Opinionated History of Mathematics: Recent Episodes

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Archimedes’s emblematic death makes sense psychologically and embodies a rich historical picture in a single scene.

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Archimedes died mouthing back at an enemy soldier: “Don’t disturb my circles.”

Or that’s how the story goes. Is this fact or fiction? We have third-hand accounts at best so there is plenty of room for doubt. But I’m putting my money on fact nonetheless. I think this standard story makes sense. I think it works psychologically with what little we know about Archimedes as a person, and I think it fits contextually with what we know about Archimedes’s era and circumstances. So let’s investigate this, and let’s use the death of Archimedes to reflect on these broader themes.

Archimedes was killed when the Romans invaded his city, Syracuse. There is little doubt about that. The precise details are less clear. There are various versions of the story from several ancient authors. These passages are all conveniently collected at the Archimedes website by Chris Rorres, which I highly recommend.

Let’s quote the standard version from Plutarch: “Archimedes was working out some problem by a diagram, and having fixed his mind and his eyes alike upon the subject of his speculation, he never noticed the incursion of the Romans, nor that the city was taken. In this transport of study and contemplation, a soldier, unexpectedly coming up to him, commanded him to follow him. Archimedes declined to do so before he had worked out his problem to a demonstration. The soldier, enraged, drew his sword and ran him through.”

It is quite popular to cast doubt on the story of Archimedes’s death. One example is the recent biography “Archimedes: Fulcrum of Science” by Nicholas Nicastro (pages 43-44). This biography argues that the standard story “doesn’t pass the smell test” to use Nicastro’s words. Because “any properly self-interested soldier would know the reward for capturing Archimedes.” Indeed, Archimedes was famous and the Roman commander wanted him captured alive, it is said.

So the idea that “the soldier recognizes Archimedes but simply liquidates a valuable prisoner – indeed one who amounted to a strategic asset for Rome – simply because he was lackadaisical in responding to orders doesn’t pass the smell test,” according to Nicastro’s biography.

I’m not so sure about that. We know about police brutality. We know for example that George Floyd was killed by police while being apprehended, after being suspected of using a counterfeit twenty-dollar bill. And that was on an ordinary Monday in a peaceful, prosperous country.

The soldier who killed Archimedes was not having a normal Monday dealing with petty delinquents. This soldier was in enemy territory in an active war zone. You would think that this soldier would have been on high alert against ambushes and sudden movements, quite rightly.

And let’s consider what the soldier’s opinion of Archimedes would have been. Archimedes was well known and famously led the military engineering efforts that fended off the Romans for years. What would the soldier think of the figurehead of the enemy? Would he find that such a great geometer must be spared for the greater good? Or would he think that Archimedes was a terrorist responsible for the deaths of his friends?

This soldier may very well have seen first hand the death and suffering inflicted by Archimedes’s famous warfare machines. Maybe for example a friend of his drowned when Archimedes sunk a Roman ship during one of the previous invasion attempts. Or maybe his brother had his legs crushed by one of Archimedes’s catapults, and returned home as a cripple, which made such an impression on the younger brother that an unstoppable hatred festered in him and he swore to dedicate his life to revenge against this evil Greek insurgent.

Indeed, maybe on this very day, the day that he came to stand before Archimedes, this soldier has already had to watch helplessly as a close friend and brother in arms died a gruesome death. Such things can happen in war.

So I don’t think we can say: the soldier wouldn’t have killed Archimedes because he had orders not to and, rationally speaking, it would have been in his best interest to obey. This soldier may very well have been under immense and acute psychological pressure and trauma at this moment, when he happened to come face to face with the very symbol of everything he had been taught to hate. That’s what I think about this so-called “smell test.”

But that’s the soldier’s psychology. Now let’s consider it from Archimedes’s point of view. Would Archimedes be calm and collected and compliant when the soldier comes to arrest him? No, he would not.

The invasion is even more traumatic for Archimedes. Archimedes was born in Syracuse and spent his life there. There is every reason to think that these roots meant a lot to Archimedes. Archimedes was famous already in his lifetime. No doubt he had generous offers to go elsewhere, just like superstar academics today. But Archimedes stayed.

And he wrote his treatises in the local dialect of Greek, rather than adapting to the more prestigious version of Greek spoken in Athens and Alexandria. Perhaps again a sign of local pride.

Archimedes also mentions his father, who was apparently an astronomer. So that’s another sign that Archimedes attached some importance to his heritage.

And of course Archimedes was heavily involved in the defense of the city as a military engineer for many years. Obviously another sign of considerable patriotism.

And now, all of that is being destroyed. Archimedes’s birthplace, his home for his entire life, burnt and ransacked by a heartless military force.

If Archimedes looks out his window all he sees is everyone he ever loved being slaughtered, and generations of cultural heritage being sadistically trampled to dust by soldiers’ boots.

This would be heartbreak and trauma enough. But it’s worse. It’s worse for Archimedes because he was in charge of the defense. It’s his fault. All this blood is on his hands. Or so it would seem to him.

Archimedes was given every resource to orchestrate the Syracusan defense. All those notorious warfare machines that held the Romans at bay for so long: that’s not something you throw together in your basement. Archimedes must have been entrusted with massive resources and he must have had considerable manpower under his command.

His friends and brothers had put their faith in him in their hour of need, and he failed. Archimedes has let them all down. He has let his father down, and his forefathers.

Not only is Archimedes watching his city burn. He is also overcome by the crushing guilt that this is all because of his personal failure.

How do you think this guy is going to react when an enemy soldier comes to take him away? He’s not in a mood to be read his Miranda rights, is he?

It was time for Archimedes to go. Shot down on the pavement. It was the only honorable option left.

Most of the historical accounts frame the death of Archimedes in terms of the trope of the absent-minded professor, lost in a diagram, oblivious to the world around him. I imagine that this is a sanitized account. Most of the historical accounts were written under Roman rule.

