**Transcript**

Let’s read Euclid together. Euclid’s Elements, one of the most important and influential works in human history, who wouldn’t want to read that? “Euclid alone has looked on beauty bare,” as the poets say.

Let’s do some episodes on this where we go through Euclid’s Elements Book I. And here’s the first twist: Let’s read it backwards. Well, not quite. But it’s a good idea to start at the end. Book I of the Elements ends with the Pythagorean Theorem and its converse. It’s not a murder mystery, it won’t spoil the fun to know the ending.

I will explain why I think this is a good idea. This has to do with appreciating the refined goals of the Elements. It’s a very subtle work, in ways that are easy to miss. So I will use this idea of starting at the end as a way of highlighting some things to keep in mind in that regard, so that we approach the text with appreciation of these subtleties.

It might be a bit dry to do only that, so I will also mix it up with some lighter things. Some stories related to the Pythagorean Theorem. Did the Egyptians use the Pythagorean Theorem to build the pyraminds, for example? Is that how they got the angles just right? We will discuss that soon. And I will also play a clip of RoboCop.

I will try to do this for the Elements as a whole: a serious discussion of its finer points, as well as some entertaining tangents exploring the many cultural links of the various parts of the Elements.

So here we go. My first goal is to outline the mindset with which we must approach Euclid’s text.

If you’re a young person, you may look at Euclid’s Elements and say: yeah yeah, triangles and stuff, I saw all of that in high school too; our textbook had proofs just like this thing by Euclid; it’s pretty much the same thing. No, no, no. That’s like listening to Mozart and saying: yeah yeah, big deal, music is music.

Forget it. There’s a world of difference. Euclid is on a whole other level of sophistication than some crappy high school textbook. You wouldn’t know it just by looking at the text though. The text looks the same as any other geometry text. Triangle ABC blah blah blah. It’s the same with musical scores, isn’t it? They all look the same when you just glance at the pages. You can’t tell Mozart from some hack.

We must look deeper to appreciate the subtlety and genius of Euclid. The text itself doesn’t spell that out, just as a Mozart quartet doesn’t have a narrator telling you what’s great about it. But great works reward reflection. The more you study Euclid, the more you interrogate the text, the more you puzzle over its oddities, the more you come to appreciate the mastery that went into crafting everything just right. Euclid knew exactly what he was doing. His work is orders of magnitude more sophisticated than other superficially similar works in the same genre.

The exercise of reading backwards is one angle we can use to start getting a handle on this. If we read Euclid from cover to cover, in the order it’s written, we get a strictly “bottom-up” perspective: we start with the most basic things and gradually get to higher and higher levels of sophistication. That’s how mathematics is typically written down. And with good reason. But the way mathematics comes into being is much more bidirectional. Mathematics grows like a tree: as the branches extend, so do the roots. Starting our Euclid adventure with the Pythagorean Theorem is a way of making us think about this.

Of course when we read Euclid’s proof of the Pythagorean Theorem we find that it is based on earlier results. So you might say: Obviously you have to read those first before you can understand this proof. But that’s a bit simplistic. You could also say: Actually you need to look at the Pythagorean Theorem first because only then can you understand what the purpose is of those earlier propositions. From a purely logical perspective you have to read it linearly from start to finish, but to understand the meaning and purpose of these logical constructions you have to take a step back and interrogate the text from other angles as well. For a dogmatic understanding, it is enough to read it linearly, and parse the logical steps like a machine. But for a critical, independent understanding you want to not only verify the logic but also see how one could arrive at such logical constructions organically.

That goes for any formal mathematics text, still to this day. Or maybe even more so today than ever. The definitions and axioms are the starting points of the way mathematics is written, but often they are almost the end product of the actual creative thought process. Only after you have figured out the hard parts of your theory do you know what the starting points need to be. Or at least there’s an interaction, a back-and-forth negotiation between the top and the bottom of the theory. Each is adapted to the other.

So that’s one reason to read Euclid backwards. It’s a reason that applies to any formal mathematical theory, because they all have this element of bidirectionality.

Actually geometry might be among the more unidirectional formal mathematical theories in how it was conceived, because the results of geometry were known in great detail, long before they were formalised. The tree came before the roots, so to speak.

Here’s another way of visualising it. Think of the Pythagorean Theorem as the apex of a pyramid. The proof reveals which lower, more foundational stones it rests on. Those stones in turn rest on other stones, and so on. Something has to be the bedrock that is considered solid enough not to need any further support beneath it. Euclid’s Elements can be read in two directions: as a way of building up a more and more elaborate structure on top of solid foundations, or as a way of reducing advanced results to their basic components. So when we read the proof of the Pythagorean Theorem, one of the perspectives we should use is to think of it as “boiling down” this somewhat advanced result to more basic ones. This will help us appreciate the purpose and achievement of the more fundamental parts of the Elements when we get to those.

Indeed, by the time Euclid wrote the Elements, the theorems themselves—such as the Pythagorean Theorem—had been known for hundreds or even thousands of years. Even proving the theorem wasn’t all that new. There were plenty of proofs. I bet Euclid knew two dozen proofs of the Pythagorean Theorem.

We shouldn’t think of Euclid as saying: Hey guys, I discovered some things about triangles and stuff; check out this book where I explain how I came up with these theorems.

No, no, no. That’s not at all what Euclid is doing. We must understand, when we read the Elements, that we’re way beyond that.

If you just wanted to convince a random person that the Pythagorean Theorem is true, then there are much better proofs than Euclid’s. Simpler ones. More intuitive, based on simple diagrams. If all you want is a psychologically compelling argument that the Pythagorean Theorem is true then there are better options than Euclid.

Euclid knew all of that, and he chose his proof very deliberately. Because it’s the best proof for his purposes. Namely the purpose of carefully analysing how the truth of the Pythagorean Theorem can be broken down into smaller truths. And more generally to do the same thing for all the truths of geometry in a comprehensive and systematic manner.

So the proof of the Pythagorean Theorem isn’t so much about showing that the theorem is true. It’s more about showing what its ultimate foundations are.

Here’s another metaphor for this. Think of a mathematical theorem as a dish that you cook. The Pythagorean Theorem is like a soup, let’s say. You can whip it up very quickly with store-bought ingredients like stock cubes or just microwaving something from a can. But Euclid doesn’t do store-bought. He’s going to do everything from scratch. And I mean really from scratch. If there’s going to be carrots in there, then Euclid is going to grow his own carrots.

In fact you might say that Euclid is not so interested in cooking at all, even though a proof is like a recipe. Euclid is like a cookbook author who doesn’t like cooking and has no interest in feeding anyone.

Instead he’s more like a chemist who is analyzing the molecular composition of foods. His recipes are not meant as a practical cooking guide but as an analysis of what the core ingredients of the dish are if you deconstruct the recipe as far as you possibly can.

Here we have the idea of reading backwards again: Euclid isn’t really interested in making Pythagorean Theorem soup, but in starting with Pythagorean Theorem soup and taking it apart in the lab. Put it on the Bunsen burner. Different ingredients have different boiling points and so on, so you can carefully separate them out again.

There was already plenty of geometry before Euclid. If theorems are food, everyone was already well fed, so to speak. Everyone already had their favourite dishes and neither they nor Euclid were looking to replace the traditional menus. What Euclid is bringing to the table is not new food but a refined theoretical perspective that stands apart from actual cooking.

The idea of reading Euclid backwards is also related to a famous anecdote recorded about Thomas Hobbes, the 17th-century philosopher. Here’s what it says about Hobbes:

“He was 40 years old before he looked on geometry; which happened accidentally. Being in a gentleman’s library, Euclid’s Elements lay open, and ‘twas the [47th Proposition of Elements Book I, the Pythagorean Theorem]. He read the proposition. By God, sayd he, this is impossible! So he reads the Demonstration of it, which referred him back to such a Proposition; which proposition he read. That referred him back to another, which he also read. [And so on], that at last he was demonstratively convinced of that trueth. This made him in love with geometry.”

It is interesting that Hobbes ended up reading Euclid backwards by accident like this. Precisely what I recommended as a deliberate strategy. But he doesn’t seem to have appreciated the point of doing so the way I have described it. Maybe he could just as well have read the book forwards and had the same experience, as far as this anecdote goes.

Hobbes fell “in love with geometry” by reading it backwards, but others had the same experience reading it forwards. Bertrand Russell, another famous philosopher, read Euclid the conventional way, starting at the beginning, and he still found it, as he later said, “as dazzling as first love”: “I had not imagined there was anything so delicious in the world.” Bertrand Russell was eleven at the time, while Hobbes was 40 when he stumbled upon Euclid. They lived almost three centuries apart. So these anecdotes speak to the universality of Euclid’s text: young or old, forwards or backwards, conservative or socialist, in a society of cars or one of horses—the one thing they have in common is the love that Euclid stirred up in them.

That’s all very nice, but it kind of misses the point in terms of what I have tried to argue was the goal of Euclid’s Elements. What Hobbes and Russell fell in love with was the idea of geometrical proof, it seems. Historically, those epiphanies are better associated with a pre-Euclidean period. We discussed Thales before, and there were plenty of others in the centuries between him and Euclid.

So when you read Euclid, by all means, do fall in love. Be seduced like so many others have been. But also keep in mind that these charms are only part of the greatness of Euclid. Euclid’s Elements can be as good a vehicle as any to have that epiphany of the beauty of mathematics. But to Euclid and many of his readers that was old news.

Euclid wanted to do more than that. He didn’t want to just show how cool it is to prove stuff, although that is lovely. More than that, he wanted to explore the very essence of geometrical knowledge. What are its preconditions, and the source of its certainty? Just as a chemist seeks to decompose any substance into the elements of the periodic table, so Euclid sought to find the “periodic table” of geometry, so to speak: he wanted to uncover the ultimate building blocks of this entire branch of knowledge.

Ok, so that’s my lesson one in how to read Euclid. Start at the back and keep in mind this theme of distillation into ultimate foundations.

So I urge you to go read Euclid that way. I’m not going to go through the proof here; you’ll have to follow along in your own copy of the Elements. I recommend my own edition, for which I added illustrations for each step of the proofs. It’s a joy to read, in my opinion. But it’s too visual to translate into this medium, so I’ll leave that to you to pursue.

Now I wanted to take this opportunity to think about the origin of the Pythagorean Theorem. Part of the appeal of reading Euclid’s Elements is how embedded it is many aspects of human culture and history. So in parallel with our reading of Euclid I wanted to bring up such themes as well.

The Pythagorean Theorem has little to do with Pythagoras. It was discovered independently in several cultures, some of them long before Pythagoras. But never mind the name. The more interesting question is: Why were people interested in this theorem? Why would anybody want to calculate a bunch of hypothenuses?

If you look in a modern geometry textbook, you won’t find any good answers. The book will give you the formula and ask you to apply it in all kinds supposedly real-world cases, but they are all fake and transparently ridiculous. How to calculate the diagonal of a field when you know the lengths of the sides: When would you ever use this? Why wouldn’t you just measure the diagonal then if that’s what you want to know?

Ladder problems is another one of those fake classics. The foot of the ladder is so-and-so far from the wall, and the ladder is so-and-so long, will it reach to such-and-such a height, maybe for instance the ladder of a fire truck to save someone from a burning building? Not a very realistic scenario. Wouldn’t you just try it and see if it worked? Wouldn’t that be just as easy as sitting around making calculations? And why would the distance from the wall to the foot of the ladder be some exact given number? And so on.

It doesn’t make sense that people discovered the Pythagorean Theorem because they were wrestling with practical problems like those. They would not have needed mathematics for that. If they wanted to solve those problems they would have used trial and error and direct measurements.

Unfortunately, ancient textbooks are as ridiculous as modern ones in this regard. Here’s an example from a Chinese text from about the time of Euclid. A 10-feet-high stem of bamboo broke in the wind. It broke into two straight prices. One part remains upright, perpendicular to the ground. But the other part, that broke off but is still attached, tipped over and is now touching the ground, 3 feet away from the base of the stem. How high up the stem did the break occur?

You can calculate this with the Pythagorean Theorem, sure enough, but of course there is no way anyone would ever do something so absurd in the real world. Just measure it, if you want to know. You apparently already measure the distance along the ground and the full height somehow, so why couldn’t you just as well measure this thing? Doesn’t make any sense.

Here’s another scenario some have claimed involves the Pythagorean Theorem. On the Greek island of Samos, there’s an ancient tunnel, which was dug in fact right in the lifetime of Pythagoras.

This tunnel is a marvelous thing, a tribute to the engineering skills of the Greeks. It’s still there today. The tunnel is over one kilometer in length through a big mountain. It was dug to supply the capital with fresh water.

Digging the tunnel was certainly a geometrical project. In fact, the walls still have letters on them, like the lettering of a geometrical diagram. Evidently there was a plan of the tunnel in the form of a drawn diagram, with points makes by letters, and then as it was dug these letters were inscribed on the wall to keep track of how the actual tunnel corresponded to the geometrical plan.

This was all the more essential since the tunnel had to be dug from both ends, in order to complete it in half the time. So the diggers had to be coordinated to ensure they met in the middle. A highly non-trivial problem, which the Greek geometers solved flawlessly.

In fact, at some point the plan even had to change because the rock was becoming to porous. So there was a risk that tunnel would collapse. Therefore it was necessary to make a bend in the tunnel that took it more toward the core of the mountain, which had harder rock. The geometers dealt with this flawlessly as well. They added a shallow isosceles triangle to the diagram. So each digging team had started out along straight lines that would have met in the middle, but halfway through both teams were instructed to make a slight turn which was specified with geometrical precision. So the whole tunnel has a kind of V-shaped bend in the middle. But it still worked. The two digging teams met just as the geometers had calculated.

That’s great stuff, but is it the Pythagorean Theorem? Let me play to you a clip from the History Channel documentary series Engineering an Empire, which claims that it is.

“Eupalinos dug tunnels from each side of the mountain, until they met in the middle. To succeed, Eupalinos had to make sure that each tunnel started at the same vertical height., on opposite sides of the mountain. The tunnels also had to match up on a horizontal plane. Otherwise, they would pass each other like ships in the night.”

“By forging a path from the spring to the city, in short perpendicular lines, Eupalinos could measure each small length in order to calculate two sides of a right triangle. With two known sides of the triangle, the hypothenuse became the path of the tunnel through the mountain.”

So according to the History Channel, the plan for the tunnel was based on the Pythagorean Theorem. The History Channel are not even taking into account the alterations of the plans midway through, by the way. They just discuss the problem of making a straight tunnel.

The presenter of this documentary is Peter Weller, who is also the actor who played RoboCop in the 1987 movie. Turns out he’s also a historian.

I must say though that I disagree with RoboCop’s analysis. The tunnel of Samos was great geometry but it wasn’t the Pythagorean Theorem. The way RoboCop puts it in the documentary, it sounds as if the point was to calculate the length of the tunnel. That’s the hypothenuse that RoboCop is talking about in that clip. But of course the real problem is the coordination of the two digging teams, so they won’t miss each other “like ships in the night,” as RoboCop himself said. How is the length of the hypothenuse supposed to be useful for this? Knowing how long the tunnel is supposed to be doesn’t help you determine the direction of digging.

So I don’t think this tunnel stuff is a great example of real-world motivation for the Pythagorean Theorem. We have to keep looking for where ancient man could have had reason to discover or apply this theorem.

Here’s another such scenario. Did the Egyptians use the Pythagorean Theorem to build the pyramids? I’ll play another clip from another documentary series that claims: yes. This is from The Story of Maths, a BBC documentary presented by Marcus du Sautoy.

“The most impressing thing about the pyramids in the mathematical brilliance that went into making them. Including the first inkling of one of the great theorems of the ancient world: Pythagoras’s Theorem. In order to get perfect right-angled corners on their buildings and pyramids, the Egyptians would have used a rope with knots tied in it. At some point, the Egyptians realised that if they took a triangle with sides marked with 3 knots, 4 knots, and 5 knots, it guaranteed them a perfect right angle.”

The theorem involved here is not the Pythagorean Theorem itself, but the converse of it, which is Proposition 48 in Euclid.

In terms of historical evidence, we really don’t know if the Egyptians did this or not. It’s plausible that they knew this but there’s very little documentary evidence from way back then.

Obviously you can’t believe anything just because Marcus du Sautoy said it in a BBC documentary. Marcus du Sautoy is not a historian, he’s just clowning around. But let’s see, if we’re serious about it, does it make any sense?

I used to be skeptical about this, but I have come to think maybe it’s not so bad. I think the standard formulation about a rope with 3+4+5 equally spaced knots on it is a bit silly. Seems very complicated to get the knots just right.

But you don’t really need one triangular rope. Instead you can just use three separate ropes, of lengths 3, 4, and 5. That’s easy to make. Then when you need to make a right triangle you stretch the 3 and 4 ropes along the intended sides, and you check if the 5 rope fits between their endpoints. Then you have the guy holding the end of the 4 rope move a bit this way or that until it lines up perfectly.

I have to admit, if I was building a pyramid I would probably go with this method. Especially because of the scale of the project. The base of the pyramid is enormous. You would use ropes with lengths 3, 4, 5, but not in feet or meters but some bigger unit. Maybe 30 meters, 40 meters, 50 meters. The ratio is all that matters of course. The longer the ropes, the less significant the measurement error becomes. So it’s a pretty good method I think.

Let me read you a quote here from the book Euclid’s Window by Leonard Mlodinow. I thought it was quite funny.

“Picture a windswept, desolate desert, the date, 2580 B.C. The architect had laid out a papyrus with the plans for your structure. His job was easy—square base, triangular faces—and, oh yeah, it has to be 480 feet high and made of solid stone blocks weighing over 2 tons each. You were charged with overseeing completion of structure. Sorry, no laser sight, no fancy surveyor’s instruments at your disposal, just some wood and rope. As many homeowners know, marking the foundation of a building or the perimeter of even a simple patio using only a carpenter’s square and measuring tape is a difficult task. In building this pyramid, just a degree off from true, and thousands of tons of rocks, thousands of person-years later, hundreds of feet in the air, the triangular faces of your pyramid miss, forming not an apex but a sloppy four-pointed spike. The Pharaohs, worshipped as gods, with armies who cut the phalluses off enemy dead just to help them keep count, were not the kind of all-powerful deities you would want to present with a crooked pyramid. Applied Egyptian geometry became a well-developed subject.”

So that’s a quite comical way of putting it, but the point is well taken, I think. Indeed it does make some sense, this whole thing. The historical and societal context, the mathematics available at that time, the need to make exact right angles, the method for doing so using strings and a Pythagorean triple: that is all quite plausible, I would say.

It’s hardly plausible that they would have discovered the Pythagorean Theorem this way, by starting with the problem of making right angles. But it is plausible that may have used knowledge of the 3-4-5 special case of the converse of the Pythagorean Theorem to make right angles.

Here’s another proposal for the possible origins of the Pythagorean Theorem. This proposal is from van der Waerden’s book Geometry and Algebra in Ancient Civilizations. He proposes that the original motivation for the discovery of the Pythagorean Theorem might have been related to eclipses. Namely, calculating the duration of a lunar eclipse.

Indeed, astronomy was important to many ancient peoples. You know the Stonehenge, Maya temples aligned with solstices and so on. People cared a lot about the sky back then.

Eclipses were a big deal. Probably they were often seen as having some kind of theological significance, some sort of omen, and so on. They were also scientifically important, for instance for exact calendar keeping.

So what do eclipses have to do with the Pythagorean Theorem? Mathematically, this is a neat example. Fun to use in a geometry class.

A lunar eclipse occurs when the moon passes through the shadow cast by the earth. The earth’s shadow is about twice the size of the moon, at that distance. So the moon is approaching this dark spot, it enters it, and keeps moving through it, and comes out at the other side. The whole thing takes maybe an hour or two, it differs.

We can predict in advance how long a particular lunar eclipse is going to last. The determining factor is whether the path of the moon goes right through the middle of the earth’s shadow, or cuts across it off center. The moon’s orbit is complicated and it’s different each time. Sometimes it’s coming in a bit high and sometimes a bit low. We can see this by comparing its position to the stars.

So this means that the problem of calculating the duration of an eclipse comes down to calculating the length of a line cutting through a circle, not necessarily through the middle. We assume that the moon’s speed is constant throughout the eclipse. So the duration of the eclipse is determined by how big of a segment of the moon’s path is in the circular shadow cast by the earth.

This indeed becomes a Pythagorean Theorem problem. You can picture it like this. Draw a circle. That’s the shadow cast by the earth. Now draw a line cutting through the circle, but not through the midpoint. That’s the path the moon is moving along. We want to know the length of the segment inside the circle. This is what determines the duration of the eclipse.

Find the midpoint of this segment. Connect it to the center of the circle. This is a known length, because it corresponds to how far off-center the moon was in its approach, which we can determine by comparing its position to the stars. So the distance from the midpoint of the segment to the center of the circle was known before the eclipse began.

Let’s add one more line to the diagram: the line from the center of the circle to the point where the moon’s path entered the circle. That’s of course a radius of the circle, which is known because the size of the earth’s shadow is known.

So now you see why it’s a Pythagorean Theorem problem. The two knowns are two sides of a right-angle triangle, and the sought length is the remaining side.

Could this be how ancient man discovered the Pythagorean Theorem? This hypothesis has one thing going for it, namely that the sought quantity cannot be measured directly in advance of the eclipse. You genuinely need the Pythagorean Theorem to do this. It’s not one of those fake ones where you could just as easily have measured the side you are looking for, instead of measuring the sides you don’t want and then calculating the you do want, as in those fake textbook problems.

Mathematically, that’s all very satisfying. Unfortunately this hypothesis is not very plausible historically. In the Babylonian tradition, mathematics came long before mathematical astronomy. Serious mathematical astronomy such as this, with detailed eclipse calculations and so on, was a preoccupation of the second flowering of ancient Babylonian mathematics. That’s about a thousand years after the first golden age of Babylonian mathematics.

Already the older period had excellent mathematics, including something like the Pythagorean Theorem. One of the most famous old Babylonian clay tablets states the ratio between the side and the diagonal of a square. So it’s essentially a numerical approximation of the square root of 2, in other words. The numerical value the tablet states is very nearly accurate to six decimal places. That’s very accurate indeed. Suppose you used it to compute the diagonal of a square field with a side of a hundred meters. So a football field, basically. Then the Babylonian approximation is off from the exact answer by less than one millimeter.

That’s more than a thousand years before Babylonian priests became obsessed with eclipses for the sake of ensuring the calendric accuracy of their rituals. So the mathematically pleasing hypothesis about the Pythagorean Theorem being discovered to calculate eclipse durations doesn’t really fit the historical record unfortunately.

So what can we conclude from all this? I think it’s safe to say that practical need was never the main driver of mathematics that goes even a bit beyond the basics. The Pythagorean Theorem was discovered because people were fascinated by mathematics for its own sake, not because they needed to calculate stuff. The Chinese didn’t need to know the breaking points of bamboos, the Babylonians didn’t need to know the diagonal of a football field with millimeter accuracy. They were fascinated by the power of mathematical reasoning to discover hidden relationships, and that’s why they explored these things.

This is also how we should read Euclid. The proof of the Pythagorean Theorem is not so much about proving that the theorem is true. It’s more about exploring the basis for this knowledge. Mathematics was always explored for this reason.

Discovering mathematics was like discovering magic. It impresses us as a powerful force that can do incredible things. We want to understand it: How is this possible? What makes this magic tick? It is so unlike anything else we are familiar with, it’s like a portal to a divine realm. We feel a spiritual imperative to understand it.

Already ancient civilisations started along this path, and Euclid does the same. If mathematics is magic, Euclid’s Elements is not a book of spells, but a scientific investigation of how there can be such a thing as magic at all.

Or to use another metaphor, we have to dissect mathematics like an alien corpse to discover the secrets of it mysterious inner workings. The Pythagorean Theorem is the alien: a weird thing that seems to have superhuman powers. Euclid’s proof is not a recipe to give you alien abilities; rather, it is the result of his through dissection of an alien he found in the wild.

