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

Transcript

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.

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.

Copernicus’s planetary models contain elements also found in the works of late medieval Islamic astronomers associated with the Maragha School, including the Tusi couple and Ibn al-Shatir’s models for the Moon and Mercury. On this basis many historians have concluded that Copernicus must have gotten his hands on these Maragha ideas somehow or other, even though no direct evidence for such transmission has been found. Let us consider the evidence as to whether Copernicus plagiarized these Arabic sources or not.

See PDF slides for figures and references.

Einstein’s theory of special relativity defines time and space operationally, that is to say, in terms of the actions performed to measure them. This is analogous to the constructivist spirit of classical geometry.

Transcript

Oh no, we are chained to a wall! Aaah! This is going to mess up our geometry big time. Remember what Poincaré said: self-motion is the essence of geometry. We understand that part of the environment to be geometrical that we can cancel through self-motion, through a change of perspective.

Suppose you are looking at a chair, let’s say, and somebody tips it over so that it’s laying on its side, or somebody moves it to the other end of the room. Those are geometrical transformations: rotations and displacements in space. They are the equivalence relations of space; the isometries: things you can do without changing metric relationships.

You know that these are geometrical equivalence transformations because you can cancel them through self-motion. When the guy knocks the chair over, you can tilt your head 90 degrees, and you have restored the original visual impression of the chair. And if the guy moves the chair five meters that way, then you yourself can move five meters in the same direction and once again the chair makes precisely the same visual impression on your retina as it did before.

This is how you know that rotations and displacements are geometrical equivalence transformations. The more you accumulate experience with these kinds of scenarios, the more you begin to grasp the group of geometrical transformations as a whole. You get a global sense of what kinds of transformations are possible, how they combine and interact, and so on. This process might lead you to Euclidean or non-Euclidean conceptions of space depending on your experiences. You get to know space and what kind of geometry it has by getting to know its transformation group: that is to say, what kinds of rotations and displacements exist, what happens if you do one after the other, and so on.

Now, what about the scenario when we are chained up? We must imagine that we have been chained to this wall for life. We don’t know any other reality than this.

Our sense of what geometrical transformations are possible will be very different. There is still geometry because there are still visual impressions that we can cancel through self-motion. If an object is moving across our field of view, we can keep the retinal impressions the same by tracking it with a motion of our eyes. So we understand the geometry of sideways motion well since we can move our eyes from left to right, or point our gaze in different directions.

We also understand the geometry of depth to some extent. If an object is moving away from us, we can keep track of that through self-motion also, but of a very different kind. They eye has a lens in it. The curvature of the lens is variable and is controlled by a muscle. Depending on whether you need to focus on objects that are near or far, the muscle will pinch or pull the lens so that it is more round or more flat in order to have the right focal distance for the object you are looking at. In this way you can keep track of how much an object has moved in depth by recording how much the lens needs to be adjusted to restore focus. So this gives you the data to develop a geometry of depth.

So our chains do not deprive us of geometry altogether. We can still develop the geometry of width and the geometry of depth. But these are separate geometries to us. A free person will know that width and depth are merely two dimensions of the same kind of thing. They are both spatial dimensions. They are interchangeable and homogenous. The free person will know that since they can turn width into depth by self-motion. They just need to go stand over there and the old width is the new depth and vice versa.

But we who are chained are deprived of this experience. So to us width and depth remain qualitatively different kind of things altogether. Indeed, we measure distance in width and distance in depth completely different units. We count distance in width by the direction in which our eyes are pointing, so the unit is degrees for example. An object is 30 degrees to the left of another, for example, we might say. But we count depth by how much the lens needs to be bent to achieve focus. So the unit is something like a unit of force corresponding to the muscular effort involved. That’s a completely different kind of thing altogether, and cannot be compared with our degree measures that we used to quantify position in the width direction.

It’s not so strange that width and depth would be qualitatively different things. You already treat various measurements of the same object as qualitatively different in your everyday life. For example, suppose somebody asked you: Is this building wider than it is old? Of course that doesn’t make any sense. You cannot compare a distance in space with a duration in time. Because those quantities are determined in fundamentally different kind of ways, they are measured in completely different kinds of units, and so on. Well, just as you think time and space are not comparable, so the chained person thinks depth and width are not comparable. Samesies.

In fact, maybe you are are just as delusional as the chained guy, and for much the same reason. Actually time and space are a lot more comparable and interchangeable than you think, as Einstein’s theory of relativity says. We don’t realise this in our everyday experience, because relativistic effects become significant only at high speeds, somewhat close to the speed of light. Compared to the speed of light you have practically been standing still your whole life, even when flooring it on the highway. So you might as well have been chained to a wall. The sum total of all your visual and sensory impressions are severely and systematically impoverished just like the guy chained to a wall. Just as he doesn’t realise the fundamental unity of width and depth, so you don’t realise the fundamental unity of time and space. And for the same reason: you are both essentially standing still.

I took this example from Feynman’s famous lectures on physics. Why don’t we listen to his version as well? The classic Feynman lectures on physics are nowadays available for free at a Caltech website, audio recordings and all.

“When we look at an object, there is an obvious thing we might call the ‘apparent width’, and another we might call the ‘depth’. But the two ideas, width and depth, are not fundamental properties of the object, because if we step aside and look at the same thing from a different angle, we get a different width and a different depth, and we may develop some formulas for computing the new ones from the old ones and the angles involved. … If it were impossible ever to move, and we always saw a given object from the same position, then this whole business would be irrelevant—[width and depth] would appear to have quite different qualities, because one appears as a subtended optical angle and the other involves some focusing of the eyes …; they would seem to be very different things and would never get mixed up. It is because we can walk around that we realize that depth and width are, somehow or other, just two different aspects of the same thing.

[In Einstein’s theory of special relativity] also we have a mixture---of positions and the time. … In the space measurements of one man there is mixed in a little bit of the time, as seen by the other. Our analogy permits us to generate this idea: The ‘reality’ of an object that we are looking at is somehow greater (speaking crudely and intuitively) than its ‘width’ and its ‘depth’ because they depend upon how we look at it; when we move to a new position, our brain immediately recalculates the width and the depth. But our brain does not immediately recalculate coordinates and time when we move at high speed, because we have had no effective experience of going nearly as fast as light to appreciate the fact that time and space are also of the same nature. It is as though we were always stuck in the position of having to look at just the width of something, not being able to move our heads appreciably one way or the other.” (I.17-1)

I love this thought experiment with the chained guy. Plato’s cave 2.0. And it is perfect for our purposes today. This is going to be the concluding episode of my history and philosophy of geometry story arc, and the theme will be how everything goes full circle and the beautiful ideas from days of old are as relevant as ever to us self-absorbed moderns as well. The guy chained to a wall is a perfect backward-looking example, and a perfect forward-looking example. Back to the operationalism of Greek geometry, and forward to Einsteinian modernity.

We started, way back when, with the Greeks and their ubiquitous ruler and compass. Always with the making, those guys. Lines and circles are nothing but the things you get when you draw with these tools. Not abstract things, not axiomatically defined things. Lines and circles are operations. They are things you do.

The Greeks realised that this was the rigorous way to do mathematics. The epistemological humility of the maker is far superior to hubris of the philosopher who think they can concoct a perfect theoretical system in the abstract using the power of their mind alone. People are not as good at that as they think. Time and time again, somebody’s pretentious abstract theory has proved to contain various unintended contradictions and unnoticed assumptions. As the Greeks knew all too well: the works of Plato and Aristotle do little else than poke holes in other people’s bad theories. So we should stop trying to philosophise about essences, which we are so bad at, and instead roll up our sleeves and build stuff. Clear the junk off the table in your garage, put your tool belt on, and let’s double some cubes.

To know is to do. And this runs all the way through history. In physics, we can only know relative space, not absolute space, said Descartes and Leibniz, because we can measure the distances between things, but we cannot measure any such thing as the absolute “coordinates” of any one thing in itself without reference to anything else. So, if we believe in the virtue of the humble maker and the hubris of the speculative philosopher, then it follows that we must base our physics on relative space, not absolute space. A very reasonable conclusion, which Newton abandoned to the dismay of many at the time.

So if we stick to the classical point of view then space is what you can make, and what you can measure. What you can experience, in other words. This is a good philosophy of space. And no wonder. The Greeks, Descartes, Leibniz—back then mathematical and philosophical sophistication went hand in hand to a rare degree. So it’s no wonder they had some good ideas.

But don’t take my word for it. What makes a philosophy good? Not the say-so of some podcaster, that’s for sure. But we can prove that the classical operationalist perspective was good philosophy, by considering how it fared in the face of entirely new developments. Bad philosophy is always back-pedalling. As soon as new facts come in you have to go: uh, well, actually what I mean was… Or just descriptive: some people think they have a philosophy of something when they are just describing its basic features and making up a name for each part. But good philosophy is not that. Good philosophy is a perspective that makes you think in new ways. It gives you tools that you can use to try to understand conceptually challenging new problems. Philosophy is good if it is a fruitful way to think in challenging new situations.

Such as non-Euclidean geometry, for example. A rather counterintuitive new world; we could really use some philosophy to find our feet here. What philosophy is going to help us? Maybe Aristotle’s four different names for four different kinds of causes? Yeah right. But thinking of space and geometry in terms of operations: now that’s a philosophy. And it will prove its worth by the way it interacts with these new developments.

How do we know whether we live in a Euclidean space or a hyperbolic space? Not by developing these two geometries abstractly and axiomatically, and then testing them by their angle sum theorems or whatever. No thank you, that would be that hubristic assumption again, that we could develop geometry purely in the abstract, in the mind alone. Geometry should come from experience. But how? Modern mathematics has told us exactly how. A geometry is defined by its group of equivalence transformations, as Felix Klein said in his famous Erlanger Program. And a group of equivalence transformations can be defined in terms of experience. That is what Poincaré explained: equivalence transformations are the transformations you can cancel through self-motion. Perfect! In this way the difference between Euclidean and hyperbolic geometry emerges organically from experience itself. There is no need to postulate a hubristic ability of the human mind to develop axiomatic systems in the abstract.

Later we can go on to do more conventional abstract axiomatic mathematics as well, of course, but we do that by building on the concrete substrate developed first. We are not born with general-purpose abstract reasoning skills. We have domain-specific innate abilities such as that of acquiring a geometry by extracting the group of equivalence transformations of the space we live in from our sensory experience. And, insofar as we eventually succeed at general abstract reasoning, that is because we have mobilised our domain-specific skills and modes of thought to simulate abstract general-purpose thought. This is the point of view that I associated with Poincaré and Chomsky if you recall.

The chained guy is a perfect example to illustrate this entire tradition on geometry going all the way back to the Greeks. Restrict the operations a guy can perform, and you restrict his geometry.

I think maybe Feynman didn’t realise that his thought experiment perfectly illustrates this historically rich point of view. If we assume that Feynman came up with this through experiment himself, it seems that he started with Einstein’s relativity theory and asked himself how he could illustrate it using an analogy. Then the idea that the concepts of a physical theory depend on the kinds of experience one has, or the kinds of measurements one can make, comes off looking a bit like a kind of quirky side-effect of relativity theory. Rather than a methodological axiom built in to it from the very beginning, and indeed an axiom already strongly established long before relativity theory was even conceived.

We can see this in another one of Feynman’s remarks, in another lecture. Let’s listen to this, and pay attention to what causes what. What comes first: the physics or the philosophy?

“One of the consequences of relativity was the development of a philosophy which said, ‘You can only define what you can measure! Since it is self-evident that one cannot measure a velocity without seeing what he is measuring it relative to, therefore it is clear that there is no meaning to absolute velocity.’” (I.16-1)

One could argue that it was the other way around. This way of thinking was not a consequence of relativity theory, as Feynman says.

“One of the consequences of relativity…”

If anything, relativity theory was a consequence of this way of thinking.

“The physicists should have realized that they can talk only about what they can measure.”

Yes, they should have realized that, and they did! Not from Einstein but thousands of years before.

Indeed, Einstein read a lot of that stuff in his youth, including Ernst Mach and Poincaré. And he made no secret of how much those things influenced him. Relativity theory was a philosophy-driven scientific development to an unusual degree.