Maybe the real events were quite a bit uglier and a lot less flattering for Roman historians to repeat. Maybe Archimedes was not so cartoonishly lost in geometrical thought at that moment as story-tellers pretend. Maybe he knew full well what was going on, like any normal person would. Especially since he was obviously very well aware of the prospect of Roman military invasion, and he would understand very well what it meant when Roman soldiers had reached his house.

Archimedes was an experienced military engineer who had lived under the immediate threat of military attack for years. Is it too much to imagine that such a person would carry a weapon, perhaps a small dagger?

Well, now is the time to use it. If not now, when?

Of course by the time it comes to that you have already lost. You don’t have a dagger because you think you will be able to fight your way out. You carry the dagger because when the time comes to use it your choices are: die on your knees or take one ------ down with you.

That’s how I would write Archimedes: The Gritty Reboot.

If that’s what happened then Roman historians would hardly want to admit it. It doesn’t do their self-image any favors that the great Archimedes would rather die than be taken alive by Romans. So the literary cliché of a philosopher so absorbed in thought that he does not notice his surroundings is a welcome euphemism readily at hand.

I quoted earlier the standard story from Plutarch, which leans into this cliché very heavily. Actually Plutarch also goes on to give two other versions of the death of Archimedes. “Others write”, he says. And then he says for instance that Archimedes was killed because a soldier mistook his astronomical instruments for gold trinkets and killed him to plunder his valuables. I don’t think so, but even this version clearly has some elements of truth. Namely, there was indeed plundering by the soldiers and some flashy-looking astronomical instruments made by Archimedes were indeed stolen by the Romans and publicly displayed in Rome. So this would have given some credence to the story.

Maybe Plutarch is relieved that there is some ambiguity regarding the death of Archimedes. Maybe he knew full well that these alternative stories are not true. Indeed, he first tells the standard story as if it was unequivocal fact, and then he adds the qualifier “others write” when telling the other versions. As if he knew they were false.

Since the real version is so embarrassing for the Romans, muddying the waters with some misleading alternatives is a convenient way to trick the reader into thinking that no one really knows for sure what happened. Throw a little fake news in there to dilute the facts.

It is very much possible that the circumstances of Archimedes’s death were very well known and documented at the time and for generations afterwards. It is reported that a personal friend of Archimedes wrote a biography. Which is now lost but which could have been a very excellent and reliable source available for some time.

Another credible source is the Greek historian Polybius. He writes about Archimedes’s military machines but he doesn’t mention the death of Archimedes in surviving texts. Some parts of his works have been lost, perhaps very conveniently for the Romans. Polybius was writing not too long after the fact. He could have spoken to eyewitnesses who were actually there on the ground during the siege of Syracuse.

So, we should not say: It’s all just a bunch of legends made up hundreds of years later. There were better sources. There were serious historians who tried to keep a record of these things. We should not be so pessimistic.

The existence of these better sources may have acted as a deterrent on historians like Plutarch and Livy, whose works are all we have now. They mention the death of Archimedes in passing. Their main concern is not to preserve a maximally accurate record of exactly what happened to Archimedes. They retell the story because it suits their purposes. Because it is vivid and gripping story as well as an occasion to make a moral point. And they are probably not opposed to tweaking the story to those ends.

Nevertheless they would not want to be caught saying something that is provably false. The possibility that some readers may have access to quite reliable historical accounts in other sources could very well be a check on the freedom that these other writers could afford to allow themselves.

And presumably they had some professional integrity as well. Sometimes I wonder about the modern historians who are so quick to dismiss ancient writers as if they just wrote fiction and legends and made up whatever they thought sounded cool. I wonder what it says about our modern colleagues that they find this kind of behavior from a history writer to be plausible and in character.

Of course third-hand and forth-hand accounts are distorted. Of course we should be mindful of what layers of biases and hidden agendas that these accounts have been subjected to. But that’s very different from making stuff up out of thin air.

The scholarly norms back then were different as well. Historians were expected to be storytellers with flair. Not be academic bores like today. So history writers back then would allow themselves more literary leeway that allowed them to add some stylistic embellishments. But the game was to do that will remaining faithful to the basic facts.

For example, according to one modern analysis (Archimedes and the Roman Imagination, 92), one version of the story of the death of Archimedes elegantly frames it in terms of concentric circles. First the walls of the city are breached, then the soldier breaks into Archimedes’s house, and finally, as a last layer of concentric fortifications, Archimedes wraps his arms around his precious diagram. So the theme of geometricity is echoed in the narrative structure itself. Exquisite. Things like that are fun to play with as a writer. And that’s the kind of embellishment that you can add without doing any harm to the essence of the historical facts.

Here’s another in my opinion misguided objection to the historical reliability of the story. I quote from the MAA Press book “Archimedes: What Did He Do Besides Cry Eureka?” Regarding Archimedes’s last words, this book writes:

“Who would have reported them? Would a soldier who had killed Archimedes, against orders from his commanding general, offer this incriminating evidence?” (3)

Well, yes. Yes, he would. The Roman army routinely tortured enemies for information. The solider would surely know that very well, perhaps first hand. This is the same Roman army who have given us the word “decimate”: that is to say, in case of disobedience, kill one in every ten of your own soldiers just to make a point and maintain discipline.

Obviously the soldier is at the mercy of the army. They know where he lives. They know where his family lives. Of course they can easily apply any amount of pressure. Of course the soldier will talk. How could he not?

Besides, the soldier would not have been alone, would he? I would think that, when clearing enemy territory in an active battlefield, soldiers would presumably prefer to stick together in groups rather than wander off on their own. So why wouldn’t the other soldiers report what happened? Of course they would.

That doesn’t mean that the reported last words of Archimedes are historically accurate, of course. I don’t think they are. And indeed the sources do not agree about it verbatim anyway. But the problem is not that it would not have been knowable or that actual facts were not available to historical writers. Things like last words were precisely the kind of thing that an ancient history writer will embellish a little bit for style and flair and drama and narrative when writing their own as it were reboot of this established story. But they were not fiction writers and their literary freedom was checked by professional integrity.