So let’s read Euclid this way, as an exploration into the inner mechanisms—the heartbeat—of these strange entities, these superhuman theorems, that have impressed mankind with their seemingly magical and divine aura for many thousands of years.

]]>**Transcript**

Greek geometry is oral geometry. Mathematicians memorised theorems the way bards memorised poems. Euclid’s Elements was almost like a song book or the script of a play: it was something the connoisseur was meant to memorise and internalise word for word. Actually we can see this most clearly in purely technical texts, believe it or not. It is the mathematical details of Euclid's proofs that testify to this cultural practice. That sounds almost paradoxical, but I’m sure I will convince you.

The surviving documentation about ancient Greek geometry consists almost entirely of formal treatises. Very stilted and dry texts. Definition, theorem, proof. Pedantically written. Highly standardised, formalised. Completely void of any kind of personality. Where is the flesh and blood, the hopes and dreams, the lived experience of the ancient geometer? It’s as if they were determined to erase any traces of all of those things, and leave only a logical skeleton.

But it’s not as hopeless as it seems. At first glance it looks as if these texts have been scrubbed of all humanity. But, in fact, if we read between the lines we can extract quite a bit of information. There are implicit clues in these texts that reveal more than the authors intended.

That’s our topic for today: How these seemingly purely logical texts actually say quite a lot about the social context in which they were produced.

One thing we learn this way is that we should think of the Greek geometrical tradition as spoken geometry, not written geometry. Today we think of written texts as the primary manifestation of mathematics. When mathematicians disseminate their ideas, the published article is the official, definitive, primary expression of those ideas. The mathematician crafts a written document with the expectation that reading the text on paper is going to be the primary way in which people will access this material.

Not so in antiquity. Oral transmission was considered the primary mode of explaining mathematics. Written documents were a last resort when personal contact was not possible. And the written document was not meant to be a primary exposition in its own right. Writing was merely the oral explanation put down on paper (or papyrus, rather).

At least it must have been like that in the early days. Many conventions of Greek mathematical writing only make sense from this point of view. They must have been formed in an oral mathematical culture. Probably in later antiquity the situation was not so clear cut. Writing probably gradually became more of a thing in its own right, rather than merely a record of oral exposition. But even then, the conventions of written mathematics remained largely fixed. Greek mathematics never liberated itself from these conventions that had been set in an oral culture. They lived on. Perhaps in part due to tradition and conservatism, but probably also because the oral element remained a significant part of mathematical culture, perhaps especially in teaching.

Here’s an example of this, which I have taken from Reviel Netz’s book The Shaping of Deduction in Greek Mathematics. Consider the equation A+B=C+D. Here’s how the Greeks expressed this in writing: THEAANDTHEBTAKENTOGETHERAREEQUALTOTHECANDTHED. This is written as one single string of all-caps letters. No punctuation, no spacing, no indication of where one word stops and the next one begins.

A Greek text is basically a tape recording. It records the sounds being spoken. There is a letter of the alphabet for each sound one makes when speaking. The scribe just stenographically puts them down one after the other. From this point of view there is no distinction between upper or lower case letters: a letter just stands for a sound and that’s it. And there is no punctuation or separation of words, because those are not spoken sounds. And of course no mathematical symbols such as plus or equal signs, because that also does not exist in spoken discourse.

The only way to understand a text like that is to read it out loud. You have to read it like a child who is just learning to read: you sound it out letter by letter, and then interpret the sounds, rather than interpret the writing directly.

So the Greeks had a very limited conception of writing. They thought of writing only as a way of recording speech. They completely missed the opportunities that writing provides when embraced as a primary medium in its own right. Writing is a better way of representing equations, for example, than speech. But the Greeks completely missed that opportunity because they were stuck with the limited notion of writing as merely recorded sounds.

I like to compare this with early movies. Think of those classic movies from, say, the 1950s or so. They are basically recorded stage plays. There are limitations inherent in the medium of theatre. The actors have to speak quite loudly, articulately, to be heard by the audience in the back of the theatre. And the scenery on stage cannot easily be changed or moved. In a play you better stick to one or two sets, such as the interior of a room. That you can set up carefully with furniture and all kind of stuff on the walls and so on. But because you can’t change it easily, you have to have to have large parts of the play take place in that single setting.

These technical limitations constrain the artistic freedom of the playwright. You have to come up with a story where all the various characters have some reason or other to come and go into a single room, and once there to have loud conversations that drive the plot. All emotional depth and so on must be conveyed in this particular form.

These things became second nature to writers. So when film came around they kept doing the same thing even though that was no longer necessary. Many treated film as simply a way of recording plays. So in early movies you still have a lot of these static scenes with a fixed camera at one end of a room, and characters coming and going, having loud conversations.

Film affords new artistic possibilities. You are no longer limited to a static camera showing a fixed set, the way the audience of a theatre would be looking through the “fourth wall” of a room. You have many more options to convey things visually, instead of being limited to strongly articulated stage dialogs as the only driver of the plot.

But many early movies didn’t take advantage of that. They just kept doing what they had always been doing at the theatre and just recorded that. They saw the new medium of film merely as a way of “bottling” existing practice. It’s just a storage medium. They didn’t consider that the new medium was in some ways better than the old one and enabled you to do completely new things.

It was the same with writing in antiquity. Writing was merely for storing speech. They failed to take advantage of the ways in which writing could not only preserve existing cognitive practice but in fact transform it and improve it. Such as working with equations symbolically.

Here is another consequence of this: the absence of cross-referencing. If a mathematical text is like a tape recording, you can’t easily access a particular place in the tape. The only way to make sense of the text is to “hit play,” so to speak, and translate it back into sounds. Only then can it be understood. You can “fast forward” and “rewind”—that is to say, start reading at any point in the manuscript. But you can’t turn to a particular place, such as Theorem 8.

Modern editions of Euclid’s Elements are full of cross-references. Each step of a proof is justified by a parenthetical reference to a previous theorem or definition or postulate. But that’s inserted by later editors.

There is no such thing in the original text. Because it’s a tape recording of a spoken explanation. Referring back to “Theorem 8” is only useful if the audience has a written document in front of them. If they are merely listening to a long lecture, or a tape recording of a lecture, then there is no use referring back to “Theorem 8”, because the audience has no way of going back specifically to that particular place in the exposition.

For this reason, oral mathematics involves committing a lot of material to memory. In the arts, people memorise poems and song lyrics. Actors memorise the dialogues of plays. Ancient mathematics was like that as well. You would learn to recite theorems the same way you learn to sing along to your favourite song.

This aspect of the oral culture thoroughly shaped the way ancient mathematical texts are written. Euclid’s Elements and many other texts follow a certain stylistic template that at first sight seems quite irrational, but which starts to make sense once we consider the oral context.

Consider for example Proposition 4 of Euclid’s Elements. This is the side-angle-side triangle congruence theorem. It’s completely typical, I’m just picking a theorem at random. Let’s look at the text of this proposition. First we have the statement of the theorem in purely verbal terms. It goes like this:

“If two triangles have two sides equal to two sides, respectively, and have the angle enclosed by the equal straight lines equal, then they will also have the base equal to the base, and the triangle will be equal to the triangle, and the remaining angles subtended by the equal sides will be equal to the corresponding remaining angles.”

Ok, so: two triangles have side-angle-side equal, the it follows that they also have all the other things equal. Namely the remaining side, the remaining angles, and the area. “The triangle will be equal to the triangle,” says Euclid: this is his way of saying that they have equal area.

After Euclid has stated this, he goes on to re-state the same thing, but now in terms the diagram. “Let ABC and DEF be two triangles having the two sides AB and AC equal to the two sides DE and DF, respectively. AB to DE, and AC to DF. And the angle BAC equal to the angle EDF. I say that the base BC is also equal to the base EF, and triangle ABC will be equal to triangle DEF, and the remaining angles subtended by the equal sides will be equal to the corresponding remaining angles. ABC to DEF, and ACB to DFE.”

This is exactly the same thing that he just said in words. But now he’s saying it with reference to the diagram. He always does this. He always has these two version of every proposition: the purely verbal one, and the one full of letters referring to the diagram.

For simple propositions you can understand the value of both formulations. But quite soon, when the material gets more technical, it often happens that the verbal version becomes so abstract that it’s quite impossible to follow. This happens quite soon already in Euclid. Ken Saito has a recent paper on this, “traces of oral teaching in Euclid’s Elements.” He takes as an example Proposition 37 from Book 3 of the Elements. I’ll read it to you just to convince you how convoluted and unnatural it is to state theorems in this purely verbal form. Here it is, Euclid’s statement of this proposition:

“If a point be taken outside a circle and from the point two straight lines fall on the circle, and if one of them cut the circle, and the other fall on it, and if further the rectangle contained by the whole of the straight line which cuts the circle and the straight line intercepted on it outside between the point and the convex circumference be equal to the square on the straight line which falls on the circle, the straight line which falls on it will touch the circle.”

That’s very difficult to follow. Of course, as always, Euclid immediately goes on to state the same thing, but in terms of the diagram. That part is much easier to follow, and it turns out to be a pretty straightforward claim. The theorem is a kind of formula for the length of a tangent; how far it is to the point of tangency from a given point outside the circle. But you would hardly know that by reading the verbal statement only.

For some reason the Greeks insisted that the verbal formulation should be one single, rambling sentence. No matter how complicated your theorem is, you have to cram all the conditions and all the consequences, everything you want to say, into one single sentence.

This is taken to absurd lengths in Apollonius for example. Let me read to you an example from the Conics of Apollonius. This is Proposition 15: one of the earliest. It only gets worse from there, but this is bad enough, I’m sure you will agree when I read it to you. The proposition is a kind of change-of-variables theorem for ellipses: it tells you the equation for an ellipse in a new coordinate system conjugate to the first. So it has to specify what the equation of the ellipse was in the first coordinate system and what the assumptions for that was, then how the change of coordinates is defined, and then what the equation of the ellipse is in the new coordinate system. And it has to do all of that purely verbally, and in one single sentence, one big “if ... then ...” statement. So you get this crazy monstrosity of a sentence, it goes like this:

“If in an ellipse a straight line, drawn ordinatewise from the midpoint of the diameter, is produced both ways to the section, and if it is contrived that as the produced straight line is to the diameter so is the diameter to some straight line, then any straight line which is drawn parallel to the diameter from the section to the produced straight line will equal in square the area which is applied to this third proportional and which has as breadth the produced straight line from the section to where the straight line drawn parallel to the diameter cuts it off, but such that this area is deficient by a figure similar to the rectangle contained by the produced straight line to which the straight lines are drawn and by the parameter.”

What’s going on with this crazy stuff? Were the Greeks some kind of aliens with brains that could understand that type of thing? No. When encountering a theorem like this, they surely did not try to parse a sentence like that in the abstract. Instead they would turn to the diagram explication for help. Just as Euclid always does, so also Apollonius always goes on to restate the theorem in terms of labelled point in a diagram. And this explanation is not one big crazy sentence, but nicely broken into small steps. Much easier to follow.

At a certain point you may ask yourself: Why even include the purely verbal formulation at all? It’s so abstract, so difficult to follow. Surely any reader or listener will be lost before you have even gotten halfway through a sentence like that. And since you’re going to restate the theorem immediately anyway, why bother? You might as well only do the diagram version of the theorem. That’s the one you are going to use for the proof anyway.

That’s something of a puzzle in itself, but here’s the real kicker though. Not only does Euclid insist on including the abstruse verbal formulation of every theorem, he actually includes it twice! This is because, at the end of the proof, his last sentence is always “therefore ...” and then he literally repeats the entire verbal statement of the theorem. It is literally the exact same statement, word for word, repeated verbatim. You say the exact same thing when you state the proposition and then again when you conclude the proof. Copy-paste. The exact same text just a few paragraphs apart.

Astonishing. What a waste of papyrus and scribal effort. This was an enormous cost back then. There were no printing presses. You had to copy all of this by hand. Writing materials were expensive, copying was expensive, preservation was expensive. They had every incentive to cut and keep things minimal, yet they included this massive redundancy of repeating the rambling verbal statement of every proposition twice in short succession.

You may recall that an important treatise by Archimedes was scrubbed off its parchment because the parchment itself was so valuable even when recycled. And medieval scribes were big on minimising writing. Think of “etc.”, “e.g.”, “i.e.”: we still use those shortened versions of Latin expressions. They were invented back when people were writing and copying manuscripts by hand. Very understandable.

Yet despite all of that, for some strange reason, including the entire verbal statement of the proposition twice was somehow found valuable enough to warrant the enormous cost.

In the case of the side-angle-side theorem for example, the verbal statement of the theorem takes up about 15% of the total text of the proposition and proof. And then another 15% for the redundant recapitulation. So that’s 30% of the total text that could simply be cut. The remaining 70% of the text would still contain the full statement of the theorem in its diagram form, and the complete proof.

You’d think the temptation would be great to cut at least those last 15% of pure recapitulation. Even the standard English edition of the Elements by Heath simply writes “therefore etc.” at the end of the proofs, instead of repeating the full statement like the original did.

So what was the value of this very expensive business of repeating the statement of the proposition? The oral tradition explains it. The verbal statement of the proposition is like the chorus of a song. It’s the key part, the key message, the most important part to memorise. It is repeated for the same reason the chorus of a song is repeated. It’s the sing-along part.

In a written culture you can refer back to propositions and expect the reader to have the text in front of them. Not so in an oral culture. You need to evoke the memory of the proposition to an audience who do not have a text in front of them but who have learned the propositions by heart, word by word, exactly as it was stated, the way you memorise a poem or song.

This is why, anytime Euclid uses a particular theorem at a particular point in a proof, he doesn’t says “this follows by Theorem 8” or anything like that. He doesn’t refer to earlier theorems by number or name. Instead he evokes the earlier theorem by mimicking its exact wording. Just as you just have to hear a few words of your favourite chorus and you can immediately fill in the rest. So also the reader, or listener, of a Euclidean proof would immediately recognise certain phrasings as corresponding word for word to particular earlier propositions. They would have memorised the earlier propositions not only in terms of content but in terms of the exact verbal phrasing, almost melodically, rhythmically. Just hearing the first few words of such a formula repeated would trigger the full memory to flow out naturally and unstoppably, like singing along to the chorus of a song you love.

You can see an example of this already in Euclid’s Proposition 5. We already discussed his Proposition 4, the side-angle-side triangle congruence theorem. Euclid applies this result twice in the course of the proof of Proposition 5. However, he really only needs part of the theorem. Remember that Proposition 4 concluded several things: that the remaining sides were equal, that the remaining angles were equal, and that the areas of the two triangles are equal. Areas are completely irrelevant to Proposition 5, which is a statement purely about angles. Yet each time Euclid applies the side-angle-side theorem he spells out the full conclusion. Including the needless remark that the areas are equal.

In one case it is even irrelevant that the remain sides are equal as well, but Euclid still needlessly remarks on this pointless information in the course of the proof of Proposition 5 even though it has no logical bearing on the proof. Go look up Euclid’s proof if you want to see this nonsense for yourself. Ask yourself why Euclid points out that “the base BC is common” to both triangles the second time he applies the side-angle-side theorem in the proof of Proposition 5. It’s completely redundant and worthless. He could have just omitted that remark, and it wouldn’t have affected the logic of the proof at all.

But from the oral point of view it makes sense. Applying a theorem is a kind of package deal. You get the whole thing whether you need it or not. Once you’ve triggered the memory of the previous theorem with the appropriate key phrases, then the whole conclusion comes blurting out. Once you’ve committed to singing the chorus there’s no going back. You can’t sing only the part of the chorus you need. The whole thing goes together. You have memorised it in one flow. Once you hit play on that memory you automatically run through the whole thing.

This is why Euclid is needlessly talking about areas in the proof of Proposition 5, even though that serves no logical function whatsoever. He is mimicking word for word the phrasing of the previous proposition, filling in the specifics of the case at hand as he goes along. You sing the “chorus” of the side-angle-side theorem and you “fill in the blanks” as it were. The purely verbal statement of the side-angle-side theorem spoke of sides and angles and so on in the abstract. To apply the theorem is to repeat that exact same phrasing, but inserting AB, BCF, and so on, into that formula to specify what the sides and angles are in the particular case at hand.

It’s like singing “happy birthday”: it has a fill-in-the-blank part. Just as you would go: “Happy birthday dear Euclid”, so also you would go: “If the side AB equals CD, then the angle is ...” and so on, something like that.

Here’s maybe another consequence of this: Euclid’s odd formulation of the side-side-side triangle congruence theorem. This is Euclid’s Proposition 8. As we saw, in the side-angle-side case, Euclid drew all the possible conclusion: about sides, about angles, about area. So the theorem became a mouthful, and led to the introduction of superfluous remarks any time the theorem is applied, because you have to repeat all the conclusions whether you need them or not.

To avoid this problem it might be tempting to state theorems in less general form. And this is exactly what Euclid does with the side-side-side theorem. He introduces an asymmetry in the statement of the theorem. Instead of three sides, he speaks of two sides and a base. And his statement of the conclusion is that one particular angle (the angle between the two “sides”) is equal in both triangles. Of course it is completely arbitrary which side you designate as the “base.” And of course you could just as well have concluded that the other angles too correspond to each other in these congruent triangles. Yet Euclid choses to arbitrarily limit the generality of his theorem, and introduce arbitrary specificity and asymmetry. You’d think that would be anathema to a mathematician.

But if we think of the downsides of the way he formulated the side-angle-side case, we can understand why he went with this non-general formulation in the side-side-side case. Any time you are going to apply the side-side-side theorem, you probably want to conclude something about a specific angle, not all three angles of a triangle. So if you formulated the theorem generally, then every time you applied it you couldn’t stop yourself of course from reciting the entire chorus and hence you would end up with one conclusion that you actually needed, about one angle, and then needless spelling out two other conclusions about the other two angles that you don’t want at all. So this way you will only clutter your proofs with needless and irrelevant remarks. So the strangely specific, non-general formulation of the side-side-side theorem is actually well chosen given this constraint that you have to repeat the full theorem verbatim any time you apply it.

It’s pretty fascinating, I think, how textual aspects that appear to be purely technical and mathematical, such as a few barely noticeable superfluous bits of information in the proof of Proposition 5, can open a window like this into an entire cultural practice. The oral tradition must have been there, and the best proof of this is hiding in the ABCDs of Euclid’s formal text. It’s the beauty of history that historical texts can be read on so many levels. They carry so much hidden information about the culture that produced them. You would think Euclid’s ultra-formalised proofs would be the last place to find such clues, but here they are. We’re just a few proposition into the Elements and from the smallest technical quirks we have already recreated a rich picture of the ancient singing geometers and the strange culture in which they worked.

**Transcript**

How did proofs begin? It’s like a chicken-or-the-egg conundrum. Why would anyone sit down and say to themselves “I’m gonna prove some theorems today” when nobody had ever done such a thing before? How could that idea enter someone’s mind out of the blue like that?

In fact, we kind of know the answer. The Greek tradition tells us who had this lightbulb moment: Thales. Around the year -600 or so. Hundreds of years before we have any direct historical sources for Greek geometry. But we still sort of know what Thales proved, more or less. Later sources tell us about Thales. History is perhaps mixed with legend in those kinds of accounts, but key aspects are likely to be quite reliable. More fact than fiction. Let’s analyse that question, the credibility question, in a bit more depth later, but first let’s take the stories at face value and see how we can relive the creation of deductive geometry as it is conveyed in these Greek histories.

So, here we go: What was the first theorem ever proved? What was the spark that started the wildfire of axiomatic-deductive mathematics? The best guess, based on historical evidence, goes like this. That love-at-first-sight moment, that theorem that opened our eyes to the power of mathematical proof, was: That a diameter cuts a circle in half.

Pretty disappointing, isn’t it? What a lame theorem. It’s barely even a theorem at all. How can you fall in love with geometry by proving something so trivial and obvious?

But don’t despair. It is nice, actually. It’s not about the theorem, it’s about the proof.

Here’s how you prove it. Suppose not. This is going to be a proof by contradiction. Suppose the diameter does not divide the circle into two equal halves. Very well, so we have a line going through the midpoint of a circle, and it’s cut into two pieces. And we suppose that those two pieces are not the same. Take one of the pieces and flip it onto the other. Like you fold an omelet or a crepe. The pieces were not equal, we assumed, so when you flip one on top of the other they don’t match up. So there must be some place where one of the two pieces is sticking out beyond the other. Now, draw a radius in that direction, from the midpoint of the circle to the place on the perimeter where the two halves don’t match up. Then one radius is longer than the other. But this means that the thing wasn’t a circle to start with. A circle is a figure that’s equally far away from the midpoint in all directions. That’s what being a circle means.

So we have proved that two things are incompatible with one another: You can’t be both a circle, and have mis-matched halves. Because if you have mis-matched halves you also have “unequal radii” and that means you’re not a circle.

So a circle must have equal halves. Bam. Theorem. It’s a boring result but a gorgeous proof. Or a suggestive proof. It’s a proof that hints at a new world.

Thales must have felt like a wizard who just discovered he had superpowers. “Woah, you can do that?!” By pure reasoning, by drawing out consequences of a definition, one can prove beyond any shadow of a doubt that certain statements could not possibly be wrong? That’s a thing? That’s something one can do? Wow. Let’s do that to everything! Right?

So that’s how Thales discovered proof. As best as we can guess.

A few other theorems are attributed to Thales as well. I want to bring up one in particular that I think is also a kind of archetype of what mathematics is all about.

The theorem we just saw, about the diameter bisecting the circle, perfectly embodies one prototypical mode of mathematical reasoning. The pure mathematics paradigm, you might call it. Logical consequences of definitions, proofs by contradiction. That kind of thing. Thales’s proof really hits the nail on the head with that whole aesthetic. We’ve been doing the same thing over and over ever since. A modern course in, say, group theory, for example, is just Thales’s proof idea applied five hundred times over, basically.

Now I want to take another one of the results attributed to Thales, and I want to argue that it is emblematic of another mode of mathematical thought. It’s a second road to proof. This second way is based more on play, exploration, discovery, rather than logic and definitions.

The example I want to use to make this point is what is indeed often called simply “Thales’s Theorem.” Which states that any triangle raised on the diameter of a circle has a right angle. So, in other words, picture a circle. Cut it in half with a diameter. Now raise a triangle, using this diameter as one of its sides, and the third vertex of the triangle is on the circle somewhere. So it looks like a kind of tent, sticking up from the diameter. And it could be an asymmetrical tent that is pointed more to one side or the other. No matter how you pitch this tent, as long as the tip of it is any point on the circle, then the angle between the two walls of the tent at that point, at the tip, is going to be a right angle, 90 degrees. That’s Thales’s Theorem.

How might Thales have proved this theorem? We don’t really know that based on historical evidence unfortunately. But let’s consider one hypothesis that makes sense contextually.

We must imagine that Thales would have stumbled upon the proof somehow. We are not trying to explain how someone might think of a proof of this theorem per se. That’s the wrong perspective because it takes for granted that in mathematics one tries to prove things. What we need to explain is where this vision to prove everything in geometry came from in the first place. How could someone have struck upon Thales’s Theorem unintentionally, as it were, and through that accident become aware of the idea of deductive geometry?

Indeed Thales’s Theorem is not terribly interesting or important in itself. If you had this vision of subjecting all of geometry to systematic proofs, why would you start with this theorem, or make this theorem such a center piece, as Thales supposedly did? You wouldn’t.

The interesting thing about Thales’s Theorem is not that is was one of the first results to which mathematicians applied deductive proof. Rather, the interesting thing about it is that it was the occasion for mathematicians to stumble upon the very idea of proof itself, unintentionally.

There’s a story about Thales falling into a well because he got so caught up in astronomical reasoning that he forgot his surroundings. It’s recorded in Plato: “While he was studying the stars and looking upwards, he fell into a pit. Because he was so eager to know the things in the sky, he could not see what was before him at his very feet.”

A legend maybe, but the discovery of Thales’s Theorem must have been a little bit like that too. Discovering mathematical proof must have been like falling into a pit. You are looking in one direction, and boom, suddenly you find yourself having accidentally smashed face first into this completely unrelated new thing that you didn’t know existed.