Without Poincaré’s beautiful philosophy of space, no Einstein maybe. Feynman thought the guy chained to a wall was a perfect thought experiment to describe Einstein’s theory. Yes, but it is equally perfect to describe Poincaré’s philosophy of space, which came before Einstein’s theory.

Actually Feynman was right too, when he said that the success of Einstein’s theory led to these operationalist philosophical conclusions. It did indeed. But not because science developed in its own autonomous, technical way, and then after the fact people went: huh, I wonder if we can draw some philosophical conclusion from this new science? It wasn’t like that. It was a revival of old philosophy rather than a new start stimulated by new science. Einstein’s theory didn’t so much rectify the course of philosophy, as much as it showed that the philosophers had been right all along, somewhat embarrassingly.

Remember how Newton’s absolute space was criticised. By Leibniz for example, but also others at the time. Only relative space makes epistemological sense. Only relative space is knowable. Because only relative space is measurable. Or in other words, only relative space can be operationalised.

Operationalisation is a way to ensure consistency, as the classical constructivist tradition in geometry knew. There are two ways to introduce objects in mathematics: construction, or wishlist to Santa Claus.

“Let ABC be the figure you get when you …” This is how to introduce objects by construction. “First I raise this perpendicular to that line, then I cut off a length here equal to that length over then, then I connect these two points” etc. That is the honest way to do things. The object is defined by the recipe for making it. An object is nothing but the outcome of certain operations that you perform yourself.

The other way is lazier and easier. “Let ABC be a figure such that…” This is a wishlist to Santa Claus. You state what it is that you want: “Let ABC be an equilateral triangle.” “Let ABC be a triangle with three right angles.” “Let me have a flying car and unicorn pony.” You state the properties that you want an object to have, and like a spoiled child you assume that you are thereby entitled to the object in question.

Newton was like the spoiled child asking for a unicorn. His new physics demanded absolute space and time, which were merely postulated, or wishlisted really, and cannot be operationalised.

So people like Leibniz objected, very reasonably. Newton’s physics is built on concepts that are unknowable. And it is exposed to the risk of containing inconsistencies and contradictions, since it is not susceptible to operationalisation, which has been the best way to ensure consistency since the days of Euclid. There are no unicorn ponies or triangles with three right angles, but a spoiled child wouldn’t know that, would he? Because he is not constrained by what is actually doable. So maybe Newton’s physics is ultimately incoherent since it has not taken steps to ensure otherwise.

This critique of Newton was philosophically sound, but it soon looked absolutely ridiculous. Newton’s physics was a runaway success like the world had never seen. And then you had these ridiculous little philosophers going: “well, actually, that’s actually bad science because blah blah blah.” Who would listen to such clueless nitpicks? Read the room, nerds. Newton has already won. Nobody cares about your stupid “well, actually.”

Let’s quote David Hume, for example, who was one of those philosophy losers in the 18th century. “[A] notion … beyond what we have instruments and art to make, is a mere fiction of the mind, and useless as well as incomprehensible.” Such as absolute space, for example. We have no “instruments and art”—that is to say, no physical experiments or observations—that can detect absolute space. So it must be a “useless fiction of the mind.”

Here’s another passage where Hume says the same thing: “When we entertain, therefore, any suspicion, that a philosophical term is employed without any meaning or idea (as is but too frequent), we need but enquire, from what impression is that supposed idea derived? And if it be impossible to assign any, this will serve to confirm our suspicion.” Indeed, we cannot assign any sensory impressions to the notion of absolute space, so therefore the “term is employed without any meaning.”

Hume also explains why we should insist on this criterion of meaning. “If we carry our enquiry beyond the appearances of objects to the senses, I am afraid, that most of our conclusions will be full of scepticism and uncertainty.” Meanwhile, “As long as we confine our speculations to the appearances of objects to our senses, … we are safe from all difficulties, and can never be embarrass’d by any question.” In particular, we can never run into any self-contradictions stemming from “fictions of the mind.” To be “embarrass’d” is to have contradictions in your thinking exposed to you. But just stick to the senses and “what we have instruments and art to make” and you will be fine.

Actually I kind of hate David Hume. Hume is the Galileo of philosophy: an overrated false idol who erroneously gets credit for trivial ideas that had long been obvious to mathematically and scientifically competent people.

These quotes that I just read from Hume, they are fine philosophy, or rather, they were fine philosophy a hundred years before Hume, when the same ideas were advocated by better philosophers. I have argued that those notions were well known already to Greek mathematicians. Well, we can’t prove that, because we don’t have the source evidence to know for sure one way or the other what the Greek mathematicians thought about such things. But in any case, those ideas were obviously well understood by Descartes and Leibniz for example, which is why they insisted that all geometry must be constructive, and also why they insisted that only relative space makes any philosophical sense and absolute space is a cardinal sin that must be banished from the face of the earth.

When Descartes and Leibniz said these things, they were scientifically viable ideas. Descartes and Leibniz put their money where their mouth was. They backed up their philosophy with detailed, technical scientific works, that contained both technical progress on advanced mathematical problems as well as a programmatic vision of how scientific and mathematical practice can move forward in harmony with philosophical and epistemological principles.

After Newton, that dream is dead. Philosophy lost. And all scientifically competent people knew as much. So only the scientifically ignorant, such as David Hume, kept beating this dead horse. As one historian has put it, Hume was a “dour scientific dilettante” with “almost unparalleled ignorance of the science of his day.”

Indeed, in the 18th century, only people like that could still defend this old philosophy. People with scientific integrity knew that could not in good conscience advocate for such a philosophy anymore, because that would mean that they would have to give a philosophically coherent physics that worked as well as Newton’s absolute-space-based physics, which no one could do.

So the only people who could still repeat those old philosophies that were no longer scientifically credible were now the scientifically airheaded like Hume who wouldn’t know good mathematics if it hit them in the face. That was the sad state of this once proud philosophy in the 18th century.

No wonder this anti-Newtonian philosophy became a laughing stock for centuries. And then, plot twist. They were right! Einstein’s theory exactly vindicates what these people had been saying for more than two hundred years. If you try to do science with Newton’s Santa Claus concepts of space and time, then you are doomed to run into inconsistencies. Exactly as these guys had been warning. And the way out of these hopeless inconsistencies is: operationalise everything! Exactly what these philosophy nerds had been saying all along.

Unbelievable. Imagine insisting that the most successful scientific theory of all time, that has proved itself again and again for centuries, is bound to lead to inconsistencies and self-contradictions any day now. That must be one of dumbest predictions of all time, you would think. What stubborn and oblivious people would keep embarrassing themselves by saying such silly things? And then, those guys, those very archetypes of the utter irrelevance and pointlessness of philosophy—those guys of all people—hit the cleanest home run you will ever see, with Einstein’s relativity theory. Insane.

Let’s do a bit of relativity theory here to show this. We’re on a cruise ship now. Let’s go below deck. Here there is a tennis court. Oh boy, tennis is fun! Time flies though. While we’ve been playing, has the ship reached its destination and anchored in port? Or are we still moving?

**

You can’t tell. That is the principle of relativity. Everything in the tennis room will the same whether the ship is moving at a constant speed or standing still. Even though one of us may be playing against the direction of travel and the other with it, that doesn’t mean that our serves and smashes will be boosted one way and slowed down the other. If we hit equal serves at the same time, they will reach the net in the center of the court at exactly the same time.

That feels natural to us in the room because we are so absorbed in the game and we are not paying any attention to whether the ship is moving or not. We are using the room as our frame of reference, or coordinate system, so to speak. That is the center of our universe at the moment.

For example, let’s say we can hit tennis serves of 50 meters per second, and it’s about 12.5 meters to the net, so it will take a quarter second for the serve to get there. That’s the science of tennis that is relevant to us when we are absorbed in the game, not whatever the ship is doing.

But the same thing works also if seen from the outside. Some guy is standing on the shore, watching us go by. And he’s a science nerd, it turns out. He happens to have one of those speed cameras that the police use to catch cars going above the speed limit.

So, according to his measurements, the ship is going 10 meters per second, and he also measures the speed of our serves somehow. Those were all 50 meters per second according to ourselves, but that’s not what the readings will say on the speed camera of the guy on the shore.

The velocities will behave additively: the speed of a projectile = the speed at which the projecter is moving + the firing speed. So when I’m serving with the ship, in the direction of travel, that’s 10+50 meters per second. So 60 is the speed measured by the observer on the shore with his speed camera. And the serve going the other way will have velocity 10-50, so 40 meters per second in the opposite direction.

So that guy disagrees with us about the speeds, but he still agrees with us that the two tennis serves will reach the net at the same time. Because the one that’s faster has further to go, since the ship is moving while the serve is in the air. Remember, it took a quarter second for the serves to reach the net. So the ship will have moved 2.5 meters in that time. So the 60 meters per second serve will have 12.5+2.5 meters to go, that’s 15 meters. And the slower serve of 40 meters per second, it has the net coming toward it so it only has to go 12.5-2.5 meters, so 10 meters before reaching the net. Indeed, going 15 meters at a speed of 60 takes the same time as going 10 meters at a speed of 40.

So the fact that the motion of the ship is undetectable to us inside the room is basically equivalent to the principle of additivity of velocities as seen from some other fixed vantage point.

Ok, yeah yeah, boring old high school physics. Who cares? That’s all trivial, right? Not really. It’s not so trivial. In fact, it’s wrong and inconsistent with other parts of physics, as we shall see.

So-called trivial things can be quite profound. We remember this from Euclid, for example. Recall for instance the construction of a square in Proposition 46. At first sight you might go: “What a big commotion about nothing. I guess Euclid wrote for kids in middle school or something. Obviously a research mathematician does not need to have squares explained to them step by step. That’s silly and trivial.”

You might have thought that, but you would have been wrong. In fact, there are no squares in spherical or hyperbolic geometry, so carefully tracking the fundamental assumptions on which the existence of squares is based is deep theoretical question. Euclid knew that. He didn’t write for middle-schoolers. He wrote for highly sophisticated mathematicians who had thought a lot about the foundations of geometry.

It’s the same with the so-called trivial physics in our tennis room. Let’s see how we can re-analyse this “trivial” situation in operationalist terms. “Let two tennis balls be fired at the same time…” Oh no, you don’t. That’s like saying “let ABCD be a square.” We can’t have that. We have to operationalise it.

It was all very easy with Newton’s absolute time, or unicorn time if you like. If this one universal absolute time is given to you by Santa Claus, then the tennis players can just use that to coordinate their serves, no problem.

But if we don’t want to rely on Santa we have to coordinate the tennis serves ourselves. Ok, so I’ll just count it off, right? “1… 2… 3… go!” Then we both serve at the same time. No, not really, because I hear “go” the instant I say it but you have to wait for the sound to travel across the room before you hear it. So we’re not really synchronised that way after all.

And it’s the same with any light-based signal of coordination, like a green light switching on, or both of us looking at the clock on the scoreboard. The speed of light is not instantaneous either, so it matters whether the sources of the signal is closer to one of us than the other.

So it looks like we are going to have to make our definition of same time depend on distance. Indeed, Einstein does precisely this. I will quote to you from the book Relativity: The Special and General Theory, of 1916. This is Einstein’s popularised presentation of his theory. Einstein says:

“We … require a definition of simultaneity such that this definition supplies us with the method by means of which, … [we] can decide by experiment whether or not [two events] occurred simultaneously. As long as this requirement is not satisfied, I allow myself to be deceived …, when I imagine that I am able to attach a meaning to the statement of simultaneity.”

Right, indeed. We can’t just say: we both checked our iPhones and it said the same time, so it was simultaneous. Who gave you those iPhones anyway? Santa Claus again. We need a do-it-yourself option. And Einstein has one for you. Here is what he says:

“[We] offer the following suggestion with which to test simultaneity [of two events, one occurring at point A and one at point B]. … The connecting line AB should be measured up and an observer placed at the mid-point M of the distance AB. This observer should be supplied with an arrangement ([such as] two mirrors …) which allows him visually to observe both places A and B at the same time. If the observer perceives the two flashes of lightning at the same time, then they are simultaneous.”