Nor was there necessarily any language barrier preventing Archimedes and the soldier from understanding one another, if we assume that the soldier was Latin-speaking. Before it came to war these regions had been close partners in diplomacy and trade. People may have known quite a bit of each other’s languages. Archimedes, who was highly educated and part of the king’s entourage, may very well have been able to express himself in Latin. (Ivo Schneider, Archimedes, 2nd ed., xvi)

In any case, the orders were supposedly to capture Archimedes alive. That’s what the Romans wanted. But what did Archimedes want? Did Archimedes want to be paraded around Rome like a trophy of war? So that tipsy dinner party guests could make fun of the freak with the big brain? Or did Archimedes want to sell his engineering skills and warfare know-how to his mortal enemies while the bodies of his childhood friends and neighbors were still warm?

I don’t think so. I think Archimedes would rather spit this soldier in the face and die a martyr’s death.

Not unlike Socrates two hundred years earlier. As Archimedes would have been well aware, Socrates basically chose death. Socrates was sentenced for corrupting the minds of young people with dangerous ideas.

But Socrates’s death sentence could easily have been avoided, it seems. First the trial itself, a democratic jury trial, had the possibility of bargaining built in. Socrates could have proposed a realistic alternative to the death penalty as a compromise, which could very well have worked. But he refused to do so out as a matter of principle.

Then even after being sentenced Socrates still had the chance to escape. He had powerful friends, rich friends. Some of them could have pulled some strings and made some bribes and probably Socrates could have been able to escape. Then he would have had to leave Athens and start over a couple of islands down, but alive.

Socrates wanted to make a point instead, and it worked. Maybe if he had let the Athenians boss him around then that would only have emboldened this mob to go after the next guy in the same way. Instead, Socrates died, to the shame of Athens. Right after this Plato and Aristotle thrived in Athens for many decades. Perhaps not a little thanks to Socrates’s sacrifice and moral victory in death.

In a better world, Archimedes’s death could have had much the same effect. After Socrates’s death, the Athenians had enough moral backbone to realize that they had screwed up and they got their act together. No such luck for Archimedes. The Romans were unfortunately beyond redemption. Even Archimedes’s martyr death was not enough to stem the greed and cruelty of these militaristic imperialists, unfortunately.

We know that now, after the fact. But Archimedes could not have known that. Archimedes could very reasonably have felt that dying like a man of principle and honor was the only remaining gift he could give to his countrymen.

Let’s look at the Romans now. There’s the invading general, Marcellus. The sources would have us believe that he was ever so noble, and he was ever so concerned that Archimedes not be harmed, and after this unfortunate death Marcellus paid his respects to Archimedes’s surviving relatives etc., etc., blah, blah, blah. You can decide for yourself how much of this transparent propaganda you want to believe. To me it sounds more like a slimy politician’s talking points at a staged photo op.

Of course even propaganda carries some information. In this case we see what the Romans actually cared about. They were very preoccupied with honor. They go to great lengths to explain how Marcellus’s actions were so honorable and noble. Remember, this was still good Rome. Republican Rome, democratic Rome. They still have some integrity in their own way. Before long it was to get a lot worse, and wannabe-emperors didn’t even need to pretend to be honorable anymore.

Anyway, the Romans cared about honor but not about science. Archimedes died working out some theorem. What theorem? Nobody could care less among the Roman writers.

Marcellus stole some of Archimedes’s instruments. Planetaria and spheres. Model representations of the universe. Archimedes’s planetarium perhaps used intricate combinations of cogwheels to represent the motions of the planets mechanically. These instruments don’t exist anymore, of course. No scientific interments from that era have survived the centuries. But the written sources speak in some detail about how Marcellus brought back these Archimedean devices.

Of course the Romans didn’t know what to do with these scientific instruments, since they didn’t have any scientific tradition. No academy or museum or library that could do anything with these Archimedean masterpieces.

So Marcellus just kept one of them at home, in his living room, like a hunting trophy stuffed animal head.

Another one of these models was put in the Temple of Virtues in Rome. They just stuck this valuable scientific device in a Sunday church so the plebs could gawk at it. Because no one in Rome had the competence to do anything better with it. That is what happened with important scientific artifacts in this barbaric culture.

Then there is Cicero, another one of these pseudo-intellectuals. Cicero was a career politician and his first appointment was in provincial Sicily, Archimedes’s home. That was 137 years after Archimedes was killed and this formerly Greek-speaking territory was absorbed by the Romans.

Cicero bragged that “I managed to track down [Archimedes’s] grave. The Syracusians knew nothing about it,” in Cicero’s words. Yes, they were all savages, you see, and Cicero, the white savior, is here to singlehandedly rescue mankind’s cultural heritage. According to himself.

Cicero claims that the tomb was “completely surrounded and hidden by bushes of brambles and thorns” and when he discovered it “I immediately said to the Syracusans, some of whose leading citizens were with me at the time, that I believed this was the very object I had been looking for. Men were sent in with sickles to clear the site, and when a path to the monument had been opened we walked right up to it.”

Right, so as you can see Cicero could hardly contain himself when he found the grave. He was so excited that he immediately went and sat down in a shaded area and had some chilled wine with his very important friends. Clearing a path to the grave was a top priority, so they only kept, like, two or three slaves at most to fan them with palm leaves while they waited for the other slaves to cut the path. My God.

You know Archimedes’s saying: Give me place to stand and I shall move the earth. If it was Cicero it would go: Give me a place to sit and I shall order some slaves to move the earth. Oh, and did I mention that my friends happen to be very important dignitaries by the way?

But Cicero is not done yet. Apparently he thought his bragging up to this point was too subtle for you dimwits, because now he’s going to spell it out for you:

“So one of the most famous cities in the Greek world, and in former days a great centre of learning, would have remained in total ignorance of the tomb of the most brilliant citizen it had ever produced, had [I] not come and pointed it out!”

That’s the great Cicero for you, supposedly a master of rhetoric and style. He apparently found that the very obvious moral of the story would have been too obscure without him simply stating directly that he was saving people from “total ignorance.” Apparently that’s what passes for a great orator in this lousy age. Not only telling a story blatantly designed for self-aggrandizement, but then, as if that was not enough, he just turns and looks directly into the camera and just flat out brags explicitly. What an absolute windbag.