How could Thales’s Theorem be like that? Among all the world’s theorems, what makes Thales’s Theorem particularly conducive to this kind of fortuitous discovery of proof?

Here’s my hypothesis. In this age of innocence, before anyone knew anything about proof, people still liked shapes. The had ruler and compass. They used these tools for measuring fields and whatnot, but they also liked the aesthetic of it.

They were playing around with ruler and compass. Playing with shapes. After five minutes of playing with a compass you discover how to draw a regular hexagon. Remember? You probably did this as a kid. Draw a circle, and then, without changing the compass opening, run the compass along the circumference. It fits exactly six times. A very pleasing shape.

We know for a fact that people did this before Thales. There are hexagonal tiling patterns in Mesopotamian mosaics from as early as about -700.

Dodecahedra are another one of those things. The dodecahedron is like those twelve-side dice that you use in Dungeons and Dragons and stuff like that. Do-deca-hedron, it’s literally: two-ten-sided. So twelve-sided, in other words. Twelve faces, each of which is a regular pentagon. These things are in the archeological record. People made them of stone and bronze. A couple of dozen of dodecahedra from antiquity have been found, the oldest ones even predating Thales. They were used perhaps for oracular purposes, like tarot cards or something. Or maybe for board games, who knows?

In any case, my point is that people were interested in geometrical designs for various purposes: artistic, cultural, and so on. Not just measuring fields for tax purposes. And they were clearly working with instruments such as ruler and compass to make these things.

It’s easy to arrive at Thales’s Theorem by just playing around with ruler and compass, trying to draw pretty things. Start with a rectangle. Draw its diagonals. Put the needle of a compass where they cross, right in the midpoint of the rectangle. Set the pen of the compass to one of the corners of the rectangle. Now spin it. You get a circle that fits perfectly, snugly, around the rectangle.

But look what emerged. A diagonal of the rectangle becomes a diameter of the circle. And the rectangle pieces sticking out from it are precisely those kind of “tent” triangles that Thales’s Theorem is talking about. This suddenly makes the theorem obvious.

Why is Thales’s Theorem true? Why does any of those “tents” raised on the diameter of a circle have a right angle? It’s because it comes from a rectangle. Any such tent is half a rectangle. This is a powerful shift of perspective. By looking at the triangle this way we reveal hidden relationships, a hidden order in the nature of things. Certain angles must always be right angles by a sort of metaphysical necessity, as it were. Our eyes have been opened, maybe for the first time, to the existence of these kinds of necessities, these kinds of hidden relationships that are out there for the thinking person to uncover.

So the key is this shift of perspective that the triangle is “really” half a rectangle. Suppose instead that we had been stuck in the point of view is that we are staring at a triangle inscribed in a circle. Then the kinds of associations and ideas that suggest themselves to us are not so useful for proving this theorem. From that point of view, if you were looking for a proof, what would you do? Maybe you would for example connect the midpoint of the circle to the tip of the triangle. So now you have two smaller triangles. What are you going to do with those? Something with angle sums and so on? Or maybe you would be tempted to drop the perpendicular instead from the tip of the triangle, and then you can use the Pythagorean Theorem of the two small triangles you get.

These kinds of things are not what we want. Those kinds of approaches quickly become too technical. This was supposed to be the beginnings of geometry, remember. You are not supposed to use a bunch previous results for the proof. It should be a proof from first principles. A proof before all other proofs.

The idea that the triangle is “really” half a rectangle is different. It transforms how we look at the diagram. It changes the emphasis. It changes what we think of as primary. Now the rectangle comes first, and the triangle second, and the circle last. The theorem actually isn’t so much about circles at all, so to speak, from this point of view. The circle is just a kind of secondary artefact.

With this proof we are like artists. We take a step back from the canvas and tilt our heads and have this epiphany. And the epiphany was made possible by the way we had played with these ideas previously. We were just playing around with ruler and compass, we explored triangles and rectangles and circles with an open-minded affection. Epiphanies like Thales’s Theorem emerge from this play. Inspiration comes naturally in that context.

Unlike those other boring proofs I alluded to, that were based on cutting the triangle up and throwing the book at it: angle sums, Pythagorean Theorem, everything we can think of. That’s an uninspired approach, a brute force approach. It lacks that aesthetic inspiration, that epiphany of revealing the true nature of the triangle, and its other half that it was destined to be reunited with.

Geometry could not have started with these kinds of by-the-book proofs, because they only make sense after there is a geometry book to begin with. But geometry could have started with the epiphany type of proof. So that’s a way in which someone like Thales might have arrived at the idea of proof through playing around with ruler and compass.

Perhaps you are familiar with “Lockhart’s Lament”: a great essay on what is wrong with mathematics education. Go read it, it’s available online. It is interesting that Lockhart uses this very example to make his point. He describes how his students discovered Thales’s Theorem basically the way I’m saying that Thales might have done so. He also eloquently captures how this is so much more satisfying than a dry by-the-book proof.

It’s not for nothing that history and education go together on this point. Proof must have started with a compelling aesthetic experience or wow moment. There was no other way at the time. There was no one to force Thales to memorise facts for an exam. Discovery compelled him to value mathematics. If we want to foster intrinsic motivation in our students, it’s a good idea to consider what made people fall in love with these ideas in the first place. First love is always the purest and most innocent. Modern textbooks are like arranged marriages forced upon the students. But history always has the true love story.

Nevertheless, for all this, you might still think that Thales’s Theorem is a bit boring. Something something is always a right angle. So what? Who cares?

As I tried to argue, it was probably not the theorem per se that was impressive to Thales and his contemporaries, but rather the idea that there is such a thing as theorems and proofs at all. There are hidden truths out there that can be uncovered through reasoning. Remarkable.

But in fact even the theorem itself is quite interesting. Let me show you something cool you can do with Thales’s Theorem.

There’s an ancient legend about Queen Dido. Daughter of the king of Tyre, a major city in antiquity. You can still see the ruins of this ancient city in present-day Lebanon. At a certain point Dido had to flee, because of court intrigues. Murders and betrayals and so on. So she grabs a couple of diadems off her nightstand, maybe a chest of gold she put aside for a rainy day, and hastily sails off into the night. With hardly a friend left in the world.

She has to go all the way to present-day Tunisia, thousands of kilometers away, and try to start over somehow, in a manner befitting a royal. Using her treasure chest, she strikes a bargain to buy some land. As much land as she can enclose with the skin of an ox, the story goes. So she cuts the ox hide into thin strips and ties them together, and now what? So now she has this long string, which she can use as a kind of fence to seal off the land she wants.

But what shape to make it? A square, a rectangle, a triangle? No. Dido knows better. Perhaps her royal education included mathematics. Make it round. That’s the best way. The circle has the maximal area among all figures with a given perimeter. Or in this case, since she was by the ocean: a semi-circle, with the shoreline as a natural boundary on the other side.

Let’s prove this. That the semi-circle is the best choice. I’m going to prove this by contradiction: Suppose somebody has fenced in an area that is not semi-circular; then I can show how to make it better: how to move the fence so that the area becomes even bigger, without adding any more fence.

Ok, so you have the shoreline, that’s a straight line. And from one point on the shore, going inland you have this fence which then comes back down and meets the shore again in some other point. So together with the shoreline it closes off a certain area.

Suppose this shape is not a semi-circle. If it was a semi-circle, Thales’s Theorem would apply. And it would tell you that this angle, what I called the tent angle, at any point along the fence would be a right angle. So if the shape is not a semi-circle, there must be some point along the fence where this angle is not a right angle.

I say that making this angle a right angle improves the amount of area covered. You can picture it like this. So you have this shape enclosed by the fence: imagine that you have that cut out of cardboard. And on the perimeter you have some point marked where the tent angle is not a right angel. So on your cardboard you have that triangle drawn: a triangle consisting of the straight shoreline on one side, and the two lines from its endpoints going up to meet at the tent point on the perimeter.

Let’s cut that triangle out of the cardboard. So you’re left with two pieces: whatever bits that were sticking out from the triangle sides. Now move those two pieces so that you make the tent angle a right angle. This means moving the endpoints along the shoreline. As you move the two points on the shoreline, you change the angle at which the two cardboard pieces meet. The two cardboard pieces meet in a single point, the tent point, and that’s like a hinge that can open or close to a bigger or smaller angle. So you slide these things around until that hinge angle becomes 90 degrees.

Note that you didn’t change the perimeter this way. You just moved the same amount of fence around.

But you did increase the area enclosed, in fact. Because if you have two sticks of fixed length, and you want to make the biggest triangle you can with those sticks, the best way is to make the angle between them a right angle. That’s quite clear intuitively. You know that the area of a triangle is base times height over two. So if one of your sticks is the base, then to maximise the area you want to maximise the height, that is to say the perpendicular height going up from the base, which is obviously done by pointing the other stick straight up at right angles.

So what this proves is that, for any fence enclosure that is not a semi-circle, you can make a better one. You can move the fence around and make the area bigger. So the semi-circle is the best solution, and all other ones are less good.

I don’t know if you could visualise all of that. But maybe try reconstructing this argument for yourself later. It really is very intuitive and beautiful.

So what’s the moral of the story then? Mathematically, it is an answer to the “so what?” question regarding Thales’s Theorem. It may have seemed like a boring enough theorem, but here we see it in action in a beautiful and unexpected way, as a key ingredient in this proof about how to enclose land. Who would have seen that coming?

This suggests that mathematics has a kind of snowballing or self-fertilising aspect to it. Thales’s Theorem, what’s the big deal? Just some boring observation about a triangle in a circle. May not seem like much. But one thing leads to another. Once Thales’s Theorem is a thing to you, you start seeing it in other places, unexpected places. Like this problem about area. You wouldn’t think it was related, but the more mathematics you do, the more connections you find.

Pick any theorem, no matter how boring, like Thales’s Theorem, and you can find these amazing things where the boring theorem is actually a key insight that opens entirely new ways of thinking about seemingly unrelated problems. That’s mathematics for you. No wonder it caught on like a bug among the Greeks, once they got the ball rolling. One moment you stumble upon some random result like Thales’s Theorem, and the next thing you know you’re seeing mathematics everywhere.

So that’s the mathematical moral of the story. Now we must go back and say something about the historical side of all this. What do we really know about Thales and his theorems and Queen Dido and all that? How much is history and how much is legend?

If we start with Dido, that story comes to use primarily through Virgil. The Aeneid, the famous epic poem. That was written in Roman times, around the year -20. But it is referring to historical, or supposedly historical, events that took place even before Thales, maybe two centuries before Thales, so -800-ish. We have Virgil’s version, that’s what has come down to us, but he is just stealing an older story. These things would have been around for centuries in Greek culture, in various literary and historical retellings that are now lost.

It is perfectly plausible that there really was such a historical queen, who really did flee her royal home in Tyre, and really did land on the north shores of Africa where she founded this new settlement, which was to become the great city of Carthage. Maybe indeed she even made the city walls semi-circular, who knows? It is perfectly conceivable that she might have wanted to minimise the perimeter for whatever reason, and that she might have known that a semi-circular shape was optimal for this purpose.

But at that time there would not have been any mathematical proofs of this, like the one I sketched above. The proof I outlined is from Jakob Steiner, in the early 19th century. From Greek times we have a different proof of this result. So they were certainly very much aware of the result, that the semi-circle is optimal, if perhaps not the particular proof I suggested.

If the story of Queen Dido says anything about the history of mathematics, it probably illuminates most neither the time when the events took place, around -800, nor the time when the sources we have were written, around year 0. But maybe it says something about the centuries in between, where the story would have been passed on and reworked.

The story was marinated, as it were, in Greek culture. Maybe they were the ones who gave it a mathematical flavour. The shoe fits: The Greeks valued wise, aristocratic, well-educated rulers, who design rational policy for the common good informed by reason and mathematics. Maybe they let these ideals colour the way they retold the story of Queen Dido and her round city.

From this point of view we could also speculate that by the time Virgil comes around and writes the Roman version of the story, this appreciation of mathematics is no longer what it once was. Indeed Virgil doesn’t really spell out the mathematical optimisation aspect of the story. Dido is just a side character altogether. His epic is about Aeneas, who is on a quest that will eventually lead to the founding Rome.

Aeneas is shipwrecked and blown ashore at Carthage, Dido’s round city. Dido falls in love with him, but he does not return her love. He sails away and Dido kills herself because of her broken heart. Morris Kline concludes the story: “And so an ungrateful and unreceptive man with a rigid mind caused the loss of a potential mathematician. This was the first blow to mathematics which the Romans dealt.” Sure enough there’s plenty more where that came from.

One can view this story as symbolic of this transition from the wise philosopher kings (or queens in this case) of the Greek world, who cherished mathematics and used it to improve the world. The transition from that to the heartless Roman, who only think of themselves and couldn’t care less about Thales’s Theorem. In the Greek world math nerds were considered attractive, but somehow these ignorant Romans didn’t think a geometer queen was girlfriend material at all evidently.

Ok, so the story about Dido and the round city and the optimisation proof and all that, it is very interesting in terms of the broader mathematical and cultural points its connects to, but in and of itself its is not directly history per se.

It’s different with Thales. That’s more fact than legend. As best as we can determine, Thales really did prove that diameter bisects a circle, most likely with the proof discussed above.

The sources that we have for this are far from perfect. Primarily Proclus, who was writing in year 450 or so, basically one thousand years after Thales lived. These kinds of late sources are hit and miss. They have no authority in and of themselves. Proclus was nobody. His own understanding of history and mathematics is very poor. A mediocre thinker, a mediocre scholar, living in a mediocre age.

Those are the kinds of sources that we have. Basically as authoritative as a factoid you read on the back of a cereal box or something.

But there is hope. Back in its glory days, Greece was just an outstanding intellectual culture. And some of the stuff about for example Thales can be traced back to then, which makes it highly credible. Aristotle’s student, Eudemus, wrote a history of geometry. It’s no longer with us alas. Ignorant ages neglected it and now it’s gone. But what a work that would have been.

These people knew what they were doing. Later people like Proclus are like some online rando posting half-baked ideas on blogspot or poorly informed comments on Facebook. That’s how credible they are.

But people like Eudemus is a very different story. That is more like a first-rate scholar at a research institution with all the infrastructure one could dream of: libraries, extremely knowledgable and intelligent colleagues with a range of expertises, broad financial and cultural support from the public and from politicians, and so on. Eudemus’s History of Geometry would be a proper “University Press” book, peer-reviewed to the teeth and with a nice dust-jacket blurb by Aristotle.

People like Eudemus were not in the business of passing on random gossip and unchecked factoids because they sound cool. They were proper scholars and intellectuals.

And indeed a lot of the stuff about Thales can be traced back to this lost source. When Proclus says that Thales was the first to prove that a circle is bisected by its diameter, the source of this is Eudemus. Hence it is very credible. This Thales stuff really happened. Actually that part about the diameter bisecting the circle is more certain than the part about Thales’s Theorem. Was Thales’s Theorem really Thales’s? Maybe. But we cannot trace that part specifically back to the best sources. Unlike the diameter bisection one and some other details. But contextually it makes sense.

The stories of Thales and the origin of geometry were evidently well known not only to specialised scholars but to the general Athenian public. Aristophanes the playwright uses the name of Thales as a symbol of geometry a few times in his plays. Just as today one might use the name of Einstein for instance to evoke the image of a scientist. Aristophanes has one speakers in a dialogue say: “The man is a Thales.” Meaning that the person is a geometer. Evidently the theatre-going public in classical Athens could be expected to understand this reference. Every educated person would know about Thales and the origins of geometry.

In fact, public respect for geometry and its history was apparently so great that Aristophanes even has one of his characters lament it as excessive, saying: “Why do we go on admiring old Thales?” What a time to be alive that would have been. When playwrights had to tackle issues such as there being too much respect and interest in mathematics among the general public. “Hey guys, maybe we need to cool it with how much we love geometry.” What a luxury problem. Hardly one that Hollywood blockbusters today have to grapple with.

Anyway, we should maybe not read too much into those isolated quotes. But the general intellectual credibility of this age is important. These very intelligent and serious people recorded in scholarly histories the accounts about Thales founding deductive geometry and proving that a circle is bisected by its diameter. That’s only some two or three hundred years after Thales, and in a direct lineage from him, probably with entire works by Thales still around in libraries and so on.

So there you go. The origins of proof and deductive geometry. We really do know quite a bit about it, and it’s a story worth knowing if you ask me.

]]>**Transcript**

How did geometry start? Who was doing it, and why, in early civilisations? The Greeks invented theorem and proof, but long before them there was geometry in Egypt and Mesopotamia. So that’s practical geometry, applied geometry.

Or is it? Actually even the oldest sources have lots of pseudo-applications in them. Such as: Find the sides of a rectangular field if you know the perimeter and the diagonal. Or: I have two fields, and I know how much grain each field produces per unit area, and I know the total grain produced by both of them, and I know the difference between their areas, now tell me how to find the area of each field.

Not the kind of situations you find yourself in every day exactly. You can judge for yourself if that deserves to be called applied mathematics. Given obscure and convoluted information, find something that should have been much easier to measure directly than this artificial data you somehow had access to.

In any case, geometry like that, whatever you want to call it, was highly developed almost four thousand years ago. Why? What made people do this? Let’s try to find out.

Early mathematics emerged where there was fertile soil. Rivers that made this possible. Agricultural abundance meant resources enough to expend some people specialising in mathematics instead of having all hands on the ploughs.

Look at a modern population density map of Egypt. You will find that virtually the entire population is concentrated along the Nile; all the rest is pretty much desert. That’s still today. Even with the assistance of modern technologies the river area is by far the most liveable. Even more so back then when geometry started, thousands of years ago.

It was the same in Mesopotamia, present-day Iraq. Also a river civilisation with very good agricultural conditions. They had legendary gardens that were praised in ancient sources. Google it: The Hanging Gardens of Babylon. You will see some nice pictures of what these luxurious gardens might have looked like. That’s a nice visual for this idea that it was agricultural abundance that made a specialised pursuit like mathematics possible in those societies.

So that explains why they had the resources to support mathematics. But why would they want to? What did they stand to gain from geometry?

Basically, mathematics was for a long time about commerce and taxes; bureaucratic management of workers and produce; inheritance law. Those kinds of things.

Eleanor Robson’s book is very illuminating about this. “Mathematics in Ancient Iraq: A Social History”, the book is called. She emphasises especially that mathematics was very strongly associated with justice. A society without a functioning justice system is hampered by constant disputes about land, taxes, inheritance. Everybody is fighting with everybody. Like the old American West, you board yourself up and mind your own business and if there’s a disagreement, well, that’s what guns are for, isn’t it?

Mathematics is the way out of this primitive state. Mathematics is objective. It can settle these disputes in a fair way. If everybody is wasting a huge amount of effort and resources on petty disputes in a lawless no-man’s land, who you gonna call? The mathematicians, that’s who. That’s how it went in ancient Iraq.

A specialised, highly trained mathematician would come in and delineate all the plots of land, compute all the taxes owed, and distribute every inheritance. All according to exact calculations. This stuff used to be ruled by emotions, personal animosity, and the law of the jungle. But now, thanks to mathematics, that is replaced by objective rules. Who can argue with a calculation? Mathematics takes the worst sides of human nature out of the equation.

When society is run by fair, universal rules, people no longer have to constantly look over their shoulder and fear that some lawless eruption of force could destroy everything they have at any moment. A functioning justice system enables people to work for the collective good and to plan for the long term.

It is the authority of mathematics that makes this possible. These skilled mathematical technocrats had great credibility because people recognised that they were above the subjective and the emotional. They were bound by dispassionate calculation. Mathematics compelled them to be fair and rational.

Indeed they explicitly said so themselves. As one mathematical scribe put it: “When I go to divide a plot, I can divide it; So that when wronged men have a quarrel I soothe their hearts. Brother will be at peace with brother.” That’s a quote by one of those mathematical technocrats, explaining what geometry accomplishes. Note that it has both of those elements I emphasised. Mathematics is the opposite of emotional disputes. It soothes heated hearts, it creates peace between warring brothers. And the quote also highlights that this happens because of the expertise of the mathematician: I know how to do this kind of thing, the technocrat is saying. It takes special training.

The quote is from Eleanor Robson’s book. Here’s another thing she points out that is yet more evidence of the importance of mathematics in this context. The Sumerian word for justice literally means straightness, equality, squareness. Also in Akkadian: justice is the “means of making straight.”

Again, another major indicator of this: “the royal regalia of justice were the measuring rod and rope.” Think of those Lady Justice statues that you see sometimes. She’s blindfolded because that shows that she’s unbiased, and she has these scales, showing that she’s considering both sides and weighing them carefully and fairly. That’s the symbol of justice in our society. But, in ancient Babylon, the symbols of justice were not a blindfold and a set of scales. Instead, Lady Justice was a geometer. She held her land-measuring tools. Those were the instruments of justice in ancient society.

Maybe it’s pretty much the same today, four thousand years later. Back then, the trustworthiness of mathematics was a cornerstone of society. If people didn’t trust mathematics, there could be no law and order, no state bureaucracy, no complex economy, no civilisation. Today, that link is perhaps less evident. But perhaps no less crucial. We have added many layers of complexity to our society, but perhaps looking back at historical societies is the same thing as looking into the inner essence of our own. Maybe without faith in mathematics the entire fabric of our society would unravel. Maybe without mathematicians mediating their disputes, “brother would be at war with brother” as that ancient scribe feared.

It is interesting also that this role of mathematics that I have outlined is really as much psychological as it is scientific. What makes this whole system work is not only that mathematics can give useful answers to certain technical problems. The psychological side is equally essential: mathematics has a kind of aura of objectivity, of trustworthiness, of professional expertise. That goes well beyond merely calculating the taxation rate of some field, or how many goats you can buy for a silver shekel. The system rests on a more nebulous trust in the mathematician class by the population at large. The idea of mathematics, the image of mathematics, is more important than the sum of its actual applications.

That’s an important conclusion because it explains that striking feature of ancient mathematics: namely that many of the problems the ancients texts solve are super fake. They are pseudo-applications.

For instance: Find the two sides of a rectangle, given that the sum of the length and the width is 24, and that the area plus the length minus the width = 120. So in other words, you basically have two equations in x and y, and if you solve for y in one and plug it into the other you have a quadratic equation in x. Lots and lots and lots of problems like that in Babylonian mathematics.

Obviously nobody would ever face a problem like that in any real-word situation. It’s very often like this: you are looking for something simple, like the sides of a rectangle x and y, and you are given some super weird, like some convoluted combination of x and y is three eights of some other convoluted combination of x and y.

Here’s another actual one: The width of a rectangle is a quarter less than the length. The diagonal is 40. What are the length and the width?

In what real-world scenario can you realistically end up knowing the diagonal of a rectangle, and the difference between the sides, but not the sides themselves? And why couldn’t you just measure the sides? Someone did measure the diagonal, apparently, so why not the sides?

Sometimes these texts hardly even try to hid how fake they are. One problem goes: I found a stone, but did not weigh it. I cut away one-seventh and then one-thirteenth, and then it weighed so-and-so much. What was the original weight of the stone?

Who among us has not “found” whatever random stone, then chipped away an extremely exact ratio of it, and then suffered some kind of stone-cutter’s remorse I guess, and tried to reconstruct the original weight of the stone for some reason.

Very relatable, isn’t it? Actually it kind of is. Not because we are sitting around cutting one-thirteenth out of random stones, or because we are running around measuring the diagonals of various fields and then later wish we had measured the sides instead. That never happens to any sane person in the real world. But it does happen in math books. Still today, we torture our students with such questions, one more artificial and unrealistic than the other.

Some people think that kind of thing is modern pedagogy run amok. They see these kinds of problems in modern textbooks and they think: How silly modern pedagogy has become! These naive educators are bending over backwards to make math “relevant” to kids, but the just end up with silly fake problems.

