Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries | Lex Fridman Podcast #190

The following is a conversation with Jordan Ellenberg, a mathematician at University of Wisconsin, and an author who masterfully reveals the beauty and power of mathematics in his 2014 book, How Not to Be Wrong, and his new book, just released recently, called Shape, The Hidden Geometry of Information, Biology, Strategy, Democracy, and Everything Else. Quick mention of our sponsors, Secret Sauce, ExpressVPN, Blinkist, and Indeed. Check them out in the description to support this podcast. As a side note, let me say that geometry is what made me fall in love with mathematics when I was young. It first showed me that something definitive could be stated about this world through intuitive visual proofs. Somehow, that convinced me that math is not just abstract numbers devoid of life, but a part of life, part of this world, part of our search for meaning. This is the Lex Friedman Podcast, and here is my conversation with Jordan Ellenberg. Jordan, I'm curious, what do you think is the most fundamental thing in mathematics? Is it the idea that the whole universe is built on top, extra layers, maybe the icing on the cake, maybe just, maybe consciousness is just like a cherry on top. Where do you put in this cake mathematical thinking? Is it as fundamental as language? In the Chomsky view, is it more really tightly interconnected? That's a really interesting question. You're getting me to reflect on this question of whether the feeling of producing mathematical output, if you want, is like the process of, you know, uttering language or producing linguistic output. I think it feels something like that, and it's certainly the case. Let me put it this way, it's hard to imagine doing mathematics in a completely non-linguistic way. It's hard to imagine doing mathematics without talking about mathematics and sort of thinking in propositions. But, you know, maybe it's just because that's the way I do mathematics, so maybe I can't imagine it any other way, right? It's a... Well, what about visualizing shapes, visualizing concepts, to which language is not obviously attachable? Ah, that's a really interesting question, and, you know, one thing it reminds me of is one thing I talk about in the book is dissection proofs, these very beautiful proofs of geometric propositions. There's a very famous one by Bhaskara of the Pythagorean theorem, proofs which are purely visual, proofs where you show that two quantities are the same by taking the same pieces and putting them together one way, and making one shape, and putting them together another way, and making a different shape, and then observing those two shapes must have the same area because they were built out of the same pieces. You know, there's a famous story, and it's a little bit disputed about how accurate this is, but that in Bhaskara's manuscript, he sort of gives this proof, just gives the diagram, and then the entire verbal content of the proof is he just writes under it, behold, like that's it. There's some dispute about exactly how accurate that is. But so then there's an interesting question. If your proof is a diagram, if your proof is a picture, or even if your proof is like a movie of the same pieces, like coming together in two different formations to make two different things, is that language? I'm not sure I have a good answer. What do you think? I think it is. I think the process of manipulating the visual elements is the same as the process of manipulating the elements of language. And I think probably the manipulating, the aggregation, the stitching stuff together is the important part. It's not the actual specific elements. It's more like, to me, language is a process, and math is a process. It's not just specific symbols. It's in action. It's ultimately created through action, through change, and so you're constantly evolving ideas. Of course, we kind of attach, there's a certain destination you arrive to that you attach to, and you call that a proof. But that doesn't need to end there. It's just at the end of the chapter, and then it goes on and on and on, in that kind of way. But I gotta ask you about geometry, and it's a prominent topic in your new book, Shape. So for me, geometry is the thing, just like as you're saying, made me fall in love with mathematics when I was young. So being able to prove something visually just did something to my brain that, it planted this hopeful seed that you can understand the world, like perfectly. Maybe it's an OCD thing, but from a mathematics perspective, like humans are messy, the world is messy, biology is messy, your parents are yelling, making you do stuff, but you can cut through all that BS and truly understand the world through mathematics, and nothing like geometry did that for me. For you, you did not immediately fall in love with geometry. So how do you think about geometry? Why is it a special field in mathematics? And how did you fall in love with it, if you have? Wow, you've given me a lot to say, and certainly the experience that you describe is so typical, but there's two versions of it. One thing I say in the book is that geometry is the cilantro of math. People are not neutral about it. There's people who, like you, are like, the rest of it I could take or leave, but then at this one moment, it made sense. This class made sense. Why wasn't it all like that? There's other people, I can tell you, because they come and talk to me all the time, who are like, I understood all the stuff where you're trying to figure out what x was, there's some mystery, you're trying to solve it, x is a number, I figured it out. But then there was this geometry, like what was that? What happened that year? I didn't get it. I was like lost the whole year, and I didn't understand why we even spent the time doing that. But what everybody agrees on is that it's somehow different. There's something special about it. We're going to walk around in circles a little bit, but we'll get there. You asked me how I fell in love with math. I have a story about this. When I was a small child, I don't know, maybe like I was six or seven, I don't know. I'm from the 70s. I think you're from a different decade than that. But in the 70s, we had a cool wooden box around your stereo. That was the look, everything was dark wood. And the box had a bunch of holes in it to let the sound out. And the holes were in this rectangular array, a six by eight array of holes. And I was just kind of like, you know, zoning out in the living room as kids do, looking at this six by eight rectangular array of holes. And if you like, just by kind of like focusing in and out, just by kind of looking at this box, looking at this rectangle, I was like, well, there's six rows of eight holes each. But there's also eight columns of six holes each. So eight sixes and six eights. It's just like the dissection proofs we were just talking about. But it's the same holes. It's the same 48 holes. That's how many there are, no matter whether you count them as rows or count them as columns. And this was like, unbelievable to me. Am I allowed to cuss on your podcast? I don't know if that's, are we FCC regulated? Fuck yes. Okay, it was fucking unbelievable. Okay, that's the last time. Get it in there. This story merits it. So two different perspectives on the same physical reality. Exactly. And it's just as you say, you know, I knew the six times eight was the same as eight times six, right? I knew my times tables, like I knew that that was a fact. But did I really know it until that moment? That's the question. Right? I knew that I sort of knew that the times table was symmetric. But I didn't know why that was the case until that moment. And in that moment, I could see like, oh, I didn't have to have somebody tell me that. That's information that you can just directly access. That's a really amazing moment. And as math teachers, that's something that we're really trying to bring to our students. And I was one of those who did not love the kind of Euclidean geometry ninth grade class of like, prove that an isosceles triangle has equal angles at the base like this kind of thing. It didn't vibe with me the way that algebra and numbers did. But if you go back to that moment, from my adult perspective, looking back at what happened with that rectangle, I think that is a very geometric moment. In fact, that moment exactly encapsulates the intertwining of algebra and geometry, this algebraic fact that, well, in the instance, eight times six is equal to six times eight. But in general, that whatever two numbers you have, you multiply them one way. And it's the same as if you multiply them in the other order. It attaches