I am in good company with these condemnations. Let me quote the great Heiberg (Mathematics and Physical Science in Classical Antiquity). Johan Ludvig Heiberg, the great classical scholar who published the definitive editions of the works of Archimedes and Euclid and so on, more than a hundred years ago. A legend in the field.

Like me, Heiberg laments the harm done to science by what he calls “the cold breath of Rome” (73). And he has some choice words for Cicero in particular. Here’s what he writes:

“The Romans, with their narrow, rustic horizon, had always in their heart of hearts that mixture of suspicion and contempt for pure science which is still the mark of the half-educated --- and sometimes bragged of it. Cicero, the arch-dilettante, boasts that his countrymen, God be thanked, are not as these Greeks are, but restrict the study of mathematics to what is useful and practically applicable.” (80)

By contrast, Heiberg is full of enthusiasm for “the Ionian school in the full blaze of its glory,” as he writes, when scholars “breathe a spirit of exact, critical, keen observation” and “attack charlatans and speculative theorists with a vigorous and often fiercely sarcastic polemic.” (24)

“One longs to recover something of that robust Ionian criticism” (37), says Heiberg. Yes! Let’s bring it back indeed. Sarcastic polemic and all. Then Archimedes will not have died in vain.

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There is nothing counterintuitive about an infinite shape with finite volume, contrary to the common propaganda version of the calculus trope known as Torricelli’s trumpet. Nor was this result seen as counterintuitive at the time of its discovery in the 17th century, contrary to many commonplace historical narratives.

Transcript

Torricelli’s trumpet is not counterintuitive. Your calculus textbook lied to you.

You’ve probably heard of this cliché, Torricelli’s trumpet: an allegedly “paradoxical” shape that has infinite area but finite volume. It’s staple example in calculus textbooks. Well, there’s nothing to it, in my opinion. It’s a propaganda lie. Let’s revisionist-history the heck out this thing.

I will tell you why there’s nothing counterintuitive about this result. Then I’ll argue that it was not seen as counterintuitive at the time, in the 17th century, contrary to what everyone tells you. Then I will explain, in terms of the sociology of the mathematical community, why this myth is still so popular. That is to say, why it is such a comforting myth to so many people, despite being wrong.

So, the trumpet. You take the hyperbola y=1/x and you rotate it about the x-axis. It makes a trumpet shape, a kind of funnel that becomes infinitely narrow the further you go.

Also known as “Gabriel’s Horn.” Actually I tried to look up the origin of this silly name but I couldn’t find it. I guess it was perhaps coined for the American market? “Torricelli” is bit too “Euro” isn’t it? Now, “Gabriel’s Horn” on the other hand, there you have a nice pious biblical name.

Anyway, whatever you want to call it. The volume of the funnel is finite. For example from x=1 onwards. Despite its area and its length being infinite.

There are two supposed contradictions here. On the one hand, finite volume with infinite extent (infinite length) could be regarded as two contradictory properties that it would be surprising to find in the same solid, allegedly. Alternatively, finite volume with infinite area could also be seen as a clash of two incompatible properties: a shape having those two properties at the same time is supposed to be contrary to “intuition,” allegedly. Sometimes it is put like this: Such a solid cannot be painted, since it has infinite area, yet it can be filled with paint, since it has finite volume. But I think this can be a misleading move that mixes the issue of Torricelli’s trumpet with general issues of how infinite processes correspond to everyday experience, which is perhaps a separate source of so-called “counterintuitive” phenomena. The fact is that if Torricelli’s trumpet is supposed to be a surprising result, then the source of the contradiction is supposed to be the properties of volume, area, and length of this shape, and not some secondary intuitions about infinite processes in general.

In my opinion, the properties of Torricelli’s trumpet are not “counterintuitive.”

First, is it counterintuitive for a finite amount of paint to cover an infinite surface? Of course not. Let’s put it like this. Suppose you have a can of paint. Obviously it contains a finite volume of paint, such as one liter. Now, open the can and pour the paint on the floor. Think of a big floor, like a basketball court. Get a spatula and start spreading the paint across as much area as you can.

Where do you think this process will stop? This is mathematical paint. You can spread it as thin as you like. How much area can you cover, if you can spread the paint thinner and thinner and thinner?

What does your “intuition” tell you? Does your “intuition” say that this spreading process with terminate after a certain number of square meters of the floor painted? Of course not. That would be idiotic.

And yet that is precisely what the standard account of Torricelli’s trumpet would have you believe. It is supposed to be “counterintuitive”, the story goes, for a finite volume of paint to cover an infinite area. Well, we have just seen that that premise is idiotic. Obviously a finite amount of paint can spread further and further, as long as you make it thinner and thinner. There is nothing “counterintuitive” about that.

Why would “intuition” say that the process of spreading the paint thinner and thinner would suddenly terminate at some finite bound? Why on earth would it? What would be the “intuitive” reason for why you could spread paint thinner, and then thinner, and then thinner, and then thinner, and then not thinner all of a sudden? Where would this invisible upper bound come from? Why would “intuition” stipulate the existence of such a ghost? It makes no sense.

Here’s another thought experiment that proves the same thing. We’re not using paint anymore, but rather a cube. You have a cube of unit volume in front of you. You cut it in half, horizontally. Like a sandwich. Then you place the top half side-by-side with the bottom half. Next cut the top half in half again in the same manner, and bring the new top slice down next to the bottom slice. So now you have a kind of stairway with three steps.

Continue in the same way: you keep bisecting the last piece and placing all the pieces in a row. All the slices have a length and a width of 1, the original dimensions of the cube, because you are always cutting horizontally. You are cutting the heights in half, leaving the width and the breadth of all the pieces the same throughout.

So you get a row of blocks that each have a length and width of 1, and whose heights are 1/2, 1/4, 1/8, 1/16, and so on. The total volume remains 1 throughout the process of course, because you only moved the existing volume around without adding to it or taking anything away. Meanwhile, both the length and the area of the combined shape clearly approaches infinity, as is intuitively clear.