History offers a different perspective. The problems may be silly, but the cause is not a misguided obsession with real-world relevance among modern educators. Fake problems are as old as written mathematics itself. For as long as there has been mathematics education, students have been forced to go through page after page after page of pseudo-problems that only superficially, or linguistically, appear to be talking about real-world things, while actually corresponding to absurd scenarios that would never happen.

In a way one might argue that history vindicates these problems. They are not so silly after all, if we consider them in the light of the role of mathematics in ancient Babylonian society. Mathematics doesn’t support the economy merely by keeping the account books. It’s more than that. Mathematics is what instills confidence in monetary law and order, without which any kind of complex economy would be impossible in the first place.

For this system to work, there needs to be a specialised class of number-crunching technocrats. These people need to embody logic and reason and objectivity. They need to be math machines, detached from politics and emotion. A long schooling in artificial pseudo-problems makes some sense as a means of creating this class.

From this point of view, it is even a strength that these problems are artificially divorced from real-world problems, because the mathematical technocrat is supposed to be detached from such concerns anyway. Mathematicians are valuable to society precisely because they are so disinterested in the needs of people of flesh and blood. It is this disinterestedness that makes people willing to trust the mathematicians to be the arbiters of disputes.

The sheer volume of training in pointless problems also has its point. It is not enough that people at large know some mathematics: they could use mathematics as a tool for evil, as just one more incidental weapon in a society still ruled by greed and conflict. For a complex economy to take off, there needs to be faith that the law and the state administrative bureaucracy are fair and consistent. This faith comes from the credibility of mathematics. The mathematical technocrats need to be proper experts to justify the confidence placed in them. They need to embody mathematics; they need to single-mindedly look at any situation or conflict and see only the mathematics in it.

Society needs the mathematicians to not only get the right answer, but to have great authority as proper experts. And it needs them to be “nerds,” so to speak, who are so one-sidedly developed that they can only see mathematics anywhere they look, and not let emotions or politics influence their work. A long and rigorous training in fake applied problems is not a bad recipe for bringing this about. Arguably, we pretty much still use the same recipe to the same end today, thousands of years later.

So that’s the Babylonian tradition. We know quite a bit about it because they wrote on clay which is pretty durable. In Egypt, mathematicians was recorded on papyrus, which isn’t going to survive for thousands of years normally. So we only have two or three or maybe four papyri that beat the odds and were conserved. But it seems the Egyptian situation may very well have been quite similar to the Mesopotamian one in terms of the role of the mathematicians.

“Geometry” means “earth-measurement.” That’s from the Greek: geo metria. The ancient Egyptians had the same idea but their word for it was more concrete: a geometer was literally a “rope-stretcher.” A land surveyor stretches ropes to measure distances and delineate fields.

A rope is pretty much equivalent to a ruler and compass. Pull the ends of the rope and you have a straight line. Hold one end fixed and move the other one while keeping the rope stretched: now you have a circle.

Euclid explains how to make a square with ruler and compass. That’s Proposition 46 of the Elements. The Egyptians would have done that long before with their stretched ropes. Try it for yourself, it’s fun: go out into a field with a friend and try to make a perfect square using nothing but a piece of string. You will see why geometers were called rope-stretchers.

Do you think you could make a square? Do you think anyone could? Back in the day, this skill could have given you a leg up in life. Suppose you make one square field, and then a rectangular field with the same perimeter. The square field will have greater area. But you could trick those less knowledgeable in mathematics. You could say: you get that field and I get this one, fair and square. Just try it for yourself, you would say, let’s walk around the fields and count the number of steps. 400 steps around my field, 400 steps around yours: aha, our fields are the same size. That’s what you tell the other guy, who isn’t such a math person. But you know that of course 100*100 is way more that 50*150. So later you get a much greater harvest. But of course you would pretend that that’s because you worked so hard while the other guy was lazy. Maybe that’s another way in which ancient society is like ours: privileged people use their privilege to rig the game in their favour, and then pretend it was all due to merit.

According to Proclus, this kind of mathematical deceit did indeed happened: “The participants in a division of land have sometimes misled their partners. Having acquired a lot with a longer periphery, they later exchanged it for lands with a shorter boundary and so, while getting more than their fellow colonists, have gained a reputation for superior honesty.”

Here’s how Thomas Heath paraphrases this in his History of Greek Mathematics: “Proclus mentions certain members of communistic societies who cheated their fellow members by giving them land of greater perimeter but less area than the plots which they took themselves, so that, while they got a reputation for greater honesty, they in fact took more than their share of the produce.”

A dubious paraphrase, in my opinion. Can you spot the suspicious part of it? Good old Heath put something in there that was not in the original source. Hint: turn to the title page of Heath’s book. There are some clues there. The book was published by Oxford University Press in 1921. Heath’s name comes with some bells and whistles: it’s Sir Thomas Heath, in fact, and then K.C.B, K.C.V.O. That’s Knight Commander of the Royal Victorian Order etc.

Titles upon titles. It’s an establishment guy, this Sir Thomas. A gentleman scholar, who was a civil servant as his day job at the Treasury.

What part of Sir Thomas’s paraphrase of the ancient mathematical land deceit reflects his own social context more than that of the ancients he is trying to describe? I’m thinking of his phrase that these were “communistic societies.” The original source says nothing at all about this having anything to do with communism. But you can understand how Sir Thomas would have been concerned about communism at this time. The Russian Revolution started in 1917, Heath’s book is published in 1921. While writing the book, Heath was a secretary at the British Treasury. He would have read all about Lenin and Bolsheviks in The Times while having his afternoon tea. And those worries would have been at the top of his mind when he sat down in his study to do his scholarly work in the evening. It didn’t take much provocation, one imagines, for him to have a swing at how “communistic societies” were dreadful and corrupt.

We must always read historical sources this way. Context matters.

Now, the “original” in this case was Proclus. But that’s not much of an “original” to speak of. Proclus is nobody. He’s not particularly trustworthy. He was writing in the year 450 or so, thousands of years after the historical events he is talking about. So it’s anybody’s guess how much truth there is in what he is saying. And in any case, like so many other mediocre writers, both ancient and modern, Proclus is just copying what others had said.

Let’s illustrate this point. Let’s see what we can learn by looking at Proclus’s account of the origins of geometry in Egypt. Here’s what Proclus says:

“Geometry was first discovered by the Egyptians and originated in the remeasuring of their lands. This was necessary for them because the Nile overflows and obliterates the boundary lines between their properties. It is not surprising that the discovery of this and the other sciences had its origin in necessity, since everything in the world of generation proceeds from imperfection to perfection. Thus they would naturally pass from sense-perception to calculation and from calculation to reason. Just as among the Phoenicians the necessities of trade gave the impetus to the accurate study of number, so also among the Egyptians the invention of geometry came about from the cause mentioned.”

Ok, sounds pretty plausible. But it’s worth running Proclus through a plagiarism checker, just as we do with modern student essays these days. Cutting-and-pasting from Wikipedia is nothing new. Proclus had many Wikipedia equivalents available to him. Perhaps he stole the whole thing for example from the Geography of Strabo, which was written more than 400 years before. Here’s what Strabo says:

“An exact and minute division of the country was required by the frequent confusion of boundaries occasioned at the time of the rise of the Nile, which takes away, adds, and alters the various shapes of the bounds, and obliterates other marks by which the property of one person is distinguished from that of another. It was consequently necessary to measure the land repeatedly. Hence it is said geometry originated here, as the art of keeping accounts and arithmetic originated with the Phoenicians, in consequence of their commerce.”

Basically a dead ringer for the Proclus passage. Plagiarism detected, SafeAssign™ would say.

Actually Proclus has added something that is not in Strabo, namely the claim that this historical episode illustrates how human though passes from the world of the senses to the higher realm of reason. This is card-carrying Platonism. Proclus is a sycophantic follower of Plato. He sees everything through Plato-coloured glasses. Which is not helpful if we want to use him as a source of historical information. As Heath had his anti-communism, so Proclus has his Platonic axe to grind and it infects everything he says.

Actually we can go back even earlier than Strabo. Let’s take an equal jump back in time again: another 450 years still. From Roman Strabo to classical Greek Herodotus. He too speaks of the origins of geometry in Egypt. Let’s listen to his account:

“This king [Sesostris] also (they said) divided the country among all the Egyptians by giving each an equal parcel of land, and made this his source of revenue, assessing the payment of a yearly tax. And any man who was robbed by the river of part of his land could come to Sesostris and declare what had happened; then the king would send men to look into it and calculate the part by which the land was diminished, so that thereafter it should pay in proportion to the tax originally imposed. From this, in my opinion, the Greeks learned the art of measuring land.”

Ok, I have to admit that this makes Thomas Heath look a bit better. “The king gave to each an equal parcel of land”: That is a bit more like communism. Heath said he was paraphrasing Proclus where there is no such phrase about equality. But Herodotus, the better source, kind of vindicates him a bit. You could imagine, in the scenario that Herodotus describes, that certain administrators in charge of implementing the king’s decree might secure a nice big square plot for themselves and trick the mathematically illiterate into a smaller plot with the perimeter trick. Perhaps not entirely unlike how corrupt middle-managers in the Soviet bureaucracy might manipulate the system for personal gain. But be that as it may.

I think there’s another interesting thing about Herodotus’s description compared to Strabo’s. Strabo and Proclus give a cleaner and simpler account: the flooding of the Nile obliterates everything and you have to start afresh each year with the drawing of boundaries. Herodotus’s account is much less dramatic: some parts of properties might become damaged by the floods, and the task of the mathematician is not to redraw the entire agricultural map each year but rather to calculate what proportion of area has been lost in each case for taxation purposes.

One can easily imagine how a desire to simplify and tell a clear and dramatic story might have led authors like Strabo and Proclus to prefer their version. The older source is a bit more “boring” but perhaps that makes it more credible.

Indeed, Herodotus’s account fits better with what we said about the role of mathematics in Mesopotamian society. In Herodotus’s version, the mathematicians task is more technical, more specialised, more bureaucratic. Note his phrase: “the king would send men” to do the calculations. You have to send mathematicians. They are a small, specialised class of technocratic experts that are dispatched to solve disputes with authority and objectivity. That’s precisely the main point I have made today, so let us end there.

]]>**Transcript**

Why the Greeks, of all people? Why did mathematics start there, on a few scrawny little islands in the Mediterranean?

The very idea that mathematics is about systematically proving things is an exclusively Greek invention. Axiomatic-deductive mathematics has been discovered only once in human history. No other culture independently developed anything like it.

The lettered diagram is another uniquely Greek invention. Triangle ABC, the line AB, stuff like that. Geometrical diagrams with the points denoted by letters. Only in Greece did they feel the need to do geometry this way. If you find it elsewhere, it’s because they copied it from the Greeks.

Not that the lettered diagram is a big deal in itself, of course. But it’s a symbol; it’s emblematic of how so many aspects of mathematics that we now consider so essential and indispensable were in fact discovered once and only once in human history, at a particular time and place.

So what was it about that time and that place that made it explode with intellectual progress?

You can make a pretty good case for geographical determinism. The seeds of excellence was not in the blood or the genes of these people, but it was in the land and the sea.

Islands. That is the key. Greece is a country of a thousand islands. In fact, you can hear this in the very names of the great mathematicians of that time.

Consider Pythagoras, for example. More fully you often see his name given as Pythagoras of Samos, his place of birth. Which is an island. One of those typical picturesque Greek islands.

The same goes for other great Greek mathematicians. Hippocrates of Chios, Aristarchus of Samos, Archimedes of Syracuse, Hipparchus of Rhodes: island, island, island, island. Everybody is an islander in Greek mathematics. There’s also Eudoxus of Cnidus, and Diocles of Carystus: those are technically peninsulas, but pretty nearly islands basically.

What’s with all these islands? Let’s see where this geographical argument leads us.

First of all, islands are excellent for trade. Back then, it was a thousand times easier to transport goods by water than by land. Even the Romans, centuries later, used to import huge amounts of grain from Egypt for example. And that’s the Romans, who are famous for their excellent roads. Even to them it was much more of a hassle to get grain from mainland Europe than to swish it across the sea with some efficient ships.

So the Greeks became tradespeople. Because they had so much access to the water.

And what did they have to trade? Think of the typical landscape of a Greek island. It’s hilly and full of slopes and kind of dry, rocky soil. Not the typical agriculture landscape you would have on the irrigated flats of mainland Europe or America. That kind of stuff would slide right off the Greek hills. In Greece you need tougher plants with roots that really dig in and hang on for their life as a rain shower threatens to wash the whole thing down with it down the hill.

Hence: olives and grapes. These plants love a good slope. They thrive there.

And what luck for the Greeks! These plants are perfect for trade. Think about it. You use them to make olive oil and wine: expensive, non-perishable luxury products.

Vegetable and fruit is highly perishable: by the time you get to your destination to sell it, half of it is rotten or eaten by worms. And it’s also very bulky: a big barrel of cabbage isn’t going to fetch you a whole lot of cash. It doesn’t have many calories. So it’s a lot of work to transport for so little payoff. The cabbage business isn’t very lucrative.

But olive oil and wine is perfect. Olive oil is a calorie bomb: a little goes a long way, so it’s easy to transport a fortune’s worth of it. And these products don’t mind being stored. Just stick them in an urn with a good cork on it and you’re set. Wine can even get better by sitting around. Unlike a sack of cucumbers that will spoil before you put your sandals on.

Olive oil and wine are also highly processed. A lot of work goes into the production. What are you gonna do with a bag of cucumbers? They are what they are, you just eat them. But the grapes and olives are processed by expert artisans. Lots of added value. The labor theory of value, you know, that Marx talked about and so on.

So the Greek islands are a recipe for wealth. Perfect products for trade, and perfect access to the sea for trading. This creates wealth, which creates a large middle class with lots of leisure time. That is certainly a precondition for intellectual culture.

Maybe also trade is itself a recipe for a certain open-mindedness and diversity of thought. There was no Internet back then. Travel was a good way to get exposed to other ideas, other ways of doing things. And therefore to start thinking more critically about the idiosyncrasies of your own habits and worldview.

Plus, a merchant needs to trade with whoever is paying. That may be people of different religions and so on. So you get used to dealing with people different than yourself. You develop and kind of tolerance for differences of opinion, and strategies for reasoning with people you disagree with.

All that from trade. But there is a second big consequence of the islands: independent city states. Islands are naturally isolated units. It will be much harder for a single despot to impose a unified rule on a bunch of scattered islands than on a solid land mass.

This is the geography of democracy. And democracy means debate. You don’t have “do this because I’m the king and I’ll chop your head off.” Instead you have one guy presenting reasons for this, the other guy presenting reasons for that, and people are weighing the arguments and making up their own minds.

This is going to be the setting that gives birth to mathematics and philosophy. Geography created this rich, democratic, cosmopolitan people who fell in love with clashes of ideas and took that concept to the extreme.

Geoffrey Lloyd the Cambridge professor has written good stuff about this. I’m going to quote extensively from his works.

“The level of technology and economic development” in ancient Greece was high indeed. In fact, it was “far in advance of many modern non-industrialised societies” today. And “Aristotle [explicitly] associated the development of speculative thought with the leisure produced by wealth.” And not for nothing.

However, “Egypt and Babylonia were, economically, incomparably more powerful than any of the Greek city-states.” So the explanation for the “additional distinctively Greek factor” of “generalised scepticism” and “critical inquiry directed at fundamental issues” must be something other than wealth alone.

The answer may lie in “a particular social and political situation in ancient Greece, especially the experience of radical political debate and confrontation in small-scale, face-to-face societies. The institutions of the city-state put a premium on skill in speaking and produced a public who appreciated and the exercise of that skill. Claims to particular wisdom and knowledge in other fields besides the political were similarly liable to scrutiny, and in the competition between many and varied new claimants to such knowledge those who deployed evidence and argument were at an advantage compared with those who did not.”

The Greeks were so fond of debates and clashes of ideas that they developed a refined social machinery for it. They ritualised and institutionalised the concept of a philosophical debate. “Public debates between contending speakers in front of a lay audience” was a prominent part of ancient Greek culture. Science and philosophy were born on this stage. Many otherwise peculiar characteristics of Greek thought are explained by this format.

For example, the stage debate requires the speakers to proclaim bold and provocative theses, and to strive to avoid reconciliation with other viewpoints at all costs. This is why early Greek thought is rife with crackpot claims such as that motion is impossible or “that man is all air, or fire, or water, or earth.” Indeed, the format demands a multiplicity of such viewpoints in competition with one another, whence “the remarkable proliferation of theories dealing with the same central issues” that “may well be considered one of the great strengths of Presocratic natural philosophy.”

Indeed, this used to always puzzle me. How can anyone in their right mind genuinely believe themselves to have discovered that “all is fire” or “all is water”? What were these people smoking, right? And that’s just a couple of generations before peak Greek philosophy and its many very refined insights in mathematics and science. How can they have been such crackpots and then gone from 0 to 100 in the blink of an eye?

But in fact it makes sense in the stage debate setting. “All is fire” is perfect for that. It’s like a dangerous stunt. Jumping across a ravine with a motorcycle, or juggling with three chainsaws. To go on stage and say “all is fire,” now try to prove me wrong, I will answer any counterargument. If somebody pulls that off, credit to them. The crazier and the more implausible their initial thesis is, the more impressive it is if they manage to parry objections and defend their thesis with clever arguments.

Nobody ever actually believed that “all is fire,” but they admired the guts of someone who was prepared to argue as if they did believe it. They glorified the ability to argue unconventional ideas well. This was a great move for stimulating philosophy.

The stage debate setting also explains why these kinds of crazy theses were always defended by abstract deductive reasoning, not empirical investigation. “Given an interested but inexpert audience, technical detail, and even careful marshalling of data, might well be quite inappropriate, and would, in any event, be likely to be less telling than the well-chosen plausible—or would-be demonstrative—argument.” Hence we understand why “with the Eleatics logos—reasoned argument—comes to be recognised explicitly as *the* method of philosophical inquiry.” This “notion of the supremacy of pure reason may be said to have promoted some of the triumphs of Greek science.”

However, these triumphs of reason “were sometimes bought at the price of a certain impoverishment of the empirical content of the inquiry.” In early Greek science, “observations are cited to illustrate and support particular doctrines, almost, we might say, as one of the dialectical devices available to the advocates of the thesis in question.” Also, “observations and tests could be deployed destructively [to disprove an opponent’s thesis], as they were by Aristotle especially, with great effect.”

These uses of observation fit well within the stage debate format. However, “theories were not put at risk by being checked against further observations carried out open-endedly and without prejudice as regards the outcome.” We can understand why since “The speaker’s role was to advocate his own cause, to present his own thesis in as favourable a light as possible. It was not his responsibility to scrutinise the weaknesses of his own case with the same keenness with which he probed those of his opponent.”

Of course, everyone was well aware of the deceptive potential of sly rhetoric for “making the worse argument appear the better.” So much so, in fact, that “early on it became a commonplace to insist on your own lack of skill in speaking.” But the Greeks did not see this problem, the rhetoric problem, as a reason to abandon the stage disputation format altogether. Instead they focussed on explicating “the correct rules of procedure for conducting a dialectical inquiry,” to ensure the intellectual integrity of the debates.

What I just described is basically a summary of Geoffrey Lloyd’s book “Magic, Reason and Experience,” about the origin of Greek scientific thought. Also very illuminating is Lloyd’s later book contrasting the Greek contrarian climate of thought with its opposite paradigm: reverential, conservative thought, typified for instance by the ancient Chinese tradition. The book title hints at this division: “Adversaries and Authorities,” it is called. Here is the argument.

“Any acquaintance with early Greek natural philosophy immediately brings to light a very large number of instances of philosophers criticising other thinkers.” Being a philosopher means being “subjected to blistering attack.” That could pretty much be considered the definition of philosophy in Greek antiquity. “From the list of occasions when philosophers are attacked by name, one could pretty well reconstruct the main lines of the development of Hellenistic philosophy itself.” Nor is this limited specifically to philosophy only. On the contrary, “hard-hitting polemic” is the name of the game in mathematics, medicine, and art as well. There is a “lack of great authority figures”; even Homer “is attacked more often than revered.”

This Greek style of philosophy is connected to its social context. “Greek pupils could and did pick and choose between teachers. Direct criticism of teachers is possible, and even quite common. Argument and debate are one of the means of attracting and holding students, and secondly they serve to mark the boundaries of [schools of thought].” “The Greek schools were there not just, and not even primarily, to hand on a body of learned texts, but to attract pupils and to win arguments with their rivals. They may even be said to have needed their rivals, the better to define their own positions by contrast with theirs.” “Dialectical debate, on which the reputations of philosophers and scientists alike so often depended, stimulated, when it did not dictate, confrontation. The recurrent confrontations between rival masters of truth left little room for the development of a consensus, let alone an orthodoxy; [and] little sense of the need or desirability of a common intellectual programme.”

“It was the rivalry between competing claimants to intellectual leadership and prestige in Greece, that stimulated the analysis of proving and of proof.” “Many have assume that the internal dynamic of the development of mathematics itself would, somehow inevitably, eventually lead to a demand for strict axiomatic-deductive demonstration, and that there is accordingly no need to pustulate any external stimulus such as [this.] Yet the difficulty for that view is [that] other, non-Greek, ancient mathematical traditions — Babylonian, Egyptian, Hindu, Chinese — all got along perfectly well without any notion corresponding to axioms and the particular notion of strict demonstration that went with it.”

The underlying cause is perhaps captured by the dichotomy between “adversarial Greeks and irenic, authority-bound, Chinese.”

The different philosophical styles of ancient Greece and China reflect differences in their political systems. “Extensive political and legal debates, in the assemblies, councils and law-courts, were a prominent feature of the life of Greek citizens.” Democracy primes people for debate, for listening to and assessing different points of view and conflicting claims.

“Greek philosophical and medical schools used, as the chief means for the expression of their own ideas and theories, both lectures and open, often public debates, sometimes modelled directly on the adversarial exchanges so familiar in Greek law-courts and political assemblies.” They imported democratic practices and put them to work in the sciences.

It was very different in China. “Many Greeks seem to have positively delighted in litigation; [they developed] taste for confrontational argument in that context and became quite expert in [evaluating such arguments]. [The Chinese, by contrast,] avoided any brush with the law as far as they could. Disputes that could not be resolved by arbitration were felt to be a breakdown of due order and as such reflect unfavourably on both parties, whoever was in the right.”

“The typical target audience envisaged in Greek rhetoric is some group of fellow citizens,” just as “in Greek law-courts the decisions rested with [peers] chosen by lot [who] combined the roles of both judge and jury.” “In China, the [intended] audience for much philosophical and scientific work was very different: the ruler or emperor himself.” “The Chinese were never in any doubt that the wise and benevolent rule of a monarch is the ideal.”

“We often find Greek philosophers adopting a stance of fierce independence vis-a-vis rulers. With this independence came a disadvantage. Compared with their Chinese counterparts, Greek philosophers and scientists had appreciably less chance of having their ideas put into practice. Autocrats — as in China — could and did move swiftly from theoretical approval to practical implementation.” Not so in Greece. Greek philosophers had little hope of real power, and perhaps that’s why they liked to pretend that they didn’t want any anyway. “The superiority of theory to practice is a theme repeatedly taken up by scientists as well as philosophers in Greece: but that was sometimes to make a virtue out of necessity.”

“Unlike in classical Greece, the bid to consolidate a comprehensive unified world-view was largely successful in China.” “The prime duty of members of a Chinese Jia was the preservation and transmission of a received body of texts. In that context, pupils did not criticise teachers, and any given Jia did not see it as a primary task to take on and defeat other Jia in argument.”

While the Greeks “adopted a stance of aggressive egotism in debate, the tactics of Chinese advisers was rather to build on what could be taken as common ground, [and] certainly on what could be represented as sanctioned by tradition.” “The emphasis is not on points at which [earlier philosophers] disagreed, but rather on what each of them had positively to contribute, how each succeeded, at least in part, in grasping some part of the Dao,” the true or right way.