So there you have an operational definition of what it means for two things to happen at the same time. It is interesting that this operationalisation of simultaneity involved finding the midpoint of a line segment. Einstein didn’t need to explain that any further since that part was itself operationalised by Euclid, if you recall. Proposition 10 of the Elements: to find the midpoint of a given line segment.

So Einstein’s definition of simultaneity, of what it means for two things to happen at the same time, is very much in the spirit of the operationalist tradition, and the kind of physics advocated by Descartes and Leibniz. Fundamental physical notions need to be formulated in terms that show how they are knowable. Such as simultaneity being knowable or accessible to observation by defining it in terms of looking at two events using mirrors and seeing if the two events coincide or not observationally.

If it’s just about tennis, then this doesn’t really matter. That’s indeed why this way of doing physics didn’t really go anywhere for two hundred years. People just used Newton’s Santa Claus time so they didn’t have to worry about any of that stuff with the mirrors and so on.

The philosophical subtleties about what simultaneous means only become relevant at speeds comparable to the speed of light. The speed of light does not behave additively. Unlike tennis balls. If I’m standing on a moving ship and fire a tennis ball, then I add speed to the ball in addition to the speed that it already had from moving along with the ship. But if I turn on a flashlight it will go at exactly the speed of light, a natural constant, regardless of however I was traveling. It’s not the speed at which it was already going plus the new speed. It’s always the same speed of light, regardless of whether it is going with or against whatever motion of some ship or whatever.

That light behaves this way seen experimentally, not long before Einstein’s theory. The constancy of the speed of light is also embedded in the theory of electromagnetism. Maxwell’s equations of electromagnetism from the mid-19th century were hugely successful and remain a cornerstone of physics today. As Maxwell said, “light is an electromagnetic disturbance, propagated in the same medium through which other electromagnetic actions are transmitted” (Treatise on Electricity and Magnetism, 786). So light is the same kind of thing as the WiFi signal for your phone and so on.

And that was a discovery and not an assumption. As Maxwell said: “I made out the equations in the country, before I had any suspicion of the nearness between the two values of the velocity of propagation of magnetic effects and that of light.” He did that “in the country”, that is to say, at the rural family estate in the Scottish countryside. So working in isolation, in other words, and only later being able to test the theory against lab data and such things. So the fact that light can be absorbed into the theory of electromagnetism was not an assumption built in to the theory but rather something that was independently confirmed later, when Maxwell returned from “the country.”

For our purposes the point is that the constancy of the speed of light was not just an isolated experimental fact: it was a result intertwined with core physical theory already before Einstein. Indeed the title of Einstein’s famous 1905 paper on special relativity is “On the Electrodynamics of Moving Bodies”, for precisely this reason.

So it’s not so strange to give the constancy of the speed of light a big role in operationalising time and space. This axiom that the speed of light is constant is indeed built into Einstein’s definition of simultaneity. It takes light equal time to travel equal distances. If the speed of a light ray depended on the speed of the object it was coming from, then Einstein’s definition of simultaneity wouldn’t make much sense. But because of the constancy of the speed of light we can count on equal distances in equal times.

Ok, but throwing this new ingredient into the mix seems to ruin everything we said before with the tennis stuff. We already saw that the principle of relativity—that we can’t tell in the tennis room whether we are moving or not—goes hand in hand with the principle of additivity of speeds, because it was the additivity principle that made the calculations come out equivalently for different observers. So, if light speed does not behave additively, that should mean that relativity will fall as well. We should be able to exploit the constancy of the speed of light to detect the motion of the ship from inside the tennis room. Precisely what was impossible using tennis balls should now be possible using light.

Namely: You and I stand at opposite ends of the tennis court and we each have a flashlight. We turn on our flashlights at the same time, and we see which light ray reaches the net in the middle of the court first. Since the speed of light is a universal constant regardless of whether it was fired with or against the ship’s motion, the two light rays will take a different amount of time to get to the net, since the net will will have travelled some millimeters along with the ship while the light rays are in the air.

So the principle of relativity is false: it is detectable by physical experiment from within a closed room whether the ship is moving or not. Right?

No! This is precisely an example of how the naive assumptions of Newtonian physics leads to errors and inconsistencies. We turn on our flashlights “at the same time,” I said. Here I was thinking like a Newtonian. The “same time,” according to the absolute time given to us by Santa Claus. But there is no such thing. You can’t just say: let these things be done at the same time.

It is precisely by trying to operationalise the concept of time that we see how naive this Newtonian Santa Claus perspective is. From the perspective of Newtonian absolute time, either the light rays were fired at the same time or not, and either they reach the net at the same time or not. Because of absolute time, those are straightforward raw facts as it were. And they are two independent facts: fired at the same time; reaches the goal at the same time. Two separately things. Hence it makes sense to test experimentally whether those two things are coordinated or not.

But when we operationalise we see how naive we have been. As we saw above, when we followed Einstein, the only way we could define the concept of two things happening “at the same time” operationally was to say: two events happened at the same time if the light signals from them coincide when they reach the midpoint between the two locations. So we cannot independently check whether two light rays fired at the same time reach the midpoint at the same time or not. If they reach reach the midpoint at the same time, then they were fired at the same time, by definition. If they don’t reach the midpoint at the same time, they were not fired at the same time, by definition. With the humble and honest operational notion of time, that’s all we can say.

So the principle of relativity remains valid after all. There is no experiment we can do in the tennis room that shows whether the ship is moving or not. As we see when we think operationally. Despite the fact that light speed does not behave additively, which we thought was equivalent to the principle of relativity before, when we were thinking classically, in terms of Newtonian absolute time.

So how does that work in terms of the outside observer then? The guy on the shore. When we compared these two classically it was the additivity of speeds that made everything work: the speed of the ship plus the speed of the tennis serve. But now we don’t have additivity anymore since we’re dealing with light. So what happens then?

From the perspective inside the room, we fired two light rays that hit the midpoint at the same time. We concluded that they were fired simultaneously. By definition: that was the only way we were able to define the concept of simultaneity operationally.

Now, the guy outside the ship, looking at the same thing, he’s going to come to a different conclusion. Because he will have a different opinion about what the midpoint is between the two firing positions. To us on the ship, the midpoint was the net in the middle of the tennis court. But the guy on the shore thinks the net is moving, so he doesn’t think it makes any sense to use that as a reference point. Instead he will put his finger at a point that is stationary with respect to the shore. This point will initially coincide with the position of the net but as the ship moves the net will move away from this fixed point.

Because of this, events that are simultaneous as seen from within the ship are not simultaneous as seen from outside the ship. With the net on the court as the reference midpoint, the light rays arrived at the same time, and hence were by definition fired simultaneously. That was the point of view of us on the ship. But if the light rays reach that point at exactly the same moment, then they cannot also reach the different reference point selected by the outside observer at the same moment also. So they must reach that guy’s reference point at different moments, and hence they must by definition have been fired at different times, not simultaneously, according to the outside observer’s point of view.

So, when time is defined operationally, time is different to different observers. There is no God-given absolute time that tells you whether two events really happened at the same time or not. One observer thinks one thing, another observer something else. And there is no way to decide who is “right.”

If you are reading along in Einstein’s book that I mentioned, what I have just described is his section “IX. The Relativity of Simultaneity.” After that comes section “X. On the Relativity of the Conception of Distance.” If two observers disagree about time they also disagree about distance, as we shall see. In order to be able to tackle such questions we first need to define distance operationally. Let’s see how Einstein does this.

In order to do this we have to switch our imagery. Instead of people playing tennis on a ship we are going to go with lightning hitting a train, which is Einstein’s example. The thing with people playing tennis inside a ship is a good picture in some respects. Better than the train. The idea that the motion of the ship is undetectable inside the room is quite intuitive in the case of a very large ship going at a perfectly steady speed. More so than the same thing for a train, I think. Also it was nice that a tennis court has its midpoint conveniently landmarked with a net. That helped us as well to make the prominent role of the midpoint in Einstein’s definition of simultaneity more vivid in our minds, I think.

But actually this image started to break down a bit when we tried to explain the simultaneity experiment from the point of view of the guy on the shore. He was supposed to select a midpoint between the two tennis players that was stationary with respect to the shore. Which is not very practical at all. How would you hold such a point fixed, and keep your mirrors there? And besides, how does the guy on the shore determine the midpoint anyway? I mentioned that you can find the midpoint in the manner of Euclid, for instance. Fine, that works for the observer on the ship. But not the guy on the shore. He has to measure two moving targets at the same time, and somehow fix their midpoint in space at some instant, and he doesn’t even know yet what the same instant for each point even means because that is what the experiment with the mirrors placed at the midpoint is supposed to reveal.

So the ship thing doesn’t really work from a practical point of view. But we can fix these problems by switching to the train scenario instead. Alright, a train is travelling at a constant speed. It is struck by lightning at two places. Did the two lightning strikes occur at the same time or not? Well, we know how to check that. They occurred at the same time if the light from both of them reaches the midpoint at the same time. Of course, in order to check this you would have had to have your double mirrors already set up at the midpoint to begin with, as the lightnings struck. So you had to know in advance where the lightning was going to strike. But maybe that’s doable. Let’s say that the train has two lighting rods, so lightning will only strike where the lighting rods are.

In fact, we don’t really need a train as such. We can take away the walls and the roof of the train. The train is just a moving floor and it has one lightning rod at either end, and it has a pair of mirrors set up exactly halfway between them where we can see both lightning rods at the same time and judge whether the light signals from both coincide in time or not.

It doesn’t have to be lightning either. It can be a man-made thing like a light bulb, or an explosion going off. But lightning is a pretty good image in some ways. Because it conveys the idea that it is the light from the event that is the important thing, and it also conveys that the event happens at one particular instance. It is not an ongoing thing, like a light bulb staying on. It’s bam, split second, done.

And here’s another great thing about lightning that will help us a lot. It leaves a mark, a kind of burn mark, an imprint, where it struck. We don’t need that on the train, because we already have the lightning rods marking those positions. But what about the positions of the lighting strikes as seen from a stationary outsider? This gave us quite a bit of trouble with our ship example. The guy on the shore was supposed to reference things to a stationary reference frame from his point of view, but it was hard to make that concrete in that case: how do you freeze certain positions on a moving tennis court in space? You can’t. So we could only imagine that theoretically.

Now with the lightning marks this is going to work better. The train is passing a stationary train platform. A train ”station” indeed: it is stationary. When lightning strikes, it leaves a burn mark with a certain impact radius, so it also burns the platform right next to the lightning rod at that instant. Perfect! Now we have the positions where the events took place burn-marked into the stationary platform itself. So now there is no problem anymore to talk about the midpoint of the two events with respect to the stationary frame of reference. It is just the midpoint between the two burn marks. We don’t have to chase any moving targets to determine that.

So, as we said, the two observers will disagree about whether the lightning strikes were simultaneous or not. We can now quote Einstein’s explanation of this, which is the same as what I already said about the ship, but expressed in terms of the train and lightning.

“When we say that the lightning strikes A and B are simultaneous with respect to the platform, we mean: the rays of light emitted at the places A and B, where the lightning occurs, meet each other at the mid-point M of the length AB of the platform. But the events A and B also correspond to positions A and B on the train. Let M’ be the mid-point of the distance AB on the travelling train. When the flashes of lightning occur, this point M’ naturally coincides with the point M but it moves … with the velocity v of the train. If an observer sitting in the position M’ in the train did not possess this velocity, then he would remain permanently at M, and the light rays emitted by the flashes of lightning A and B would reach him simultaneously, that is, they would meet just where he is situated. Now in reality (considered with reference to the railway platform) [this observer on the train] is hastening towards the beam of light coming from B, whilst he is riding on ahead of the beam of light coming from A. Hence the observer will see the beam of light emitted from B earlier than he will see that emitted from A. Observers who take the railway train as their reference-body must therefore come to the conclusion that the lightning flash B took place earlier than the lightning flash A. We thus arrive at the important result: Events which are simultaneous with reference to the platform are not simultaneous with respect to the train, and vice versa (relativity of simultaneity). Every reference-body (co-ordinate system) has its own particular time; unless we are told the reference-body to which the statement of time refers, there is no meaning in a statement of the time of an event.”

