”).
Let us not radically inflate our esteem for the Middle Ages by anachronistically praising them for pointing out a trivial thing that centuries later took on a significance of which they had no inkling. Let us instead recognise the theorem for the trifle that it is. Then we shall also not have any need to be surprised when it turns out that Babylonian astronomers assumed it without fanfare thousands of years earlier still. The utterly trivial “mean speed theorem” was implicitly taken for granted in Babylonian astronomy. They were too good mathematicians to make a big fuss about something so evident, unlike the medieval philosophers who sat around a proved this at length. They were so bad at mathematics that this trivial thing was the cutting edge to them, in their ignorance.
Galileo owes other debts to previous philosophical tradition as well, according to many historians. For example, we are told that there are “unmistakeable Jesuit influences in Galileo’s work”: “Above all Galileo was intent in following out Clavius’s program of applying mathematics to the study of nature and to generating a mathematical physics.” That’s a quote from Wallace. The preposterous notion that this was “Clavius’s” program can only enter one’s mind if one only reads philosophy. It was obviously Archimedes’s program, except, unlike Clavius, he proved his point by actually carrying it out instead of sermonising about what one ought to do in philosophical prose. Philosophers (ancient and modern alike) have a tendency to place disproportionate value on explaining something conceptually as opposed to actually doing it. After all, that is virtually the definition of philosophy. Hence they praise certain Aristotelians for explaining some supposedly profound principles of scientific method even when “it is quite clear that [none of them] ever applied his advocated methods to actual scientific problems.”
Descartes—a mathematically creative person—knew better: “we ought not to believe an alchemist who boasts he has the technique of making gold, unless he is extremely wealthy; and by the same token we should not believe the learned writer who promises new sciences, unless he demonstrates that he has discovered many things that have been unknown up till now.” Unfortunately, such basic common sense is often lacking among historians and philosophers assigning credit for basic principles of the scientific method.
There is a contradiction in the way modern historians try to trace many aspects of the scientific revolution to roots in the middle ages. On the one hand these historians like to claim that the traditional view of the scientific revolution is ahistorical and based on an anachronistic mindset, whereas their own account that sees continuity with the middle ages is more sensitive to how people actually thought at the time itself. Ironically, however, their view, which is supposed to be more true to the historical actors’ way of thinking, is actually all the more blatantly at odds with how virtually all leaders of the scientific revolution thought of the middle ages. One historian summarises it accurately: “The scientific achievement of the Middle Ages was held in unanimous contempt from Galileo’s time onward by those who adhered to the new science. Leibniz’ scathing verdict ‘barbaric physics’ neatly encapsulates the reigning sentiment.” This was not for nothing. Leibniz was an erudite scholar well versed in the philosophy of the schools. But he was also an excellent mathematician. The latter enabled him to pass a sound judgement on the “barbaric” science of the middle ages.
What was Galileo’s great innovation in science? To give practical experience more authority than philosophical systems? To insist on mechanical as opposed to teleological or supernatural explanations of natural phenomena? To take mathematical physics as our best window into the fundamental nature of reality as opposed to just a computational tool for a small set of technical problems? No, none of the above. All of these things had been old hat for thousands of years.
Transcript
To say that Galileo is “the father of modern science” is to say that he made some kind of unique contribution, something unprecedented, that was the starting point of science as we know it. So what would that have been? What was that uniquely Galilean ingredient, that made science appear out of thin air for the first time in human history? We spoke about this before. People have tried to pinpoint it in various ways. I refuted the main attempts: mathematisation of nature, empiricism, experimental method. Basically, those things were all commonplace already in Greek times. That’s what I argued last time. But the list goes on. There are other things that Galileo was allegedly “the first” to do. Let’s have a look at those.
Here’s one: Galileo’s greatness consists in bringing together abstract mathematics and science with concrete technology and practical know-how of craftsmen and workers in mechanical fields. Here are some quotes from various historians expressing this idea:
“Real science is born when, with the progress of technology, the experimental method of the craftsmen overcomes the prejudice against manual work and is adopted by rationally trained university-scholars. This is accomplished with Galileo.”
“[Galileo was able] to bring together two once separate worlds that from his time on were destined to remain forever closely linked—the world of scientific research and that of technology.”
“Galileo may fruitfully be seen as the culmination point of a tradition in Archimedean thought which, by itself, had run into a dead end. What enabled Galileo to overcome its limitations seems easily explicable upon considering Galileo’s background in the arts and crafts.”
“The separation between theory and practice, imposed by university professors of natural philosophy, was repeatedly exposed as untenable. Of course the greatest figure in this movement is Galileo.”
So those are four historians all saying basically the same thing.
And Galileo himself eagerly cultivated this image. The very first words of his big book on mechanics are devoted to extolling the importance for science of observing “every sort of instrument and machine” in action at the “famous arsenal” of Venice. He praises the experiential knowledge of the “truly expert” workmen there. Galileo loves these workers and craftsmen in inverse proportion to how much he hates philosophers.
It is true that universities were filled with many blockheads who foolishly insisted on keeping intellectual work aloof from such connections to the real world. For example, when Wallis went to Oxford in 1632 there was no one at the university who could teach him mathematics. As he says in his autobiography: “For Mathematicks, (at that time, with us) were scarce looked upon as Accademical studies, but rather Mechanical; as the business of Traders, Merchants, Seamen, Carpenters, Surveyors of Lands, or the like.”
That was indeed a lamentable state of affairs. But it would be mistake to infer from this that Galileo’s step was an innovation. The stupidity of the university professors was the doing of one particular clique of mathematically ignorant people. Their attitude is not natural or representative of the state of human knowledge. Galileo is not a brilliant maverick thinking outside the box. Rather, he is merely doing what had, among mathematically competent people, been recognised as the natural and obviously right way to do science for thousands of years. Galileo is not taking a qualitative leap beyond limitations that had crippled all previous thinkers. Rather, he is merely reversing the obvious cardinal error of one particularly dumb philosophical movement that had happened to gain too much influence at the time, because people were too ignorant to recognise the evident superiority of more mathematical and scientific schools of thought that had already proven their worth in a large body of ancient works available to anyone who cared to read.
In order to defend the misconceived idea of Galileo the trailblazing innovator one must ignore the large body of obvious precedent for his view in antiquity, and project the foolish nonsense of medieval universities onto the Greeks. Indeed, historians have concocted a false narrative to this effect. Here are some typical quotes:
“Greek technology and science were rigidly separated.”
“The Greek hand worker was considered inferior to the brain worker or contemplative thinker. So, despite the fact that the philosophers derived some of their conclusions as to how nature behaved from the work of the craftsmen, they rarely had experience of that work. What is more, they were seldom inclined to improve it, and so were powerless to pry apart its potential treasure of knowledge that was to lead to the scientific revolution in the Renaissance.”
Others have argued that “the fundamental brake upon the further progress of science in antiquity was slave labour [which precluded any] meaningful combination of theory and practice.”
More specialised scholarship knows better. The recent Oxford Handbook of Engineering and Technology in the Classical World is perfectly clear on the matter:
“Many twentieth-century scholars hit upon [snobbish contempt for manual labour] as an ‘explanation’ for a perceived blockage of technological innovation in the Greco-Roman world. The presence of slave labor was felt to be a related, concomitant factor. [But] this now discredited interpretation [should be rejected and we should] put an end to the myth of a ‘technological blockage’ in the classical cultures.”
This is the view of experts on the matter, while the false narrative is promulgated by scholars who focus on Galileo, take it for granted that he is “the Father of Modern Science,” and postulate such nonsense about the Greeks because that’s the only way to craft a narrative that fits with this false assumption.
Promulgators of the nonsense about practice-adverse Greeks have evidently not bothered to read mathematical authors. Pappus, for example, explains clearly that mathematicians enthusiastically embrace practical and manual skills:
“The science of mechanics has many important uses in practical life, and is zealously studied by mathematicians. Mechanics can be divided into a theoretical and a manual part; the theoretical part is composed of geometry, arithmetic, astronomy and physics, the manual of working in metals, architecture, carpentering and painting and anything involving skill with the hands.”
Pappus praises the interaction of geometry with practical fields or “arts” as beneficial to both:
“Geometry is in no way injured, but is capable of giving content to many arts by being associated with them, and, so far from being injured, it is obvious, while itself advancing those arts, appropriately honoured and adorned by them.”
These were no empty words. The Greeks had an extensive tradition of studying “machines,” meaning devices based on components such as the lever, pulley, wheel and axle, winch, wedge, screw, gear wheel, and so on. The primary purpose of these machines was that of “multiplying an effort to exert greater force than can human or animal muscle power alone.” Such machines were “used in construction, water-lifting, mining, the processing of agricultural produce, and warfare.”
The Greeks also undertook advanced engineering projects, such as digging a tunnel of more than a kilometer through a mountain, the planning of which involved quite sophisticated geometry to enable the tunnel to be dug from both ends, with the diggers meeting in the middle. In short, “while it is crucial to distinguish between theoretical mechanics and practitioners’ knowledge, there is substantial evidence of a two-way interaction between them in Antiquity.”
Mathematicians were very much involved with such things. There are many testimonies attributing to Archimedes various accomplishments in engineering, such as moving a ship singlehandedly by means of pulleys, destroying enemy ships using machines, building a screw for lifting water, and so on. Apollonius wrote a very advanced and thorough treatise on conic sections, which is studiously abstract and undoubtedly “art for the sake of art” pure mathematics if there ever was such a thing. Yet the same Apollonius “besides writing on conic sections produced a now lost work on a flute-player driven by compressed air released by valves controlled by the operation of a water wheel.” The title page of the Arabic manuscript that has preserved this work for us reads: “by Apollonius, the carpenter, the geometer.” The cliche of Greek geometry as nothing but abstruse abstractions divorced from reality is a modern fiction. The sources tell a different story. It is not for nothing that one of the most refined mathematicians of antiquity went by the moniker “the carpenter.”
Unfortunately, as Russo has observed in his excellent book, “Renaissance intellectuals were not in a position to understand Hellenistic scientific theories, but, like bright children whose lively curiosity is set astir by a first visit to the library, they found in the manuscripts many captivating topics, especially those that came with illustrations. The most famous intellectual attracted by all these ‘novelties’ was Leonardo da Vinci. Leonardo’s ‘futuristic’ technical drawings … was not a science-fiction voyage into the future so much as a plunge into a distant past. Leonardo’s drawings often show objects that could not have been built in his time because the relevant technology did not exist. This is not due to a special genius for divining the future, but to the mundane fact that behind those drawings there were older drawings from a time when technology was far more advanced.”
The false narrative of the mechanically ignorant, anti-practical Greeks has obscured this fact, and led to an exaggerated evaluation of Renaissance technology, such as instruments for navigation, surveying, drawing, timekeeping, and so on. Here for example is the view of Jim Bennett, a former Director of the Museum of the History of Science in Oxford:
“Renaissance developments in practical mathematics predated the intellectual shifts in natural philosophy. Historians of the early modern reform of natural philosophy have failed to appreciate the significance of the prior success of the practical mathematical programme, [which] must figure in an explanation of why the new dogma of the seventeenth century embraced mathematics, mechanism, experiment and instrumentation.”
Bennett proves at length that the practical mathematical tradition had much to commend it, which I do not dispute. But then he casually asserts with hardly any justification that there was nothing comparable in Greek times. This is typical of much scholarship of this period. The deeply entrenched standard view of the Galilean revolution is basically taken for granted and subsequent work is presented as emendations to it. For instance, if you want to prove the importance of a Renaissance pre-revolution in practical mathematics, you need to prove two things: first that it was relevant to the scientific revolution, and second that it was not present long before. It is a typical pattern to see historians put all their efforts toward proving the first point, and glossing over the second point in sentence or two. They can get away with this since the alleged shortcomings of the Greeks is supposedly common knowledge, while the first point is the one that departs from the standard narrative. Hence, if the standard narrative is misconceived in the first place, so is all this more specialised research, which, although it ostensibly departs from the standard view, actually retains its most fundamental errors in the very framing of its argument.
It is right to emphasise that the practical mathematical tradition stood for a much more fruitful and progressive approach to nature than that dominant among the philosophy professors of the time. But it is a mistake to believe that these professors represented the considered opinion of the best minds, while the mathematical practitioners were oddball underdogs whose pioneering success eventually proved undeniable to the surprise of everyone. The mathematical practitioners stood for simple common sense, not renegade iconoclasm. They practiced the same common sense that their peers had in antiquity, with much the same results. The university professors, meanwhile, should not be mistaken for a neutral representation of the state of human knowledge at the time. Rather, they formed one particular philosophical sect which retained its domination of the universities not because of the preeminence of its teachings but because of the conservative appointment practices and obsequiousness of academics.
So that’s my take on the role of practical mathematics in the scientific revolution.
Now let’s turn to another issue, a more philosophical one: instrumentalism versus realism.
A standard view is that “the Scientific Revolution saw the replacement of a predominantly instrumentalist attitude to mathematical analysis with a more realist outlook.” Instrumentalism means the following; I’m quoting Simplicius the ancient commentator:
“An explanation which conforms to the facts does not imply that the hypotheses are real and exist. [Astronomers] have been unable to establish in what sense, exactly, the consequences entailed by these arrangements are merely fictive and not real at all. So they are satisfied to assert that it is possible, by means of circular and uniform movements, always in the same direction, to save the apparent movements of the wandering stars.”
Instrumentalism, as opposed to realism, was supposedly the accepted philosophy of science among “the Greeks,” according to many historians. Here’s what Pierre Duhem had to say about it for example:
“[Ancient Greek astronomers] balked at the idea that the eccentrics and epicycles are bodies, really up there on the vaults of the heavens. For the Greeks they were simply geometrical fictions requisite to the subjection of celestial phenomena to calculation. If these calculations are in accord with the results of observation, if the ‘hypotheses’ succeed in ‘saving the phenomena’, the astronomer’s problem is solved.”
“An astronomer who understands the true purpose of science, as defined by men like Posidonius, Ptolemy, Proclus, and Simplicius, … would not require the hypotheses supporting his system to be true, that is, in conformity with things. For him it will be enough if the results of calculation agree with the results of observation—if appearances are saved.”
That’s Duhem, in the early 20th century. But plenty of modern historians agree as well. Here are some examples, I quote:
“The Greek geometer in formulating his astronomical theories does not make any statements about physical nature at all. His theories are purely geometrical fictions. That means that to save the appearances became a purely mathematical task, it was an exercise in geometry, no more, but, of course, also no less.”
Galileo, by contrast, brought “a radically new mode of realist-mathematical nature knowledge.”
In other words:
“Galileo endorsed a view that was [contrary to] that of the Greeks but was also much more creative … It is a crippling restriction to hold that no theory about reality can be in mathematical form; the Renaissance rejected this restriction, holding that it was a worthwhile enterprise to search for mathematical theories which also—by metaphysical criteria—could be supposed ‘real’. … The most eloquent and full defence of this process was given by Galileo.”
Hence the Scientific Revolution owes much to “the novel quality of realism that the abstract-mathematical mode of nature-knowledge acquired in Galileo’s hands.”
All of that are quotations from mainstream historical scholarship. I of course disagree with them, as you might imagine.
In reality, no mathematically competent Greek author ever advocated instrumentalism. The notion that “the Greeks” were instrumentalists relies exclusively on passages by philosophical commentators. The notion that Ptolemy believed his planetary models were “fictional combinations of circles which could never exist in celestial reality” is demonstrably false.
First of all Ptolemy opens his big book with physical arguments for why the earth is in the center of the universe. This is a blatantly realist justification for this aspect of his astronomical models.
Furthermore, Ptolemy has a detailed discussion of the order and distances of the planets that obviously assumes that the planetary models, epicycles and all, are physically real. “The distances of the … planets may be determined without difficulty from the nesting of the spheres, where the least distance of a sphere is considered equal to the greatest distance of a sphere below it.” That is to say, according to Ptolemy’s epicyclic planetary models, each planet sways back and forth between a nearest and a furthest distance from the earth. The “sphere” of each planet must be just thick enough to contain these motions. Ptolemy assumes that “there is no space between the greatest and least distances [of adjacent spheres],” which “is most plausible, for it is not conceivable that there be in Nature a vacuum, or any meaningless and useless thing.”
Clearly this is based on taking planetary models to be very real indeed, and not at all mathematical fictions invented for calculation. Nor was Ptolemy an exception in his realism. His colleague Geminos “was a thoughtful realist” too, as the translators of his surviving astronomical work have observed.
Hipparchus too evidently chose models for planetary motion on realist grounds. His works are lost, but we know that he proved the mathematical equivalence of epicyclic and equant motion. In other words, he showed that two different geometrical models of planetary motion are observationally equivalent; they lead to the exact same visual impressions seen from earth, but they are brought about by different mechanisms. How should one choose between the two models in such a case? If Hipparchus was an instrumentalist, he wouldn’t care one way or the other, or he would just pick whichever was more mathematically convenient. But if he was a realist he would be interested in which model could more plausibly correspond to actual physical reality. So what did he do? Here is what Theon says: “Hipparchus, convinced that this is how the phenomena are brought about, adopted the epicyclic hypothesis as his own and says that it is likely that all the heavenly bodies are uniformly placed with respect to the center of the world and that they are united to it in a similar way.” So Hipparchus decided between equivalent models based on physical plausibility. This is quite clearly a realist argument.
Historians have brought up other “evidence” that “the Greeks” were instrumentalists. One thing they point to is the alleged compartmentalisation of Greek science. I quote a modern historian:
“Phenomena [such as] consonance, light, planetary trajectories and the two states of equilibrium [i.e., statics and hydrostatics] are investigated separately. There is no search for interconnections, let alone for an overarching unity.”
This attitude would indeed make sense if mathematical science was just instrumental computation tools with no genuine anchoring in reality. The only problem is that the claim is false. Greek science is in fact full of interconnections, just as one would expect if they were committed realists. Ptolemy uses mechanics to justify geocentrism; Archimedean hydrostatics explains shapes of planets and “casts light on the earth’s geological past”; Archimedes used statical principles to compute areas in geometry. Ptolemy applies “consonance” (that is, musical theory) to “the human soul, the ecliptic, zodiac, fixed stars, and planets,” as he says in his book on astrology. Ptolemy also applies the law of refraction of optics to atmospheric refraction, noting its importance for astronomical observations.
In Galileo’s time, the same pattern prevails: mathematically competent people are unabashed realists, while philosophers and theologians often find instrumentalism more appealing for reasons that have nothing to do with science. Copernicus’s book, for example, is unequivocally realist. Spineless philosophers and theologians could not accept this. One even resorted to the ugly trick of inserting an unsigned foreword in the book without Copernicus’s authorisation, in which they espoused instrumentalism. Here’s what is says:
“It is the job of the astronomer to use painstaking and skilled observation in gathering together the history of the celestial movements, and then—since he cannot by any line of reasoning reach the true causes of these movements—to think up or construct whatever causes or hypotheses he pleases such that, by the assumption of these causes, those same movements can be calculated from the principles of geometry for the past and for the future too. … It is not necessary that these hypotheses should be true. … It is enough if they provide a calculus which fits the observations.”
This foreword was left unsigned so that it was easy to assume that it was written by Copernicus himself. This surely fooled no one who actually read the book, with all its blatant realism. Giordano Bruno, for one, thought “there can be no question that Copernicus believed in this motion [of the earth],” and hence concluded that the timid foreword must have been written “by I know not what ignorant and presumptuous ass.” That’s Bruno’s opinion, a early reader of Copernicus. Other mathematical readers presumably felt the same way. But then again the mathematically incompetent people whom the instrumentalist foreword was designed to appease could not read the book anyway.
In medieval and renaissance philosophical texts it is not hard to find many assertions to the effect that “real astronomy is nonexistent” and what passes for astronomy “is merely something suitable for computing the entries in astronomical almanacs.” There were many instrumentalists at the time, to be sure, but the challenge is to find a single serious mathematical astronomer among them. They were exclusively theologians and philosophers.
All historians nowadays recognise that “Copernicus clearly believed in the physical reality of his astronomical system,” but their inference that he “thus broke down the traditional disciplinary boundary between astronomy (a branch of mixed mathematics) and physics (or natural philosophy)” is dubious. This was “the traditional” view only in a very limited sense. It was traditional among the particular sect of Aristotelians that occupied the universities, but outside this narrow clique it had no credibility or standing whatsoever. Among mathematicians, Copernicus’s view was exactly the traditional one.
All mathematically competent people continued in the same vein, long before Galileo entered the scene. Already in the 16th century, “Tycho and Rothman, Maestlin, and even Ursus openly deploy a wide range of physical arguments in debating the issue between the rival world-systems.” Kepler puts the matter very clearly:
“One who predicts as accurately as possible the movements and positions of the stars performs the task of the astronomers well. But one who, in addition to this, also employs true opinions about the form of the universe performs it better and is held worthy of greater praise. The former, indeed, draws conclusions that are true as far as what is observed is concerned; the latter not only does justice in his conclusions to what is seen, but also in drawing conclusions embracing the inmost form of nature.”
As Kepler notes, this was all obviously well-known and accepted since antiquity, for “to predict the motions of the planets Ptolemy did not have to consider the order of the planetary spheres, and yet he certainly did so diligently.”
So, in conclusions, mathematicians were always realists. Galileo had nothing new to contribute on that matter. So we have refuted that as well as one of the possible Galilean innovations that caused the scientific revolution.
Here’s another of the big themes in the scientific revolution: the “mechanical philosophy.”
Some say that “the mechanization of the world-picture” was the defining ingredient of “the transition from ancient to classical science.” A paradigm conception at the heart of the new science was that of the world as a machine: a “clockwork universe” in which everything is caused by bodies pushing one another according to basic mechanical laws, as opposed to a world governed by teleological purpose, divine will and intervention, anthropomorphised desires and sympathies ascribed to physical objects, or other supernatural forces. Galileo was supposedly a pioneer in how he always stuck to the right side in this divide. Here is one historian arguing as much:
“Galileo possessed in a high degree one special faculty. That is the faculty of thinking correctly about physical problems as such, and not confusing them with either mathematical or philosophical problems. It is a faculty rare enough still, but much more frequently encountered today than it was in Galileo’s time, if only because nowadays we all cope with mechanical devices from childhood on.”
Of course, this “special faculty” is precisely what led Galileo to reject as occult the correct explanation of the tides and propose his own embarrassing nonstarter of a tidal theory based on an analogy with “mechanical devices,” as we have discussed before. But let’s put that aside.
There is nothing modern about the mechanical philosophy. “We all cope with mechanical devices from childhood on,” the quote says, but so did the Greeks. They built automata such as entirely mechanical puppet-theatres, self-opening temple doors, a coin-operated holy water dispenser, and so on. Pappus notes that “the science of mechanics” has many applications “of practical utility,” including machines for lifting weights, warfare machines such as catapults, water-lifting machines, and “marvellous devices” using “ropes and cables to simulate the motions of living things.”
Clearly, then, “Ancient Greek mechanics offered working artifacts complex enough to suggest that organisms, the cosmos as a whole, or we ourselves, might ‘work like that’.” Thus we read in ancient sources that “the universe is like a single mechanism” governed by simple and deterministic laws that ultimately lead to “all the varieties of tragic and comedic and other interactions of human affairs.” This line of reasoning soon lead to a secularisation of science. “Bit by bit, Zeus was relieved of thunderbolt duty, Poseidon of earthquakes, Apollo of epidemic disease, Hera of births, and the rest of the pantheon of gods were pensioned off” in the same manner.
Mechanical explanations are widespread in Greek science. The Aristotelian Mechanics uses the law of lever to explain “why rowers who are in the middle of the ship move the ship the most,” and “how it is that dentists extract teeth more easily by a tooth-extractor [or forceps] than with the bare hand only.” Greek scientists explained perfectly clearly that sound is a “wave of air in motion,” comparable to the rings forming on a pond when when one throws in a stone. Atomism—a widely espoused conception of the world in Greek antiquity—is of course in effect a plan to “make material principles the basis of all reality.”
Greek astronomy went hand in hand with mechanical planetaria that directly reproduced a scale model of planetary motion. And not just basic toy models, but “complex and scientifically ambitious instruments” that could generate all heavenly motions mechanically from a single generating motion (the turn of a crank, as it were).
The possibility that even biological phenomena worked on the same principle immediately suggested itself and was eagerly pursued. Here’s what Galen says, the ancient physician:
“Just as people who imitate the revolutions of the wandering stars by means of certain instruments instill a principle of motion in them and then go away, while [the devices] operate just as if the craftsman was there and overseeing them in everything, I think in the same way each of the parts in the body operates by some succession and reception of motion from the first principle to every part, needing no overseer.”
Indeed, ancient medical research put this vision into practice. “The use of what we should call mechanical ideas to explain organic processes”—such as digestion and other physiological functions—is “the most prominent feature” of the work of Erasistratus in medicine, who also tested his ideas experimentally.
In conclusion, then, the world did not need Galileo to tell them about the mechanical philosophy, since it had been widely regarded as common sense already in antiquity.
The scientific revolution did not come about by any innovative or groundbreaking insights of Galileo. It came about by simply listening to what the mathematicians had been saying for thousands of years.
Was Galileo “the father of modern science” because he was the first to unite mathematics and physics? Or the first to base science on data and experiments? No. Galileo was not the first to do any of these things, despite often being erroneously credited with these innovations.
Transcript
Galileo is “the father of modern science,” people would have you believe. But why? What exactly did he do that was so new that he fathered the entire concept of science? Was Galileo the first to bring together physics and mathematics? Was he the first to base science on data and experiments, or to give practical experience more authority than philosophical systems?
The answer to these questions is: no, no, no. Galileo was nowhere near the first to do any of these things. But he is still often credited with these innovations, even in scholarly sources. So I’m going to run down the list and prove point by point why these people are wrong.
The notion that Galileo was somehow “the father of modern science” remains a standard view among modern historians. For instance, the Oxford Companion to the History of Modern Science published in 2003 flat out says that Galileo “may properly be regarded as the ‘father of modern science’.” This view is considered so unassailable that even the very Pope once conceded that Galileo “is justly entitled the founder of modern physics.” Pope John Paul II said this is 1979.
But there is less agreement on what exactly Galileo did to deserve this epithet. As Dijksterhuis says in his classic history of mechanics: “No one indeed is prepared to challenge [Galileo’s] scientific greatness or to deny that he was perhaps the man who made the greatest contribution to the growth of classical science. But on the question of what precisely his contribution was and wherein his greatness essentially lay there seems to be no unanimity at all.”
So let’s go though all major attempts at capturing Galileo’s alleged greatness, and criticise them one by one.
First: Mathematics and nature.
It is a common view that Galileo was the first to bring together mathematics and the study of the natural world. I could give you long list of scholars who have said exactly this. For this to make sense, one must obviously maintain that, before Galileo, mathematics and natural science were fundamentally disjoint. This assumption is plainly and unequivocally false. In Greek works by mathematically competent authors, there is zero evidence for this assumption and a mountain of evidence to the contrary. “We attack mathematically everything in nature” said Iamblichus of Greek science, and he was right. This is a commonplace, explicit methodological program in Greek science, as the The Cambridge Companion to the Hellenistic World points out: “Hellenistic natural philosophers often took mathematics as the paradigm of science and sought to mathematize their study, that is, to ground all its claims in mathematical theorems and procedures, a goal shared by modern scientists.” This is the exact opposite of the claim that the ancients were unable to conceive the unity of mathematics and science.
How can so many historians get it exactly backwards? By ignoring the entire corpus of Greek mathematics and instead relying exclusively on philosophical authors. Thus we are told that, following “the classification of philosophical knowledge deriving from Aristotle,” a sharp division prevailed among “the Greeks” between “natural science (or ‘physics’), which studied the causes of change in material things,” and “mathematics, which was the science of abstract quantity.” Well, this was perhaps a problem for philosophers who spent their time trying to classify scientific knowledge instead of contributing to it. But I challenge you to produce one single piece of evidence that this division had any impact whatsoever on any mathematically creative person in antiquity.
The alleged divide doesn’t exist in Aristotle’s own works either, for that matter. Aristotle lived well before the glory days of Greek science, and he was clearly no mathematician. But even Aristotle lists mechanics, optics, harmonics, and astronomy as fields based on mathematical demonstrations. He even explicitly calls them “branches of mathematics.” How can anyone infer from this that Aristotle saw the very notion of mathematical science as a conceptual impossibility? That’s nuts. But historians in fact do so, by insisting that these fields are mere exceptions. Here’s a typical quote, from A Short History of Scientific Thought published by Palgrave Macmillan in 2012:
“Previous assumptions [before Galileo], encouraged by Aristotle and scholastic philosophers, held that mathematics was only relevant to our understanding of very specific aspects of the natural world, such as astronomy, and the behaviour of light rays ([that is to say] optics), both of which could be reduced to exercises in geometry. Otherwise, mathematics was just too abstract to have any relevance to the physical world.”
The implausibility of this view is obvious. If, as Aristotle himself clearly states, mechanics, optics, harmonics, and astronomy are four entire fields of knowledge that successfully use mathematics to understand the natural world, who in their right mind would then categorically insist that, nevertheless, other than that mathematics surely has nothing to contribute to science. It makes no sense. If mathematics has already given you four entire branches of science, why close your mind to the possibility of any further success along similar lines? It is hard to think of any reason for taking such a stance, except perhaps for someone who themselves lack mathematical ability and want to justify their neglect of this field.
The strange habit of writing off the numerous branches of mathematical science in antiquity as so many exceptions is necessary to maintain triumphalist narratives of the great Galilean revolution. For example, we are told that “it was Galileo who first subjected other natural phenomena to mathematical treatment than the Alexandrian ones.” In other words, except mechanics, astronomy, optics, music, statics, and hydrostatics, Galileo was the very first to take this step. That is to say, if you ignore all previous mathematicians who did this exact thing in great detail, Galileo’s step was completely revolutionary.
Another strategy for explaining away the obvious fact of extensive mathematical sciences in antiquity is to discount them as genuine science on the grounds that they were abstractions. Thus some claim that, despite ostensible applications of mathematics in numerous fields, “mathematical theory and natural reality remained almost entirely separate entities” due to the “high level of abstraction” of the mathematical theories, which meant that they were “barely connected with the real world.”
Supposedly, Galileo broke this spell — an absurd claim since this critique is all the more true for his science: even Galileo’s supposedly “best” discoveries are often way out of touch with reality: his law of fall, his law of parabolas, they obviously fail experimentally. Not to mention Galileo’s many erroneous theories, which were even more disconnected from reality for obvious reasons. Meanwhile, Greek scientific laws of statics, optics, hydrostatics, and harmonics concern everyday phenomena that can be verified by anyone in their own back yard using common household items. Indeed, they are still part of modern physics textbooks — and high school laboratory demonstrations — to this day. Take optics, for example. Heron of Alexandria proved the law of reflection, which anyone with a mirror can readily check, using the distance-minimisation argument still found in every textbook today. Light travels along the shortest path from point A to point B via the mirror. Diocles demonstrated the reflective property of the parabola and used it to “cause burning” by concentrating the rays of the sun with a paraboloid mirror: a principle still widely applied today, for example in satellite dishes and flashlights. Ptolemy demonstrated the magnifying property of concave mirrors, such as modern makeup mirrors. These kinds of results, which are not atypical, are clearly not disconnected from reality by any means.
The false notion of a divide between mathematics and science also rests on a conception of mathematics itself as a purely abstract field. Here’s a quote expressing a typical view:
“Traditionally, geometry was taken to be an abstract inquiry into the properties of magnitudes that are not to be found in nature. Dimensionless points, breadthless lines, and depthless surfaces of Euclidean geometry were not traditionally taken to be the sort of thing one might encounter while walking down the street. Whether such items were characterized as Platonic objects inhabiting a separate realm of geometric forms, or as abstractions arising from experience, it was generally agreed that the objects of geometry and the space in which they are located could not be identified with material objects or the space of everyday experience.”
This is again a view expressed by philosophers only. Nothing of the sort is ever stated by any mathematically competent author in antiquity. On the contrary, mathematicians routinely take the exact opposite for granted. Allegedly “abstract” geometry is constantly applied to physical objects in Greek mathematical works without ado. The long list of Greek mathematicians who studied the natural world always took for granted the identification of geometry with the space and material objects around us. And why shouldn’t they? For thousands of years geometry had been used to delineate fields, draw up buildings, measure volumes of produce, and a thousand other practical purposes — exactly “the sort of thing one might encounter while walking down the street.” Every single theorem of Euclid’s geometry can be verified by concrete measurements and constructions with physical tools and materials. So why would mathematicians suddenly insist that their field is completely divorced from reality? What could possibly be their motivation for doing so? It accomplishes nothing and creates tons of obvious problems when one wants to apply mathematics far and wide in numerous areas, as mathematicians always did. The only people with any motive to take such an extremist stance are philosophers with an axe to grind.
Only those ignorant of the vast tradition of Greek mathematical science can maintain that the unity of mathematics and science in the 17th century was in any way revolutionary. However, even if one accepts this completely wrongheaded view, credit still should not go to Galileo. Some recent historians have begun to stress that “the mathematization of the sublunary world begins not with Galileo but with Alberti,” who wrote on the geometrical principles of perspective in painting in the 15th century.
“The invention of perspective by the Renaissance artists, by demonstrating that mathematics could be usefully applied to physical space itself, [constituted] a momentous step toward the general representation of physical phenomena in mathematical terms.”
These historians correctly challenge the narrative of Galileo as the heroic visionary who united mathematics and the physical world, but they retain the erroneous underlying assumption that this unification was revolutionary to begin with. Perspective painting is fine mathematics, but it wasn’t a “momentous step” “demonstrating” that mathematics could be applied to the world, because that had already been demonstrated over and over again thousands of years before. Vitruvius, to take just one example, had pointed out the obvious: “an architect should be instructed in geometry,” which “is of much assistance in architecture.” Certainly a strange thing to say if the “momentous” insight that geometry is relevant to “the space of everyday experience” is still more than a thousand years in the future! No, the absurd notion that the application of geometry to physical space was somehow a Renaissance revolution can only occur to those who spend too much time reading philosophical authors pontificating about the divisions of knowledge instead of reading authors actually active in those fields.
The restriction to “the sublunary world” in the above quotation is also telling. The allegedly profound conceptual divide between heaven and earth in this period is a standard trope among historians, as we have discussed before. Of course, the Greeks mathematised the sublunary world too, but you have to read specialised works to find out much about that. Astronomy, on the other hand, is such an obvious example of an extremely successful and detailed mathematisation of one aspect of reality that even philosophers and historians cannot ignore this elephant in the room. Hence they rely on the qualifier that the allegedly revolutionary step was “the mathematization of the sublunary world.”
Aristotle did indeed make much of the difference between the earthly, sublunary world and the world of heavenly motions. But this is one particular dogma of one particular school of philosophy. There is no reason for any mathematician to accept it, nor is there any evidence that any mathematically competent person in the golden age of Greek science did so. The Aristotelian dichotomy is far from natural or necessary: in fact, “Aristotle argues, against his predecessors, that the celestial world is radically different from the sublunary world,” as one historian has observed. For that matter, even if Aristotle’s dogmatic and arbitrary dichotomy is accepted, it would still be madness to acknowledge the undeniable success of mathematics on one side of the divide, yet consider its application on the other side of the divide a conceptual impossibility.
Ptolemy, the ancient astronomer, speaks in Aristotelian terms when he contrasts astronomy with physics. The subject matter of astronomy is “eternal and unchanging,” while physics “investigates material and ever-moving nature situated (for the most part) amongst corruptible bodies and below the lunar sphere.” This is arguably more of a fact than a philosophical commitment: planetary motions are regular and periodic, whereas falling bodies, projectile motion, and other phenomena of terrestrial physics are inherently fleeting and limited to a short time span. It is conceivable that someone might seize on this dichotomy to “explain” why mathematics is suitable for the heavens only, and not for the sublunary world. This, however, is definitely not Ptolemy’s stance. He unequivocally expresses the exact opposite view: “as for physics, mathematics can make a significant contribution” there too.
In sum, the Aristotelian dichotomy between heaven and earth was never an obstacle to mathematicians. And this with good reason. The whole business of emphasising the dichotomy in the context of the mathematisation of the world is a figment of the imagination of historians, who find themselves having to somehow explain away astronomy as irrelevant when they want to claim that there was a mathematical revolution in early modern science. We do not need to resort to such fictions if we instead accept that the unity of mathematics and science had been obvious since time immemorial.
Another argument for Galileo as the unifier of physics and mathematics consists in stressing that other mathematicians of his day were often more concerned with pure geometry than with projectile motion and the like. For instance, in France there were highly capable “new Archimedeans” like Descartes, Roberval, and Fermat, but their focus differed from that of Galileo. Here’s a quote from a recent book expressing this view:
“They were indeed good mathematicians, but they did not consider mathematics as a method for understanding physical things. Mathematical constructions were only abstractions to them, with which it was fun to play, but which were not to be confused with what really happened in nature. Moreover, they were not interested in the ways in which motion intervened in natural processes.”
In my view, Galileo would have loved to have been this kind of “new Archimedean” too if only he had been capable of it. And it is not true that these Frenchmen ignored motion and the mathematisation of nature. We have already noted that Descartes studied the law of fall, and that Fermat corrected Galileo on the path of a falling object in absolute space. Both Descartes and Fermat also wrote on the law of refraction of optics, deriving it from physical considerations regarding the speed of light in different media. Also, Descartes explained the motion of the planets, and the fact that they all revolve in the same direction about the sun, by postulating that they were carried along by a vortex. So these mathematicians were clearly not ignorant of or averse to studying how “motion intervened in natural processes.”
So it is not attention to motion per se, but the study of projectile motion specifically, that sets Galileo apart from these mathematical contemporaries. Does Galileo deserve great credit in this regard? I don’t think so. Why is projectile motion important? With Newton, projectile motion took on a fundamental importance because he saw that planetary motion was governed by the same principles. Galileo had no inkling of this insight. With Newton, projectile motion is also fundamental as a paradigm illustration of the principles — such as inertia and Newton’s force law — that govern all other mechanics. In Newtonian mechanics this is the basis for understanding phenomena such as pendulum motion. Galileo, however, got this wrong, so he cannot be celebrated for this insight either.