So with this simple intuitive argument we get infinite extent with finite volume. So clearly infinite extent with finite volume is by no means contrary to intuition.

Note that this simple thought experiment also shows that Torricelli’s trumpet is not a case of a technical mathematical result being philosophically or qualitatively different from simple common-sense examples, contrary to what calculus teachers like to pretend. So the standard story is wrong in two ways. It claims that advanced calculus proves intuition wrong. But that’s doubly a lie. First, intuition is not wrong, and furthermore you don’t need fancy calculus to show any of this anyway. The simple thought experiments of the repeatedly bisected cube or the painted basketball court, which you can explain to a 5-year-old, contain everything that is relevant. There is nothing qualitatively new added by the use of calculus.

These things were also well understood historically. For example, Isaac “Barrow … clearly saw that [Torricelli’s] theorem can be intuitively explained by the fact that ‘the infinite diminution of one dimension compensates the infinite increase of the other’.” This fits with the basketball court example. Although Barrow was talking about the relationship between volume and length, not volume and area. But the point is the same.

The equivalent of the bisected cube example was also well understood. For example, Leibniz rightly remarked that “there is nothing more extraordinary about [Torricelli’s result] than about infinite series, where we find that 1/2 + 1/4 + 1/8 + 1/16 + 1/32 etc. = 1.” Exactly. Nothing to see here. Just as Leibniz says.

Let’s turn to history then. Let’s see what people thought back then, besides Barrow and Leibniz. Torricelli published his result in 1641. In fact, all this stuff about area plays no part in the 17th century. The whole thing is about volume only. Torricelli’s trumpet has finite volume and infinite extent, that is to say, infinite length. That’s it. No area, no paint, no painting an infinite area. Nobody said anything about that in the 17th century. Just volume and length.

Still, even then many people today say that this was a big paradox back then. Lots of mainstream accounts of the history of mathematics repeat this story. For example, here is a quote from Simmons’s book Calculus Gems: “That a solid can have finite volume even though it has infinite extent … caused great astonishment at the time.” Another example, from Burton’s History of Mathematics: “The result seemed so counterintuitive and astonishing that at first some of the leading European mathematicians thought it impossible.” And here’s another example, from the book When Least Is Best by Nahin: “Prior to this discovery it was commonly accepted that a surface extended to infinity would necessarily have to be of infinite size, that is, enclose infinite volume.”

So, that’s what everybody says. But here at this podcast we are not in the business of repeating boring clichés like some people, so let’s have a go at it. All of these people are wrong, in my opinion.

I’m going to base my discussion on Paolo Mancosu’s book Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century. This is by far the most complete and authoritative historical account of all of this, of the reception of Torricelli’s trumpet in the 17th century. Mancosu covers the relevant historical sources quite exhaustively. I am going to use the same sources, but I am going to interpret them differently.

Mancosu himself is of the same opinion as all these other guys I quoted, team cliché. According to Mancosu, I quote him, Torricelli’s result “seemed so counterintuitive and astonishing that, at first, some of the leading mathematicians thought it impossible.” “There are good intuitive grounds for holding that an infinitely long solid cannot have a finite volume,” and that is what gave rise to “the intuitive, widespread, and erroneous inference from infinite length to infinite volume.”

So Mancosu is wrong like everybody else, in my view. Let’s look at his evidence and refute it all one by one.

First, Mancosu’s chapter on this stuff is off to an interesting start. He says: “Even eighty years later Bernard de Fontenelle commented: ‘One apparently expected, and should have expected, to find [Torricelli’s solid] infinite [in volume].’”

Ha! Did you catch that rhetorical trick? “Even eighty years later …” “Even”! In fact, instead of “even eighty years later” it would be more accurate to say: not until eighty years later is this allegedly common view actually explicitly asserted, and then by an essayist and not a mathematician. It’s neither here nor there what some non-mathematician wrote 80 years later. He could very well have been subject to the same biases as the modern proponents of the story.

So, let’s look further back in time. What were Torricelli’s own words? Indeed, Torricelli called his result “as it were paradoxical,” in Mancosu’s translation. The Latin is “ut ita dicam paradoxicum.” I think maybe a more literal translation would be “so to speak paradoxical.” “So to speak”: that is to say, not actually.

It’s not a paradox but it’s kind of like a paradox if you squint a bit, Torricelli seems to be saying. Of course that seems to give some credence to the standard interpretation, perhaps the strongest evidence that exists for this view. Fair enough. There is some connection to paradox, like the standard story says. On the other hand, Torricelli’s phrasing is also a notably guarded formulation coming from the discoverer himself. You would think that Torricelli, the very discoverer of this result, would be inclined to play up, if anything, the philosophical significance of his own theorem. Despite this incentive, Torricelli evidently felt the need for the qualifier “so to speak paradoxical.” The result is not actually paradoxical, in other words.

Torricelli then goes on to explain why the result is “so to speak” paradoxical: “The reason is that ‘if one proposes to consider a solid … infinitely extended, everybody immediately thinks that such a figure must be of infinite size’.” Again, this fits to some extent with the standard narrative. But on the other hand it is striking how much more restrained this wording is. The Latin for “immediately” is “statim,” which arguably means something like “the very instant that,” or in other words: before thinking about it for more than a few seconds. Quite plausibly, Torricelli has chosen his words carefully since he is well aware of examples such as that of the repeatedly bisected cube that I described before. Perhaps Torricelli knows well that if he asked a mathematician colleague whether a solid with infinite length must necessarily have infinite volume then they would likely very soon think of such counterexamples and realise that the answer is no. Perhaps that is why Torricelli does not say that his theorem contradicts what everybody believes, or that it contradicts intuition. Instead he merely says, much more cautiously, that his result is “so to speak” surprising since it is different from the first example of a solid with infinite length that pops into someone’s head the very split second that they are asked to imagine such a shape.

Quick! Think of a shape with infinite length! Ok, maybe then you think of a shape that also has infinite volume. But that is not the same as to say that “intuition” says that it must have infinite volume.