So there you have it. The source of Greek exceptionalism in intellectual history comes down to this: to glorifying extreme adversarialism; to waking up in the morning and going “today I’m gonna point out errors in other people’s arguments.” The Greeks lived for that stuff. And it was this that made them mathematicians, eventually. But that was not a planned child. Geography led to democracy, which led to this combative philosophical climate.

When some fragments of mathematics from Egypt and the orient were dropped into this petri dish, the reaction was explosive. These two were made for each other. Mathematics and argumentative debate was match made in heaven. The Greek philosophical context triggered an avalanche of mathematical progress that took geometry from a set of obscure calculation rules to mankind’s best exemplar of perfect knowledge.

]]>**Transcript**

I’m going to conclude my case against Galileo with this final episode on this subject. Here’s a little anecdote I found that can be used to frame the overall point that I have made. Galileo was sentenced by the church in 1633. And to go along with this there was a bit of a crackdown on Galileo sympathisers. Somebody in Florence was going to publish a book that made reference to the “most distinguished Galileo.” But the Inquisition intervened and demanded that this phrase should be changed. Instead of “most distinguished Galileo,” the phrase should be changed to: “Galileo, man of noted name.” I am not generally on the side of the Inquisition, but I have come to the conclusion that this particular decree is sound. Instead of “Galileo, father of modern science,” we would be better off saying “Galileo, man of noted name.”

That’s what I have argued before. Today I will offer a bit of a roundup with some new perspectives on these issues. And that will be the end of my 18-episode rant against Galileo.

My main claim has been that Galileo was a poor mathematician. Historians are still blind to this fact. People still speak of “Galileo’s mathematical genius.” That persistent myth must certainly die. John Heilbron, the UC Berkeley historian of science, published an authoritative biography of Galileo in 2010. There Galileo is called “the greatest mathematician in Italy, and perhaps the world” in his time.

Galileo was no such thing. In reality, tell-tale signs of mathematical mediocrity permeate all his works. Many pages of Galileo would not be out of place somewhere in the middle of the piles of slipshod student homework that some of us grade for a living.

A number of Galileo’s numerous mathematical errors even concern some of his core achievements. I have discussed all of his notable scientific contributions and found much to object in every single case. For instance, Galileo uses “his” law of fall erroneously on a number of occasions: when he tries to explain the orbital speeds of the planets, when he tries to calculate how long it would take for the moon to fall to the earth, when erroneous claims to have proved that centrifugal whirling could never throw objects off the earth regardless of speed, and when he erroneously describes the path of a falling object in a reference frame not rotating with the earth. Galileo is praised for having discovered the law of fall, but the fact is that he derived as many false conclusions from it as correct ones.

He also not infrequently presents arguments that are demonstrably inconsistent with his core beliefs, such as his tidal theory contradicting his own principle of relativity, his Joshua argument contradicting his own principle of inertia, and his objection to the geocentric explanation of sunspots being inconsistent with his own heliocentrism.

I should say that, of course, other people made mistakes too. It was the early days of science after all. Suppose I concede that everyone has an equal comedy of errors to their name. Even so, this would still prove my point that Galileo was a dime a dozen scientist and not at all a singular “father of modern science.” But I do not in fact need to concede this much. Galileo’s sum of errors are not just par for the course. They are exceptionally poor, and in matters of mathematics especially they are astonishing.

We have seen time and time again that virtually all of “Galileo’s” achievements were either anticipated or at least made independently by others. To name just the most striking case:

“Let us hypothetically assume that a scholar contemporary to Galileo pursued experiments with falling bodies and discovered the law of fall as well as the parabolic shape of the projectile trajectory, that he found the law of the inclined plane, directed the newly invented telescope to the heavens and discovered the mountains on the moon, observed the moons of the planet Jupiter and the sunspots, that he calculated the orbits of heavenly bodies using methods and data of Kepler with whom he corresponded, and that he composed extensive notes dealing with all these issues. In short, let us assume that this man made essentially the same discoveries as Galileo and did his research in precisely the same way with only one qualification: he never in his life published a single line of it. As a matter of fact, the above description refers to a real person, Thomas Harriot.”

Actually these discoveries are not identical with those of Galileo but rather go beyond them, because Galileo never “calculated the orbits of heavenly bodies using methods and data of Kepler,” as Harriot did, who was a better mathematician.

So the history of science would have been much the same without Galileo, because people like Harriot and others were doing all of that stuff independently anyway.

It’s instructive to compare Galileo to Kepler in these kinds of terms. We can find independent contemporary discoveries for almost everything Galileo did, but not so for Kepler’s achievements, even though many of them are still central in modern science. Harriot was a “second Galileo” and you could go on to a third or a fourth stand-in without much loss. It would be much harder to find a “second Kepler.”

In my view it is not hard to see why: Kepler was an excellent mathematician who worked on difficult things, while Galileo didn’t know much mathematics and therefore focussed on much easier tasks. The standard story has it that Galileo’s insights were more “conceptual,” yet at least as deep as technical mathematics. On this account it is imagined that basic conceptions of science that we consider commonsensical today were once far from obvious: we greatly underestimate the magnitude of the conceptual breakthroughs required for these developments because we are biased our modern education and anachronistic perspective.

But if this is true, how come that Galileo’s ideas—for all their alleged “conceptual” avant-gardism—spontaneously sprung up like mushrooms all over Europe? And how come all of those ideas can easily be explained to any high school student today, if they are supposedly so profound and advanced? The same cannot be said for Kepler’s ideas. They were neither simultaneously developed by dozens of scientists, nor can they be taught to a modern student without years of specialised training. Perhaps this contrast between Galileo and Kepler says something about what genuine depth in the mathematical sciences looks like.

In my opinion, mathematicians at the time realised this perfectly well. I have already spoken before about the very harsh words that Descartes and sometimes Kepler had for Galileo. “He is eloquent to refute Aristotle but that is not hard,” as Descartes said. There are a number of quotes like that from mathematicians. And of course they spotted numerous mathematical blunders in Galileo, which they condemned.

Let’s look at what some other competent mathematicians thought of Galileo.

Christiaan Huygens was perhaps the greatest physicist and mathematician of the generation between Galileo and Newton. He is often portrayed as continuing the scientific program of Galileo. Huygens’s collected works is 22 thick volumes. Go ahead and try to find any strong praise of Galileo in there, let alone anything remotely like calling him a “father of science.” Somehow Huygens never got around to saying any such thing, in these tens of thousands of pages on physics and mathematics and astronomy that we wrote. Hmm, what a mystery.

The closest Huygens ever gets to mentioning Galileo favourably is in the context of a critique of Cartesianism. In the late 17th century, the teachings of Descartes had attracted a strong following. In the eyes of many mathematicians, the way Cartesianism had become an entrenched belief system was uncomfortably similar to how Aristotelianism had been an all too dominant dogma a century before. Huygens makes this parallel explicit:

“Descartes had a great desire to be regarded as the author of a new philosophy [and] it appears that he wished to have it taught in the academies in place of Aristotle. [Descartes] should have proposed his system of physics as an essay on what can be said with probability. That would have been admirable. But in wishing to be thought to have found the truth, he has done something which is a great detriment to the progress of philosophy. For those who believe him and who have become his disciples imagine themselves to possess an understanding of the causes of everything that it is possible to know; in this way, they often lose time in supporting the teaching of their master and not studying enough to fathom the true reasons of this great number of phenomena of which Descartes has only spread idle fancies.”

It is in direct contrast with this that Huygens slips in a few kind words for Galileo: “[Galileo] had neither the audacity nor the vanity to wish to be the head of a sect. He was modest and loved the truth too much.” Historians have observed that Huygens in all likelihood quite consciously intended this passage to apply to himself as much as to Galileo. Perhaps this is why Huygens is surely too generous in praising Galileo’s alleged “modesty.” Galileo was anything but modest, of course.

In any case, it is very interesting to see what Huygens says about Galileo’s actual science in this passage. Let us read it, and keep in mind that this is as close as Huygens ever gets to praising Galileo, and that the context of the passage—a scathing condemnation of Cartesianism—gives Huygens a notable incentive to put Galileo’s scientific achievements in the most positive terms for the sake of contrast.

In light of this, Huygens’s ostensible praise for Galileo is most remarkable, I think, for how qualified and restrained it is. Huygens’s praise begins like this:

“Galileo had, in spirit and awareness of mathematics, all that is needed to make progress in physics …”

Interesting phrasing. Huygens seems to be saying: Galileo said all the right things about about mathematics and scientific method, but he didn’t actually carry through on it. Given Galileo’s rhetoric, he ought to have been able to do it, but be didn’t.

Interestingly, Huygens does not say that Galileo had great mathematical ability or demonstrable achievements, only that he was “aware” that mathematics is necessary for physics. In this respect, Galileo “had all that is needed to make progress in physics,” Huygens says. Why not simply say that Galileo *made* great progress in physics, instead of this convoluted and qualified “he had what was needed to do so”? So really Huygens’s ostensible praise for Galileo is actually quite backhanded. At least that’s how it seems to me.

Let’s continue reading because the Huygens quote goes on. Here is the rest of the sentence:

“… and one has to admit that he was the first to make very beautiful discoveries concerning the nature of motion …”

Galileo wasn’t the first, as we now know. Huygens didn’t know about the unpublished work of Harriot etc., so he is overly generous in that regard. But never mind that. Huygens’s formulation is still very restrained in an interesting way: did you notice that strange phrase “one has to admit”? “One has to admit” that Galileo was the first to make certain discoveries. Who speaks of their greatest hero in such terms? One “has to admit” that he made some discoveries? That seems more like the kind of phrasing you use to describe the work of someone who is overrated, not someone you esteem as the founder of science.

Huygens wrote in French. The phrase is “il faut avouer.” I’m not a linguist but I think the translation I gave is the most natural one. “Il faut avouer”: “one has to admit”; it suggests a reluctance to concede the point. I’m not sure if it’s possible to argue that taken in context it could also be construed as “even a Cartesian would have to admit” or something like that. If you’re an expert of 17th-century French I would like to hear your opinion about this.

Let’s see, the Huygens quote continues even further and here’s how it concludes. After this remark about Galileo having made discoveries concerning motion, Huygens adds:

“ … although he left very considerable things to be done.”

Well, yes. That’s my point exactly. What is most striking and remarkable about the work of Galileo is not the few discoveries he “admittedly” made, but how very little he actually accomplished despite all his posturing about mathematics and scientific method. It seems to me that Huygens and I agree on this. Even in his most pro-Galilean sentence in all his works, Huygens is undermining Galileo as much as he is praising him.

What about Isaac Newton? What did he think of Galileo?

Newton famously said that “if I have seen further it is by standing on the shoulders of giants.” Many have erroneously assumed that Galileo was one of these “giants.” One scholar even proposes to explain that “when Newton credits Galileo with being one of the giants on whose shoulders he stood, he means …” blah blah blah. We do not need to listen to what this philosopher thinks Newton meant, because the first part of the sentence is false already. The assumption that Galileo was one of the scientific giants in question has no basis in fact.

The closest Newton gets to praising Galileo is in the Principia, his most important work. After introducing his laws of motion, Newton adds some notes on their history.

“The principles I have set forth are accepted by mathematicians and confirmed by experiments of many kinds. By means of the first two laws and the first two corollaries Galileo found that the descent of heavy bodies is in the squared ratio of the time and that the motion of projectiles occurs in a parabola.”

The laws and corollaries in question are: the law of inertia, which Galileo did not know, as we have seen; then Newton’s second law, the force law F=ma, which Galileo also did not know; and the composition of forces and motions, which was established in antiquity.

Note that Newton doesn’t say that Galileo was the discoverer of these laws. All Newton says is that Galileo used these laws to find the path of projectiles. Indeed, as one historian has pointed out, “Newton’s Latin contains some ambiguity” for it “can have two very different meanings: that the two laws were completely accepted by Galileo before he found that projectiles follow a parabolic path, or that these two laws were already generally accepted by scientists at the time that Galileo made his discovery of the parabolic path.”

Either way, Newton is wrong. Of course, once you are looking at the world though Newtonian mechanics it is natural to think that surely Galileo must have had these laws, because that is so obviously the right way to think about parabolic motion. Therefore, as Dijksterhuis says, “[according to] the myth in which he appears as the founder of classical dynamics, [Galileo] must surely have known the proportionality of force and acceleration. But to those who have become acquainted with Galileo through his own works, not at second hand, there can be no doubt that he never possessed this insight.”

Quite so, and indeed Newton was not acquainted with Galileo’s work directly. As I.B. Cohen says, “Newton almost certainly did not read [Galileo’s] Discorsi until some considerable time after he had published the Principia,” if ever. On the other hand “early in his scientific career, [Newton] had read [Galileo’s] Dialogo”—but that is of course his work on Copernicanism, not his work on mechanics and the laws of motion, which is what Newton is referencing in the Principia.

“Hence Newton (rather too generously, for once!) allowed to Galileo the discovery of the first two laws of motion.” And the reason for Newton’s excessive charity is not hard to divine. To quote I.B. Cohen again, Newton’s Principia is marked by an obvious and vehement “anti-Cartesian bias.” “Because of his strongly anti-Cartesian position, Newton might have preferred to think of Galileo rather than Descartes as the originator of the First Law.” Whereas, “in point of fact, the Prima Lex [that is, the law of inertia] of Newton’s Principia was derived directly from the Prima Lex of Descartes’s Principia”—that is, the correct law of inertia. Descartes stated the correct law of inertia with crystal clarity in this published key work, while Galileo never stated it anywhere, nor believed it.

Clearly, then, Newton’s attribution of these laws to Galileo means next to nothing. Galileo demonstrably did not know these laws; Newton hadn’t read Galileo anyway; and Newton had an obvious bias and incentive to overstate Galileo’s importance in order to belittle the influence of Descartes which he did not want to admit.

Newton’s words aren’t high praise in any case. In fact, that becomes ever clearer if we read on in Newton’s text. For when Newton continues his historical discussion he says on the very same page:

“Sir Christopher Wren, Dr. John Wallis, and Mr. Christiaan Huygens, easily the foremost geometers of the previous generation, independently found the rules of the collisions and reflections of hard bodies.”

So evidently Newton was in the mood when writing this to point out who “the foremost geometers” of the past were. Yet on the very same page he had no such words for Galileo. A telling omission. Altogether there is no evidence that Newton regarded Galileo particularly highly, let alone considered him anywhere near a “father of modern science.”

The time has come to wrap up my Galileo story. Perhaps I can sum it up like this.

Say you go to the library and find the shelves with philosophical texts ordered chronologically. You pick the books up one by one and see what they have to say about science. Century after century you find the same thing. Aristotle, Aristotle, Aristotle. Then commentaries on Aristotle. Then commentaries on commentaries on Aristotle. Then people who ostensibly try to think more independently, yet cling desperately to Aristotelian concepts and terminology as if their life depended on it, even when they try to challenge isolated claims of Aristotelian dogma. Then, suddenly: Galileo. What a breath of fresh air this is. The Aristotelian shackles are emphatically discarded, and all the nowadays familiar principles of modern science are articulated in lucid and entertaining prose. At once after him everyone is a scientist. The Aristotelianism that ran rampant for centuries had suddenly stopped dead in its tracks. How can one not admire this singular father of the scientific worldview, this pivotal hero who divides the entire history of thought into two disjoint worlds separated by such an abyss?

Alas, you made one mistake. You went to the philosophy shelves. You should have gone to subbasement 3, where the mathematics books are kept. This may not have been an obvious choice. Perhaps you were educated in the humanities and therefore naturally drawn to the sprawling and well-attended shelves in your part of the library. The out-of-the-way mathematics section never caught anyone’s eye. Isn’t it just for nerds in training who need to double-check their formulas? Apparently there are a few books there from Greek times, but blink and you miss them between thick modern textbooks on algebraic topology and partial differential equations. And if you do open one of those old math books, it’s full of technical diagrams and equations anyway. Who would ever think to look there for man’s view of the world? Surely that’s what philosophy is for?

In reality, we sent Galileo to your shelves because he wasn’t good enough for ours. Galileo wasn’t the first to do anything except explain what mathematicians had always known in such basic terms that even philosophers could understand. Galileo once wrote to a fellow philosopher:

“If philosophy is that which is contained in Aristotle’s books, you would be the best philosopher in the world. But the book of [natural] philosophy [or science] is that which is perpetually open to our eyes. But being written in characters different from those of our alphabet, it cannot be read by everyone; the characters of this book are triangles, squares, circles, spheres, cones, pyramids and other mathematical figures, the most suited for this sort of reading.”

That is Galileo’s advice to the philosophers of his day. I say much the same thing to modern scholars. If the history of science is that which is contained in philosophical books, you would be the best historians in the world. But the real truth is perpetually open to our eyes, if only we take the trouble to read mathematics.

]]>**Transcript**

To praise Galileo is to criticise the Greeks. The contrast class of “Aristotelian” science is constantly invoked to explain Galileo’s alleged greatness, both in Galileo’s own works and in modern scholarship. But this narrative gets it all wrong, in my opinion. It is based on a caricature of Greek science that effectively ignores the Greek mathematical tradition.

Francis Bacon put it well: when “human learning suffered shipwreck” with the death of the classical world, “the systems of Aristotle and Plato, like planks of lighter and less solid material, floated on the waves of time and were preserved,” while treasure troves of much more mathematically advanced works were lost forever.

Aristotelian science is not the pinnacle of Greek scientific thought. Far from it. It is not the best part of Greek science, but the part of Greek science that was most accessible and appealing to the generations of mathematically ignorant people who populated the universities in medieval Europe for hundreds of years. And perhaps some generations who still do.

Mathematicians have always felt differently. “So many great findings of the Ancients lie with the roaches and worms,” said Fermat. They are lost, in other words, these mathematical masterpieces that once existed. That’s how Fermat put it, and all his mathematical colleagues agreed. And they were right.

In the 20th century a few such masterpieces were recovered. So these 17th-century mathematicians were proven right in their intuition that great works were forgotten and hidden away among “roaches and worms” indeed.

In 1906, a work of Archimedes that had been lost since antiquity was rediscovered in a dusty Constantinople library. The valuable parchment on which it was written had been scrubbed and reused for some religious text. But the original could still just about be made out underneath it. As one historian put it: “Our admiration of the genius of the greatest mathematician of antiquity must surely be increased, if that were possible,” by this “astounding” work, which draws creative inspiration from the mechanical law of the lever to solve advanced geometrical problems. If even this brilliant work by antiquity’s greatest geometer only survived by the skin of its teeth and dumb luck, just imagine how many more works are lost forever.

Also in the 20th century, divers chanced upon an ancient shipwreck, which turned out to contain a complex machine (the so-called Antikythera mechanism). Again historians were astonished: “From all we know of science and technology in the Hellenistic age we should have felt that such a device could not exist.” “This singular artifact is now identified as an astronomical or calendrical calculating device involving a very sophisticated arrangement of more than thirty gear-wheels. It transcends all that we had previously known from textual and literary sources and may involve a completely new appraisal of the scientific technology of the Hellenistic period.”

Another example. The Greeks appear to have been much further ahead than conventional sources would lead one to believe in a number of mathematical fields. One example is combinatorics. Of this entire mathematical field little more survives than one stray remark mentioned parenthetically in a non-mathematical work by Plutarch:

“Chrysippus said that the number of intertwinings obtainable from ten simple statements is over one million. Hipparchus contradicted him, showing that affirmatively there are 103,049 intertwinings.”

“This passage stumped commentators until 1994,” when a mathematician realised that it corresponds to the correct solution of a complex combinatorial problem worked out in modern Europe in 1870, thereby forcing “a reevaluation of our notions of what was known about combinatorics in Antiquity.” It is undeniable from this evidence that this entire field of mathematics must have reached an advanced stage, yet not one single treatise on it survives.

These are just a few striking examples illustrating an indisputable point: the Hellenistic age was extremely sophisticated mathematically and scientifically, and we don’t even know the half of it.

Scores of key treatises are lost, and we are forced to rely on later commentators and compilers for accounts of the works of Hellenistic authors. It’s like trying to understand modern science and mathematics from popularisations in the Sunday newspaper. It’s vastly oversimplified and dumbed-down. It reduces complex science to one or two simplistic ideas while conveying nothing whatsoever of the often massive technical groundwork that it is based on. That’s the state of our sources for much Greek science: all that has come down to use are some clickbait headlines and blurbs by people who are themselves not scientists and wouldn’t understand the first thing about the technical details of the works they are trying to summarise.

Actually this is a misleading analogy. The situation is even worse than this. Here is how one historian puts it:

“Nearly all that we know on observations and experiments among the Greeks comes from compilations and manuals composed centuries later, by men who were not themselves interested in science, and for readers who were even less so. Even worse, these works were to a great extent inspired by the desire to discredit science by emphasizing the way in which men of science contradicted each other, and the paradoxical character of the conclusions at which they arrived. This being the object, it was obviously useless, and even out of place, to say much about the methods employed in arriving at the conclusions. It suited Epicurean and Sceptic, as also Christian, writers to represent them as arbitrary dogmas. We can get a slight idea of the situation by imagining, some centuries hence, contemporary science as represented by elementary manuals, second- and third-hand compilations, drawn up in a spirit hostile to science and scientific methods. Such being the nature of the evidence with which we have to deal, it is obvious that all the actual examples of the use of sound scientific methods that we can discover will carry much more weight than would otherwise be the case. If we can point to indubitable examples of the use of experiment and observation, we are justified in supposing that there were others of which we know nothing because they did not happen to interest the compilers on whom we are dependent. As a matter of fact, there are a fair number of such examples.”

In previous episodes we have discussed the many ways in which Greek sources already showed full awareness of many things often attributed to Galileo. Taking this context of filtering and lost sources into account means that we should give all the more weight to those arguments.

Sadly, however, the lack of appreciation for science among these ignorant commentators continues among scholars today. I collected some quotes on this by some very respectable classicists of today.

“The state of editions and translations of ancient scientific works as a whole remains scandalous by comparison with the torrent of modern works on anything unscientific — about 100 papers per year on Homer, for example. An embarrassingly large number of classicists are ignorant of Greek scientific works.”

“Classicists include many who have chosen Latin and Greek precisely to escape from science at the very early stage of specialisation that our schools’ curricula permit: and often a very successful escape it is, to judge from the depth of ignorance of science ancient and modern that it often secures.”

It is remarkable how strongly these authors make this point. The first quote is from Lloyd, the Cambridge professor. It takes a lot for people like that to almost condemn their colleagues to their face. They wouldn’t do this if it wasn’t serious.

Little wonder then that Greek science is systematically misunderstood and undervalued, and that simplistic ideas of philosophical authors and commentators are substituted for the real thing.

Galileo’s relation to the preceding philosophical tradition has been systematically misunderstood because of this.

How did modern science grow out of mathematical and philosophical tradition? The humanistic perspective is that science needed both: it was born through the unification of the technical but insular know-how of the mathematicians with the conceptual depth and holistic vision of the philosophers. The mathematical perspective is that science is what the mathematicians were doing all along. Science did not need philosophy to be its eye-opener and better half; it merely needed the philosophers to step out of the way and let the mathematicians do their thing. So which is it?

Many historians have tried to stress commonalities between Galileo and the Aristotelian philosophers who preceded him. That is to say, they argue for the “continuity thesis” which says that the so-called “Scientific Revolution” was not a radical or revolutionary break with previous thought. Here is what they say:

“Galileo essentially pursued a progressive Aristotelianism [during the first half of his life—the period of] positive growth that laid the foundation for the new sciences.”

“A particular school of Renaissance Aristotelians, located at the University of Padua, constructed a very sophisticated methodology for experimental science; … Galileo knew this school of thought and built upon its results; this goes a long way toward explaining the birth of early modern science.”

“The mechanical and physical science of which the present day is so proud comes to us through an uninterrupted sequence of almost imperceptible refinements from the doctrines professed within the Schools of the Middle Ages.”

“Galileo was clearly the heir of the medieval kinematicists.”

I agree with these authors that “those great truths for which Galileo received credit” are not his. But the notion that they were first conceived in Aristotelian schools of philosophy is wrongheaded.