it to this geometric fact about a rectangle, which in some sense, makes it true. So you know, who knows, maybe I was always fated to be an algebraic geometer, which is what I am as a, as a researcher. So that's the kind of transformation and you talk about symmetry in your book. What the heck is symmetry? What the heck is these kinds of transformation on objects that once you transform them, they seem to be similar? What do you make of it? What's its use in mathematics, or maybe broadly in understanding our world? Well, it's an absolutely fundamental concept. And it starts with the word symmetry in the way that we usually use it when we're just like talking English and not talking mathematics, right? Sort of something is when we say something is symmetrical, we usually means it has what's called an axis of symmetry, maybe like the left half of it looks the same as the right half, that would be like a left right axis of symmetry, or maybe the top half looks like the bottom half, or both, right? Maybe there's sort of a fourfold symmetry where the top looks like the bottom and the left looks like the right, or more. And that can take you in a lot of different directions, the abstract study of what the possible combinations of symmetries there are a subject, which is called group theory was actually one of my first loves in mathematics, what I thought about a lot when I was in college. But the notion of symmetry is actually much more general than the things that we would call symmetry if we were looking at like a classical building or a painting or, or something like that. You know, nowadays, in in math, we could use a symmetry to refer to any kind of transformation of an image or a space or an object. You know, so what I talk about in in the book is take a figure and stretch it vertically, make it twice as big vertically, and make it half as wide. That I would call a symmetry, it's not a symmetry in the classical sense. But it's a well defined transformation that has an input and an output, I give you that output, I give you some shape. And it gets kind of, I call this in the book a scrunch, I just made I had to make up some sort of funny sounding name for it, because it doesn't really have a name. And just as you can sort of study which kinds of objects are symmetrical, under the operations of switching left and right, or switching top and bottom, or rotating 40 degrees, or what have you, you could study what kinds of things are preserved by this kind of scrunch symmetry. And this kind of more general idea of what a symmetry can be. Let me put it this way. A fundamental mathematical idea, in some sense, I might even say the idea that dominates contemporary mathematics, or by contemporary, by the way, I mean, like the last like 150 years, we're on a very long timescale. In math, I don't mean like yesterday, I mean, like a century or so, up till now, is this idea that's a fundamental question of when do we consider two things to be the same? That might seem like a complete triviality. It's not. For instance, if I have a triangle, and I have a triangle of the exact same dimensions, but it's over here. Are those the same? Or different? Well, you might say like, well, look, there's two different things. This one's over here, this one's over there. On the other hand, if you prove a theorem about this one, it's probably still true about this one, if it has like all the same side lanes and angles and like looks exactly the same. The term of art, if you want it, you would say they're congruent. But one way of saying it is there's a symmetry called translation, which just means move everything three inches to the left. And we want all of our theories to be translation invariant. What that means is that if you prove a theorem about a thing that's over here, and then you move it three inches to the left, it would be kind of weird if all of your theorems like didn't still work. So this question of like, what are the symmetries and which things that you want to study are invariant under those symmetries is absolutely fundamental. Boy, this is getting a little abstract, right? It's not at all abstract. I think this is completely central to everything I think about in terms of artificial intelligence. I don't know if you know about the MNIST dataset, what's handwritten digits. And I don't smoke much weed or any really, but it certainly feels like it when I look at MNIST and think about this stuff, which is like, what's the difference between one and two? And why are all the twos similar to each other? What kind of transformations are within the category of what makes a thing the same? And what kind of transformations are those that make it different? And symmetry is core to that. In fact, whatever the hell our brain is doing, it's really good at constructing these arbitrary and sometimes novel, which is really important when you look at like the IQ test or they feel novel ideas of symmetry of like what, like playing with objects, we're able to see things that are the same and not and construct almost like little geometric theories of what makes things the same and not and how to make programs do that in AI is a total open question. And so I kind of stared and wonder how, what kind of symmetries are enough to solve the MNIST handwritten digit recognition problem and write that down. And exactly. And what's so fascinating about the work in that direction from the point of view of a mathematician like me and a geometer is that the kind of groups of symmetries, the types of symmetries that we know of are not sufficient. So in other words, like, we're just going to keep on going into the weeds on this. The deeper the better. You know, a kind of symmetry that we understand very well is rotation. So here's what would be easy. If humans, if we recognize the digit as a one, if it was like literally a rotation by some number of degrees with some fixed one in some typeface like Palatino or something, that would be very easy to understand. It would be very easy to like write a program that could detect whether something was a rotation of a fixed digit one. Whatever we're doing when you recognize the digit one and distinguish it from the digit two, it's not that. It's not just incorporating one of the types of symmetries that we understand. Now, I would say that I would be shocked if there was some kind of classical symmetry type formulation that captured what we're doing when we tell the difference between a two and a three, to be honest. I think what we're doing is actually more complicated than that. I feel like it must be. They're so simple, these numbers. I mean, they're really geometric objects. Like we can draw one, two, three. It does seem like it should be formalizable. That's why it's so strange. Do you think it's formalizable when something stops being a two and starts being a three, where you can imagine something continuously deforming from being a two to a three? Yeah, but there is a moment. I have myself written programs that literally morph twos and threes and so on. And you watch, and there is moments that you notice, depending on the trajectory of that transformation, that morphing, that it is a three and a two. There's a hard line. Wait, so if you ask people, if you show them this morph, if you ask a bunch of people, do they all agree about where the transition happened? That's an interesting question. Because I would be surprised. I think so. Oh my god, okay, we have an empirical dispute. But here's the problem. Here's the problem. That if I just showed that moment that I agreed on... Well, that's not fair. No, but say I said, so I want to move away from the agreement, because that's a fascinating, actually, question that I want to backtrack from, because I just dogmatically said, because I could be very, very wrong. But the morphing really helps. That the change, because I mean, partially, it's because our perception systems, see, it's all probably tied in there. Somehow the change from one to the other, like seeing the video of it, allows you to pinpoint the place where a two becomes a three much better. If I just showed you one picture, I think you might really, really struggle. You might call it a seven. I think there's something also that we don't often think about, which is, it's not just about the static image, it's the transformation of the image, or it's not a static shape, it's the transformation of the shape. There's something in the movement that seems to be not just about our perception system, but fundamental to our cognition, like how we think about stuff. Yeah, and that's part of geometry too. And in fact, again, another insight of modern geometry is this idea that maybe we would naively think we're going to study, I don't