Let’s listen to Torricelli’s words again: “if one proposes to consider a solid infinitely extended, everybody immediately thinks that such a figure must be of infinite size.” That is to say, in that moment, when you are asked to picture such a shape, your “immediate” impulse is to think of one that has infinite size. That’s your “immediate” reaction. Not your considered reaction, not your mathematician’s intuition, not what you think after a bit of thought. Only “immediately.” Torricelli has chosen his words wisely, so that it seems like he is saying a lot but in fact he is carefully choosing a formulation that is not subject to refutation based on thought experiments like the ones I gave before, because Torricelli is only talking about “immediate”, split-second visualisation, and he wisely says nothing about what somebody who thought about it for two minutes would think. In this way Torricelli stops well short of asserting the standard view that I quoted from so many modern historians. Torricelli seems to be going in that direction but he doesn’t go anywhere near as far as those statements from the modern literature that I quoted.

That’s all we have to go on in terms of Torricelli’s own words. Now let’s turn to the reaction of Torricelli’s contemporaries.

It is true, as the standard story says, that a number of mathematicians at the time reacted with some surprise to Torricelli’s result. But here we must ask ourselves: Why were they surprised? The standard story assumes that they were surprised because the result was counterintuitive. Therefore contemporary expressions of surprise are often cited as self-evidently supporting the standard story. But in fact there are other good reasons for contemporary mathematicians to have been surprised that have nothing to do with intuitions about infinite shapes.

Torricelli’s work showed that infinite curvilinear figures are mathematically tractable. That in itself was novel and surprising, quite apart from the specific result and whether it was intuitive or not. Indeed, Torricelli explicitly makes this point himself: “Among all the solids of which ancient and modern authors have determined the measures with much effort, none, as far as I know, has an infinite extension.”

Exactly. It was rare indeed for geometers at this time to be able to go qualitatively beyond what the ancient masters such as Archimedes had done, as opposed to supplementing and extending their results with more of the same. In fact, the very nature of the ancient methods for determining areas and volumes, which was still the gold standard in Torricelli’s time, and which he still relied on for proving his result in the most rigorous way, would seem to preclude application to infinite figures, as Torricelli himself noted: “In introducing the proof by exhaustion, Torricelli remarked that it would seem an impossible task to inscribe in another figure a figure of infinite length, or to circumscribe another figure around such a figure.”

In fact, therefore, when we have historical evidence that certain 17th-century mathematicians found Torricelli’s result to be “impossible,” we cannot infer that they therefore regarded the volume result as counterintuitive. Instead, the impossibility in question could be what Torricelli himself called the seemingly “impossible task” of applying rigorous exhaustion methods to infinite figures. And that is a methodological matter entirely independent of the specific content of Torricelli’s theorem and its intuitiveness or lack thereof.

There are indeed some historical testimonies that are ambiguous in this respect. For example, according to Torricelli himself, “Roberval considered this proposition false and impossible; actually, when Father Mersenne came by here he told me that Roberval, having thought about it for some time wrote I don’t know what demonstrations or speeches in order to prove that my proposition was absurd and impossible.” This is as consistent with the standard intuition-centred narrative as with the alternative intuition-independent narrative that I have proposed. Maybe Roberval thought the result was impossible because of his intuition about infinite shapes, or because he though it was not methodologically possible to apply rigorous geometry to such shapes. After all, why had Archimedes not done it, if it could be done?

Other 17th-sources are inconclusive in the same way. For example, Cavalieri said about Torricelli’s result that it is “extraordinary that that could be.” Again a generic expression of surprise. That is not enough to establish a link to any intuition or common conviction that solids of infinite extent ought to have infinite volume. Indeed, Cavalieri wrote to Torricelli that “I do not know how you fished out its measure so easily in the infinite depth of that solid.” So here Cavalieri is clearly highlighting that it is the new reach of the mathematical methods that is surprising rather than the result itself.

There is another sense too in which Torricelli’s result could have been seen as surprising in a way that has nothing to do with geometrical intuitions about infinite volumes. Namely, that the theorem invokes the notion of an actual infinity. Classically, statements involving infinity were often studiously avoided in favour of equivalent finitistic formulations. For example, instead of “there are infinitely many primes,” Euclid says “prime numbers are more than any assigned multitude of prime numbers” (Elements IX.20). But no such finitistic paraphrase was forthcoming in the case of Torricelli’s theorem. As Mancosu says, “The notion of actually infinite length is present in the very statement of the theorem. Moreover, the proofs themselves, as they stand, make sense only if the hyperbolic solid is given as infinitely long in actu.” But “Torricelli’s result is boldly infinitistic,” and efforts by purists “at reducing it to a finitistic framework appear destined to fail.”

One such finitism purist was Hobbes. In his view, as summarised by Mancosu, “any consideration of infinity, both geometrical and arithmetical, ought to be banished. Hobbes claimed that when mathematicians use the word ‘infinite’, what they usually mean, or ought to mean, is ‘indefinite’, that is, as great, or small, as one pleases.” “In Hobbes’ mathematical universe …, to talk about a solid ‘whose sides are extended out to infinity is absurd’.”

This is certainly a crucial background to keep in mind when interpreting Hobbes’s comments on Torricelli’s result. Hobbes claimed that “to understand [Torricelli’s theorem] for sense, it is not required that a man should be a geometrician or a logician, but that he should be mad.”

This striking quote is often connected with the intuition narrative. For example, The Cambridge Companion to Hobbes says: No, it is not necessary to be mad to accept Torricelli’s result. “Rather, of course, it is required that one should see, as Hobbes sometimes failed to do, that mathematics can raise and resolve problems where physical experience is lacking and physical intuition may deceive.” That’s the opinion of The Cambridge Companion to Hobbes, which is in line with the standard view.

But note indeed that Hobbes is not claiming that the theorem is contrary to “physical intuition.” Instead he says that it has no “sense.” This is in line with the standard finitistic position in philosophy that statements involving infinity have meaning (or “sense”) only when they can be, and are, reformulated in finitistic terms.