Thus we see that praising Galileo for studying projectile motion is anachronistic. Galileo got lucky: the topic he studied later turned out to be very important for reasons he did not perceive, so that in retrospect his work seems much more prescient and groundbreaking than it really was. He himself in fact motivates the theory of projectile motion almost exclusively in terms of practical ballistics — a nonsensical application of zero practical value, which one cannot blame other mathematicians for ignoring.
So those are my rebuttals of the various ways in which Galileo has been praised for mathematising nature in innovative ways.
Another way in Galileo was supposedly innovative is in his emphasis on an empirical scientific approach.
The Cambridge Companion to Galileo expresses this view clearly: “Galileo became (and still is) the model for the empiricist scientist who, unlike the natural philosophers of his day, sought to answer questions not by reading philosophical works, but rather through direct contact with nature.” This is an image Galileo eagerly (but dishonestly) sought to promote, as we have seen. Recall the story of the Babylonian eggs cooked in a sling for example, and also Galileo’s rhetoric against Aristotle on the law of fall.
Praise for Galileo in this regard naturally goes hand in hand with “the verdict that Greek science suffered from an overdose of rash generalizations at the expense of a careful scrutiny, whether experimental or observational, of the relevant facts.” In other words, “Greek thinkers generally overrated the power of unchecked, speculative thought in the natural sciences.” So many people have claimed.
In reality, an empirical approach to the study of nature is not a newfangled invention by Galileo but just common sense. It was obviously adopted by the Greeks, especially the mathematicians. Even Aristotle, who practiced “speculative thought in the natural sciences” to a much greater extent than mathematicians, was a keen empiricist, and his followers insisted on this as one of the key principles of his philosophy. Aristotle’s zoology largely follows a laudable empirical method quite modern in spirit, such as braking open lots of bird eggs at different stages to study the development of the embryo and many other things like that. The same approach was applied by his immediate followers in botany and petrology, including for example cataloging extensive empirical data on how a wide variety of minerals react to heating.
This was far from forgotten in Galileo’s day, where one often encounters passages like these from committed Aristotelians:
“We made use of a material instrument to establish by means of our senses what the demonstration had disclosed to our intellect. Such an experimental verification is very important according to [Aristotelian] doctrine.” That’s Piccolomini, an Aristotelian philosopher, writing well before Galileo, in the 16th century.
Not infrequently, Galileo’s Aristotelian opponents attacked him for being too speculative while they saw themselves as representing the empirical approach. For example, one critic writes to Galileo:
“At the beginning of your work, you often proclaim that you wish to follow the way of the senses so closely that Aristotle (who promised to follow this method and taught it to others) would have changed his opinion, having seen what you have observed. Nonetheless, in the progress of the book you have always been so much a stranger to this way of proceeding that all your controversial conclusions go against our sense knowledge, as anyone can see by himself, and as you expressly say yourself, speaking of the theory of Copernicus, which was rendered plausible and admirable to many by abstract reasoning although it was against all sensory experience.”
It is true that there were also many spineless “Aristotelians” in Galileo’s day who preferred hiding behind textual studies rather than engaging with actual science. But this was one perverse sect of scholasticism, not the overall state of human knowledge before Galileo. A contemporary colleague of Galileo put is well:
“The Science of Nature has been already too long made only a work of the Brain and the Fancy: It is now high time that it should return to the plainness and soundness of Observations on material and obvious things.”
That’s Robert Hooke. Note that word choice: “return” — “return to observation.” Not: Galileo invented this new thing, empiricism. Rather: empiricism is the natural and obvious way to study nature, and the departure from it in certain philosophical circles is a corrupt aberration.
The misconception that the Greeks were anti-empirical stems from a foolish reading of the mathematical tradition. Galileo fan Stillman Drake put it like this:
“Archimedes never appealed to actual measurements in any of his proofs, or even in confirmation of his theorems. The idea that actual measurement could contribute anything of real value was absent from physics for two millennia.”
Or again:
“The mathematics of Euclid and the physics of Archimedes were necessary, but not sufficient, for Galileo’s science. They leave unexplained Galileo’s repeated appeals to sensate experience.”
On a superficial reading this may indeed appear so. Open, say, Archimedes’s treatise on floating bodies and you will find no mention of any measurement or experiment or data of any kind, only theorems and proofs. It may seem natural to infer from this that Archimedes was doing speculative mathematics divorced from reality, and that he had no understanding of the importance of empirical tests. This is what it looks like to historians who insist on an overly literal reading of the text and lack a sympathetic understanding of how the mathematical mind works. The fact of the matter is that Archimedes’s theorems are empirically excellent. It makes no sense to imagine that Archimedes was reasoning about abstractions as an intellectual game, and that his extremely elaborate and detailed claims about the floatation behaviour of various bodies given their shapes and densities just happened to align exactly with reality by pure chance. Archimedes doesn’t have to point out that he made very careful empirical investigations, because it is obvious from the accuracy of his results that he did.
Here is a better way of putting the relation between mathematics and empirical data, from The Oxford Handbook of the History of Physics:
“Mixed mathematics were often presented in axiomatic fashion, following the Archimedean tradition. In this tradition, experiments were often conceived of as inherently uncertain and therefore they could not be placed at the foundation of a science, lest that science too be tainted with that same degree of uncertainty. To be sure, experiments were still used as heuristic tools, for example, but their role often remained private, concealed from public presentations.”
So the point is not that empirical data is neglected, but that it is a mere preliminary step. Anyone can make measurements and collect data. Self-respecting mathematicians do not publish such trivialities. Instead they go on to the really challenging step of synthesising it into a coherent mathematical theory. Galileo did not have the ability to do the latter, so he had to stick with the basics, and pretend, nonsensically, that this was somehow an important innovation. Then as now, there were enough non-mathematicians in the world for his cheap charade to be successful.
What about the experimental method? Was that Galileo’s special contribution and insight?
Some say so. Empiricism, which we just discussed, is mere passive observation. The real innovation was active experiment. A famous supporter of this view is Immanuel Kant, who wrote as follows in the Critique of Pure Reason:
“When Galileo caused balls to roll down an inclined plane, a light broke upon all students of nature. Reason must approach nature in order to be taught by it. It must not, however, do so in the character of a pupil who listens to everything that the teacher chooses to say, but of an appointed judge who compels the witnesses to answer questions which he has himself formulated.”
Modern historians have expressed the same idea. Here is one example:
“The originality of Galileo’s method lay precisely in his effective combination of mathematics with experiment. The distinctive feature of scientific method in the seventeenth century, as compared with that in ancient Greece, was its conception of how to relate a theory to the observed facts and submitting them to experimental tests. [This feature] transformed the Greek geometrical method into the experimental science of the modern world.”
In reality, the use of experiment in Greek science is abundantly documented to anyone who bothers to read mathematical authors.
Greek scientists knew perfectly well that “it is not possible for everything to be grasped by reasoning, many things are also discovered through experience,” as Philon said. This quote refers to the precise numerical proportions needed for the spring in a stone-throwing engine. The same author also offered an experimental demonstration that air is corporeal. Ptolemy experimented with balloons (or “inflated skins” as he says) to investigate whether air or water has weight in their own medium. Does a balloon full of water sink in water, or float or what? Indeed, Ptolemy “performed the experiment with the greatest possible care,” according to Simplicius. Heron of Alexandria gives a detailed description of an experimental setup to prove the existence of a vacuum. He explicitly states that “referring to the appearances and to what is accessible to sensation” trumps abstract arguments that there can be no vacuum. Such arguments had been given by Aristotle, but here we have a mathematically minded author saying “no way, that’s nonsense” and proving as much with experiment. In optics, Ptolemy explicitly verified the law of reflection by experiment. He also studied refraction experimentally, giving tables for the angle of refraction of a light ray for various incoming angles in increments of 10 degrees for passages between air, water, and glass.
Archimedes caught a forger who tried to pass off as pure gold a crown that was actually gold-coated silver. By an experiment based on hydrostatic principles, he was able to expose the crown as a knock-off without damaging it in any way. This discovery was the occasion for him to reportedly run naked through the streets yelling “eureka” in excitement. Such was his love of empirical, experimental science — yet many scholars keep insisting that, like a second Plato, all he really cared about was abstract geometry. Evidently, even running naked through the streets and screaming at the top of one’s lungs is not enough for some people to open their eyes. It is hard to imagine what else one can do to draw their attention to the obvious: namely that Greek mathematicians embraced experimental method through and through.
Ok, so I have argued that Galileo wasn’t the first to apply mathematics to nature, nor the first to base science on data, nor on experiment. So we’ve ruled out those three but we’re still only halfway down the list of things that Galileo supposedly pioneered. We will have to go through the other ones next time.
Galileo’s sentencing by the Inquisition was avoidable. The Church had no interest in prosecuting mathematical astronomers, but since Galileo had so little to contribute in that domain he foolishly got himself involved with Biblical interpretation. His scriptural interpretations not only got him into hot water: they are also scientifically unsound and blatantly inconsistent with his own science.
Transcript
The Bible says basically nothing about astronomy. It has a lot more to say about righteous war. And it is in this context only that it has occasion to speak of the motions of heavenly bodies. In the Book of Joshua, we find our hero with the upper hand in battle, but alas dusk is drawing close. What a pity if some of the enemies “delivered up before the children of Israel” should be able to get away under the cover of darkness. “Then spake Joshua to the Lord,” and he said: “Sun, stand thou still.” “And the sun stood still until the people had avenged themselves upon their enemies.” That is what the Bible tells us. “The sun stood still in the midst of heaven,” so that Joshua and the chosen people could keep slaughtering infidels all night long.
This is the full extent of astronomy in the holy book. No further detail is provided anywhere in the Bible regarding the astronomical constitution of the universe or the motions of the heavenly bodies.
Obviously, serious scientists have little reason to engage with this passing and tangential allusion to cosmology in the Book of Joshua. But Galileo’s philosophical enemies saw an opportunity. By persistently and prominently accusing Galileo of proposing theories contrary to scripture they forced him into a dilemma: either let the argument stand unopposed, and hence let his enemies have the last word, or else get involved with the very dangerous matter of scriptural interpretation. Galileo foolishly took the bait. Now all the Aristotelians had to do was to sit back and watch Galileo march to his own ruin in this minefield.
So let’s see how Galileo proposes that we interpret the Biblical passage about the sun standing still. His interpretation is nuts. It is a prime example of his shameless drive to score rhetorical points at any cost. It is perfectly reasonable to argue that the phrase about the sun “standing still” should not be taken too literally. Indeed, it is commonly accepted, as Galileo observes, that various things in the Bible “were set down in that manner by the sacred scribes in order to accommodate them to the capacities of the common people, who are rude and unlearned.”
Indeed, if the Bible is read literally, “it would be necessary to assign to God feet, hands, and eyes,” as Galileo says. But those passages are only figures of speech, according to orthodox Christian understanding. When the Old Testament says that the commandments handed to Moses were “written with the finger of God,” the intended takeaway is of course not that God has an actual physical finger and that he needs it to write. It doesn’t make a whole lot of sense that he could create the entire universe in under a week, or flood the entire earth at will, yet if he has to write something down he has to painstakingly trace it out in clay with his finger.
So perhaps it is the same with the sun “standing still.” It’s just a phrase adapted to everyday speech, not a scientific account. In fact, even Copernicus himself speaks of “sunrise” and “sunset,” as Galileo points out, even though the sun doesn’t move in his system. So it is hardly unreasonable to think that “the sacred scribes” used this kind of common parlance as well, even if they knew that the sun is always stationary.
That’s all fine and well. But Galileo does not stop with this balanced and reasonable point. Instead he makes the outlandish claim that the Joshua passage in fact literally agrees best with heliocentrism rather than geocentrism:
“If we consider the nobility of the sun I believe that it will not be entirely unphilosophical to say that the sun, as the chief minister of Nature and in a certain sense the heart and soul of the universe, infuses by its own rotation not only light but also motion into other bodies which surround it. So if the rotation of the sun were to stop, the rotations of all the planets would stop too. [Therefore,] when God willed that at Joshua’s command the whole system of the world should rest and should remain for many hours in the same state, it sufficed to make the sun stand still. In this manner, by the stopping of the sun, the day could be lengthened on earth—which agrees exquisitely with the literal sense of the sacred text.”
This is a terrible argument. It is so unscrupulous that its absurdity can be exposed simply by quoting the words of Galileo himself, written in another context, in his Dialogue:
“If the terrestrial globe should encounter an obstacle such as to resist completely all its whirling motion and stop it, I believe that at such a time not only beasts, buildings, and cities would be upset, but mountains, lakes, and seas, if indeed the globe itself did not fall apart. This agrees with the effect which is seen every day in a boat travelling briskly which runs aground or strikes some obstacle; everyone aboard, being caught unawares, tumbles and falls suddenly toward the front of the boat.”
So in this manner “Joshua would have destroyed not only the Philistines, but the whole earth,” if stopping the sun meant stopping the motion of the earth, as Galileo claims. Not to mention that the idea that the sun’s rotation on its axis is the only thing moving the planets is completely unsubstantiated in the first place. It seems that Galileo pretended to believe in this principle on this occasion solely for the sake of being able to make this scriptural argument. The hypocrisy and unbridled opportunism of Galileo’s forays into biblical interpretation are plain to see.
It is very difficult, if not impossible, to see his interpretation of the Joshua passage as a scientific argument that Galileo genuinely believed. The second quote I read, about everything collapsing like a house of cards if the earth stopped, that is from 15 years later. But surely Galileo realised this all along. If he didn’t, he was stupid. If he did, then he was clearly perfectly happy to fabricate scientifically nonsensical lies as long as it helped him score a satisfying rhetorical point.
This just goes to show how little all of this had to do with science. Galileo’s interpretation of the Joshua passage is terrible science, and he probably knew that perfectly well. This was a conflict between science and religion if by “science” you mean the ludicrous idea that stopping the sun’s rotation would immediately stop the earth dead in its tracks, and that the people on the earth would suffer no consequences of this whatsoever except that the day would became longer. This is the “science” in science versus religion, if we go by what Galileo wrote.
It was only because Galileo got involved with biblical interpretation that he ended up in the crosshairs of the Inquisition. Nobody minded mathematical astronomy, but the question of who has the right to interpret the Bible was the stuff that wars were made of. Luther challenged church authority and emphasised personal understanding of the Bible—“sola scriptura,” as the motto went. This was the core belief of protestantism, and eradicating protestantism was top of the agenda for the catholic church. This is right in the middle of the Thirty Years’ War, which was centered on this core conflict between protestantism and catholicism. A devastating war, comparable to the world wars in terms of per capita deaths.
Once Galileo’s enemies baited him into commenting on the Bible, it was all too easy for them to connect Galileo’s otherwise harmless dabbling to this heresy du jour. Because of the war and raging conflict, this was a matter on which the church could not afford to show any weakness.
There is only one mystery: Why did Galileo walk straight into such an obvious trap? The answer lies, as ever, in his mathematical ineptitude. Galileo was told by church authorities that “if he spoke only as a mathematician he would have nothing to worry about.” Galileo would presumably have followed this advice if he could. The problem, of course, was that he did not have anything to contribute “as a mathematician.” Since a mathematical defence of heliocentrism was beyond his abilities, Galileo was left with no other recourse than to roll the dice and try his luck in the dangerous and unscientific game of scriptural interpretation.
So the church was reluctantly drawn into these astronomical squabbles and had to do something. The Inquisition settled for a slap on the wrist: in the future, Galileo must not “hold, teach or defend [the Copernican system] in any way whatever,” they decided. They also ordered mild censoring of Copernicus’ book, namely the removal of a brief passage concerning the conflict with the Bible and a handful expressions which insinuated the physical truth of the theory. That was it. No book bans, no imprisonments. And Galileo got away with just a warning.
Galileo did indeed keep quiet for a number of years after being ordered to do so by the Inquisition. But times changed. After waiting for over a decade, Galileo felt it was safe to try the waters again. A new Pope was in power, Urban VIII, who was quite liberal. He even said of the 1616 censoring of Copernicus that “if it had been up to me that decree would never have been issued.” Galileo had good personal relations with this new open-minded Pope. So Galileo sensed an opening and obtained a permission to publish the Dialogue in 1632. Or rather, as the Inquisition would later put it, he “artfully and cunningly extorted” this permission to publish. For when the permission was granted the Pope did not know about the private injunction of 1616 for Galileo to keep off the subject. When this came to light the Pope was outraged and felt, with good cause, that Galileo had been deliberately deceitful and reportedly stated that “this alone was sufficient to ruin [Galileo] now.”
So the wheels of the Inquisition were in motion again. A special commission was appointed. It found many inappropriate things in the Dialogue, but this was not a major issue, they noted, for such things “could be emended if the book were judged to have some utility which would warrant such a favor.” The real problem was instead that Galileo “overstepped his instructions” not to treat heliocentricism.
The same report also points out that Galileo had disrespected the Pope on another point as well. The Pope had asked Galileo to include the argument that since God is omnipotent he could have created any universe, including a heliocentric one. So even though the church does not agree with Copernicus, their own logic, namely belief in God’s omnipotence, can be used to legitimate at least considering the possibility of this hypothesis. So that’s a useful argument that Galileo could have used to try to find at least a little bit of common ground with his opponents. But instead of using it for such purposes of reconciliation as intended, Galileo used it to fuel the fires of conflict even more. He made had placed the Pope’s favourite argument “in the mouth of a fool,” the commission observed. He made Simplicio, the dumb character in the Dialogue who constantly expresses the wrong ideas and is proven wrong at every turn, be the one who spoke the Pope’s words. He hardly did himself any favours with this disrespectful move.
Following these findings, the second Inquisition proceedings took place in 1633: 17 years after the first Inquisition where Galileo had gotten off easy, and the year after the publication of his inflammatory Dialogue in defence of Copernicanism. The outcome was a forgone conclusion. Galileo’s defence was transparently dishonest. He pretended that, in the Dialogue, “I show the contrary of Copernicus’s opinion, and that Copernicus’s reasons are invalid and inconclusive.” This is of course pure nonsense. In private correspondence shortly before, Galileo had spoken more honestly, and stated that the book was “a most ample confirmation of the Copernican system by showing the nullity of all that had been brought by Tycho and others to the contrary.” But now before the Inquisition he had to pretend otherwise. In light of the accusations, Galileo continued, “it dawned on me to reread my printed Dialogue,” and “I found it almost a new book by another author.” These transparent lies did little to save him. He was forced to abjure. The Dialogue was prohibited, but not for its contents but rather, in the words of the Inquisition’s sentence, “so that this serious and pernicious error and transgression of yours does not remain completely unpunished” and as “an example for others to abstain from similar crimes.”
There is a popular myth that Galileo muttered “eppur si muove”—”yet it moves” (the earth moves, that is)—as he rose from his knees after abjuring before the Inquisition. But this is certainly false. Obviously the Inquisition would not have tolerated such insubordination, especially since the whole point the trial in the first place was to punish Galileo for his defiance. Galileo had been shown the instruments of torture, and such a rebellious exclamation would have been the surest way to have them dusted off for the occasion. Today no historian believes the myth that Galileo mumbled these words before the Inquisition. Yet it remains instructive in warning us of the lengths many Galilean idol worshippers are willing to go to, who do not want to admit the many ignominious historical facts about their hero. The sheer multitude of such myths now universally regarded as busted should leave us open to the distinct possibility that we have not gotten to the end of them yet.
A similar myth, which has been appealing to anti-religion ideologues, is that “the great Galileo groaned away his days in the dungeons of the Inquisition, because he had demonstrated the motion of the earth.” That’s a quote from Voltaire. But in reality Galileo was sentenced more for his provocateurism than for his science, and furthermore he was never imprisoned in any “dungeon.” He was sentenced to house arrest. A visitor “reported that [Galileo] was lodged in rooms elegantly decorated with damask and silk tapestries.” Soon thereafter he retired to “this little villa a mile from Florence,” where “nearby I had two daughters whom I much loved” and where he also received many friends and guests. Many today would pay dearly for such a retirement. Galileo got it as a so-called “punishment.”
So that’s the story of the Inquisition proceedings. Let’s look at some lessons from this.
Galileo’s conflict with the church was entirely unnecessary. It arose precisely because Galileo was a lampooning populariser rather than a mathematical astronomer and scientist. “[Galileo] was far from standing in the role of a technician of science; had he done so, he would have escaped all trouble,” as Santillana says in his book, The Crime of Galileo. The church establishment had no interest in prosecuting geometers and astronomers. Copernicus’ book had long been permitted, and Galileo’s own Letters on Sunspots of 1613 had been censored only where it referred to scripture, not where it asserted heliocentrism. In reality, “a major part of the Church intellectuals were on the side of Galileo, while the clearest opposition to him came from secular ideas” and philosophical opponents.
Today many take for granted that a fundamental rift between science and religion was unavoidable. Some have imagined for instance that Galileo defied the worldview of the church by demoting the earth from its supposedly “privileged” position. 20th-century playwright Bertolt Brecht appreciated the dramatic flare of framing the conflict in such terms when he wrote a play about Galileo. He has one of the characters argue the privilege point passionately:
“I am informed that Signor Galilei transfers mankind from the center of the universe to somewhere on the outskirts. Signor Galilei is therefore an enemy of mankind and must be dealt with as such. Is it conceivable that God would trust this most precious fruit of his labor to a minor frolicking star? Would He have sent His Son to such a place? The earth is the center of all things, and I am the center of the earth, and the eye of the Creator is upon me.”
But historically this is nonsense, to be sure. Nobody was concerned about this at the time. In fact, classical cosmology clearly stipulated that the Earth was not at all in a privileged position but rather condemned to its very lowly place in the universe. Doesn’t everybody know that hell is just below the surface of the earth, while heaven is way up above? Clearly, then, being at the center of the universe is nothing to be proud of.
It was a commonplace argument in Galileo’s time “that the earth is located in the place where all the dregs and excrements of the universe have collected; that hell is located at the centre of this collection of refuse; and that this place is as far as possible from the outermost empyrean heaven where the angels and blessed reside.” That’s a quote from a book review in the latest issue of the Journal for History of Astronomy. You can go there and find entire books about this.
Even Galileo himself added to the pile of such descriptions. Here is what he says: “after the marvellous construction of the vast celestial sphere, the divine Creator pushed the refuse that remained into the center of that very sphere and hid it there lest it be offensive to the sight of the immortal and blessed spirits.”
Many of Galileo’s contemporaries reasoned alike. Let me quote one more such example: considering “the Vileness of our Earth,” it “must be situated at the center, which is the worst place, and at the greatest distance from those Purer and incorruptible Bodies, the Heavens.” That’s a quote from John Wilkins, an Anglican bishop. This is obviously the very opposite of the argument retrospectively imagined by Brecht and other modern minds, about the supposedly privileged position of the earth.
Here’s another take you sometimes hear: Maybe Galileo brought revolutionary progress by outlining the modern conception of the relation between science and religion. Was it Galileo who showed how faith and science can coexist? How they need not undermine or conflict with one another since one is about the spiritual and the other about the physical? Galileo indeed makes such a case. But those points are common-sense platitudes, not a new vision for the place of science in human thought.
Let’s listen to Galileo’s words from his famous and widely circulated Letter to Duchess Christina of 1615. Here is what Galileo says:
“Far from pretending to teach us the constitution and motions of the heavens and the stars, the authors of the Bible intentionally forbore to speak of these things, though all were quite well known to them. The Holy Spirit has purposely neglected to teach us propositions of this sort as [they are] irrelevant to the highest goal (that is, to our salvation). The intention of the Holy Ghost is to teach us how one goes to heaven, not how heaven goes.”
Even a recent Pope praised Galileo for his supposed insight on this subject: “Galileo, a sincere believer, showed himself to be more perceptive [in regard to the criteria of scriptural interpretation] than the theologians who opposed him.” That’s Pope John Paul II, who said this in 1992.
I disagree with this papal statement on two grounds. First of all, Galileo was not pioneering a new vision for the roles of science and religion more perceptively than anyone else. Rather, he was merely recapitulating elementary ideas that were virtually as old as organised Christianity itself. McMullin has a chapter on this in the Cambridge Companion to Galileo. He concludes that: “[Galileo’s] exegetical principles were not in any sense novel, as he himself went out of his way to stress. They were all to be found in varying degrees of explicitness in Augustine”—twelve centuries before Galileo—”and, separately, they could call on the support of other [even] earlier theologians.” Galileo indeed quotes at great length from Augustine and the church fathers. Not that Galileo knew anything about the history of biblical interpretation: “He had no expertise whatever in that area, so he evidently asked his Benedictine friend, Castelli, to seek out references that would support the exegetical principles he had outlined.” So there was no novelty or insight in Galileo’s treatment of the relation between science and religion.
And here’s a second reason to disagree with the Pope. It is highly doubtful whether Galileo genuinely was “a sincere believer,” as he purported to be. David Wootton has made a compelling case for “two Galileos, the public Catholic and the private sceptic.” Here’s his argument:
“The only decisive document we have [is a 1639 letter to Galileo from] Benedetto Castelli, Galileo’s old friend, former pupil and long-time intellectual companion. … If anyone was in a position to know if Galileo was or was not a believer it was Castelli. … [Castelli writes in his letter that he] has heard news of Galileo that has made him weep with joy, for he has heard that Galileo has given his soul to Christ [in his old age—Galileo was 75 at this point]. Castelli immediately refers to the parable of the labourers in the vineyard: even those who were hired in the last hour of the day received payment for the whole day’s work. … Then … he turns to the crucifixion, and in particular to the two thieves crucified on either side of Christ. One confessed Christ as his saviour and was saved; the other did not and was damned. … Castelli’s [point] is clear and unambiguous. He believes Galileo is coming to Christianity at the last moment, but not too late to save his soul. There is no conceivable interpretation of this letter which is compatible with the generally held view that Galileo was, throughout his career, a believing Catholic.”
That’s David Wootton’s argument in his book on Galileo. It is not a mainstream view but I am inclined to believe it.
The Cambridge Companion to Galileo poses for itself the question: “What did Galileo actually do that made his image so great and so long-standing?” Its answer is not a list of great scientific accomplishments but rather: “Certainly his was the first main effort that fired the vision of science and the world that went well beyond limited intellectual circles.” Galileo was a populariser, in other words. “It was to the man of general interests that Galileo originally addressed his works,” as Stillman Drake says. Indeed, Galileo embraced this role, praising himself for “a certain natural talent of mine for explaining by means of simple and obvious things others which are more difficult and abstruse.”
I agree with these learned authors that Galileo wrote for the vulgar masses. I must add only one point, which they omit, namely that Galileo was driven to turn to popularisation because he was so bad at mathematics. “Galileo scarcely ever got around to writing for physicists,” Drake says. Yes, and he was scarcely able to do so either. The two are not unrelated.
Take for instance the “new stars” (or supernovas, as they would be called today) that appeared in Galileo’s lifetime. One appeared in 1572. It was studied with great care by Tycho Brahe. Another appeared in 1604, when Galileo was 40 years old and an established professor of mathematics. But Galileo didn’t make a contribution based on serious astronomy as Tycho had done. Instead he gave public lectures on the nova to a layman audience totalling more than a thousand people. This is precisely the difference between Galileo and the mathematicians. In modern terms, Galileo is less of a scientist and more of a presenter of TV specials.
Galileo’s little science extravaganzas were a hit at bourgeois dinner parties. Here’s how a contemporary witness describes it:
“We have here Signor Galileo who, in gatherings of men of curious mind, often bemuses many concerning the opinion of Copernicus, which he holds for true. He discourses often amid fifteen or twenty guests who make hot assaults upon him. But he is so well buttressed that he laughs them off; and although the novelty of his opinion leaves people unpersuaded, yet he convicts of vanity the greater part of the arguments with which his opponents try to overthrow him. What I liked most was that, before answering the opposing reasons, he amplified them and fortified them himself with new grounds which appeared invincible, so that, in demolishing them subsequently, he made his opponents look all the more ridiculous.”
Again: Galileo’s speciality is burlesque astronomical road shows, not serious science. If you are an Italian aristocrat who enjoys seeing the learned establishment lose face but don’t want to rock the boat yourself, then you can live vicariously through Galileo’s snappy comebacks and provocations. To this end it matters little whether they are scientifically sound or not.
This is the context in which we must understand Galileo’s conflict with the Church. If we want a parallel of the Galileo trials today we should not think of some totalitarian regime imprisoning intellectuals. A better parallel is cancel culture in popular media. Galileo is a charismatic TV personality. Many enjoy listening to him make fun of the other team. But sometimes he is politically incorrect. So his enemies organise a social media campaign, making a lot of noise. And Galileo is too hot-headed for his own good so he joins in the mud-fight with a bunch of @-replies on Twitter that he didn’t vet with his legal department first. That’s exactly what his opponents were fishing for, and now they got their gotcha quotes that they can take to the network executives and get Galileo cancelled.
Altogether a regrettable spectacle, but one that has not all that much to do with science.
Galileo thought comets were an atmospheric phenomenon, not physical bodies in outer space. How could he be so wrong when all his colleagues got it right? Perhaps because his theory was a convenient excuse for not doing any mathematical astronomy of comets. We also discuss his unsavoury ways of dealing with data in the case of double stars and the rings of Saturn.
Transcript
“Have you seen the fleeting comet with its terrifying tail?” That was the question on everyone’s lips in 1618. In that year a comet appeared that was “of such brightness that all eyes and minds were immediately turned toward it.” “Suddenly, men had no greater concern than that of observing the sky. Great throngs gathered on mountains and other very high places, with no thought for sleep and no fear of the cold.” “That stellar body with its menacing rays was considered a monstrous thing.” According to some prophets, the comet was a cosmic omen foretelling imminent disaster.
I quoted these vivid descriptions from Orazio Grassi: a contemporary of Galileo. These two had a big fight about comets. Grassi was a fine scientist. He was basically right about comets. Galileo, on the other hand, was way wrong on this. His theory of comets is extremely poor. However, Galileo managed to spin this somehow and still come out on top, in the eyes of many modern readers, despite being absolutely wrong as a matter of scientific fact.
This is quintessential Galileo: wrong on science, but a rhetorical master. Galileo could write a self-help book called “How to appear to win any debate even when you’re wrong from start to finish on every single point of substance.” If Galileo is the father of anything it is this art form. So you’re looking to pick up some tricks from that playbook then Galileo is your guy, and the comets dispute is the place to start.
Galileo skilfully caricatures his opponent as an obstinate enemy of science who relies on books and the words of authorities instead of using facts and reason and observation. People eat this up, this propaganda. Galileo is like a populist politician. He’s giving people a pleasing narrative that flatters and validates their worldview. Truth has little to do with anything.
That’s an overview of the story. Now let’s look at the details.
The science of comets. Like Grassi says, “the single role of the mathematician” is merely to “explain the position, motion, and magnitude of those fires,” that is to say the comets. So none of that superstition nonsense, just calculate the paths and distances and speeds and so on. Indeed, this is what mathematicians had been doing for generations. Tycho Brahe, for instance, worked extensively on comets in the generation before Galileo. He gave thorough mathematical analyses of their motions, as a mathematician should.
Now, of course, it would be difficult for Galileo to enter this game, since he was such a poor mathematician, as I have argued before. If Galileo had been honest he would have said: frankly, all those detailed calculations that Tycho Brahe and the other big-boy mathematicians are doing, that’s all too technical for me to follow.
But of course he doesn’t want to say that. He needs to save face. He needs an excuse for ignoring what all serious mathematical astronomers were saying about comets. Sure enough, he is quick to offer such excuses. First he claims that mathematical accounts of comets are hopelessly inconsistent. Here are his own words:
“Observations made by Tycho and many other reputable astronomers upon the comet’s parallax vary among themselves. If complete faith be placed in them, one must conclude that the comet was simultaneously below the sun and above it,” for example.
So the mathematical astronomy of comets is just a bunch of useless nonsense, you see. In fact Galileo has an even more fundamental argument for this. Namely that comets are not physical bodies travelling through space at all. Rather comets are nothing but a chimerical atmospheric phenomena. “In my opinion,” says Galileo, comets have “no other origin than that a part of the vapour-laden air surrounding the earth is for some reason unusually rarefied, and … is struck by the sun, and made to reflect its splendour.” A comet is like the northern lights. Galileo specifically makes this comparison.
So that’s Galileo’s very convenient excuse for why he doesn’t engage with the best mathematicians working on comets. This way he is able to pretend that: well, you see, it’s not that I can’t do these calculations, it’s just that I don’t want to, because they all just contradict themselves anyway, and it’s all nonsense in the first place because you can’t do mathematical astronomy of some vapour-cloud optical illusion thing. That’s a futile as chasing a rainbow.
That’s textbook Galileo. If you don’t believe my thesis that Galileo was a poor mathematician, the you tell me a better explanation for this. Why did Galileo propose such an idiotic theory of comets, that is dead wrong and obviously way worse than the common-sense standard opinion among all mathematical astronomers at the time? I gave you one explanation. I don’t think you can come up with a better one. Nobody has so far.
Galileo’s claim that the mathematical astronomy of comets was incoherent and self-contradictory did not convince anybody. Kepler was flabbergasted that someone who calls himself a geometer could write such drivel. Here are Kepler’s words:
“Galileo, if anyone, is a skilled contributor of geometrical demonstrations and he knows what a difference there is between the incredible observational diligence of Tycho and the indolence common to many others in this most difficult of all activities. Therefore, it is incredible that he would criticize as false the observations of all mathematicians in such a way that even those of Tycho would be included.”
Indeed. It is “incredible” that a “skilled geometer” could make such ludicrous claims. But of course the paradox disappears if one recognises that Galileo is not a skilled geometer after all.
Galileo also offered another very poorly considered argument against the correct view of comets as orbiting bodies. The orbits of comets are clearly much bigger than that of the planets in our solar system. Galileo tries to argue that this is unrealistic. Here is what he says: “How many times would the world have to be expanded to make enough room for an entire revolution [of a comet] when one four-hundredth part of its orbit takes up half of our universe?” This is a poor argument, because the universe must indeed be very big and then some according to Copernican theory. This is because of the absence of stellar parallax, as we have discussed before. Since the earth’s motion is observationally undetectable, the orbit of the earth must be minuscule in relation to the distance to the stars. That means there is plenty of room for comets. But Galileo conveniently pretends otherwise in his argument against comets. Evidently Galileo “was so intent on refusing Tycho[’s treatment of comets] that he failed to notice that he was pleading for a universe in which there would be no room for the heliocentric theory” either.
Galileo’s vapour theory of comets, meanwhile, is inconsistent with basic observations, as he himself admits. If comets are nothing but “rarefied vapour”---that is to say, some kind of pocket of thin gas---then you’d imagine that their natural motion would be straight up, like a helium balloon. Indeed Galileo does propose that comets have such paths. But then he at once admits that this doesn’t fit the facts: “I shall not pretend to ignore that if the material in which the comets takes form had only a straight motion perpendicular to the surface of the earth …, the comet should have seemed to be directed precisely toward the zenith, whereas, in fact, it did not appear so. … This compels us either to alter what was stated, … or else to retain what has been said, adding some other cause for this apparent deviation. I cannot do the one, nor should I like to do the other.” Bummer, it doesn’t work. But Galileo sees no way out, so he just leaves it at that.
Galileo’s contemporaries were not impressed. “[Grassi’s] criticism of Galileo is on the whole penetrating and to the point. He was quick to spot Galileo’s inconsistencies. Grassi produced an impressive array of arguments to show that vapours could not explain the appearance and the motion of the comets [as Galileo had claimed].” For instance, the speeds of comets do not fit Galileo’s theory. According to Galileo’s theory, the vapours causing the appearance of comets rise uniformly from the surface of the earth straight upwards. Therefore the comet should appear to be moving fast when it is close to the horizon, and then much slower when it is higher in the sky. Just imagine a red helium balloon released by a child at a carnival: it first it shoots off quickly, but soon you can barely tell if it’s rising anymore, even though it keep going up at more or less the same speed, because your distance and angle of sight is so different. But comets do not behave like that. Detailed observations of the comet of 1618 showed a much more constant speed than Galileo’s hypothesis requires.
Now let’s see how Galileo responded to this. Not by improving the scientific quality of his arguments, mind you. But with some clever rhetorical tricks that has many readers fooled to this day. Many find Galileo’s rousing mockery of his opponent so satisfying that they are seduced into celebrating it as proof of Galileo’s philosophical acumen. You can read Galileo’s triumphant put-downs of his opponent and go “yeah, crush him!” It’s the same kind of pleasure as watching the villain get punched in the face in an action movie. But a little reflection shows that this hero-versus-villain dynamic that Galileo tries to cultivate is a dishonest fiction that has very little to do with reality.
One of Galileo’s most celebrated passages concerns eggs. The context is this. Grassi makes the absolutely correct point that comets, if they entered the earth’s atmosphere, would quickly heat up to very great temperatures due to the friction of the air. In support of this point, Grassi quotes a 10th-century Byzantine author, Suidas, who claimed that “The Babylonians whirl[ed] about eggs placed in slings … [and] by that force they also cooked the raw eggs.” Grassi also quotes passages describing similar phenomena in Ovid, Lucan, Lucretius, Virgil, and Seneca. And then he says: “For who believes that men who were the flower of erudition and speak here of things which were in daily use in military affairs would wish egregiously and impudently to lie? I am not one to cast this stone at those learned men.”
