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Intellectual Mathematics
Hosted by Intellectual Mathematics · EN
Cracking tales of historical mathematics and its interplay with science, philosophy, and culture. Revisionist history galore. Contrarian takes on received wisdom. Implications for teaching. Informed by current scholarship. By Dr Viktor Blåsjö.
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Death of Archimedes
Jul 1500:26:21Tap to summarizeArchimedes’s emblematic death makes sense psychologically and embodies a rich historical picture in a single scene. Transcript Archimedes died mouthing back at an enemy soldier: “Don’t disturb my circles.” Or that’s how the story goes. Is this fact or fiction? We have third-hand accounts at best so there is plenty of room for doubt. But I’m putting my money on fact nonetheless. I think this standard story makes sense. I think it works psychologically with what little we know about Archimedes as a person, and I think it fits contextually with what we know about Archimedes’s era and circumstances. So let’s investigate this, and let’s use the death of Archimedes to reflect on these broader themes. Archimedes was killed when the Romans invaded his city, Syracuse. There is little doubt about that. The precise details are less clear. There are various versions of the story from several ancient authors. These passages are all conveniently collected at the Archimedes website by Chris Rorres, which I highly recommend. Let’s quote the standard version from Plutarch: “Archimedes was working out some problem by a diagram, and having fixed his mind and his eyes alike upon the subject of his speculation, he never noticed the incursion of the Romans, nor that the city was taken. In this transport of study and contemplation, a soldier, unexpectedly coming up to him, commanded him to follow him. Archimedes declined to do so before he had worked out his problem to a demonstration. The soldier, enraged, drew his sword and ran him through.” It is quite popular to cast doubt on the story of Archimedes’s death. One example is the recent biography “Archimedes: Fulcrum of Science” by Nicholas Nicastro (pages 43-44). This biography argues that the standard story “doesn’t pass the smell test” to use Nicastro’s words. Because “any properly self-interested soldier would know the reward for capturing Archimedes.” Indeed, Archimedes was famous and the Roman commander wanted him captured alive, it is said. So the idea that “the soldier recognizes Archimedes but simply liquidates a valuable prisoner – indeed one who amounted to a strategic asset for Rome – simply because he was lackadaisical in responding to orders doesn’t pass the smell test,” according to Nicastro’s biography. I’m not so sure about that. We know about police brutality. We know for example that George Floyd was killed by police while being apprehended, after being suspected of using a counterfeit twenty-dollar bill. And that was on an ordinary Monday in a peaceful, prosperous country. The soldier who killed Archimedes was not having a normal Monday dealing with petty delinquents. This soldier was in enemy territory in an active war zone. You would think that this soldier would have been on high alert against ambushes and sudden movements, quite rightly. And let’s consider what the soldier’s opinion of Archimedes would have been. Archimedes was well known and famously led the military engineering efforts that fended off the Romans for years. What would the soldier think of the figurehead of the enemy? Would he find that such a great geometer must be spared for the greater good? Or would he think that Archimedes was a terrorist responsible for the deaths of his friends? This soldier may very well have seen first hand the death and suffering inflicted by Archimedes’s famous warfare machines. Maybe for example a friend of his drowned when Archimedes sunk a Roman ship during one of the previous invasion attempts. Or maybe his brother had his legs crushed by one of Archimedes’s catapults, and returned home as a cripple, which made such an impression on the younger brother that an unstoppable hatred festered in him and he swore to dedicate his life to revenge against this evil Greek insurgent. Indeed, maybe on this very day, the day that he came to stand before Archimedes, this soldier has already had to watch helplessly as a close friend and brother in arms died a gruesome death. Such things can happen in war. So I don’t think we can say: the soldier wouldn’t have killed Archimedes because he had orders not to and, rationally speaking, it would have been in his best interest to obey. This soldier may very well have been under immense and acute psychological pressure and trauma at this moment, when he happened to come face to face with the very symbol of everything he had been taught to hate. That’s what I think about this so-called “smell test.” But that’s the soldier’s psychology. Now let’s consider it from Archimedes’s point of view. Would Archimedes be calm and collected and compliant when the soldier comes to arrest him? No, he would not. The invasion is even more traumatic for Archimedes. Archimedes was born in Syracuse and spent his life there. There is every reason to think that these roots meant a lot to Archimedes. Archimedes was famous already in his lifetime. No doubt he had generous offers to go elsewhere, just like superstar academics today. But Archimedes stayed. And he wrote his treatises in the local dialect of Greek, rather than adapting to the more prestigious version of Greek spoken in Athens and Alexandria. Perhaps again a sign of local pride. Archimedes also mentions his father, who was apparently an astronomer. So that’s another sign that Archimedes attached some importance to his heritage. And of course Archimedes was heavily involved in the defense of the city as a military engineer for many years. Obviously another sign of considerable patriotism. And now, all of that is being destroyed. Archimedes’s birthplace, his home for his entire life, burnt and ransacked by a heartless military force. If Archimedes looks out his window all he sees is everyone he ever loved being slaughtered, and generations of cultural heritage being sadistically trampled to dust by soldiers’ boots. This would be heartbreak and trauma enough. But it’s worse. It’s worse for Archimedes because he was in charge of the defense. It’s his fault. All this blood is on his hands. Or so it would seem to him. Archimedes was given every resource to orchestrate the Syracusan defense. All those notorious warfare machines that held the Romans at bay for so long: that’s not something you throw together in your basement. Archimedes must have been entrusted with massive resources and he must have had considerable manpower under his command. His friends and brothers had put their faith in him in their hour of need, and he failed. Archimedes has let them all down. He has let his father down, and his forefathers. Not only is Archimedes watching his city burn. He is also overcome by the crushing guilt that this is all because of his personal failure. How do you think this guy is going to react when an enemy soldier comes to take him away? He’s not in a mood to be read his Miranda rights, is he? It was time for Archimedes to go. Shot down on the pavement. It was the only honorable option left. Most of the historical accounts frame the death of Archimedes in terms of the trope of the absent-minded professor, lost in a diagram, oblivious to the world around him. I imagine that this is a sanitized account. Most of the historical accounts were written under Roman rule. Maybe the real events were quite a bit uglier and a lot less flattering for Roman historians to repeat. Maybe Archimedes was not so cartoonishly lost in geometrical thought at that moment as story-tellers pretend. Maybe he knew full well what was going on, like any normal person would. Especially since he was obviously very well aware of the prospect of Roman military invasion, and he would understand very well what it meant when Roman soldiers had reached his house. Archimedes was an experienced military engineer who had lived under the immediate threat of military attack for years. Is it too much to imagine that such a person would carry a weapon, perhaps a small dagger? Well, now is the time to use it. If not now, when? Of course by the time it comes to that you have already lost. You don’t have a dagger because you think you will be able to fight your way out. You carry the dagger because when the time comes to use it your choices are: die on your knees or take one ------ down with you. That’s how I would write Archimedes: The Gritty Reboot. If that’s what happened then Roman historians would hardly want to admit it. It doesn’t do their self-image any favors that the great Archimedes would rather die than be taken alive by Romans. So the literary cliché of a philosopher so absorbed in thought that he does not notice his surroundings is a welcome euphemism readily at hand. I quoted earlier the standard story from Plutarch, which leans into this cliché very heavily. Actually Plutarch also goes on to give two other versions of the death of Archimedes. “Others write”, he says. And then he says for instance that Archimedes was killed because a soldier mistook his astronomical instruments for gold trinkets and killed him to plunder his valuables. I don’t think so, but even this version clearly has some elements of truth. Namely, there was indeed plundering by the soldiers and some flashy-looking astronomical instruments made by Archimedes were indeed stolen by the Romans and publicly displayed in Rome. So this would have given some credence to the story. Maybe Plutarch is relieved that there is some ambiguity regarding the death of Archimedes. Maybe he knew full well that these alternative stories are not true. Indeed, he first tells the standard story as if it was unequivocal fact, and then he adds the qualifier “others write” when telling the other versions. As if he knew...