Right, so that’s the relativity of simultaneity, or relativity of time. Relativity is a consequence of operationalisation. And the relativity of time is in turn going to imply the relativity of space, or relativity of length, as we said. Let’s now look at that.

Let’s quote Einstein’s words here about how to compare distances across the two different reference frames. First Einstein gives a very operational definition of how distance is defined within a given reference frame, for example within the train.

“Let us consider two particular points on the train travelling along the platform with the velocity v, and inquire as to their distance apart. … An observer in the train measures the interval by marking off a measuring-rod in a straight line (… along the floor …) as many times as is necessary to take him from the one marked point to the other. Then the number which tells us how often the rod has to be laid down is the required distance.”

Great to see Einstein crawling around on all fours, laying down measuring rods. Remember Euclid in Rafael’s fresco of the School of Athens? Hunched-over with his geometry tools. We have gone full circle with Einstein. Not only does he have the same hairstyle as Rafael’s Euclid, he also has the same philosophy. If you want to understand the geometry of space you have to get operationalising, tools to the ground.

So that’s how you define distance, or operationalise distance. How many sticks long is it? How many times do I have to put a stick down to cover the whole thing, to get from one end to the other? That number is the length. And half of that number is the midpoint.

Now it gets trickier if the guy on the platform wants to measure this length. How are you supposed to do this thing with the sticks if everything is moving? One stick, two sticks, … Oh crap, the train is already long gone. That didn’t work.

The solution is to transfer out an imprint of the train, a kind of freeze-frame version of it imprinted on the platform, which we can then measure in peace and quiet after the train has left.

Kind of like the lightning marks, except the marks have to be made at the same time, and that’s precisely where it gets complicated. The same time according to whom? Were the lightning strikes simultaneous or not? Different observers disagreed about that. So when we say “make two marks on the platform corresponding to the endpoints of the train” we have to be careful about how we time the two marking events so that they are simultaneous.

Here is how Einstein puts it:

“The following method suggests itself. If we call A’ and B’ the two points on the train whose distance apart is required, then both of these points are moving with the velocity v along the platform. In the first place we require to determine the points A and B of the platform which are just being passed by the two points A’ and B’ at a particular time t, judged from the platform. These points A and B of the platform can be determined by applying the definition of time given [above]. The distance between these points A and B is then measured by repeated application of the measuring-rod along the platform. It is by no means certain that this last measurement will supply us with the same result as the [measurement performed by the observer on the train]. Thus the length of the train as measured from the platform may be different from that obtained by measuring in the train itself.”

We can picture it like this. At each end of the train there’s a guy with a paint brush. As the train passes the platform, the two guys each draw a mark on the platform at the same time. Then the length between the marks is the length of the train, surely.

But now we used the idea of “at the same time” again. So if the painters on the train draw the marks “at the same time” according to themselves, then the observer standing on the platform will go: “Noo! What are you doing?! You mistimed it! The guy at the front of the train drew his mark later than the guy at the back of the train.”

This is the direction in which the two observers disagree about simultaneity, as we saw already in the lightning examples. If the light rays from the two events to reach the midpoint of the train at the same time, then, at that moment, the guy standing at the midpoint between the two marks on the platform will already have seen the signal from the event at the back of the train, but not yet the signal from the front of the train. So simultaneous according to “train time” means that the event at the front of the train was delayed, according to “platform time.”

And the faster the train is going, the greater the delay will be. So the distance between the two painted marks will be greater for a faster train. I mean the distance between the marks drawn simultaneously according to train time. So in other words, the length of the train takes up a bigger portion of the platform. If the train covered 50% the platform when it was at rest, it will cover 80% of the platform when swishing by at a large constant speed. According to the observer on the train, that is, because we made the marks according to his simultaneity.

So “the train has grown,” you might say. Or rather, the platform has shrunk, of course. That’s how the guy in the train will interpret it. In his view, the train is the thing that remains fixed. To him the train of course has the same length all the time. For example, it takes him the same amount of effort to walk from one end of the train to the other. And it is as many sticks long, the same stick he used to measure it when the train was parked.

So when he’s swishing by the platform and makes his “simultaneous” paint marks corresponding to the length of the train, and the marks take up a greater part of the platform than before, he will say: “Huh, I guess the platform has shrunk since last time I was here.”

That is the famous length contraction phenomenon. Things that move shrink. Or rather: things that move relative to an observer appear to shrink according to that observer. The train guy thinks the platform shrunk, as we saw.

And the platform guy thinks the train shrunk, which we didn’t see directly the way we described the experiment but it must be that way because the roles of the platform and the train are interchangeable. The guy on the train can say: “I’m not moving. You’re moving!” And no physical experiment can prove him wrong. That is an axiom of relativity theory. So therefore if one guy thinks the other one shrunk, then the second one must think the first one shrunk. Because there can’t be any asymmetry, as long as they are moving at constant relative speed.

But ok, enough physics. We’re not going to become a physics podcast. My point was not to do an intro lecture on special relativity. My point was to emphasise how operational that theory is.

Time is something you do. Just as a triangle is something you do. I put a ruler on a piece of paper, I draw three intersecting lines, bam bam bam, that’s a triangle. “Oh no, that can’t be a triangle because it’s not perfect, actually. A perfect triangle must be a Platonic blah blah blah.” No. You haven’t learned the lesson of relativity theory.

Einstein didn’t use operational definitions of time and distance because he wanted his theory to be applicable in practice. He defined everything in terms of doing because the more theoretical alternative was philosophically naive and untenable.

For the same reason the Greeks reasoned in terms of constructions. Not because they were practically oriented and childlike and pre-theoretical, but precisely because they were more theoretically sophisticated, not less. Just as Einstein’s mirrors and sticks are more foundationally sophisticated than the precious abstract theory of Newton.

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” theory.

Transcript

A new book just appeared: A New History of Greek Mathematics, by Stanford Professor Reviel Netz, Cambridge University Press. Let’s do a book review.

It will be a critical review. The main theme will be the sciences versus the humanities. Note the title of the book: “a New History.” Netz’s “New History” represents the new humanities-centred dominance in the field. As opposed to the “old” histories written by more mathematically oriented people. In my opinion, “new” does not mean better in this case. And I will tell you why.

Let’s start by attacking a city. The enemy are hunkering down behind their city walls. We are going to have to scale the walls with ladders. How long should we make the ladders? The ancient historian Polybius has the answer:

“The method of discovering right length for ladders is as follows. … If the height of the wall be, let us say, ten of a given measure, the length of the ladders must be a good twelve. The distance from the wall at which the ladder is planted must, in order to suit the convenience of those mounting, be half the length of the ladder, for if they are placed farther off they are apt to break when crowded and if set up nearer to the perpendicular are very insecure for the scalers. … So here again it is evident that those who aim at success in military plans and surprises of towns must have studied geometry.”

Great stuff. But Netz gets it wrong, in my opinion. Here is how he concludes:

“And then, of course, we are supposed to apply – Polybius leaves this implicit – Pythagoras’s theorem.” (223)

I don’t think so. I don’t think that’s what Polybius intended.

Sure enough, you can solve for the length of the ladder using the Pythagorean Theorem, but that is a clumsy and inefficient way to do it. If you did this the modern way you would need to do some algebra followed by some calculation involving a square root. They didn’t have calculators on their phones back then, you know. Do you expect carpenters in the military to be able to calculate square roots by hand?

In fact, Polybius has already told you everything you need to know with his numerical example. If the wall is 10, the ladder should be 12, he says. But it scales! So what Polybius is really saying is that, whatever the height of the wall is, the ladder is always 20% longer than that. That’s all you need to know. No Pythagorean Theorem needed.

Those numbers are a rule of thumb. You can also do it more exactly if you want, according to Polybius’s more theoretical characterisation of the optimal length. But you don’t need the Pythagorean Theorem for that either. There’s a much better way, that you can easily teach to an illiterate carpenter in five minutes.

Draw an equilateral triangle, just as Euclid does in Proposition 1 of the Elements. Cut it down the middle. Now you have a right-angled triangle, where the base is exactly half of the hypothenuse. This corresponds precisely to Polybius’s rule: the distance along the ground is half the length of the ladder.

So now we have a scale model of what we want. The height down the middle of the equilateral triangle represents the city wall; the side of the equilateral triangle represents the ladder, and it is precisely half its own length from the foot of the wall, exactly as Polybius says it should be for optimal stability.

So if we are given that the height of the wall is for example 10 meters, then we divide the height of the triangle into ten equal parts. We take a blank ruler and mark those ten marks on it. Then we take this ruler, with this length unit, and measure the hypothenuse of the triangle. However many marks long it is, that’s how many meters our ladder needs to be.

Piece of cake. Easy to improvise in the field without any specialised knowledge or tools. While Netz is busy trying to teach his carpenters the algebra of quadratic expressions and how to extract square roots, I have already scaled his walls using my much quicker methods. That is what you get when you put humanities people in charge of mathematics.

So I wouldn’t trust Netz when it comes to mathematics, even when he says “of course,” as he does here.

Here is another example: Did you know that parabolas are pointier than hyperbolas? At least if we are to believe Professor Netz. This claim occurs in a discussion of Archimedes. Archimedes studied solids of revolution obtained by rotating a conic section around its axis. Here are Netz’s words:

“In the case of a parabola, this will be of a more pointed shape; in the case of the hyperbola, this may be more bowl-like.” (140)

This is BS. Parabolas are not “more pointed” than hyperbolas.

This is clear for example from the following fact: you can draw a hyperbola having any two given lines as asymptotes and passing through any given point. So in other words, you can draw a V, an arbitrarily pointy letter V, and then pick an arbitrary point inside that V, for instance a point super close to the vertex of the V. Then there is always a hyperbola that fits inside the V and that passes through the designated point. You can hardly get any pointier than that, now can you? Yet parabolas are nevertheless “more pointed”, somehow, Netz apparently believes.

By the way, this fact I just mentioned, about constructing a hyperbola within a given V (that is to say, with given asymptotes), that is Proposition 4 of Book II of the Conics of Apollonius.

Or is it? Here we have another interesting point. It seems that this proposition was actually not in the original version of the Conics. Because Eutocius, in late antiquity, needs this theorem at a certain point and he says he better prove it since it’s not in the Conics of Apollonius. But then in the text we have of the Conics, what we call Apollonius’s Conics today, this proposition clearly is there, with the exact same proof.

And in fact the standard text that we call Apollonius’s Conics today comes to us only through that very same author, Eutocius, who wrote a commentary on the Conics and also preserved the text at the same time. So it seems that Eutocius inserted this proposition into Apollonius’s original text, because he had noticed in other works that it was a useful thing to prove.

Netz describes this correctly, which is all the more reason why he should know that a hyperbola can be as pointy as you’d like, since this follows immediately from this proposition that he discusses at length.

But anyway, there is another kind of error here in Netz’s discussion of this. The point that this proposition of the Conics is an insertion by Eutocius — that insight, says Netz, is due to Wilbur Knorr, Netz’s predecessor as a classics professor at Stanford.

“No one noticed that prior to Knorr” (431-432), says Netz.

But that is not true. Wilbur Knorr was not the first to discover this. In fact, Knorr clearly says so in his own article, the very article cited by Netz, which Netz has evidently not read very carefully. Already in the 16th century, Commandino, in his Latin edition of the Conics, very clearly and explicitly made the exact same point as Knorr, using the exact same evidence and arguments. And this in turn was cited in a 19th-century German edition of the Conics, as Knorr himself says. So Knorr didn’t discovery anything except what people had already known for hundreds of years.

This is not such an innocent mistake. How are we supped to trust anything Netz says if he makes blatantly false statements that are clearly and unequivocally seen to be factually incorrect by simply glancing at the very article that Netz himself cites in support of his own claims?

But it’s even more problematic than that. Because it’s clearly not just a random mistake. It is an ideologically driven error. By saying that Stanford humanities professor Wilbur Knorr was the first to make this important scholarly discovery, Netz is obviously indirectly boosting the impression that his own claims are important and novel, since he too is a Stanford humanities professor.