know, like Poincaré, we're going to study the three-body problem. We're going to study three objects in space moving around subject only to the force of each other's gravity, which sounds very simple, right? And if you don't know about this problem, you're probably like, okay, so you just put it in your computer and see what they do. Well, guess what? That's a problem that Poincaré won a huge prize for, making the first real progress on in the 1880s, and we still don't know that much about it 150 years later. I mean, it's a humongous mystery. You just opened the door, and we're going to walk right in before we return to symmetry. So, let's talk about geometry. Who's Poincaré, and what's this conjecture that he came up with? Why is it such a hard problem? Okay, so Poincaré, he ends up being a major figure in the book, and I didn't even really intend for him to be such a big figure, but he's first and foremost a geometer, right? So, he's a mathematician who kind of comes up in late 19th century France at a time when French math is really starting to flower. Actually, I learned a lot. I mean, you know, in math, we're not really trained on our own history. We get a PhD in math, learn about math. So, I learned a lot. There's this whole kind of moment where France has just been beaten in the Franco-Prussian War, and they're like, oh my god, what did we do wrong? And they were like, we got to get strong in math like the Germans. We have to be more like the Germans, so this never happens to us again. So, it's very much, it's like the Sputnik moment, you know, like what happens in America in the 50s and 60s with the Soviet Union. This is happening to France, and they're trying to kind of like, instantly like modernize. That's fascinating that the humans and mathematics are intricately connected to the history of humans. The Cold War is, I think, fundamental to the way people saw science and math in the Soviet Union. I don't know if that was true in the United States, but certainly it was in the Soviet Union. It definitely was, and I would love to hear more about how it was in the Soviet Union. I mean, there's, and we'll talk about the Olympiad. I just remember that there was this feeling like the world hung in a balance, and you could save the world with the tools of science, and mathematics was like the superpower that fuels science. And so, like, people were seen as, you know, people in America often idolize athletes, but ultimately the best athletes in the world, they just throw a ball into a basket. So, like, there's not, what people really enjoy about sports, and I love sports, is like excellence at the highest level, but when you take that with mathematics and science, people also enjoyed excellence in science and mathematics in the Soviet Union, but there's an extra sense that that excellence will lead to a better world. So, that created all the usual things you think about with the Olympics, which is like extreme competitiveness, right? But it also created this sense that in the modern era in America, somebody like Elon Musk, whatever you think of him, like Jeff Bezos, those folks, they inspire the possibility that one person or a group of smart people can change the world, like not just be good at what they do, but actually change the world, and mathematics was at the core of that. And I don't know, there's a romanticism around it too, like, when you read books about, in America, people romanticize certain things like baseball, for example, there's like these beautiful poetic writing about the game of baseball. The same was the feeling with mathematics and science in the Soviet Union, and it was in the air. Everybody was forced to take high-level mathematics courses, like you took a lot of math, you took a lot of science and a lot of like really rigorous literature. Like, the level of education in Russia, this could be true in China, I'm not sure, in a lot of countries, is in whatever that's called, it's K-12 in America, but like young people education, the level they were challenged to learn at is incredible. It's like America falls far behind, I would say. America then quickly catches up and then exceeds everybody else at the, like, as you start approaching the end of high school to college, like the university system in the United States arguably is the best in the world. But like, what we challenge everybody, it's not just like the good, the A students, but everybody to learn in the Soviet Union was fascinating. I think I'm going to pick up on something you said. I think you would love a book called Duel at Dawn by Amir Alexander, which I think some of the things you're responding to what I wrote, I think I first got turned on to by Amir's work. He's a historian of math, and he writes about the story of Evariste Galois, which is a story that's well known to all mathematicians, this kind of very, very romantic figure who, he really sort of like begins the development of this, well, this theory of groups that I mentioned earlier, this general theory of symmetries, and then dies in a duel in his early 20s, like all this stuff, mostly unpublished. It's a very, very romantic story that we all learn. And much of it is true. But Alexander really lays out just how much the way people thought about math in those times in the early 19th century was wound up with, as you say, romanticism. I mean, that's when the romantic movement takes place. And he really outlines how people were predisposed to think about mathematics in that way, because they thought about poetry that way. And they thought about music that way. It was the mood of the era to think about, we're reaching for the transcendent, we're sort of reaching for sort of direct contact with the divine. And so, part of the reason that we think of Galois that way was because Galois himself was a creature of that era. And he romanticized himself. I mean, now, you know, he wrote lots of letters. And he was kind of like, I mean, in modern terms, we would say he was extremely emo. We wrote all these letters about his florid feelings and the fire within him about the mathematics. And, you know, so he, so it's just as you say, that the math history touches human history. They're never separate, because math is made of people. LUKE Yeah. DAVE I mean, that's what it's, it's people who do it, and we're human beings doing it. And we do it within whatever community we're in. And we do it affected by the mores of the society around us. LUKE So the French, the Germans and Poincaré. DAVE Yes. Okay, so back to Poincaré. So he's, you know, it's funny, this book is filled with kind of, you know, mathematical characters who often are kind of peevish or get into feuds or sort of have like, weird enthusiasms. Because those people are fun to write about. And they sort of like say, very salty things. Poincaré is actually none of this as far as I can tell. He was an extremely normal dude. He didn't get into fights with people. And everybody liked him. And he was like pretty personally modest. And he had very regular habits. You know what I mean? He did math for like, four hours in the morning and four hours in the evening. And that was it. Like he had his schedule. I actually it was like, I still am feeling like somebody's gonna tell me now the book is out like, Oh, didn't you know about this like incredibly sordid episode of Mr. Poincaré? As far as I could tell, a completely normal guy, but he just kind of, in many ways creates the geometric world in which we live. And you know, his first really big success is this prize paper he writes for this prize offered by the King of Sweden for the study of the three body problem. The study of what we can say about, yeah, three astronomical objects moving in what you might think would be this very simple way. Nothing's going on except gravity. So what's the three body problem? Why is that a problem? So the problem is to understand when this motion is stable and when it's not so stable, meaning they would sort of like end up in some kind of periodic orbit, or I guess it would mean sorry, stable would mean they never sort of fly off far apart from each other. And unstable would mean like eventually they fly apart. So understanding two bodies is much easier. Yeah, exactly. Third wheel is always a problem. This is what Newton knew. Two bodies, they sort of orbit each other in some kind of, either in an ellipse, which is the stable case, you know, that's what the planets do that we know. Or one travels on a hyperbola around the other, that's the unstable case. It sort of like zooms in from far away, sort of like whips around the heavier thing and like zooms out. Those are basically the two options. So it's a very simple and easy to classify story. With three bodies, just