Transcribe →Torricelli’s trumpet is not counterintuitive
Dec 30, 202400:56:39Tap to summarizeThere is nothing counterintuitive about an infinite shape with finite volume, contrary to the common propaganda version of the calculus trope known as Torricelli’s trumpet. Nor was this result seen as counterintuitive at the time of its discovery in the 17th century, contrary to many commonplace historical narratives. Transcript Torricelli’s trumpet is not counterintuitive. Your calculus textbook lied to you. You’ve probably heard of this cliché, Torricelli’s trumpet: an allegedly “paradoxical” shape that has infinite area but finite volume. It’s staple example in calculus textbooks. Well, there’s nothing to it, in my opinion. It’s a propaganda lie. Let’s revisionist-history the heck out this thing. I will tell you why there’s nothing counterintuitive about this result. Then I’ll argue that it was not seen as counterintuitive at the time, in the 17th century, contrary to what everyone tells you. Then I will explain, in terms of the sociology of the mathematical community, why this myth is still so popular. That is to say, why it is such a comforting myth to so many people, despite being wrong. So, the trumpet. You take the hyperbola y=1/x and you rotate it about the x-axis. It makes a trumpet shape, a kind of funnel that becomes infinitely narrow the further you go. Also known as “Gabriel’s Horn.” Actually I tried to look up the origin of this silly name but I couldn’t find it. I guess it was perhaps coined for the American market? “Torricelli” is bit too “Euro” isn’t it? Now, “Gabriel’s Horn” on the other hand, there you have a nice pious biblical name. Anyway, whatever you want to call it. The volume of the funnel is finite. For example from x=1 onwards. Despite its area and its length being infinite. There are two supposed contradictions here. On the one hand, finite volume with infinite extent (infinite length) could be regarded as two contradictory properties that it would be surprising to find in the same solid, allegedly. Alternatively, finite volume with infinite area could also be seen as a clash of two incompatible properties: a shape having those two properties at the same time is supposed to be contrary to “intuition,” allegedly. Sometimes it is put like this: Such a solid cannot be painted, since it has infinite area, yet it can be filled with paint, since it has finite volume. But I think this can be a misleading move that mixes the issue of Torricelli’s trumpet with general issues of how infinite processes correspond to everyday experience, which is perhaps a separate source of so-called “counterintuitive” phenomena. The fact is that if Torricelli’s trumpet is supposed to be a surprising result, then the source of the contradiction is supposed to be the properties of volume, area, and length of this shape, and not some secondary intuitions about infinite processes in general. In my opinion, the properties of Torricelli’s trumpet are not “counterintuitive.” First, is it counterintuitive for a finite amount of paint to cover an infinite surface? Of course not. Let’s put it like this. Suppose you have a can of paint. Obviously it contains a finite volume of paint, such as one liter. Now, open the can and pour the paint on the floor. Think of a big floor, like a basketball court. Get a spatula and start spreading the paint across as much area as you can. Where do you think this process will stop? This is mathematical paint. You can spread it as thin as you like. How much area can you cover, if you can spread the paint thinner and thinner and thinner? What does your “intuition” tell you? Does your “intuition” say that this spreading process with terminate after a certain number of square meters of the floor painted? Of course not. That would be idiotic. And yet that is precisely what the standard account of Torricelli’s trumpet would have you believe. It is supposed to be “counterintuitive”, the story goes, for a finite volume of paint to cover an infinite area. Well, we have just seen that that premise is idiotic. Obviously a finite amount of paint can spread further and further, as long as you make it thinner and thinner. There is nothing “counterintuitive” about that. Why would “intuition” say that the process of spreading the paint thinner and thinner would suddenly terminate at some finite bound? Why on earth would it? What would be the “intuitive” reason for why you could spread paint thinner, and then thinner, and then thinner, and then thinner, and then not thinner all of a sudden? Where would this invisible upper bound come from? Why would “intuition” stipulate the existence of such a ghost? It makes no sense. Here’s another thought experiment that proves the same thing. We’re not using paint anymore, but rather a cube. You have a cube of unit volume in front of you. You cut it in half, horizontally. Like a sandwich. Then you place the top half side-by-side with the bottom half. Next cut the top half in half again in the same manner, and bring the new top slice down next to the bottom slice. So now you have a kind of stairway with three steps. Continue in the same way: you keep bisecting the last piece and placing all the pieces in a row. All the slices have a length and a width of 1, the original dimensions of the cube, because you are always cutting horizontally. You are cutting the heights in half, leaving the width and the breadth of all the pieces the same throughout. So you get a row of blocks that each have a length and width of 1, and whose heights are 1/2, 1/4, 1/8, 1/16, and so on. The total volume remains 1 throughout the process of course, because you only moved the existing volume around without adding to it or taking anything away. Meanwhile, both the length and the area of the combined shape clearly approaches infinity, as is intuitively clear. So with this simple intuitive argument we get infinite extent with finite volume. So clearly infinite extent with finite volume is by no means contrary to intuition. Note that this simple thought experiment also shows that Torricelli’s trumpet is not a case of a technical mathematical result being philosophically or qualitatively different from simple common-sense examples, contrary to what calculus teachers like to pretend. So the standard story is wrong in two ways. It claims that advanced calculus proves intuition wrong. But that’s doubly a lie. First, intuition is not wrong, and furthermore you don’t need fancy calculus to show any of this anyway. The simple thought experiments of the repeatedly bisected cube or the painted basketball court, which you can explain to a 5-year-old, contain everything that is relevant. There is nothing qualitatively new added by the use of calculus. These things were also well understood historically. For example, Isaac “Barrow … clearly saw that [Torricelli’s] theorem can be intuitively explained by the fact that ‘the infinite diminution of one dimension compensates the infinite increase of the other’.” This fits with the basketball court example. Although Barrow was talking about the relationship between volume and length, not volume and area. But the point is the same. The equivalent of the bisected cube example was also well understood. For example, Leibniz rightly remarked that “there is nothing more extraordinary about [Torricelli’s result] than about infinite series, where we find that 1/2 + 1/4 + 1/8 + 1/16 + 1/32 etc. = 1.” Exactly. Nothing to see here. Just as Leibniz says. Let’s tur...
Transcribe →Did Copernicus steal ideas from Islamic astronomers?
Nov 29, 202301:27:04Tap to summarizeCopernicus’s planetary models contain elements also found in the works of late medieval Islamic astronomers associated with the Maragha School, including the Tusi couple and Ibn al-Shatir’s models for the Moon and Mercury. On this basis many historians have concluded that Copernicus must have gotten his hands on these Maragha ideas somehow or other, even though no direct evidence for such transmission has been found. Let us consider the evidence as to whether Copernicus plagiarized these Arabic sources or not. See PDF slides for figures and references.