Netz is not only saying that Wilbur Knorr was the first to discover this particular thing. He is implicitly saying that earlier generations of scholars missed important insights, and that only people like him — Stanford humanities professors — are true experts.

That is of course the point of the title of the book: A *New* History of Greek Mathematics. In the past everybody did it wrong, and we need people like Netz to finally do it right. There is indeed a lot of explicit posturing to this effect throughout the book.

Let’s look at another example of this. Let me read a passage where Netz is attacking Thomas Kuhn’s account of the history of astronomy. Thomas Kuhn wrote in the mid-20th century and he worked on the history of science even though his PhD was in physics. So that is exactly the kind of people Netz wants to denigrate. He wants to say that only specialised humanities professors, with their “new” histories, are actual experts in the field.

Here is what Netz says about Kuhn: “Like most nonspecialists, Kuhn supposed …” See? I told you. It’s not just that Kuhn was wrong. It is that Kuhn epitomises the kind of people (people with a PhD in physics, for example) who need to be eliminated from the field because they make so many hopelessly naive assumptions without even realising it. Anyway, let’s continue with the quote:

“Like most nonspecialists, Kuhn supposed that Aristotle was broadly canonical from the beginning and that although the ancients offered various astronomical variations, these had all to agree with the Aristotelian framework. … This is wrong. In fact, Aristotle was not canonized throughout most of antiquity; Greek philosophers were in continuous, ever-shifting debate; the very practices of astronomy went through several stages in antiquity before they became stabilized through the ultimate canonization of Ptolemy – and of Aristotle – in Late Antiquity.” (487)

Indeed, I agree with Netz that mathematicians and scientists would have ignored Aristotle. Netz says it very well:

“In the second century BCE itself, Aristotle was marginal even within philosophy, let alone to a scientist such as Hipparchus. It is quite likely that Hipparchus never even read Aristotle’s Physics.” (346) Reassessing ancient science in this light, “we come close to imagining a very Galilean Hipparchus” (347).

Yes, perfect, I agree. That is exactly what I have said before about ancient science as well. Go Team Netz on that one.

But what about poor Kuhn whom Netz uses as a punching bag? Was he really so stupid? No.

I went to my copy of Kuhn’s book on the Copernican Revolution to check Netz’s accusations, and here is what I found. Here is a quote from Kuhn’s book:

“The great Greek philosopher and scientist, Aristotle, whose immensely influential opinions *later* provided the starting point for most medieval and much Renaissance cosmological thought.” (Kuhn, Copernican Revolution, 78)

So Kuhn says exactly the opposite of what Netz accuses him of “supposing”. *Later* Aristotle provided the starting point of scientific thought. Not “from the beginning.” Later. Exactly as Netz himself argues.

Here is another quote from Kuhn’s book that says the same thing:

“Aristotle said a great many things which later philosophers and scientists did not have the least difficulty in rejecting. In the ancient world there were other schools of scientific and cos­mological thought, apparently little influenced by Aristotelian opinion. Even in the late centuries of the Middle Ages, when Aristotle did become the dominant authority on scientific matters, learned men did not hesitate to make drastic changes in many isolated portions of his doctrine.” (Kuhn, Copernican Revolution, 83)

There is no way you can read this and say that “Kuhn supposed that Aristotle was broadly canonical from the beginning,” that is to say, from his own lifetime onwards. Kuhn clearly says the opposite.

Netz’s accusation is just slander. So it’s the same in both the Knorr case and the Kuhn case: Netz makes false assertions and then cites sources that clearly and explicitly say the exact opposite of what Netz alleges.

At least in these cases Netz bothered to provide references at all. More often he doesn’t even do that. He allows himself the licence to make assertions at will, which readers are supposed to accept on his authority alone. Consider for example the following rant about the alleged bias of some unnamed “past scholarship”:

“In past scholarship, this Babylonian achievement [in astronomy] was sometimes dismissed as ‘merely’ practical, the Babylonians unfavorably compared with the Greeks in that they did not produce a geometrical account of the sky, hence no physical model, so, unlike the Greeks, ‘not real science’. This is obviously an absurd special pleading, where one defines as scientific whatever it is that the Greeks do and then reprimands the non-Greeks for failing to be Greek. The Babylonian theory is in fact directly analogous to the Greek mathematical theory of music – whose scientific significance no one doubts.” (326)

Well, no wonder that we need a “new” history of Greek mathematics then, amirite? That darn “past scholarship,” you know, they couldn’t think straight back then because they were so biased in favour of the Greeks. Or why sugarcoat it, why not just come out and say it: They were all racist back then, weren’t they? Thank God we have proper humanities-trained experts like Netz at last to save us from all of that. “A New History of Greek Mathematics”. Basically code for: The First non-Racist History of Greek Mathematics.

Well, yes, the argument that Netz refutes is indeed idiotic. But what is this so-called “past scholarship” that allegedly made this idiotic and basically racist assertion that Babylonian astronomy is “not real science” because it’s not geometrical? Who ever said that? No one I ever heard of.

Maybe Thomas Heath? If Netz is the “new” history of Greek mathematics, then Heath’s famous book is obviously the old one, written more than a hundred years ago.

But no. I looked it up. Even old Heath explicitly uses the phrase “Babylonian science” with approval (History I, 8). Of course it was “science”. Perhaps Thales, in his travels, learned of “Babylonian science”, for example, Heath says (Aristarchus of Samos, 18), in exactly those words.

So who, then, is Netz arguing against, except straw men that he has made up to present himself as the anti-racist saviour? I don’t know.

But enough bickering about that. Let’s turn to a big issue of major interpretative importance.

According to Netz, “Thales and Pythagoras did no mathematics whatsoever” (17). According to Netz, earlier generations of scholars naively believed in such fairy tales because they blindly trusted a single source:

“My predecessor Heath and many historians – up until the last generation – gave credence to the view according to which Thales, and then Pythagoras, made lasting contributions to mathematics. This derives almost entirely from Proclus’s commentary, which, because of its overall sobriety, was taken seriously even for such obviously unfounded assertions.” (423)

First of all, it is not true that this “derives almost entirely from Proclus’s commentary.” It is disturbing that Netz makes this false and self-serving statement. Just read Heath, whom Netz names in this very rant. Read Heath’s chapter on Thales. Heath goes through the sources explicitly. There are several sources about Thales as a mathematician that predate Proclus. And several of those testimonies, as well as passages in Proclus, are explicitly attributed to various specific earlier authors.

So it is not the case that earlier generations of scholars uncritically and blindly relied “almost entirely” on a single biased source, as Netz dishonestly and falsely claims.

Let’s look at Thales and Pythagoras in turn. Let’s start with Thales.

I have spoken before about how the idea of Thales as the originator of formal geometry makes good sense. The way I told it was based on two theorems attributed to Thales.

The first theorem is that a diameter cuts a circle in half. I described how one can show that using a very neat proof by contradiction. The appeal, obviously, would not have been the theorem as such, but the realisation that that kind of thing can be established by a very elegant and satisfying type of reasoning, namely a rigorous argument based on paying careful attention to the definitions of concepts such as circle and diameter, and the remarkable power of proofs by contradiction for proving this kind of thing. That is exactly the same aesthetic that one finds on the first pages of almost any modern mathematics textbook in abstract algebra, for example: proofs of basic results driven by carefully formulated definitions and tidy proofs by contradiction. It makes sense that people would fall in love with this aesthetic that has stood the test of time, and it makes sense that it would have begun with a basic theorem such as that the diameter bisects a circle. Just as ancient sources suggest.

A second theorem attributed to Thales is that a triangle inscribed in a circle with the diameter as one of its sides must be a right triangle. It is natural to arrive at this insight by playing around with ruler and compass. And the aha-moment would then have been that one can prove such things. Make a rectangle, draw a diagonal, draw the circumscribing circle. Now you are in business. From playing with shapes, you have arrived at a proof of a universal truth. Pretty cool. It makes sense that the idea of proving geometrical theorems might have started with something like that, as some ancient sources suggest.

I told my own version of this story, but in broad outline something like that is a pretty standard and well-known point of view. But Netz acts as if he has never heard of any of that. He pretends that people who believe that Thales initiated geometry are simply blindly taking Proclus’s word for it without having thought it through at all. Netz says so explicitly. Here are his words:

“I suggest here that Hippocrates’s works were among the earliest pieces of Greek mathematics ever to be written.” (48)

Ok, so that’s Hippocrates, considerably later than Thales, famous for a very technical and detailed argument about the areas of lunes, a kind of shape composed from circles. This looks a lot more like a specialised piece of technical geometry from a quite mature geometrical tradition. It seems like a very odd and obscure place to start with geometry altogether. In reply to this, Netz says:

“This might seem surprising. Could mathematics emerge like that – springing forth from Zeus’s head? Would we not expect mathematics to emerge in a more rudimentary form? In fact, I think this is precisely how we should expect mathematics to emerge: from Zeus’s head, fully armed. What would be the alternative? … Of course the very first mathematical works in circulation would contain remarkable, surprising results. Why else would you even bother to circulate them? I suspect that the counterfactual is sometimes not sufficiently carefully thought through here. Just what would a more rudimentary piece of mathematics look like? Would it prove some truly elementary results, such as, say, the equality of the angles at the base of an isosceles triangle? Why would anyone care about such a treatise, proving such a result?” (48)

It is baffling that Netz allows himself to make this lazy argument, as if no one had ever though those things through. He states these rhetorical questions as if no one had ever thought of any of that. But of course people have thought about that and they have compelling answers to Netz’s questions.

I just told you what the alternative to Netz’s narrative is and why it would make sense. And I am not the first person to say this. But Netz is too lazy to engage with alternative views seriously, so instead he dishonestly says that no one has ever thought through any alternative to his view.

So that’s Thales. Netz rejects a plausible interpretation of the Thales testimonies in ancient sources by dishonestly mischaracterising as hopelessly naive any scholars who adhere to such views.

Now Pythagoras. “Heath … had three full chapters on the mathematics of Thales and Pythagoras!” (22), Netz says triumphantly, suggesting that this is proof that his “new” history of Greek mathematics is sorely needed.

Anyone who believes in Pythagorean mathematics is stupid, according to Netz, and for this he relies on a famous book by Burkert. Here is how Netz describes it:

“[Burkert’s book] Lore and Science in Early Pythagoreanism … was a more careful, professionalized classical philology, keen to understand the authors we read not as mere parrots, repeating their sources, but instead as thoughtful agents who shape and retell the evidence as suits their agenda. Pythagoras, under such a reading, crumbles to the ground: almost everything … comes to be seen as the making of later authors from Aristotle on. Never mind: the historians of mathematics went on as before.” (23)

We hear the ideological overtones here. Burkert is Netz’s kind people: he is hailed as “professionalized.” By contrast, “the historians of mathematics went on as before”. That is to say, the mathematically trained people working on history of mathematics were a bunch of fools who didn’t even realise what fools they were, and we would be much better off if “professionalized” experts such as Burkert and, presumably, Netz himself, would be given a monopoly on expertise status in the field.

I do not agree with this, neither in terms of content nor ideology.

Regarding Pythagorean mathematics, since Netz doesn’t go into any more depth, I will now analyse Burkert’s book itself, which Netz accepts as gospel truth. A book review within a book review! Here we go.

According to Burkert, “the apparently ancient reports of the importance of Pythagoras and his pupils in laying the foundations of mathematics crumble on touch”. Not that phrase: “the foundations of mathematics.” I am going to criticise Burkert, and I am going to say that Burkert makes a naive and anachronistic assumption about what “the foundations of mathematics” are. (For page references for the quotes from Burkert, see my Operationalism article.)

When Burkert speaks of “the foundations of mathematics,” he takes for granted the traditional view that a core pillar of Greek geometry was its Platonist detachment from the physical world. As Burkert says, “Greek geometry assumed its final form in the context of [Plato’s] Academy … after Plato had … fixed its position as a discipline of pure thought.”