a small switch from two to three, it's a complete zoo. What we would say now is it's the first example of what's called chaotic dynamics, where the stable solutions and the unstable solutions, they're kind of like wound in among each other. And a very, very, very tiny change in the initial conditions can make the long-term behavior of the system completely different. So Poincaré was the first to recognize that that phenomenon even existed. What about the conjecture that carries his name? Right. So he also was one of the pioneers of taking geometry, which until that point had been largely the study of two and three dimensional objects, because that's like what we see, right? That's the objects we interact with. He developed a subject we now call topology. He called it analysis situs. He was a very well-spoken guy with a lot of slogans, but that name did not, you can see why that name did not catch on. So now it's called topology now. Sorry, what was it called before? Analysis situs, which I guess sort of roughly means like the analysis of location or something like that. It's a Latin phrase. Partly because he understood that even to understand stuff that's going on in our physical world, you have to study higher dimensional spaces. How does this work? And this is kind of like where my brain went to it because you were talking about not just where things are, but what their path is, how they're moving when we were talking about the path from two to three. He understood that if you want to study three bodies moving in space, well, each body, it has a location where it is. So it has an X coordinate, a Y coordinate, a Z coordinate, right? I can specify a point in space by giving you three numbers, but it also at each moment has a velocity. So it turns out that really to understand what's going on, you can't think of it as a point, or you could, but it's better not to think of it as a point in three dimensional space that's moving. It's better to think of it as a point in six dimensional space where the coordinates are, where is it and what's its velocity right now? That's a higher dimensional space called phase space. And if you haven't thought about this before, I admit that it's a little bit mind bending, but what he needed then was a geometry that was flexible enough, not just to talk about two dimensional spaces or three dimensional spaces, but any dimensional space. So the sort of famous first line of this paper where he introduces analysis is, no one doubts nowadays that the geometry of N dimensional space is an actually existing thing. Right. I think that maybe that had been controversial. And he's saying like, look, let's face it just because it's not physical, doesn't mean it's not there. It doesn't mean we shouldn't study it. Interesting. He wasn't jumping to the physical interpretation. Like it can be real, even if it's not perceivable to human cognition. I think that's right. I think, don't get me wrong, Poincaré never strays far from physics. He's always motivated by physics, but the physics drove him to need to think about spaces of higher dimension. And so he needed a formalism that was rich enough to enable him to do that. And once you do that, that formalism is also going to include things that are not physical. And then you have two choices. You can be like, oh, well, that stuff's trash. Or, but this is more of the mathematician's frame of mind. If you have a formalistic framework that like seems really good and sort of seems to be like very elegant and work well, and it includes all the physical stuff, maybe we should think about all of it. Like maybe we should think about it, thinking, oh, maybe there's some gold to be mined there. And indeed, guess what? Before long, there's relativity and there's space-time. And all of a sudden it's like, oh yeah, maybe it's a good idea. We already had this geometric apparatus set up for how to think about four-dimensional spaces. Turns out they're real after all. This is a story much told in mathematics, not just in this context, but in many. I'd love to dig in a little deeper on that, actually, because I have some intuitions to work out in my brain. But— Well, I'm not a mathematical physicist, so we can work it out together. Good. We'll together walk along the path of curiosity. But Poincaré conjecture, what is it? Poincaré conjecture is about curved three-dimensional spaces. So I was on my way there, I promise. The idea is that we perceive ourselves as living in— we don't say a three-dimensional space, we just say three-dimensional space. You know, you can go up and down, you can go left and right, you can go forward and back. There's three dimensions in which we can move. In Poincaré's theory, there are many possible three-dimensional spaces. In the same way that going down one dimension to sort of capture our intuition a little bit more, we know there are lots of different two-dimensional surfaces, right? There's a balloon, and that looks one way, and a donut looks another way, and a Mobius strip looks a third way. Those are all like two-dimensional surfaces that we can kind of really get a global view of, because we live in three-dimensional space, so we can see a two-dimensional surface sort of sitting in our three-dimensional space. Well, to see a three-dimensional space whole, we'd have to kind of have four-dimensional eyes, right? Which we don't, so we have to use our mathematical eyes, we have to envision. The Poincaré conjecture says that there's a very simple way to determine whether a three-dimensional space is the standard one, the one that we're used to. And essentially, it's that it's what's called fundamental group has nothing interesting in it. And that I can actually say without saying what the fundamental group is, I can tell you what the criterion is. This would be good. Oh, look, I can even use a visual aid. So for the people watching this on YouTube, you'll just see this. For the people on the podcast, you'll see this. So I'm going to use a visual aid. For the people on the podcast, you'll have to visualize it. So Lex has been nice enough to like, give me a surface with some interesting topology. It's a mug. Right here in front of me. A mug, yes. I might say it's a genus one surface, but we could also say it's a mug, same thing. So if I were to draw a little circle on this mug, which way should I draw it so it's visible? Like here, okay. If I draw a little circle on this mug, imagine this to be a loop of string. I could pull that loop of string closed on the surface of the mug, right? That's definitely something I could do. I could shrink it, shrink it, shrink it until it's a point. On the other hand, if I draw a loop that goes around the handle, I can kind of zhuzh it up here and I can zhuzh it down there and I can sort of slide it up and down the handle. But I can't pull it closed, can I? It's trapped. Not without breaking the surface of the mug, right? Not without like going inside. So the condition of being what's called simply connected, this is one of Poincaré's inventions, says that any loop of string can be pulled shut. So it's a feature that the mug simply does not have. This is a non-simply connected mug and a simply connected mug would be a cup, right? You would burn your hand when you drank coffee out of it. So you're saying the universe is not a mug? Well, I can't speak to the universe, but what I can say is that regular old space is not a mug. Regular old space, if you like sort of actually physically have like a loop of string, You can always close it. You can pull it shut. You can always pull it shut. But what if your piece of string was the size of the universe? Like what if your piece of string was like billions of light years long? How do you actually know? I mean, that's still an open question of the shape of the universe. Exactly. Whether it's, I think there's a lot, there is ideas of it being a torus. I mean, there's some trippy ideas and they're not like weird out there, controversial. There's legitimate at the center of cosmology debate. I mean, I think most people think it's flat. I think there's somebody who thinks that there's like some kind of dodecahedral symmetry or I mean, I remember reading something crazy about somebody saying that they saw the signature of that in the cosmic noise or what have you. I mean. To make the flat earthers happy, I do believe that the current main belief is it's flat. It's flat-ish or something like that. The shape of the universe is flat-ish. I don't know what the heck that means. I think that has like a very, I mean, how are you even supposed to think about the shape of a thing that doesn't have any thing outside of it? I mean. Ah, but that's exactly what topology does. Topology is what's