The argument of these historians is based on a simple logic. First they show that various concepts of “Galilean” science are prefigured in earlier sources. Then they want to infer from this that these sources marked the true beginning of the scientific revolution. But in order to draw this inference they need two assumptions: first, that Galileo was the father of modern science; and second, that the Greeks were nowhere near the same accomplishments. These two assumptions are simply taken for granted by these authors, as a matter of common knowledge. But in reality both assumptions are dead wrong, and therefore the inference to the significance of the Aristotelian sources is unwarranted.

It is interesting that the continuity thesis on the one hand devalues the contributions of Galileo, yet at the same time desperately needs to reassert the traditional view that “Galileo has a clear and undisputed title as the ‘father of modern science’,” as one of these historians puts it. They need to say this because this is what gives them the one point of connection they are able to establish between medieval and modern science. The entire argument stands and falls with this false premiss. Therefore, if one proves, as I have done before, that Galileo was a mediocre scientists of negligible importance to the mathematically competent people who actually achieved the scientific revolution, then the continuity thesis collapses like a house of cards.

The defenders of the continuity thesis are equally ineffectual in establishing the second false premiss of their argument, namely the alleged absence of these “new” ideas in Greek thought. In fact, even continuity thesis advocates make no secret of the fact that the medieval tradition was built on “remnants of Alexandrian science.” For example, “although we are left with few monuments from the profound research of the Ancients into the laws of equilibrium, those few are worthy of eternal admiration.” Obviously, “masterpieces of Greek science [such as the works of] Pappus, and especially Archimedes, are proof that the deductive method can be applied with as much rigor to the field of mechanics as to the demonstrations of geometry.” All of that are quotes form Pierre Duhem, a passionate advocate of the continuity thesis.

How can people like Duhem acknowledge these “masterpieces” “worthy of eternal admiration” from antiquity, yet at the same time attribute the scientific revolution to medieval or renaissance philosophers? Here’s how. By writing off those ancient works as minor technical footnotes to an otherwise thoroughly Aristotelian paradigm. Only if this picture is accepted can any kind of greatness be ascribed to the pre-Galileans, as is evident from passages such as these:

“Some philosophers in medieval universities were teaching ideas about motion and mechanics that were totally non-Aristotelian [and] were consciously based on criticisms of Aristotle’s own pronouncements.”

“Admittedly, most of these significant medieval mechanical doctrines were formed within the Aristotelian framework of mechanics. But these medieval doctrines contained within them the seeds of a critical refutation of that mechanics.”

“The medieval mechanics occupied an important middle position between Aristotelian and Newtonian mechanics. [Hence it was] an important link in man’s efforts to represent the laws that concern bodies at rest and in movement.”

“The impressive set of departures from Aristotelianism achieved by medieval science nevertheless failed to produce genuine efforts to reconstruct, or replace, the Aristotelian world picture.”

If Aristotle is taken as the baseline, this looks quite impressive indeed. But why should Aristotle be accepted as the default opinion? Aristotle was one particular philosopher who was a nobody in mathematics and lived well before the golden age of Greek science. Medieval and renaissance thinkers indeed mustered up the courage to challenge isolated claims of his teachings almost two thousand years later, while mostly retaining his overall outlook. This does not constitute great open-mindedness and progress. Rather it is a sign of small-mindedness that these people paid so much attention to Aristotle at all in the first place. In my view, it is not so much impressive that they deviated a bit from Aristotle as it is deplorable that they framed so much of what they did relative to Aristotle, even when they disagreed with him. This is very different from post-Aristotelian thought in Greek times, where there is no evidence that any mathematician paid any attention to Aristotle’s mechanics.

In any case, “extravagant claims for the modernity of medieval concepts” suffer from “serious defects.” One historian has summarised it well:

“There was no such thing as a fourteenth-century science of mechanics in the sense of a general theory of local motion applicable throughout nature, and based on a few unified principles. By searching the literature of late medieval physics for just those ideas and those pieces of quantitative analysis that turned out, three centuries later, to be important in seventeenth-century mechanics, one can find them; and one can construct a “medieval science of mechanics” that appears to form a coherent whole and to be built on new foundations replacing those of Aristotle’s physics. But this is an illusion, and an anachronistic fiction, which we are able to construct only because Galileo and Newton gave us the pattern by which to select the right pieces and put them together.”

The main piece of such precursorism is the so-called “mean speed theorem.” This is a completely trivial result. You can visualise it in terms of a graph with time on the x-axis and velocity on the y-axis. Suppose you plot the graph of a uniformly accelerated motion, such as a freely falling object. It makes a straight line going from the bottom left to the to right. It starts from no velocity and goes to a certain final velocity. How far did the thing travel? Distance travelled is the area under the graph. So it’s the area of a triangle. Base times height over 2. That is to say, the time of fall, times half the final velocity. Or another way of putting it is that half the final velocity is the same thing as the average velocity. The triangle has the same area as a rectangle with the same base and half the height. The “mean speed theorem” is just this. In terms of distance covered, a uniformly accelerated motion is equivalent to a constant-speed motion with the same average speed. A very simple thing to see.

Some people praise this as an “impressive” achievement of the middle ages—”probably the most outstanding single medieval contribution to the history of physics,” derived by “admirable and ingenious” reasoning, according to one historian. Even though these medieval authors did absolutely nothing with this trivial theorem and only deduced it to illustrate the notion of uniform change abstractly within Aristotelian philosophy. Later the theorem became central in “Galilean” mechanics since free fall is uniformly accelerated. But it “was, in fact, never applied to motion in fall from rest during the 14th, or even in the 15th century” (only in the mid-16th century there is a passing remark to this effect within the Aristotelian tradition, “without any accompanying evidence”).

Let us not radically inflate our esteem for the Middle Ages by anachronistically praising them for pointing out a trivial thing that centuries later took on a significance of which they had no inkling. Let us instead recognise the theorem for the trifle that it is. Then we shall also not have any need to be surprised when it turns out that Babylonian astronomers assumed it without fanfare thousands of years earlier still. The utterly trivial “mean speed theorem” was implicitly taken for granted in Babylonian astronomy. They were too good mathematicians to make a big fuss about something so evident, unlike the medieval philosophers who sat around a proved this at length. They were so bad at mathematics that this trivial thing was the cutting edge to them, in their ignorance.

Galileo owes other debts to previous philosophical tradition as well, according to many historians. For example, we are told that there are “unmistakeable Jesuit influences in Galileo’s work”: “Above all Galileo was intent in following out Clavius’s program of applying mathematics to the study of nature and to generating a mathematical physics.” That’s a quote from Wallace. The preposterous notion that this was “Clavius’s” program can only enter one’s mind if one only reads philosophy. It was obviously Archimedes’s program, except, unlike Clavius, he proved his point by actually carrying it out instead of sermonising about what one ought to do in philosophical prose. Philosophers (ancient and modern alike) have a tendency to place disproportionate value on explaining something conceptually as opposed to actually doing it. After all, that is virtually the definition of philosophy. Hence they praise certain Aristotelians for explaining some supposedly profound principles of scientific method even when “it is quite clear that [none of them] ever applied his advocated methods to actual scientific problems.”

Descartes—a mathematically creative person—knew better: “we ought not to believe an alchemist who boasts he has the technique of making gold, unless he is extremely wealthy; and by the same token we should not believe the learned writer who promises new sciences, unless he demonstrates that he has discovered many things that have been unknown up till now.” Unfortunately, such basic common sense is often lacking among historians and philosophers assigning credit for basic principles of the scientific method.

There is a contradiction in the way modern historians try to trace many aspects of the scientific revolution to roots in the middle ages. On the one hand these historians like to claim that the traditional view of the scientific revolution is ahistorical and based on an anachronistic mindset, whereas their own account that sees continuity with the middle ages is more sensitive to how people actually thought at the time itself. Ironically, however, their view, which is supposed to be more true to the historical actors’ way of thinking, is actually all the more blatantly at odds with how virtually all leaders of the scientific revolution thought of the middle ages. One historian summarises it accurately: “The scientific achievement of the Middle Ages was held in unanimous contempt from Galileo’s time onward by those who adhered to the new science. Leibniz’ scathing verdict ‘barbaric physics’ neatly encapsulates the reigning sentiment.” This was not for nothing. Leibniz was an erudite scholar well versed in the philosophy of the schools. But he was also an excellent mathematician. The latter enabled him to pass a sound judgement on the “barbaric” science of the middle ages.

]]>**Transcript**

To say that Galileo is “the father of modern science” is to say that he made some kind of unique contribution, something unprecedented, that was the starting point of science as we know it. So what would that have been? What was that uniquely Galilean ingredient, that made science appear out of thin air for the first time in human history? We spoke about this before. People have tried to pinpoint it in various ways. I refuted the main attempts: mathematisation of nature, empiricism, experimental method. Basically, those things were all commonplace already in Greek times. That’s what I argued last time. But the list goes on. There are other things that Galileo was allegedly “the first” to do. Let’s have a look at those.

Here’s one: Galileo’s greatness consists in bringing together abstract mathematics and science with concrete technology and practical know-how of craftsmen and workers in mechanical fields. Here are some quotes from various historians expressing this idea:

“Real science is born when, with the progress of technology, the experimental method of the craftsmen overcomes the prejudice against manual work and is adopted by rationally trained university-scholars. This is accomplished with Galileo.”

“[Galileo was able] to bring together two once separate worlds that from his time on were destined to remain forever closely linked—the world of scientific research and that of technology.”

“Galileo may fruitfully be seen as the culmination point of a tradition in Archimedean thought which, by itself, had run into a dead end. What enabled Galileo to overcome its limitations seems easily explicable upon considering Galileo’s background in the arts and crafts.”

“The separation between theory and practice, imposed by university professors of natural philosophy, was repeatedly exposed as untenable. Of course the greatest figure in this movement is Galileo.”

So those are four historians all saying basically the same thing.

And Galileo himself eagerly cultivated this image. The very first words of his big book on mechanics are devoted to extolling the importance for science of observing “every sort of instrument and machine” in action at the “famous arsenal” of Venice. He praises the experiential knowledge of the “truly expert” workmen there. Galileo loves these workers and craftsmen in inverse proportion to how much he hates philosophers.

It is true that universities were filled with many blockheads who foolishly insisted on keeping intellectual work aloof from such connections to the real world. For example, when Wallis went to Oxford in 1632 there was no one at the university who could teach him mathematics. As he says in his autobiography: “For Mathematicks, (at that time, with us) were scarce looked upon as Accademical studies, but rather Mechanical; as the business of Traders, Merchants, Seamen, Carpenters, Surveyors of Lands, or the like.”

That was indeed a lamentable state of affairs. But it would be mistake to infer from this that Galileo’s step was an innovation. The stupidity of the university professors was the doing of one particular clique of mathematically ignorant people. Their attitude is not natural or representative of the state of human knowledge. Galileo is not a brilliant maverick thinking outside the box. Rather, he is merely doing what had, among mathematically competent people, been recognised as the natural and obviously right way to do science for thousands of years. Galileo is not taking a qualitative leap beyond limitations that had crippled all previous thinkers. Rather, he is merely reversing the obvious cardinal error of one particularly dumb philosophical movement that had happened to gain too much influence at the time, because people were too ignorant to recognise the evident superiority of more mathematical and scientific schools of thought that had already proven their worth in a large body of ancient works available to anyone who cared to read.

In order to defend the misconceived idea of Galileo the trailblazing innovator one must ignore the large body of obvious precedent for his view in antiquity, and project the foolish nonsense of medieval universities onto the Greeks. Indeed, historians have concocted a false narrative to this effect. Here are some typical quotes:

“Greek technology and science were rigidly separated.”

“The Greek hand worker was considered inferior to the brain worker or contemplative thinker. So, despite the fact that the philosophers derived some of their conclusions as to how nature behaved from the work of the craftsmen, they rarely had experience of that work. What is more, they were seldom inclined to improve it, and so were powerless to pry apart its potential treasure of knowledge that was to lead to the scientific revolution in the Renaissance.”

Others have argued that “the fundamental brake upon the further progress of science in antiquity was slave labour [which precluded any] meaningful combination of theory and practice.”

More specialised scholarship knows better. The recent Oxford Handbook of Engineering and Technology in the Classical World is perfectly clear on the matter:

“Many twentieth-century scholars hit upon [snobbish contempt for manual labour] as an ‘explanation’ for a perceived blockage of technological innovation in the Greco-Roman world. The presence of slave labor was felt to be a related, concomitant factor. [But] this now discredited interpretation [should be rejected and we should] put an end to the myth of a ‘technological blockage’ in the classical cultures.”

This is the view of experts on the matter, while the false narrative is promulgated by scholars who focus on Galileo, take it for granted that he is “the Father of Modern Science,” and postulate such nonsense about the Greeks because that’s the only way to craft a narrative that fits with this false assumption.

Promulgators of the nonsense about practice-adverse Greeks have evidently not bothered to read mathematical authors. Pappus, for example, explains clearly that mathematicians enthusiastically embrace practical and manual skills:

“The science of mechanics has many important uses in practical life, and is zealously studied by mathematicians. Mechanics can be divided into a theoretical and a manual part; the theoretical part is composed of geometry, arithmetic, astronomy and physics, the manual of working in metals, architecture, carpentering and painting and anything involving skill with the hands.”

Pappus praises the interaction of geometry with practical fields or “arts” as beneficial to both:

“Geometry is in no way injured, but is capable of giving content to many arts by being associated with them, and, so far from being injured, it is obvious, while itself advancing those arts, appropriately honoured and adorned by them.”

These were no empty words. The Greeks had an extensive tradition of studying “machines,” meaning devices based on components such as the lever, pulley, wheel and axle, winch, wedge, screw, gear wheel, and so on. The primary purpose of these machines was that of “multiplying an effort to exert greater force than can human or animal muscle power alone.” Such machines were “used in construction, water-lifting, mining, the processing of agricultural produce, and warfare.”

The Greeks also undertook advanced engineering projects, such as digging a tunnel of more than a kilometer through a mountain, the planning of which involved quite sophisticated geometry to enable the tunnel to be dug from both ends, with the diggers meeting in the middle. In short, “while it is crucial to distinguish between theoretical mechanics and practitioners’ knowledge, there is substantial evidence of a two-way interaction between them in Antiquity.”

Mathematicians were very much involved with such things. There are many testimonies attributing to Archimedes various accomplishments in engineering, such as moving a ship singlehandedly by means of pulleys, destroying enemy ships using machines, building a screw for lifting water, and so on. Apollonius wrote a very advanced and thorough treatise on conic sections, which is studiously abstract and undoubtedly “art for the sake of art” pure mathematics if there ever was such a thing. Yet the same Apollonius “besides writing on conic sections produced a now lost work on a flute-player driven by compressed air released by valves controlled by the operation of a water wheel.” The title page of the Arabic manuscript that has preserved this work for us reads: “by Apollonius, the carpenter, the geometer.” The cliche of Greek geometry as nothing but abstruse abstractions divorced from reality is a modern fiction. The sources tell a different story. It is not for nothing that one of the most refined mathematicians of antiquity went by the moniker “the carpenter.”

Unfortunately, as Russo has observed in his excellent book, “Renaissance intellectuals were not in a position to understand Hellenistic scientific theories, but, like bright children whose lively curiosity is set astir by a first visit to the library, they found in the manuscripts many captivating topics, especially those that came with illustrations. The most famous intellectual attracted by all these ‘novelties’ was Leonardo da Vinci. Leonardo’s ‘futuristic’ technical drawings … was not a science-fiction voyage into the future so much as a plunge into a distant past. Leonardo’s drawings often show objects that could not have been built in his time because the relevant technology did not exist. This is not due to a special genius for divining the future, but to the mundane fact that behind those drawings there were older drawings from a time when technology was far more advanced.”

The false narrative of the mechanically ignorant, anti-practical Greeks has obscured this fact, and led to an exaggerated evaluation of Renaissance technology, such as instruments for navigation, surveying, drawing, timekeeping, and so on. Here for example is the view of Jim Bennett, a former Director of the Museum of the History of Science in Oxford:

“Renaissance developments in practical mathematics predated the intellectual shifts in natural philosophy. Historians of the early modern reform of natural philosophy have failed to appreciate the significance of the prior success of the practical mathematical programme, [which] must figure in an explanation of why the new dogma of the seventeenth century embraced mathematics, mechanism, experiment and instrumentation.”

Bennett proves at length that the practical mathematical tradition had much to commend it, which I do not dispute. But then he casually asserts with hardly any justification that there was nothing comparable in Greek times. This is typical of much scholarship of this period. The deeply entrenched standard view of the Galilean revolution is basically taken for granted and subsequent work is presented as emendations to it. For instance, if you want to prove the importance of a Renaissance pre-revolution in practical mathematics, you need to prove two things: first that it was relevant to the scientific revolution, and second that it was not present long before. It is a typical pattern to see historians put all their efforts toward proving the first point, and glossing over the second point in sentence or two. They can get away with this since the alleged shortcomings of the Greeks is supposedly common knowledge, while the first point is the one that departs from the standard narrative. Hence, if the standard narrative is misconceived in the first place, so is all this more specialised research, which, although it ostensibly departs from the standard view, actually retains its most fundamental errors in the very framing of its argument.

It is right to emphasise that the practical mathematical tradition stood for a much more fruitful and progressive approach to nature than that dominant among the philosophy professors of the time. But it is a mistake to believe that these professors represented the considered opinion of the best minds, while the mathematical practitioners were oddball underdogs whose pioneering success eventually proved undeniable to the surprise of everyone. The mathematical practitioners stood for simple common sense, not renegade iconoclasm. They practiced the same common sense that their peers had in antiquity, with much the same results. The university professors, meanwhile, should not be mistaken for a neutral representation of the state of human knowledge at the time. Rather, they formed one particular philosophical sect which retained its domination of the universities not because of the preeminence of its teachings but because of the conservative appointment practices and obsequiousness of academics.

So that’s my take on the role of practical mathematics in the scientific revolution.

Now let’s turn to another issue, a more philosophical one: instrumentalism versus realism.

A standard view is that “the Scientific Revolution saw the replacement of a predominantly instrumentalist attitude to mathematical analysis with a more realist outlook.” Instrumentalism means the following; I’m quoting Simplicius the ancient commentator:

“An explanation which conforms to the facts does not imply that the hypotheses are real and exist. [Astronomers] have been unable to establish in what sense, exactly, the consequences entailed by these arrangements are merely fictive and not real at all. So they are satisfied to assert that it is possible, by means of circular and uniform movements, always in the same direction, to save the apparent movements of the wandering stars.”

Instrumentalism, as opposed to realism, was supposedly the accepted philosophy of science among “the Greeks,” according to many historians. Here’s what Pierre Duhem had to say about it for example:

“[Ancient Greek astronomers] balked at the idea that the eccentrics and epicycles are bodies, really up there on the vaults of the heavens. For the Greeks they were simply geometrical fictions requisite to the subjection of celestial phenomena to calculation. If these calculations are in accord with the results of observation, if the ‘hypotheses’ succeed in ‘saving the phenomena’, the astronomer’s problem is solved.”

“An astronomer who understands the true purpose of science, as defined by men like Posidonius, Ptolemy, Proclus, and Simplicius, … would not require the hypotheses supporting his system to be true, that is, in conformity with things. For him it will be enough if the results of calculation agree with the results of observation—if appearances are saved.”

That’s Duhem, in the early 20th century. But plenty of modern historians agree as well. Here are some examples, I quote:

“The Greek geometer in formulating his astronomical theories does not make any statements about physical nature at all. His theories are purely geometrical fictions. That means that to save the appearances became a purely mathematical task, it was an exercise in geometry, no more, but, of course, also no less.”

Galileo, by contrast, brought “a radically new mode of realist-mathematical nature knowledge.”

In other words:

“Galileo endorsed a view that was [contrary to] that of the Greeks but was also much more creative … It is a crippling restriction to hold that no theory about reality can be in mathematical form; the Renaissance rejected this restriction, holding that it was a worthwhile enterprise to search for mathematical theories which also—by metaphysical criteria—could be supposed ‘real’. … The most eloquent and full defence of this process was given by Galileo.”

Hence the Scientific Revolution owes much to “the novel quality of realism that the abstract-mathematical mode of nature-knowledge acquired in Galileo’s hands.”

All of that are quotations from mainstream historical scholarship. I of course disagree with them, as you might imagine.

In reality, no mathematically competent Greek author ever advocated instrumentalism. The notion that “the Greeks” were instrumentalists relies exclusively on passages by philosophical commentators. The notion that Ptolemy believed his planetary models were “fictional combinations of circles which could never exist in celestial reality” is demonstrably false.

First of all Ptolemy opens his big book with physical arguments for why the earth is in the center of the universe. This is a blatantly realist justification for this aspect of his astronomical models.

Furthermore, Ptolemy has a detailed discussion of the order and distances of the planets that obviously assumes that the planetary models, epicycles and all, are physically real. “The distances of the … planets may be determined without difficulty from the nesting of the spheres, where the least distance of a sphere is considered equal to the greatest distance of a sphere below it.” That is to say, according to Ptolemy’s epicyclic planetary models, each planet sways back and forth between a nearest and a furthest distance from the earth. The “sphere” of each planet must be just thick enough to contain these motions. Ptolemy assumes that “there is no space between the greatest and least distances [of adjacent spheres],” which “is most plausible, for it is not conceivable that there be in Nature a vacuum, or any meaningless and useless thing.”

Clearly this is based on taking planetary models to be very real indeed, and not at all mathematical fictions invented for calculation. Nor was Ptolemy an exception in his realism. His colleague Geminos “was a thoughtful realist” too, as the translators of his surviving astronomical work have observed.

Hipparchus too evidently chose models for planetary motion on realist grounds. His works are lost, but we know that he proved the mathematical equivalence of epicyclic and equant motion. In other words, he showed that two different geometrical models of planetary motion are observationally equivalent; they lead to the exact same visual impressions seen from earth, but they are brought about by different mechanisms. How should one choose between the two models in such a case? If Hipparchus was an instrumentalist, he wouldn’t care one way or the other, or he would just pick whichever was more mathematically convenient. But if he was a realist he would be interested in which model could more plausibly correspond to actual physical reality. So what did he do? Here is what Theon says: “Hipparchus, convinced that this is how the phenomena are brought about, adopted the epicyclic hypothesis as his own and says that it is likely that all the heavenly bodies are uniformly placed with respect to the center of the world and that they are united to it in a similar way.” So Hipparchus decided between equivalent models based on physical plausibility. This is quite clearly a realist argument.

Historians have brought up other “evidence” that “the Greeks” were instrumentalists. One thing they point to is the alleged compartmentalisation of Greek science. I quote a modern historian:

“Phenomena [such as] consonance, light, planetary trajectories and the two states of equilibrium [i.e., statics and hydrostatics] are investigated separately. There is no search for interconnections, let alone for an overarching unity.”

This attitude would indeed make sense if mathematical science was just instrumental computation tools with no genuine anchoring in reality. The only problem is that the claim is false. Greek science is in fact full of interconnections, just as one would expect if they were committed realists. Ptolemy uses mechanics to justify geocentrism; Archimedean hydrostatics explains shapes of planets and “casts light on the earth’s geological past”; Archimedes used statical principles to compute areas in geometry. Ptolemy applies “consonance” (that is, musical theory) to “the human soul, the ecliptic, zodiac, fixed stars, and planets,” as he says in his book on astrology. Ptolemy also applies the law of refraction of optics to atmospheric refraction, noting its importance for astronomical observations.

In Galileo’s time, the same pattern prevails: mathematically competent people are unabashed realists, while philosophers and theologians often find instrumentalism more appealing for reasons that have nothing to do with science. Copernicus’s book, for example, is unequivocally realist. Spineless philosophers and theologians could not accept this. One even resorted to the ugly trick of inserting an unsigned foreword in the book without Copernicus’s authorisation, in which they espoused instrumentalism. Here’s what is says:

“It is the job of the astronomer to use painstaking and skilled observation in gathering together the history of the celestial movements, and then—since he cannot by any line of reasoning reach the true causes of these movements—to think up or construct whatever causes or hypotheses he pleases such that, by the assumption of these causes, those same movements can be calculated from the principles of geometry for the past and for the future too. … It is not necessary that these hypotheses should be true. … It is enough if they provide a calculus which fits the observations.”