Transcribe →Operational Einstein: constructivist principles of special relativity
Jul 23, 202301:16:38Tap to summarizeEinstein’s theory of special relativity defines time and space operationally, that is to say, in terms of the actions performed to measure them. This is analogous to the constructivist spirit of classical geometry. Transcript Oh no, we are chained to a wall! Aaah! This is going to mess up our geometry big time. Remember what Poincaré said: self-motion is the essence of geometry. We understand that part of the environment to be geometrical that we can cancel through self-motion, through a change of perspective. Suppose you are looking at a chair, let’s say, and somebody tips it over so that it’s laying on its side, or somebody moves it to the other end of the room. Those are geometrical transformations: rotations and displacements in space. They are the equivalence relations of space; the isometries: things you can do without changing metric relationships. You know that these are geometrical equivalence transformations because you can cancel them through self-motion. When the guy knocks the chair over, you can tilt your head 90 degrees, and you have restored the original visual impression of the chair. And if the guy moves the chair five meters that way, then you yourself can move five meters in the same direction and once again the chair makes precisely the same visual impression on your retina as it did before. This is how you know that rotations and displacements are geometrical equivalence transformations. The more you accumulate experience with these kinds of scenarios, the more you begin to grasp the group of geometrical transformations as a whole. You get a global sense of what kinds of transformations are possible, how they combine and interact, and so on. This process might lead you to Euclidean or non-Euclidean conceptions of space depending on your experiences. You get to know space and what kind of geometry it has by getting to know its transformation group: that is to say, what kinds of rotations and displacements exist, what happens if you do one after the other, and so on. Now, what about the scenario when we are chained up? We must imagine that we have been chained to this wall for life. We don’t know any other reality than this. Our sense of what geometrical transformations are possible will be very different. There is still geometry because there are still visual impressions that we can cancel through self-motion. If an object is moving across our field of view, we can keep the retinal impressions the same by tracking it with a motion of our eyes. So we understand the geometry of sideways motion well since we can move our eyes from left to right, or point our gaze in different directions. We also understand the geometry of depth to some extent. If an object is moving away from us, we can keep track of that through self-motion also, but of a very different kind. They eye has a lens in it. The curvature of the lens is variable and is controlled by a muscle. Depending on whether you need to focus on objects that are near or far, the muscle will pinch or pull the lens so that it is more round or more flat in order to have the right focal distance for the object you are looking at. In this way you can keep track of how much an object has moved in depth by recording how much the lens needs to be adjusted to restore focus. So this gives you the data to develop a geometry of depth. So our chains do not deprive us of geometry altogether. We can still develop the geometry of width and the geometry of depth. But these are separate geometries to us. A free person will know that width and depth are merely two dimensions of the same kind of thing. They are both spatial dimensions. They are interchangeable and homogenous. The free person will know that since they can turn width into depth by self-motion. They just need to go stand over there and the old width is the new depth and vice versa. But we who are chained are deprived of this experience. So to us width and depth remain qualitatively different kind of things altogether. Indeed, we measure distance in width and distance in depth completely different units. We count distance in width by the direction in which our eyes are pointing, so the unit is degrees for example. An object is 30 degrees to the left of another, for example, we might say. But we count depth by how much the lens needs to be bent to achieve focus. So the unit is something like a unit of force corresponding to the muscular effort involved. That’s a completely different kind of thing altogether, and cannot be compared with our degree measures that we used to quantify position in the width direction. It’s not so strange that width and depth would be qualitatively different things. You already treat various measurements of the same object as qualitatively different in your everyday life. For example, suppose somebody asked you: Is this building wider than it is old? Of course that doesn’t make any sense. You cannot compare a distance in space with a duration in time. Because those quantities are determined in fundamentally different kind of ways, they are measured in completely different kinds of units, and so on. Well, just as you think time and space are not comparable, so the chained person thinks depth and width are not comparable. Samesies. In fact, maybe you are are just as delusional as the chained guy, and for much the same reason. Actually time and space are a lot more comparable and interchangeable than you think, as Einstein’s theory of relativity says. We don’t realise this in our everyday experience, because relativistic effects become significant only at high speeds, somewhat close to the speed of light. Compared to the speed of light you have practically been standing still your whole life, even when flooring it on the highway. So you might as well have been chained to a wall. The sum total of all your visual and sensory impressions are severely and systematically impoverished just like the guy chained to a wall. Just as he doesn’t realise the fundamental unity of width and depth, so you don’t realise the fundamental unity of time and space. And for the same reason: you are both essentially standing still. I took this example from Feynman’s famous lectures on physics. Why don’t we listen to his version as well? The classic Feynman lectures on physics are nowadays available for free at a Caltech website, audio recordings and all. “When we look at an object, there is an obvious thing we might call the ‘apparent width’, and another we might call the ‘depth’. But the two ideas, width and depth, are not fundamental properties of the object, because if we step aside and look at the same thing from a different angle, we get a different width and a different depth, and we may develop some formulas for computing the new ones from the old ones and the angles involved. … If it were impossible ever to move, and we always saw a given object from the same position, then this whole business would be irrelevant—[width and depth] would appear to have quite different qualities, because one appears as a subtended optical angle and the other involves some focusing of the eyes …; they would seem to be very different things and would never get mixed up. It is because we can walk around that we realize that depth and width are, somehow or other, just two different aspects of the same thing. [In Einstein’s theory of special relativity] also we have a mixture---of positions and the time. … In the space measurements of one man there is mixed in a little bit of the time, as seen by the other. Our analogy permits us to generate this idea: The ‘reality’ of an object that we are looking at is somehow greater (speaking crudely and intuitively) than its ‘width’ and its ‘depth’ because they depend upon how we look at it; when we move to a new position, our brain immediately recalculates the width and the depth. But our brain does not immediately recalculate coordinates and time when we move at high speed, because we have had no effective experience of going nearly as fast as light to appreciate the fact that time and space are also of the same nature. It is as though we were always stuck in the position of having to look at just the width of something, not being able to move our heads appreciably one way or the other.” (I.17-1) I love this thought experiment with the chained guy. Plato’s cave 2.0. And it is perfect for our purposes today. This is going to be the concluding episode of my history and philosophy of geometry story arc, and the theme will be how everything goes full circle and the beautiful ideas from days of old are as relevant as ever to us self-absorbed moderns as well. The guy chained to a wall is a perfect backward-looking example, and a perfect forward-looking example. Back to the operationalism of Greek geometry, and forward to Einsteinian modernity. We started, way back when, with the Greeks and their ubiquitous ruler and compass. Always with the making, those guys. Lines and circles are nothing but the things you get when you draw with these tools. Not abstract things, not axiomatically defined things. Lines and circles are operations. They are things you do. The Greeks realised that this was the rigorous way to do mathematics. The epistemological humility of the maker is far superior to hubris of the philosopher who think they can concoct a perfect theoretical system in the abstract using the power of their mind alone. People are not as good at that as they think. Time and time again, somebody’s pretentious abstract theory has proved to contain various unintended contradictions and unnoticed assumptions. As the Greeks knew all too well: the works of Plato and Aristotle do little else than poke holes in other people’s bad theories. So we should stop trying to philosophise about ess...
Transcribe →Review of Netz’s New History of Greek Mathematics
Oct 11, 202200:52:04Tap to summarizeReviel Netz’s New History of Greek Mathematics contains a number of factual errors, both mathematical and historical. Netz is dismissive of traditional scholarship in the field, but in some ways represents a step backwards with respect to that tradition. I argue against Netz’s dismissal of many anecdotal historical testimonies as fabrications, and his “ludic proof” theory. Transcript A new book just appeared: A New History of Greek Mathematics, by Stanford Professor Reviel Netz, Cambridge University Press. Let’s do a book review. It will be a critical review. The main theme will be the sciences versus the humanities. Note the title of the book: “a New History.” Netz’s “New History” represents the new humanities-centred dominance in the field. As opposed to the “old” histories written by more mathematically oriented people. In my opinion, “new” does not mean better in this case. And I will tell you why. Let’s start by attacking a city. The enemy are hunkering down behind their city walls. We are going to have to scale the walls with ladders. How long should we make the ladders? The ancient historian Polybius has the answer: “The method of discovering right length for ladders is as follows. … If the height of the wall be, let us say, ten of a given measure, the length of the ladders must be a good twelve. The distance from the wall at which the ladder is planted must, in order to suit the convenience of those mounting, be half the length of the ladder, for if they are placed farther off they are apt to break when crowded and if set up nearer to the perpendicular are very insecure for the scalers. … So here again it is evident that those who aim at success in military plans and surprises of towns must have studied geometry.” Great stuff. But Netz gets it wrong, in my opinion. Here is how he concludes: “And then, of course, we are supposed to apply – Polybius leaves this implicit – Pythagoras’s theorem.” (223) I don’t think so. I don’t think that’s what Polybius intended. Sure enough, you can solve for the length of the ladder using the Pythagorean Theorem, but that is a clumsy and inefficient way to do it. If you did this the modern way you would need to do some algebra followed by some calculation involving a square root. They didn’t have calculators on their phones back then, you