Galileo is unable to answer the substantive point. Indeed, he thinks comets entering the atmosphere would cool down because of the wind rather than heat up because of friction. Galileo is wrong and Grassi is right about the actual scientific issue about comets. But that’s nothing Galileo’s trademarked sophistry can’t work around. Galileo finds a way to “win” the debate anyway, without actually offering any correct scientific claim regarding the actual subject of comets. He does this by gloatingly attacking Grassi for relying on books rather than experimental evidence:
“If [Grassi] wants me to believe that the Babylonians cooked their eggs by whirling them in slings, … I reason as follows: If we do not achieve an effect which others formerly achieved, then it must be that in our operations we lack something that produced their success. And if there is just one single thing we lack, then that alone can be the true cause. Now we do not lack eggs, nor slings, nor sturdy fellows to whirl them; yet our eggs do not cook, but merely cool down faster if they happen to be hot. And since nothing is lacking to us except being Babylonians, then being Babylonians is the cause of the hardening of eggs, and not friction of the air. … Is it possible that [Grassi] has never observed the coolness produced on his face by the continual change of air when he is riding post? If he has, then how can he prefer to believe things related by other men as having happened two thousand years ago in Babylon rather than present events which he himself experiences?”
Like I said, not a few modern philosophers blindly and uncritically fall for Galileo’s rhetoric. Here’s a typical quote on this. It’s from the Wiley-Blackwell book “Philosophy of Science: An Historical Anthology.” Here’s what the editors of this popular textbook say about Galileo’s argument: “Galileo shot back with a blistering critique in which he pillories [Grassi] and articulates a tough-minded empiricism as an alternative to the mere citation of venerable authority.”
Galileo would no doubt be very pleased that so many readers still to this day come away with the impression that “tough-minded empiricism” is what sets him apart from his opponents. That is precisely the intended effect of his ploy. It has very little basis in reality, however. Just a few pages earlier in the same treatise, Grassi describes extensively various laboratory experiments he has carried out himself with regard to another point. “I decided that no industry or labor ought to be spared in order to prove this by many and very careful experiments,” says this supposed obstinate enemy of empirical science. So the notion that Galileo is the only one “tough-minded” enough to reject authority in favour of experiment is very far off the mark.
Even in the passage criticised, Grassi is clearly not engaged in “the mere citation of venerable authority.” Rather he honestly and openly cites sources purporting to truthfully report empirical information, just like any scientist today cites previous works without re-checking all the experiments personally. Grassi does not believe that these authors are automatically right because they are “venerable authorities.” Rather he explicitly considers the possibility that they are wrong, but estimates, quite reasonably, that they are probably right.
For that matter, Galileo himself was not above believing falsehoods on the basis of “venerable authorities.” We have seen him make an error of this type in his theory of tides. He had heard somewhere that high and low tide in Lisbon occurred twelve hours apart rather than six, and jumped at the chance to cite this false information as “evidence” for his erroneous theory. To take another example, Galileo also believed the ancient myth of Archimedes setting fire to enemy ships by means of mirrors focussing the rays of the sun. This myth is “credible,” Galileo says. Descartes sensibly took the opposite view.
Altogether, the simplistic contrast between Grassi the credulous believer in authority and Galileo the experimenter has little basis in fact. Galileo is scoring easy points with his taunts about the eggs, by dishonestly pretending that a simplistic point about empiricism was the crux of the matter.
It is worth keeping the context of the passage in mind. Indeed, the pro-Galileo interpretation I quoted above from the Wiley-Blackwell textbook comes with its own origin story:
“In the course of his career [Galileo] engaged in many controversies and made powerful enemies. One of those enemies was the Jesuit Grassi, who published an attack on some of Galileo’s works.”
This framing goes well with the notion of the “tough” Galileo bravely defending himself against “attacks” from the “powerful” establishment. But the reality is quite different. Grassi was not a “powerful enemy”: he was a middling college professor just like Galileo. And the conflict did not start with Grassi “attacking” Galileo, but precisely the other way around. Grassi published a fine lecture on comets in which he argued, correctly, that the absence of parallax shows that comets are beyond the moon. Galileo is not mentioned in this work. Galileo read Grassi’s lecture and filled the margins, as one scholar has observed, with an entire vocabulary’s worth of savage expletives. Buffoon, bumbling idiot, piece of utter stupidity, and so on.
Galileo then published an attack on Grassi which was not much more restrained than these marginal notes. Grassi replied to it. It is this reply that is called “an attack on some of Galileo’s works” in the pro-Galilean quotation above.
So, to sum up, Galileo’s celebrated “pillorying” of Grassi was not a “tough” defence against an “attack” on “some of his works” by “powerful enemies.” The “enemy” was not a “powerful” arm of “authority,” but a conscientious scholar who was right about comets based on good scientific arguments that Galileo rejected. And the enemy was not a cruel aggressor going after “some works” by Galileo unprovoked; rather, the “some works” in question was an aggressive attack initiated by Galileo in the first place. Furthermore, Galileo’s enemy did not favour venerable authority over empiricism, but rather based his analysis of comets on much more thorough empirical work than Galileo did.
Ok, that’s what I had to say about comets.
Let me tell you another story: Double stars. The telescope revealed the existence of “double stars,” meaning stars that had appeared as just a single point of light to the naked eye but then when you looked at them with good magnification in a telescope they turned out to consist of two separate stars.
Double stars had the potential to prove Copernicus right. This was pointed out to Galileo by his friend Castelli. Castelli was excited about double stars, because he hoped they could be used to prove that the earth moves around the sun because of how the double star would change appearance in the course of a year.
The idea is the following. You look at the double star in your telescope. You see that it is not one star but two: one bigger and one smaller. Now you make the assumption that probably all stars are pretty much the same. They are all just so many suns, as it were. So the smaller-looking one is probably about the same size, in reality. It’s just further away.
Now let’s see what happens when the earth moves. Let’s try to picture this. You can use your index fingers. Hold up one finger in front of you. Now put your other index finger further away from you but aligned with the first one in a single line of sight. Now if you move your head slightly to one side, you will see the two fingers “move apart,” so to speak. And if you move your head to the other side, they will move apart in the other direction. So the closer finger, which corresponds to the bigger star, is sometimes to the left and sometimes to the right of the other one. Moving your head means moving the earth. If the earth is truly moving like Copernicus said then we should be able to observe this kind of thing: stars “switching places” in this way. This would certainly not happen if the earth was stationary, so we have striking and undeniable evidence for the motion of the earth.
This is a parallax effect. We spoke about parallax before. Astronomers had failed to detect parallax in the past, even though Copernican theory predicts that parallax must be a thing. The traditional method to look for parallax was based on trying to detect subtle shifts in the relative position of stars using tricky precision measurements of angles. The double star case would prove the matter in a much more striking and immediate way, without the need for technical measurements: anyone would be able to see with their own eyes the undeniable fact the the two stars switched places in the course of a year. And since with this method everything takes place within the field of view of the telescope, there was reason to hope that this new technology could enable success where conventional naked-eye astronomy had failed.
Castelli urged Galileo to make observations of double stars for this purpose, as indeed Galileo did in 1617, when he made detailed observations of the double star Mizar. Galileo used the above principle that however many times smaller a star is, it is that many times further away. With this method Galileo estimated that Mizar A and B were 300 and 450 times further away than the sun, respectively. This means the above effect should easily be noticeable: “Mizar A and Mizar B should have swung around each other dramatically as Galileo observed them over time.” But that didn’t happen. They didn’t change position at all. Everything remained exactly stationary, as if the earth did not move.
Today we know that all the stars in the night sky are much further away than Galileo estimated, and much too far away for any effects of this sort to be detectable with the telescopes of Galileo’s time. Galileo’s distance estimates were way off because of certain optical effects that make it impossible to judge the distances of stars in the manner outlined above. It would be anachronistic to blame Galileo for not knowing these things, which were only understood much later.
But Galileo’s way of discussing the matter in the Dialogue is not above reproach. He describes the above procedure but frames it hypothetically: “if some tiny star were found by the telescope quite close to some of the larger ones,” they would, if the above effect could be observed, “appear in court to give witness to such motion … of the earth.” “This is the very idea that later won Galileo renown and for which he was to be remembered by parallax hunters in the centuries that followed. While it is generally thought that Galileo never tried to detect stellar parallax himself, he is credited with this legacy to future generations.” In reality he deserves no renown, because the idea was not his own. It had already been explained to him in detail not only by Castelli, who discovered the double star Mizar and explained its importance for parallax to Galileo, but also even earlier by Ramponi in 1611. There is no indication that Galileo had though of any of this before his friends explained it to him.
Furthermore, Galileo’s discussion in the Dialogue is deceitful. He didn’t want to state the truth, of course, which is that he tried the experiment and it came out the wrong way; the data said that the earth did not move. But that’s only important if you are an honest scientist concerned with objectively evaluating the evidence. Galileo instead finds it more convenient to pretend that this falsifying data doesn’t exists. Instead he presents the double star idea as a suggestion for further research, and pretends that he hasn’t already carried it out. That way he doesn’t have to explain actual data or engage seriously with actual current astronomy like the system of Tycho for instance which agreed better with this data. It was much easier for Galileo to suppress his data and disingenuously insinuate that the outcome of the observation would be the opposite of what he knew it to be.
Now I will turn to another topic. The rings of Saturn. We all know that iconic cartoon-planet look. But that image only became clear some twenty years after Galileo’s death. Christiaan Huygens published a book on Saturn in 1659 where the rings are depicted with perfect clarity just as we are used to seeing it.
But the telescopes of Galileo’s day were not good enough to show the rings of Saturn with any clarity. Instead Galileo thinks the rings are actually two moons. Saturn is “made of three stars,” says Galileo. The planet has two “ears,” as it were. We can’t blame Galileo for limitations that were inherent to his time. It was no fault of his that he didn’t discern the rings of Saturn. Neither did any of his contemporaries.
However, we can blame Galileo for his lack of balance in evaluating the evidence. He does not say, as an honest scientist might, that his theory about Saturn’s “companion stars” is the best guess on the available evidence and that we can’t know for sure until we have better telescopes. Instead he boldly proclaims it as certainty that Saturn is “accompanied by two stars on its sides,” “as perfect instruments reveal to perfect eyes.” Those are Galileo’s words. And they are of course very hubristic. But that’s Galileo for you, always overstating his case, not least when he is wrong.
In the same vein, Galileo overconfidently declared that the appearance of Saturn’s companions would never change:
“I, who have examined [Saturn] a thousand times at different times, with an excellent instrument, can assure you that no change at all is perceived in him: and the same reason … can render us certain that, likewise, there will be none.”
Bombastic certainty as usual. All the more embarrassing then when in fact the appearances did change radically soon thereafter. Here’s Galileo again, just a few months later:
“I found [Saturn] solitary without the assistance of the supporting stars. … Now what is to be said about such a strange metamorphosis? Perhaps the two smaller stars … have vanished and fled suddenly? Perhaps Saturn has devoured his own children?”
This is a reference to classical mythology. Saturn the god “devoured his newborn children to forestall a prophecy that he would be overthrown by one of his sons.”
In any case, one moment Galileo says that “thousands” of observations prove that Saturn’s companion stars will never change, and then just months later he has to admit that, whoops, it turns out that that exact thing he said would never happen actually took place almost right away. That was some bad publicity, especially at a time when many doubted the reliability of his telescope.
The so-called disappearance of Saturn’s ring was due to the earth passing through the plane of the ring, so that a line of sight from earth was parallel to the plane of the ring. This made the ring invisible, just like a sheet of paper becomes vanishingly thin if you look at it exactly sideways.
But Galileo did not interpret it that way. Instead, he proposed what he considered to be some “probable conjectures” about the future appearance of Saturn’s companion stars. This theory was based on attributing to them a slow revolution, like very slow-moving moons. Later he praised himself for “thinking in my own special way” and marvelled at how “I took the courage” to make such brave conjectures. Those are Galileo’s own words, praising himself.
Indeed, Galileo liked his model so much that he also “took the courage” to lie about having made an observation verifying it. He claims that he “saw Saturn triple-bodied this year [1612], at about the time of the summer solstice.” But modern calculations show that the ring of Saturn would have been vanishingly thin at this time. There was a paper on this in the Journal for History of Astronomy not long ago. Here is the conclusion from the paper: “Clearly [Galileo] could not have observed the ring at the summer solstice of 1612. … Yet the picture of the Saturnian system that was accepted by Galileo implied that the ring should have been visible, so much so that he made a claim to this effect that we know must have been untrue.” Oh well. That’s business as usual in Galileo land.
This concludes our discussion of Galileo’s work with the telescope. Next time I believe we shall have to get to the real hot potato: Galileo and the church.
Telescopic observations of Venus provided evidence for the Copernican view of the solar system. But was Galileo the first to see this, as he claims? Or did he steal the idea from a colleague and lie about having made the observations months before?
Transcript
Galileo and the phases of Venus: it’s a plot that mirrors that of a murder mystery. Some scholars accuse Galileo of a crime—of falsifying data. The circumstantial evidence is enough to make anyone suspicious. If he’s innocent, he had remarkably bad luck with the timing of certain events; the worst possible coincidences for him. But he has an alibi! Huh, conflicting evidence. It’s a head-scratcher. Most historians these days believe Galileo. They think he’s innocent. But let’s see if we can’t poke some holes in their story.
First some background. The planet Venus is our closest neighbour in the solar system. But what kind of thing is Venus actually? To the naked eye, all we see is a dot of light. Is it a fire? A big thing of metal or diamond maybe? Some kind of mirror? It doesn’t seem likely to be just a plain old rock. Rocks don’t sparkle like that, do they? It’s so bright, you’d think it would have to be its own light source. You could speculate all day long. All of that was anyone’s guess before the telescope.
Now let’s look at Venus through a telescope. Aha! It’s a rock after all! It actually doesn’t shine with its own light, only reflected light from the sun. It’s a big round thing, a sphere, and only half of it is bright at any given moment, namely the half facing the sun.
This says a lot about the geometry of the universe. It’s telling us the relative spatial position of Venus and the sun. Back in the old days there was no way you could tell just by looking which thing in the sky was closer than the other. But now you can. Is Venus on the far side of the sun or nearer to us? Just look at which way the bright half of it is pointing and you have the answer. If you look at Venus and it’s like a new moon—mostly dark but with a crescent sliver of light on one side—then the bright side of Venus is evidently facing away from us, and hence Venus is between us and the sun. Sometimes it’s the other way around. Venus is on the far side, and then the bright half of it is facing us, so we see it almost completely lit up, like a full moon.
Those are the phases of Venus. They are just like the phases of the moon. Just like you have new moon, half moon, full moon, so you have “half Venus,” “full Venus,” and so on.
This could never happen in the Ptolemaic, geocentric system of the cosmos. In that system, the orbit of Venus is enclosed within the orbit of the sun. But the phases of Venus show that Venus is sometime on our side of the sun and sometimes on the far side, interchangeably. That’s impossible in the old system. So the phases of Venus are a great argument for the new Copernican system. Although they are also consistent with the hybrid system of Tycho Brahe.
But now, Galileo. What was his role in this? Of course he tried to claim credit for all of this. But does he deserve any? Most scholars think: yes. But I’m going to challenge conventional wisdom on this point.
Here are the key facts, which are very intriguing. The phases of Venus were discovered in 1610. But look at the timeline, it raises a lot of questions.
The first documented record we have is from December 5th. It’s a letter from Castelli to Galileo. Castelli was Galileo’s former student and a close friend. Castelli’s letter explains perfectly clearly the idea of using the phases of Venus to confirm heliocentrism, just as I outlined above. If Copernicus was right and Ptolemy was wrong, then it should be possible to prove this by means of the phases of Venus. But Castelli is not making observations himself, he is suggesting that Galileo make them. Remember, the telescope is almost brand new. Telescopic astronomy is less than a year old at this point.
There is no record whatsoever that Galileo knew anything about the phases of Venus before being told about it by Castelli on December 5th. But look what happens next: on December 11th, less than one week later, Galileo suddenly announces his great discovery, the phases of Venus. Galileo’s own words are that this something “just observed by me which involves the outcome of the most important issue in astronomy and, in particular, contains in itself a strong argument for the … Copernican system.” “By me”: Galileo is unequivocally claiming credit for himself, even though the timing is super suspicious. Did Galileo in fact steal the idea from Castelli? Very possibly.
But it gets more complicated. On December 30th, so another three weeks later, Galileo gave for the first time an account of his observations of Venus. At this point he claims to have observed Venus for about three months, and gives an accurate and fairly detailed description of its appearance during this period. So either he’s telling the truth and he already knew about the very important phases of Venus and he was just sitting on it for a while and just happened to receive Castelli’s letter at just about the time he was about to go public anyway. Or else he’s lying and he secretly made up those observations to boost his case.
There’s another letter, from November 13th of the same year, where Galileo seems to state expressly that he had no new planetary discoveries to report, which implies that he did not yet know about the phases of Venus, contrary to his later assertions. His defenders have a way of explaining this. When Galileo says “I haven’t discovered anything new about the planets,” these modern scholars are saying: yes, but nothing new “about” the planets really means nothing new “around” the planets, which means no new moons. So even though Galileo had made a very important new discovery concerning the planets, namely the phases of Venus, that was not a discovery “about” the planets, you see. So it comes down to a point of Italian linguistics how tenable that interpretation is. I’m not sure about it.
But it would make sense for Galileo to be on the lookout for moons. He had found moons around Jupiter and Saturn, and that was a huge deal. But that’s all the more reason for him to miss Venus’ phases. At this time, during these fall and winter months, Venus was well over half full. Modern astronomers can calculate backwards and know exactly the state of the planets so we know what Galileo would have seen. The shape of Venus would not have been very remarkable at this time, at the time when Galileo says he started to observe it. Unless you were specifically paying attention to it you may well think is was just a round blob. Galileo could easily have missed it; he could have failed to see that the shape was a bit off from a perfect circle. Especially if he was too busy moon hunting and looking only “around” the planets.
Another thing we have to take into account here are postal service delivery times. Castelli signed his letter December 5th and Galileo’s letter announcing the same idea to various colleagues is dated six days later. Is that enough time? Historians have argued about it. According to Westfall, it is “easily possible” that Galileo could have received the letter before December 11, while Stillman Drake, on the other hand, finds the probability of this “vanishingly small.” I guess we will never know. It is also possible that Galileo would have postdated his latter by a day or two just in case. He obviously realised the urgency of being first with these kinds of things.
Now, what about the details of Galileo’s observational report? If Galileo had not observed the phases of Venus before Castelli’s letter, then how could he later give an accurate description of their appearance dating back two months before this letter? Of course he could have fabricated data and passed it off as actual observations. We know that he did on many occasions, as we have discussed before. That’s business as usual for Galileo. It’s a fact that nobody disputes that Galileo published fake data on several occasions.
And if there ever was a time to fake data it was now. Galileo was of course concerned to get the important pro-Copernican argument from the phases of Venus on the record as quickly as possible and claim it for himself. And for this purpose it would be important to have observed the “full” appearance of Venus in the fall. Toward the end of the year it is transitioning to a crescent phase, which is no good, because that’s consistent with Ptolemy. The next opportunity to observe it in its full phase would be months away.
So Galileo certainly had a strong motive to fabricate this data. Making observations throughout most of December, after receiving Castelli’s letter, would also have been enough to give him great confidence that the heliocentric explanation for the phases of Venus was right. So faking the data was not risky.
Galileo’s defenders have a counterargument to this. They claim that Galileo could not have fabricated the data in question even if he had wanted to. According to them, the changes in appearance of Venus during these months were so complex and “non-linear” that Galileo could never have given such an accurate account if he had not if fact made these observations. Specifically, Galileo correctly describes the fact that the transition from a full to semicircular phase is quite rapid, while a roughly semicircular phase lingers for a considerable time. Here is what Palmieri says in a rather recent paper:
“Castelli’s letter cannot have been the spark that ignited Galileo’s programme of observation of Venus. It was simply too late. If he only then had started observing Venus, he would have seen it already nearing the exact semicircular phase, thus completely missing the non-linear patterns of change. And he could not possibly have been able to calculate the duration of one month for the “lingering” phenomenon. In other words, Galileo cannot have predicted Venus’s non-linear patterns of behaviour by re-constructing them ‘backwards’. For a Copernican it might have been easy to predict that Venus should display phases. However, it is one thing to predict this type of behaviour qualitatively and quite another to predict the non-linear patterns of change of Venus’s phases. A quantitative analysis would have required of Galileo a sophisticated mathematical theory that he did not have. There remains only one possibility, namely, that Galileo really did observe Venus’s non-linear patterns of behaviour.”
I say: this is wrong. On the contrary, Galileo could easily have reconstructed these phenomena. He would not have needed any sophisticated mathematics as all. All he would have had to do would have been to simulate the appearance of Venus with a simple physical model. Just take a sphere and paint half of it black and half of it white, and then look at it from different vantage points corresponding to the Earth’s position relative to Venus. That way you can simulate looking at Venus through a telescope very well without any need for sophisticated calculations.
I carried out such a simulation myself, using very simple means. I went to IKEA and bought a white spherical lamp. That was my model Venus. Then I had some black masking tape that I used to cover half of the sphere, to represent the half not illuminated by the Sun. I went to the office on a weekend and built a solar system in an empty parking lot. I picked one point to represent the sun, and then I put my Venus model in a certain position and then for the earth I used a camera so that I could take pictures. So I positioned these things according to a simplistic Copernican theory, just like Galileo could have done. I assumed for simplicity that the orbits of Venus and the earth are perfect circles and that their orbital speeds are constant. Estimates for these distances and speeds were common knowledge in Galileo’s time. Venus was seen exactly semicircular on December 18, right before Galileo’s observational report, which is very convenient for calibrating the initial position of this setup. Then you can just calculate backwards from there and move Venus and the earth back however many degrees they would cover in one month, two months, and so on. Just put a protractor at the sun and mark off those degrees.
Galileo could easily have completed such a simulation from start to finish in just a few hours, just as I did. Galileo would not have needed much imagination to come up with this scheme. The idea of illustrating the phases of the moon by an illuminated or half-painted sphere had been commonplace since antiquity, of course. It is a very obvious idea.
The results of my simple simulation are very close to the true appearances. Modern astronomers have calculated these appearances with great precision. If you put their figures next to my photos from the parking lot and you see that it’s the same thing. And crucially, the simulation is easily sufficient to reproduce the allegedly so unpredictable “non-linear” phenomena that Galileo got right in his December 30 report. So the claim that it would have been impossible for Galileo to recreate these appearances after the fact is definitely false. It would have been very much possible, in fact easy, for Galileo to recreate these appearances that he had not actually observed.
Another article on this has argued that Galileo’s account has “the ring of a record of visual impressions rather than an account coloured by calculations” in that it “has a highly visual character.” That’s supposed to show that Galileo did make actual observations. But obviously these markers of actual visual experience are consistent with my simulation hypothesis just as well as with actual observations.
In fact, my hypothesis makes a lot of sense if we compare it with Galileo’s treatment of sunspots. If my hypothesis is correct, these two cases are strikingly similar in several key respects. We discussed sunspots before. The following are the key facts for now. Galileo realised that sunspots constituted an important pro-Copernican argument only quite late, based on the input of others. He needed to act fast in writing something about it without having the time for thorough observations. I suggest that this is an exact parallel of what happened also in the case of the phases of Venus.
This parallel undermines the common assumption that Castelli’s idea must already have been obvious to Galileo. One scholar, for example, thinks it “would be to dignify the idea beyond reasonable measure” to view Castelli’s suggestion as a significant insight; rather, “the thought that Venus might have phases was ‘in the air’” and hence Castelli’s contribution is to be considered quite trifling. Another historian argues along similar lines that Galileo had no need to be spurred to action by Castelli’s letter, only by news of others making advanced telescopic observations. Around a day or two before hearing from Castelli, Galileo had received another letter, reporting that Clavius and his assistants at Rome had observed the moons of Jupiter. So Galileo now had serious competitors in the realm of advanced telescopic observations, or so it would have seemed to him. Presumably they would turn to the other planets next, and perhaps anticipate the discovery of the phases of Venus, so that would explain Galileo’s sudden urgency. According to this scholar, “the problem was to have a good telescope, not to posses reasoning power that astronomers had never lacked.”
These two scholars are wrong. The sunspots case is a counterexample to their claim. If there was no shortage of “reasoning power,” as they maintain, then Galileo should have realised the potential importance of sunspots much earlier and not let himself be beaten to the punch about their curved appearance by his arch-rival Scheiner. The fact of the matter is that the sunspots argument for heliocentrism eluded Galileo for twenty years, despite the fact the was passionately committed to proving heliocentrism in novel ways, and despite the fact that he himself had written specifically and in detail about the very phenomenon at stake, and despite the fact that the argument is very simple.
By analogy, this suggests that Castelli’s idea about Venus could very well have been news to Galileo. If Galileo could somehow miss the sunspots argument for twenty years despite all of this, then he could certainly have failed to think of the Venus argument during his one initial frantic year of telescopic observations, when he had a myriad other novelties and issues to deal with all at once.
But perhaps the most interesting aspect of the parallel between the two cases is the possibility that they both involved the use of physical models to simulate celestial appearances. In Galileo’s Dialogue, one speaker reports regarding the appearances of the paths traced on the surface of the sun by sunspots as seen from the Earth that Galileo “assisted my understanding by representing the facts for me upon a material instrument, which was nothing but an astronomical sphere, making use of some of its circles—though a different use from that which they ordinarily serve.”
The same sentiment is repeated later in the Dialogue: the appearances of the sunspot paths “will become better fixed in my mind when I examine them by placing a globe at this tilt and then looking at it from various angles.” This is very closely analogous to the Venus simulation I outlined above, suggesting that the latter would have been quite natural to Galileo, and in keeping with his style of reasoning.
So, to sum up, the following are generally accepted facts about the sunspots case: Galileo claimed to have conducted careful observations when he had not; according to his own account, Galileo simulated observations by looking at a physical sphere from a variable vantage point corresponding to the position of the Earth; Galileo failed to see an important pro-Copernican argument for a long time, despite it being simple and very naturally connected to his own work. These things did happen in the sunspots case. That’s a fact. Which obviously suggests that they very well could have happened also in the Venus case.
In conclusion, if Galileo had wanted to fabricate or reconstruct Venus observations he had not made, he could easily have done so. His December 30 account, where he describes appearances of Venus going back to October, is perfectly consistent with the hypothesis that he only started serious observations after receiving Castelli’s letter in December and simulated earlier observations using a simple physical model. There are, as we have seen, furthermore a number of circumstantial indications that this would have been very much in keeping with his character and habits.
Galileo thought sunspots were one of the three best arguments for heliocentrism. He was wrong.
Transcript
The early days of telescopic astronomy were exhilarating. Listen to this anecdote by Kepler. He is writing in 1610, right after the appearance of Galileo’s first telescope reports. Here’s what Kepler says: “My friend the Baron Wakher von Wachenfels drove up to my door and started shouting excitedly from his carriage: ‘Is it true? Is it really true that he has found stars moving around stars?’ I told him that it was indeed so, and only then did he enter the house.”
Those were the good old days. People shouting in the streets in excitement over scientific discoveries. “Stars moving around stars,” the quote says: that’s moons moving around planets, as we would say today. The moons of Jupiter, in fact. And the “he” who has discovered such things, that’s Galileo.
It’s a big deal, “stars moving around stars,” because it proves that there are several centers of motion in the universe. Not everything revolves around the earth. The earth is not the midpoint of every motion, as the old geocentric vision of the cosmos would have it. Of course in the heliocentric system of Copernicus there are already different centers of motion, because the moon goes around the earth while the planets go around the sun. If you wanted to stick to the old view you could try to argue that it made more sense for everything to have one center, for some metaphysical reason or other. But now that there are moons of Jupiter, that ruins that. Multiple centers of motion are a fact, so this can no longer be used as an argument against Copernicus.
That’s all very important and exciting, but what exactly was Galileo’s role in this? Well, it seems Galileo was indeed the first to observe the moons of Jupiter, but only by the smallest possible margin. His competitor Simon Marius observed them the very next day. In any case one hardly qualifies as the “Father of Modern Science” just by looking.
Nor does Galileo’s account stand out for its scientific excellence. For instance, he tries to “correct” Marius regarding the inclination of the orbits of the moon of Jupiter. Marius found that these orbits sit at an angle to the orbital plane of Jupiter itself. Galileo claimed, no no, they are actually perfectly parallel to Jupiter’s own orbit. But Galileo is wrong, and his opponent is right. As so often happens in such matters. Galileo has no patience for painstaking observations. He oversimplifies, relies on rhetorical points. He is more interested in writing polemics, claiming credit, proving others wrong. He doesn’t have so much time for actual science.
Galileo also tried to do mathematics to the moons of Jupiter, and failed at that too. Amazingly, “Galileo’s first calculations [of the orbital periods of Jupiter’s moons] were geocentric, not heliocentric. … Galileo was treating Jupiter as if it revolved around the Earth, not the Sun. How he ever came to make such an error … is an interesting question,” says one historian. It is interesting, indeed. And we know the answer, don’t we? Galileo couldn’t calculate himself out of a paper bag.
Kepler, meanwhile, of course understood the matter perfectly and realised at once that you need a heliocentric calculation for this to work. The fact that the calculations only work this way is in fact another good argument against the geocentric system.
One Galileo supporter offers a very charitable interpretation of why Galileo didn’t see this: “this throws in doubt the view that by 1611 Galileo was already a Copernican zealot anxious to find every possible argument for the Earth’s motion.” Right. So Galileo didn’t use heliocentric calculations because he was so open-minded, you see. That makes no sense, of course. Galileo was most definitely a “zealot.” A more plausible explanation, in my opinion, is that it was not lack of desire to prove the earth’s motion that made Galileo miss the point, it was lack of ability in mathematical astronomy.
Anyway, let’s change the subject and look at another important novelty discovered when astronomers first turned telescopes to the heavens: sunspots. The sun, the image of brilliance and clarity, turns out to have black spots on it when you study it under the telescope. “Filth on the cheeks of the Sun,” as one contemporary called it. Another was equally disturbed: “Who does not blush that we see the sun occasionally disfigured?”
In Latin there is a saying: adversus solem ne loquitor. Do not speak against the sun. That is, do not argue against what is the clearest and most perfect thing imaginable. How disturbing then that the time had come to in fact argue against the sun. Nothing is sacred as science advances, apparently.
The earliest recorded telescopic observation of sunspots are by Harriot. Before Galileo. Soon many more astronomers across Europe joined in the craze. Of course it is dangerous to stare at the sun, and all the more so through a telescope. Indeed, some people “neglected to observe them, being afraid, … that the image might burn my eye,” as one contemporary put it. But others figured that God had given them two eyes for a reason and “burned” them alternately in the name of science. Thus Harriot “saw it twise or thrise, once with the right ey & [the] other time with the left,” before “the Sonne was to cleare.”
Soon a method was developed for projecting the telescopic image of the sun onto a piece of paper so that no burning of the eyes was needed. That’s convenient enough, but even without this trick there would have been no shortage of martyrs of science willing to pay with their eyes for wisdom.
“Galileo insisted to his dying day that he was the first to have seen” sunspots. But in reality he was “probably preceded by [others, including] the Dutch astronomer Johann Fabricius, who was the first to publish information about them.” To boost his priority case, Galileo later claimed he had seen sunspots already in 1610, rather than in 1611 as documented, but this is almost certainly a lie, as even Galileo’s supporters admit. For that matter, even pre-telescopic astronomers had noticed the phenomenon of sunspots. Already in antiquity there are some allusion to sunspots. It is not impossible to see large sunspots without a telescope.
In any case, the game was now on to explain the nature of the spots. Galileo’s main competitor on this point was Christoph Scheiner, a German astronomer. Scheiner was concerned “to liberate the Sun’s body entirely from the insult of spots,” as he said, for “who would dare call the Sun false?” He found a way of accomplishing this by arguing that the sunspots were “many miniature moons,” rather than blemishes on the sun itself. So, perfection restored.
Galileo on the other hand eagerly embraced sunspots as an opportunity to stick it to his Aristotelian enemies, “for this novelty appears to be the final judgement of their philosophy,” as he said. Thus Galileo placed the spots on the sun itself, arguing that “clouds about the Sun” was the most plausible explanation, for one “would not find anything known to us that resembles them more.” It is true that sunspots are on the surface of the sun—a conclusion, incidentally, which Kepler had already reached before reading Galileo. However, sunspots are not clouds above the surface of the sun, as Galileo believed, but rather dark pits or depressions in the solar surface. “Scheiner … entertained the possibility of this hypothesis, while Galileo resolutely discarded it as unworthy of serious consideration.” Oh well.
Galileo loved claiming new discoveries as his own and using them as ammunition in his philosophical disputes. But he soon lost interest when it came to the detailed work of actual science. Again and again he makes careless errors and jumps to conclusions with premature confidence, while his competitor Scheiner does the meticulous observational work that Galileo had no patience for. For instance, Galileo erroneously claimed—supposedly based on “a great number of most diligent observations of this particular”—that all sunspots had the same orbital period, regardless of latitude. In fact, sunspots near the sun’s equator orbit quicker than those near the poles—a difference of a few days. Galileo was corrected by Scheiner about this.
Galileo also did not miss the opportunity to make some mathematical errors as usual. He tried to compute the perspective aspect of the sunspots’ motion: how does their apparent speed along the sun’s disc vary, given that their actual direction of motion turns more away from us the closer they get to the edge? Galileo’s attempted demonstration covers three pages and contains at least as many errors. Just basic geometry mistakes.
But let’s turn to the most important thing about sunspots. Namely that they can be used as evidence that the earth moves around the sun. In his Dialogue, Galileo considered this one of his three best arguments in favour of Copernicus. Here’s what he says: “The sun has shown itself unwilling to stand alone in evading the confirmation of so important a conclusion [that is, the conclusion that the earth orbits the sun], and instead wants to be the greatest witness of all to this.”
The Copernican argument from sunspots goes as follows. Imagine a standard globe of the earth standing on a table. Its axis is a bit tilted, of course—the north pole is not pointing straight up. Have a seat at one side of the table and face the globe. What do you see? Focus on the equator. What kind of shape is it? If the north pole is facing in your direction, the equator will make a “happy mouth” or U shape. If you move to the opposite side of the table, where you see mostly the southern hemisphere, the equator is instead a sad mouth shape. From the sides, the equator looks like a diagonal line.
Now, suppose the sun had its equator marked on it. And suppose that in the course of a year we would see it as alternately as a happy mouth, straight diagonal, sad mouth, straight diagonal, etc. That would correspond exactly to us moving around the table, looking at a stationary globe from different vantage points. In the same way, if the sun’s equator exhibited those appearances, the most natural explanation would be that the earth is moving around it and we are looking at its equator slightly from above, from the side, from below, etc., just like looking at the globe from different sides of the table.
The sun does not have the equator conveniently marked on its surface, but not far from it. The sun is spinning rather quickly. It makes a full turn in less than a month. As it spins, a point on its surface traces out an equatorial or at least latitude circle. So by tracking the paths of sunspots over the course of a few weeks, we in effect see equatorial and other latitude circles being marked on the surface of the sun.
So the shapes of those paths traced by sunspots show that we are indeed looking at the sun from alternating vantage points. But this does not necessarily mean that we are moving around the sun. The same phenomena could be accounted for from a geostatic or Ptolemaic point of view by saying that the sun is so to speak wobbling. It is showing us different sides of itself in the course of a year, not because we are moving around it but because it is turning different parts of itself in our direction.
You can see the same thing with your globe on the table. Instead of moving around the table and looking at the globe from different sides, you can have a friend tilt the globe, pointing its axis now this way and now that. If you let the axis spin around in a conical motion, this will produce the exact same visual impressions for you as if you had moved around the table.
In order to use the sunspot paths as evidence for Copernicus, then, Galileo needed to dismiss this alternative explanation. He did so by attacking it as physically implausible. To account for the sunspots phenomena from a Ptolemaic point of view, the sun had to orbit the earth, and spin on its own axis, and have its axis wobble in a conical motion. These diverse motions, says Galileo are “so incongruous with each other and yet necessarily all attributable to the single body of the sun.” Surely this is a geometrical fiction that would never happen in an actual physical body.
Actually, such an “incongruous” combination of motions is not only possible but a plain fact. The earth, in fact, has exactly such a combination of motions, as had been known since Copernicus. The earth has a conical wobbling motion which means that the north pole is pointing to a slightly different spot among the stars from year to year. It circles back to its original spot after 26,000 years. This is the explanation for the so-called precession of the equinoxes, an important technical aspect of classical astronomy. So if Galileo’s argument about “incongruous” motions disproves the Ptolemaic explanation of sunspots, it also disproves Copernicus’s correct explanation of the precession of the equinoxes.
Galileo conveniently neglects to bring up this rather obvious problem with his argument. Whether he did so out of ignorance or dishonesty is hard to say, but either way is none too flattering. Any serious mathematical astronomer was well acquainted with the precession of the equinoxes and of course considered it an essential requirement that any serious astronomical system account for this phenomenon. Galileo, though, is not a serious mathematical astronomer. He is a simplistic populariser who simply ignores technical aspects like these. And it is only because of this oversimplification that he is able to maintain his argument against the Ptolemaic interpretation of sunspots.
Or if you like here’s another way we can counter Galileo’s argument. Is it in fact really necessary at all to say that sun’s axis is wobbling in the geocentric explanation of sunspots? Yes and no. It depends on what you mean when you say that one body orbits another. We can picture this with the globe on the table that we considered before. Let’s say you climb onto the table and go sit in the middle of it. Now you grab the globe and make it orbit around you. So you are simulating the hypothesis that the earth is moving around the sun, let’s say. But how exactly do you move the earth? In what way? This will turn out to be more subtle than you might think. You just move the globe around in a big circle, what’s the problem, right? No. Big problem.
Let’s say that you start out with the globe in front of you with its north pole pointing slightly away from you. Remember, the axis is not purely vertical, it’s at an angle. So it’s pointing away from you. So you’re seeing more of Australia than of Siberia or something like that. Now move the globe in a circular orbit around you. Imagine doing this physically. What happens to the axis? Is it still pointing the same way? When you’ve gone halfway around, is the axis pointing toward you now? Or still away from you? Both are quite reasonable ways of conceiving orbital motion.