Indeed, Burkert’s arguments against Pythagoras’s mathematical significance are really arguments that he did not advocate a proto-Platonist philosophy of mathematics. Burkert’s overall thesis is that “that which was later regarded as the philosophy of Pythagoras had its roots in the school of Plato.” And indeed he proves convincingly that there was a clear tendency to distort history in this way in Platonic sources that is not consistent with more reliable sources outside this tradition.

For example, Burkert shows that when Proclus mentions Pythagoras in his “catalogue of geometers,” and attributes to him “a nonmaterialistic procedure” in mathematics, this, unlike the rest of the catalogue of geometers, is not based on the highly credible Eudemus. Instead it is copied from Iamblichus, that is to say, from the biased Platonic tradition.

From this it does not follow, as Burkert tries to argue, that Eudemus did not mention Pythagoras as a geometer at all. It follows only that Eudemus in this place likely did not associate Pythagoras with proto-Platonic views. This is enough to give Proclus the motivation to supplement his account with phrases from Iamblichus, even if Eudemus had mentioned Pythagoras in the original.

Burkert also observes that “Aristotle [says] expressly of the Pythagoreans [that] ‘they apply their propositions to bodies’---bringing out the distinction, in this regard, between them and all genuine Platonists.” Eudemus and Aristotle are clearly much more credible than the much later, more biased, and less intellectually accomplished Iamblichus and Proclus.

Thus Burkert’s arguments that Pythagoras’s alleged proto-Platonist philosophy of geometry is a fabrication of biased sources are quite convincing. However, it does not follow from this that the Pythagoreans did not take a profound theoretical and foundational interest in geometry altogether.

Burkert conflates these two conclusions, because he sees no alternative path to theoretical mathematics than through Platonic-style abstraction and detachment from physical considerations. Burkert believes that early work on geometrical constructions “is still not doing mathematics for its own sake”; rather, the “discovery of pure theory” was a later “accomplishment,” in his words.

If you have followed what I have said in the past you know that I reject this. Burkert is naive to assume a dichotomy between constructions and “pure theory.” Constructions were not the opposite of theory, and hence the opposite of “the foundations of mathematics,” as Burkert erroneously assumes. On the contrary, constructivism *was* the foundations of mathematics.

Once we admit that possibility, there is every reason to think that earlier mathematicians, such as the Pythagoreans, could very well have made profound and foundationally sophisticated contributions, while at the same time rejecting Platonising tendencies in the philosophy of geometry.

Indeed, when going beyond his convincing case against Pythagoras the Platonist, to the more general case of trying to minimise the significance of Pythagoras and his followers in the history of geometry, Burkert find himself on the back foot. He is forced to try to explain away Aristotle’s compelling statement that “the so-called Pythagoreans were the first to take up mathematics; they advanced this study, and having been brought up in it they thought its principles were the principles of all things.”

Burkert’s thesis leaves him little choice but to dismiss the centrality of mathematics implied by this statement as “a psychological conjecture of Aristotle, which the historian is not obliged to accept.” That Proclus was wrong is plausible enough, but having to postulate that Aristotle was wrong comes at a considerably higher cost. And while Burkert was able to discredit Proclus’s mention of Pythagoras in the catalogue of geometers, he cannot deny that numerous attributions of mathematical discoveries to Pythagoreans made by Proclus are indeed based on Eudemus and hence credible, by Burkert’s own admission. Thus even Burkert must admit that “Pythagoreans made significant contributions to the development of Greek geometry.” Yet he hastens to add: “but the thesis of the Pythagorean foundation of Greek geometry cannot stand.”

Once again Burkert’s argument is based on tacitly assuming a monolithic conception of what “the foundations of Greek geometry” consisted in. The constructivist reading of Greek geometry problematises this assumption. It shows that one cannot simply take for granted that “the foundations of geometry” means what modern authors think it should mean. Constructivism offers an alternative vision, according to which much early Greek geometry may very well have been eminently foundational, but in a sense different from that commonly assumed by modern observers. This at the very least raises the possibility that early traditions such as that of the Pythagoreans may have been more foundationally significant than Burkert’s argument admits.

So much for Burkert, whose judgement Netz accepts unconditionally. Far from being an unequivocal triumph of “professionalized” expertise over previous naiveté, as Netz would have it, Burkert’s account is itself naive and by no means unquestionable.

So Netz is fond of dismissing what the ancient sources say. All the stories about Thales and Pythagoras, that’s just so much fiction. To be sure, the sources are highly imperfect and definitely contain a lot of misinformation. Nevertheless, it is surely better to try to save some meaning in these stories than to almost take it as a point of pride to dismiss as much of it as possible, as if the more sources you dismiss the more sophisticated a historian you are.

In fact, Netz continues in the same vein for later Greek geometry as well. “The stories [about Archimedes] probably are fabricated,” (128) we are told.

Stories such as Archimedes’s use of the principles of hydrostatics to detect a fake gold crown, because it did not have the right density properties. That is the “Eureka!” story.

“Biographers concoct anecdotes, based on the contents of the authors’ works. This is clearly the case here. The story of the crown is a clear echo of Archimedes’s study of solids immersed in liquids, On Floating Bodies.” (129)

Now, how would this work exactly? Let us “think through the counterfactual,” as Netz admonished others to do above.

Ok, so Archimedes wrote a sophisticated technical work on floating bodies. For some reason. Certainly not because of fake gold crowns and such things, because those are just “concocted anecdotes.” I guess Archimedes just woke up one day as said to himself: I think I will prove a bunch of theorems about hydrostatics, which nobody has done before, because I’m a mathematician and I just do things arbitrarily for no reason with no connection to the real world.

So he wrote a detailed, hyper-mathematical treatise on floating bodies. Theorem-proof, theorem-proof.

And then, maybe hundreds of years later or whatever, another guy told himself: Hey hey, I’m a writer! I’m going to write about the history of mathematics, but I won’t find out actual facts about the history of mathematics. Instead I’m going to pour over these extremely technical treatises that very few people can understand, and I’m going to master their content in great depth, to the point where I will be able to invent out of thin air real-world scenarios that involve realistic, sophisticated applications of the complicated technical results found in these treatises. And my goal in doing so is to concoct a one-paragraph anecdote about for example Archimedes making a discovery in the bath that made him run naked through the streets. Haha, what a funny image to imagine him running and screaming eureka like that. Totally worth all those probably hundreds of hours that I had to spend studying very complicated mathematics and then designing and working out my own research-level applied mathematics problem just so that I could make this little joke about Archimedes running from the bath.

Well, that’s apparently what happened if we are to believe Netz.

I very much doubt that story tellers were ever that good. The story about Archimedes and the crown is really very good scientifically. The connection with the technical details of Archiemdes’s treatise is the real deal. If this is a “fabricated” anecdote “concocted” by a biographer, as Netz says, then that biographer was not only a story teller but one of the leading scientists of their age.

Look, I teach calculus regularly, and I always try to get students to think about the physical meaning of mathematical notions and interpret results in the context of a real-world scenario. And I can tell you that that is an uphill battle to say the least.

I don’t think Netz teaches calculus so I think he underestimates how hard it is to make up stories that simultaneously make perfect scientific sense.

It is quite easy, on the other hand, to make up stories that do *not* make scientific sense. And that bring us to another one of Netz’s theories.

Netz has another book called “Ludic Proof”. Ludic as in play, playfulness. According to this theory, mathematicians borrowed stylistic approaches from poets. Poets had a fondness for cleverly constructing narratives that led to surprising twist reveals. Mathematicians shared the same aesthetic, according to Netz.

Netz, in all seriousness, proposes that this could be the main reason why Archimedes did calculus-style calculations of areas at all, and why he even turned to mathematical physics at all. The root cause is supposed to be not ordinary scientific or mathematical motivations, but Archimedes’s desire to do mathematics in the style of the poets: mathematics was “written, always, against the background of wider literary currents, emphasizing subtlety and surprise” (218).

According to Netz this is why Archimedes did calculus-style calculations of areas and volumes:

“Archimedes … picked up a particular technique, first offered by Eudoxus, because its subtlety … made a certain kind of surprise especially satisfying. Hence the infinitary methods.” (218)

And this is also what made Archimedes apply mathematics to physics:

“[Archimedes] saw the possibilities of applying geometry to a seemingly unrelated field – the study of centers of the weight in solids … – because there was a particular payoff of subtlety and surprise to be obtained by the bringing together of apparently irreconcilable, maximally distinct fields of study. This was rather like Callimachus’s poetry! Hence the mathematization of physics.” (218)

So there you go, calculus and mathematical physics are just side effects of mathematicians pursuing their true goal, which was to imitate the poets. That is some tin-foil-hat level of crackpottery, in my opinion.

It is one thing that Netz previously advanced his bizarre theory in a specialised monograph. Of course it must be possible for scholars to try out unconventional ideas. But to put this crazy stuff in a survey history with a straight face, as if this was objective information that any beginner in the field needs to learn, that is quite irresponsible, in my opinion. Certain chunks of this book are not an introduction to the history of Greek mathematics, but an introduction to the pet theories of Reviel Netz that no one but him believes.

Let’s look at some specific mathematical examples that are allegedly all about surprise, according to Netz.

For example, Archimedes found the area of one revolution of the Archimedean spiral. How do you think he’s going to prove this? Well, you have probably already seen how Archimedes found the area of a circle. Naturally readers of his more advanced treatise on spirals would already have read his more basic treatise on the circle.

Archimedes found the area of a circle by cutting it into wedges, as it were. Equal-angle pizza slices all the way around.

Naturally it makes a lot of sense to try the same idea for the spiral. The Archimedean spiral is like a circle but with a linearly growing radius. In polar coordinates, the radius r is proportional to the angle theta.

So when we apply the method we used for the circle to the spiral we get a bunch of equal-angle wedges that gradually get bigger and bigger. The radius grows linearly with the angle, so the radii of the wedges form an arithmetic progression. For every equal increment of the angle, the radii increase by the same amount, let’s say alpha. And the Archimedean spiral starts with radius zero, so the radii go: alpha, 2 alpha, 3 alpha, etc.

To get the area of the spiral we have to add up all the wedges. Obviously the areas scale like the square of the radii. Linear scaling of distances means square scaling of areas. So since the radii went alpha, 2 alpha, 3 alpha, the areas will be proportional to alpha^2, (2 alpha)^2, (3 alpha)^2, and so on.

So to get the area we have to add up a series of squares, the squares of numbers in an arithmetic progression. Indeed, Archimedes has a theorem that does exactly this. That is his Proposition 10.

Did you find any of this “surprising”? Hardly. It was a predictable extension of the idea used for the circle. And the trick of getting a complicated area or volume by an infinite series sum of simpler components is also a well established trick. Archimedes used the same trick for the area of a parabolic segment, for example, and Euclid used it too, for example for the volume of a tetrahedron. The sum of a geometric series was the key ingredient in those cases, and now for the spiral we need the same kind of theorem but for the squares of numbers in an arithmetic progression. Very predictable and business as usual for a Greek geometer.

But Netz doesn’t think so. According to Netz, the reader of Archimedes’s treatise is not supposed to have been able to see those things and instead they are supposed to have been baffled by the introduction of Proposition 10, that is to say, the sum of the series. They are not supposed to have been able to realise that this series is obviously the same kind of area calculation by series that had been well-known at least since Euclid, and that the particular terms of the series obviously correspond to the most natural way of cutting up the spiral area.

Here is what Netz says:

“Archimedes aims at surprise. The key point is that as proposition 10 is introduced, Archimedes makes all efforts to disguise its potential application. … The key observation – that the sectors in a circle behave as the series of squares on an arithmetical progression – is not asserted in advance. Instead, the application of proposition 10 is postponed and revealed only at the very last minute when, introduced in the middle of proposition 24, it finally makes sense of the argument. … Everything is designed for the sake of this denouement where, finally, the narrative of the treatise would make sense in a surprising turn. Ugly, misshapen proposition 10 is really about sectors in spirals: the duckling was a swan all along!” (149)

I think this is nonsense. I don’t think Archimedes’s readers would have been surprised at all by any of this.

Today we teach our mathematics students: when you read a theorem, before you look at the proof, take a few minutes to think about how you would prove it. Then when you read the proof you will understand it much better. You will know which parts are easy and obvious, because you have already thought of those yourself. And you will appreciate the difficult parts because you have realised when trying to prove it yourself that certain steps would have to involve some real work.