know. Do you expect carpenters in the military to be able to calculate square roots by hand? In fact, Polybius has already told you everything you need to know with his numerical example. If the wall is 10, the ladder should be 12, he says. But it scales! So what Polybius is really saying is that, whatever the height of the wall is, the ladder is always 20% longer than that. That’s all you need to know. No Pythagorean Theorem needed. Those numbers are a rule of thumb. You can also do it more exactly if you want, according to Polybius’s more theoretical characterisation of the optimal length. But you don’t need the Pythagorean Theorem for that either. There’s a much better way, that you can easily teach to an illiterate carpenter in five minutes. Draw an equilateral triangle, just as Euclid does in Proposition 1 of the Elements. Cut it down the middle. Now you have a right-angled triangle, where the base is exactly half of the hypothenuse. This corresponds precisely to Polybius’s rule: the distance along the ground is half the length of the ladder. So now we have a scale model of what we want. The height down the middle of the equilateral triangle represents the city wall; the side of the equilateral triangle represents the ladder, and it is precisely half its own length from the foot of the wall, exactly as Polybius says it should be for optimal stability. So if we are given that the height of the wall is for example 10 meters, then we divide the height of the triangle into ten equal parts. We take a blank ruler and mark those ten marks on it. Then we take this ruler, with this length unit, and measure the hypothenuse of the triangle. However many marks long it is, that’s how many meters our ladder needs to be. Piece of cake. Easy to improvise in the field without any specialised knowledge or tools. While Netz is busy trying to teach his carpenters the algebra of quadratic expressions and how to extract square roots, I have already scaled his walls using my much quicker methods. That is what you get when you put humanities people in charge of mathematics. So I wouldn’t trust Netz when it comes to mathematics, even when he says “of course,” as he does here. Here is another example: Did you know that parabolas are pointier than hyperbolas? At least if we are to believe Professor Netz. This claim occurs in a discussion of Archimedes. Archimedes studied solids of revolution obtained by rotating a conic section around its axis. Here are Netz’s words: “In the case of a parabola, this will be of a more pointed shape; in the case of the hyperbola, this may be more bowl-like.” (140) This is BS. Parabolas are not “more pointed” than hyperbolas. This is clear for example from the following fact: you can draw a hyperbola having any two given lines as asymptotes and passing through any given point. So in other words, you can draw a V, an arbitrarily pointy letter V, and then pick an arbitrary point inside that V, for instance a point super close to the vertex of the V. Then there is always a hyperbola that fits inside the V and that passes through the designated point. You can hardly get any pointier than that, now can you? Yet parabolas are nevertheless “more pointed”, somehow, Netz apparently believes. By the way, this fact I just mentioned, about constructing a hyperbola within a given V (that is to say, with given asymptotes), that is Proposition 4 of Book II of the Conics of Apollonius. Or is it? Here we have another interesting point. It seems that this proposition was actually not in the original version of the Conics. Because Eutocius, in late antiquity, needs this theorem at a certain point and he says he better prove it since it’s not in the Conics of Apollonius. But then in the text we have of the Conics, what we call Apollonius’s Conics today, this proposition clearly is there, with the exact same proof. And in fact the standard text that we call Apollonius’s Conics today comes to us only through that very same author, Eutocius, who wrote a commentary on the Conics and also preserved the text at the same time. So it seems that Eutocius inserted this proposition into Apollonius’s original text, because he had noticed in other works that it was a useful thing to prove. Netz describes this correctly, which is all the more reason why he should know that a hyperbola can be as pointy as you’d like, since this follows immediately from this proposition that he discusses at length. But anyway, there is another kind of error here in Netz’s discussion of this. The point that this proposition of the Conics is an insertion by Eutocius — that insight, says Netz, is due to Wilbur Knorr, Netz’s predecessor as a classics professor at Stanford. “No one noticed that prior to Knorr” (431-432), says Netz. But that is not true. Wilbur Knorr was not the first to discover this. In fact, Knorr clearly says so in his own article, the very article cited by Netz, which Netz has evidently not read very carefully. Already in the 16th century, Commandino, in his Latin edition of the Conics, very clearly and explicitly made the exact same point as Knorr, using the exact same evidence and arguments. And this in turn was cited in a 19th-century German edition of the Conics, as Knorr himself says. So Knorr didn’t discovery anything except what people had already known for hundreds of years. This is not such an innocent mistake. How are we supped to trust anything Netz says if he makes blatantly false statements that are clearly and unequivocally seen to be factually incorrect by simply glancing at the very article that Netz himself cites in support of his own claims? But it’s even more problematic than that. Because it’s clearly not just a random mistake. It is an ideologically driven error. By saying that Stanford humanities professor Wilbur Knorr was the first to make this important scholarly discovery, Netz is obviously indirectly boosting the impression that his own claims are important and novel, since he too is a Stanford humanities professor. Netz is not only saying that Wilbur Knorr was the first to discover this particular thing. He is implicitly saying that earlier generations of scholars missed important insights, and that only people like him — Stanford humanities professors — are true experts. That is of course the point of the title of the book: A *New* History of Greek Mathematics. In the past everybody did it wrong, and we need people like Netz to finally do it right. There is indeed a lot of explicit posturing to this effect throughout the book. Let’s look at another example of this. Let me read a passage where Netz is attacking Thomas Kuhn’s account of the history of astronomy. Thomas Kuhn wrote in the mid-20th century and he worked on the history of science even though his PhD was in physics. So that is exactly the kind of people Netz wants to denigrate. He wants to say that only specialised humanities professors, with their “new” histories, are actual experts in the field. Here is what Netz says about Kuhn: “Like most nonspecialists, Kuhn supposed …” See? I told you. It’s not just that Kuhn was wrong. It is that Kuhn epitomises the kind of people (people with a PhD in physics, for example) who need to be eliminated from the field because they make so many hopelessly naive assumptions without even realising it. Anyway, let’s continue with the quote: “Like most nonspeciali...
Transcribe →The “universal grammar” of space: what geometry is innate?
May 20, 202200:32:22Tap to summarizeGeometry might be innate in the same way as language. There are many languages, each of which is an equally coherent and viable paradigm of thought, and the same can be said for Euclidean and non-Euclidean geometries. As our native language is shaped by experience, so might our “native geometry” be. Yet substantive innate conceptions may be a precondition for any linguistic or spatial thought to be possible at all, as Chomsky said for language and Kant for geometry. Just as language learning requires singling out, from all the sounds in the environment, only the linguistic ones, so Poincaré articulated criteria for what parts of all sensory data should be regarded as pertaining to geometry. Transcript The discovery of non-Euclidean geometry in the early 19th century was quite a wake-up call. It showed that everybody had been a bit naive, you might say. Here’s an analogy for this. Suppose we had all been speaking one language, let’s say English. And we were all convinced that English is the only natural language. In fact, that question didn’t even arise to us; we simply assumed that English and language is the same thing. We even had philosophers “explaining why” English is a priori necessary. These philosophers had “proved,” they thought, that without English the very notion of linguistic communication or thought is impossible. And then we discovered that there are French speakers and Chinese speakers. Oops. Very embarrassing. English is not necessary after all. It is not innate, it is not synonymous with language itself. For thousands of years we made those embarrassing mistakes because we were not aware of the existence of other languages. That’s how it was with geometry. What I said about English corresponds to Euclidean geometry. For thousands of years, nobody thought of Euclidean geometry as one kind of geometry. Everybody thought of it as THE geometry. Geometry and Euclidean geometry was the same thing. Just as an isolated linguistic community thinks their language is THE language. And the philosophers I spoke of, Kant is an example of that. He argued that Euclidean geometry was a necessary precondition for having spatial experience or spatial perception at all, which is like saying that English is necessary for any kind of linguistic expression. And already long before Kant, many people had been convinced by how intuitively natural and obvious the axioms of Euclidean geometry feel. Descartes for instance and many others made a lot of this fact. Remember how important it was to Descartes that our intuitions were truths implanted by God in our minds. Well, we all think our native language is intuitive. And we think other people’s languages are not intuitive. This feeling is so strong that we think it must be objective. When we try to learn a foreign language, it feels impossible that anyone could think that was intuitive. And yet they do. So apparently our intuitions can deceive us. We feel that our native grammar is much more natural than everyone else’s, but that’s a delusion. It felt like an objective fact, but it turned out to be subjective. Could it be the same with geometry? Could the alleged naturalness and intuitiveness of Euclidean geometry turn out to be just an arbitrary cultural bias, like thinking English feels more natural than French? So, ouch, we took quite a hit there with the discovery on non-Euclidean geometry. It exposed our insularity. It showed that things we had thought we had proven to be impossible were in fact perfectly possible and every bit as viable as what we had thought was the only way to do geometry. And yet there is hope as well. The language analogy doesn’t just expose what an embarrassing mistake we made, or how non-Euclidean geometry hit us where it hurts. The language analogy also suggests a way out; a way to rise from the ashes. We were wrong about the specific claim that Euclidean geometry is innate, because that’s like saying that English is innate. Nevertheless we were right too, one could argue. Language is impossible without something innate. Everybody learns their native language with incredible fluency. Every child learns it somehow, with hardly any systematic teaching; they just pick it up naturally. And they do so at a very early age, when their general intelligence is still very limited. Meanwhile, no animal comes anywhere near achieving the same feat. Nor any adult human for that matter. I’m better than a three-year-old child at any intellectual task, except learning French. Somehow the child is super good at that. So clearly something about language is innate. The ability of language acquisition is innate. Some kind of general principles of language are innate. Perhaps geometry is like language in this way. We have some innate geometry. Not specific Euclidean propositions or