This foreword was left unsigned so that it was easy to assume that it was written by Copernicus himself. This surely fooled no one who actually read the book, with all its blatant realism. Giordano Bruno, for one, thought “there can be no question that Copernicus believed in this motion [of the earth],” and hence concluded that the timid foreword must have been written “by I know not what ignorant and presumptuous ass.” That’s Bruno’s opinion, a early reader of Copernicus. Other mathematical readers presumably felt the same way. But then again the mathematically incompetent people whom the instrumentalist foreword was designed to appease could not read the book anyway.

In medieval and renaissance philosophical texts it is not hard to find many assertions to the effect that “real astronomy is nonexistent” and what passes for astronomy “is merely something suitable for computing the entries in astronomical almanacs.” There were many instrumentalists at the time, to be sure, but the challenge is to find a single serious mathematical astronomer among them. They were exclusively theologians and philosophers.

All historians nowadays recognise that “Copernicus clearly believed in the physical reality of his astronomical system,” but their inference that he “thus broke down the traditional disciplinary boundary between astronomy (a branch of mixed mathematics) and physics (or natural philosophy)” is dubious. This was “the traditional” view only in a very limited sense. It was traditional among the particular sect of Aristotelians that occupied the universities, but outside this narrow clique it had no credibility or standing whatsoever. Among mathematicians, Copernicus’s view was exactly the traditional one.

All mathematically competent people continued in the same vein, long before Galileo entered the scene. Already in the 16th century, “Tycho and Rothman, Maestlin, and even Ursus openly deploy a wide range of physical arguments in debating the issue between the rival world-systems.” Kepler puts the matter very clearly:

“One who predicts as accurately as possible the movements and positions of the stars performs the task of the astronomers well. But one who, in addition to this, also employs true opinions about the form of the universe performs it better and is held worthy of greater praise. The former, indeed, draws conclusions that are true as far as what is observed is concerned; the latter not only does justice in his conclusions to what is seen, but also in drawing conclusions embracing the inmost form of nature.”

As Kepler notes, this was all obviously well-known and accepted since antiquity, for “to predict the motions of the planets Ptolemy did not have to consider the order of the planetary spheres, and yet he certainly did so diligently.”

So, in conclusions, mathematicians were always realists. Galileo had nothing new to contribute on that matter. So we have refuted that as well as one of the possible Galilean innovations that caused the scientific revolution.

Here’s another of the big themes in the scientific revolution: the “mechanical philosophy.”

Some say that “the mechanization of the world-picture” was the defining ingredient of “the transition from ancient to classical science.” A paradigm conception at the heart of the new science was that of the world as a machine: a “clockwork universe” in which everything is caused by bodies pushing one another according to basic mechanical laws, as opposed to a world governed by teleological purpose, divine will and intervention, anthropomorphised desires and sympathies ascribed to physical objects, or other supernatural forces. Galileo was supposedly a pioneer in how he always stuck to the right side in this divide. Here is one historian arguing as much:

“Galileo possessed in a high degree one special faculty. That is the faculty of thinking correctly about physical problems as such, and not confusing them with either mathematical or philosophical problems. It is a faculty rare enough still, but much more frequently encountered today than it was in Galileo’s time, if only because nowadays we all cope with mechanical devices from childhood on.”

Of course, this “special faculty” is precisely what led Galileo to reject as occult the correct explanation of the tides and propose his own embarrassing nonstarter of a tidal theory based on an analogy with “mechanical devices,” as we have discussed before. But let’s put that aside.

There is nothing modern about the mechanical philosophy. “*We* all cope with mechanical devices from childhood on,” the quote says, but so did the Greeks. They built automata such as entirely mechanical puppet-theatres, self-opening temple doors, a coin-operated holy water dispenser, and so on. Pappus notes that “the science of mechanics” has many applications “of practical utility,” including machines for lifting weights, warfare machines such as catapults, water-lifting machines, and “marvellous devices” using “ropes and cables to simulate the motions of living things.”

Clearly, then, “Ancient Greek mechanics offered working artifacts complex enough to suggest that organisms, the cosmos as a whole, or we ourselves, might ‘work like that’.” Thus we read in ancient sources that “the universe is like a single mechanism” governed by simple and deterministic laws that ultimately lead to “all the varieties of tragic and comedic and other interactions of human affairs.” This line of reasoning soon lead to a secularisation of science. “Bit by bit, Zeus was relieved of thunderbolt duty, Poseidon of earthquakes, Apollo of epidemic disease, Hera of births, and the rest of the pantheon of gods were pensioned off” in the same manner.

Mechanical explanations are widespread in Greek science. The Aristotelian Mechanics uses the law of lever to explain “why rowers who are in the middle of the ship move the ship the most,” and “how it is that dentists extract teeth more easily by a tooth-extractor [or forceps] than with the bare hand only.” Greek scientists explained perfectly clearly that sound is a “wave of air in motion,” comparable to the rings forming on a pond when when one throws in a stone. Atomism—a widely espoused conception of the world in Greek antiquity—is of course in effect a plan to “make material principles the basis of all reality.”

Greek astronomy went hand in hand with mechanical planetaria that directly reproduced a scale model of planetary motion. And not just basic toy models, but “complex and scientifically ambitious instruments” that could generate all heavenly motions mechanically from a single generating motion (the turn of a crank, as it were).

The possibility that even biological phenomena worked on the same principle immediately suggested itself and was eagerly pursued. Here’s what Galen says, the ancient physician:

“Just as people who imitate the revolutions of the wandering stars by means of certain instruments instill a principle of motion in them and then go away, while [the devices] operate just as if the craftsman was there and overseeing them in everything, I think in the same way each of the parts in the body operates by some succession and reception of motion from the first principle to every part, needing no overseer.”

Indeed, ancient medical research put this vision into practice. “The use of what we should call mechanical ideas to explain organic processes”—such as digestion and other physiological functions—is “the most prominent feature” of the work of Erasistratus in medicine, who also tested his ideas experimentally.

In conclusion, then, the world did not need Galileo to tell them about the mechanical philosophy, since it had been widely regarded as common sense already in antiquity.

The scientific revolution did not come about by any innovative or groundbreaking insights of Galileo. It came about by simply listening to what the mathematicians had been saying for thousands of years.

]]>**Transcript**

Galileo is “the father of modern science,” people would have you believe. But why? What exactly did he do that was so new that he fathered the entire concept of science? Was Galileo the first to bring together physics and mathematics? Was he the first to base science on data and experiments, or to give practical experience more authority than philosophical systems?

The answer to these questions is: no, no, no. Galileo was nowhere near the first to do any of these things. But he is still often credited with these innovations, even in scholarly sources. So I’m going to run down the list and prove point by point why these people are wrong.

The notion that Galileo was somehow “the father of modern science” remains a standard view among modern historians. For instance, the Oxford Companion to the History of Modern Science published in 2003 flat out says that Galileo “may properly be regarded as the ‘father of modern science’.” This view is considered so unassailable that even the very Pope once conceded that Galileo “is justly entitled the founder of modern physics.” Pope John Paul II said this is 1979.

But there is less agreement on what exactly Galileo did to deserve this epithet. As Dijksterhuis says in his classic history of mechanics: “No one indeed is prepared to challenge [Galileo’s] scientific greatness or to deny that he was perhaps the man who made the greatest contribution to the growth of classical science. But on the question of what precisely his contribution was and wherein his greatness essentially lay there seems to be no unanimity at all.”

So let’s go though all major attempts at capturing Galileo’s alleged greatness, and criticise them one by one.

First: Mathematics and nature.

It is a common view that Galileo was the first to bring together mathematics and the study of the natural world. I could give you long list of scholars who have said exactly this. For this to make sense, one must obviously maintain that, before Galileo, mathematics and natural science were fundamentally disjoint. This assumption is plainly and unequivocally false. In Greek works by mathematically competent authors, there is zero evidence for this assumption and a mountain of evidence to the contrary. “We attack mathematically everything in nature” said Iamblichus of Greek science, and he was right. This is a commonplace, explicit methodological program in Greek science, as the The Cambridge Companion to the Hellenistic World points out: “Hellenistic natural philosophers often took mathematics as the paradigm of science and sought to mathematize their study, that is, to ground all its claims in mathematical theorems and procedures, a goal shared by modern scientists.” This is the exact opposite of the claim that the ancients were unable to conceive the unity of mathematics and science.

How can so many historians get it exactly backwards? By ignoring the entire corpus of Greek mathematics and instead relying exclusively on philosophical authors. Thus we are told that, following “the classification of philosophical knowledge deriving from Aristotle,” a sharp division prevailed among “the Greeks” between “natural science (or ‘physics’), which studied the causes of change in material things,” and “mathematics, which was the science of abstract quantity.” Well, this was perhaps a problem for philosophers who spent their time trying to classify scientific knowledge instead of contributing to it. But I challenge you to produce one single piece of evidence that this division had any impact whatsoever on any mathematically creative person in antiquity.

The alleged divide doesn’t exist in Aristotle’s own works either, for that matter. Aristotle lived well before the glory days of Greek science, and he was clearly no mathematician. But even Aristotle lists mechanics, optics, harmonics, and astronomy as fields based on mathematical demonstrations. He even explicitly calls them “branches of mathematics.” How can anyone infer from this that Aristotle saw the very notion of mathematical science as a conceptual impossibility? That’s nuts. But historians in fact do so, by insisting that these fields are mere exceptions. Here’s a typical quote, from A Short History of Scientific Thought published by Palgrave Macmillan in 2012:

“Previous assumptions [before Galileo], encouraged by Aristotle and scholastic philosophers, held that mathematics was only relevant to our understanding of very specific aspects of the natural world, such as astronomy, and the behaviour of light rays ([that is to say] optics), both of which could be reduced to exercises in geometry. Otherwise, mathematics was just too abstract to have any relevance to the physical world.”

The implausibility of this view is obvious. If, as Aristotle himself clearly states, mechanics, optics, harmonics, and astronomy are four entire fields of knowledge that successfully use mathematics to understand the natural world, who in their right mind would then categorically insist that, nevertheless, other than that mathematics surely has nothing to contribute to science. It makes no sense. If mathematics has already given you four entire branches of science, why close your mind to the possibility of any further success along similar lines? It is hard to think of any reason for taking such a stance, except perhaps for someone who themselves lack mathematical ability and want to justify their neglect of this field.

The strange habit of writing off the numerous branches of mathematical science in antiquity as so many exceptions is necessary to maintain triumphalist narratives of the great Galilean revolution. For example, we are told that “it was Galileo who first subjected other natural phenomena to mathematical treatment than the Alexandrian ones.” In other words, except mechanics, astronomy, optics, music, statics, and hydrostatics, Galileo was *the very first* to take this step. That is to say, if you ignore all previous mathematicians who did this exact thing in great detail, Galileo’s step was completely revolutionary.

Another strategy for explaining away the obvious fact of extensive mathematical sciences in antiquity is to discount them as genuine science on the grounds that they were abstractions. Thus some claim that, despite ostensible applications of mathematics in numerous fields, “mathematical theory and natural reality remained almost entirely separate entities” due to the “high level of abstraction” of the mathematical theories, which meant that they were “barely connected with the real world.”

Supposedly, Galileo broke this spell — an absurd claim since this critique is all the more true for his science: even Galileo’s supposedly “best” discoveries are often way out of touch with reality: his law of fall, his law of parabolas, they obviously fail experimentally. Not to mention Galileo’s many erroneous theories, which were even more disconnected from reality for obvious reasons. Meanwhile, Greek scientific laws of statics, optics, hydrostatics, and harmonics concern everyday phenomena that can be verified by anyone in their own back yard using common household items. Indeed, they are still part of modern physics textbooks — and high school laboratory demonstrations — to this day. Take optics, for example. Heron of Alexandria proved the law of reflection, which anyone with a mirror can readily check, using the distance-minimisation argument still found in every textbook today. Light travels along the shortest path from point A to point B via the mirror. Diocles demonstrated the reflective property of the parabola and used it to “cause burning” by concentrating the rays of the sun with a paraboloid mirror: a principle still widely applied today, for example in satellite dishes and flashlights. Ptolemy demonstrated the magnifying property of concave mirrors, such as modern makeup mirrors. These kinds of results, which are not atypical, are clearly not disconnected from reality by any means.

The false notion of a divide between mathematics and science also rests on a conception of mathematics itself as a purely abstract field. Here’s a quote expressing a typical view:

“Traditionally, geometry was taken to be an abstract inquiry into the properties of magnitudes that are not to be found in nature. Dimensionless points, breadthless lines, and depthless surfaces of Euclidean geometry were not traditionally taken to be the sort of thing one might encounter while walking down the street. Whether such items were characterized as Platonic objects inhabiting a separate realm of geometric forms, or as abstractions arising from experience, it was generally agreed that the objects of geometry and the space in which they are located could not be identified with material objects or the space of everyday experience.”

This is again a view expressed by philosophers only. Nothing of the sort is ever stated by any mathematically competent author in antiquity. On the contrary, mathematicians routinely take the exact opposite for granted. Allegedly “abstract” geometry is constantly applied to physical objects in Greek mathematical works without ado. The long list of Greek mathematicians who studied the natural world always took for granted the identification of geometry with the space and material objects around us. And why shouldn’t they? For thousands of years geometry had been used to delineate fields, draw up buildings, measure volumes of produce, and a thousand other practical purposes — exactly “the sort of thing one might encounter while walking down the street.” Every single theorem of Euclid’s geometry can be verified by concrete measurements and constructions with physical tools and materials. So why would mathematicians suddenly insist that their field is completely divorced from reality? What could possibly be their motivation for doing so? It accomplishes nothing and creates tons of obvious problems when one wants to apply mathematics far and wide in numerous areas, as mathematicians always did. The only people with any motive to take such an extremist stance are philosophers with an axe to grind.

Only those ignorant of the vast tradition of Greek mathematical science can maintain that the unity of mathematics and science in the 17th century was in any way revolutionary. However, even if one accepts this completely wrongheaded view, credit still should not go to Galileo. Some recent historians have begun to stress that “the mathematization of the sublunary world begins not with Galileo but with Alberti,” who wrote on the geometrical principles of perspective in painting in the 15th century.

“The invention of perspective by the Renaissance artists, by demonstrating that mathematics could be usefully applied to physical space itself, [constituted] a momentous step toward the general representation of physical phenomena in mathematical terms.”

These historians correctly challenge the narrative of Galileo as the heroic visionary who united mathematics and the physical world, but they retain the erroneous underlying assumption that this unification was revolutionary to begin with. Perspective painting is fine mathematics, but it wasn’t a “momentous step” “demonstrating” that mathematics could be applied to the world, because that had already been demonstrated over and over again thousands of years before. Vitruvius, to take just one example, had pointed out the obvious: “an architect should be instructed in geometry,” which “is of much assistance in architecture.” Certainly a strange thing to say if the “momentous” insight that geometry is relevant to “the space of everyday experience” is still more than a thousand years in the future! No, the absurd notion that the application of geometry to physical space was somehow a Renaissance revolution can only occur to those who spend too much time reading philosophical authors pontificating about the divisions of knowledge instead of reading authors actually active in those fields.

The restriction to “the sublunary world” in the above quotation is also telling. The allegedly profound conceptual divide between heaven and earth in this period is a standard trope among historians, as we have discussed before. Of course, the Greeks mathematised the sublunary world too, but you have to read specialised works to find out much about that. Astronomy, on the other hand, is such an obvious example of an extremely successful and detailed mathematisation of one aspect of reality that even philosophers and historians cannot ignore this elephant in the room. Hence they rely on the qualifier that the allegedly revolutionary step was “the mathematization of the *sublunary* world.”

Aristotle did indeed make much of the difference between the earthly, sublunary world and the world of heavenly motions. But this is one particular dogma of one particular school of philosophy. There is no reason for any mathematician to accept it, nor is there any evidence that any mathematically competent person in the golden age of Greek science did so. The Aristotelian dichotomy is far from natural or necessary: in fact, “Aristotle argues, *against his predecessors*, that the celestial world is radically different from the sublunary world,” as one historian has observed. For that matter, even if Aristotle’s dogmatic and arbitrary dichotomy is accepted, it would still be madness to acknowledge the undeniable success of mathematics on one side of the divide, yet consider its application on the other side of the divide a conceptual impossibility.

Ptolemy, the ancient astronomer, speaks in Aristotelian terms when he contrasts astronomy with physics. The subject matter of astronomy is “eternal and unchanging,” while physics “investigates material and ever-moving nature situated (for the most part) amongst corruptible bodies and below the lunar sphere.” This is arguably more of a fact than a philosophical commitment: planetary motions are regular and periodic, whereas falling bodies, projectile motion, and other phenomena of terrestrial physics are inherently fleeting and limited to a short time span. It is conceivable that someone might seize on this dichotomy to “explain” why mathematics is suitable for the heavens only, and not for the sublunary world. This, however, is definitely not Ptolemy’s stance. He unequivocally expresses the exact opposite view: “as for physics, mathematics can make a significant contribution” there too.

In sum, the Aristotelian dichotomy between heaven and earth was never an obstacle to mathematicians. And this with good reason. The whole business of emphasising the dichotomy in the context of the mathematisation of the world is a figment of the imagination of historians, who find themselves having to somehow explain away astronomy as irrelevant when they want to claim that there was a mathematical revolution in early modern science. We do not need to resort to such fictions if we instead accept that the unity of mathematics and science had been obvious since time immemorial.

Another argument for Galileo as the unifier of physics and mathematics consists in stressing that other mathematicians of his day were often more concerned with pure geometry than with projectile motion and the like. For instance, in France there were highly capable “new Archimedeans” like Descartes, Roberval, and Fermat, but their focus differed from that of Galileo. Here’s a quote from a recent book expressing this view:

“They were indeed good mathematicians, but they did not consider mathematics as a method for understanding physical things. Mathematical constructions were only abstractions to them, with which it was fun to play, but which were not to be confused with what really happened in nature. Moreover, they were not interested in the ways in which motion intervened in natural processes.”

In my view, Galileo would have loved to have been this kind of “new Archimedean” too if only he had been capable of it. And it is not true that these Frenchmen ignored motion and the mathematisation of nature. We have already noted that Descartes studied the law of fall, and that Fermat corrected Galileo on the path of a falling object in absolute space. Both Descartes and Fermat also wrote on the law of refraction of optics, deriving it from physical considerations regarding the speed of light in different media. Also, Descartes explained the motion of the planets, and the fact that they all revolve in the same direction about the sun, by postulating that they were carried along by a vortex. So these mathematicians were clearly not ignorant of or averse to studying how “motion intervened in natural processes.”

So it is not attention to motion per se, but the study of projectile motion specifically, that sets Galileo apart from these mathematical contemporaries. Does Galileo deserve great credit in this regard? I don’t think so. Why is projectile motion important? With Newton, projectile motion took on a fundamental importance because he saw that planetary motion was governed by the same principles. Galileo had no inkling of this insight. With Newton, projectile motion is also fundamental as a paradigm illustration of the principles — such as inertia and Newton’s force law — that govern all other mechanics. In Newtonian mechanics this is the basis for understanding phenomena such as pendulum motion. Galileo, however, got this wrong, so he cannot be celebrated for this insight either.

Thus we see that praising Galileo for studying projectile motion is anachronistic. Galileo got lucky: the topic he studied later turned out to be very important for reasons he did not perceive, so that in retrospect his work seems much more prescient and groundbreaking than it really was. He himself in fact motivates the theory of projectile motion almost exclusively in terms of practical ballistics — a nonsensical application of zero practical value, which one cannot blame other mathematicians for ignoring.

So those are my rebuttals of the various ways in which Galileo has been praised for mathematising nature in innovative ways.

Another way in Galileo was supposedly innovative is in his emphasis on an empirical scientific approach.

The Cambridge Companion to Galileo expresses this view clearly: “Galileo became (and still is) the model for the empiricist scientist who, unlike the natural philosophers of his day, sought to answer questions not by reading philosophical works, but rather through direct contact with nature.” This is an image Galileo eagerly (but dishonestly) sought to promote, as we have seen. Recall the story of the Babylonian eggs cooked in a sling for example, and also Galileo’s rhetoric against Aristotle on the law of fall.

Praise for Galileo in this regard naturally goes hand in hand with “the verdict that Greek science suffered from an overdose of rash generalizations at the expense of a careful scrutiny, whether experimental or observational, of the relevant facts.” In other words, “Greek thinkers generally overrated the power of unchecked, speculative thought in the natural sciences.” So many people have claimed.

In reality, an empirical approach to the study of nature is not a newfangled invention by Galileo but just common sense. It was obviously adopted by the Greeks, especially the mathematicians. Even Aristotle, who practiced “speculative thought in the natural sciences” to a much greater extent than mathematicians, was a keen empiricist, and his followers insisted on this as one of the key principles of his philosophy. Aristotle’s zoology largely follows a laudable empirical method quite modern in spirit, such as braking open lots of bird eggs at different stages to study the development of the embryo and many other things like that. The same approach was applied by his immediate followers in botany and petrology, including for example cataloging extensive empirical data on how a wide variety of minerals react to heating.

This was far from forgotten in Galileo’s day, where one often encounters passages like these from committed Aristotelians:

“We made use of a material instrument to establish by means of our senses what the demonstration had disclosed to our intellect. Such an experimental verification is very important according to [Aristotelian] doctrine.” That’s Piccolomini, an Aristotelian philosopher, writing well before Galileo, in the 16th century.

Not infrequently, Galileo’s Aristotelian opponents attacked him for being too speculative while they saw themselves as representing the empirical approach. For example, one critic writes to Galileo:

“At the beginning of your work, you often proclaim that you wish to follow the way of the senses so closely that Aristotle (who promised to follow this method and taught it to others) would have changed his opinion, having seen what you have observed. Nonetheless, in the progress of the book you have always been so much a stranger to this way of proceeding that all your controversial conclusions go against our sense knowledge, as anyone can see by himself, and as you expressly say yourself, speaking of the theory of Copernicus, which was rendered plausible and admirable to many by abstract reasoning although it was against all sensory experience.”

It is true that there were also many spineless “Aristotelians” in Galileo’s day who preferred hiding behind textual studies rather than engaging with actual science. But this was one perverse sect of scholasticism, not the overall state of human knowledge before Galileo. A contemporary colleague of Galileo put is well:

“The Science of Nature has been already too long made only a work of the Brain and the Fancy: It is now high time that it should *return* to the plainness and soundness of Observations on material and obvious things.”

That’s Robert Hooke. Note that word choice: “return” — “return to observation.” Not: Galileo invented this new thing, empiricism. Rather: empiricism is the natural and obvious way to study nature, and the departure from it in certain philosophical circles is a corrupt aberration.

The misconception that the Greeks were anti-empirical stems from a foolish reading of the mathematical tradition. Galileo fan Stillman Drake put it like this:

“Archimedes never appealed to actual measurements in any of his proofs, or even in confirmation of his theorems. The idea that actual measurement could contribute anything of real value was absent from physics for two millennia.”

Or again:

“The mathematics of Euclid and the physics of Archimedes were necessary, but not sufficient, for Galileo’s science. They leave unexplained Galileo’s repeated appeals to sensate experience.”

On a superficial reading this may indeed appear so. Open, say, Archimedes’s treatise on floating bodies and you will find no mention of any measurement or experiment or data of any kind, only theorems and proofs. It may seem natural to infer from this that Archimedes was doing speculative mathematics divorced from reality, and that he had no understanding of the importance of empirical tests. This is what it looks like to historians who insist on an overly literal reading of the text and lack a sympathetic understanding of how the mathematical mind works. The fact of the matter is that Archimedes’s theorems are empirically excellent. It makes no sense to imagine that Archimedes was reasoning about abstractions as an intellectual game, and that his extremely elaborate and detailed claims about the floatation behaviour of various bodies given their shapes and densities just happened to align exactly with reality by pure chance. Archimedes doesn’t have to point out that he made very careful empirical investigations, because it is obvious from the accuracy of his results that he did.