We can put it this way. Suppose you are not actually touching the globe at all. Instead the globe is standing on a big sheet of paper that covers the entire table. If you want to simulate the orbital motion of the globe around you, how do you move the paper? There are two ways. You can spin the paper around the midpoint, like a roulette wheel. So the midpoint of the paper, where you’re sitting, remains fixed and everything else is spinning around it. Think about what this means for the axis of the globe. If this is how the globe is moved, the axis will keep pointing away from you throughout the entire motion. If the north pole is pointing away or outwards to begin with, it will keep pointing outwards in all the other positions along the orbit as well.
But then there’s a second way you can move the paper. You can slide the entire paper around, without keeping the midpoint fixed. Think of the way you wipe the table with a dishcloth. You put your hand palm down on the table and you make circles with it. You could move the paper that way. Then the globe would go in circles. But if you do it this way, the way the axis of the globe behaves is completely different. If you slide the paper around this way, the axis keeps pointing toward the same end of the room, not the same direction relative to the midpoint of the table as before. So now when you’ve completed half the orbit the north pole is pointing towards you. Unlike before, when it kept pointing away from you.
So those are the two options. What does it mean for a globe to orbit a certain central point? Does it mean that the axis is always pointing the same way with respect to the central point? That’s the roulette wheel case. Or does it mean that the axis is always pointing the same way with respect to the walls of the room? That’s the dishcloth case.
Now, the sunspots. Galileo’s argument that I discussed above assumes that orbital motion is roulette wheel motion and not dishcloth motion. If we put a globe on a roulette wheel and spit it around, it will always show us the same part of itself. So you wouldn’t see that sad-mouth, happy-mouth alteration that happens when you are looking at the equator from above or from below alternately. Instead you would just see for example the sad-mouth equator all the time. But the sunspots show that we do see the equator from different angles, so if we assume that the sun is orbiting the earth, and that orbiting means orbiting like on a roulette wheel, then yes, we must indeed attribute one more motion to the sun; a wobbling of its axis in addition to its orbital motion. This is what Galileo attacks: the multitude of different motions needed. This is what we refuted with the argument about the precession of the equinoxes.
But if we are prepared to say that orbital motion means dishcloth-style motion then Galileo’s objection goes away. In this case the sun will indeed show us happy and sad mouths automatically, without the need for any additional motions. So that makes refuting Galileo even easier.
Now, from a modern point of view, dishcloth motion is a good way to think about orbital motion. The seasons—spring, summer, winter—they are caused by the earth’s axis pointing sometimes toward the sun and sometimes away from the sun. If we were on a roulette wheel the axis would always point for example away and there would always be summer in Australia for example year round. So it makes more sense to think of orbital motion as dishcloth motion. Also from the point of view of Newtonian mechanics, with inertia and stuff, that makes more sense. Gravity doesn’t “turn” things, so to speak.
But in Galileo’s time roulette-wheel motion was the default assumption. Even Copernicus stuck with this, which was a missed opportunity really. It was a kind of relic of older conceptions. Remember the crystalline spheres. Planets are embedded in solid, translucent, spherical shells. The planets each have one, and they fit together like the layers of an onion. The orbital motions of the planets are just a side-effect of the rotations of these entire shells in which they are embedded. So indeed that implies roulette-wheel motion. So before you had Newtonian mechanics this was the most reasonable way of conceptualising the physics of planetary motion.
Anyway, that’s fun to think about but it doesn’t change our point regarding the sunspots. The important point is the one regarding the precession of the equinoxes. This certainly disproves Galileo’s argument. Messing around with the distinction between roulette-wheel and dishcloth motion is more anachronistic and a bit of a tangent that I include for completeness and intrinsic interest. And in any case, bringing that in certainly wouldn’t save Galileo but on the contrary it would completely remove the entire basis for his argument altogether. Either way Galileo loses.
Ok, so we sorted that out. Now let’s get back to history. It is striking that, in the 1610s, when he was first studying sunspots, Galileo completely missed all of this stuff. I don’t mean the point about the precession of the equinoxes, which he never acknowledged at any stage. But even the very idea that sunspots can support heliocentric theory. That entire idea was missed by Galileo for decades. It just didn’t occur to him. And it was his own sloppiness that cost him this discovery. Let’s see how.
As I said, Galileo lacked the disposition to do painstaking scientific research like Scheiner. Instead, with premature hubris, he soon imagined that he had “looked into and demonstrated everything that human reason could attain to” regarding sunspots. Those are his own bombastic words. Many years later he was still convinced that his was the last word on the matter: as one historian observes, “writing … apropos of recent news that Scheiner would soon publish a thick folio volume on sunspots, [Galileo] remarked that any such book would surely be filled with irrelevancies, as there was no more to be said on the subject than he had already published in his Letters on Sunspots.”
But Galileo’s arrogance proved unfounded. When Scheiner’s much better work on sunspots came out, Galileo realised he had to completely reverse his earlier proclamations, even though he had stated those things with such confidence. With unwarranted pomposity, he had claimed to offer “observations and drawings of the solar spots, ones of absolute precision, in their shapes as well as in their daily changes in position, without a hairsbreadth of error.” According to Galileo, the sunspots were “describing lines on the face of the sun”: “they travel across the body of the sun … in parallel lines.” In fact, “I do not judge that the revolution of the spots is oblique to the plane of the ecliptic, in which the earth lies.” In other words, every sunspot path is straight as an arrow, just as the equator of a globe would be from every side if its axis was perfectly vertical.
But Scheiner showed that the sun’s axis in fact does have a inclination of just over 7 degrees and that the paths exhibit exactly those alternating diagonal and U shapes that we discussed before. He published this result in 1630, in the folio Galileo had mocked as bound to be superfluous. It was from this work that Galileo now realised his error.
When Galileo finally realised that inclined sunspot paths spoke in favour of heliocentrism, he immediately threw all his old observations out the window. These were the observations he had called “without a hairsbreadth of error,” if you recall. Galileo had been so proud of those observations for decades, but now they contradicted the point he wanted to make regarding heliocentrism, so he pretended they didn’t exist. Galileo has a very lax relation to empirical data as usual. One minute his observations are “without a hairsbreadth of error” but the next thing you know the Facts According to Galileo have changed radically into something completely different that is in direct contradiction with his own explicit statements just moths before.
Anyway, now that he had made up his mind about which way he was supposed to fudge the data, Galileo rushed the pro-Copernican argument into print without making any new observations. This is clear from the fact that the published argument “displays entire ignorance or complete neglect of the observational data,” his vague descriptions being “utterly wrong” and “almost the exact opposite” of the careful data published by Scheiner. Those are quotes from Stillman Drake, Galileo’s greatest admirer. “Ignorant” and “utterly wrong”—those are very harsh words from your greatest supporter!
But that actually is the most charitable reading of Galileo. Stillman Drake, who is always trying to save Galileo, is driven to call Galileo ignorant in order to avoid an even greater disgrace: namely, that Galileo plagiarised Scheiner’s book and then tried to pass these things off as his own discoveries. So Galileo’s defence lawyer is saving him from the charge of plagiarism by pointing out that Galileo’s account has a thousand errors in it, while Scheiner’s does not. If Galileo was a plagiarist, how come his book stinks and gets all the facts wrong? That’s some “defence,” isn’t it? But there you have it.
Indeed, Galileo did not want to admit his debt to Scheiner, so he pretended that he had come upon this discovery independently. He lied that he had made, as he says, “very careful observations for many, many months, and noting with consummate accuracy the paths of various spots at different times of the year, we found the results to accord exactly with the predictions.” In reality, says one modern scholar, “the evidence is unequivocal: Galileo … must have had a copy of Scheiner’s book in front of him as he wrote this section.” By pretending otherwise, “Galileo has deliberately set out to efface Scheiner from the historical record and to deny his debt to him. It is impossible to find any excuse for this behaviour.”
So let’s stop there with those apt words from David Wootton. But there’s plenty more “inexcusable behaviour” coming up; we haven’t even discussed all of Galileo’s shenanigans with the telescope yet. Until next time.
The telescope offered a shortcut to stardom for Galileo. We offer some fun cynical twists on the standard story.
Transcript
The year is 1609. What a time to be alive. In London you can go to the theatre and catch the fresh new play Macbeth. In Amsterdam you can make a quick buck trading in stocks—a brand new invention. Science is on fire as well. Kepler’s Astronomia Nova is published this very year—an exquisite masterpiece demonstrating that planets move in elliptical orbits among other things. Quite clearly the single greatest scientific work since the time of Archimedes.
So many exciting things happening. How will you keep up with this whirlwind of innovations and fascinating developments? Perhaps with the aid of another newfangled invention: the newspaper. The first of which comes out right this year, promising in its title to cover for its German readers all “gedenckwürdigen Historien”—thoughtworthy events. Truly it is a time of the new. The winds of change are blowing throughout Europe. Thoughtworthy events are everywhere you look.
What about our friend Galileo? What is he up to at this time? You won’t find him in any of those chronicles of thoughtworthy events. Galileo is already well into his middle age. He is a frail man of 45, not infrequently bedridden with rheumatic or arthritic pains. He is stuck teaching basic geometry for a pittance of a salary in some backwater town. Had Galileo died from his many ailments in this year, 1609, he would have been all but forgotten today. He would have been an insignificant footnote in the history of science, no more memorable than a hundred of his contemporaries. It has often been said that mathematics is a young man’s game. Newton had his annus mirabilis in his early twenties—”the prime of my age for invention,” as he later said. Kepler was the same age when he finished his first masterpiece, the Mysterium Cosmographicum of 1596. Galileo was already nearly twice this age, and he had nothing to show for it but some confused piles of notes of highly dubious value. In short, as a mathematician the ageing Galileo had proved little except his own mediocrity.
It is this middle-aged, run-of-the-mill nobody that first hears of a new invention: the telescope. Now here was his chance at last. He only had to point this contraption to the skies and record what he saw. No need anymore for mathematical talent or painstaking scientific investigations. For twenty years Galileo had tried and failed to gain scientific fame the hard way, but now a bounty of it lay ripe for the plunder. All you needed was eyes and being first.
The mysterious new “optical tube for seeing things close,” as it was called, was the talk of the town at the time. Galileo first hears about it in July 1609. A week or two later a traveller offered one for sale in Padua and Venice at an outrageous price—about twice Galileo’s yearly salary. This enterprising salesman found no takers for his offer. But the sense of opportunity remained in the air. And it was an opportunity tailor-made for Galileo: finally a path to scientific fame that required only handiwork and none of that tiresome thinking in which he was so deficient.
The design of telescopes was still a trade secret among the Dutch spectacle-makers who had stumbled upon the discovery. But acting fast was of the essence. Making a basic telescope is not rocket science. Soon many people figured out how to make their own. “It took no special talent or unique inventiveness to come up with the idea that combining two different lenses … would create a device allowing people to see faraway objects enlarged.” Reading glasses and magnifying glasses were already in common use: they obviously made text and other things appear bigger. They were used for thread count in the cloth business for example. So it was not a far-fetched idea to lenses them to magnify more distant objects as well. And the external shape of the telescopes people reported seeing suggested that at least two lenses were combined in a long cylinder. It didn’t take a genius, therefore, to soon strike upon the simple recipe Galileo found: take one convex and one concave lens and stick them in a tube, and look through the concave end. That’s it. No theoretical knowledge of optics played any part in this; it was purely a matter of hands-on craftsmanship and trial-and-error. As Galileo himself basically admitted.
About a month after first hearing of the telescope, Galileo has managed to build his own, with 8 times magnification. A bit later, maybe 12 times magnification, eventually 16 or so. If you go to a modern toy store or sports good store and but whatever cheapest binoculars they have, that will have the same magnification as a Galilean telescope basically. So if you have an old pair of field binoculars lying around, you basically have a Galilean telescope. So dust it off, why don’t you, and follow along with your own observations as we describe what Galileo found.
In any case, Galileo’s first goal is to leverage the telescope into a more lucrative appointment for himself. He gives demonstrations to various important dignitaries—”to the infinite amazement of all,” according to himself. So on the basis of this Galileo enters multiple negotiations about improved career prospects. Between hands-on optical trials and lens grinding, these showmanship demonstrations, shrewd self-marketing and hyperbole about how “infinitely amazed” everyone is by him, Galileo must have had a busy couple of months indeed. And on top of this marketing campaign and juggling potential job offers, his regular teaching duties were just starting again in the fall.
So we can easily understand why, in these hectic days, the scientific importance of the new instrument for astronomy was not realised right away. At first neither Galileo nor anyone else thought of the telescope as primarily an astronomical instrument. Galileo instead tried to market it as “a thing of inestimable value in all business and every undertaking at sea or on land,” such as spotting a ship early on the horizon. But the moon does make an obvious object of observation, especially at night when there is little else to look at. Perhaps indeed moon observations were part of Galileo’s sales pitch routine more or less from the outset, though as a gimmick rather than science.
But this was soon to change. In the dark of winter, the black night sky is less bashful with its secrets than in summer. It monopolises the visible world from dinner to breakfast; it seems so eager to be seen that it would be rude not to look. In January, Galileo takes up the invitation and spots moons around Jupiter. Yikes! Other planets have moons?! This changes everything. Suddenly it is clear that the telescope is the key to a revolution in astronomy. Eternal scientific fame is there for the taking for whoever is the first to plant his flag on the shores of this terra incognita.
For the next two months Galileo goes on a frenzied race against the clock. He writes during the day and raids of the heavens for one precious secret after another at night. In early March he has cobbled together enough to claim the main pearls of the heavens for himself. He rushes his little booklet into print with the greatest haste: the last observation entry is dated March 2, and only ten days later the book is coming off the presses. Remarkable. It’s a turnaround time modern academic publishers can only dream of, even though they do not have to work with hand-set metal type and copper engravings for the illustrations.
It was a race against the clock and Galileo won. “I thank God from the bottom of my heart that he has pleased to make me the sole initial observer of so many astounding things, concealed for all the ages.” So wrote Galileo, and his palpable relief is fully justified. Little more than dumb luck—or, as he would have it, the grace of God—separated Galileo from numerous other telescopic pioneers who also produced telescopes and made the same discoveries independently of Galileo. For example, Simon Marius in Germany who discovered the moons of Jupiter one single day later than Galileo. As one historian observes, “a delay of only three or four months would have set [Galileo] behind several of his rivals and undercut his claim to priority regarding several key discoveries with the telescope.” Perhaps it was not the grace of God, but Galileo’s desperation, born of decades of impotence as a mathematician, that drove him to publish first. Being incapable of making any contribution to the mathematical side of science and astronomy, Galileo needed and craved this shortcut to stardom more than anyone else.
Accordingly, Galileo greedily sought to milk every last drop of fame he could from the telescope. “I do not wish to show the proper method of making them to anyone”; rather “I hope to win some fame.” Those are Galileo’s own words. His competitors quickly realised that, as one contemporary says, “we must resign ourselves to obtaining the invention without [Galileo’s] help.” Still six years after his booklet of discoveries, people who thought science should be a shared and egalitarian enterprise were rightly upset by Galileo’s selfish quest for personal glory. One writes as follow to Galileo: “How long will you keep us on the tenterhooks? You promised in your Sidereal Message to let us know how to make a telescope so that we could see all the things that are invisible to the naked eye, and you haven’t done it to the present day.”
Meanwhile Galileo never missed a chance to mock stuffy Aristotelian professors for thinking “that truth is to be discovered, not in the world or in nature, but by comparing texts”, Galileo wrote in scorn, adding that “I use their own words.” His opponents themselves had stated that “comparing texts” was their methodology. But if Galileo genuinely wanted them to turn to nature he could have shared his techniques for telescope construction. In truth it served his own interests very well that these people were left with no choice but “comparing texts” while he claimed the novelties of the heavens for himself.
Let’s look at Galileo’s professional situation in a bit more detail. You may have heard that Galileo was a “Professor of Mathematics.” Indeed he was, for twenty years. But we must not let the title fool us. The position had nothing to do with the research frontier in the field. In modern terms Galileo’s position was more comparable to that of high school teacher. Galileo taught very basic and practical courses. His lectures were unoriginal and usually cribbed from standard sources. His mathematical lectures went no further than elementary Euclidean geometry. He also had to teach a basic astronomy course “mainly for medical students, who had to be able to cast horoscopes.” They “needed it to determine when [and when] not to bleed a patient” and the like. Perhaps Galileo didn’t mind, for he seems to have been quite open to astrology judging by the fact that he cast horoscopes for his own family members and friends without renumeration. Alas, he did not enjoy much success as an astrologer. Here’s a quote from The Cambridge Companion to Galileo: “In 1609, Galileo … cast a horoscope for the Grand Duke Ferdinand I, foretelling a long and happy life. The Duke died a few months later.” That’s great, isn’t it? Such a nice bit of deadpan there by The Cambridge Companion.
Galileo was eager to get out of his lowly university post. Now with the telescope he was in a decent negotiating position. After much scheming he resigned from the university and took up a court appointment. You would rather work for some rich guy, a patron, than at a university. That was how it went at the time. Some decades later Leibniz for example did the same thing. He could easily have taken a university job but who wants to be an academic when you can be the resident scholar in the gilded halls of some prince?
So Galileo got his wish. His new appointment freed him from teaching duties and boosted his finances. But Galileo also had an additional demand. Here is what he says:
“I desire that in addition to the title of mathematician His Highness will annex that of philosopher; for I may claim to have studied more years in philosophy than months in pure mathematics.”
This is traditionally taken as a request for a kind of promotion: in addition to being a great mathematician, Galileo also wanted recognition in philosophy, which in some circles was considered more prestigious and in any case included what today is called science (then “natural philosophy”). But I think a more literal reading of Galileo’s request is in order. Galileo is not only declaring himself a philosopher; he is also confessing his ignorance in mathematics. Taken literally, his statement that he has spent “more years in philosophy than months in mathematics” implies that he could not have spent more than two or three years at most on mathematics—which indeed sounds about right considering his documented mediocrity in this field.
Anyway, back to the telescope. So Galileo had some success with it clearly, but not everyone was convinced.
Some believed “the telescope carries spectres to the eyes and deludes the mind with various images … bewitched and deformed.” Perhaps these peculiar “Dutch glasses” were but a cousin of the gypsy soothsayer’s crystal ball? The “transmigration into heaven, even whil’st we remain here upon earth in the flesh,” as Robert Hooke put it, may indeed seem like so much black magic. Add to this the very numerous imperfections of early telescopes, which often made it very difficult even for sympathetic friends to confirm observations, not to mention gave ample ammunition to outright sceptics.
Indeed, we find Galileo on the defensive right from the outset, just a few pages into his first booklet. Seen through the telescope, the moon appeared to have enormous mountains and craters—a big deal, allegedly one of “Galileo’s” monumental discoveries. This was based on shadow effects. Looking at the moon when it’s half full, you see that the surface is uneven because of the shadows cast by mountains and craters. But already there are big problems. The boundary of the moon was still perfectly smooth. A crazy inconsistency. How can there be big mountains in the middle of the moon, but none along the edge? It doesn’t make any sense, yet that’s what it looked like.
Here are Galileo’s own words in the Sidereus Nuncius of 1610, his famous booklet and first claim to fame. “I am told that many have serious reservations on this point”: for if the surface of the moon is “full of … countless bumps and depressions,” then “why is the whole periphery of the full Moon not seen to be uneven, rough and sinuous?” Galileo replies that this is because the Moon has an atmosphere, which “stop[s] our sight from penetrating to the actual body of the Moon” at the edge only, since there “our visual rays cut it obliquely.” So when we look at the edge of the moon our line of sight spends more time passing through the atmosphere of the moon and that’s why it’s blurred. Hence it is “obvious,” says Galileo, that “not only the Earth but the Moon also is surrounded by a vaporous sphere.” This is of course completely wrong.
So already we see that there were serious problems with the telescope. It’s not as simple as saying: the telescope showed everyone new facts. What was a fact and what was an inference or an illusion? Not a trivial question, and as we see Galileo himself got it wrong right off the bat.
And there’s plenty more where that came from. Another puzzling fact was that the planets were magnified by the telescope, but not the stars. The stars remained the same point-sized light spots no matter what the strength the telescope. Some even mistook this for a “law that the enlargement appears less and less the farther away [the observed objects] are removed from the eye.” Galileo tried to explain these things, but once again he gets it completely wrong. A correct explanation was given in 1665, it’s a technical optics thing.
Clearly, then, in light of all these challenges to the reliability and consistency of the telescope, it was important to understand its basis in theoretical optics. That is why, presumably, Galileo felt obliged to swear at the outset, in the Sidereus Nuncius, that “on some other occasion we shall explain the entire theory of this instrument.” To those aware of his mathematical shortcomings, it will come as no surprise that Galileo never delivered on this promise. Kepler—a competent mathematician—took up the task instead, and in the process came up with a fundamentally new telescope design better than that of Galileo. That’s in Kepler’s Dioptrice of 1611. Kepler’s telescope uses two convex lenses instead of Galileo’s pair of one convex and one concave lens. According to Galileo, Kepler’s work was, in his own words, “so obscure that it would seem that the author did not understand it himself.” A modern scholar comments that “this is a curious statement since the Dioptrice, unlike other works by Kepler, is remarkably straightforward.” Apparently still not straightforward enough for a simpleton like Galileo, however. Indeed, Galileo’s naive conception of optics was still mired in the old notion that seeing involved rays of sight spreading outward from the eye rather than conversely. He repeatedly gave statements to this effect.
Regarding the mountains on the moon, let’s look a bit more at the significance of that, which has often been overstated. So Galileo’s famous discovery is, as he puts it, that “The moon is not robed in a smooth and polished surface but is in fact rough and uneven, covered everywhere, just like the earth’s surface, with huge prominences, deep valleys, and chasms.” Now, it is all too easy to cast this report by Galileo as a revolutionary discovery. The “Aristotelian” worldview rested on a sharp division between the sublunar and heavenly realm. Our pedestrian world is one of constant change—a bustling soup of the four elements (earth, water, air, fire) mixing and matching in fleeting configurations. The heavens, by contrast, were a pristine realm of perfection and immutability, governed by its very own fifth element entirely different from the physical stuff of our everyday world. If we are predisposed to view Galileo as the father of modern science, a pleasing narrative readily suggests itself: With his revolutionary discovery of mountains on the moon, Galileo disproved what “everybody” believed. Indeed this is a standard story peddled by many scholars. Let me quote two of them.
“Every educated person in the sixteenth century took as well-established fact … that the Moon was a very different sort of place from the Earth. … The lunar surface, according to the common wisdom, was supposed to be as smooth as the shaven head of a monk.”
Here’s another quote to the same effect. This one is from a Harvard University Press book from 2015, “Galileo’s Telescope”, so this stuff is mainsteam modern scholarship. Here is the quote:
“In those years virtually no one questioned the ontological difference between heaven and earth. … The difference between Earth and the heavenly bodies was an absolute truth for astrologers and astronomers, theologians and philosophers of every ilk and school. … If the Moon turned out to be covered with mountains, just like Earth, a millenary representation of the sky would be shattered.”
So in other words, Galileo sent an entire worldview crashing down by using data and hard facts to expose its prejudices.
The problem with this narrative lies in one word: “everybody.” The Aristotelian worldview is not what “everybody” believed. It is what one particular sect of philosophers believed. As ever, Galileo’s claim to fame rests on conflating the two. If we compare Galileo to this sect of fools—as Galileo wants us to do—then indeed he comes out looking pretty good. Members of this sect did indeed try to deny the mountainous character of the moon in back-pedalling desperation. For instance by arbitrarily postulating that the mountains were not on the surface of the moon at all but rather enclosed in a perfectly round, clear crystal ball. So that way the surface was smooth after all, even though there were shadows and stuff, because the shadows were in the interior of this glass sphere. If we mistake this kind of rubbish for the state of science of the day, then indeed Galileo will appear a great revolutionary hero.
But to anyone outside of that particular sect blinded by dogma, the idea of a mountainous moon had been perfectly natural for thousands of years. It is obvious to anyone who has ever looked at the moon that its surface is far from uniform. Clearly it has dark spots and light spots. If one wanted to maintain the Aristotelian theory one could try to argue, as many people indeed did, that this is perhaps some kind of marbling effect. The moon is still perfectly spherical, only it has some differential colouration like a smooth piece of marble. Or maybe it’s a reflection thing: perhaps the moon is so polished that it is reflective like a mirror. So the light and dark areas are not actual irregularities in the moon itself but just the mirror image of oceans and stuff on earth.
Whatever one thinks of the plausibility of such arguments, they are certainly defensive in nature: the Aristotelian theory is on the back foot trying to explain away even the most rudimentary phenomena that any child is familiar with. The idea of an irregular moon is an obvious and natural alternative explanation. Which is why we find for instance in Plutarch, a millennium and a half before Galileo, the suggestion that “the Moon is very uneven and rugged.” That’s a literal quote, from antiquity.
If we look to actual scientists and mathematically competent people instead of Aristotelian fools, we find that “Galileo’s” discovery of mountains on the moon was already accepted as fact long before. Kepler had already, and I quote him, “deduced that the body of the moon is dense … and with a rough surface,” or in other words the moon is “the kind of body that the earth is, uneven and mountainous.” Those are quotes from a 1604 work by Kepler. Before the telescope.
Kepler also points out that this was also the opinion of his teacher Maestlin before him, who, according to Kepler, “proves by many inferences … that [the moon] also got many of the features of the terrestrial globe, such as continents, seas, mountains, and air, or what somehow corresponds to them.” That’s from the Mysterium Cosmographicum, 1596, long before the telescope.
In a later edition of this work, Kepler added the note that “Galileo has at last throughly confirmed this belief with the Belgian telescope,” thereby vindicating “the consensus of many philosophers on this point throughout the ages, who have dared to be wise above the common herd.” Indeed, Galileo himself says his observations are reason to “revive the old Pythagorean opinion that the moon is like another earth.”
So, altogether, Galileo’s discovery of mountains on the moon was not a revolutionary refutation of what “everybody” thought they knew, but rather a vindication of what everyone with half a brain had seen for thousands of years.
The same goes for other supposed discoveries by Galileo relating to the moon. For instance the discovery of the phenomenon of “Earth shine”: like moon shine, but in the reverse direction. So reflected light from the earth lights up the moon to some extent. Galileo discusses this as one of the novelties made clear by the telescope, but in reality it had been correctly explained previously, by Kepler in 1604.
A similar reality check is in order regarding the idea one sometimes hear that Galileo’s discoveries regarding the moon instigated celestial physics. These people say: By revealing the similarity of heaven and earth, Galileo opened the door to a unification of terrestrial and celestial physics—in other words, he led us to the brink of Newton’s insight that a moon and an apple are governed by the very same gravitational force. In reality, though, a unity between terrestrial and celestial physics had been advocated since antiquity, as we have seen. You don’t need a telescope to realise that this idea makes sense.
Meanwhile, Galileo’s bumbling and superficial attempts to do celestial physics are an embarrassment to all, as we have seen. Remember his erroneous thing about planetary speeds being determined by falling from some faraway point toward the sun, or his completely wrongheaded calculation of how long it would take the moon to fall to the earth.
In fact, Kepler had already written an excellent book on celestial physics before the telescope: the Astronomia Nova of 1609. This is the work where Kepler explains the elliptical orbits of the planets (which Galileo never accepted or even mentioned). Kepler explains the elliptical orbits of the planets as the result of a quasi-magnetic force residing in the sun. So that’s certainly celestial physics in full swing before the telescope.
Ok, so that’s what I had to say about the telescope itself. Next we must turn to the impact of telescopic evidence on the debate between geocentrism and heliocentrism. That’s next time. Thank you.
Galileo is credited with defeating Ptolemaic earth-centered astronomy, but most mathematical astronomers had already abandoned this theory long before Galileo.
Transcript
Does the earth move around the sun, or is it the other way around? Copernicus worked out the right answer long before Galileo was even born, as did the best Greeks mathematicians thousands of years earlier. Yet somehow Galileo has ended up with much of the credit. Here’s a quote from the Cambridge Companion to Galileo: “If one wonders why the Copernican theory, [which had] almost no adherents at the beginning of the seventeenth century, had pretty much swept the field by the middle [of the century], the answer is above all [Galileo’s] Dialogue.” And here’s another quote from the Very Short Introduction to Copernicus: “Galileo wrote the book that won the war [and] made belief in a moving earth intellectually respectable.”
This standard story from mainstream scholarship may be right in a limited sense: maybe indeed the ignorant masses needed a book like Galileo’s to dumb it down for them before they could finally come to their senses. But mathematically competent people were already convinced long before and had no use for Galileo telling them the ABCs.
When Copernicus made the earth go around the sun, he confidently declared that “I have no doubt that talented and learned mathematicians will agree with me.” Those are his own words. And he was right. In 1600, long before Galileo has published a single word on the matter, there were already at least ten committed Copernicans, besides Copernicus himself. Westman lists them as follows: “Digges and Harriot in England; Giordano Bruno and Galileo in Italy; Diego de Zuniga in Spain; Simon Stevin in the Low Countries; and, in Germany, the largest group—Rheticus, Maestlin, Rothmann, and Kepler.”
That makes eleven in total. Eleven believers in the new astronomy. This is what the Cambridge Companion calls “almost no adherents.” But what do you expect? How many “talented and learned mathematicians” do you think there were in pest-ridden, blood-letting, witch-burning Europe of 1600? And how many of concerned themselves with the Copernican question and formed an opinion on it, even though that was a philosophical question beyond the scope of the official computational task of the astronomer? And, among those in turn, how many were prepared to declare allegiance to a flagrantly heretical opinion in an age where the religious thought-police routinely employed vicious torture and burnt dissenters alive? Including one guy on that very list I just read, in fact: Giordano Bruno, who was burned at the stake by the authorities, as punishment for his heretical views, which included heliocentrism (although it is unclear what weight that carried exactly in relation to his other thought crimes).
At any rate, given this context, I think eleven avowed Copernicans in 1600 is really quite a crowd. Indeed, consider Galileo’s own assessment of the situation in 1597: “I have preferred not to publish, intimidated by the fortune of our teacher Copernicus, who though he will be of immortal fame to some, is yet by an infinite number … laughed at and rejected … for such is the multitude of fools.”
This confirms that social pressures to avoid the issue were very real---enough to “intimidate” Galileo and surely many others. More importantly, Galileo is also making my main point for me: even at this very early stage---long before Galileo has published a single word on the matter, and well before the invention of the telescope---everyone with half a brain has rejected the old astronomy already. By Galileo’s own reckoning, there were only “fools” left to convince. I say that on this point he is exactly right.
Furthermore, there were doubtless many more closet Copernicans who figured “don’t ask, don’t tell” was the best policy to avoid needless conflicts with the intolerant. For example, Harriot had a number of followers in England who were enthusiastic believers in heliocentrism, but “the local political and intellectual milieu … forced them into what we can term preventive self-censorship,” one scholar has observed. Another example is Mersenne, in France. He does not go on the official list of Copernicans because, as one historian puts it, “at no time during his life did he find any proof so overwhelming that he felt like challenging the Church on the matter.” And this despite the fact that he was one of the most enthusiastic readers of Galileo’s Dialogue. He remained uncommitted for political reasons, it would seem.
Another indication that many were silently receptive to Copernicanism is the fact that most of the leading astronomers of the 16th century owned Copernicus’s book, and many of them wrote extensive notes in the margins, as was the habit at the time. Books were printed with enormous margins because everyone was expected to take detailed notes as they read. And indeed they did. Owen Gingerich conducted a thorough census of all surviving copies of Copernicus’s book. He looked at all this marginalia that this large group of serious, competent readers of Copernicus’s book has written. That’s a group far larger than those eleven I mentioned before. Some of them were probably secretly convinced that Copernicus was right; others studied the work because they saw it as they duty to keep up with the best technical mathematical astronomy of the day whether they agreed with it or not. Either way, they took meticulous notes as they painstakingly worked through this long and technical treatise.
But this group did not include Galileo, however. Galileo’s dilettantism is so blatant and shameless that Gingerich could hardly believe his eyes: “I had long supposed that Galileo was not the sort of astronomer who would have read Copernicus’ book to the very end. … Still, when I saw the copy in Florence, my reaction was one of scepticism that it was actually Galileo’s copy, since there were so few annotations in it. … This copy had no technical marginalia, in fact, no penned evidence that Galileo had actually read any substantial part of it. … Eventually, … I realized that my scepticism was unfounded and that it really was Galileo’s copy.” There is no need for surprise, of course. Galileo was a poor mathematician. He had neither the patience nor the ability to understand serious mathematical astronomy, let alone make any contribution to it.
Let’s have a look at what some of these more serious mathematical astronomers were up to. For example, Tycho Brahe, the Danish astronomer in the generation before Galileo. He also considered the question of whether it is the earth or the sun that moves. And he came up with a creative answer that is neither that of Ptolemy nor that of Copernicus.
He saw the strengths in the Copernican system, but he was worried about its drawbacks. Most importantly the problem of parallax, which we discussed before. If the earth moves in a big circle that implies that we are looking at the heavens from different points of view in the course of a year. This should be detectable when we study the positions of the stars. The angles between them should shrink and grow as we move around the sun, because our distance from any particular star configuration would change radically in the course of a year.
But this does not happen. The night sky is immutable. As far as 17th-century astronomers could detect, the constellations all look exactly the same throughout the year, just as if we never move an inch. Tycho Brahe was the most exacting astronomical observer in the pre-telescopic era, but even he, with his very advanced and precise observations, could find no parallactic effect.
To maintain the earth’s motion in spite of this, it is therefore necessary to postulate, as Copernicus does, that “the fixed stars … are at an immense height away.” The diameter of the earth’s orbit is so small in relation to such an astronomical distance that our feeble little motion is all but tantamount to standing still. That is why no parallax can be detected. This is the correct explanation, as we now know, but in the 16th century it didn’t sound too convincing.
Tycho Brahe was one of the sceptics. He calculated that, if Copernicus was right, the stars would have to be at least 700 times further away than Saturn. The universe would not have been designed with so much wasted space, he reasoned.
Tycho therefore devised a system of his own. In this system, the earth remained the center of the universe, while the planets orbit the sun. So it’s a hybrid of Ptolemy and Copernicus. A halfway house that takes the best of both worlds.
This solves the problem of parallax: since the earth is not moving, there shouldn’t be any parallax, so that’s that, problem solved. You can put the “sphere of the fixed stars” just beyond Saturn, like Ptolemy did. That’s how people used to view the stars: as a bunch of specks of shiny glitter glued onto the inside wall of a big ball. We can’t judge depth or distance of heavenly objects by eye anyway, so we might as well imagine everything taking place on a single surface like that. And if you adopt the cosmology of Ptolemy or Tycho, you can imagine this sphere as a natural container of the solar system. Everything fits snugly. Every bit of space has a purpose. Not so in Copernicus’s universe. If there is a glitter sphere of stars at all, it must be enormous. That’s just grotesque, isn’t it? It’s as if someone would make a huge clock face, Big Ben style, many meters across, but then put tiny wristwatch arms in the middle. It’s absurd. So point Tycho for avoiding that problem.
But what about the strengths of the Copernican system? We discussed this before. The advantages have to do with explanatory simplicity. It explains the retrograde motion of the outer planets, and the bounded elongation of the inner planets. But so does Tycho’s system. In fact, Tycho’s system is equivalent to the Copernican one as far as the relative positions of the heavenly bodies are concerned. Tycho and Copernicus describe the same planetary motions, but they choose a different reference point in terms of which to describe them. Kepler illustrates the point with an analogy: the same circle can be traced on a piece of paper by either rotating the pen arm of a compass around the fixed leg, or by keeping the compass fixed while rotating the paper underneath it. So the Copernican and Tychonic systems are by necessity on equal footing as far as the arguments regarding explanatory simplicity are concerned. Although Tycho’s system feels weirder, so to speak, there’s no denying that for all those purposes it is strictly equivalent to heliocentrism. That’s just a geometrical fact.
One might feel that the Tychonic system is less physically plausible than those of Ptolemy or Copernicus. Indeed, traditional conceptions had it that planets were enclosed in translucent crystalline spheres, like the layers of an onion. Both the Ptolemaic and Copernican systems are basically compatible with such an “onion” conception of the cosmos. The Tychonic system clearly is not: planets are crossing each other’s “orbs” all over the place. But Tycho had some good counterarguments to this. By a careful study of the paths of comets, he proved that they evidently passed through the alleged crystalline spheres with ease. Furthermore, he pointed out that these alleged spheres did not refract light, as glass and other materials had been known to do since antiquity. So Tycho has some decent arguments in defence of his system even in the domain of physics.
All in all, Tycho’s system was a serious scientific theory with good arguments to its credit. This is another reason why our headcount of Copernicans above is misleading. The number of people who rejected the Ptolemaic system was certainly greater than the number of outright Copernicans. The middle road put forth by Tycho was by no means blind conservatism but rather a viable system based on the latest mathematical astronomy.
Galileo, however, liked to pretend otherwise. The full title of his famous book reads: Dialogue Concerning the Two Chief World Systems: Ptolemaic and Copernican. Well, that certainly made Galileo’s life a lot easier. It was very convenient for him to frame his fictional debate with fictional opponents in those antiquated terms. That way he could battle two-thousand-year-old ideas instead of having to engage with the latests mathematical astronomy.
More serious and mathematically competent people had a very different view of which were “the two chief world systems.” Around 1600, long before Galileo enters the fray, Kepler considers it obvious that the Ptolemaic system is obsolete. This is Kepler writing before Galileo’s works:
“Today there is practically no one who would doubt what is common to the Copernican and Tychonic hypotheses, namely, that the sun is at the centre of motion of the five planets, and that this is the way things are in the heavens themselves---though in the meantime there is doubt from all sides about the motion or stability of the sun.”
Later, after the telescope has brought its new evidence, not much has changed. Kepler is a bit more assured that “today it is absolutely certain among all astronomers that all the planets revolve around the sun.” That’s Kepler writing in 1619, after the telescope. But even then the battle between Copernicus and Tycho remained far from settled: “either [of those two] hypotheses are today publicly accepted as most true, and the Ptolemaic as outmoded.” “The theologians may decide which of the two hypotheses … ---that of Copernicus or that of Brahe---should henceforth be regarded as valid [for] the old Ptolemaic is surely wrong.” Again, Kepler, writing more than a decade before Galileo’s famous book that dishonestly pretends that the Ptolemaic view is still one of “the two chief world systems.”