axioms maybe, but some kind of “geometryness” nonetheless. Some sort of more structural or general principles of geometry than specific propositions, just as our innate linguistic ability doesn’t contain anything specific to any one language but instead has to do with “languageness” in general. In linguistics, this is called “universal grammar.” Every language has it own grammar, of course, but there are some more general or structural principles of language that are the same for all human languages. This common core is the universal grammar. Here’s an example of such a principle that belongs to universal grammar. Consider the statement “the man is tall.” It is a declaration, an assertion. You could turn it into a question: “Is the man tall?” We turned the assertion into a question by moving the “is” to the front of the sentence. That’s how you make questions from statements: start with assertions—“the man is tall”—and move the “is” to the front—“is the man tall?” So that’s a recipe for making questions. But consider now the statement “the man who is tall is in the room.” How do you form the corresponding question? There are two “is”’s in the sentence. So the rule “to form a question, move the is to the front” is ambiguous. Which of the is’s should we move? It could become: “is the man who is tall in the room?” That works. But if we move the other is to the front we get nonsense: “is the man who tall is in the room?” Well, that didn’t work. It didn’t make a question, it just made gibberish. Even though we followed the same rule as before: move the “is” to the front. If language was nothing but a social construction then it would be perfectly reasonable for children trying to form questions to come up with the gibberish one. If the child simply extracts general rules from a bunch of examples, it would have been perfectly reasonable for the child to have guessed that the general rule is: move the first “is” to the front of the sentence to form a question. Which would lead to the nonsense question “is the man who tall is in the room?” In fact, however, “children make many mistakes in language learning, but never mistakes such as [this].” Apparently, “the child is employing a ‘structure-dependent rule’” rather than the much simpler rule to put the first “is” in front. Why? “There seems to be no explanation in terms of ‘communicative efficiency’ or similar considerations. It is certainly absurd to argue that children are trained to use the structure-dependent rule, in this case. The only reasonable conclusion is that Universal Grammar contains the principle that all such rules must be structure-dependent. That is, the child’s mind contains the instruction: Construct a structure-dependent rule, ignoring all structure-independent rules.” So although each language has its own grammar, there are some general principles like that that are universal: common to all languages. Those principles are hard-wired into the mind at birth. This stuff about there being an innate “universal grammar” is the view Chomsky, the leading 20th-century linguist. The example and explanation I just quoted are from his book Reflections on Language. There are many debates about Chomskyan linguistics, but I’m going to assume Chomsky’s point of view for the purposes of this discussion, because its parallels with geometry are very interesting. You might say that this view of language is Kantian in a way. We saw that Kant put a lot of emphasis on the necessary preconditions for certain kinds of knowledge. Geometry, for example, is not an external, free-standing theory that we can analyze with our general intellectual capacities. Rather, the fundamental concepts of geometry are bound up with the very cognitive structure of our mind itself. Some things are purely learned through experience and convention, such as how to play chess or how to dance the tango. But some things are not like that. For instance, they way we experience color. The mind is made to see red and blue and to not see infrared and so on. That’s a fixed, domain-specific property of how our mind works. You can neither learn nor unlearn that through general intelligence. Color experience is just one of those basic things hardwired right into the brain. From the Chomskyan point of view, language is like that as well. Language is not merely a social construct with man-made rules like chess or tango. Nor is it explicable in terms of general intelligence only. Chess or tango you can learn by general intelligence. That is to say, if you spend enough time looking at people playing chess or dancing, you can eventually figure out what the rules are by general rational thinking such as pattern recognition, and forming preliminary hypotheses about how you think it works and then observing some more to check if you maybe need to revise the hypothesis to take into account some other possible circumstances or cases. Color exper...
Transcribe →“Repugnant to the nature of a straight line”: Non-Euclidean geometry
Feb 20, 202200:30:40Tap to summarizeThe discovery of non-Euclidean geometry in the 19th century radically undermined traditional conceptions of the relation between mathematics and the world. Instead of assuming that physical space was the subject matter of geometry, mathematicians elaborated numerous alternative geometries abstractly and formally, distancing themselves from reality and intuition. Transcript The mathematician has only one nightmare: to claim to have proved something that later turns out to be false. There are thousands of theorems in Greek geometry, and every last one of them is correct. That’s what mathematical proofs are supposed to do: eliminate any risk of being wrong. If this armor is compromised, if proofs are fallible, then what’s left of mathematics? It would ruin everything. But the nightmare came true in the 19th century. What had been thought to have been proofs were exposed as fallacies. Top mathematicians had made mistakes. Mistakes! Like some commoner. It’s going to be hell to pay for this, as you can imagine. I’m referring to Euclid’s fifth postulate, the parallel postulate. Euclid’s postulate had rubbed a lot of people the wrong way even since antiquity. It sounds more like a theorem. The earlier postulates were very straightforward: there’s a line between any two points, stuff like that. Very primitive truths. It makes sense that the bedrock axioms of geometry should be the simplest possible things, such as the existence of a line between any two points. The parallel postulate, by contrast, is not very simple at all. It’s not a primordial intuition like the other postulates. It states that two lines will cross if a rather elaborate condition is met. That’s the kind of thing theorems say. This particular type of configuration has such-and-such a particular property. That’s how theorems go in Euclid. Very different from the other postulates, which are things like: any line segment can be spun around to make a circle. So, people tried to prove the parallel postulate as a theorem. Many felt that it should not be necessary to assume the parallel postulate as Euclid had done. The parallel postulate should be a consequence of the core notions of geometry, not a separate assumption. Many people tried to “improve” on Euclid in this way. From antiquity all the way to the 19th century. Around 1800, people like Lagrange and Legendre proposed so-called proofs of the parallel postulate. Those are big-name mathematicians. Their names are engraved in gold on the Eiffel Tower. Lagrange was even buried in the Panthéon in Paris. Elite establishment stuff. But even these bigwigs were wrong. Their proofs contain hidden mistakes. It’s astonishing that this was more than 2000 years after Euclid. People tried to improve on Euclid for millennia. And not a few claimed to have succeeded. But the fact is that Euclid was right all along. The parallel postulate really does need to be a separate assumptions, just as Euclid had made it. It cannot be proved from the other axioms, as so many mathematicians during those millennia had mistakenly believed. The Greeks, you know, they were really something else. It’s so easy to make subtle mistakes in the theory of parallels. History shows that there are a hundred ways to make tiny invisible mistakes that fool even the best mathematicians. Top mathematicians who were never wrong about anything else stumbled on this one issue. Somehow Euclid got it exactly right. He didn’t make any of those hundred mistakes that later mathematicians did. That’s not luck, in my opinion. Arguably, the Greeks were more sophisticated foundational geometers than even the Paris elite in 1800. Unbelievable but true. Euclid’s Elements is not a classic for nothing. Euclid is not a symbol of exact reasoning because of some lazy Eurocentric birth right. Euclid’s Elements really is that good. When Euclid made the parallel postulate an axiom, he seems to be suggesting that it cannot be proved from the other axioms. And he was right. But, as I said, many people had a hunch that he was wrong about this. They thought it would be impossible for the other axioms to be true and the parallel postulate not true. So many mathematicians figured they could prove this by contradiction: Suppose the parallel postulate is false. If we could show that that assumption would contradict other geometrical truths, then the assumption must be false. So this way we could prove that the parallel postulate must be true, by showing that it would be incoherent or impossible for it to be false. Indeed, it was found that negating the parallel postulate had various strange consequences. For example, if the parallel postulate is false then squares do not exist. Suppose you try to make a square. So you have a base segment, and you raise two perpendiculars of equal length from the two endpoints of the segment. Then you connect the two top points of these two perpendiculars. That ought to make a square. In Euclid’s world it does. But proving that this really makes a square requires the parallel postulate. If the parallel postulate is false, one can instead prove that this construction does not make a square but rather a weirdly disfigured quadrilateral. Because the last side of the “square” doesn’t make right angles with the other sides. So even though you made sure you had right angles at the base of the quadrilateral, and that the perpendicular sides were equal, the fourth and final side still somehow manages to “miss the mark” so to speak. It makes non-right angles. It’s as if the sides are sort of bent. It’s as if you had four perfectly equal sticks of wood, but then you stored them carelessly and they were exposed to humidity and so on and they were warped. So now they’re kind of mismatched in terms of length and straightness, and when you try to piece them together to make a square they don’t fit right. They make some wobbly not-quite-square shape. Doing geometry without Euclid’s parallel postulate feels a bit like that. It’s sort of bent out of shape and nothing fits the way it should anymore. One person who investigated this was Saccheri. He wrote a big book discussing this misshaped square and other things like that, in 1733. Saccheri felt that he had justified Euclid’s parallel postulate by examples such as theses. The square that’s not a square and other such deformities, Saccheri declared to be “repugnant to the nature of the straight line.” But one might say that he used this emotional language to compensate or cover up a shortcoming in the mathematical argument. He had indeed showed that if the parallel postulate is false then geometry is weird. Then you have squares that don’t fit, and other things that feel like doing carpentry with crooked wood. But weird is not the same as self-contradictory. Despite their best efforts, mathematicians could not find a clear-cut proof that negating the parallel postulate led to directly contradictory conclusions. This is why Saccheri had to say “repugnant” rather than contradictory. You only get “repugnantly” deformed squares, not direct contradictions such as 2=1 or a part being greater than the whole. Those things would be logical contradictions and you wouldn’t need emotions like repugnance. In fact, a hundred years after Saccheri, mathematicians