But, this point aside, there is one final passage in Hobbes that Mancosu regards as an outright explicit affirmation of the standard intuition-centred interpretation, leading Mancosu to confidently assert that “Hobbes believed that natural light teaches us that an infinitely long solid, however subtle, must exceed in volume any finite solid.” “Natural light” is indeed a synonym for intuition, so, if substantiated, this would make a strong case for the standard narrative. The Hobbes passage in question is the following.

“It has been demonstrated, they say, by Torricelli that a certain acute hyperbolic solid, in this sense of infinite [namely, greater than any assignable measure], is equal to a [finite] cylinder. … The distance which Torricelli supposes infinite is to be understood as indefinite. Nor could it be understood differently by him, who in quite many demonstrations uses Cavalieri’s principle of indivisibles, which are such that their aggregate can be equated to whatsoever given magnitude. Therefore, a proposition as absurd as this that the infinite is equal to the finite, must not be attributed to Torricelli. In fact, as it is clear by the natural light, there cannot be a solid so subtle which does not infinitely exceed every finite solid.”

So those are Hobbes’s words, and according to Mancosu’s interpretation he is saying that Torricelli’s result is counterintuitive.

But if you parse the quote you see that there a number of problems with Mancosu’s reading of this passage. For one thing, note that Hobbes clearly connects his conclusion to Torricelli’s use of Cavalieri’s principle, that is to say, his method of proof, by means of infinitesimals or indivisibles. But if Hobbes is trying to say that the result itself is counterintuitive, then why does it matter what method was used to prove it? Hobbes is saying that something is contrary to “natural light,” which means contrary to intuition, but what exactly? The passage is not so clear, but apparently the point is connected to the method of proof, and not merely to the content of the theorem itself.

Indeed, the crucial final sentence of Hobbes’s passage is difficult to parse altogether. It appears to be nonsensical on the face of it. The thing that is evident by “natural light” or intuition according to Hobbes is that: “there cannot be a solid so subtle which does not infinitely exceed every finite solid.”

This sounds crazy. It appears to say that any “subtle” (that is to say, small) solid must be infinite, which is obviously nonsense. I checked Hobbes’s original Latin but unfortunately that’s just as ambiguous and it doesn’t clarify anything.

For Mancosu’s reading to work, it seems that this crucial sentence, about any “subtle” body being infinite, must be interpreted as follows.

The passage starts:

“There cannot be a solid…”

Here, for Mancosu’s reading, we should think of Torricelli’s trumpet as the solid in question.

“There cannot be a solid so subtle…”

“Subtle” here then means thinning out or tapering off, as the trumpet does when it goes to infinity.

“There cannot be a solid so subtle which does not infinitely exceed every finite solid.”

So, first of all, for this not to be nonsense we must, for Mancosu’s reading to work, we must insert an extra clause that must be regarded as implicit in Hobbes:

Instead of: “There cannot be a solid so subtle which does not infinitely exceed every finite solid.”

We must read: “There cannot be a solid [that is infinitely extended and yet is] so subtle which does not infinitely exceed every finite solid.”

If we twist the passage like this then indeed it does say that the volume of Torricelli’s trumpet must exceed any finite volume. That is to say, it must be infinite. That is how Mancosu wants to read it.

I don’t agree. I will propose a different interpretation. But first let’s summarise the problems with Mancosu’s interpretation. First there is the connection to the method of proof, that is to say to the use of infinitesimal slices. Hobbes says: Torricelli used Cavalieri’s method and “therefore” etc. Hobbes’s conclusion is a consequence of the method of proof used. Which is not reflected in Mancosu’s interpretation of the concluding sentence. Mancosu’s interpretation only has to do with the statement of the theorem and not with its proof.

Furthermore, for Mancosu’s reading to work we had to insert that the solid Hobbes was talking about had to be infinite in extent even though Hobbes doesn’t say that. We would have to regard that as implied by context. Which is certainly questionable.

And one more thing: I don’t know if you caught this but Hobbes used the phrase “infinitely exceed every finite solid.” In Mancosu’s reading he’s saying: Any infinitely extended solid, no matter how thin it gets or how much it tapers off, will “infinitely exceed every finite solid.”

Why “infinitely”? That’s completely redundant isn’t it? You could just as well have just said: it exceeds every finite solid. If you exceed every finite solid you are infinite already. Of course if you only finitely exceed some finite solid then you are obviously yourself finite. So of course you cannot then exceed every finite solid. So if you exceed every finite solid, then you also automatically exceed every finite solid by infinitely much, so to speak. Because if you exceeded one of them only by a finite amount then you couldn’t also exceed all the other ones.

So Mancosu’s reading doesn’t explain why Hobbes uses this redundant word “infinitely” which doesn’t add anything and doesn’t change the meaning in any way. This word could have been omitted and the sentence would have meant exactly the same thing, according to Mancosu’s interpretation. Although Mancosu himself doesn’t mention that, of course.

Now I am going to propose my alternative reading, which avoids all of these problems.

Hobbes’s sentence starts:

“There cannot be a solid…”

Here I claim that we should not in fact think of Torricelli’s trumpet. Instead, the solids Hobbes is thinking of here are the infinitesimal components dV, the Cavalieri slices, of the trumpet volume. In other words, the slices that are used to calculate the volume, basically calculus-style. These dV’s, these infinitesimal components of volume, are what Hobbes is speaking about, I claim.

Let’s continue reading Hobbes’s sentence:

“There cannot be a solid so subtle…”

That is to say, so small: dV = epsilon. We are doing a finitistic paraphrase of a calculus argument. A limit-style paraphrase. Where the volume element dV is some very small real number epsilon.

Back to Hobbes:

“There cannot be a solid so subtle which does not infinitely exceed every finite solid.”

There cannot be any epsilon that is so small that it does not, when taken infinitely, that is to say added to itself any number of times, does not eventually exceed any given real number. This is the Archimedean property of the real numbers, in other words.

This is why we need the word “infinitely” that didn’t serve a purpose in Mancosu’s interpretation. The little number epsilon, the dV, is, taken singly, just a very small thing of course, but taken “infinitely” (that is to say added together over and over and over again) it eventually becomes greater than anything.