Nor was this merely Kepler’s opinion. Historians who study much more minor figures have also concluded that, indeed, “the Ptolemaic system already had been set aside, at least among mathematical astronomers,” well before Galileo wrote his Dialogue. But Galileo, in his great book, like a schoolyard bully secretly too scared to pick on someone his own size, preferred to pretend that the old Ptolemaic system was still the enemy of the day. To be sure, there were still a “multitude of fools” left to convince, and perhaps indeed Galileo did so more effectively than anyone else. But that proves at most that Galileo should be praised as a populariser, not as a scientist. To mathematically competent astronomers he was beating a dead horse.
Some scholars have made too much of the above arguments in favour of the Tychonic system, however. They have concluded that “it is fair to say that, contrary to [the standard view], science backed geocentrism.” That’s a quote from Christopher Graney’s book. Indeed, if you take the petty works of Galileo and his immediate opponents to be the extent of “science” then this conclusion does make some sense. But the conclusion is false if you include genuinely talented scientists like Kepler. Unlike Galileo, Kepler dared to take on the Tychonic system and he gave a long list of compelling arguments against it.
Tycho’s system is equivalent to the Copernican one in terms of relative position of the planets, but “that Copernicus is better able than Brahe to deal with celestial physics is proven in many ways,” says Kepler. And he’s right. He has a range of physics arguments, including the one I discussed before about the implausibility of a heavier body orbiting a lighter one.
Let’s consider in some more detail the relation between Kepler and Galileo. Kepler was the best mathematical astronomer in Galileo’s day. They were contemporaries. Kepler was seven years younger than Galileo. What did they think of each other? What were their relationship? Certainly not as substantive as one might expect.
As one historian says: “One wonders why these two great men, who were both present and actual participants at the very birth of some of the most world-shaking scientific events, and who apparently were very much in accord in their astronomical views, did not engage in a more on-going correspondence over these years.”
This is a puzzle and a paradox if one accepts the standard view of Galileo. But of course it becomes perfectly understandable as soon as one realises that Kepler, who was a brilliant mathematician, had very little to learn from a dilettante such as Galileo.
Their correspondence began when, “in 1597, as a lowly high-school teacher of mathematics and a fledgling author, Kepler … vainly implored Galileo, the established university professor, to give him the benefit of a judgment of his first major work.” Galileo replied briefly, declaring himself in agreement with the Copernican standpoint of Kepler’s book, and then this is where he gives that statement I already quoted, that “I have preferred not to publish, intimidated by the fortune of our teacher Copernicus, who though he will be of immortal fame to some, is yet by an infinite number … laughed at and rejected … for such is the multitude of fools.”
Kepler, in his reply, is happy to hear that Galileo, “like so many learned mathematicians,” has joined in supporting “the Copernican heresy,” as Kepler calls it. Those are Kepler’s words: “like so many learned mathematicians”. Galileo is late to the party. “So many learned mathematicians” got there before him.
Kepler goes on to say, in reply to Galileo’s admission of being “intimidated” by opponents, that Galileo should really grow a backbone. Here is what Kepler says: “For it is not only you Italians who do not believe that they move unless they feel it, but we in Germany, too, in no way make ourselves popular with this idea.” Kepler urges Galileo to focus on compelling mathematics instead of on the number of fools: “Not many good mathematicians in Europe will want to differ from us; such is the power of truth.”
This is very typical and illustrative of the outlooks of Kepler and Galileo. Galileo has his attention turned to “the multitude of fools,” as he calls them. The uneducated masses and their naive beliefs. From that point of view, fighting for heliocentrism is a huge uphill battle. Kepler on the other hand is more concerned with what other mathematicians think, what intelligent people who studied the matter is a serious way believe about the motion of the earth. From that point of view there is every reason to be very optimistic, as Kepler says, because all competent people had rejected Ptolemy’s astronomy already.
At this point, Kepler naively mistook Galileo for a serious scientific interlocutor. In connection with their discussion of Copernicanism, Kepler noted the importance of parallax and asked Galileo if he could help him with observations for this, adding detailed instructions regarding the exact nature and timing of the required measurements. Kepler also sent additional copies of his book, as Galileo had requested, and “asked only for a long letter of response as payment---which was, however, never forthcoming.” Galileo stopped replying, presumably since this kind of actual, substantive mathematical astronomy was beyond his abilities.
Kepler’s was not the only scientific correspondence Galileo shrunk from. He also neglected to reply to all three letters he received from Mersenne, for example, offering only “the rather limp excuse that he found Mersenne’s handwriting too hard to read.” It seems he had a point, for others complained similarly of Mersenne’s letters that “his hande is an Arabicke character to me,” as Cavendish said. Nevertheless those are further instances of Galileo failing to reply to a serious scientific interlocutor.
The tables were turned in 1610. While Galileo had not seen the greatness in Kepler’s book, more mathematically competent people had, and consequently Kepler had succeeded Tycho Brahe as the Imperial Mathematician of the self-declared Holy Roman Emperor in Prague. “In that capacity Kepler’s help was sorely needed by Galileo in 1610, when his momentous telescopic discoveries were being received on all sides with skepticism and hostility.” “To Kepler’s credit … he … swallowed his justifiable resentment” and “ungrundingly gave Galileo the authoritative support he could find nowhere else.” “In spite of Galileo’s earlier silence after his own request in 1597, Kepler quickly and enthusiastically responded to Galileo’s findings, within 11 days.”
Galileo surely had this in mind when, in reply, he praised Kepler for “your uprightness and loftiness of mind”---”you were the first one, and practically the only one, to have complete faith in my assertions” regarding the telescopic discoveries. Kepler’s support was indeed crucial, and Galileo keenly flaunted it to his advantage when promoting his work to others.
Galileo did not take the occasion to revive their scientific discussion or comment on Kepler’s brilliant new book, the Astronomia Nova of 1609, which proved the law of ellipses among other things. Instead Galileo only wanted to make fun of dumb philosophers:
“Oh, my dear Kepler, how I wish that we could have one hearty laugh together! Here at Padua is the principal professor of philosophy, whom I have repeatedly and urgently requested to look at the moon and planets thorough my glass, which he … refuses to do. Why are you not here? What shouts of laughter we should have at this glorious folly!”
Kepler wasn’t there because he was busy doing real science. He ignored idiotic philosophers, as all mathematically competent people had done for thousands of years. Galileo, however, has nothing better to do than to sit around and laugh at idiots, because he is not a serious scientist. He is a salon scientific poseur, to whom the most desirable application of science is a clever put-down and the last laugh.
Kepler eventually grew weary of Galileo’s dilettantism. When, in later years, he found himself having to correct errors in Galileo’s superficial writings, he fully justifiably took a patronising tone. Here is a quote from 1625:
“Galileo rejects Tycho’s argument that there are no celestial orbs with definite surfaces because there are no refractions of the stars. … Rays reach the earth perpendicular to the spheres, says Galileo, and perpendicular rays are not refracted. But oh, Galileo, if there are orbs, it is necessary that they be eccentric. Therefore, no rays perpendicular to the spheres reach to the earth except at apogee and perigee. Hence, Tycho’s argument is a strong one, if you are willing to listen.”
Indeed. So Galileo ignored all the mathematical details as usual and instead only addressed a simplistic straw-man version of Tycho’s actual argument. That’s the way it always goes with Galileo, and by now Kepler has realised as much.
Here’s another example of the same thing. Again Kepler has lost patience with Galileo’s shameless and self-serving rhetoric and sets the record straight as follows. This is a long quote from Kepler:
“Galileo denies that the Ptolemaic hypothesis could be refuted by Tycho, Copernicus, or others, and says that it was refuted only by Galileo [himself] through the use of the telescope for observation of the variation of the discs of Mars and Venus. … Nothing is more valuable than that observation of yours Galileo; nothing is more advantageous for the advancement of astronomy. Yet, with your indulgence, if I may state what I believe, it seems to me that you would be well advised to collect those thoughts of yours that go wandering from the course of reason …. This observation of yours … does not refute the very distinguished system of Ptolemy nor add to it. Indeed, this observation of yours refutes not the Ptolemaic system but rather, I say, it refutes the traditions of the Ptolemaics regarding the least difference of planetary diameters. … Your own observation of the discs confirms the proportion for the eccentric to the epicycle in Ptolemy, as it does the orbit of the sun in Tycho or of the orbis magnus in Copernicus.”
So even though Kepler favours the Copernican conclusions just like Galileo does, he, unlike Galileo, is honest when the evidence is consistent with either of the hypotheses. Kepler thinks it is the obligation of the scientist to consider alternative hypotheses seriously and to engage with them on their own terms, instead of opportunistically twisting everything to confirm one’s own preferred conclusion.
In both of these cases that I quoted, Kepler exposes Galileo’s true colours. Galileo doesn’t treat the matter as a serious mathematical astronomer, but rather as a superficial and unscrupulous rhetorician. Kepler is quite right to scold him as he does.
Two thousand years before Galileo, Greek astronomers argued that the heavenly bodies revolve around the sun. Their reasoning involved sophisticated mathematics and sound physical considerations.
Transcript
Is the earth the center of the universe? Or does it orbit around the sun? The Greeks had some interesting ideas about this. For example, the earth is basically a big rock, right? Well, have you ever seen a rock just hovering in empty space without being supported by anything? Of course not. That would be crazy. On the other hand, think of a rock in a sling. Like a slingshot, the ancient weapon. Like what David used against Goliath, for example. So the rock is at the end of a string and you are whirling it about in a circle. In this way you can in fact keep a heavy rock suspended in the air indefinitely. Without there being any ground or support on which it is resting.
This is a powerful argument in favour of the hypothesis that the earth revolves around the sun. Of course this theory requires that the sun, on the other hand, is in fact hovering in empty space. So aren’t we back to square one, the same problem we started with? No, not at all. Because the sun is made of fire, isn’t it? Fire has no problem levitating. It’s a weightless substance, as everybody knows. Just light a match and see how the flame rises completely unencumbered by gravity and feel how no weight was added to the match even though the flame has considerable volume.
So on the basis of this we can conclude that the earth moves around the sun. Because that agrees much better with everyday physical experience than the outlandish hypothesis that a massive chunk of rock is just sitting there in nothingness without falling.
As you may know, this is not the “official” Greek position, so to speak. It’s not Aristotle’s opinion, and it’s not Ptolemy’s opinion. Those are the main authorities that have come down to us on these issues. Aristotle the philosopher, Ptolemy the astronomer. They are usually taken to express the party line, as it were, of “the Greeks.” These people did indeed put the earth at the center of the universe.
But I want to do some revisionist history here and speak for the underdogs of Greek cosmology. Those guys had some good ideas. Unfortunately their works are lost. We have only some scraps to go by. But the indications we have are very intriguing.
In fact, this idea that I just described, that it’s unrealistic for a heavy rock to simply levitate in empty space, is mentioned by Ptolemy. He brings it up only in order to dismiss it, of course. But more interesting than Ptolemy’s counterarguments is what he is implicitly telling us about his opponents. We must reverse-engineer his text: What does Ptolemy’s dismissals of alternative views tell us about what those views must have been?
For example, Ptolemy tries to argue that, if the earth moved, any loose objects would be thrown off. Just like objects placed on a sleeping animal would immediately fall off as soon as it woke up and started running about. But from the way Ptolemy presents this argument it is clear that he by no means thinks this is self-evident and will be taken for granted by his readers. On the contrary, he addresses counterarguments to this view that show that his opponents, whoever they were, had a well articulated theory based on something like inertia or relativity of motion quite similar to how we today would explain why a moving earth does not throw things off.
It is interesting also that Ptolemy acknowledges the multitude of his opponents. To “all those who” believe in the earth’s motion, I reply as follows, he says. “All those”! Isn’t that an interesting phrase? Evidently there were lots of people who believed the earth moved. That’s what Ptolemy’s own words are saying. What a pity that we don’t have any books by “all those” rebels anymore.
Here’s another important conclusion we can draw from the above example: these forgotten astronomers clearly believed that the heavens should be explained in terms of everyday physics. The earth is like a rock in a sling, for example. That way of thinking was largely rejected in the Aristotelian tradition, where the heavens are seen as a profoundly different kind of thing altogether than what we encounter here on earth. The heavens are some kind of quasi-divine realm, made of some sublime fifth element, and so on. That’s the basic Aristotelian story. But the mathematicians did not care much for that fairytale.
Actually even the Aristotelian tradition made more concessions to heavenly physics than some people like to admit, in my opinion. The Aristotelian story is that the planets are enclosed in enormous crystalline spherical shells that fill up the entire universe like the layers of an onion. The planet itself is just a little speck stuck inside this translucent spherical shell like tiny imperfection in hand-blown glass.
Why did the Aristotelians feel the need for this fiction? Well, they didn’t believe vacuum could exist, so that’s one reason to fill the heavens with some material or other. But I think a more compelling reason was the physics of perpetual motion. Think about it: What kinds of sustained circular motions are you familiar with from everyday experience? Have you ever seen a rock just go in a circle over and over again spontaneously, without being guided by any other material objects? Of course not.
You have, however, seen many sustained circular motions where the object moving circularly is materially connected to the midpoint, such as a rock in a sling or a wheel. Or a millstone disc that has been set in rotational motion and then keeps going by inertia even after you have stopped applying force. A sphere behaves the same way. In fact, it can be a hollow sphere. Like a basketball spinning on your finger. You set in in motion and it keeps going, seemingly forever if it wasn’t for outside resistance slowing it down.
The Aristotelian “onion” universe fits with that physical intuition. The translucent layers of the onion spin on their axis forever just like a basketball. That is why the planets, which are embedded in the shells, go around and around. So it’s a model of how planetary motions work based on a mechanism that is familiar to everyone from everyday experience.
So that’s some physics even in the conservative Aristotelian view. Let’s see how far the more mathematically creative Greeks pushed the idea of a physical analysis of the heavens.
First of all it is quite evident that the spherical shape of the earth can be explained as a consequence of gravitational forces. In fact, Archimedes proves as a theorem in his hydrostatics that a spherical shape is the result or equilibrium outcome of basic gravitational assumptions. This idea, that gravitational forces are the cause of the spherical shape of the earth, is explicitly stated in ancient sources.
But of course other heavenly bodies are round too, such as the moon for example. So that very naturally suggests that they have their own gravity just like the earth. This conclusion too is explicitly spelled out in ancient sources. Here’s Plutarch: “The downward tendency of falling bodies is evidence not of the earth’s centrality but of the affinity and cohesion to earth of those bodies which when thrust away fall back again. … The way in which things here [fall] upon the earth suggests how in all probability things [on the moon] fall … upon the moon and remain there.”
Now, from this way of thinking, it is a short step to the idea that the heavenly bodies pull not only on nearby objects but also on each other. This is again explicit in ancient sources. This is why Seneca, for example, says that “if ever [these bodies] stop, they will fall upon one another.” That is correct, of course. The planets would “fall upon one another” if it wasn’t for their orbital speed.
This point of view explains the motions of the planets in terms of physical forces. It’s not that the planets have circularity of motion as an inherent attribute imbedded in their essence, as Aristotle would have it. Rather, circularity is a secondary effect, the result of the interaction of two primary forces: a tangential force from motion and a radial force from gravity. There are clear indications that ancient astronomers worked out such a theory, including a mathematical treatment.
Here is Vitruvius for example: “the sun’s powerful force attracts to itself the planets by means of rays projected in the shape of triangles; as if braking their forward movement or holding them back, the sun does not allow them to go forth but [forces them] to return to it.” Pliny says the same thing: planets are “prevented by a triangular solar ray from following a straight path.”
All this talk of triangles, in both of these authors, certainly suggests an underlying mathematical treatment. Indeed, the Greeks knew very well the parallelogram law for the composition of forces or displacements, and in fact explicitly used this to explain circular motion as the net result of a tangential and a radial motion.
Isn’t it beautiful how coherently all of that fits together and how naturally we were led from one idea to the other? Just like the water of the oceans naturally seeks a spherical shape, so the spherical shape of the earth has been formed by the same forces. And just as gravity explains why the earth is round, so it must explain why other planets are round. Hence they have gravity. But just as they attract nearby objects, so they attract each other. So the heavens have a perpetual tendency to lump itself up, except this tendency is counterbalanced by the tendency of speeding objects to shoot off in a straight line.
This physical view of the heavens clearly had much to commend it. And clearly the Greeks saw this, even though the original works are lost and we are left with only the kinds of scraps I quoted from the Roman era.
Let’s take a closer look at one of the earlier lost works in particular. Aristarchus. He was a quality mathematician. We know for a fact that Aristarchus wrote a treatise advocating the motion of the earth about the sun. Archimedes, who was a contemporary of Aristarchus, mentions this work, basically with tacit approval.
Archimedes brings this up in connection with a discussion of the size of the universe. Aristarchus’s theory implies that the universe must be very big. This is because of parallax, which means the following. If you move from one side of room to another, your view of everything on the walls will change. The wall you are approaching will appear to “grow,” so to speak, while the wall behind you will shrink and occupy a smaller part of your field of vision. This is what is called parallax. If the earth moves in an enormous circle around the sun, we should be at one moment close to some particular constellation of stars, and then half a year later much further away from them. Hence we should see them sometimes big and “up close,” and sometimes shrunk into a small area, like a faraway wall at the end of a long corridor. But no such effect is observed. The only way to reconcile this with the motion of the earth is to stipulate that the stars are so far away that the diameter of the earth’s orbit around the sun is insignificant by comparison. So the universe must be very big indeed for this theory to work.
From the remark about this in Archimedes we learn quite a lot, even though Aristarchus’s treatise is lost. We learn that Aristarchus’s theory was worked out in some detail. It was a serious scientific proposal. It grappled with nontrivial observational and theoretical implications in a substantive way. And it evidently did so quite convincingly. For why else would Archimedes take the theory seriously?
Many people refuse to believe this. For example, there was a paper on Aristarchus in the January 2018 issue of the Archive for History of Exact Sciences. According to this paper, “pre-Copernican heliocentrisms (that of Aristarchus, for example) have all the disadvantages and none of the advantages of Copernican heliocentrism,” because they postulated only that the earth revolves around the sun, not, as has commonly been assumed, that all the other planets do so as well. This supposedly “explains why Copernicus’s heliocentrism was accepted …, while pre-Copernican heliocentrism” was not.
This is completely wrong, in my opinion. And for an obvious reason. Namely: Why would Aristarchus have affirmed and written a treatise on heliocentrism if it had nothing but disadvantages? What possible reason could he have had done for doing so? None, in fact. Yet this is exactly what this recent article proposes.
It is a fact that Aristarchus asserted the physical reality of his hypothesis. And it is a fact that he recognised the parallax argument against it. Even the recent article I cited admits this. So why, then, would Aristarchus write a treatise proposing this bold hypothesis, discuss a major argument against it (namely the parallax argument) and no arguments in favour of it, and then nevertheless conclude that the hypothesis is true? And, furthermore, why would Archimedes, perhaps the greatest mathematician of all time, cite this treatise with tacit approval as a viable description of physical reality? None of that makes any sense.
The only reasonable explanation is that Aristarchus did in fact recognise an advantage of placing the sun in the center. And the obvious guess for what this was is that he saw the same advantages as Copernicus did. Including the more natural explanations of the retrograde motion of the outer planets and the bounded deviation from the sun of the inner planets that I have discussed before.
Now, Aristarchus’s treatise on heliocentrism is lost, as I said. However, another astronomical treatise by Aristarchus has survived. In this work Aristarchus calculates the relative distances and sizes of the sun, the earth, and the moon. This treatise shows that Aristarchus was at any rate a highly competent mathematician. But I think it shows much more than that. I think it feeds directly into his heliocentrism.
An important argument for heliocentrism is this: Smaller bodies orbit bigger ones. Not the other way around. This conforms with everyday experience. For instance, take a lead ball and a ping-pong ball, and tie them together with a string. If you flick the ping-pong ball it will start spinning around the lead ball. But if you flick the lead ball it will roll straight ahead without any regard for the ping-pong ball, which will simply be dragged along behind it. So the lighter object adapts its motion to the heaver one but not conversely. The planets are not tied to the sun with a string but the point generalises. You can observe the same principle with a big and a small magnet for example: the little one is moved by the bigger, not the other way around.
Kepler used this argument in the 17th century. Here is how he put it: “Just as Saturn, Jupiter, Mars, Venus, and Mercury are all smaller bodies than the solar body around which they revolve; so the moon is smaller than the Earth … [and] so the four [moons] of Jupiter are smaller than the body of Jupiter itself, around which they revolve. But if the sun moves, the sun which is the greatest … will revolve around the Earth which is smaller. Therefore it is more believable that the Earth, a small body, should revolve around the great body of the sun.”
The moons of Jupiter were not known in antiquity but other than that this is a basic idea that fits very well with their extensive attention to the physics of the heavens.
Surely this must have occurred to Aristarchus. Or are we supposed to believe that Aristarchus calculated the sizes of heavenly bodies just for kicks in one treatise and did not see any connection with the heliocentrism he advanced in another treatise even though the obvious connection was right under his nose? What is the probability that he suffered from such schizophrenia? Virtually zero, in my opinion.
In fact there are certain aspects Aristarchus’s treatise on sizes and distances that make much more sense when you read it this way. On its own it is a weird treatise. On the one hand it calculates the sizes and distances of the sun, moon, and earth in a mathematically sophisticated manner. Very detailed, technical stuff, including the completely pointless complication that the sun does not quite illuminate half the moon but only maybe 49.9% of its surface or something like that [correction: this should be ever so slightly more than 50%, not less, since the sun is larger than the moon]. This is “pure mathematical pedantry,” as Neugebauer calls it. It makes the geometrical calculations ten times more intricate while having only the most minuscule and completely insignificant impact on the final results.
On the other hand, the observational data that Aristarchus uses for his calculations are extremely crude. He says that the angular distance between the sun and the moon at half moon is 87 degrees. A pretty terrible value. The real value is more like 89.9 degrees. Because of this his results are way off. For instance, his calculated distance to the sun is off by a factor of 20 or so.
So what’s going on? Why do such intricate mathematics with such worthless data? Did he just care about the mathematical ideas and not about the actual numbers? I think it would be a mistake to jump to that simplistic conclusion, even though many people have done so.
In fact, it is easy to see how Aristarchus had a purpose in underestimating the angle. His purpose with the treatise, I propose, is to support his heliocentric cosmology based on the principle that smaller bodies orbit bigger ones.
This hypothesis fits very well with the structure of Aristarchus’s treatise. The treatise has 18 propositions. Proposition 16 says that the sun has a volume about 300 times greater than the earth, and Proposition 18, the very last proposition, says that the earth has a volume about 20 times greater than the moon. These are exactly the propositions you need to explain which body should orbit which. And that is exactly where Aristarchus chooses to end his treatise.
Many commentators have been puzzled by why Aristarchus ends in that strange place. In particular, many have been baffled by why he does not give distances and sizes in terms of earth radii. This seems like the natural and obvious thing to do, and doing so would have been easily within his reach. Many modern commentators add the small extra steps along the same lines needed to fill this obvious “gap.”
Except it’s not a gap at all and there is no need to be puzzled by Aristarchus’s choices. If we accept my hypothesis, everything he does makes perfect sense all of a sudden. He carries his calculations precisely as far as he needs for this purpose, and no further.
And my hypothesis also explains why he chose such a poor value for the angular measurement. He has every reason to purposefully use a value that is much too small. Underestimating this angle means that the size of the sun will underestimated. And his goal is to show that the sun is much bigger than the earth. So he has shown that even if we grossly underestimate the angle, the sun is still much bigger than the earth. So he has considered the worst case scenario for his desired conclusion, and he still comes out on top. That just makes his case all the stronger, of course.
Clearly my interpretation requires that Aristarchus knew that 87 degrees was an underestimate. The standard view in the literature is that he could not have known this. Aristarchus’s numerical data are “nothing but arithmetically convenient parameters, chosen without consideration for observational facts”. That’s a quote from Neugebauer. And he continues:
“It is obvious that [Aristarchus’s] fundamental idea … is totally impracticable. … 87 degrees is a purely fictitious number. … The actual value … must … [have] elude[d] direct determination by methods available to ancient observers.”
Neugebauer tries to prove this as follows. You are trying to measure the angle between the sun and the moon at half moon. But to do that you need to pinpoint the moment of half moon, which can only be done with an accuracy of maybe half a day. But in half a day the moon has moved six degrees, and therefore radically changed the angle you are trying to measure. So your observational value is going to have a margin of error of 6 degrees, which is enormous and makes the whole thing completely pointless.
That’s Neugebauer’s opinion. But I’m not convinced that it’s as hopeless as all that. One way to work around the problem would be to use not one single observation, but the average of many observations. There is little evidence that the Greeks ever made use of averaging that way, but the idea is simple enough.
I did a bit of statistics to see if this would be viable. Let’s assume that our angular measurements are normally distributed about the true value. Neugebauer says that one would be lucky to get the moment of half moon correct to the day. So we can tell it’s today rather than yesterday, but we can’t tell at what exact hour the moon is exactly half full. Let us roughly translate this into statistical terms by saying that the observations have a standard deviation of 12 hours, or six degrees.
Now, an astronomer active for, let’s say, two decades would have occasion to observe about 500 half moons. So say he makes 500 angular measurements and then average them. This would produce an estimate of the true value with a 95% confidence interval of plus/minus about half a degree. A margin of error of half a degree is easily enough to support my interpretation that 87 is a conscious underestimate.
Naturally, Aristarchus would not have reasoned in such terms exactly, but it is not necessary to know any statistical theory to get an intuitive sense of the order of magnitude of the error in such an estimate. As you keep adding observations, and keep averaging them, you will see the average stabilising over time, of course. So certainly it will become clear after a while that, whatever the true value is, it must be greater than 87 degrees at any rate.
It is certainly extremely speculative to imagine that Aristarchus might have had something like this in mind. But in any case my argument shows that it cannot be ruled out as out of the question that Aristarchus could in principle have had solid empirical evidence that his value of 87 degrees was certainly an underestimate.
So, in conclusion: Aristarchus was a good mathematician. He proposed a heliocentric theory that was taken very seriously by Archimedes. There was a long tradition in Greek thought of trying to account for the motions of the planets in terms of everyday physics. This is naturally connected to heliocentrism because of the natural idea that smaller bodies orbit bigger ones. Aristarchus in fact wrote a major treatise devoted specifically to comparing the sizes of the sun and the earth, and the earth and the moon. Several otherwise peculiar aspects of the treatise fit like a glove the idea that it was written precisely to lend credibility to heliocentrism.
On the whole, non-Ptolemaic Greek astronomy was fascinating. It was full of interesting and correct ideas. Nowadays we are stuck with Ptolemy as the canonical source for Greek astronomy. But Ptolemy lived hundreds of years after the golden age of Greek science. It is likely that he was not the pinnacle of Greek astronomy, but rather a regressive later author who perhaps took astronomy backwards more than anything else. Let us keep that in mind as we turn to Galileo’s Dialogue, in which Ptolemy is the designated punching-bag and symbol of stale received wisdom.
Galileo dismissed the notion that the moon influences the tides as “childish” and “occult.” Instead he argued that tides are a kind of sloshing due to the motion of the earth. This very poor theory is inconsistent with several of his own scientific principles.
Transcript
At the end of his famous Dialogue, Galileo lists what he considers to be his three best arguments for proving that the earth moves around the sun. One of these is his argument “from the ebbing and flowing of the ocean tides”. High and low tidal water. Galileo believed the tides were caused by the motion of the earth. This is truly one of his very worst theories. He was so proud of it. But it stinks. And I will tell you why.
First things first. How do the tides work? As we know today, “the ebb and flow of the sea arise from the action of the sun and the moon.” That’s Newton’s accurate statement of the correct explanation for the tides. They are a consequence of gravitational forces, as Newton proved. The moon, and to a lesser extent the sun, pull water towards them, causing our oceans to bulge now in one direction, then the other. This was clearly understood already in Galileo’s time. Kepler explained it perfectly, and many others likewise proposed lunar-attraction theories of the tides. In fact, the lunisolar theory of tides is found already in ancient sources, including the causal role of the sun and moon, and descriptions of the effects in extensive and accurate detail.
Galileo, however, got all of this completely wrong. Why should “the tides of the seas follow the movements of the fireballs in the skies,” as Kepler had put it? Galileo considered the very notion “childish” and “occult,” and declared himself “astonished” that “Kepler, enlightened and acute thinker as he was, … listened and assented to the notion of the Moon’s influence on the water.” Those are Galileo’s words. Continuing in his treatise, he writes: “There are many who refer the tides to the moon, saying that this has a particular dominion over the waters … [and] that the moon, wandering through the sky, attracts and draws up toward itself a heap of water which goes along following it.”
Yes, many indeed believed such things. And they were right. But Galileo would have none of it. This theory is not “one which we can duplicate for ourselves by means of appropriate devices,” he objects. How indeed could we ever “make the water contained in a motionless vessel run to and fro, or rise and fall”? Certainly not by moving about some heavy rock located many miles away. “But if,” says Galileo, “by simply setting the vessel in motion, I can represent for you without any artifice at all precisely those changes which are perceived in the waters of the sea, why should you reject this cause and take refuge in miracles?”
That’s Galileo’s objection to the lunar theory of tides: It’s hocus-pocus. It assumes the existence of mysterious forces that we cannot otherwise observe or test. Proper science should be based on stuff we can do in a laboratory, like shaking a bowl of water.
Actually I think this argument is not a half bad. How is the moon supposed to influence the oceans across thousands of miles of empty space? What reason do we have to believe that such a force exists? It doesn’t fit with any common-sense knowledge. It doesn’t fit with any empirical experience. It’s a wild idea. A kind of mysticism.
But maybe that’s the lesson of the story. Sometimes wild ideas are right. Rational scientific prudence is not all it’s cracked up to be. It’s a good thing we had some Keplers who said “why not?” instead of only Galileos who said “don’t be silly.”
A “childish,” “occult,” theory based on “miracles.” That’s what Galileo thought of the lunar theory of tides. He put it in so many words, literally. And with good reason. But he was wrong. The world is “occult,” it turns out.
Nowadays we’ve been brainwashed into thinking that the moon’s pull on the oceans is just “science” and there’s nothing weird about it. But isn’t it just as occult as it ever was? Isn’t that what science is, actually? A bunch of occult stuff that we’ve gotten so used to that we’ve given it another name?
By the way, if I may editorialise a bit further, don’t we have a lot of mini Galileos running around today? All these self-proclaimed rational disciples of science, who are out to tell you about all your dumb beliefs being nothing but a “childish” faith in “occult” “miracles”? Think of vaccines causing autism, homeopathy, creationism. There’s an army of little Galileo clones waging war on those occult things. Perhaps it is worthwhile to remember, on such occasions, that Galileo was just as sure with his argument about tides in a bucket.
But enough about that. Now, if Galileo didn’t believe any of that stuff, then what did he believe? Let’s turn to Galileo’s own theory of the tides. I like to think of it in terms of a torus, as we say in mathematics. That’s a ring, like a hollow donut, or one of those inflatable rings that children put around their waists when they play in the pool.
To understand Galileo’s theory, picture one of those rings, a torus, filled halfway with water. So it’s lying flat on the ground and the bottom half of it is filled with water. This represents the water encircling the globe of the earth. Now spin it. Spin the torus in place, around its midpoint, like a steering wheel. This represents the rotation of the earth.
For symmetry reasons it is clear that the water will not move. Because there is no reason for the surface of the water to become higher or lower in one place of the tube than another. Every part is rotating equally, everything is symmetrical, and therefore no asymmetrical distribution of water could arise from this process.
But now picture the torus sitting on a merry-go-round. The represents the earth orbiting the sun. And also at the same time it is still spinning around its own midpoint like before. Now there is asymmetry in the configuration. The torus has one part facing inward toward the middle of the merry-go-round, and one part facing outward. As far as the rotation of the torus around its midpoint is concerned, these two parts of the torus are moving in opposite directions. But as far as the rotation of the merry-go-round is concerned they are both going the same way. So one part of the torus spins along with the orbital motion, and one part against the orbital motion. Therefore one part of the water moves faster than the other. It is boosted by the merry-go-round rotation helping it along in the direction it was already going, while the other part of the torus is slowed down; the merry-go-round cancelling its efforts by going in the opposite direction.
So you have fast-moving water and slow-moving water. The fast-moving water will catch up with the slow water and pile up on top of it, creating a high tide. And the space it vacated will not be replenished because the slow water behind it isn’t keeping up. Hence the low tide. Let’s hear Galileo’s own words; here’s how he puts it: “Mixture of the annual and diurnal motions causes the unevenness of motion in the parts of the terrestrial globe. … Upon these two motions being mixed together there results in the parts of the globe this uneven motion, now accelerated and now retarded by the additions and subtractions of the diurnal rotation upon the annual revolution.”
Thus, in fact, “the flow and ebb of the seas endorse the mobility of the earth.” That’s Galileo’s conclusion. This was one of his favourite arguments for the motion of the earth around the sun.
Unfortunately, Galileo’s theory is completely out of touch with even the most rudimentary observational facts about tidal waters. High and low water occur six hours apart. In the lunisolar theory this is explained very naturally as an immediate consequence of its basic principles. Namely as follows. The rotation of the earth takes 24 hours. There’s a wave of high water pointing toward the moon, basically (give or take a bit of lag in the system), and then another high water mark diametrically opposite that that on the other side of the earth. So that’s two highs and two lows in 24 hours, so 6 hours apiece.
Galileo’s theory, on the other hand, implies that high and low water should be twelve hours apart rather than six: “there resides in the primary principle no cause of moving the waters except from one twelve-hour period to another.” Those are Galileo’s own words in his big treatise. So Galileo immediately finds himself on the back foot. He has to somehow talk himself out of this obvious flaw of his theory. To this end he alleges that “the particular events observed [regarding tides] at different times and places are many and varied; these must depend upon diverse concomitant causes,” such as the size, depth and shape of the sea basin, and the internal forces of the water trying to level itself out. The fact that everyone could observe two high and two low tides per day Galileo thus wrote off as purely coincidental. He explicitly says so:
“Six hours … is not a more proper or natural period for these reciprocations than any other interval of time, though perhaps it has been the one most generally observed because it is that of our Mediterranean.”
Galileo even has some fake data to prove his erroneous point: namely that tides twelve hours apart are “daily observed in Lisbon,” he believes, even though that is completely false.
There is a further complication involved in Galileo’s theory, which “caused embarrassment to his more competent readers.” The inclination of the earth’s axis implies that the effects that Galileo describes should be strongest in summer and winter. Unfortunately for Galileo’s theory, the reverse is the case. Actually the tides are most extreme in spring and fall because they receive the maximum effects of the sun’s gravitational pull.
Galileo got himself confused on this point because he was again relying on false data. Galileo—the self-declared enemy of relying on textual authority, who often mocked his opponents for believing things simply because it said so in some book—he was the one in this case who found in some old book the claim that tides are greatest in summer and winter, took this for fact and derived this supposed effect from his own theory.
The mismatch between Galileo’s theory and basic facts is on display in another episode as well: “Galileo … attacked [those who] postulated that an attractive force acted from the Moon on the ocean for failing to realize that water rises and falls only at the extremities and not at the center of the Mediterranean. [But his opponents] can hardly be blamed for failing to detect this [so-called] phenomenon: it only exists as a consequence of Galileo’s own theory.” In other words, Galileo was so biased by his wrongheaded theory that he used its erroneous predictions as so-called “facts” with which to attack his opponents who were actually right.
But all of the above is actually still not yet the worst of it, believe it or not. There is an even more fundamental flaw in Galileo’s theory. Namely that it is inconsistent with the principle of relativity that he himself espoused. Think back to his scenario of the scientist locked in a cabin below deck of a ship that may or may not be moving. Galileo’s conclusion on that occasion was that no physical experiment could detect whether the ship was moving or not. But the torus tidal simulation, if it really worked as Galileo claimed, certainly could detect such a motion. If we put the torus on the floor of the cabin and spun it, one part would be spinning with the direction of motion of the ship, and another part against it. Hence high and low water should arise, by the same logic as in Galileo’s tidal theory. If the ship stood still, on the other hand, no such effect would be observed, of course. So we have a way of determining whether the ship is moving, which is supposed to be impossible. And indeed it is impossible, but if that’s so then Galileo’s theory of the tides cannot possibly work because it is inconsistent with this principle.
This objection against Galileo’s theory was in fact raised immediately already by contemporary readers. Here’s what one contemporary wrote to Galileo in 1633. He is reporting the reflections on Galileo’s book by a group of scholars who studied it:
“They draw attention to a difficulty raised by several members about the proposition you make that the tides are caused by the unevenness of the motion of the different parts of the earth. They admit that these parts move with greater speed when they [go] along [with] the annual motion than when they move in the opposite direction. But this acceleration is only relative to the annual motion; relative to the body of the earth as well as to the water, the parts always move with the same speed. They say, therefore, that it is hard to understand how the parts of the earth, which always move in the same way relative to themselves and the water, can impress varying motions to the water.”
That is to say, picture the earth moving along its orbit and also rotating around its axis at the same time. Hit pause on this animation and mark two diametrically opposite spots on the equator. Then hit play, let it run for a second or two, and then pause it again. Now, compare the new positions of the two marked spots with their original position. One will have moved further than the other. But that’s in a coordinate system that doesn’t move with the earth. That type of inequality of speed is irrelevant.
What is needed to created tides is a different kind of inequality of speed of the water. Namely, a difference in speed relative to the earth itself and to the other water. Tides arise when a fast-moving part of the water catches up with a slow-moving part of the water. But that is to say, these waters are fast and slow in their speed of rotation about the earth. So inequality of speed in a coordinate system centred on the earth. But no inequality of this type arises from the motion of the earth about the sun.
Galileo had no solution to this accurate objection. He’s just wrong. His theory is dumb.