Here is a better way of putting the relation between mathematics and empirical data, from The Oxford Handbook of the History of Physics:

“Mixed mathematics were often presented in axiomatic fashion, following the Archimedean tradition. In this tradition, experiments were often conceived of as inherently uncertain and therefore they could not be placed at the foundation of a science, lest that science too be tainted with that same degree of uncertainty. To be sure, experiments were still used as heuristic tools, for example, but their role often remained private, concealed from public presentations.”

So the point is not that empirical data is neglected, but that it is a mere preliminary step. Anyone can make measurements and collect data. Self-respecting mathematicians do not publish such trivialities. Instead they go on to the really challenging step of synthesising it into a coherent mathematical theory. Galileo did not have the ability to do the latter, so he had to stick with the basics, and pretend, nonsensically, that this was somehow an important innovation. Then as now, there were enough non-mathematicians in the world for his cheap charade to be successful.

What about the experimental method? Was that Galileo’s special contribution and insight?

Some say so. Empiricism, which we just discussed, is mere passive observation. The real innovation was active experiment. A famous supporter of this view is Immanuel Kant, who wrote as follows in the Critique of Pure Reason:

“When Galileo caused balls to roll down an inclined plane, a light broke upon all students of nature. Reason must approach nature in order to be taught by it. It must not, however, do so in the character of a pupil who listens to everything that the teacher chooses to say, but of an appointed judge who compels the witnesses to answer questions which he has himself formulated.”

Modern historians have expressed the same idea. Here is one example:

“The originality of Galileo’s method lay precisely in his effective combination of mathematics with experiment. The distinctive feature of scientific method in the seventeenth century, as compared with that in ancient Greece, was its conception of how to relate a theory to the observed facts and submitting them to experimental tests. [This feature] transformed the Greek geometrical method into the experimental science of the modern world.”

In reality, the use of experiment in Greek science is abundantly documented to anyone who bothers to read mathematical authors.

Greek scientists knew perfectly well that “it is not possible for everything to be grasped by reasoning, many things are also discovered through experience,” as Philon said. This quote refers to the precise numerical proportions needed for the spring in a stone-throwing engine. The same author also offered an experimental demonstration that air is corporeal. Ptolemy experimented with balloons (or “inflated skins” as he says) to investigate whether air or water has weight in their own medium. Does a balloon full of water sink in water, or float or what? Indeed, Ptolemy “performed the experiment with the greatest possible care,” according to Simplicius. Heron of Alexandria gives a detailed description of an experimental setup to prove the existence of a vacuum. He explicitly states that “referring to the appearances and to what is accessible to sensation” trumps abstract arguments that there can be no vacuum. Such arguments had been given by Aristotle, but here we have a mathematically minded author saying “no way, that’s nonsense” and proving as much with experiment. In optics, Ptolemy explicitly verified the law of reflection by experiment. He also studied refraction experimentally, giving tables for the angle of refraction of a light ray for various incoming angles in increments of 10 degrees for passages between air, water, and glass.

Archimedes caught a forger who tried to pass off as pure gold a crown that was actually gold-coated silver. By an experiment based on hydrostatic principles, he was able to expose the crown as a knock-off without damaging it in any way. This discovery was the occasion for him to reportedly run naked through the streets yelling “eureka” in excitement. Such was his love of empirical, experimental science — yet many scholars keep insisting that, like a second Plato, all he really cared about was abstract geometry. Evidently, even running naked through the streets and screaming at the top of one’s lungs is not enough for some people to open their eyes. It is hard to imagine what else one can do to draw their attention to the obvious: namely that Greek mathematicians embraced experimental method through and through.

Ok, so I have argued that Galileo wasn’t the first to apply mathematics to nature, nor the first to base science on data, nor on experiment. So we’ve ruled out those three but we’re still only halfway down the list of things that Galileo supposedly pioneered. We will have to go through the other ones next time.

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The Bible says basically nothing about astronomy. It has a lot more to say about righteous war. And it is in this context only that it has occasion to speak of the motions of heavenly bodies. In the Book of Joshua, we find our hero with the upper hand in battle, but alas dusk is drawing close. What a pity if some of the enemies “delivered up before the children of Israel” should be able to get away under the cover of darkness. “Then spake Joshua to the Lord,” and he said: “Sun, stand thou still.” “And the sun stood still until the people had avenged themselves upon their enemies.” That is what the Bible tells us. “The sun stood still in the midst of heaven,” so that Joshua and the chosen people could keep slaughtering infidels all night long.

This is the full extent of astronomy in the holy book. No further detail is provided anywhere in the Bible regarding the astronomical constitution of the universe or the motions of the heavenly bodies.

Obviously, serious scientists have little reason to engage with this passing and tangential allusion to cosmology in the Book of Joshua. But Galileo’s philosophical enemies saw an opportunity. By persistently and prominently accusing Galileo of proposing theories contrary to scripture they forced him into a dilemma: either let the argument stand unopposed, and hence let his enemies have the last word, or else get involved with the very dangerous matter of scriptural interpretation. Galileo foolishly took the bait. Now all the Aristotelians had to do was to sit back and watch Galileo march to his own ruin in this minefield.

So let’s see how Galileo proposes that we interpret the Biblical passage about the sun standing still. His interpretation is nuts. It is a prime example of his shameless drive to score rhetorical points at any cost. It is perfectly reasonable to argue that the phrase about the sun “standing still” should not be taken too literally. Indeed, it is commonly accepted, as Galileo observes, that various things in the Bible “were set down in that manner by the sacred scribes in order to accommodate them to the capacities of the common people, who are rude and unlearned.”

Indeed, if the Bible is read literally, “it would be necessary to assign to God feet, hands, and eyes,” as Galileo says. But those passages are only figures of speech, according to orthodox Christian understanding. When the Old Testament says that the commandments handed to Moses were “written with the finger of God,” the intended takeaway is of course not that God has an actual physical finger and that he needs it to write. It doesn’t make a whole lot of sense that he could create the entire universe in under a week, or flood the entire earth at will, yet if he has to write something down he has to painstakingly trace it out in clay with his finger.

So perhaps it is the same with the sun “standing still.” It’s just a phrase adapted to everyday speech, not a scientific account. In fact, even Copernicus himself speaks of “sunrise” and “sunset,” as Galileo points out, even though the sun doesn’t move in his system. So it is hardly unreasonable to think that “the sacred scribes” used this kind of common parlance as well, even if they knew that the sun is always stationary.

That’s all fine and well. But Galileo does not stop with this balanced and reasonable point. Instead he makes the outlandish claim that the Joshua passage in fact literally agrees best with heliocentrism rather than geocentrism:

“If we consider the nobility of the sun I believe that it will not be entirely unphilosophical to say that the sun, as the chief minister of Nature and in a certain sense the heart and soul of the universe, infuses by its own rotation not only light but also motion into other bodies which surround it. So if the rotation of the sun were to stop, the rotations of all the planets would stop too. [Therefore,] when God willed that at Joshua’s command the whole system of the world should rest and should remain for many hours in the same state, it sufficed to make the sun stand still. In this manner, by the stopping of the sun, the day could be lengthened on earth—which agrees exquisitely with the literal sense of the sacred text.”

This is a terrible argument. It is so unscrupulous that its absurdity can be exposed simply by quoting the words of Galileo himself, written in another context, in his Dialogue:

“If the terrestrial globe should encounter an obstacle such as to resist completely all its whirling motion and stop it, I believe that at such a time not only beasts, buildings, and cities would be upset, but mountains, lakes, and seas, if indeed the globe itself did not fall apart. This agrees with the effect which is seen every day in a boat travelling briskly which runs aground or strikes some obstacle; everyone aboard, being caught unawares, tumbles and falls suddenly toward the front of the boat.”

So in this manner “Joshua would have destroyed not only the Philistines, but the whole earth,” if stopping the sun meant stopping the motion of the earth, as Galileo claims. Not to mention that the idea that the sun’s rotation on its axis is the only thing moving the planets is completely unsubstantiated in the first place. It seems that Galileo pretended to believe in this principle on this occasion solely for the sake of being able to make this scriptural argument. The hypocrisy and unbridled opportunism of Galileo’s forays into biblical interpretation are plain to see.

It is very difficult, if not impossible, to see his interpretation of the Joshua passage as a scientific argument that Galileo genuinely believed. The second quote I read, about everything collapsing like a house of cards if the earth stopped, that is from 15 years later. But surely Galileo realised this all along. If he didn’t, he was stupid. If he did, then he was clearly perfectly happy to fabricate scientifically nonsensical lies as long as it helped him score a satisfying rhetorical point.

This just goes to show how little all of this had to do with science. Galileo’s interpretation of the Joshua passage is terrible science, and he probably knew that perfectly well. This was a conflict between science and religion if by “science” you mean the ludicrous idea that stopping the sun’s rotation would immediately stop the earth dead in its tracks, and that the people on the earth would suffer no consequences of this whatsoever except that the day would became longer. This is the “science” in science versus religion, if we go by what Galileo wrote.

It was only because Galileo got involved with biblical interpretation that he ended up in the crosshairs of the Inquisition. Nobody minded mathematical astronomy, but the question of who has the right to interpret the Bible was the stuff that wars were made of. Luther challenged church authority and emphasised personal understanding of the Bible—“sola scriptura,” as the motto went. This was the core belief of protestantism, and eradicating protestantism was top of the agenda for the catholic church. This is right in the middle of the Thirty Years’ War, which was centered on this core conflict between protestantism and catholicism. A devastating war, comparable to the world wars in terms of per capita deaths.

Once Galileo’s enemies baited him into commenting on the Bible, it was all too easy for them to connect Galileo’s otherwise harmless dabbling to this heresy du jour. Because of the war and raging conflict, this was a matter on which the church could not afford to show any weakness.

There is only one mystery: Why did Galileo walk straight into such an obvious trap? The answer lies, as ever, in his mathematical ineptitude. Galileo was told by church authorities that “if he spoke only as a mathematician he would have nothing to worry about.” Galileo would presumably have followed this advice if he could. The problem, of course, was that he did not have anything to contribute “as a mathematician.” Since a mathematical defence of heliocentrism was beyond his abilities, Galileo was left with no other recourse than to roll the dice and try his luck in the dangerous and unscientific game of scriptural interpretation.

So the church was reluctantly drawn into these astronomical squabbles and had to do something. The Inquisition settled for a slap on the wrist: in the future, Galileo must not “hold, teach or defend [the Copernican system] in any way whatever,” they decided. They also ordered mild censoring of Copernicus’ book, namely the removal of a brief passage concerning the conflict with the Bible and a handful expressions which insinuated the physical truth of the theory. That was it. No book bans, no imprisonments. And Galileo got away with just a warning.

Galileo did indeed keep quiet for a number of years after being ordered to do so by the Inquisition. But times changed. After waiting for over a decade, Galileo felt it was safe to try the waters again. A new Pope was in power, Urban VIII, who was quite liberal. He even said of the 1616 censoring of Copernicus that “if it had been up to me that decree would never have been issued.” Galileo had good personal relations with this new open-minded Pope. So Galileo sensed an opening and obtained a permission to publish the Dialogue in 1632. Or rather, as the Inquisition would later put it, he “artfully and cunningly extorted” this permission to publish. For when the permission was granted the Pope did not know about the private injunction of 1616 for Galileo to keep off the subject. When this came to light the Pope was outraged and felt, with good cause, that Galileo had been deliberately deceitful and reportedly stated that “this alone was sufficient to ruin [Galileo] now.”

So the wheels of the Inquisition were in motion again. A special commission was appointed. It found many inappropriate things in the Dialogue, but this was not a major issue, they noted, for such things “could be emended if the book were judged to have some utility which would warrant such a favor.” The real problem was instead that Galileo “overstepped his instructions” not to treat heliocentricism.

The same report also points out that Galileo had disrespected the Pope on another point as well. The Pope had asked Galileo to include the argument that since God is omnipotent he could have created any universe, including a heliocentric one. So even though the church does not agree with Copernicus, their own logic, namely belief in God’s omnipotence, can be used to legitimate at least considering the possibility of this hypothesis. So that’s a useful argument that Galileo could have used to try to find at least a little bit of common ground with his opponents. But instead of using it for such purposes of reconciliation as intended, Galileo used it to fuel the fires of conflict even more. He made had placed the Pope’s favourite argument “in the mouth of a fool,” the commission observed. He made Simplicio, the dumb character in the Dialogue who constantly expresses the wrong ideas and is proven wrong at every turn, be the one who spoke the Pope’s words. He hardly did himself any favours with this disrespectful move.

Following these findings, the second Inquisition proceedings took place in 1633: 17 years after the first Inquisition where Galileo had gotten off easy, and the year after the publication of his inflammatory Dialogue in defence of Copernicanism. The outcome was a forgone conclusion. Galileo’s defence was transparently dishonest. He pretended that, in the Dialogue, “I show the contrary of Copernicus’s opinion, and that Copernicus’s reasons are invalid and inconclusive.” This is of course pure nonsense. In private correspondence shortly before, Galileo had spoken more honestly, and stated that the book was “a most ample confirmation of the Copernican system by showing the nullity of all that had been brought by Tycho and others to the contrary.” But now before the Inquisition he had to pretend otherwise. In light of the accusations, Galileo continued, “it dawned on me to reread my printed Dialogue,” and “I found it almost a new book by another author.” These transparent lies did little to save him. He was forced to abjure. The Dialogue was prohibited, but not for its contents but rather, in the words of the Inquisition’s sentence, “so that this serious and pernicious error and transgression of yours does not remain completely unpunished” and as “an example for others to abstain from similar crimes.”

There is a popular myth that Galileo muttered “eppur si muove”—”yet it moves” (the earth moves, that is)—as he rose from his knees after abjuring before the Inquisition. But this is certainly false. Obviously the Inquisition would not have tolerated such insubordination, especially since the whole point the trial in the first place was to punish Galileo for his defiance. Galileo had been shown the instruments of torture, and such a rebellious exclamation would have been the surest way to have them dusted off for the occasion. Today no historian believes the myth that Galileo mumbled these words before the Inquisition. Yet it remains instructive in warning us of the lengths many Galilean idol worshippers are willing to go to, who do not want to admit the many ignominious historical facts about their hero. The sheer multitude of such myths now universally regarded as busted should leave us open to the distinct possibility that we have not gotten to the end of them yet.

A similar myth, which has been appealing to anti-religion ideologues, is that “the great Galileo groaned away his days in the dungeons of the Inquisition, because he had demonstrated the motion of the earth.” That’s a quote from Voltaire. But in reality Galileo was sentenced more for his provocateurism than for his science, and furthermore he was never imprisoned in any “dungeon.” He was sentenced to house arrest. A visitor “reported that [Galileo] was lodged in rooms elegantly decorated with damask and silk tapestries.” Soon thereafter he retired to “this little villa a mile from Florence,” where “nearby I had two daughters whom I much loved” and where he also received many friends and guests. Many today would pay dearly for such a retirement. Galileo got it as a so-called “punishment.”

So that’s the story of the Inquisition proceedings. Let’s look at some lessons from this.

Galileo’s conflict with the church was entirely unnecessary. It arose precisely because Galileo was a lampooning populariser rather than a mathematical astronomer and scientist. “[Galileo] was far from standing in the role of a technician of science; had he done so, he would have escaped all trouble,” as Santillana says in his book, The Crime of Galileo. The church establishment had no interest in prosecuting geometers and astronomers. Copernicus’ book had long been permitted, and Galileo’s own Letters on Sunspots of 1613 had been censored only where it referred to scripture, not where it asserted heliocentrism. In reality, “a major part of the Church intellectuals were on the side of Galileo, while the clearest opposition to him came from secular ideas” and philosophical opponents.

Today many take for granted that a fundamental rift between science and religion was unavoidable. Some have imagined for instance that Galileo defied the worldview of the church by demoting the earth from its supposedly “privileged” position. 20th-century playwright Bertolt Brecht appreciated the dramatic flare of framing the conflict in such terms when he wrote a play about Galileo. He has one of the characters argue the privilege point passionately:

“I am informed that Signor Galilei transfers mankind from the center of the universe to somewhere on the outskirts. Signor Galilei is therefore an enemy of mankind and must be dealt with as such. Is it conceivable that God would trust this most precious fruit of his labor to a minor frolicking star? Would He have sent His Son to such a place? The earth is the center of all things, and I am the center of the earth, and the eye of the Creator is upon me.”

But historically this is nonsense, to be sure. Nobody was concerned about this at the time. In fact, classical cosmology clearly stipulated that the Earth was not at all in a privileged position but rather condemned to its very lowly place in the universe. Doesn’t everybody know that hell is just below the surface of the earth, while heaven is way up above? Clearly, then, being at the center of the universe is nothing to be proud of.

It was a commonplace argument in Galileo’s time “that the earth is located in the place where all the dregs and excrements of the universe have collected; that hell is located at the centre of this collection of refuse; and that this place is as far as possible from the outermost empyrean heaven where the angels and blessed reside.” That’s a quote from a book review in the latest issue of the Journal for History of Astronomy. You can go there and find entire books about this.

Even Galileo himself added to the pile of such descriptions. Here is what he says: “after the marvellous construction of the vast celestial sphere, the divine Creator pushed the refuse that remained into the center of that very sphere and hid it there lest it be offensive to the sight of the immortal and blessed spirits.”

Many of Galileo’s contemporaries reasoned alike. Let me quote one more such example: considering “the Vileness of our Earth,” it “must be situated at the center, which is the worst place, and at the greatest distance from those Purer and incorruptible Bodies, the Heavens.” That’s a quote from John Wilkins, an Anglican bishop. This is obviously the very opposite of the argument retrospectively imagined by Brecht and other modern minds, about the supposedly privileged position of the earth.

Here’s another take you sometimes hear: Maybe Galileo brought revolutionary progress by outlining the modern conception of the relation between science and religion. Was it Galileo who showed how faith and science can coexist? How they need not undermine or conflict with one another since one is about the spiritual and the other about the physical? Galileo indeed makes such a case. But those points are common-sense platitudes, not a new vision for the place of science in human thought.

Let’s listen to Galileo’s words from his famous and widely circulated Letter to Duchess Christina of 1615. Here is what Galileo says:

“Far from pretending to teach us the constitution and motions of the heavens and the stars, the authors of the Bible intentionally forbore to speak of these things, though all were quite well known to them. The Holy Spirit has purposely neglected to teach us propositions of this sort as [they are] irrelevant to the highest goal (that is, to our salvation). The intention of the Holy Ghost is to teach us how one goes to heaven, not how heaven goes.”

Even a recent Pope praised Galileo for his supposed insight on this subject: “Galileo, a sincere believer, showed himself to be more perceptive [in regard to the criteria of scriptural interpretation] than the theologians who opposed him.” That’s Pope John Paul II, who said this in 1992.

I disagree with this papal statement on two grounds. First of all, Galileo was not pioneering a new vision for the roles of science and religion more perceptively than anyone else. Rather, he was merely recapitulating elementary ideas that were virtually as old as organised Christianity itself. McMullin has a chapter on this in the Cambridge Companion to Galileo. He concludes that: “[Galileo’s] exegetical principles were not in any sense novel, as he himself went out of his way to stress. They were all to be found in varying degrees of explicitness in Augustine”—twelve centuries before Galileo—”and, separately, they could call on the support of other [even] earlier theologians.” Galileo indeed quotes at great length from Augustine and the church fathers. Not that Galileo knew anything about the history of biblical interpretation: “He had no expertise whatever in that area, so he evidently asked his Benedictine friend, Castelli, to seek out references that would support the exegetical principles he had outlined.” So there was no novelty or insight in Galileo’s treatment of the relation between science and religion.

And here’s a second reason to disagree with the Pope. It is highly doubtful whether Galileo genuinely was “a sincere believer,” as he purported to be. David Wootton has made a compelling case for “two Galileos, the public Catholic and the private sceptic.” Here’s his argument:

“The only decisive document we have [is a 1639 letter to Galileo from] Benedetto Castelli, Galileo’s old friend, former pupil and long-time intellectual companion. … If anyone was in a position to know if Galileo was or was not a believer it was Castelli. … [Castelli writes in his letter that he] has heard news of Galileo that has made him weep with joy, for he has heard that Galileo has given his soul to Christ [in his old age—Galileo was 75 at this point]. Castelli immediately refers to the parable of the labourers in the vineyard: even those who were hired in the last hour of the day received payment for the whole day’s work. … Then … he turns to the crucifixion, and in particular to the two thieves crucified on either side of Christ. One confessed Christ as his saviour and was saved; the other did not and was damned. … Castelli’s [point] is clear and unambiguous. He believes Galileo is coming to Christianity at the last moment, but not too late to save his soul. There is no conceivable interpretation of this letter which is compatible with the generally held view that Galileo was, throughout his career, a believing Catholic.”

That’s David Wootton’s argument in his book on Galileo. It is not a mainstream view but I am inclined to believe it.

The Cambridge Companion to Galileo poses for itself the question: “What did Galileo actually do that made his image so great and so long-standing?” Its answer is not a list of great scientific accomplishments but rather: “Certainly his was the first main effort that fired the vision of science and the world that went well beyond limited intellectual circles.” Galileo was a populariser, in other words. “It was to the man of general interests that Galileo originally addressed his works,” as Stillman Drake says. Indeed, Galileo embraced this role, praising himself for “a certain natural talent of mine for explaining by means of simple and obvious things others which are more difficult and abstruse.”

I agree with these learned authors that Galileo wrote for the vulgar masses. I must add only one point, which they omit, namely that Galileo was driven to turn to popularisation because he was so bad at mathematics. “Galileo scarcely ever got around to writing for physicists,” Drake says. Yes, and he was scarcely able to do so either. The two are not unrelated.

Take for instance the “new stars” (or supernovas, as they would be called today) that appeared in Galileo’s lifetime. One appeared in 1572. It was studied with great care by Tycho Brahe. Another appeared in 1604, when Galileo was 40 years old and an established professor of mathematics. But Galileo didn’t make a contribution based on serious astronomy as Tycho had done. Instead he gave public lectures on the nova to a layman audience totalling more than a thousand people. This is precisely the difference between Galileo and the mathematicians. In modern terms, Galileo is less of a scientist and more of a presenter of TV specials.

Galileo’s little science extravaganzas were a hit at bourgeois dinner parties. Here’s how a contemporary witness describes it:

“We have here Signor Galileo who, in gatherings of men of curious mind, often bemuses many concerning the opinion of Copernicus, which he holds for true. He discourses often amid fifteen or twenty guests who make hot assaults upon him. But he is so well buttressed that he laughs them off; and although the novelty of his opinion leaves people unpersuaded, yet he convicts of vanity the greater part of the arguments with which his opponents try to overthrow him. What I liked most was that, before answering the opposing reasons, he amplified them and fortified them himself with new grounds which appeared invincible, so that, in demolishing them subsequently, he made his opponents look all the more ridiculous.”

Again: Galileo’s speciality is burlesque astronomical road shows, not serious science. If you are an Italian aristocrat who enjoys seeing the learned establishment lose face but don’t want to rock the boat yourself, then you can live vicariously through Galileo’s snappy comebacks and provocations. To this end it matters little whether they are scientifically sound or not.

This is the context in which we must understand Galileo’s conflict with the Church. If we want a parallel of the Galileo trials today we should not think of some totalitarian regime imprisoning intellectuals. A better parallel is cancel culture in popular media. Galileo is a charismatic TV personality. Many enjoy listening to him make fun of the other team. But sometimes he is politically incorrect. So his enemies organise a social media campaign, making a lot of noise. And Galileo is too hot-headed for his own good so he joins in the mud-fight with a bunch of @-replies on Twitter that he didn’t vet with his legal department first. That’s exactly what his opponents were fishing for, and now they got their gotcha quotes that they can take to the network executives and get Galileo cancelled.

Altogether a regrettable spectacle, but one that has not all that much to do with science.

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