came to accept that this strange non-Euclidean world of the warped wood is not contradictory. It is coherent and consistent. It is merely another kind of geometry. An alternative to Euclid. People used to shout and scream that all kinds of things were repugnant, such as homosexuality, for instance. That doesn’t really prove anything except the narrow-mindedness of those accusers. Mathematicians had been equally narrow-minded. They had tried to justify the status quo for thousands of years. They had tried to prove that their way of doing things–their geometry–was the only right way. Only in the 19th century did they finally realize that it was much more productive to embrace diversity, to accept all the geometries of the rainbow. For so many years mathematicians could not get away from the idea that the “straight” squares of Euclid were the only “normal” ones, and that the “repugnant” alternative squares of non-Euclidean geometry were birth defects. But they were wrong. Non-Euclidean geometry is as legitimate as any other. It was a creative watershed shift in perspective in mathematics to finally accept this instead of trying to prove the opposite. Here’s how Gauss, the greatest mathematician at the time, put it in the early 19th century. Negating Euclid’s parallel postulate “leads to a geometry quite different from Euclid’s, logically coherent, and one that I am entirely satisfied with. The theorems [of this non-Euclidean geometry] are paradoxical but not self-contradictory or illogical.” “The necessity of our [Euclidean] geometry cannot be proved. Geometry must stand, not with arithmetic which is pure a priori, but with mechanics.” Geometry has become like mechanics in the sense that it is empirically testable. The theorems of geometry are not absolute truths but hypotheses like the hypotheses of physics that have to be checked in a lab and perhaps corrected if they don’t agree with measurements. For example, Euclid proves that the angle sum of a triangle is 180 degrees. But this theorem depends on the parallel postulate, just as Euclid’s proof reveals it to do. In non-Euclidean geometries, angle sums of triangles will be different. So that’s something testable. Measure some triangles to see which geometry is right, just as you drop some weights or whatever in a physics lab to see which law of gravity is right. Let me quote Lobachevsky, one of the other discoverers of non-Euclidean geometry. Here’s how he makes this point in his book o...
Transcribe →Rationalism 2.0: Kant’s philosophy of geometry
Nov 17, 202100:30:00Tap to summarizeKant developed a philosophy of geometry that explained how geometry can be both knowable in pure thought and applicable to physical reality. Namely, because geometry is built into not only our minds but also the way in which we perceive the world. In this way, Kant solved the applicability problem of classical rationalism, albeit at the cost of making our perception of the world around us inextricably subjective. Kant’s theory also showed how rationalism, and philosophy generally, could be reconciled with Newtonian science, with which it had been seen as embarrassingly out of touch. In particular, Kant’s perspective shows how Newton’s notion of absolute space, which had seemed philosophically repugnant, can be accommodated from an epistemological point of view. Transcript Rationalism says that geometrical knowledge comes from pure thought. Empiricism says that it comes from sensory experience. Neither is very satisfactory, because geometry is clearly both: it’s too good a fit for the physical world to be only thought, and it relies too much on abstract proofs to be only experience. So it seems the “right” philosophy of mathematics must be a little bit of both. But how? Rationalism and empiricism mix like oil and water. They are so different, so opposite, that it seems impossible to find any sort of middle ground that has half of each. Nevertheless there is a golden mean of sorts. The philosophy of Kant. Immanuel Kant, the late 18th-century philosopher. His view of geometry is in a way the best of both worlds: combining the best rationalism with the best of empiricism. Let’s see how he pulled that off. Kant’s theory is another innateness theory. Geometry is innate; it is hardwired into our minds. That’s what the rationalists said too, of course. Innateness was why the uneducated boy in Plato’s Meno could reach substantial geometrical insights without instruction. The innate intuitions of geometry are reliable because God is not a deceiver, said Descartes. And they apply to the physical world, because the Creator put into our minds the same ideas he had used to design the universe, said Kepler and others. But Kant’s innateness is different. He lived a century and a half after Kepler and Descartes, and the reliance on God to support the rationalist worldview had become much less fashionable during the intervening years. And indeed Kant does away with it. By taking geometry to be innate, Kant automatically inherits the best of rationalism. That is to say, he is able to account for the prominent role of pure reason in geometry in a convincing way, just as the earlier rationalists had done before him. But Kant solves the weakness of rationalism very differently. If geometry is innate and susceptible to purely theoretical elaboration, then why does it always agree so well with experience? Not because “God was a geometer,” as Plato and Kepler said. No, Kant’s solution is: Not only are geometrical principles innate in our thoughts, they are also innate in our perceptions. Aha, plot twist! The so-called success of mathematics in the real world is no miracle. It’s a rigged game. Our eyes and our senses are biased. They can only see Euclidean geometry. It’s an illusion to think that we can compare our mathematical deductions with reality. We think we can “test” whether theoretically established theorems are true or not in the physical world; for example, by measuring the sides of right-angle triangles and comparing the results to what Euclid says it should be. But we were naive to think that these two things were really independent. Just as our thoughts are shaped by our innate intuitions, so also our perceptions of physical reality are shaped by the same intuitions. We don’t have direct access to “raw data” about the physical world. When we think we observe the world, we really observe an interpreted version of the world. The mind can only process information that is interpreted or converted to fit a particular format. In terms of geometry, that means Euclidean geometry. We see the world through Euclid-colored glasses. Like those pink sunglasses that John Lennon used to wear. Of course if you wear pink glasses, then everything looks pink. But of course John Lennon could not say: aha, I told you, the world is rosy; I argued theoretically that the world should be rosy, and now I have confirmed by observation of the actual world is in fact rosy, because look how everything is pink. John Lennon would be fooling himself if he argued that way. Euclidean geometry is exactly like that, according to Kant. With experience and observation, we do not discover that Euclidean geometry applies to the real world. What we really observe is our own biases. We discover not facts about the world, but facts about what kind of glasses we are wearing. John Lennon made the world rosy by putting on his glasses, and in the same way we make the world Euclidean by a hidden lens in the mind’s eye. Think of a chili pepper. This pepper proves that Kant was right. How so? Think about it. What are the characteristics of a chili pepper? It’s red. What is red? Is the pepper “really” red? That is to say, is its redness an objective property of reality? Well, yes and no, right? The way science accounts for colors is in terms of wave length of light. What we in our minds regard as red correspond in reality to a particular frequency of light waves. So there’s something there corresponding to red, but our minds put a major interpretative spin on it. Red is really just a wavelength, “just as number,” so to speak. But our minds turn it into something more qualitative. And the mind is very selective as well. The remote control of your TV sends its signal using infrared light, which the minds chooses not to see at all. Even though infrared light is exactly the same kind of thing as the color red from the pepper. They are exactly the same kind of waves, only with slightly different frequencies. “Objectively” they are very similar. But subjectively, in our minds, they couldn’t be more different. One is the color red that is one of the fundamental categories in terms of which we navigate reality, and the other is completely invisible. So perception is very far from direct access to “objective” raw data. The mind interferes very heavily: it selects, transforms, distorts. Maybe when we think Euclidean geometry fits reality it’s only because of this. Only because things that don’t fit a Euclidean mold are as invisible to us as infrared light. The chili pepper is also very hot to taste. Again, is that an objective property of pepper itself, or is it just a subjective matter of how it interacts with our tongues? In fact, birds eat chili peppers and to them they are not hot. Chili peppers evolved to be hot because it’s better to be eaten by birds than to be eaten by mammals. The seeds spread better that way. So chili peppers developed this characteristic of being repellently hot to mammals like us, but but perfectly appetizing to birds. Could it be the same with our geometrical intuitions? “Two lines cannot enclose a space,” Euclid says. The very thought is repellent! Well, maybe that’s just our subjective experience, just as we find it repellent to bite into a hot pepper. Maybe other creatures have completely different geometrical intuitions. Just as birds think pepper are not hot, maybe they also think space has five dimensions or whatever. Who knows. Different animals can have all kinds of different intuitions and modes of perception that fits the way they navigate the world, the way they find food, and so on. Maybe Euclidean geometry is just something that happened to be convenient to us, just as night vision is convenient to a cat, or a heightened sense of smell is to a dog. That would make geometrical experience quite subjective. Or at least subjectively contaminated, or entangled with subjective factors. This is how Kant is able to bridge the gap between rationalism and empiricism. We can sit in a closed room and figure out in our heads geometry that then turns out to be true in the real world. Because, says Kant, what is really happening is that the mind is analyzing itself. What we discover through geometrical meditation is not objective facts, but facts about the geometrical preconceptions hardwired into our minds and the consequences of those conceptions. And these results apply to the world not because they are really out there but because our perception actively processes and converts and interprets any sensory input in Euclidean terms. As Kant himself says: “Space is not something objective and real; instead, it is subjective and ideal, and originates from the mind’s nature as a scheme, as it were, for coordinating everything sensed externally.” “It is from the human point of view only that we can speak of space, extended objects, etc.” In the introduction I framed the issue in a way that is flattering to Kant. I said: Everybody knew that you needed some kind of middle ground between rationalism and empiricism, but no one could figure out how to do it. Until Kant, he finally cracked the nut. But maybe that’s the wrong way to look at it. It often is, this kind of narrative. “Everybody tried to do such-and-such but no one could do it, until finally one guy was smart enough to figure it out.” That’s not usually how history works. If we look into the details we often find that contextual factors explain why key steps happened at certain times. It’s true that everyone knew that neither rationalism nor empiricism were perfect for hundreds of years. And Kant’s proposal is certainly a very clever way of tackling that problem. So why did it take all the way to the late 18th century before these ideas were proposed? <p...