called an intrinsic theory. That's what's so great about it. This question about the mug, you could answer it without ever leaving the mug, right? Because it's a question about a loop drawn on the surface of the mug and what happens if it never leaves that surface. So it's like always there. See, but that's the difference between the topology and say, if you're like trying to visualize a mug, that you can't visualize a mug while living inside the mug. Well, that's true. The visualization is harder, but in some sense, no, you're right. But if the tools of mathematics are there, I don't want to fight, but I think the tools of mathematics are exactly there to enable you to think about what you cannot visualize in this way. Let me give, let's go, always to make things easier, go down a dimension. Let's think about we live on a circle. Okay? You can tell whether you live on a circle or a line segment, because if you live on a circle, if you walk a long way in one direction, you find yourself back where you started. And if you live in a line segment, you walk for a long enough one direction, you come to the end of the world. Or if you live on a line, like a whole line, an infinite line, then you walk in one direction for a long time. And like, well, then there's not a sort of terminating algorithm to figure out whether you live on a line or a circle, but at least you sort of, at least you don't discover that you live on a circle. So all of those are intrinsic things, right? All of those are things that you can figure out about your world without leaving your world. On the other hand, right? Now we're going to go from intrinsic to extrinsic. Why did I not know we were going to talk about this, but why not? Why not? If you can't tell whether you live in a circle or a knot, like imagine like a knot floating in three dimensional space, the person who lives on that knot to them, it's a circle. They walk a long way, they come back to where they started. Now we with our three dimensional eyes can be like, oh, this one's just a plain circle and this one's knotted up. But that has to do with how they sit in three dimensional space. It doesn't have to do with intrinsic features of those people's world. We can ask you one ape to another. How does it make you feel that you don't know if you live in a circle or on a knot, in a knot, inside the string that forms the knot? I don't even know how to say that. I'm going to be honest with you. I don't know if like, I fear you won't like this answer, but it does not bother me at all. I don't lose one minute of sleep over it. So like, does it bother you that if we look at like a Mobius strip, that you don't have an obvious way of knowing whether you are inside of cylinder, if you live on a surface of a cylinder or you live on the surface of a Mobius strip? No, I think you can tell if you live. Which one? Because if what you do is you like, tell your friend, hey, stay right here. I'm just going to go for a walk. And then you like walk for a long time in one direction and then you come back and you see your friend again. And if your friend is reversed, then you know you live on a Mobius strip. Well, no, because you won't see your friend, right? Okay, fair point. Fair point on that. But you have to believe the stories about, no, I don't even know. Would you even know? Would you really? Oh, no, your point is right. Let me try to think of a better, let's see if I can do this on the vlog. It may not be correct to talk about cognitive beings living on a Mobius strip, because there's a lot of things taken for granted there. And we're constantly imagining actual like three dimensional creatures, like how it actually feels like to live in a Mobius strip is tricky to internalize. I think that on what's called the real projective plane, which is kind of even more sort of like messed up version of the Mobius strip, but with very similar features, this feature of kind of like only having one side, that has the feature that there's a loop of string, which can't be pulled closed. But if you loop it around twice along the same path, that you can pull closed. That's extremely weird. Yeah. But that would be a way you could know without leaving your world that something very funny is going on. You know, what's extremely weird, maybe we can comment on, hopefully it's not too much of a tangent is, I remember thinking about this. This might be right. This might be wrong. But if we now talk about a sphere and you're living inside a sphere that you're going to see everywhere around you, the back of your own head. That I was, cause like, I was, this was very counterintuitive to me to think about, maybe it's wrong. But cause I was thinking of like earth, you know, your 3D thing on sitting on a sphere. But if you're living inside the sphere, like you're going to see, if you look straight, you're always going to see yourself all the way around. So everywhere you look, there's going to be the back of your own head. I think somehow this depends on something of like how the physics of light works in this scenario, which I'm sort of finding it hard to bend my... That's true. The sea is doing a lot of work, like saying you see something's doing a lot of work. People have thought about this. I mean, this, this metaphor of like, what if we're like little creatures in some sort of smaller world? Like how could we apprehend what's outside? That metaphor just comes back and back. And actually I didn't even realize like how frequent it is. It comes up in the book a lot. I know it from a book called Flatland. I don't know if you ever read this when you were a kid or an adult, you know, this, this sort of, this sort of comic novel from the 19th century about an entire two-dimensional world. It's narrated by a square, that's the main character. And the kind of strangeness that befalls him when, you know, one day he's in his house and suddenly there's like a little circle there and they're with him. And then the circle, but then the circle like starts getting bigger and bigger and bigger. And he's like, what the hell is going on? It's like a horror movie, like for two-dimensional people. And of course, what's happening is that a sphere is entering his world. And as the sphere kind of like moves farther and farther into the plane, it's cross-sectioned, the part of it that he can see. To him, it looks like there's like this kind of bizarre being that's like getting larger and larger and larger until it's exactly sort of halfway through. And then they have this kind of like philosophical argument with the sphere, it's like, I'm a sphere, I'm from the third dimension. The square is like, what are you talking about? There's no such thing. And they have this kind of like sterile argument where the square is not able to kind of like follow the mathematical reasoning of the sphere until the sphere just kind of grabs him and like jerks him out of the plane and pulls him up. And it's like, now, like now do you see, like now do you see your whole world that you didn't understand before? So do you think that kind of process is possible for us humans? So we live in the three-dimensional world, maybe with the time component four-dimensional, and then math allows us to go into high dimensions comfortably and explore the world from those perspectives. Is it possible that the universe is many more dimensions than the ones we experience as human beings? So if you look at the, especially in physics theories of everything, physics theories that try to unify general relativity and quantum field theory, they seem to go to high dimensions to work stuff out through the tools of mathematics. Is it possible? So like the two options are, one is just a nice way to analyze a universe, but the reality is as exactly we perceive it, it is three-dimensional. Or are we just seeing, are we those flatland creatures that are just seeing a tiny slice of reality and the actual reality is many, many, many more dimensions than the three dimensions we perceive? Oh, I certainly think that's possible. Now, how would you figure out whether it was true or not is another question. And I suppose what you would do, as with anything else that you can't directly perceive, is you would try to understand what effect the presence of those extra dimensions out there would have on the things we can perceive. Like what else can you do, right? And in some sense, if the answer is they would have no effect, then maybe it becomes like a little bit of a sterile question because what question are you even asking, right? You can kind of posit however many entities that you want. Well, is it possible to intuit how to mess with the other dimensions while living in a three-dimensional world? I mean, that seems like