So, to sum up, Galileo small-mindedly rejected the correct theory of the tides, based on the sun and the moon, even though this was widely understood by his contemporaries. He then proposed a completely wrongheaded theory of his own, which is based on elementary errors of physical reasoning that are inconsistent with his own principles. These flaws were readily spotted by his contemporaries. Furthermore, his theory is fundamentally at odds with the most basic phenomena, which he tried to explain away by attributing them to untestable, ad hoc secondary effects. He also adduced several false observational so-called “facts” in support of his theory.
No wonder many have felt that Galileo’s “ill-fated theory of the tides is a skeleton in the cupboard of the scientific revolution,” as one historian puts it. But this is a problem only if one assumes that Galileo was science personified. If we accept instead that Galileo was an exceptionally mediocre mind, who constantly got wrong what mathematically competent people like Kepler got right, then we see that Galileo’s skeletons belong only to himself, not to the scientific revolution. So once again we solve a problem of this type by throwing Galileo under the bus. It’s not that the scientific revolution was flawed. It’s just that Galileo was. If we restrict ourselves to mathematically competent people then we don’t have to deal with this kind of nonsense.
I’m going to use the tides to make the transition from Galileo’s physics, which we have discussed at length, to his astronomy. Galileo, as we saw, wanted to use the tides as a proof that the earth is moving, both rotationally and also orbitally around the sun.
Other people were not fooled by this poor argument. And yet leading mathematicians did have the sense to accept Copernicus’s vision. Why? Why did they become Copernicans? Why did Copernicus, why did Kepler? Certainly not because of Galileo, of course, but what then?
Arguably the most compelling reason was that the Copernican system explains complex phenomena in a simple, unified way. Here are some basics observational facts: The planets sometimes move forwards and sometimes backwards. Also, Mercury and Venus always remain close to the sun. In a geocentric system, with the earth in the center, there is no inherent reason why those things should be like that. Nevertheless these are the most basic and prominent facts of observational astronomy. In the Ptolemaic system, the geocentric system of antiquity, these facts are accounted for by introducing complicated secondary effects and coordinations beyond the basic model of simple circles. So planetary orbits are not just simple circles but combinations of circles in complicated ways that also happen to be coordinated with one another in particular patterns. In the Ptolemaic system there is no particular reason why these complicated constructions should be just so and not otherwise. We have to accept that it just happens to be that way.
So Ptolemy could account for, or accommodate, the phenomena, but he can hardly have been said to have explained them. The basic idea, that planets move in circles around the earth, is on the back foot from the outset. It is inconsistent with the most basic data and is therefore forced to add individual quick-fixes for these phenomena. In the end there are so many layers of patches that it’s like a house that consists more of duct-taped emergency fixes than that the original foundation.
In the Copernican system, it’s the opposite. The phenomena I mentioned are here instead an immediate, natural consequence of the motion of the earth. It becomes obvious and unavoidable that outer planets appear to stop and go backwards as the earth is speeding past them in its quicker orbit. It becomes obvious and unavoidable that Mercury and Venus are never seen far from the sun, since, like the sun, they are always on the “inside” of the earth’s orbit. Ad hoc secondary causes and just-so numerical coincidences in parameter values are no longer needed to accommodate these facts; instead they follow at once from the most basic assumptions of the system. They are built right into the foundation, no duct tape needed.
Galileo makes this point in his Dialogue: “You see, gentlemen, with what ease and simplicity the annual motion … made by the earth … lends itself to supplying reasons for the apparent anomalies which are observed in the movements of the five planets. … It removes them all. … It was Nicholas Copernicus who first clarified for us the reasons for this marvelous effect. … This alone ought to be enough to gain assent for the rest of the [Copernican] doctrine from anyone who is neither stubborn nor unteachable.”
This is all good and well. Galileo is absolutely right. Although of course this point was already a hundred years old and common knowledge by the time Galileo repeated it.
But here now is my fun twist on the story. Compare this story with Galileo’s theory of the tides. In fact, the correct, lunisolar theory of the tides explains the basic phenomena in a simple and natural way as immediate consequences of the first principles of the theory. That’s exactly the same point that we made about the Copernican system. The correct theory of the tides thus has the same kind of credibility as the Copernican system. So, by Galileo’s logic, this “ought to be enough to gain assent.” But in the case of the tides it seems Galileo was the “stubborn and unteachable” one. He insisted on a theory which—like that of Ptolemy—could only account for basic facts by invoking arbitrary and unnatural secondary causes unrelated to the primary principles of the theory.
It’s a sign that your theory has poor foundations if the foundations themselves are good for nothing and all the actual explanatory work is being done by emergency extras duct-taped on later to specifically fix obvious problems with the foundations. Intelligent people realised this, which is why they turned to the sun-centered view of the universe. Galileo paid lip service to the same principle when he wanted to ride on the coattails of their insights. But, if he had been consistent in his application of this principle, he should have used it to reject his foolish theory of the tides.
Galileo committed scores of errors in his physics. These are bad in themselves and also undermine Galileo’s claim to credit for the things he did get right.
Transcript
Nostradamus published a famous book of prophesies in 1555. Some people like to praise him for having predicted the future. Allegedly he foresaw all kinds of things about world history. With a bit of imagination you can see him speaking about Napoleon and Hitler and god knows what else. Of course for every such pseudo-truth he also said a hundred things that turned out dead wrong. But people are less excited about that. It’s easy to be a prophet if you’re allowed a thousand guesses and people only count the few that came true.
Galileo is another Nostradamus. He too threw a thousand guesses out there and hoped that one or two would stick. Like Nostradamus, Galileo’s reputation rests on his admirers having selective amnesia, and remembering only the rare occasions when he got something right.
That’s our thesis for today. So, let’s have a go at Galileo’s catalogue of errors. A bunch of them have to do with the law of fall. Supposedly Galileo’s strong point. But in fact if we judge Galileo’s grasp of the law of fall by the way he applied it, then we must conclude that he did not understand it very well at all. He gets it wrong more often than not when he tried to apply his own law.
All objects fall with the same acceleration. But how fast is that exactly? What is this same universal acceleration that every object shares? The answer is well-known to any student of high school physics: the constant of acceleration, that famous lower case g, it is approximately 9.8 meters per second squared. But Galileo messed this up. He gives values equivalent to less than half of the true value. According to his defenders, and I quote, “clearly, round figures were taken here in order to make the ensuing calculation simple.” In other words, Galileo “used arbitrary data.” That’s again a quote. And that’s what the people trying to defend Galileo are saying.
Isn’t the law of fall supposed to be one of Galileo’s greatest discoveries? Why did he use fake data? Why not use real data? It was readily doable. His contemporaries did it. Why not do a little work to get the details right when you are publishing your supposed key results in your mature treatise? Is that really too much to ask? Instead of reporting make-believe evidence with a straight face, as Galileo does.
Competent and serious readers, like Mersenne and Newton, were all in disbelief at Galileo’s inaccurate data. They certainly did not think it was fine to “use arbitrary data” in order to get ”round” numbers for simplicity. Nor did they think it was “clear” that this is what Galileo was doing, contrary to what Galileo’s defenders are forced to argue when they try to excuse his inexcusable behaviour.
Mersenne put it clearly: “I doubt whether Mr Galileo has performed the experiment on free fall on a plane, since … the intervals of time he gives often contradict experiment.” Indeed. Mersenne was a serious and diligent scientist. He did the work to find the correct value, unlike Galileo. As usual, Galileo’s supposedly scientific treatises are popularising polemics and little else. To actual scientists, what he has to say is disappointingly shallow and lacks serious follow-through.
Here’s another error Galileo made with his law of fall: his so-called “Pisan Drop” theory of planetary speeds. The planets orbit the sun at different speeds. Mercury has a small orbit and zips around it quickly. Saturn goes the long way around in a big orbit and it is also moving very slowly. Galileo imagines that these speeds were obtained by the planets falling from some faraway point toward the sun, and then being somehow deflected into their circular orbits at some stage during this fall. That supposedly explains why the planets have the speeds they do.
I’ll read Galileo’s description in the Dialogue. “Suppose all the [planets] to have been created in the same place … descending toward the [sun] until they had acquired those degrees of velocity which originally seemed good to the Divine mind. These velocities being acquired, … suppose that the globes were set in rotation [around the sun], each retaining in its orbit its predetermined velocity. Now, at what altitude and distance from the sun would have been the place where the said globes were first created, and could they all have been created in the same place? To make this investigation, we must take from the most skilful astronomers the sizes of the orbits in which the planets revolve, and likewise the times of their revolutions.” Using this data and “the natural ratio of acceleration of natural motion” (that is, the constant g), one can compute “at what altitude and distance form the center of their revolutions must have been the place from which they departed.” According to Galileo this shows that indeed all the planets were dropped from a single point and their orbital data “agree so closely with those given by the computations that the matter is truly wonderful.”
Galileo omits the details though. Galileo has one of the characters in his dialogue say: “Making these calculations … would be a long and painful task, and perhaps one too difficult for me to understand.” Galileo’s mouthpiece in the dialogue confirms that “the procedure is indeed long and difficult.”
Actually there is nothing “difficult” about it. At least not to mathematically competent people. Mersenne immediately ran the calculations and found that Galileo must have messed his up, because his scheme doesn’t work. There is no such point from which the planets can fall and obtain their respective speeds. Later Newton made the same observation. Galileo’s precious idea is so much nonsense, which evidently must have been based on an elementary mathematical error in calculation.
Here’s another example. Again involving the law of fall. Galileo wished to refute this ancient argument: “The earth does not move, because beasts and men and buildings” would be thrown off. Picture an object placed at the equator of the earth, such as a rock lying on the African savanna. Imagine this little rock being “thrown off” by the earth’s rotation. In other words, the rock takes the speed it has due to the rotation of the earth, and shoots off with this speed in the direction of a tangent line of its motion. The spectacle will be rather underwhelming at first: since the earth is so large, the tangent line is almost parallel to the ground, and since the speed of the rock and of the earth are the same they will keep moving in tandem. So rather than shooting off into the air like a canon ball, the rock will slowly begin to hover above the ground, a few centimeters at a time.
Of course this is not what happens to an actual rock, because gravity is pulling it back down again. The rock stays on the ground since gravity pulls it down faster than it rises due to the tangential motion. How can we compare these two forces quantitatively? Since we know the size and rotational speed of the earth, it is a simple task (suitable for a high school physics test) to calculate how much the rock has risen after, say, one second. This comes out as about 1.7 centimeters. We need to compare this with how far the rock would fall in one second due to gravity. Again, this is a standard high school exercise (equivalent to knowing constant of gravitational acceleration g). The answer is about 4.9 meters. This is why the rock never actually begins to levitate due to being “thrown off”: gravity easily overpowers this slow ascent many times over.
But of course this conclusion depended on the particular size and speed and mass of the earth. We could make the rock fly by spinning the earth fast enough. For example, if we run the above calculations again assuming that the earth rotates 100 times faster, we find that, instead of rising a measly 1.7 centimeters above the ground in one second, the rock now soars to 170 meters in the same time. The fall of 4.9 meters due to gravity doesn’t put much of dent in this, so indeed the rock flies away.
Galileo, alas, gets all of this horribly wrong. Even though we are supposed to celebrate Galileo as the discoverer of the law of fall, it is apparently too much to ask that he work out this very basic application of it. As we noted, Galileo did not offer a serious estimate for the constant of gravitational acceleration g, unlike his contemporaries who were proper scientists. Therefore he did not have the quantitative foundations to carry out the above analysis, which high school students today can do in five minutes.
Worse yet, Galileo maintains that no such analysis is needed in the first place, because he can “prove” that the rock will never be thrown off regardless of the rotational velocity. “There is no danger,” Galileo assures us, “however fast the whirling and however slow the downward motion, that the feather (or even something lighter) will begin to rise up. For the tendency downward always exceeds the speed of projection.” Galileo even offers us “a geometrical demonstration to prove the impossibility of extrusion by terrestrial whirling.” Those are quotes from his big treatise on this.
Galileo’s claim to fame as a “mathematiser of nature” is certainly done no favours by this episode. He doesn’t know how to quantify his own law of fall, and doesn’t understand basic implications of it. His physical intuition is categorically wrong on a qualitative level, and worse than that of the ancients he is trying to refute (whose stance was quite reasonable and would be accurate if the earth was spinning faster). Galileo even offers a completely wrongheaded geometrical “proof” that the ancients’ conception is impossible, even though so-called “Galilean” physics leads to the opposite conclusion in an elementary way.
Galileo’s error in effect amounts to assuming that speed in free fall is proportional to distance rather than time. This is a crucial distinction in “Galileo’s” law of fall, which Galileo and others at times got wrong. By messing up this very point in his mature work, Galileo is undermining his claim to being the rightful father of the correct law of fall.
Galileo also makes another, independent error in this connection when he claims that the earth has the same whirling potential as a wheel with the same rotational period. In other words, that centrifugal force doesn’t depend on radius, only period. “Anyone familiar with simple merry-go-rounds will know that this is false,” as one scholar observes. Is it really just as hard to “hang on” to the speeding carousel whether you are close to the center or right out at the periphery? Alternatively, think of the pottery wheel used when shaping the clay when making ceramic pots. Is a piece of clay equally likely to be thrown off the wheel by its rotation if it sits near the middle as if it is placed near the edge? According to Galileo, yes.
Another example. Galileo tried to compute how long it would take for the moon to fall to the earth, if it was robbed of its orbital speed. “Making the computation exactly,” according to himself, he finds the answer: 3 hours, 22 minutes, and 4 seconds. This is way off the mark because Galileo assumes that his law of fall (that is, constant gravitational acceleration) extends all the way to the moon, which of course it does not. Ironically, Galileo’s purpose with this calculation was to refute the claim of another scholar that the fall would take about six days, which is a much better value: in fact it would take the moon almost five days to fall to the earth. That’s Galileo, the great hero of quantitative science, in action for you: bombastically claiming to refute others with his “exact calculations,” only to make fundamental mistakes and err orders of magnitude worse than his opponents did.
Another example. A rock dropped from the top of a tower falls in a straight line to the foot of the tower. But its path of fall is not actually straight if we take into account the earth’s rotation. Seen from this point of view—that is to say, in a coordinate system that doesn’t move with the rotation of the earth—what kind of path does the rock trace? Galileo answers, erroneously, that it will be a semicircle going from the top of the tower to the center of the earth. Here’s what he says: “If we consider the matter carefully, the body really moves in nothing other than a simple circular motion, just as when it rested on the tower it moved with a simple circular motion. … I understand the whole thing perfectly, and I cannot think that … the falling body follows any other line but one such as this. … I do not believe that there is any other way in which these things can happen. I sincerely wish that all proofs by philosophers had half the probability of this one.”
This is inconsistent with Galileo’s own law of fall. Once again he doesn’t understand basic implications of his own law. Mersenne readily spotted Galileo’s error. And Fermat observed that the path should be a spiral, not a semi-circle. When his embarrassing error was pointed out to him, Galileo replied that “this was said as a jest, as is clearly manifest, since it is called a caprice and a curiosity.” Some defence this is! Far from offering exonerating testimony, Galileo actually openly pleads guilty to the main charge: namely, that his science is a joke.
And if he meant is as “a caprice and a curiosity” then why did he say, in the quote I just read, that he “considered the matter carefully” and “sincerely wished that all proofs by philosophers had half the probability of this one” and so on? He always says cocky stuff like that. Just look back at the errors I already mentioned today. All of them came with bombastic claims where Galileo is editorialising about how remarkably convincing his own arguments are.
Isn’t that convenient? Throw out a bunch of half-baked guesses, and when they turn out right you can claim credit for stating it with such confidence while a more responsible scientist may have been exercising prudent caution. And when the guesses turn out wrong, you can apparently just write it off as a “joke” and pretend that that was what you intended all along, even though you published it with all those extremely assertive phrases right in the middle of your big definitive book on the subject. It’s easy to be “the father of science” if you can count on posterity to play along with this double standard.
So much for the law of fall. Let’s look at some of Galileo’s other physics errors. As we have seen, Aristotelians were often as inclined to experiment as Galileo—a point obscured by Galileo’s pretences to the contrary when it suited his purposes. Elsewhere it suited Galileo better to feign other straw men. As Butterfield has observed, “in one of the dialogues of Galileo, it is Simplicius, the spokesman of the Aristotelians—the butt of the whole piece—who defends the experimental method of Aristotle against what is described as the mathematical method of Galileo.”
Consider for example the question of the resistance of the medium (such as air or water) on a moving object. Aristotle had stated a law regarding how a body moves faster in a rarer medium than in a dense one. Galileo, in an early text, criticises Aristotle for accepting this “for no other reason than experience”; instead one must “employ reasoning at all times rather than examples,” “for we seek the causes of effects, and these are not revealed by experience.” Alas, despite his avowed allegiance to “reasoning,” Galileo’s own law as to how resistance depends on density of the medium is itself “incompatible with classical mechanics” as one study puts it—a polite, scholarly way of saying that it’s wrong.
Employing some more “reasoning” along the same lines, Galileo decided that air resistance doesn’t really increase appreciably with speed: “The impediment received from the air by the same moveable when moved with great speed is not very much more than that with which the air opposes it in slow motion.” A surprising conclusion to modern bicyclists, among others. Yet “experiment gives firm assurance of this,” Galileo promises. Alas, once again “the statement is false, and the experiment adduced in its support is fictitious.” So more fake data, in other words. This is quickly becoming a pattern.
The pendulum is a case that is similar in this regard. “With regard to the period of oscillation of a given pendulum, [Galileo] asserted that the size of the arc [that is, the height of the starting position of the pendulum] did not matter, whereas in fact it does.” Galileo’s allegedly experimental report on pendulums in the Discourse is clearly fabricated—or “exaggerated,” to use the diplomatic term preferred by some scholars. Mersenne did the experiment and rejected Galileo’s claim. Galileo’s friend Guidobaldo del Monte did the same, but when he told Galileo of his error Galileo rejected the experiment and insisted in his claim. Instead of admitting what experiments made by sympathetic and serious scientists showed, Galileo preferred to defend his false theory with what one historian calls “conscious deception.” Perhaps more commonly known as lying.
Another example. The shape of a hanging chain, like a neckless suspended from two points, looks deceptively like a parabola. It is not, but Galileo fell for the ruse. As he says: “Fix two nails in a wall in a horizontal line … From these two nails hang a fine chain … This chain curves in a parabolic shape.” More competent mathematicians proved him wrong: Huygens demonstrated that the shape was not in fact parabolic. Admittedly, his proof is from 1646, which is four years after Galileo’s death. So one may consider Galileo saved by the bell, as it were, on this occasion, since he was proved wrong not by his contemporaries but only by posterity. It is not fair to judge scientists by anachronistic standards. On the other hand, Huygens was only seventeen years old when he proved Galileo wrong. So another way of looking at it is that a prominent claim in Galileo’s supposed masterpiece of physics was debunked by a mere boy less than a decade after its publication.
In any case, Galileo thus ascribed to the catenary the same kind of shape as the trajectory of a projectile. He considered this to be no coincidence but rather due to a physical equivalence of the forces involved in either case. Indeed, Galileo made much of this supposed equivalence and “intended to introduce the chain as an instrument by which gunners could determine how to shoot in order to hit a given target.”
Galileo also tried to test experimentally whether the catenary is indeed parabolic. To this end he drew a parabola on a sheet of paper and tried to fit a hanging chain to it. His note sheets are preserved and still show the needle holes where he nailed the endpoints of his chain. The fit was not perfect, but Galileo did not reject his cherished hypothesis. Instead of questioning his theory, he evidently reasoned that the error was due merely to a secondary practical aspect, namely the links of the chain being too large in relation to the measurements. Therefore he tried it with a longer chain, and found the fit to be better. In this way he evidently convinced himself that he was right after all.
The catenary case thus undermines two of Galileo’s main claims to fame. First it bring his work on projectile motion into disrepute. The composition of vertical and horizontal motions that we are supposed to admire in that case looks less penetrating and perceptive when we consider that Galileo erroneously believed it to be equivalent to the vertical and horizontal force components acting on a catenary. Secondly, Galileo’s reputation as an experimental scientists par excellence is not helped by the fact that his experiments in this case led him to the wrong conclusion, apparently because his love of his pet hypothesis led him to a biased interpretation of the data and a sweeping under the rug of an experimental falsification. So those are two conclusions we can draw from the hanging chain episode.
Another example. The brachistochrone problem. This is the challenge to find the path along which a ball rolls down the quickest from one given point to another. Galileo believed himself to have proved that the optimal curve was a circular arc. “The swiftest movement of all from the terminus to the other is … through the circular arc.” That’s Galileo. Actually the fastest curve is not a circle but rather a cycloid. But this was only proved in the 1690s, using quite sophisticated calculus methods. We cannot blame Galileo for not possessing advanced mathematical tools developed only half a century after his death.
Nevertheless it is one more on his pile of erroneous assertions about various physical curves and problems. We are supposed to celebrate him for being the first to discover the parabolic path of projectile motion, and conveniently forget that at the same time he was wrong on the brachistochrone, wrong on the catenary, wrong on the isochrone, and so on. With all these errors stacking up, one may be forgiven for beginning to wonder whether the one thing he got right was any more than dumb luck. Galileo’s accounts of his correct discoveries may sound very convincing and emphatic, but knowing that he was equally sure of a long list of errors gives us reason to suspect that some of the things he got right are to some extent guesswork propped up with overconfident rhetoric in the hope that readers will mistakenly take his case to be stronger than it is.
That’s it for today. And I have still only dealt with physics. Galileo’s astronomy is a whole other can of worms with a parade of blunders all of its own. But we’ll get to that another time.
Galileo gets credit he does not deserve for the parabolic nature of projectile motion, the law of inertia, and the “Galilean” principle of relativity. In reality, his treatments of all of these matters were riddled with errors and fundamental misunderstandings.
Transcript
Pick up a rock and throw it in front of you. It makes a parabola. The path of its motion is parabolic. That’s Galileo’s great discovery, right? Well, not really. Galileo does claim this but he doesn’t prove it. Even Galileo’s own follower Torricelli acknowledged this. The result is “more desired than proven,” as he says, very diplomatically.
And the reason why Galileo doesn’t prove it is a revealing one. It is due to a basic physical misunderstanding.
The right way to understand the parabolic motion of projectiles like this is to analyse it in terms of two independent components: the inertial motion and the gravitational motion. If we disregard gravity, the rock would keep going along a straight line forever at exactly the same speed. That’s the law of inertia. But gravity pulls it down in accordance with the law of fall. The rock therefore drops below the inertial line by the same distance it would have fallen below its starting point in that amount of time if you had simply let it fall straight down instead of throwing it. A staple fact of elementary physics is that the resulting path composed of these two motions has the shape of a parabola.
Galileo does not understand the law of inertia, and that is why he fails on this point. If the projectile is fired horizontally, like for instance a ball rolling off a table, then Galileo does prove that it makes a parabola. He proves it the right way, they way I just outlined, by composition of inertial and gravitational motion.
But if you throw the rock at some other angle, not horizontally, then Galileo doesn’t dare to give such an analysis. This is because he thinks the law of inertia is maybe not true for such motions. He thinks, if you throw a rock at an upward angle, then maybe the rock won’t have such an inertial disposition to keep going in that direction with that speed. Instead, we thinks maybe the motion is going to slow down gradually, like a ball struggling to roll up a hill or an inclined plane.
Galileo asserts neither this wrong form of inertia nor the right one. He equivocates and never takes a stand, because he isn’t sure. And this is why he cannot give a correct proof of the theorem of parabolic motion. Even though such a proof was very much within his reach. In fact, Cavalieri, who was a better mathematician, had already published this proof, the correct analysis of parabolic motion, before Galileo wrote his book.
So it’s not that this stuff was beyond the reach of the mathematical and scientific methods of the time by any means. On the contrary, it was already explicitly spelled out completely and correctly in a published book that Galileo was aware of. And still Galileo gets it wrong in his famous work. He’s just not a very good physicist.
Ok, so that’s the big picture on parabolic motion. Now I want to go into more detail on these things. First let’s take a step back and look at inertia generally.
Here is Newton’s law of inertia: “Every body perseveres in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by forces impressed.” That’s from Newton’s great Principia of 1687. It’s Law 1 of that work. A cornerstone of the whole thing.
In Galileo there’s nothing like that. Even the most ardent Galileo admirers admit this. Here’s Stillman Drake, Galileo’s great defender. Even he, and I quote, “freely grant that Galileo formulated only a restricted law of inertia” and that “he neglected to state explicitly the general inertial principle” that everyone knows today, which was instead correctly “formulated two years after his death by Pierre Gassendi and René Descartes.”
That’s the charitable interpretation. That’s the view of Galileo’s most committed supporters. And it is rather too kind, in my opinion. Trying to attribute to Galileo some kind of “restricted law of inertia” is a dubious business. Stillman Drake tries to do so, and here is what he says: “in my opinion the essential core of the inertial concept lies in the ideas of a body’s indifference to motion or to rest and its continuance in the state it is once given. This idea is, to the best of my knowledge, original with Galileo.”
You could very well argue that that’s not really inertia at all because it doesn’t involve the straightness of the direction of the motion, nor does it explicitly say that the motion keeps going at a perpetual uniform speed. It only focusses on indifference of motion versus rest and preservation of the state of motion.
So that’s “the essential core of the inertial concept” according to Galileo’s defenders. That’s very convenient. Galileo got half the properties of inertia right and half wrong, so his supporters try to spin it and say that the parts he did get right are “the essential core”, you see, and the other stuff is just secondary anyway so it doesn’t really matter that Galileo was wrong about all of that.
Sure enough, if you’re allowed to pick and choose like this which half of inertia you think is important then you can find some evidence for that part is Galileo. For example, Galileo says quite correctly: “No one could say why a thing once set in motion should stop anywhere; for why should it stop here rather than there? So that a thing will either be at rest or must be moved ad infinitum, unless something more powerful get in its way.”
Sure enough, that’s indifference of motion versus rest and preservation of the state of motion, the alleged “core” of the inertial concept. How much credit do you think Galileo deserves for this? For getting half of inertia right? Maybe you think that was the difficult step, the conceptual revolution, and then it was easy for Newton and others to fill in the details by just continuing what Galileo started.
Actually I tricked you. The quote I just read is not from Galileo at all. I lied. The quote is from Aristotle. It’s from Aristotle’s physics, written two thousand years before Galileo. So if you think that’s “the essential core of the inertial concept,” then Aristotle was the pioneering near-Newtonian who conceived it, not Galileo.
This claim is rather isolated in Aristotle and didn’t really form part of a sustained and coherent physical treatment of motion comparable to how we use inertia today. Aristotle as usual is focussed on much more philosophical purposes. So you might say: that’s a one-off quote taken out of context which sounds much more modern than it really is.
Indeed. But then again the same could be said for Aristotle’s so-called law of fall that Galileo refuted with so much fanfare. This too is only mentioned in passing very briefly and plays no systematic role in Aristotle’s thought. Yet Galileo takes great pride in defeating this incidental remark, and his modern fans applaud him greatly for it. So if we want to dismiss Aristotle’s inertia-like statement as insignificant, then, by the same logic, we ought to likewise dismiss all of Galileo’s exertions to refute his law of fall as completely inconsequential as well. If we argue that statements such as those of Aristotle don’t count as scientific principles unless they are systematically applied to explain various natural phenomena, then we would have to conclude that there was no Aristotelian science of mechanics at all. This, of course, would be a disastrous concessions to make for advocates of Galileo’s greatness, since so much of Galileo’s claim to fame is based on contrasting his view with so-called “Aristotelian” science.
So take your pick. Here are the three options:
Option 1. Galileo’s understanding of inertia was very poor.
Option 2. Galileo’s understanding of inertia was pretty good, but so was Aristotle’s.
Option 3. Galileo’s understanding of inertia was pretty good, but not Aristotle’s, because Aristotle’s statements, even though they say pretty much what Galileo says, should be disqualified because they are philosophy rather than science.
I, of course, advocate the first solution: throw Galileo under the bus. He and Aristotle were both stupid. Problem solved.
If you want to preserve Galileo’s reputation you’re in a trickier position. Are you going to admit that Aristotle understood inertia? But then what was Galileo’s contribution, and how could it be revolutionary, if that kind of stuff was already well understood two thousand years before? Or do you want to say: No, Aristotle didn’t really understand this, because his text wasn’t meant as science anyway. Well, then what is the value in Galileo spending hundreds of pages of his most important works arguing against Aristotle?
You tell me how you’re gonna solve these puzzles. Trying to maintain Galileo’s alleged greatness, it just doesn’t add up. You’re left having to bend over backwards with these inconsistent rationalisations.
What about the rectilinear character of inertia? The thing keeps going straight. Is that in Galileo? The following passage may appear to suggest as much. Quote: “A projectile, rapidly rotated by someone who throws it [like a rock in a sling], upon being separated from him retains an impetus to continue its motion along the straight line touching the circle described by the motion of the projectile at the point of separation. The projectile would continue to move along that line if it were not inclined downward by its own weight. The impressed impetus, I say, is undoubtedly in a straight line.”
That’s Galileo, and it’s straight up rectilinear inertia, right? Done and dusted. No, not so. It’s not inertia, it’s impetus. That’s what Galileo calls it. The projectile has “impetus” to go straight. But what does that mean? What is “impetus”? Is it the same thing as inertia? Will “impetus” run out, for example? Is the motion caused by the “impetus” perpetual and uniform? Galileo doesn’t say, and most likely he didn’t believe so.
In many other sources at the time, “loss of impetus by projectiles was likened to the diminution of sound in a bell after it is struck, or heat in a kettle after it is removed from the fire,” as Drake remarks. This conception is perfectly compatible, to say the least, with what Galileo writes. In fact, Galileo nowhere asserts the eternal conservation of rectilinear motion. On the contrary, he explicitly rejects it: “Straight motion cannot be naturally perpetual.” That’s an exact quote form his major work. “It is impossible that anything should have by nature the principle of moving in a straight line.” Again, a literal quotation right out of Galileo’s main work. It is easy to understand, then, why Galileo’s defenders are so eager to insist on characterising “the essential core of the inertial concept” in a way that does not involve its rectilinear character, since Galileo clearly and explicitly rejected rectilinear inertia.
If there’s any inertia in Galileo it is horizontal rather than rectilinear inertia. Here are some quotes from Galileo.
“To some movements [bodies] are indifferent, as are heavy bodies to horizontal motion, to which they have neither inclination or repugnance. And therefore, all external impediments being removed, a heavy body on a spherical surface concentric with the earth will be indifferent to rest or to movement toward any part of the horizon. And it will remain in that state in which it has once been placed; that is, if placed in a state of rest, it will conserve that; and if placed in movement toward the west, for example, it will maintain itself in that movement. Thus a ship, for instance, having once received some impetus through the tranquil sea, would move continually around our globe without ever stopping.”
Another one:
“Motion in a horizontal line which is tilted neither up nor down is circular motion about the center; once acquired, it will continue perpetually with uniform velocity.”
Again, as with the sling and the projectile, one can debate whether this is inertia per se. In Newtonian mechanics too a hockey puck on a spherical ice earth would glide forever in a great circle, even though this is not inertial motion. But this agreement with Newtonian mechanics only holds if the object is prevented from moving downward, as the puck is by the ice, or the ship by the water. Galileo seems to have believed horizontal inertia to hold also for objects travelling freely through the air, which is of course not compatible with Newtonian mechanics. For example, Galileo says:
“I think it very probable that a stone dropped from the top of the tower will move, with a motion composed of the general circular movement and its own straight one.”
Once again it is not entirely clear that this is supposed to represent inertia at all. It is conceivable that, in Galileo’s conception, the circular motion itself is not a force-free, default motion, but rather a motion somehow caused or contaminated by gravity-type forces. Who knows? Galileo just isn’t clear about these kinds of things. Newton and Descartes, like the good mathematicians that they are, state concisely and explicitly what the exact fundamental assumption of their theory of mechanics are. Their laws of inertia are crystal clear and specifically announced to be basic principles upon which the rest of the theory is built. Galileo never comes close to anything of this sort. He uses the casual dialogue format of his books to hide behind ambiguities. One moment he seems to be saying one thing, then soon thereafter something else, like an opportunist who doesn’t have a systematically worked out theory but rather adopts whatever assumptions are most conducive to his goals in any given situation.
Let’s get back to parabolic motion. Some have tried to argue that “if Galileo never stated the law [of inertia] in its general form, it was implicit in his derivation of the parabolic trajectory of a projectile.” That’s a quote from Stillman Drake. It would have been a very good argument if Galileo had treated parabolic trajectories correctly. But he didn’t, so the evidence goes the other way: Galileo’s bungled treatment of parabolic motion is actually yet more proof that he did not understand inertia.
His restriction of inertia to horizontal motion only is clear in his treatment of projectiles. He speaks unequivocally of “the horizontal line which the projectile would continue to follow with uniform motion if its weight did not bend it downward.” But he does not make the same claim for projectiles fired in non-horizontal directions. Rather he studiously avoided committing himself on that point because he was afraid it wasn’t true, like we said.
Since he only trusted the horizontal case, Galileo tried to analyse other trajectories in terms of this case. To this end he assumed, without justification, that a parabola traced by an object rolling off a table would also be the parabola of an object fired back up again in the same direction. In other words, “he takes the converse of his proposition without proving or explaining it.” That judgement is in fact a quote from Descartes, a mathematically competent reader who immediately spotted this blatant flaw in Galileo’s book.
Here’s another interesting point that Descartes makes: Galileo “seems to have written [this theory] only to explain the force of cannon shots fired at different elevations.” That is to say, Galileo made no theoretical use of his theory of projectile motion whatsoever. For example, he makes no connection to the motion of planets, the moon, comets; nothing like that. That’s a huge missed opportunity.
Instead Galileo erroneously claimed that his theory would be practically useful for people who were firing cannons. That’s quite naive, as Descartes pointed out. Here’s a quote on this by the historian A. Rupert Hall: “In many passages Galileo remarks that the theory of projectiles is of great importance to gunners. He made little or no distinction between his theory and useful ballistics; he believed—though without experiment—that he had discovered methods sufficiently accurate within the limitations of military weapons to be capable of direct application in the handling of artillery.”
This belief, however, was completely wrong. A contemporary put the matter to experimental test, and reported as follow: “I was astonished that such a well-founded theory responded so poorly in practice. If the authority of Galileo, to which I must be partial, did not support me, I should not fail to have some doubts about the motion of projectiles, and whether it is parabolical or not.” That’s a follower of Galileo writing shortly after his work was published.
Galileo foolishly thought his theory would work without testing it. This is evident for example from the extensive tables that he printed as an appendix to his big book: ballistic range tables based on his theory. These long tables make no sense at all other than as a practical guide for firing cannons. So clearly Galileo thought his theory was practically viable, which it is absolutely not.
Here’s a more theoretical issue related to inertia: the relativity of motion.
When teaching basic astronomy at Padua, Galileo explained to his students that Copernicus was undoubtedly wrong about the earth’s motion. The earth doesn’t move, Galileo explained. Because, if the earth moved, a rock dropped from a tower would strike the ground not at its foot but some distance away, since the earth would have moved during the fall. In support of this claim, “Galileo observed that a rock let go from the top of a mast of a moving ship hits the deck in the stern.” This had indeed been reported as an experimental fact by people who actually carried it out.
Of course this is completely backwards and the opposite of Galileo’s later views that he is famous for. To be sure, these lectures do not necessarily say anything about Galileo’s personal beliefs. In all likelihood he simply taught the party line because it was the easiest way to pay the bills. But at least the episode does show that the simplistic narrative that “the experimental method” forced the transition from ancient to modern physics is certainly wrong. On the contrary, experimental evidence was among the standard arguments for the conservative view well before Galileo got into the game.
In his later works Galileo of course affirms the opposite of what he said in those lectures: the rock will fall the same way relative to the ship regardless of whether the ship is standing still or travelling with a constant velocity. He gives a very vivid and elaborate description of this principle. I’ll quote in it full, it’s a long quote but it’s quite fun:
“Shut yourself up with some friend in the main cabin below decks on some large ship, and have with you there some flies, butterflies, and other small flying animals. Have a large bowl of water with some fish in it; hang up a bottle that empties drop by drop into a wide vessel beneath it. With the ship standing still, observe carefully how the little animals fly with equal speed to all sides of the cabin. The fish swim indifferently in all directions; the drops fall into the vessel beneath; and, in throwing something to your friend, you need throw it no more strongly in one direction than another, the distances being equal; jumping with your feet together, you pass equal spaces in every direction. When you have observed all these things carefully (though doubtless when the ship is standing still everything must happen in this way), have the ship proceed with any speed you like, so long as the motion is uniform and not fluctuating this way and that. You will discover not the least change in all the effects named, nor could you tell from any of them whether the ship was moving or standing still. In jumping, you will pass on the floor the same spaces as before, nor will you make larger jumps toward the stern than toward the prow even though the ship is moving quite rapidly, despite the fact that during the time that you are in the air the floor under you will be going in a direction opposite to your jump. In throwing something to your companion, you will need no more force to get it to him whether he is in the direction of the bow or the stern, with yourself situated opposite. The droplets will fall as before into the vessel beneath without dropping toward the stern, although while the drops are in the air the ship runs many spans. The fish in their water will swim toward the front of their bowl with no more effort than toward the back, and will go with equal ease to bait placed anywhere around the edges of the bowl. Finally the butterflies and flies will continue their flights indifferently toward every side, nor will it ever happen that they are concentrated toward the stern, as if tired out from keeping up with the course of the ship, from which they will have been separated during long intervals by keeping themselves in the air. And if smoke is made by burning some incense, it will be seen going up in the form of a little cloud, remaining still and moving no more toward one side than the other.”
Ok, so that’s that famous passage. Galileo’s prose is as embellished with fineries as this little curiosity-cabinet of a laboratory that he envisions. But is it any good of an argument? Insofar as it is, the credit is perhaps due to Copernicus himself, who had already made much the same point a hundred years before. Here are his words:
“When a ship floats over a tranquil sea, all the things outside seem to the voyagers to be moving in a movement which is an image of their own, and they think they themselves and all the things with them are at rest. So it can easily happen in the case of the movement of the Earth that the whole world should be believed to be moving in a circle. Then what would we say about the clouds and the other things floating in the air or falling or rising up, except that not only the Earth is moved in this way but also no small part of the air [is moved along with it]?”