Transcribe →Rationalism versus empiricism
Sep 18, 202100:43:50Tap to summarizeRationalism says mathematical knowledge comes from within, from pure thought; empiricism that it comes from without, from experience and observation. Rationalism led Kepler to look for divine design in the universe, and Descartes to reduce all mechanical phenomena to contact mechanics and all curves in geometry to instrumental generation. Empiricism led Newton to ignore the cause of gravity and dismiss the foundational importance of constructions in geometry. Transcript Here’s a fundamental problem in the philosophy of mathematics. You can sit in an isolated room, in an arm chair, and prove theorems about triangles, such as the angle sum of a triangle or the Pythagorean theorem. When you do this, you have the feeling that you have established these results with absolute certainty. You feel that they must be true because of how compelling the proof is. And you feel that you have established this by thought alone, by purely intellectual means. Mathematics is unique in this respect. In other subjects, thinking is a powerful tool, but it is always supplemented by observation and experience. If you spent your whole life isolated in a locked room, you would not be able to say anything about the laws of astronomy or the anatomy of the digestive system, because without observation, with only pure thought, it is impossible to even get started in those field. But you could figure out everything about triangles. If one day you were released from your prison where you had been sitting for decades, you could go out and measure actual triangles and you would find that, indeed, their angle sum is always two right angles, the Pythagorean theorem always holds for right-angle triangles and so on. Just as you had predicted by pure thought. This is a bit of a mystery. Because it shows that there are two sides of mathematics that are difficult to reconcile. On the one hand, the internal, mental conviction that mathematics establishes absolute truths purely by reasoning. On the other hand, the external, physical fact that mathematics works in the real world. What is the bridge between these two worlds? It is as if there is a natural harmony between our minds and the outer world. What is the cause of that harmony? These two poles can be called rationalism and empiricism. Rationalism takes mathematics to be fundamentally a matter of pure thought. This fits well with the sense we have when doing mathematics, when reading Euclid, that we are establishing absolute truths by sheer reasoning. But it doesn’t explain why mathematics works so well in the physical world. We have encountered some rationalists already: Plato, Descartes. We saw how Descartes solved the problem. Mathematics is pure thought, and it works in the physical world because the Creator put mathematical ideas in our minds. As the Bible says, “God created man in his image.” That is to say, God created the world based on mathematical ideas, and then created humans and sort of pre-programmed their minds with the same kinds of ideas that he had used to create the world. So no wonder there’s a harmony between the mental and the physical worlds: they both stem from the same source, the Creator, who used the same principles when designing both. Descartes said basically this quite explicitly, as we recall. Plato pretty much hints at the same idea. God is a mathematician. That is a central belief in Platonist thought as well. And it is a necessary thesis for the rationalists to explain why mathematics works so well. We have already encountered some empiricist as well: Aristotle, Francis Bacon. They think knowledge ultimately comes from the world around us. From that point of view, it is no mystery that mathematics works on physical triangles. It stems from physical experience to begin with, so of course it conforms to physical experience. The challenge for the empiricists is instead to explain the mental experience of doing mathematics; our feeling that it brings absolute truth by pure thought in a way that no other subject does. From the empiricist point of view, this feeling is a mistake, a delusion. We think we are doing pure thought, but actually mathematical thought is generalized experience. We think we can sit in a closed room, an arm chair, and figure things out about an outside world that we have never even seen. But it only feels that way. We have seen and touched many lines and triangles and squares our entire life, since the year we were born. We have internalized this experience. It has become second nature to us. Basic truths of geometry, such as Euclid’s axioms, may feel like core intuitions that are much more pure and absolute and undoubtable than things we know from experience. But that feeling is a delusion, according to the empiricists. Our minds, our feelings have imperfect self-awareness. Just as we are not aware through introspection how our digestive system works, so we are not conscious of the psychological origins of our mathematical intuitions. I think we can agree that rationalism and empiricism both face big challenges. The challenge for rationalism is to explain why mathematics applies to the physical world. Traditional rationalism had an answer that was very compelling at the time: the explanation in terms of God, the Creator. But nowadays we may want an atheistic answer. And then rationalism is back to square one, facing the original problem all over again, without any solution in sight. Empiricism doesn’t have that problem, but it has other ones. If mathematics comes from experience, how can it seem so absolute and undoubtable? How can an exact science come from inexact sensory impressions? If mathematics is based on experience like everything else, why does it seem to be such a different kind of knowledge in so many respects? Those are challenges for the empiricist to answer. It matters how you answer these questions. It shapes the kind of science that you do. Consider for instance Kepler, the 17th-century astronomer. He was another rationalist. As Kepler says: “Nature loves [mathematical] relationships in everything. They are also loved by the intellect of man who is an image of the Creator.” That’s almost word for word how I described the rationalist position just moments ago. Kepler felt that the world was designed with the intent that we should study the universe mathematically. As he says: “Whenever I consider in my thoughts the beautiful order [of the universe] then it is as though I had read a divine text, written onto the world itself saying: Man, stretch thy reason hither, so that thou mayest comprehend these things.” In fact, scientific facts support this view, in Kepler’s opinion. For example, as he says, “Sun and moon have the same apparent sizes, so that the eclipses, one of the spectacles arranged by the Creator for instructing observing creatures in the orbital relations of the sun and the moon, can occur.” That is indeed a striking fact: that the moon is exactly the right size to precisely block out the sun at the moment of a solar eclipse. From the point of view of modern science, this is a remarkable coincidence. It’s pure chance that the moon is exactly the right size. You can understand why the explanation in terms of purpose was more compelling in Kepler’s time. Witnessing a solar eclipse is a spiritual experience. It all seems so perfect. Much too perfect to chalk it up to chance. It’s very disappointing that modern science offers nothing more than this non-explanation of such an emotionally compelling spectacle. And not just modern science. Such views were around already in Kepler’s time. Atomism is a classical worldview that is indeed happy to attribute almost everything, eclipses included, to chance and randomness. According to Kepler’s teacher, Melanchthon, such views “wage war against human nature, which was clearly founded to understand divine things.” So here we have again that double challenge to empiricism. If mathematics is just one type of knowledge among many that we pick up from experience, then, first of all, why does the universe show so many signs of being mathematically designed? Like the thing with the eclipses, but there are also countless other examples one could use to make this point. Empiricism has no answer to this. It thinks that’s all just a bunch of coincidences, and we are just fooling ourselves by looking for purpose and design that isn’t there. And secondly, if empiricism is right, and mathematics is just experiential knowledge like everything else, then why does mathematical reasoning feel so uniquely compelling and convincing? As Melanchthon says, mathematics is as natural to a human being as “swimming to a fish or singing to a nightingale.” Just as animals are born with these instincts, so our minds are innately predisposed to do mathematics. Empiricism does not explain why that is the case, or why that seems to be the case. So it’s understandable that Kepler was a convinced rationalist instead. And this conviction shaped his scientific work. Astronomers are “priests of the book of nature,” as Kepler said. So he was always looking for meaning and purpose and design. For example, the telescope was a new invention in Kepler’s time, and it was a big moment when the moons of Jupiter were discovered. Kepler immediately looked for the purpose behind the existence of these moons. He concluded that Jupiter must be inhabited. Why else would it have moons? As Kepler says: “For whose sake, the question arises, if there are no people on Jupiter to behold this wonderfully varied display with their own eyes? We deduce with the highest degree of probability that Jupiter is inhabited.” Another of Kepler’s attempts at uncovering divine design was his theory of planetary distances. According t...