a very challenging thing to do. The reason flatland could be written is because it's coming from a three-dimensional writer. Yes, but what happens in the book, I didn't even tell you the whole plot, what happens is the square is so excited and so filled with intellectual joy. By the way, maybe to give the story some context, you ask, is it possible for us humans to have this experience of being transcendentally jerked out of our world so we can truly see it from above? Well, Edwin Abbott, who wrote the book, certainly thought so because Edwin Abbott was a minister. So the whole Christian subtext of this book, I had completely not grasped reading this as a kid, that it means a very different thing, right? If a theologian is saying, oh, what if a higher being could pull you out of this earthly world you live in so that you can sort of see the truth and really see it from above, as it were. So that's one of the things that's going on for him. And it's a testament to his skill as a writer that his story just works, whether that's the framework you're coming to it from or not. But what happens in this book, and this part now, looking at it through a Christian lens, it becomes a bit subversive, is the square is so excited about what he's learned from the sphere. And the sphere explains to him what a cube would be. Oh, it's like you, but three-dimensional. And the square is very excited. And the square is like, okay, I get it now. So now that you explained to me how just by reason I can figure out what a cube would be like, like a three-dimensional version of me, let's figure out what a four-dimensional version of me would be like. And then the sphere is like, what the hell are you talking about? There's no fourth dimension? That's ridiculous. There's only three dimensions. That's how many there are. I can see. So it's this sort of comic moment where the sphere is completely unable to conceptualize that there could actually be yet another dimension. So yeah, that takes the religious allegory to like a very weird place that I don't really like understand theologically. But that's a nice way to talk about religion and myth in general, as perhaps us trying to struggle, us meaning human civilization, trying to struggle with ideas that are beyond our cognitive capabilities. But it's in fact not beyond our capability. It may be beyond our cognitive capabilities to visualize a four-dimensional cube, a tesseract as some like to call it, or a five-dimensional cube or a six-dimensional cube. But it is not beyond our cognitive capabilities to figure out how many corners a six-dimensional cube would have. That's what's so cool about us. Whether we can visualize it or not, we can still talk about it. We can still reason about it. We can still figure things out about it. That's amazing. Yeah. If we go back to this, first of all to the mug, but to the example you give in the book of the straw, how many holes does a straw have? And you, listener, may try to answer that in your own head. Yeah, I'm going to take a drink while everybody thinks about it so I can give you a moment. A slow sip. Is it zero, one, or two, or more than that maybe? Maybe you can get very creative. But it's kind of interesting to dissecting each answer as you do in the book. It's quite brilliant. People should definitely check it out. But if you could try to answer it now, think about all the options and why they may or may not be right. Yeah, it's one of these questions where people on first hearing it think it's a triviality and they're like, well, the answer is obvious. And then what happens if you ever ask a group of people this, something wonderfully comic happens, which is that everyone's like, well, it's completely obvious. And then each person realizes that half the person, the other people in the room have a different obvious answer for the way they have. And then people get really heated. People are like, I can't believe that you think it has two holes. Or like, I can't believe that you think it has one. And then you really, like people really learn something about each other. And people get heated. I mean, can we go through the possible options here? Is it 0, 1, 2, 3, 10? Sure. So I think, you know, most people, the zero holers are rare. They would say like, well, look, you can make a straw by taking a rectangular piece of plastic and closing it up. Rectangular piece of plastic doesn't have a hole in it. I didn't poke a hole in it. So how can I have a hole? It's like, it's just one thing. Okay. Most people don't see it that way. That's like, um, Is there any truth to that kind of conception? Yeah, I think that would be somebody who's account, I mean, what I would say is you could say the same thing. About a bagel, you could say I can make a bagel by taking like a long cylinder of dough, which doesn't have a hole and then smushing the ends together. Now it's a bagel. So if you're really committed, you can be like, okay, a bagel doesn't have a hole either. But like, who are you if you say a bagel doesn't have a hole? I mean, I don't know. Yeah. So that's almost like an engineering definition of it. Okay, fair enough. So what's, what about the other options? So, you know, one hole people would say, um, I like how these are like groups of people, like we've planted our foot. Yes, team one hole. There's books written about each belief. You know, once they look, there's like a hole and it goes all the way through the straw, right? There's one region of space. That's the hole. And there's one and two hole people would say like, well, look, there's a hole in the top and the hole at the bottom. I think a common thing you see, when people argue about this, they would take something like this bottle of water I'm holding, I'll open it. And they say, well, how many holes are there in this? And you say, like, well, there's one, there's one hole at the top. Okay, what if I like, poke a hole here so that all the water spills out? Well, now it's a straw. Yeah. So if you're a one hole, or I say to you, like, well, how many holes are in it now? There was a hole, there was one hole in it before and I poked a new hole in it. And then you think there's still one hole, even though there was one hole and I made one more? Clearly not. This is two holes. Yeah. Um, and yet, if you're a two hole or the one hole, or we'll say like, okay, where does one hole begin in the other hole end? Yeah, like, what's it like? And, um, and in the in the book, I sort of, you know, in math, there's two things we do when we're faced with a problem that's confusing us. We can make the problem simpler. That's what we were doing a minute ago, and we were talking about high dimensional space. And I was like, let's talk about like circles and line segments. Let's like go down a dimension to make it easier. The other big move we have is to make the problem harder and try to sort of really like face up to what are the complications. So, you know, what I do in the book is say like, let's stop talking about straws for a minute and talk about pants. How many holes are there in a pair of pants? So I think most people who say there's two holes in a straw would say there's three holes in a pair of pants. I guess I mean, I guess we're filming only from here I could take up not I'm not gonna do it. You just have to imagine the pants. Sorry. Yeah. Lex, if you want to know, okay, no. That's gonna be in the director's cut. It's a Patreon only footage. There you go. So many people would say there's three holes in a pair of pants. But you know, for instance, my daughter when I asked, by the way, talking to kids about this is super fun. I highly recommend it. What did she say? She said, well, yeah, I feel a pair of pants like just has two holes because yes, there's the waist, but that's just the two leg holes stuck together. Whoa, okay. Two leg hole. Yeah. Okay. I mean, that's a one hole for the straw. So she's a one hole for the straw too. And, and that really does capture something. It captures this fact, which is central to the theory of what's called homology, which is like a central part of modern topology that holes, whatever we may mean by them, there's somehow things which have an arithmetic to them. There are things which can be added, like the waist, like waist equals leg plus leg is kind of an equation, but it's not an equation about numbers. It's an equation about some kind of geometric, some kind of topological thing, which is very strange. And so, you know, when I come down, you know, like a rabbi, I like to kind of like, come up with these answers to somehow like dodge the original question and say, like, you're both right, my children. Okay, so. Yeah. So for this truth, for the, for the straw, I think what a modern