So Galileo’s relativity argument, like so much else he says, is old news. The primary contribution of his version is literary ornamentation. Adding some butterflies and whatnot, while saying nothing new in substance.
Perhaps one could argue that Galileo goes beyond Copernicus’s passage by asserting more definitively that no mechanical experiment of any kind could prove that the ship is moving. Today the so-called “Galilean” principle of relativity says that the phenomena in the cabin cannot be used to distinguish between the ship being at rest or moving with constant velocity in a straight line. But Galileo clearly has another scenario in mind: he sees the ship as travelling along a great circle around the globe. This is the kind of motion he believes cannot be distinguished from rest, in keeping with his misconceived idea of horizontal inertia. This principle of relativity—the actually “Galilean” one—is of course false. In fact it’s even worse than that. Galileo’s purpose with this passage about the ship is to argue, erroneously, that the rotation of the earth cannot be detected by physical experiments, which in fact it can. The Foucault pendulum is a device that can detect this.
So the attribution of the principle of relativity of motion to Galileo in modern textbooks is doubly mistaken. First of all, relativity of motion and the idea of an inertial frame had been noted long before and was invoked by Copernicus to much the same end as Galileo. Moreover, Galileo’s principle is wrong in itself (because it’s about motion in a great circle, not in a straight line), and furthermore his purpose in introducing it is to draw another false conclusion from it (namely that the earth’s motion is undetectable). So errors at every turn as usual with Galileo. And there’s plenty more where that came from.
Galileo is praised for his work on falling bodies, but his arguments were dishonest and his trifling discoveries were not new.
Transcript
In 1971, Apollo 15 astronauts conducted a famous experiment on the moon. Here’s a bit of the original recording:
“In my left hand I have a feather. In my right hand a hammer. And I’ll drop the two of them here and hopefully they will hit the ground at the same time. How about that? Mr Galileo was correct.”
Actually, no. Mr Galileo was not correct. What the astronauts should have said is: Mr Galileo was wrong. According to Galileo, the moon has an atmosphere like the earth. So the feather should fall more slowly then, just like on earth. Galileo even claimed that this is “obvious” that the moon has an atmosphere. Obvious! That’s his word. This is in the Sidereus Nuncius, one of his famous published works. It is “obvious” that “not only the Earth but also the Moon is surrounded by a vaporous sphere.” Those are Galileo’s own words.
So if the astronauts wanted to test Galileo’s theory they should not have dropped a hammer and a feather. They should have taken off their helmets and suits and tried to breathe. That would have showed you how “right” Galileo really was.
But ok, let’s put the issue of the moon’s atmosphere aside. Heavy objects fall as fast as light ones, if we ignore air resistance. We are often told that this is one of Galileo’s most fundamental discoveries, and also that he supposedly destroyed the Aristotelian theory of physics on this point by simply dropping some objects of different weight from the Leaning Tower of Pisa. That was allegedly an eye-opening moment in which the world realised that empirical science is more reliable than philosophy and the words of ancient authorities. So goes the story-book version. Let’s see how much truth there is to these things. If any.
Galileo indeed often portrays himself as defeating obstinate philosophers who would rather cling to the words of Aristotle than believe empirical evidence and experimental demonstration. He imagines his enemies to say things like: “You have made me see this matter so plainly and palpably that if Aristotle’s text were not contrary to it, I should be forced to admit it to be true.” That’s a quote from one of Galileo’s dialogues. Galileo likes to pretend that his enemies are like that. It makes life easy for him. But in reality Galileo’s “anti-Aristotelian polemics were directed only at straw men.” Galileo concocted these caricature Aristotelians in order to “let them play the buffoon in his dialogues, and thus enhance his own image in the eyes of his readers,” as one historian has aptly put it.
Galileo’s ploy was well calculated. It tricks many of his readers to this day into believing the fairytale of Galileo the valiant knight singlehandedly fighting for truth in world beset by dogmatism. But “excessive claims for Galileo the dragon slayer have to be muted,” as one overly diplomatic historian puts it. Such ridiculous characterisations of Galileo should not only be “muted” but actively reversed.
This is clearly seen in the case of that famous question: do heavy objects fall faster than lighter ones? Aristotle had answered: yes. Twice as heavy, twice as fast. According to Aristotle.
Legend has it that Galileo shocked the world when he dropped some balls of different weight from the tower of Pisa and revealed them to fall at the same speed. But the notion that it required some kind of radical conceptual innovation by a scientific genius like Galileo to realise that one could test the matter by experiment is ludicrous and idiotic. Of course one can drop some rocks and see if it works: this much has been obvious to any fool since time immemorial. In fact, Philoponus—an unoriginal commentator—had clearly and explicitly rejected Aristotle’s law of fall by precisely such an experiment more than a thousand years before Galileo. This is just a plain fact. We have the source.
So if you want to believe that experimental science and empirical verification was a radical new insight then this puts you in quite a pickle. If that was a revolution, then why is it found for the first time in this extremely mediocre commentator from the 6th century? If these people had the key to science, then why did sit around and write commentary upon commentary on Aristotle? Their contributions to mathematics and science is otherwise zero. Do you really expect anyone to believe that the principles of scientific method escaped the many first-rate minds of the age of Archimedes, only to be discovered by utter nobodies in an age of vastly, incomparably lower intellectual quality? How likely is it that such elementary scientific principles eluded generations of the best mathematical minds the world has ever seen, only to be then discovered by derivative and subservient thinkers in an age where the pinnacle of mathematical expertise extended little further than the ability to multiply 3-digit numbers?
It doesn’t make any sense. So that’s a proof by contradiction that testing things empirically was never revolutionary. It is a trivial idea that any fool has always considered obvious.
Indeed Galileo was far from original in his own age either. Have you ever heard anyone calling Benedetto Varchi “the father of modern science”? Yet here is his statement of “Galileo’s” great insight, expressed two decades before Galileo was even born. Quote:
“The custom of modern philosophers is always to believe and never to test that which they find written in good authors, especially Aristotle. [But it would be] both safer and more delightful to descend to experience in some cases, as for example in the motion of heavy bodies, in which both Aristotle and all other philosophers without ever doubting the fact, have believed and affirmed that according to how much a body is more heavy, by so much more [speed] will it descend, the test of which shows it not to be true.”
All of that is a quote from a run-of-the-mill humanist writing in 1544, long before Galileo. That just goes to show what an obvious idea it was.
It is not clear whether Galileo did in fact carry out such an experiment from the tower of Pisa when he was teaching at the university there, as legend would have it. Personally I find the story plausible since a field day throwing rocks surely had great appeal to a professor who wasn’t very good at thinking. But be that as it may. It is in any case perfectly clear that many people carried out such experiments around that time, independently of Galileo. Simon Stevin, for example, in the Netherlands, certainly did, and published his results years before Galileo made his experiment, if indeed he ever did. Stevin used lead balls of different weights. He dropped them from a height of 30 feet. And on the ground he had placed some metal sheet or something that would make a lot of noise. So you could hear whether they banged down at the same time or not. So again a way of avoiding the need for stopwatches or high-speed cameras by relying on our natural hearing which is pretty good at this. And Stevin was not the only one. In Paris, Mersenne was doing the same thing. He was dropping weights out out Parisian chamber windows before he ever heard of Galileo.
Here’s a quote from Butterfield: “To crown the comedy, it was an Aristotelian who in 1612 claimed that previous experiments had been carried out from too low an altitude. In a work published in that year he described how he had improved on all previous attempts—he had not merely dropped the bodies from a high window, he had gone to the very top of the tower of Pisa. The larger body had fallen more quickly than the smaller one, and the experiment, he claimed, had proved Aristotle to have been right all along.”
Galileo himself once cited another Aristotelian philosopher who conducted similar experiments. In order to investigate whether lead falls faster than wood, “we took refuge in experience, the teacher of all things,” says this author, and hence “threw these two pieces of equal weight from a rather high window of our house at the same time. The lead descended more slowly. Not only once but many times we tried it with the same results.” That’s a work from 1575.
So much for Aristotelians hating experiments. On the contrary, appeal to experiments were commonplace long before Galileo. But these guys got the result wrong, you say. Maybe they didn’t experiment at all, or if they did they messed it up somehow.
Well, so did Galileo. You remember how he got the wrong value for the area of the cycloid? So also for falling bodies. He messed that up too at first. In his earlier notes on this he writes: “If an observation is made, the lighter body will, at the beginning of the motion, move ahead of the heavier and will be swifter,” but if the fall is long enough the heavier body will eventually overtake it. Galileo devotes a full chapter to following this up, in which, in his words, “the cause is given why, at the beginning of their natural motion, bodies that are less heavy move more swiftly than heavier ones.”
So when Galileo started experimenting on this he got the wrong result, and he also believed himself to a have a good theory “explaining” those false results. We now know that the true cause for the erroneous results was not a theoretical one like Galileo imagined, but more likely a more pedestrian circumstance. Namely, that we are not good at dropping one object from each hand at the same time. Modern experiments show that people tend to drop the lighter object sooner, even though we feel that we have dropped them at the same time. This is why Galileo and others ended up thinking that lead balls started out slower than lighter bodies and only then picked up speed.
With experiments being so inconclusive, then, it is no wonder that Galileo relied more on a theoretical argument in his published account. His supporters would have us believe that “Galileo showed that Aristotle’s rule could be refuted by logic alone.” His argument is supposedly a “splendid and incontrovertible” model example of “cast-iron reductio ad absurdum reasoning.” Those are quotes from Galileo scholars. And they are all wrong.
Here is the quote from Galileo: “By a short and conclusive demonstration, we can prove clearly that it is not true that a heavier moveable is moved more swiftly than another, less heavy. If we had two moveables whose natural speeds were unequal, it is evident that were we to connect the slower to the faster, the latter would be partially retarded by the slower. But the two stones joined together make a larger stone; therefore this greater stone is moved less swiftly than the lesser one. But this is contrary to your assumption. So you see how, from the supposition that the heavier body is moved more swiftly than the less heavy, I conclude that the heavier moves less swiftly.”
“From this we conclude that both great and small bodies, of the same specific gravity, are moved with like speeds.” Furthermore, “if one were to remove entirely the resistance of the medium, all materials would descend with equal speed.”
All of that is Galileo, his celebrated argument. With this argument Galileo allegedly exposes a fundamental logical inconsistency in the Aristotelian theory of fall. Except he doesn’t. Aristotle is perfectly clear: heavier objects fall faster. So when you put the heavy and the light together they will fall faster. The inconsistency arises only when one inserts the additional assumption that when you put two bodies together the lighter will retard the heavier and slow it down. But there is no basis whatsoever for this latter assumption in Aristotle. It is a fiction that Galileo has made up. Only by dishonestly misrepresenting the view he is trying to refute in this way is he able to draw his triumphant conclusion.
A more honest form of the argument, which doesn’t depend on misrepresenting Aristotle, is the following.
Consider two identical bricks. If you drop them at the same time they would fall side by side. Of course. If a loose string connected them that wouldn’t make any difference. What if you glued the bricks together? According to Galilean logic, “no reason appears why this double brick of double weight should fall faster than two bricks tied together—or either one alone.”
Galileo was aware of this form of the argument, it’s in some notes of his. Although he decided to use the more dishonest version in his published work.
Anyway, what this shows, then, is that the real crux of the argument is the claim that two bricks held side by side should behave the same way in terms of fall whether they are glued together or not. This is not a bad argument, but it is not a matter of “logic alone” as the Galileo fans would have us believe. For instance, imagine you are taking a basketball free throw. You can choose between trying to hit the hoop with either two bricks glued together or two bricks merely held side by side as you throw them. Would you really say that “no reason appears” why nature should treat the two cases differently, so you might as well go with the loose bricks? I don’t think so. Then why accept this assumption in Galileo’s case?
So, if we’re being honest, we are back to having to rely on experiment after all. Galileo indeed discusses the experimental side of the matter too in his treatise. He admits that actual experiments do not come out in accordance with his law because of air resistance. But, he says, the fit is much better than for Aristotle’s law. He gives specific numbers for this. Exact measurements of how much the slower ball lags behind the heavier one. But this is fake data. He cooks the numbers to sound much more convincing in favour of his theory. The actual lag or difference between the two bodies is more than 20 time greater than the fake data Galileo reports in his published so-called masterpiece. “In no case could Galileo have consistently achieved the results he reported,” as one scholar says.
Nevertheless it remains true that Galileo’s law doesn’t fare as poorly as that of Aristotle in this experiment. So wow, what a hero, the great Galileo. He managed to improve on a two-thousand-year-old claim, made by a non-mathematician, which not a single mathematician ever believed. And which Aristotle himself obviously did not intend as quantitative science. Aristotle only introduced his so-called law very passingly and parenthetically as a stepping-stone toward making the philosophical point that there can be no such thing as an object of infinite weight. That’s one paragraph buried somewhere in the middle of his voluminous metaphysics, never to be used again.
Aristotle did not claim or intend his law as a definitive or quantitative treatment of falling bodies. Only later fools who clung to his words like gospel because they could not think for themselves made a big thing of this so-called law of fall. Kind of like some people seized upon some obscure remark in the Bible and gave it all kinds of significance beyond the apparent intent of the text.
Nobody was so foolish in antiquity as far as we know. As we saw, even people like Philoponus, who were so obsessed with Aristotle that they wrote long commentaries on his every word, even people like that rejected the law. In fact, “Galileo’s” discovery, so-called, that in the absence of air resistance, all objects fall at the same speed regardless of weight—that law is in fact not first stated by Galileo, as so many people believe. Rather it is explicitly stated by Lucretius, well over a thousand years before Galileo. “Lucretius was correct”; that’s what the astronauts should have said. As ever, Galileo gets credit for elementary ideas that are thousands of years old. In his own time too, a number of people discovered “Galileo’s” law of fall independently of him: certainly Thomas Harriott and Isaac Beeckman; arguably Descartes as well.
So we see how dependent Galileo is on the framing of him versus Aristotle. No wonder he clings to this and uses it as the trope of his dialogues. Galileo’s entire case rests on his readers considering Aristotle to be a great authority. If we admit the truth—that Aristotle’s law had been refuted more than a thousand years before, and that the ridiculous idea of relying on Aristotle for quantitative science would never have entered the mind of any mathematically competent person in Galileo’s time or in antiquity—then what does Galileo have to show for himself? An unproven claim that doesn’t even fit the fake data Galileo has specifically concocted for the purpose, let alone the many experiments that proved the opposite, including his own before he knew which way he was supposed to fudge the data. Galileo likes to portray himself as doing the world a great service by defeating the rampant Aristotelianism all around him. The truth is that he is rather doing himself a great service by pretending that these Aristotelian opinions are ever so ubiquitous, so that he can inflate the importance of his own contributions, the feebleness of which would be all too evident if he addressed actual scientists instead of straw men Aristotelians.
So, with all these arguments we have done quite a bit of damage to Galileo’s claim to credit for his law of fall, I believe. But we’re only getting started. There’s plenty more to come, including several howlers Galileo made when he tried to apply his law. But that’s for another day.
Ancient Greek scientists studied the dynamics of falling bodies. Were “Galileo’s” discoveries anticipated in these treatises that have since been lost? This question leads to a bigger one regarding relativism versus universalism in the history of thought.
Transcript
Quiz! Who said the following:
“The study of mechanics is eagerly pursued by all those interested in mathematics.”
What is the source of that quote? And here is another one:
“We attack mathematically everything in nature.”
Who said that? Surely it is after Galileo anyway, right? Since he is the one who invented the mathematisation of nature?
No. The quotes are from Pappus and Iamblichus. Greek authors, ancient authors. Well over a thousand years before Galileo.
Much modern scholarship would have you believe that such attitudes never existed. This is what you get if you read too much Plato and Aristotle and not not enough mathematicians. In Plato, mathematics is purer than snow. To apply it to the physical world is to defile it. So that’s quite an obstacle to science. If you glorify pure and abstract thought as the only worthwhile pursuit of rational beings, and deride empiricism as fit only for unphilosophical beasts, then you’re not going to get a whole lot of science done.
Aristotle too has many teachings antithetical to proper science. There’s his big book on “Physics”, so called, which is really metaphysical philosophy. Aristotle doesn’t care about the phenomena or laws of motion. Instead he cares about pseudo-profound philosophy puzzles like: Does any motion have a cause? Or: Does motion presuppose the existence of a cause of that motion? And by the way, there are four different kinds of causes, you see: the efficient cause, the material cause, and blah blah blah, whatever the other ones are, who cares. And you can sit around and split hairs between them all day long. Typical philosophy boilerplate stuff. Obviously a completely wrong turn on the road to science.
So that’s Plato and Aristotle in a nutshell. If that’s all you know about Greek scientific thought then Galileo is a breath of fresh air, sure enough. But Plato and Aristotle do not speak for “the Greeks”. You don’t get a very good idea of 20th-century science by reading Sartre and Heidegger. Likewise, if you want to understand Greek scientific thought, forget about Plato and Aristotle. On mathematical subjects they are derivative commentators at best.
There is zero evidence that those Platonic and Aristotelian ideologies that I outlined had any influence whatsoever on any mathematically competent person in the classical and Hellenistic eras of antiquity. People often refuse to accept this. Leading scholars. Their entire worldview is centered around the assumption that Plato and Aristotle are the alpha and the omega of Greek thought. Any suggestion to the contrary they will fight with tooth and nail as if their life depended on it. Which it actually kind of does, as I explained before.
But to take Aristotle’s ultra-philosophical physics as the state of the art of Greek science is ridiculous. It’s like saying that Wittgenstein was a leading 20th-century quantum physicist. Read the mathematicians, if you want to know what was really going on.
We know for a fact that even Aristotle’s own successors as heads of his own Lyceum departed right away from his teachings on a number of scientific questions. And that’s philosophers from his own school. Just imagine how little the mathematicians cared about Aristotle’s ideas. Only much later, in the Middle Ages, which was a time of vastly lower intellectual quality, did people come up with the imbecile notion of taking Aristotle to be an authority on physics.
But ok, enough ranting about that. Now let’s look at some actual Greek science instead. Let’s think about some of Galileo’s main results and investigate whether there was anything like that in Greek times.
Falling bodies, for example. The laws of motion of falling bodies, that’s a big Galilean triumph, right? Well, not so fast.
Maybe the Greeks already knew those things. We don’t know because a huge part of the Greek scientific corpus is lost. Burnt down in various library fires and what not. Or just disintegrated and discarded. It was a fragile thing. These works had to be hand-copied over and over to survive. Just think about books from the 19th century and what state they are in today. They’re already falling apart. Ancient Greek works had to beat long odds to make it.
Take Euclid’s Elements, for example. The oldest manuscript we have is closer to us in time than to Euclid. From ancient times we have only the tiniest scraps. A quarter of a page here and three lines there. Basically nothing.
It was a long time ago. How many of the books we print today are going to be around in more than 2000 years? Not many, I bet.
But we know something about what was lost through references in other works. And as far as the science of falling bodies is concerned we have some very intriguing indications.
We know for a fact that Strato wrote a treatise on falling bodies. It did not survive. But here’s what we know. Strato was an avid experimenter. To prove that falling objects speed up, he said: Pour water slowly from a vessel. At first it flows in a continuous stream, but then further down its fall it breaks up into drops and trickles. This is because the water is speeding up. So the water spreads out, like cars let loose on a highway after a congested area.
And here’s another experiment Strato used. Stones dropped into a sand bed from various heights. The stone makes craters of different depths depending on height fallen. Again showing that the thing is picking up speed as it goes. Did Strato make actual numerical measurements of this? Of the sand craters and other ways of quantifying fall? Maybe. Who knows?
And here’s another guy who wrote on falling bodies: Hipparchus. That’s an author who commands respect. Probably the greatest mathematical astronomer of antiquity, though his works are lost. Certainly a way better mathematician than Galileo, that’s for sure.
So we know that he wrote a treatise on falling bodies, which is lost. Of course he didn’t follow Aristotle’s ridiculous theory of falling bodies. In fact, a commentator on Aristotle explicitly says: “Hipparchus contradicts Aristotle regarding weight.”
But what exactly did he say? The indications are that he argued that weight depend on distance from center of the earth. He seems to have been engaged in questions like: if there was a tunnel through the center of the earth and an object fell down it, what would happen as it approached the middle? Superficial commentators have picked up on these striking aspects of his work that can be presented as a kind of gimmick in isolation. In the original they would surely have been incorporated in a mathematical treatment.
These things are very much in line with 17th-century physics. Dropping stones into sand, thinking about how gravity varies on a super-terrestrial scale and inside the earth. That is literally exactly what scientists spent a lot of serious effort on in the 17th century.
So here’s what we know. Greek scientists, who were excellent mathematicians and keen experimenters, wrote several lost treatises on falling bodies. What are the chances that this included good chunks of so-called Galilean science? Maybe including for example “Galileo’s” law of fall? They are appreciable. It is squarely within the realm of possibility and then some.
Here’s what a modern scholar says about Galileo’s law of fall: “The ease of stumbling upon this discovery renders it highly improbable that natural philosophers had ever searched for the law of fall [before the 17th century].” Yes, either that, or: they did search and they did find it. If it’s easy to stumble upon, what are the chances that these very sophisticated first-rate minds, who wrote entire treatises on this exact subject, somehow missed it? It’s food for thought.
The law I’m talking about here is the one that say that the acceleration of a freely falling object is constant. Equivalently, its velocity is proportional to time, and the distance fallen is proportional to time squared. Motion on an inclined plane is closely related to this. A ball rolling down a slope is basically a slow-motion version of falling. The ball will acquire the same speed as it would have in free fall through the same vertical distance. As we would say today, in anachronistic terms, since all balls covering the same vertical distance trade in the same amount of potential energy, they get the same amount of kinetic energy out of it.
This slow-motion of falling is easier to deal with experimentally. Here’s Stillman Drake the Galileo scholar again; he says: “In a way it is surprising that the law for the spontaneous descent of heavy bodies had not been recognized long before the 17th century. Measurements sufficient to put the law within someone’s grasp are quite simple. Equipment for making them had not been lacking---a gently sloping plane, a heavy ball, and the sense of rhythm with which everyone is born.”
The sense of rhythm is like a natural clock. You can tell when a musician is off beat. We can use this innate skill to verify the law of fall. No stopwatch required.
Just get your inclined plane, and put markers at interval that the ball ought to cover in equal times according to your hypothesised law, and then put little bells there at those points, and roll the ball down the slope. Do the bells ring at equal intervals or not? That’s quite easy to tell.
I did this experiment for your benefit. I’m going to play it for you. I used a very simple and primitive setup, using just some stuff I had around the house. For my inclined plane or ramp I found a suitable piece of wood in my basement; it used to be part some furniture or other. So I used a kitchen knife and cut some markings into it at places a rolling ball should cover in equal times. That just means putting a measuring tape down and marking at all the squares. I made marks at 4, 9, 16, and 25 units from the starting point. You can use whatever unit you like. There is no need to know anything about a gravitational constant or anything like that. Just put your ruler down and make these marks at the square numbers and that’s it.
So then I put the ramp in position on my living room coffee table. Then I got four wine glasses that I placed along the ramp at the markings I had made. I just propped them up with some books as needed. Then I had a glass marble lying around that happed to be suitable to the purpose. I positioned everything so that the marble would just about touch the wine glasses as it rolled down the ramp.
So, here we go, I’m going to play it for you. The idea is that you will hear four equally spaced clinking sounds.
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It worked pretty well, I’d say. The thump at the end is the marble crashing into a bundled-up t-shirt that I put at the end of the ramp. I also made some recordings with the glasses in the wrong positions. I put them about equally spaced instead of spaced like square numbers. So I will play that recording as well and see if you can tell the difference.
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I think the difference is pretty clear. Considering that I threw this experiment together very sloppily in about ten minutes I’d say it is viable and easy to build a quantitative theory of falling bodies this way. Obviously it would be easy to improve the accuracy of the experiment a lot from my extremely primitive setup. The way I did it the marble whacked into the wine glasses pretty good. That probably slowed it down by perhaps a non-trivial amount. If I was doing this for real I would of course use a heavier ball, a longer ramp, and, for the markers, tiny bells or something that isn’t heavy enough to impact the ball.
But I think even my setup kind of worked. I made a number of takes with each configuration in fact. I’ll play a couple of them and then as a blind test we will see if you can hear which are the correct configuration with the equal intervals and which are the bad ones with unequal intervals. Here we go.
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We don’t know if the Greeks did this something like this. There is no evidence that they did. They certainly could have. But let’s put that question aside.
There’s a more general point to be made here. This stuff about the inclined plane, it’s very simple. The law is simple; the mathematics is simple; the experiment is simple; the idea is simple. You can explain everything to a child. If Galileo was the first to discover stuff like this, it must be considered a revolution or breakthrough of a conceptual nature. Certainly not a mathematical one, because mathematically it’s trivial. This goes for all or almost all of Galileo’s so-called achievements.
The underlying question is: Are there conceptual revolutions like that? Is the history of science a story of conceptual revolutions that made previously unimaginable things suddenly obvious? Some people are very willing to accept this. Many scholars are. It’s quite fashionable even, I would say, to insist on this.
Historians make a big mistake here in my opinion. They start from a reasonable premiss. They observe that those who have little or no expertise in history often interpret historical episodes in a naive way. These historical beginners take current ways of thinking for granted, and small-mindedly interpret past thought in terms of those narrow categories. That’s a massive mistake of course. And it is very stupid because it defeats the whole purpose of studying history in the first place, namely that history shows us other perspectives, other ways of thinking.
So that’s all fine and well. Modern historians are very much correct in condemning this kind of thing. However, they then go too far. They think to themselves: since it is bad to involve too much current ways of thinking when looking at the past, the best historian must be the one who does the exact opposite of that. That is to say, the best historian must be the historian who always emphasises conceptual differences whenever possible, and who always blows the tiniest indications of differences way out of proportion as if it was a very big deal. Since this is the opposite of the naive approach, it must be the pinnacle of sophistication, right? That’s what many historians seem to think.
This is of course a bad idea, since it is just as naive and dogmatic as the erroneous mentality it is supposed to counter. But it’s not my purpose to criticise it now, only to consider what it means in the case of Galileo.
I said I could explain the rolling ball and the law of fall to a child. Any child; modern or ancient. This is what the modern historians are eager to deny. They do not believe in such universalism. They say: No, the Greeks lived in a different conceptual universe; their way of thinking is just fundamentally, qualitatively different from ours. Even the most obvious or evident thing to us may very plausibly have been completely outside of their conceptual sphere. They couldn’t even think it, because the way they approached the world was just inherently and profoundly different from ours.
Galileo’s status stands and falls with our willingness to accept this radical relativism. His discoveries are so basic and obvious that the only way to consider them profound is to maintain that they were once not basic and obvious. In other words, that they are fundamentally different in character from anything the Greeks were doing, for example.
If we think there is only one common sense, and that mathematical truth and thought is the same for everyone, then we are strongly inclined to see Galileo’s achievements as quite basic, and we are strongly inclined to think that the works of for example Hipparchus on falling bodies was probably very similar to what 17th-century authors said on the same subject.
On the other hand, if we reject the very notion of a universal scientific common sense, then we are primed to think that Galileo opened up an entirely new world with his style of science.
So studying Galileo is a mirror to much larger questions. Either you are a cultural relativist and you think Galileo was a revolutionary, or you think mathematical thought is the same for you, me and everybody who ever lived, and then you think Galileo was just doing common-sense stuff. Those are the two possibilities. You have to pick sides. You can’t mix and match. You can’t have both mathematical universalism and Galileo being a revolutionary. These two basically contradict one another.
So once again Galileo is at the heart of fundamental questions about history. All the more reason to study him in detail. Which is exactly what we are going to do. Next episode it’s into the weeds on Galileo’s work on falling bodies.
Divergent interpretations of Galileo’s alleged greatness cut across disciplinary divides: mathematics versus philosophy, science versus humanities. Understanding Galileo means dealing with these fundamental tensions.
Transcript
Those who can’t do, teach. I’m sure you have heard this saying. It sums up Galileo’s role in the history of scientific thought, in my opinion. Galileo’s books are “Science for Dummies”. He drones on and on about elementary principles of scientific method because he lacks the mathematical ability to do anything more advanced.
It is precisely because he is so bad at mathematics that he is forced to waste so many words doing something so trivial.
Galileo was not much of a mathematician but he knew a thing or two about rhetoric. He saw a way to make a virtue out of necessity. He realised very well that he could not hold a candle to Archimedes. So he chose to play a different sport. He went after Aristotelian philosophers instead. Sure enough he scores some points against these fools, but that’s fish in a barrel.
Here is what Descartes said about Galileo: “He is eloquent to refute Aristotle but that is not hard.” Yes, exactly. Descartes hits the nail on the head right there. If Descartes had a podcast I wouldn’t have to make mine. Relative to the philosophers Galileo is a big step forward, yes, that’s true. But relative to the mathematicians he is just saying obvious things that everyone had already known for thousands of years.
Descartes continued in the same vein: “I see nothing in his books to make me envious, and hardly anything I should wish to avow mine.” Galileo’s mathematical demonstrations in particular did not impress Descartes: “he did not need to be a great geometer to discover those,” Descartes says.
That’s right on the money. Do you think my take on Galileo is crazy and unbalanced? Well, then you think Descartes is crazy too. Because he agreed with me. And so did other mathematically competent people at that time. People who knew Archimedes, unlike many who write on Galileo today.
Galileo’s claim to fame rests on the assumption that Archimedes does not exist and that everyone but Galileo was a raving Aristotelian. Galileo himself went out of his way to ensure this framing. His two big books are both dialogues. One character is a mouthpiece for Galileo and another an Aristotelian simpleton. This is the contrast class Galileo wants us to use when evaluating his achievements. And no wonder. Refuting Aristotle is not hard, as Descartes said, so of course Galileo can score some zingers against this feeble opposition.
Galileo tries to pass himself off as a rebel truth-teller taking on the supposedly all-pervasive Aristotelian establishment. If one buys into this deceptive framing one may very well come away with the impression that Galileo has done something of value. But no. Galileo has rigged the game. He has pitted himself against a convenient punching bag.
Here’s a quote: “The philosophers of our times philosophise as men of no intellect and little better than absolute fools.” That’s Galileo. And he’s right. Fools, the lot of them, those philosophers. But Galileo was not the only one to see this.
Descartes, for example, said of the Aristotelians that they were “less knowledgable than if they had abstained from study.” Galileo’s claim to fame is that he refuted people who were “less knowledgable than if they had abstained from study.” Hardly the pinnacle of intellectual achievement.
Another example: Tycho Brahe. A leading astronomer some decades before Galileo, and a rather conservative guy. He too complained about “the oppressive authority of Aristotle” as he called it. “Aristotle’s individual words are worshipped as though they were those of the Delphic Oracle,” he says.
This kind of attitude was universal among mathematicians. No use writing several thick books hammering home this point and little else, which is what Galileo did. That’s just beating a dead horse as far as the mathematicians are concerned.
Mathematicians had a clear sense of “us versus them” in this respect. The mathematicians versus the philosophers. It is remarkable how they have complete faith in the judgement of other mathematicians, and utter contempt for everyone else. This attitude is everywhere in Galileo’s time and even before. Probably already in Greek times, for all we know.
Consider Copernicus, for instance. That’s a hundred years before Galileo. Here’s what he says when he introduces his theory that the sun is in the center of the solar system: “I have no doubt that talented and learned mathematicians will agree with me.” For I will, he says, make everything “clearer than day—at least for those who are not ignorant of the art of mathematics.”
And those who are ignorant of mathematics? He also addresses them. “If perchance there are certain idle talkers wholly ignorant of mathematics dare to attack my work; they worry me so little that I shall scorn their judgements.” These are the kinds of people who “on account of their natural stupidity hold the position among philosophers that drones hold among bees.” “The studious need not be surprised if people like that laugh at us. Mathematics is written for mathematicians.” That’s all from the introduction to Copernicus’s great masterpiece. He doesn’t mince words, does he?
It’s the same in Kepler, Galileo’s contemporary. “Let all the skilled mathematicians of Europe come forward,” he implores. Evidently he has complete confidence that mathematical reason compels them to speak with one voice.
Like Copernicus, Kepler also addresses the non-mathematicians. In his great masterpiece, the Astronomia Nova, he puts a section in the introduction with the heading “advice for idiots.” There he says things like: “whoever is too stupid to understand astronomical science, I advise him that he mind his own business and scratch in his own dirt patch.”
This is all before Galileo has published anything. Mathematically competent people were united and had nothing but complete contempt for Aristotelian philosophy and the like. By the time Galileo comes along and belabours this point it has been old news for hundreds of years.
You gotta admire the guts of these people. These quotes from Copernicus, Kepler, they are from their scientific masterpieces. Not a passing remark in confidential personal correspondence to blow off steam, not a reply to a particular provocation, not something said in the heat of the moment. No, the decided to put this “advice for idiots” right at the heart of their scientific masterpieces; their crowning accomplishments that were written for the ages.
Galileo joked about his “lack of tact” as he called it. And he was not alone in this. Mathematicians of this age were not much for tact. “The presence of good tone means the absence of good sense,” says Schopenhauer. The mathematicians of Galileo’s time had a lot of good sense.
In fact, this has always been the way of the mathematician. Just imagine today a philosopher walking into a mathematics department, and starting telling them what to do and how to think. Of course no mathematicians listen to that. Not today, not in Galileo’s time, not ever.
In antiquity there’s Ptolemy, for example, the astronomer. Here’s what he has to say: “only mathematics can provide sure and unshakeable knowledge”; other “divisions of theoretical philosophy should rather be called guesswork than knowledge.”
Mathematicians have always taken this for granted. And this is why, from a mathematical point of view, Galileo is nobody, because he did little else than provide redundant proofs of this self-evident truth.
Now: a conspiracy theory of sorts. As we have seen, Galileo needs us to assume that Aristotelianism and philosophy was the state of scientific knowledge in his day, and that no one had ever heard of Archimedes. Only then does his so-called accomplishments come off looking any good.
Ask yourself: Who is inclined to go along with such an assumption? I’ll tell you who: Someone who doesn’t know any Archimedes but is very comfortable with Aristotle and other philosophers. People from the humanities, in other words.
So Galileo is in luck. He needs an audience with certain blind spots and predispositions, and he gets exactly that. Modern academia is set up in his favour. History of science is nowadays a humanities field. The default training of historians of science is not higher mathematics and physics; it is reading seminars based on non-mathematical authors such as Aristotle.
So the people tasked with being Galileo experts are by design the people most inclined to accept Galileo’s deceit. Pretending that Archimedes does’t exist serves both their purposes and Galileo’s. They share Galileo’s aversion to proper mathematics, so they are more than happy to write off Archimedes as a genius to be sure but a very specialised one, who is just doing some esoteric math stuff that doesn’t really matter to the history and philosophy of science.
I tried to quantify this a bit. The History of Science Society publishes an annual bibliography of works in the history of science. I thought I would use it to compare Aristotle and Archimedes. I compiled the number of entries for the past 15 years. I found the following. Archimedes: 42 entries. 42 books and articles written about him in the past 15 years. Aristotle: 482 entries. Well over ten times as many as Archimedes. This concerns Aristotle’s role in the history of science only, mind you. It’s not counting works on Aristotle altogether, of which there are many thousands more.
Actually the ratio in favour of Aristotle should be doubled because the bibliography also has another 339 for Aristotelianism. Which means dogmatic followers of Aristotle, basically. There were many of those in the middle ages and still in Galileo’s time. They get a lot of attention.
There has never been any entry on Archimedeanism in the bibliography. But if you’re writing one then count me in as a reader.
So, anyway, that’s 20 times as many works on Aristotelian thought as on Archimedes. And what would you expect? If you put the history of science in the hands of humanities people, that’s what they’re gonna do.
In fact, how could they do otherwise? Suppose I’m right and that the 20 to 1 ratio in favour of Aristotle over Archimedes is foolish and distorts the true nature of the development of scientific thought. Suppose for the sake of argument that I am right about that. Even so, the humanities people could not very well say so, even if they believed it. That would basically amount to saying: Please don’t give us any more money. You trusted us to study the development of scientific thought but we have come to the conclusion that it is best understood from a scientific and mathematical point of view than the philosophical training that we have. Please give our research money to them instead. Please fire half of the people in my department because there is too much work on Aristotle being done already. They are not going to say that, are they? Even if they believed it, they would be fools to say it.
So the way modern historical scholarship is set up plays right into Galileo’s hands. Actually that’s true in more ways than one. A heavy bias toward philosophy and away from mathematics is one thing. But here’s another one. Contextualism versus universalism.
The issue comes down to this: Do great minds think alike? I say they do. I say there is a spiritual unity of scientific thought from ancient to modern times. I say that what is obvious to us was obvious to the Greeks. I say it is ludicrous to think that generations of Greek mathematical geniuses of the first order, with their extensively documented interest science, all somehow failed to conceive basic principles of scientific method. I say these things because I can feel it in my bones. I say these things because I have spent my life in mathematics departments and experienced so many times the profound sense of thinking exactly alike with another person. Young or old, student or professor, when we talk about mathematics our minds are one. Mathematics has this power, to make brethren of us all. This is why Copernicus and Kepler had unshakeable confidence that mathematicians would ultimately agree with them. For the same reason Galileo says with conviction that “if Aristotle were now alive, I have no doubt he would change his opinion.” Such is the historiographical outlook of mathematicians.
The idea that modern science was born in a Galilean revolution, on the other hand, is based on seeing history as soaked in cultural relativism. Following this school of thought, you must approach ancient texts as if they were mysterious communiqués from an alien life form on the other side of the universe. You must banish any notion of unity in human thought and instead view old and new as worlds apart, separated by a conceptual abyss that no intuition can bridge. This is the worldview and historiographical approach of many who are far removed from mathematics. From this point of view it is completely natural to think that basic principles of scientific method that seem so obvious today were in fact once completely outside the conceptual universe of even extremely sophisticated mathematical scientists like Archimedes.