Transcribe →Cultural reception of geometry in early modern Europe
Jul 10, 202100:33:47Tap to summarizeEuclid inspired Gothic architecture and taught Renaissance painters how to create depth and perspective. More generally, the success of mathematics went to its head, according to some, and created dogmatic individuals dismissive of other branches of learning. Some thought the uncompromising rigour of Euclid went hand in hand with totalitarianism in political and spiritual domains, while others thought creative mathematics was inherently free and liberal. Transcript Gothic architecture is known for its pointed arches. Unlike round arches like a classical Roman aqueduct for example. Those are semi-circular, but Gothic arches are steeper, pointier. Gothic buildings, cathedrals, have these arches everywhere: windows, doorways, and so on. Gothic arches consist of two circular arcs. You can make it like this. First make a rectangular shape. Like a plain window or door. A boring old rectangle. Now let’s spice it up. Take out your compass, and put it along the top side of the rectangle. Draw two circular arcs going up above the rectangle. Use the top side of the rectangle as the radius, and its two endpoints as the two midpoints of the two arcs you are drawing. The two arcs make a pointed extension of the rectangle. Now you have your Gothic window. If you have your Euclid in fresh memory you will recognize at once that this is precisely the type of construction involved in Proposition 1 of the Elements. Coincidence? No, I don’t think so. The Gothic style of architecture arose in Europe in the early 12th century, within a decade or two of the first Latin translation of Euclid’s Elements. If that’s not cause and effect, it‘s an incredible coincidence. There is little direct documentation about this, but, I am quoting now from Otto von Simson’s book The Gothic Cathedral, “at least one literary document survives that explains the use of geometry in Gothic architecture: the minutes of architectural conferences held in 1391 in Milan. The question debated at Milan is not whether the cathedral is to be built according to a geometrical formula, but merely whether the figure to be used is to be the square or the equilateral triangle. The minutes of one particularly stormy session relate an angry dispute between the French expert, Jean Mignot, and the Italians. Overruled by them on a technical issue, Mignot remarks bitterly that his opponents have set aside the rules of geometry by alleging science to be one thing and art another. Art, however, he concludes, is nothing without science, ars sine scientia nihil est. This argument was considered unassailable even by Mignot’s opponents. They hasten to affirm that they are in complete agreement as regards this theoretical point and have nothing but contempt for an architect who presumes to ignore the dictates of geometry.” So the geometrical ethos was very strong indeed. This hardline view probably softened a bit over time. Renaissance art is more expressive, emotive, more alive, one might say, than this rigid late medieval stuff. That’s if we fast-forward two hundred years from these Gothic conferences about how art is nothing without geometry. Then you have people like Michelangelo who said: “the painter should have compasses in his eyes, not in his hands.” I suppose it means that art should go a little more by feeling and intuition, and not be completely dictated by mathematics. But you still have “compasses in your eyes,” so there’s still a very significant role for geometry, it seems. It is also revealing, perhaps, that Michelangelo thought it was important to point this out at all. I guess there were a lot of artists with compasses in their hands running around back then. Why else would Michelangelo feel the need to criticise that practice? In fact, geometry proved useful to art again, in new ways, in the Renaissance. At this time artists discovered (or perhaps rediscovered) the geometrical principles of perspective. Accurate representation of depth in a painting follows simple geometrical principles. The key construction is that of a tiled floor. Like a chessboard type of pattern of floor tiles, but seen in perspective, so tiles that are further away appear smaller in the picture. There are a lot of tiled floors in Renaissance art, because they are great for conveying a sense of depth. You draw it like this. Draw two horizontal lines: one is where the floor starts, and one is the horizon. Divide the floor line into many equal pieces, representing the size of the tiles. Connect all these points to a fixed point on the horizon. This is because all the parallel rows of the floor will appear to converge in one point, just as whenever you are looking at parallel lines that go off into the distance, such as railroad tracks for instance, they appear to meet at the horizon. Next, draw a second horizontal line representing the other edge of the front row of floor tiles. Now here comes the magic step. Draw the diagonal of the first tile, and extend it. Where this line cuts the other lines you have already drawn, those are all corners of other tiles. This is because of the geometrical principle that a straight line, no matter how you look at it, from whatever angle, will always still be straight. A lot of stuff looks funny in perspective: big things can look small, parallel lines appear to meet, perfectly round things appear to be oval, and so on. Perspective distorts shapes in all kind of ways. But straight lines remain straight lines. That is an invariant in all of this. This is why the diagonal of the first floor tile is also the diagonal of successive tiles. Since that makes a straight line in the real world, it must make a straight line in the picture as well. Once you have this diagonal it is easy to complete the rest of the floor. It looks great. It creates a photorealistic sense of depth. No wonder so many artists chose to set their scenes in locales that just happened to have lots of tiled floors. But the insight is much deeper than this. It’s not really about tiled floors; it’s about the correct perspective representation of depth and size generally. Even if you don’t want a tiled floor in your finished picture, it’s still very useful to draw one in pencil on your canvas as a reference grid. You can use the tiles as a guide to transfer and compare sizes at different depths and distances. Then later you can paint it over with some trees or whatever, so nobody can see the grid anymore; you just used it behind the scenes to get the proportions right. The Greeks probably knew about this stuff, but basically no paintings from antiquity have been preserved. But we know they were skilled artists. One guy is said to have painted grapes so realistically that birds came and pecked at it. It seems the Greeks knew the geometrical principles of perspective and used it to creates scenes for the theatre for example. As Vitruvius says, “by this deception a faithful representation of the appearance of buildings might be given in painted scenery, so that, though all is drawn on a vertical flat facade, some parts may seem to be withdrawing into the background, and others to be standing out in front.” This wasn’t just a party trick to the Greeks. It also had philosophical implications. To Plato they raised profound epistemological conundrums. He was concerned that optical illusion painting has “powers that are little short of magical,” “because they exploit this weakness in our nature,” bypassing “the rational part of the soul.” The solution to this problem, as Plato saw it, was a solid mathematical education. Since “sense perception seems to produce no sound result” with these illusory paintings, “it makes all the difference whether someone is a geometer or not.” “The power of appearance often makes us wander all over the place in confusion, often changing our minds about the same thing and regretting our actions and choices with respect to things large and small.” “The art of measurement,” by contrast, “would make the appearances lose their power” and “give us peace of mind firmly rooted in the truth.” Those are all Plato’s words. A rousing case for mathematics! But Plato perhaps drew his conclusions a step too far, rejecting categorically the role of observational data in science: “there’s no knowledge of sensible things, whether by gaping upward or squinting downward.” Science must be based on “the naturally intelligent part of the soul,” not observation. For example, “let’s study astronomy by means of problems, as we do geometry, and leave the things in the sky alone.” With such attitudes, perhaps it is no wonder that the Greeks excelled more in mathematics than in the sciences. But indeed the threat of optical illusions is a legitimate argument in Plato’s defense. When the principles of perspective were rediscovered in Renaissance Italy, they were again at the heart of scientific developments, but this time on the side of empirical science. Galileo looked at the moon through a telescope and concluded that it had mountains and craters. Of course the image one sees through a telescope is flat. But mountains and craters are revealed by the shadows they cast. This is not necessarily very easy to see, or not necessarily a very evident conclusion. Some scholars have argued that the artistic tradition, and its extensive study of perspective and shadows, was a necessary training for the eye to be able to correctly interpret the telescope data. Is it a coincidence that Galileo the telescopic astronomer came from the same land as the great Italian Renaissance painters? Galileo was born and raised in Tuscany, right where so many of these masters had worked. Maybe only someone immersed in this artistic culture had the right eyes to interpret the heavens. A far-fetched theory, in my opinion, but it’...
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