I bet Archimedes’s readers did the same. They get a treatise by Archimedes, a key result of which is the area of a spiral. Indeed, the treatise comes with a prefatory letter by Archimedes himself where he highlights the key results, so obviously you know where it’s heading. You don’t just start reading cold from A to Z.

And if you follow the elementary advice that we teach all our undergraduates, without which you will never get far in mathematics, to try to prove it yourself before reading the solution, then you will very quickly realise that the obvious approach is to cut the spiral area into wedges and sum the components, which will obviously lead to a series of squares of numbers in an arithmetic progression. So when you get to Archimedes’s Proposition 10 you will be far from surprised. On the contrary, you knew all along that he would have to do this sum.

Let’s look at another example of a so-called “ludic proof.” If you point a parabolic mirror at the sun, all the rays are reflected toward a single point, the focus of the parabola. Diocles proved this, and the “ludic” part is that he first proved some properties of tangents and normals of a parabola, and only then introduced a line parallel to the axis, which represent the rays of the sun. Surprise! It was about rays of the sun all along. Who could ever have guessed that saying something about the tangent first would be relevant to this! Except of course someone who has read the title of the treatise and has basic mathematical competence.

Here is how Netz describes it:

“[Diocles’s proof of the focal property of the parabola is] palpably Archimedean. The same emphasis on subtle surprise – down to the intentional delay in the construction of the parallel line, so that, throughout the argument, we do not yet see the relevance of any of it for the optics of rays of the sun.” (215)

So the “surprise” is that basic properties of the tangent of the parabola are relevant to the optics of rays of the sun. What a shocking reveal! Since the solar ray had not been drawn yet, there is no way we could have known this, according to Netz.

Once again, any mathematically competent person who looks at this problem for five seconds will realise that of course it is going to involve the tangent. The notion that mathematically competent readers would not have been able to see the relevance of theorems about tangents for the optics of rays of the sun is ridiculous. And yet that notion is the corner stone of Netz’s ludic proof interpretation of this episode.

There is another bit of nonsense here as well. Diocles talks about the tangent of a parabola, but Archimedes also talked about the tangent of a parabola. Aha! Therefore Diocles’s proof “is really a brilliant variation on an Archimedean theme” (215), in Netz’s words.

This is a way of thinking that perhaps makes sense in literary history. Poets and playwrights like to draw inspiration from earlier masterpieces and rework their themes in a new way. Netz tries to do the same thing for mathematics, but in my opinion the results are nonsensical.

What Netz is saying is like saying that if Person A gives a mathematical argument involving the derivative of a quadratic function, and then Person B gives a completely different argument that has nothing to do with the first one except that it too involves the derivative of a quadratic function, then Person B’s argument is a variation on Person A’s theme.

That’s rubbish. Of course tangents of parabolas show up regularly in mathematics. That doesn’t mean that anyone who talks about the tangent of a parabola is subtly reworking what earlier authors have done. That may be how literature works, but it is not how mathematics works.

So, in this case as in so many others, Netz’s “new history” is what you get when you look at Greek mathematics through eyes attuned to the humanities but not to mathematics. Indeed, Netz’s description of the mathematics is factually wrong as well. Archimedes and Diocles both state the tangent theorem in terms of “the intercept between tangent and ordinate” (215), according to Netz. No, that’s not right. It’s the intercept between the tangent and the axis. Not ordinate, axis.

But it is not my goal to catalogue all the mathematical errors in Netz’s book. If you take a humanities professor as your guide to mathematics then you have only yourself to blame anyway.

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 may be a precondition for any linguistic or spatial thought to be possible at all, as Chomsky said for language and Kant for geometry. Just as language learning requires singling out, from all the sounds in the environment, only the linguistic ones, so Poincaré articulated criteria for what parts of all sensory data should be regarded as pertaining to geometry.

Transcript

The discovery of non-Euclidean geometry in the early 19th century was quite a wake-up call. It showed that everybody had been a bit naive, you might say.

Here’s an analogy for this. Suppose we had all been speaking one language, let’s say English. And we were all convinced that English is the only natural language. In fact, that question didn’t even arise to us; we simply assumed that English and language is the same thing. We even had philosophers “explaining why” English is a priori necessary. These philosophers had “proved,” they thought, that without English the very notion of linguistic communication or thought is impossible.

And then we discovered that there are French speakers and Chinese speakers. Oops. Very embarrassing. English is not necessary after all. It is not innate, it is not synonymous with language itself. For thousands of years we made those embarrassing mistakes because we were not aware of the existence of other languages.

And the philosophers I spoke of, Kant is an example of that. He argued that Euclidean geometry was a necessary precondition for having spatial experience or spatial perception at all, which is like saying that English is necessary for any kind of linguistic expression.

And already long before Kant, many people had been convinced by how intuitively natural and obvious the axioms of Euclidean geometry feel. Descartes for instance and many others made a lot of this fact. Remember how important it was to Descartes that our intuitions were truths implanted by God in our minds.

Well, we all think our native language is intuitive. And we think other people’s languages are not intuitive. This feeling is so strong that we think it must be objective. When we try to learn a foreign language, it feels impossible that anyone could think that was intuitive. And yet they do.

So apparently our intuitions can deceive us. We feel that our native grammar is much more natural than everyone else’s, but that’s a delusion. It felt like an objective fact, but it turned out to be subjective.

Could it be the same with geometry? Could the alleged naturalness and intuitiveness of Euclidean geometry turn out to be just an arbitrary cultural bias, like thinking English feels more natural than French?

So, ouch, we took quite a hit there with the discovery on non-Euclidean geometry. It exposed our insularity. It showed that things we had thought we had proven to be impossible were in fact perfectly possible and every bit as viable as what we had thought was the only way to do geometry.

And yet there is hope as well. The language analogy doesn’t just expose what an embarrassing mistake we made, or how non-Euclidean geometry hit us where it hurts. The language analogy also suggests a way out; a way to rise from the ashes.

We were wrong about the specific claim that Euclidean geometry is innate, because that’s like saying that English is innate. Nevertheless we were right too, one could argue.

Language is impossible without something innate. Everybody learns their native language with incredible fluency. Every child learns it somehow, with hardly any systematic teaching; they just pick it up naturally. And they do so at a very early age, when their general intelligence is still very limited.

Meanwhile, no animal comes anywhere near achieving the same feat. Nor any adult human for that matter. I’m better than a three-year-old child at any intellectual task, except learning French. Somehow the child is super good at that.

So clearly something about language is innate. The ability of language acquisition is innate. Some kind of general principles of language are innate.

Perhaps geometry is like language in this way. We have some innate geometry. Not specific Euclidean propositions or axioms maybe, but some kind of “geometryness” nonetheless. Some sort of more structural or general principles of geometry than specific propositions, just as our innate linguistic ability doesn’t contain anything specific to any one language but instead has to do with “languageness” in general.

In linguistics, this is called “universal grammar.” Every language has it own grammar, of course, but there are some more general or structural principles of language that are the same for all human languages. This common core is the universal grammar.

Here’s an example of such a principle that belongs to universal grammar. Consider the statement “the man is tall.” It is a declaration, an assertion. You could turn it into a question: “Is the man tall?” We turned the assertion into a question by moving the “is” to the front of the sentence. That’s how you make questions from statements: start with assertions—“the man is tall”—and move the “is” to the front—“is the man tall?” So that’s a recipe for making questions.

But consider now the statement “the man who is tall is in the room.” How do you form the corresponding question? There are two “is”’s in the sentence. So the rule “to form a question, move the is to the front” is ambiguous. Which of the is’s should we move?

It could become: “is the man who is tall in the room?” That works. But if we move the other is to the front we get nonsense: “is the man who tall is in the room?” Well, that didn’t work. It didn’t make a question, it just made gibberish. Even though we followed the same rule as before: move the “is” to the front.

If language was nothing but a social construction then it would be perfectly reasonable for children trying to form questions to come up with the gibberish one. If the child simply extracts general rules from a bunch of examples, it would have been perfectly reasonable for the child to have guessed that the general rule is: move the first “is” to the front of the sentence to form a question. Which would lead to the nonsense question “is the man who tall is in the room?”

In fact, however, “children make many mistakes in language learning, but never mistakes such as [this].” Apparently, “the child is employing a ‘structure-dependent rule’” rather than the much simpler rule to put the first “is” in front. Why? “There seems to be no explanation in terms of ‘communicative efficiency’ or similar considerations. It is certainly absurd to argue that children are trained to use the structure-dependent rule, in this case. The only reasonable conclusion is that Universal Grammar contains the principle that all such rules must be structure-dependent. That is, the child’s mind contains the instruction: Construct a structure-dependent rule, ignoring all structure-independent rules.” So although each language has its own grammar, there are some general principles like that that are universal: common to all languages. Those principles are hard-wired into the mind at birth.

This stuff about there being an innate “universal grammar” is the view Chomsky, the leading 20th-century linguist. The example and explanation I just quoted are from his book Reflections on Language. There are many debates about Chomskyan linguistics, but I’m going to assume Chomsky’s point of view for the purposes of this discussion, because its parallels with geometry are very interesting.

You might say that this view of language is Kantian in a way. We saw that Kant put a lot of emphasis on the necessary preconditions for certain kinds of knowledge. Geometry, for example, is not an external, free-standing theory that we can analyze with our general intellectual capacities. Rather, the fundamental concepts of geometry are bound up with the very cognitive structure of our mind itself.

Some things are purely learned through experience and convention, such as how to play chess or how to dance the tango. But some things are not like that. For instance, they way we experience color. The mind is made to see red and blue and to not see infrared and so on. That’s a fixed, domain-specific property of how our mind works. You can neither learn nor unlearn that through general intelligence. Color experience is just one of those basic things hardwired right into the brain.

From the Chomskyan point of view, language is like that as well. Language is not merely a social construct with man-made rules like chess or tango. Nor is it explicable in terms of general intelligence only.

Chess or tango you can learn by general intelligence. That is to say, if you spend enough time looking at people playing chess or dancing, you can eventually figure out what the rules are by general rational thinking such as pattern recognition, and forming preliminary hypotheses about how you think it works and then observing some more to check if you maybe need to revise the hypothesis to take into account some other possible circumstances or cases.

Color experience is not like that. You don’t learn to experience redness by watching other people. It just is. And if you’re not born with it, then you can’t learn it by general intelligence, like you can learn chess.

Language is similar to color and not similar to chess. You don’t learn color perception by watching others and using general intelligence to figure out the patterns and rules. General intelligence is not sufficient to sustain such a thing.

Many people overestimate the potential of general-purpose intelligence. Both Kant and Chomsky agree about this. Remember the tile of Kant’s work: a critique of pure reason. “Pure reason,” or general-purpose intelligence, is not by itself capable of generating human linguistic capacity or geometric experience.

The capacities of our mind depend much more than people realize on domain-specific conceptions. It is obvious that color experience is a hardwired specific domain of our cognitive structure and isn’t merely the outcome of some pattern-recognition process of general-purpose intelligence. But it’s less obvious that geometry is like that, or that language is like that. But Kant and Chomsky maintain that they are. According to them, we underestimate the extent to which basic geometrical and linguistic conceptions are intertwined with the very nature of our mind and our cognitive capacities.

So the wrong way to think about it would be like this. The human brain is a general-purpose thinking machine. Imagine a person in a prehistoric hunter-gatherer society. This person’s general-intelligence mind might think to itself: Well, it’s great that I’m so smart. I can learn many things, like which plants are poisonous; I can figure things out like how to make fire, how to use tools and so on. But gee, wouldn’t it be handy if I could communicate my thoughts to others. Then we could organise collaborations, learn from each other’s experiences, and so on. I know, let me invent language, that will work for this.

From the Chomskyan point of view this story is wrong because it overestimates the general-purpose mind. In fact, note that I described what the pre-linguistic mind was thinking by using language. But I was talking about a hypothetical stage in history in which there was no language. Does it even make sense to imagine such a thing as thought without language? No, according to Chomsky. The very nature of thought itself cannot be separated from language like that.