This is why Galileo is the idol of the humanists and the bane of the mathematicians. The philosophers say he invented modern science; the mathematicians that he’s a poor man’s Archimedes. The issue cuts much deeper than merely allotting credit to one century rather than another. Much more than a question of the detailed chronology of obscure scientific facts, it is a question of worldview and how one should approach and understand history.
I say that the traditional view of the “Galilean” scientific revolution is not only historically wrong but fundamentally inconsistent with the nature of mathematical thought. When I say Galileo boo, Archimedes yay, my point is not who was the “first” or who should “get credit” for this or that. That’s not so interesting. But Galileo is a window into more important things.
What is the relation between mathematics and science? Was mathematics before Galileo a technocratic enterprise? Compartmentalised, limited to certain computational tasks, blind to its own potential? Was mathematics stuck in that ditch until it was liberated by a “conceptual” breakthrough from without, so to speak; from philosophy? Or was mathematics always an expansive, empirically informed, interconnected study of all quantifiable aspects of the world?
The latter, of course, if you ask me. In any case, Galileo is ground zero for grappling with these questions. And that is why we study him.
Galileo's bumbling attempts at determining the area of the cycloid suggests a radical new interpretation of his scientific opus. Archimedes's work on floating bodies is an example of excellent Greek science that has not been sufficiently appreciated.
Transcript
Galileo is the most overrated figure in the history of science. That’s the thesis of Season 1 of this podcast. It’s going to be proper revisionist history fun. If you enjoy scholarly polemics and received wisdom turned on its head, then this is the story for you.
But it’s also more than that. Galileo is at the heart of fundamental questions. What is the relation between science, mathematics, and philosophy? Between ancient and modern thought? What’s the history of our scientific worldview, of scientific method? All this big-picture stuff. Galileo is right in the thick of the action is on all of these issues. So that’s all the more reason to study him.
But let’s start small. Here’s a simple snapshot of Galileo at work.
The cycloid. It’s a famous geometrical curve. The cycloid is the path traced by a point on a rolling circle. So, in other words, say you have a bicycle wheel. And you attach a piece of chalk to the rim of it. Then you roll the wheel along a wall, so that the chalk is drawing on the wall as it’s rolling. That makes a cycloid. It’s a kind of arch shape.
What’s the area of the cycloid? That was a natural question in Galileo’s time. Finding areas of shapes like that is what geometers had been doing for thousands of years. Archimedes for instance found the area of any section of a parabola, and the area of a spiral, and so on. The cycloid was a natural next step. It fit right into this tradition.
Of course nobody cared about the area of the cycloid as such. That’s not the point. Think for example of portraits painted by great artists. Of course the value in such a painting is not that it accurately depicts some prince and whatever pompous costume he was wearing. Obviously the value of this as art is not the subject but the method. How does the artist manipulate composition and subtle detail to achieve a particular impact? How does the artist build on tradition, yet innovate beyond it? Those are the things we admire when we view the works of great artists.
And it’s the same in mathematics. Archimedes, when he found his areas, gave clever geometrical arguments; gorgeous proofs. The point is not that he gave you a “formula” to compute various areas. How often have you needed to know the area of a spiral anyway? Never, of course. The point is not the result. The point is that Archimedes took human thought to a new level. His proofs are beautiful; they are logically flawless. They give you a sense that you are at the pinnacle of what the human mind can achieve. Everybody wanted to see more that kind of thing.
So solving problems like the area of the cycloid then, in this sense, in the Archimedean sense, was the way to prove yourself a worthy geometer. So Galileo tried. And failed. All those brilliant feats of ingenuity that Archimedes and his friends had blessed us with; it just wasn’t happening for Galileo. He just wasn’t any good at it.
In fact he said so himself. Here is a quote from Galileo on Archimedes: “Those who read his works realise only too clearly how inferior are all other minds compared with Archimedes’, and what small hope is left of ever discovering things similar to those he discovered.”
That’s Galileo. And he’s quite right. Except maybe it’snot that all other minds are inferior to Archimedes. Although certainly Galileo’s is.
So picture Galileo sitting in his study, racking his brain, staring at the cycloid. The books of Archimedes lie open on his desk. Agh, all that math, it’s giving him a headache. Thinking isn’t really working out for Galileo. It’s not his strong suit.
Desperate, frustrated, he turns to the failed mathematician’s last resort: trial and error. He cuts the cycloid out of thick paper and starts weighing it. It’s much like in modern mathematics classrooms where certain students prefer pushing buttons on a calculator and have some device do the thinking for them instead of trying to figure stuff out for themselves. So also Galileo starts fiddling around with scales and cardboard cut-outs as a substitute for thinking.
It is striking to compare Galileo’s take on this problem with that of his contemporaries. They in fact did solve it. Roberval in France, Torricelli, Galileo’s countryman in Italy, and also Descartes. They all solved it. And not with some middle school cut-and-paste project. They solved it as mathematicians, using reasoning and proof. These are contemporaries of Galileo. If they could do it, why couldn’t he?
Galileo himself says the cycloid area is a “very difficult” problem. “I worked on it fruitlessly,” he says. Compare that with Descartes’s reaction. Descartes, a mathematically competent person. When Descartes receives the problem—it was being passed around in correspondence at this time, as a challenge to various mathematicians—when Descartes receives it, he at once replies: “I do not see why you attribute such importance to something so simple, that anyone who knows even a little geometry could not fail to observe, were he simply to look.” And he back this up by immediately sending his solution, which he composed on the spot.
Now, Descartes, he is not famous for his humility, to put it mildly. So we should not necessarily read too much into those words there. But even so, the contrast with Galileo is very striking. The fact is that a number of mathematicians solved the cycloid problem with relative ease, while Galileo was fumbling with scissors and glue.
And Galileo got it wrong, too. Not only did he fail to find one of those beautiful Archimedean proofs, which was the whole purpose of the exercise in the first place. Not only that, but he also got the result itself wrong. The area of the cycloid is in fact 3 times the area of the generating circle, as these mathematicians showed. Galileo through specifically concluded based on his bumbling experiments that it was not exactly 3 but a bit off from 3. By relying on experiments unchecked by proper mathematics, Galileo got the answer wrong, and not for the first time nor the last.
That just goes to show why mathematicians have little respect for experiment. Christiaan Huygens, another very competent mathematician, once said: “Do not think that I am relying on experiments, because I know they are deceitful.” That was in a different context, nothing to do with the cycloid. But all the same. This is a universal attitude among mathematicians. And for good reason, as we see in the Galileo case. Haphazard trial and error has to be superseded by rigorous demonstration, as able mathematicians have always known.
So the moral of the story is this. In the case of the cycloid, it is undeniable that Galileo turned to empiricism precisely because he lacked the mathematical ability to tackle the problem any other way. It is beyond any shadow of a doubt that if he had had the ability to compose a mathematical proof, like some of his contemporaries did, he would have loved nothing more than to do so. In the case of the cycloid, these are facts.
This leads us to my revisionist thesis. I suggest that the cycloid episode is typical of Galileo’s science altogether. As with the cycloid, so with science. Galileo was bad at mathematics. And it is precisely because he was so bad at mathematics that he was so keen on experiments. He was not a pioneer of scientific method. He was not the father of modern science. He was not a heroic knight defeating dogmas and superstitions with the light of empirical truth. No, he was none of those things. Galileo was, first and foremost, a failed mathematician. This is the key to understanding his role in the history of science, in my opinion.
Galileo’s contribution to the history of thought is to cut off mathematical reasoning at the training-wheels stage; to air in public what true mathematicians considered unworthy scratch work at best. He experiments because he cannot think. He cannot reach insights by reason, so he turns to more simplistic methods, hands-on methods. In physics this blatant shortcoming has been mistaken for methodological innovation. But in the case of the cycloid we see its true colours. We see that it is a sign of failure rather than genius. Galileo’s empiricism is the last resort of a failed mathematician. It is not science being born; it is science being dumbed down.
I will argue for this conclusion in considerable detail in many future episodes. In any case, the cycloid example obviously fits this thesis like a glove. So that’s certainly food for thought at the very least.
It is instructive to compare Galileo with Archimedes, who was a proper mathematician. Consider Archimedes’s work on floating bodies. This is science done right, almost two thousand years before Galileo. Archimedes’s treatise is an outstanding masterpiece of science by the standards of any age. Only the mathematically illiterate could fail to grasp its immense significance. As indeed they have.
Archimedes gives a thorough theory of the floatation behaviour of paraboloids. That’s the shape generated when a parabola is rotated about its axis. Think of a wine glass or a champagne glass for example. The cups of those glasses are kind of paraboloids. Suppose you put a shape like that in water. Sometimes it will float upright, sometimes it will tip over, and so on. Under certain conditions the thing is in equilibrium, in other conditions not so it will wobble one way or the other.
Three parameters determine the state of the system. One is the inclination or tilt; so the angle the axis of the paraboloid makes with the water. Another is density. These are solid paraboloids. The material they are made of is either heavier or lighter than water in some certain ratio. Then there’s the steepness of the paraboloid. So whether it is more round like a wine glass or pointy like a champagne glass.
The equilibrium conditions of floating paraboloids depend on these parameters in highly nontrivial ways. It is a complicated and complex matter to know what’s going to happen if you vary any of these values. Perhaps you are familiar with dynamical systems catastrophe theory stuff from the 20th century? It’s like that. So you have a phase space and the stable states make some complicated surface and so on.
Here’s an example of an application. Consider an iceberg. It’s floating upright, let’s say. But it’s melting. It’s becoming thinner and thinner, it goes from a wine glass to a champagne glass kind of shape. There will be a critical point when upright floating is no longer an equilibrium state. The iceberg topples over. Even though the parameter changed only very slowly, the effect on the floating position was at first nothing for a long time and then a dramatic collapse all of a sudden as some critical threshold was met.
Archimedes does all of this. Well, he doesn’t mention applications like icebergs but he has all the mathematics of the thing. He gives detailed, exact, quantitative predictions of the floatation behaviour of paraboloids. When will it float, when will it tip over. Archimedes tells you all of that.
And he’s right. All of these very specific empirical predictions that he derives, they are spot on correct.
Now, poor Archimedes, he is often misunderstood. So many people who don’t care for mathematics, they hardly even know this work exists. But if they do look at it they say: What’s this? It’s just a bunch of technical geometry about parabolas and stuff. Archimedes says not a word about any experiment, not a word about any empirical data, nothing about testing his theory, nothing science-y like that at all. It’s just a bunch of intricate geometry. You might as well have opened any work on geometry, maybe Euclid or Apollonius or whoever. It looks just like that, like more geometry. It seems to have very little to do with the real world.
Archimedes has a few very basic postulates in the beginning: some common-sense assumptions about how fluids push on submerged objects. That’s the only link to the physical world. After stating these two or three assumptions at the beginning, it’s all-out geometry from there on.
So it has been easy for people to say: Well, let some specialised historian of mathematics read that technical gibberish, I’m a historian of science, I don’t care about that because it says nothing about empirical science. The Greeks may have been excellent geometers but they didn’t really do science, you see. They were speculative thinkers, philosophers. They were great with abstract stuff but they didn’t have the sense to ground their fanciful theories in reality.
That attitude is completely wrong, in my opinion. Ask yourself: What are the odds that Archimedes got his detailed, quantitative theory of floating bodies absolutely spot-on right if he was ignorant of empiricism and experiment and scientific method? Was he just sitting around doings speculative armchair geometry and, whoops, it just happened to come out exactly equal to empirical facts in a range of far from obvious ways? Are we supposed to believe that was just dumb luck? It doesn’t make any sense.
Of course Archimedes knew about the scientific method. Of course Archimedes tested his theory by experiment. That’s obvious. His text doesn’t say that because he was too good of a mathematician to think that kind of kid’s stuff counted for much of anything. He only published the actual theory, not the obvious tests that any fool with half a brain could do for themselves.
Galileo, though, was precisely that fool with that half brain. He spent his whole life spelling out those parts that Archimedes thought were too trivial to mention. People ignorant of Archimedes are readily tricked into thinking that this was somehow profound. But mathematicians know better.
This is why we need the mathematician’s point of view represented in historical scholarship. And that’s what you will get in this podcast. And that will be the basis for my revisionist interpretation of Galileo’s contribution to the history of science.
A new paper on “The concept of given in Greek mathematics” by Nathan Sidoli contains some remarks pertaining to the issue of “geometrical algebra”. Ostensibly, Sidoli challenges what I wrote on this subject. But in fact I believe our views are not far apart.
The geometrical algebra issue, at bottom, stems from this conundrum: How different are the Greeks from us? Can we use our “mathematician’s empathy” to make sense of their work, and view differences in style and expression as superficial? Or is this the cardinal error of historical scholarship? In order to understand past thinkers, does sound historiography require us to put our own conceptions aside and start from the assumption that the Greeks inhabit a conceptual universe completely different than our own? When the Greeks do things that feel like algebra to us, are we feeling the underlying gist of their thinking, or are we merely seeing a reflection of our own ways of thinking that were not in the sources until we projected them there?
In my paper I argued for a fundamental unity rather than a fundamental chasm between ancient and modern thought. Specifically I defended these two theses:
GA1. The Greeks possessed a mode of reasoning analogous to our algebra, in the sense of a standardised and abstract way of dealing with the kinds of relations we would express using high school algebra. By and large, whenever we find it natural to interpret Greek mathematics in algebraic terms, the Greeks were capable of a functionally equivalent line of reasoning. If an algebraic interpretation of a Greek mathematical work suggests to us certain connections, strategies of proof, etc., then the Greeks could reach the same insights in a similarly routine fashion. This was an abstract, quantitative-relational mode of thought that was not confined to concrete geometrical configurations and not dependent on geometrical visualisation or formulation; in particular, it was obvious to the Greeks that the exact same kind of reasoning could just as well be applied to numerical relations as geometrical ones.
GA2. The Greeks were well aware of methods for solving quadratic problems (such as those exhibited in the Babylonian tradition). Books II and VI of the Elements contain propositions intended as a formalisation of the theoretical foundations of such methods.
In perfect agreement with these theses, Sidoli observes that many mathematicians from late antiquity onwards indeed switched back and forth between numerical and geometric perspectives, treating them as trivially equivalent:
“At least by the classical Islamicate period, and probably from much earlier, this blending of the geometrical and arithmetical readings of geometrical books of Euclidean works was commonplace.” Heron and Ptolemy did things of this nature and “there is no indication that they thought of their use of metrical analysis as in any way new, or innovative.” (§5.3)
Nevertheless Sidoli suspects that this was a later appropriation of the Euclidean material and not Euclid’s original intention:
“On balance I think [the geometrical algebra reading of Euclid] is less likely than the claim the Data was originally composed to address the needs of geometrical problem-solving and was then later repurposed as a means to justify and generalize metrical arguments.” (§6)
Sidoli’s arguments for this are as follows:
- The propositions in Euclid that can be read in a GA way have a geometrical character that does not readily match up step-by-step with the arithmetical-algorithmic procedures the are, according to the GA interpretation, supposed to formalise. E.g.: “The only way this theorem could have any use to us in metrical, or arithmetic, problem-solving, is if we already knew, through independent considerations, what sorts of arithmetic operations to carry out, but were interested in an unrelated geometric approach as a purely theoretical justification of these operations.” (§3.4)
My reply: Indeed, which is why, in GA2, I spoke of “a formalisation of the theoretical foundations of such methods,” as opposed to a practical recipe version of them. Similarly, no one would use Elements Definition V.5 to compare magnitudes or Proposition I.2 to transfer lengths in everyday situations. Of course Euclid’s treatment is a highly refined formal system, far removed from giving practical recipes and very much concerned with investigating subtleties involved in founding all of mathematics on a minimalistic set of principles. It is indeed a matter of “a purely theoretical justification” just as Sidoli says. This is not in conflict with the GA theses.
- The Data is not a systematic exposition of GA-type material. “At the very least, we must accept that if, in fact, Euclid had devised [the Data] in order to justify arithmetic procedures, he did so in a rather disorganized way.” (§6)
My reply: The Data is not a systematic exposition of anything. It’s a weird hodgepodge of theorems by anyone’s reckoning. No one, as far as I am aware, has ever made sense of the Data as a cogent treatise leading in a coherent way to any kind of natural goals, or given a good account of why Euclid selected the particular propositions he did. Even Sidoli himself admits that “the Data is more of a compilation than the Elements and it is really only the first half to three quarters of the text that can be read as articulating a single program” (source, p. 404). So the fact that the Data is not a perfect fit for the GA hypothesis is neither here nor there.
- The correspondence between the arithmetical operations and their geometric equivalents is not systematically expounded by Euclid. Thus: “While Data 3 and 4 could be taken to justify adding and subtracting for general quantities, for multiplying and squaring, taking square roots and dividing, we must turn to Data 52, 55 and 57—but, as we saw, the proof of these theorems rely on the geometric construction of a square and a similar triangle.” (§6) “Proponents of [GA2] must contend with the fact that this highly geometrical, and frankly rather peculiar, proposition [Data 52] is the only candidate in the text for propositions demonstrating that the product of two given numbers is given.” (§3.4)
My reply: Euclid does after all establish the result, even if Sidoli thinks his manner is “peculiar,” so there is perhaps not much to discuss here. Especially not since the Data is an enigma on any reading whatsoever. But let us put those points aside for the sake of argument. Even so I do not think this proves much.
GA2 has to do with the fact that the very same extremely basic ruler-and-compass methods that suffice to build up all the geometry of the Elements are also sufficient to solve any quadratic equation and carry out various other manipulations of quadratic expressions. This is a sophisticated, nontrivial point regarding the scope of ruler-and-compass constructions. I do not see why this would need to be bundled, as Sidoli’s argument assumes it should be, with a pedantic foundational account of spelling out the elementary idea that products of numbers corresponds to areas of rectangles and so on. As Sidoli himself says: “there is no clear evidence that Greek mathematicians thought that the basic arithmetical operations needed to be justified, so there is no reason for us to believe that Euclid felt the need to engage in such a project” (§6). Indeed, so why could Euclid not simply have taken this for granted? I do not see how this speaks against the claim that Euclid was also interested in proving the nontrivial result that having a ruler and a compass means being able to solve any quadratic equation.
This does not contradict my point that Euclid’s concerns are foundational, because his theory remains internally foundational through and through. The intuitive association between arithmetical operations and geometrical constructs is not needed internally but only externally to identify the formal theory with other things. Such associations take place in a different arena of reasoning, not subject to the internal stringency of the theory. Just as in modern real analysis, say, mathematicians reason with the utmost care when developing the theory internally, but then switch to much more intuitive and informal notions when they apply it to differential equations arising from a physics problem. While the theory of real analysis is built up from set theory in a stringent, formal way, its association with its physical applications is taken for granted in an intuitive, common-sense way that is not incorporated in the formal theory itself. So also Euclid can sensibly do his formalised version of the theory of quadratic equations without having to specify internally within his system how this maps to “real world” ways of doing these things.
For all of these reasons I am not convinced that any conclusions regarding GA can be drawn from Euclid’s Data.
Beyond his main focus on the Data, Sidoli makes a few very brief general points on GA that are very interesting. Basically he argues “against” GA by saying that GA1 is so great that GA2 is not needed:
“GA1 is simply another way talking about the theory of the application of areas, which is not in any dispute.” (§3.4) “Almost no one would argue that it is not possible to make a reading of Elements [II] and VI as motivated by and justifying computational problem-solving. The question is rather whether such a reading [=GA2], or that through the theory of the application of areas [=GA1], is more broadly successful in explicating the ancient sources.” (§6)
This novel take is news to me. Traditionally, GA1 and GA2 have been considered naturally and closely associated. People have either accepted them both or rejected them both. Sidoli, instead, seems to construe them as mutually exclusive hypotheses in direct competition with one another.
I’m not sure what Sidoli means when he says “GA1 is not in any dispute.” Does he mean that he does not dispute it? Or that no one disputes it? The latter is certainly false. People absolutely do dispute GA1. They give arguments of the form “if the Greeks had truly been able to think algebraically [=GA1] then they would have done so-and-so differently than they did.” Such arguments are clearly a direct denial of GA1. The bulk of my GA paper is devoted to rebutting arguments (by Unguru, Mueller, Saito, Grattan-Guinness) of exactly this type.
So it is strange for me to see Sidoli claiming to disagree with me when he happily accepts GA1. In my eyes, if everyone accepts GA1 this is basically a win for GA, regardless of whether they also agree on GA2.
I came up with this taxonomy when trying to explain certain conflicts and research choices that shaped the early history of the calculus. Below are a number of episodes that fit these categories very well. (For more details on the earlier systems, see this useful essay.)
Maestro philosophy: “Few but ripe.” It is a mark of class to focus only on elegant, simple, important, emblematic masterpieces. Write enough to give a definitive, impeccable treatment of the subject, but not more. It will be evident that this is merely the tip of an iceberg, resting on a solid body of technical expertise. But the tedious explication of the latter—the scaffolding, the tricks of the trade—is left to lesser Technocrat mathematicians.
E.g. Archimedes:
17th century Maestro examples:
Huygens: tautochrone = cycloid = evolute of cycloid.
Leibniz:
Non-examples:
Johann Bernoulli:
Not Maestro because not classically motivated and self-contained; presupposes “nerd” Technocrat interest in evaluating everything that can be symbolically formulated. Same with Euler’s so-called beautiful (actually only Technocrat) formula
A typical Maestro versus Technocrat conflict/misunderstanding: Leibniz versus the English on power series in the 1670s.
Leibniz typical Maestro, cares about singular, beautiful results: “I possess certain analytical methods, extremely general and far-reaching,” but “exquisite” series “especially is most wonderful.”
English typical Technocrat, care about plug-and-chug-ready formulas, criticise Leibniz for merely giving special cases. Collins: “infinite Series to be generally fitted to any equation proposed, so that an Algebraist being furnished with his Stock, will quickly fitt a Series.” Newton: I gave “a general Method of doing in all Figures,” whereas “Leibnitz never produced any other Series than numerical Series deduced from them in particular Cases.”
But Leibniz has no interest in Technocrat that doesn’t lead to Maestro: “I too used this method [of series inversion] at one time, but after nothing elegant had resulted in the example which I had by chance taken up, I neglected it forthwith with my usual impatience.”
Later Newton turns from Technocrat to Maestro, because more classical and elegant (and perhaps associated with a certain snobbery and sense of superiority): “He thought Huygens’s stile and manner the most elegant of any mathematical writer of modern times, and the most just imitator of the antients. Of their taste, and form of demonstration, Sir Isaac always professed himself a great admirer: I have heard him even censure himself for not following them yet more closely than he did; and speak with regret of his mistake at the beginning of his mathematical studies, in applying himself to the work of Des Cartes and other algebraic writers.”
Euler disapproves, goes back to Technocrat, values toolbox adaptability more than beauty: “I always have the same trouble, when I might chance to glance through Newton’s Principia: Whenever the solutions of problems seem to be sufficiently well understood by me, yet by making only a small change, I might not be able to solve the new problem using this method.”
Leibniz is by nature a Visionary. The Maestro tendencies in the series episode are coloured by the influence of Huygens, who, in typical Maestro manner, praised the series as “a discovery always to be remembered among mathematicians.”
Later Leibniz resisted Maestro and saw it as a distraction from his main task of Visionary. This is why, for example, he fights not to get drawn into the brachistochrone problem (a true Maestro problem): “The problem draws me reluctantly and resistingly to it by its beauty, like the apple did Eve. For it is a grave and harmful temptation to me.”
Visionary need Technocrat to spell out the details of their systems. E.g. Descartes: Van Schooten; Leibniz: l’Hôpital, Johann Bernoulli.
Leibniz: “I wish there were young people who would apply themselves to these calculations. With me it’s like the tiger who lets run whatever he does not catch in one or two or three attempts.”
Leibniz is no more than 5% Technocrat: “Had I 20 heads, or better yet 20 good friends, I would put one of them toward working out the theory of conics.”
Huygens: I will learn calculus but only for Maestro, not for Technocrat. “I still do not understand anything about ddx, and I would like to know if you have encountered any important problems where they should be used, so that this gives me desire to study them.” “[Natural] curves merit, in my opinion, that one selects them for study, but not those [curves] newly made up solely for using the geometrical calculus upon them.”
Leibniz: Agree, Technocrat calculus worth little. “You are right, Sir, to not approve if one amuses oneself researching curves invented for pleasure.” But the difference between Visionary and Maestro is that Visionary is more focussed on general methodological insights, which is why Leibniz adds: “I would however add a restriction: Except if it can serve to perfect the art of discovery.”
L’Hôpital’s Rule: typical Technocrat result of the sort condemned here.
Systematic theory of integration by partial fractions: a Technocrat topic needed for Visionary, namely “a question of the greatest importance: whether all rational quadratures can be reduced to the quadrature of the hyperbola and the circle” (Leibniz). This forces Leibniz, reluctantly and contrary to his nature, to do some Technocrat work, with poor results (Leibniz erroneously believes that “ can be reduced to neither the circle nor the hyperbola by [partial fractions], but establishes a new kind of its own”). A typical Visionary, Leibniz clearly has very little interest in actually evaluating integrals, and only cares about giving a big-picture methodological-foundational account of integration in general.
Myth: Early Leibnizian calculus driven by applications; lacks attention to rigour. Hence typically Pragmatist.
Reality: The exact opposite: Early Leibnizian calculus primarily Visionary. It was actually indifferent to applications and was consumed by rigour, only rigour meant something completely different at that time. This is one of the main points of my book.
Christiaan Huygens was the greatest mathematician in the generation before Newton and Leibniz. He spent his prime years in Paris, where he was a key member of the Academy of Sciences. Here he is with his buddies. The young Leibniz wanted nothing more than to join these esteemed gentlemen. He admired the great Huygens and followed his example in wigs and mathematics alike.
But times change. French politics soured and foreigners were chased out of the country. Leibniz had to return to Germany. Huygens withdrew to his family mansion in the Netherlands. The Academy descended into reactionary mediocrity.
But Huygens did not retire to feed the ducks in his estate gardens. Though old and frail at this point, he kept up with the latest mathematics. This meant learning the new calculus developed by his former protégé Leibniz. The student had become the master, as the saying goes. But perhaps more interestingly, the master had become the student.
What a treat of history this is. Reading the correspondence between Huygens and Leibniz during these years, we get to see learning in action. We get to see how the calculus is taught by its inventor, and how a sage mathematician of the highest credentials goes about learning it. We get to see the former director of scientific research at the Academy of Sciences take a seat in the front row of Calculus I, pencils sharpened and notebook in hand. It’s a naked view of calculus genesis, unique in history.
Huygens proved a feisty pupil. He was not the kind of student who copies down the formulas and asks questions about the homework problems he got stuck on. Proofs do not impress him much either. What he demands most of all is motivation. He wants new mathematics to be thoroughly justified, not in the narrow sense of being logically correct, but in the broader sense of being a worthy human enterprise.
Thus, after having mastered derivatives, he wonders if second derivatives are just a formalistic indulgence or if they’re actually good for something. He writes to Leibniz:
“I still do not understand anything about ddx, and I would like to know if you have encountered any important problems where they should be used, so that this gives me desire to study them.”
Tell me why I would want to study second derivatives, Huygens demands. Not the formal rules for working with them, and the proofs thereof, and artificial problems specifically invented for them. No, not that. Any mathematician can make up such mathematics ad infinitum. A new mathematical theory must prove itself not by solving its own internal problems, but by proving itself on a worthy, honest-to-god problem recognised in advance.
Leibniz understands well, and replies:
“As for the ddx, I have often needed them; they are to the dx, as the conatus to heaviness or the centrifugal solicitations are to the speed. Bernoulli employed them for the curves of sails. And I have used them for the movement of the stars.”
We don’t care about second derivatives because the symbolism suggested we could do derivatives once over. We care about them because they are the right way to tackle mathematically a rich range of fascinating and important phenomena. Do you want to understand the shape of a sail bowed by the wind? Do you want to describe how planets move around the sun? Then you want to understand second derivatives.
This is not about pure versus applied mathematics. For example, in his big book on the pendulum clock, Huygens took inspiration from this concrete situation to develop a thoroughly mathematical theory treating evolutes and involutes abstractly and exhaustively. He gives a general proof, for example, that the evolute of any algebraic curve is algebraic. This is the kind of theorem that would make even the most doggedly purist mathematician proud.
The point is not that mathematics needs to be applied. It is that it needs to be motivated. We don’t study nature because we refuse to admit value in abstract mathematics. We study nature because she has repeatedly proven herself to have excellent mathematical taste, which is more than can be said for the run-of-the-mill mathematicians who have to invent technical pseudo-problems because they can’t solve any real ones. Says Huygens:
“I have often considered that the curves which nature frequently presents to our view, and which she herself describes, so to speak, all possess very remarkable properties. Such as the circle which one encounters everywhere. The parabola, which is described by streams of water. The ellipse and the hyperbola, which the shadow of the tip of a gnomon traverses and which one also encounters elsewhere. The cycloid which a nail on the circumference of a wheel describes. And finally our catenary, which one has noticed for so many centuries without examining it. Such curves merit, in my opinion, that one selects them for study, but not those [curves] newly made up solely for using the geometrical calculus upon them.”
Leibniz agrees: “You are right, Sir, to not approve if one amuses oneself researching curves invented for pleasure.”
If only modern calculus books lived by the same rule! Flip to the problem section at the end of any chapter in any standard calculus textbook and you will find a thousand problems “made up solely for using the calculus upon them”—exactly what Huygens condemns. Perhaps it should give us pause for thought when both the inventor of the calculus and its most able student ever are in complete agreement that our way of writing textbooks is stupid.
Modern students may well sympathise with Huygens again when he makes a similar point regarding exponential expressions:
“I must confess that the nature of that sort of supertranscendental lines, in which the unknowns enter the exponent, seems to me so obscure that I would not think about introducing them into geometry unless you could indicate some notable usefulness of them.”
Leibniz shows him how such expressions can solve certain problems, but Huygens is still not impressed: “I do not see that this expression is a great help for that. I knew the curve already for a long time.” Again: first show me what your technical thing can do, or else I have no reason to study it. And if I can do the same thing by other means then you have still failed.
I for one wish we had more little Huygenses in our calculus classrooms today. And I am worried we alienate the few we have by requiring them to swallow a style of exposition that a master like Huygens unequivocally rejected as bad mathematics unworthy of his time.
A widely publicised new study claims to find gender bias against women in student evaluations of teaching. There are massive methodological problems with it.
The study concerns the evaluations of precisely two teachers: one male, one female. Obviously no sane human being would draw general conclusions about gender from two individuals. But that is apparently what passes for peer-reviewed research in this field.
In the reported data, the female instructor had 167 students and the male instructor 51, in one semester. Could it be that having a more than three times higher teaching load has a negative impact on teaching quality? And that this rather than gender was the key difference between these two instructors? This possibility is not considered by the authors.
Actually the researchers threw away at least half of the actual data. We have no idea what it said. Here is their justification for this:
Students had a tendency to enroll in the sections with the lowest number initially (merely because those sections appeared first in the registration list). This means that section 1 tended to fill up earlier than section 3 or 4. It may also be likely that students who enroll in courses early are systematically different than those who enroll later in the registration period; for example, they may be seniors, athletes, or simply motivated students. For this reason, we examined sections in the mid- to high- numerical order: sections 6, 7, 8, 9, and 10. (Supplementary materials, page 4)
This is crazy. The authors are openly admitting that they purposefully selected a non-representative sample, which by their own admission is likely to exclude certain types of students. Why on earth would you do this? Why not sample for instance all odd-numbered sections and hence get a sample that includes early- and late-registering students in representative proportions?
I can think of one reason. The authors of course knew that if they found no gender bias their study would go unpublished and would have been a waste of time, whereas if they found gender bias they could get it published in a Cambridge University Press journal and featured on Slate. So they had every incentive in the world to ensure that the data came out the way they wanted. And if you are allowed to study only two instructors, and arbitrarily discard half the data on nonsensical grounds, then it is not difficult to prove anything you want.
Also extremely problematic is that the teachers in question were the researchers themselves. This is obviously a terrible idea methodologically speaking. It is not far-fetched to think that their obvious incentive, as researchers, to find gender bias influenced their behaviour as instructors. The standard practice of keeping studies double-blind exists precisely to prevent the risk for such contamination. This study is about as far from double-blind as you can get.
For instance, the authors make a big fuss about how the evaluations more often referred to the male instructor as “professor” and the female instructor as “teacher” despite equal credentials. This is obviously the kind of thing that can be easily manipulated by the instructor. Throughout the semester they would have had every opportunity to plant the language they want the students to use.
If the authors wanted to ensure the desired outcome of the study, they could, for example, distribute the evaluations with different prompts. The male instructor can tell the students: “These evaluations are important. The university uses them to decide which professors get their positions renewed.” Now you have planted the terminology of “professor”, and also made students apprehensive about being critical since you have made them think about the possibility of you being fired. The female instructor, meanwhile, might say while handing out the evaluations: “Honest feedback is important to me as a teacher.” Now you have primed the students to refer to you as a “teacher”, and encouraged them to speak freely without holding back, since you have implied that the evaluations are for your own use and that you value honest feedback. The danger of such contamination of the data is vastly greater when the instructors in questions have a blatant vested interest in ensuring a particular outcome of the study, as in the case of this study.
[Zigerell corrected a mistake in an earlier version of the above: I reported the number of students as 1,169 and 357 respectively for the two instructors, but these were the number of data points, and there were 7 data points per students.]
The National Academies of Sciences, Engineering, and Medicine—a major lobby organisation in Washington—recently released a big report on STEM education. It seems to be basically a rehash of the usual opinions of education researchers. The tone of it is representative of the field, and in my opinion problematic. It looks at education in very black and white terms: there is traditional teaching, which is bad and only espoused by irrational reactionaries, and then there the progressive teaching espoused by edu-people, which mountains of research has proven to be The One And Only Right Way To Teach. Consequently, the way to improve education is to force everyone to adhere to the Right Way and steamroller anyone foolish enough to oppose it.
In my opinion, the field would be better off if it was open for debate and diverging viewpoints, instead of simplistically insisting that there is only one Right Way that no rational person could possibly disagree with. But the world of educational research and policy prefers the latter framing.
Edu-people are an ideologically homogenous group. They have very definite opinions on what is right and wrong in education, and virtually no one in the field ever disagrees on these core beliefs. 99 times out of a hundred their research confirms these opinions that they already held. This is either because they are brilliant and objectively right, or because they are biased and shield themselves from alternative viewpoints and critical thought because of the echo-chamber uniformity in the field. To tell which, we should look at the quality of their research, as I have done in many cases. From such investigations I have concluded that, in my view, we would do well to regard educational research with suspicion to say the least. But anyone with such opinions cannot get into the world of edu-people so the consensus stands.
A number of STEM faculty share my sceptical view of educational research. The way edu-people deal with such opposition rubs me the wrong way and only gives me all the more reason to be apprehensive about their claims.
For one thing, edu-people constantly refer to their own opinions as fact. They don’t say “we believe” but rather “research shows”. One phrase they have devised to this end is “evidence-based”. This phrase is repeated manically hundreds of times in the report. Instead of saying “that person has a different view of teaching than me”, edu-people say “that person’s views are not evidence-based”. It is hardly the hallmark of objectivity and open-mindedness to systematically use such a blatantly value-laden yet ostensibly factual term to refer to one’s own opinions.
It makes one wonder whether a statement such as the following is not basically tautological: “In the committee’s view, improving the quality of undergraduate STEM education will require wider use of evidence-based STEM educational practices and programs.” (1-7) Since “evidence-based practices” is effectively code for “our opinions”, the statement basically reads: in the committee’s view, more people need to agree with the committee.
Or put it this way: among the many critics of the edu-people consensus, has anyone ever said: “I don’t agree with you because I don’t think evidence should be taken into account when making instructional and policy decisions.” Of course not, that would be ridiculous. Yet edu-people insist on such a framing, hammered home with hundreds of repetitions of such phraseology. We may want to ask ourselves why edu-people are so attached to this rhetoric, by which they imply that anyone opposed to them must be ignorant of evidence.
Let’s keep this meaning of “evidence-based” in mind when we read the following quote from the report:
“A growing body of research indicates that many dimensions of current departmental and institutional cultures in higher education pose barriers to educators’ adoption of evidence-based educational practices.” (3-12)
Translation: Our opinions have still not achieved complete hegemony. To fix this we have spent a lot of time analysing why.
“A well-established norm in some STEM departments [is that of] allowing each individual instructor full control over his or her course. … One recent analysis found that the University of Colorado-Boulder science education initiative made little progress in shifting ownership of course content from individual instructors to the departmental level because of this dimension of departmental culture.” (3-12) Thankfully, Michigan State University offers an encouraging model where “leadership … from the provost” led to “increased coordination of instructional and assessment practices.” (3-13)
Translation: There is too much democracy and decentralised power in academia. Our sect is in control of administrative positions, but our power is not yet great enough to force everyone to agree with us, though soon we hope to achieve this goal.
Professors being in control of course content is apparently an evil that cannot be eradicated soon enough. Apparently it is better if a provost—a career bureaucrat—is in charge and bosses around the professors who are the actual experts in the field. That’s apparently what “evidence-based” “best practices” demand. I do not see the rationale for this insistence on uniformity, other than the one I have inferred in my translation. Or do these people also wish to replace all local small businesses with a McDonald’s and a Walmart, by the same logic?
The curious emphasis on instructional homogeneity is perhaps all the more jarring when juxtaposed with another perennial edu-slogan: “instructor diversity provides educational benefits to all students” (4-10); indeed, “the benefits of instructor diversity are clearly demonstrated by available research” (4-11). It serves edu-people well that the term “diversity” has been emptied of any meaning but the modern politicised one, for else this would square poorly with their explicitly announced intent to eradicate diversity of pedagogical approaches among instructors.