mathematician would say is like, the first version would be to say, like, well, there are two holes, but they're really both the same hole. Well, that's not quite right. A better way to say it is, there's two holes, but one is the negative of the other. Now, what can that mean? One way of thinking about what it means is that if you sip something like a milkshake through the straw, no matter what, the amount of milkshake that's flowing in one end, that same amount is flowing out the other end. So, they're not independent from each other. There's some relationship between them in the same way that if you somehow could like suck a milkshake through a pair of pants, the amount of milkshake, just go with me on this experiment. I'm right there with you. The amount of milkshake that's coming in the left leg of the pants, plus the amount of milkshake that's coming in the right leg of the pants is the same that's coming out the waist of the pants. So, just so you know, I fasted for 72 hours the last three days. So, I just broke the fast with a little bit of food yesterday. So, this is like, this sounds, food analogies or metaphors for this podcast work wonderfully, because I can intensely picture it. Is that your weekly routine or just in preparation for talking about geometry for three hours? Exactly. Just for this. It's hardship to purify the mind. No, it's for the first time. I just wanted to try the experience. Oh, wow. And just to pause, to do things that are out of the ordinary, to pause and to reflect on how grateful I am to be just alive and be able to do all the cool shit that I get to do. So, did you drink water? Yes, yes, yes, yes, yes, yes. Water and salt. So, like electrolytes and all those kinds of things. But anyway, so the inflow on the top of the pants equals to the outflow on the bottom of the pants. Exactly. So, this idea that, I mean, I think, you know, Poincaré really had this idea, this sort of modern idea. I mean, building on stuff other people did, Betty is an important one, of this kind of modern notion of relations between wholes. But the idea that wholes really had an arithmetic, the really modern view was really Emmy Noether's idea. So, she kind of comes in and sort of truly puts the subject on its modern footing that we have now. So, you know, it's always a challenge. You know, in the book, I'm not going to say I give like a course so that you read this chapter and then you're like, oh, it's just like I took like a semester of algebraic topology. It's not like this. And it's always a challenge writing about math because there are some things that you can really do on the page and the math is there. And there's other things which, it's too much in a book like this to like do them all the page. You can only say something about them, if that makes sense. So, you know, in the book, I try to do some of both. I try to do, I try to, topics that are, you can't really compress and really truly say exactly what they are in this amount of space. I try to say something interesting about them, something meaningful about them so that readers can get the flavor. And then in other places, I really try to get up close and personal and really do the math and have it take place on the page. LUKE RENNER. To some degree, be able to give inklings of the beauty of the subject. DAVID LEVIN. Yeah, I mean, there's, you know, there's a lot of books that are like, I don't quite know how to express this well. I'm still laboring to do it. But there's a lot of books that are about stuff. But I want my books to not only be about stuff, but to actually have some stuff there on the page in the book for people to interact with directly and not just sort of hear me talk about distant features about, distant features of it. LUKE RENNER. Right. So, not be talking just about ideas, but the actually be expressing the idea. Is there, you know, somebody in the, maybe you can comment, there's a guy, his YouTube channel is 3Blue1Brown, Grant Sanderson. He does that masterfully well. DAVID LEVIN. Absolutely. LUKE RENNER. Of visualizing, of expressing a particular idea and then talking about it as well, back and forth. What do you think about Grant? DAVID LEVIN. It's fantastic. I mean, the flowering of math YouTube is like such a wonderful thing because, you know, math teaching, there's so many different venues through which we can teach people math. There's the traditional one, right? Well, where I'm in a classroom with, you know, depending on the class, it could be 30 people, it could be 100 people, it could, God help me, be 500 people, if it's like the big calculus lecture or whatever it may be. And there's sort of some, but there's some set of people of that order of magnitude. And I'm with them, we have a long time, I'm with them for a whole semester. And I can ask them to do homework and we talk together, we have office hours, if they have one-on-one questions, blah, blah, blah. It's like a very high level of engagement. But how many people am I actually hitting at a time? Like not that many, right? And you can, and there's kind of an inverse relationship where the more, the fewer people you're talking to, the more engagement you can ask for. The ultimate, of course, is like the mentorship relation of like a PhD advisor and a graduate student where you spend a lot of one-on-one time together for like three to five years. And the ultimate high level of engagement to one person. You know, books, this can get to a lot more people than are ever going to sit in my classroom and you spend like however many hours it takes to read a book. Somebody like 3Blue1Brown or Numberphile or people like Vi Hart. I mean, YouTube, let's face it, has bigger reach than a book. Like there's YouTube videos that have many, many, many more views than like, you know, any hardback book like not written by a Kardashian or an Obama is going to sell, right? So that's, I mean, and then, you know, those are, you know, some of them are like longer, 20 minutes long, some of them are five minutes long, but they're, you know, they're shorter. And then even some, look, look, like Eugenia Chang is a wonderful category theorist in Chicago. I mean, she was on, I think, the Daily Show or is it? I mean, she was on, you know, she has 30 seconds, but then there's like 30 seconds to sort of say something about math, mathematics to like untold millions of people. So everywhere along this curve is important. And one thing I feel like is great right now is that people are just broadcasting on all the channels because we each have our skills, right? Somehow along the way, like I learned how to write books. I had this kind of weird life as a writer where I sort of spent a lot of time like thinking about how to put English words together into sentences and sentences together into paragraphs, like at length, which is this kind of like weird specialized skill. And that's one thing, but like sort of being able to make like, you know, winning good looking eye catching videos is like a totally different skill. And, you know, probably, you know, somewhere out there, there's probably sort of some like heavy metal band that's like teaching math through heavy metal and like using their skills to do that. I hope there is at any rate. Their music and so on. Yeah. But there is something to the process. I mean, Grant does this especially well, which is in order to be able to visualize something, now he writes programs, so it's programmatic visualization. So like the things he is basically mostly through his Manum library in Python, everything is drawn through Python. And you have to, you have to truly understand the topic to be able to visualize it in that way, and not just understand it, but really kind of think in a very novel way. It's funny because I've spoken with him a couple of times, spoken to him a lot offline as well. He really doesn't think he's doing anything new, meaning like he sees himself as very different from maybe like a researcher. But it feels to me like he's creating something totally new, like that act of understanding and visualizing is as powerful or has the same kind of inkling of power as does the process of proving something. You know, it just, it doesn't have that clear destination, but it's pulling out an insight and creating multiple sets of perspective that arrive at that insight. And to be honest, it's something that I think we haven't quite figured out how to value inside academic mathematics in the same way, and this is a bit older, that I think we haven't quite figured out how to value the development of computational infrastructure. You know, we all have computers as our partners now, and people build computers that sort of assist and participate in our mathematics. They build