This reading explains why the link with the Cavalierian method of proof is crucial to the point, just as Hobbes seemed to indicate. It also explains why the phrase “infinitely exceed,” as opposed to merely “exceed,” is needed. It also does not require a crucial phrase that dramatically impacts the meaning to be postulated as implicit.

According to my reading, Hobbes’s statement is in effect simply an assertion of the Archimedean property of the real numbers, which was a well-known axiom that many greater mathematicians than Hobbes did not hesitate to regard as evident by “natural light.” Therefore Hobbes’s claim, on this reading, is correct and unproblematic, and does not have anything to do with physical or geometrical intuitions about infinite volumes.

Nor is Hobbes’s point an impactful critique of Torricelli’s reasoning for that matter, since Torricelli does not rely on non-Archimedean assumptions in the manner Hobbes seems to be trying to suggest. But that is beside the point for our purposes.

This concludes my analysis of the historical reception of Torricelli’s theorem. I think there is very little evidence to say that it was regarded as counterintuitive at the time, contrary to the standard narrative in the literature. What I have discussed is essentially all the evidence. I have systematically gone through basically all passages from 17th-century sources that are commonly cited in support of the standard narrative. There’s nothing more. There is no smoking gun anywhere. There are only these passages, all of which can be interpreted very differently than in the standard story, as I have shown.

Now I will add a little speculation about why I think the myth of Torricelli’s trumpet is so popular. Basically, I believe this is ultimately because the way we teach calculus is poorly motivated.

Modern calculus textbooks are monstrosities. They mix classical calculus with modern real analysis as if they were one and the same subject, which they are not.

One moment you are doing applications of integration by parts, and the next moment you are thinking about whether continuity implies differentiability.

These are both great subjects. I love calculus: going nuts with integration by parts, applying it to a bunch of differential equations and physics and stuff. Wonderful.

And I love real analysis. Such an elegant, unified theory that brings such razor clarity to so many tricky questions. “Baby Rudin” is a work of art. It is stylistic perfection.

But these are two separate subjects. Modern calculus textbooks don't want you to know this. Modern calculus textbooks pretend that real-analysis-style rigor is the correct and appropriate lens through which to understand classical calculus. Which it is not.

Mathematicians who teach calculus pretty much know this in their heart of hearts, even if they try to repress it. They know that the need for real-analysis-style rigour in calculus serves no reasonable purpose in that course. They know that you could do the entire calculus course completely unrigorously and it would be perfectly fine.

Unlike real analysis, of course. You couldn’t teach that course intuitively or informally. Doing it formally is precisely part of the core of what makes the subject great. Formal rigour in real analysis is very well motivated. It is a beautiful course for that reason.

But formal rigour in calculus is largely pointless and serves pretty much no purpose within calculus itself. The so-called rigorous approach to calculus didn’t exist for the first 200 years of calculus, when all the best applications of calculus were worked out by all the best mathematicians of the day who were the most accomplished calculus practitioners the world has ever seen, perhaps more so than anyone alive today because there are no research professors of calculus anymore.

Mathematicians who teach calculus know all of this. The know that the calculus has always thrived without rigour. They know that every single one of the great applications of calculus were developed with complete disregard for real-analysis-style rigor. At some level they know this, at least subconsciously. They may try to suppress this knowledge because it makes their life easier.

They have to teach real-analysis-style proofs in calculus courses, so it would be easier for them if doing so was a good idea. Otherwise they are torturing students with lots of difficult technicalities that are poorly motivated.

You can see now why Torricelli’s trumpet is a godsend to these people. Supposedly, Torricelli’s trumpet shows that intuition cannot be trusted. In other words, it motivates taking much more formal and rigorous approach to calculus.

Phew! As an instructor you had seen your students suffering from the overly formal approach, and you had felt bad about it. Maybe in a moment of weakness you had even wondered to yourself why indeed we have to teach calculus this way, although you would certainly keep those kinds of speculations to yourself.

It is a weight on your conscience as a calculus instructor. Am I just torturing my students with vastly excessive technicalities for no good reason?

What you need to be able to sleep well at night is a nice self-rationalising story that tells you that you are doing the right thing. “Ah, well, you see, this rigour may be a bitter medicine for some but you need it! It’s for the best! It’s for your own good!”

And how do we prove that rigour in calculus is actually a good thing? Oh well, a good example is Torricelli’s trumpet, you see.

These people desperately need the standard story of Torricelli’s trumpet to be true. It is exactly the self-rationalising justification they need to keep teaching the way they do and convince themselves that their conscience is clear. It is a necessary myth.

That is why so many people do not want to know the real history of Torricelli’s trumpet. They would rather have the comforting myth that flatters the status quo.

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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.

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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

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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

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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?

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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 … Continue reading Read Euclid backwards: history and purpose of Pythagorean Theorem

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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 … Continue reading Singing Euclid: the oral character of Greek geometry

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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 … Continue reading First proofs: Thales and the beginnings of geometry

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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 … Continue reading Societal role of geometry in early civilisations

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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 … Continue reading Why the Greeks?

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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 … Continue reading The mathematicians’ view of Galileo

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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 … Continue reading Historiography of Galileo’s relation to antiquity and middle ages

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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 … Continue reading More things Galileo didn’t do first

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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,” … Continue reading Galileo was the first to … what exactly?

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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 … Continue reading Galileo and the Church

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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 … Continue reading Galileo’s theory of comets is hot air

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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 … Continue reading Phases of Venus

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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 … Continue reading Blemished sun

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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 … Continue reading The telescope

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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 … Continue reading Heliocentrism before the telescope

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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 … Continue reading Heliocentrism in antiquity

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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 … Continue reading Galileo’s theory of tides

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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 … Continue reading Why Galileo is like Nostradamus

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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 … Continue reading Galileo’s errors on projectile motion and inertia

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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. … Continue reading The case against Galileo on the law of fall

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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 … Continue reading Galilean science in antiquity?

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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 … Continue reading Mathematics versus philosophy, then and now

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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 … Continue reading Galileo bad, Archimedes good