The story of the hunter-gatherer inventing language is no more plausible than the story that he invented color experience by discovering that certain wavelengths of electromagnetic radiation were associated with grass, others with fruit, and so on.

Instead of thinking of the mind as starting from general-purpose intelligence and then inventing domain-specific things like color and language, we should perhaps think of it exactly the other way around. The mind is made up of the domain-specific skills. Those are the fundamental cognitive starting-points. Insofar as we have any general-purpose intelligence, that comes from piecing together the domain-specific skills. Not the other way around.

From an evolutionary point of view, the human mind perhaps evolved by adding domain-specific modules one by one: first color, then a hundred thousand years later geometry, then a hundred thousand years later language, and so on. We don’t have general-purpose intelligence. We only have the sum of our modular parts. But eventually these modules became so advanced, and combined in such fruitful and powerful ways, that we fool ourselves into thinking that we have general intelligence, “pure reason.” But at bottom our precious “pure reason” actually still depends more than we realize on domain-specific preconceptions hardwired into our cognitive capacities. That’s what Kant said about geometry and that’s what Chomsky said about language.

So in this way we can “save Kant.” The discovery of non-Euclidean geometries was a blow to Kant’s idea of the innateness of geometry. Kant associated the intuitiveness of Euclidean geometry with its innateness. But native languages are intuitive, yet they are not innate. And geometry could be the same, because just as there are many languages there are many geometries. This shows that intuitive and innate is certainly not the same thing, so it calls into question the Kantian story that the mind is constrained by pre-programmed conceptions.

We save Kant with the rebuttal that in fact language too is innate after all. Even though there are many languages that all differ in fundamental respects, nevertheless there is some universal languageness that is common to all and without which language learning would be impossible in the first place.

Same with geometry. Instead of focusing on the differences between Euclidean and non-Euclidean geometries and concluding from this that no one geometry could be a necessity of thought, we should instead focus on the more fundamental and structural preconceptions common to all geometries, without which any kind of geometry would be unthinkable at all.

Or we can put it like this. Thought presupposes language. When you think, you think in terms of words and sentences. Of course thought does not presuppose any specific language. You can think the same thing in English or German. Nevertheless thought does presuppose that you use some language. There is no “pure thought,” or hardly any, that does not involve words.

It’s funny: thought cannot exist without language, yet you can switch the entire language and still have the same thought. So there’s both dependence and independence.

Kant says basically the same thing but for geometry. You can’t have spatial perception or spatial reasoning without geometrical presuppositions. Just as you can’t think without presupposing some language, so you can’t geometrize without presupposing some geometry.

The choice of which language or which geometry you take as the basis for thought is arbitrary. As Kant says, it’s a synthetic a priori, not an analytic a priori. That is to say, it is not logically necessary that we must use Euclidean geometry as the presupposition for all our spatial experience. But it is necessary that we must make some such presupposition.

Remember, as Kant said, we don’t have direct access to objective physical reality. We only know the outside world through perception which is always necessarily interpreted. The presuppositions of that interpretation are arbitrary—in fact, it’s arbitrary in two ways one might say: one good and one bad. It’s arbitrary in a “bad” way in that it is subjective. It lacks objective justification. But it’s also arbitrary in a “good” sense, namely that it doesn’t necessarily matter all that much which interpretation we choose.

Just like language. It is arbitrary that I’m speaking English. There’s no objective or logical reason for why English is any better than any other language. But it’s also arbitrary in that it doesn’t matter. I could have said the same things in some other language. And in fact it’s only because of my choice of some arbitrary language that I am able to say anything at all.

Same with geometry. Our minds think in terms of Euclidean geometry even though that has no absolute logical justification. Yet it would be a mistake to criticize this as arbitrary subjectivity. Because it is only because I have some geometrical preconceptions at all, no matter how subjective, that I am able to reason spatially and have spatial perception and experience in the first place.

The analogy that geometry is like language is suggestive in other respects as well. Here’s one interesting question. When a child is learning their native language by picking up the speech of their parents and their environment, how does the child know which sounds are language and which sounds are other kinds of noises? It’s a pretty difficult problem, isn’t it?

Suppose you had to program a computer to detect and recognize speech. What criteria could you define by which the computer could tell if any given sound is linguistic or not? Words come in many forms: you can scream them, whisper them, sing them. Those are very different as sounds, but somehow you have to be able to tell that they are all words. And you have to be able to tell that other sounds are not linguistic, such as a doorbell, a barking dog, the sizzling of a frying pan, and so on.

You have the same problem in geometry. Among all the sensory impressions we are bombarded with every second, which ones should be regarded as geometrical, and which not? If geometry is like a language, a child must have some criteria by which to answer this. Just like the child somehow picks out linguistic sounds from the environment and lets that shape their native language, so also the child must pick out geometric features of the environment and let that shape their native geometry. This is how their intuitive geometry can become either Euclidean or non-Euclidean depending on the environment, just as their native language can become English or Russian or whatever.

So: What parts of all our sensory impressions have to do with geometry? You must know that first, before you can start thinking about whether those impressions are Euclidean or non-Euclidean.

Poincaré had a very elegant solution to this problem. Here’s his criterion for telling geometry from non-geometry. It goes like this: Among all sensory impressions, those are geometrical that you can cancel through self-motion.

Let me explain what this means by an example. I have a piece of paper. One side is white and the other side is red. I hold the paper up with the white side facing toward you. Then I rotate it so that the red side is facing you. This is a geometrical transformation: it has to do with rotation, with position. You know that it was geometrical because you could walk around and stand on the other side and then you would see the white side of the paper again. So you could cancel the transformation in impressions, you could restore the original sensory impression, through self-motion. By moving yourself. Not by manipulating the environment, but only by moving around in it.

There are many transformation of sensory impressions that are not like that. That are not cancelable or reversible through self-motion. Including other kinds of switches from white to red. Pour a white liquid, like a lemon sports drink, into a glass. And then pour in something very red, like beet juice or some strawberry syrup. The liquid in the glass went from white to red, just like the paper did when I flipped it over.

But the liquid is different, because you can’t cancel it this time by moving around and looking at it from another point of view. This is precisely why it is not geometrical. The paper example should be interpreted in terms of geometry. If someone asks: what happened? Then for the paper example you would give an explanation in geometrical terms: the object rotated 180 degrees. But for the liquid example you would give an explanation in non-geometrical terms: the red liquid “colors over” the white one by some kind of, I don’t know, chemistry somehow; not geometry anyway.

So there you have a very clear criterion for selecting from the environment which things are to be accounted for in terms of geometry and which not. Cancelability through self-motion.

Before a child can tell if their parents speak French or Russian, they must be able to distinguish which sounds are linguistic at all. And before we can tell if the space around us is Euclidean or non-Euclidean, we must first be able to distinguish which sensory impressions have to do with geometry at all. Poincaré’s criterion in terms of self-motion answers this problem.

So this suggests that it is only through motion that we can impose a geometric interpretation on our visual impressions. It may feel to us as if our sense of sight is inherently geometrical: geometry is visual, it lives in the eyes. But Poincaré’s perspective suggests that it’s more complicated than that.

Vision becomes endowed with geometry only through its interaction with self-motion. If we could not move ourselves or our eyes, our sense of sight would be as un-geometrical as our sense of taste or smell. It would be just a bunch of qualitative impressions with no particular structure.

With sense and smell, you can tell when one thing is different from another, but you can’t do much more than that. There is no “Pythagorean Theorem of taste” that allows you to calculate the taste-distance between wine and beer if you know the distances between beer and water and water and wine. Taste impressions don’t have geometrical structure or any comparable kind of structure. And if we didn’t have self-motion then sight would be like that as well.

There’s a passage in Rousseau’s Emile that fits this perspective. It goes like this:

“It is only by our own movements that we gain the idea of space. The child has not this idea, so he stretches out his hand to seize the object within his reach or that which is a hundred paces from him. You take this as a sign of tyranny, an attempt to bid the thing draw near, or to bid you bring it. Nothing of the kind, it is merely that he has no conception of space beyond his reach.”

So imperfect capacity for self-motion goes with imperfect understanding of space, it seems, in the child. Of course Rousseau was writing long before Poincaré. I used Poincaré as the point person for this perspective about the role of self-motion in geometry but indeed the basic ideas go back centuries before. Poincaré explains his view very well in his book La Valeur de la Science of 1905. But that’s the culmination of a tradition of more than two centuries.

For example, many philosophers had debated the following question: Suppose a person who has been blind all their life has an operation that makes them able to see. Can they then, from visual impressions alone, tell for example a cube from a sphere? They already knew the difference by touch, but could they then automatically make the connection between that and sight, or would they have to learn to recognize things by sight through experience?

This is the so-called “Molyneux’s question.” Molyneux raised it in 1688. Obviously it has a lot to do with the question of whether geometry is innate, or whether it is learned by experience.

This thing about a blind person becoming sighted was not just a thought experiment. It could be done through surgery in some cases. Let me read to you a report of the experiences of such a person. This is from the Philosophical Transactions of 1728. A boy who was 13 years old and had been blind all his life got his sight back through a surgical procedure. And his reactions were as follows.

“When he first saw, he was so far from making any Judgment about Distances, that he thought all Objects that he saw touch’d his Eyes, (as he express’d it) as what he felt, did his Skin.”

“He knew not the Shape of any Thing, nor any one Thing from another, however different in Shape, or Magnitude; but upon being told what Things were, whose Form he before knew from feeling, he would carefully observe them, that he might know them again; but having too many Objects to learn at once, he forgot many of them. One Particular only (tho’ it may appear trifling) I will relate; Having often forgot which was the Cat, and which the Dog, he was asham’d to ask; but catching the Cat (which he knew by feeling) he was observ’d to look at her stedfastly, and then setting her down, said, So Puss! I shall know you another Time.”

“He was very much surpriz’d, that those Things which he had lik’d best, did not appear most agreeable to his Eyes, expecting those Persons would appear most beautiful that he lov’d most, and such Things to be most agreeable to his Sight that were so to his Taste.”

“We thought he soon knew what Pictures represented, which were shew’d to him, but we found afterwards we were mistaken; for about two Months after he [became sighted], he discovered [that] they represented solid Bodies; when to that Time he consider’d them only as Party-colour’d Planes, or Surfaces diversified with Variety of Paint; but even then he was no less surpriz’d, expecting the Pictures would feel like the Things they represented, and was amaz’d when he found those Parts, which by their Light and Shadow appear’d now round and uneven, felt only flat like the rest; and ask’d which was the lying Sense: Feeling or Seeing? Being shewn his Father’s Picture in a Locket at his Mother’s Watch, and told what it was, he acknowledged a Likeness, but was vastly surpriz’d; asking, how it could be, that a large Face could be express’d in so little Room, saying, It should have seem’d as impossible to him, as to put a Bushel of any thing into a Pint.” (That is to say, a larger volume into a smaller.)

That’s quite entertaining but also quite significant evidence for the debates we have been considering. Clearly, learning the geometry of sight was a bit like learning a language for this person who became sighted. He didn’t immediately understand the geometrical structure of visual impressions, so clearly all of that is not completely innate. So it speaks against a Kantian account that takes Euclidean geometry to be a precondition of any geometrical thought or geometrical sensory perception.

But the story of the boy who became sighted fits quite well with a Poincaré-type account in which the geometry of sight can only be developed gradually through experience and coordination with self-motion.

Nevertheless, you can still say that Kant was right in a way. Poincaré is in a sense neo-Kantian. According to Poincaré, Euclidean geometry is not innate, but some geometrical notions are. The mind is predisposed to discern geometrical aspects of its surroundings. Hardwired into the mind are not all of Euclid’s axioms but still a good bit of geometry, such as the categorisation of which perceptions are related to geometry at all, and perhaps related to this some concepts such as displacement, rotation, and so on.

So, those are the ways in which geometry is like language. Both are part innate and part shaped by the environment. To adopt a particular language or a particular geometry is to fit your thoughts into an arbitrary and subjective framework. But that’s a good thing because there are no objective frameworks, and without some such conceptual framework, thinking could never even get off the ground in the first place.

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,