those systems, and that's a kind of mathematics too, but not in the traditional form of proving theorems and writing papers. But I think it's coming. Look, I mean, I think, you know, for example, the Institute for Computational Experimental Mathematics at Brown, which is like a, you know, it's a NSF-funded math institute, very much part of sort of traditional math academia, they did an entire theme semester about visualizing mathematics, like the same kind of thing that they would do for like an up-and-coming research topic, like that's pretty cool. So I think there really is buy-in from the mathematics community to recognize that this kind of stuff is important and counts as part of mathematics, like part of what we're actually here to do. Yeah, I'm hoping to see more and more of that from like MIT faculty, from faculty from all the top universities in the world. Let me ask you this weird question about the Fields Medal, which is the Nobel Prize in mathematics. Do you think, since we're talking about computers, there will one day come a time when a computer, an AI system will win a Fields Medal? No. Of course, that's what a human would say. Why not? Is that like, that's like my cap shot? That's like the proof that I'm a human? Is like, deny that I'm not? Yeah. What is, how does he want me to answer? Is there something interesting to be said about that? Yeah, I mean, I am tremendously interested in what AI can do in pure mathematics. I mean, it's of course, it's a parochial interest, right? You're like, why am I not interested in like how it can like help feed the world or help solve like real-world problems? I'm like, can it do more math? Like, what can I do? We all have our interests, right? But I think it is a really interesting conceptual question. And here too, I think it's important to be kind of historical because it's certainly true that there's lots of things that we used to call research in mathematics that we would now call computation. Tasks that we've now offloaded to machines like, you know, in 1890, somebody could be like, here's my PhD thesis, I computed all the invariants of this polynomial ring under the action of some finite group, doesn't matter what those words mean, just it's like some thing that in 1890 would take a person a year to do and would be a valuable thing that you might want to know. And it's still a valuable thing that you might want to know, but now you type a few lines of code in Macaulay or Sage or Magma, and you just have it. So we don't think of that as math anymore, even though it's the same thing. What's Macaulay, Sage and Magma? Oh, those are computer algebra programs. So those are like sort of bespoke systems that lots of mathematicians use. That's similar to Maple and Mathematica? Yeah. Oh yeah. So it's similar to Maple and Mathematica. Yeah. But a little more specialized, but yeah. It's programs that work with symbols and allow you to do, can you do proofs? Can you do kind of little leaps and proofs? They're not really built for that, and that's a whole other story. But these tools are part of the process of mathematics now? Right. They are now for most mathematicians, I would say, part of the process of mathematics. And so, you know, there's a story I tell in the book, which I'm fascinated by, which is, you know, so far, attempts to get AIs to prove interesting theorems have not done so well. It doesn't mean they can't. There's actually a paper I just saw, which has a very nice use of a neural net to find counter examples to conjecture. Somebody said, like, well, maybe this is always that. And you can be like, well, let me sort of train an AI to sort of try to find things where that's not true. And it actually succeeded. Now, in this case, if you look at the things that it found, you say, like, okay, I mean, these are not famous conjectures. Okay, so like, somebody wrote this down, maybe this is so. Looking at what the AI came up with, you're like, you know, I bet if like, five grad students had thought about that problem, they wouldn't have come up with that. I mean, when you see it, you're like, okay, that is one of the things you might try if you sort of like, put some work into it. Still, it's pretty awesome. But the story I tell in the book, which I'm fascinated by is, there is, there's a, okay, we're gonna go back to knots. There's a knot called the Conway knot, after John Conway, who maybe we'll talk about a very interesting character also. Yeah, there's a small tangent, somebody I was supposed to talk to, and unfortunately, he passed away. And he's somebody I find as an incredible mathematician, incredible human being. Oh, and I am sorry that you didn't get a chance because having had the chance to talk to him a lot when I was, you know, when I was a postdoc. Yeah, you missed out. There's no way to sugar coat it. I'm sorry that you didn't get that chance. Yeah, it is what it is. So knots. Yeah, so there was a question. And again, it doesn't matter the technicalities of the question, but it's a question of whether the knot is sliced. It has to do with something about what kinds of three dimensional surfaces in four dimensions can be bounded by this knot. But never mind what it means. It's some question. And it's actually very hard to compute whether a knot is sliced or not. And in particular, the question of the Conway knot, whether it was sliced or not, was particularly vexed. Until it was solved just a few years ago by Lisa Piccirillo, who actually now that I think of it was here in Austin, I believe she was a grad student at UT Austin at the time, I didn't even realize there was an Austin connection to this story until I started telling it. She is in fact, I think she's now at MIT. So she's basically following you around. If I remember correctly, the reverse. There's a lot of really interesting richness to this story. One thing about it is her paper was rather was very short, was very short and simple nine pages in which two were pictures. Very short for like a paper solving a major conjecture. And it really makes you think about what we mean by difficulty in mathematics. Like, do you say, Oh, actually, the problem wasn't difficult because you could solve it so simply? Or do you say like, Well, no, evidently, it was difficult because like the world's top topologist, many, you know, worked on it for 20 years, and nobody could solve it. So therefore it is difficult. Or is it that we need sort of some new category of things that about which it's difficult to figure out that they're not difficult? I mean, this is the computer science formulation, but the sort of the journey to arrive at the simple answer may be difficult. But once you have the answer, it will then appear simple. And I mean, there might be a large category, I hope there's a large set of such solutions, because, you know, once we stand at the end of the scientific process that we're at the very beginning of, or at least it feels like, I hope there's just simple answers to everything. That we'll look and it'll be simple laws that govern the universe, simple explanation of what is consciousness, of what is love, is mortality fundamental to life? What's the meaning of life? Are humans special? Or we're just another sort of reflection of all that is beautiful in the universe in terms of like life forms, all of it is life and just has different, when taken from a different perspective, is all life can seem more valuable or not, but really it's all part of the same thing. All those will have a nice, like two equations, maybe one equation. Why do you think you want those questions to have simple answers? I think just like symmetry and the breaking of symmetry is beautiful somehow. There's something beautiful about simplicity. I think it... So it's aesthetic. It's aesthetic, yeah. But it's aesthetic in the way that happiness is an aesthetic. Like, why is that so joyful that a simple explanation that governs a large number of cases is really appealing? Even when it's not, like obviously we get a huge amount of trouble with that because oftentimes it doesn't have to be connected with reality or even that explanation could be exceptionally harmful. Most of like the world's history that was governed by hate and violence had a very simple explanation at the core that was used to cause the violence and the hatred. So like, we get into trouble with that, but why is that so appealing? And in its nice forms in mathematics, in mathematics, like you look at the Einstein papers, why are those so beautiful? And why is the Andrew Wiles proof of the Fermat's last theorem not quite so beautiful? Like what's beautiful about that story is the human struggle of like the human story of perseverance, of the drama of not knowing if the proof is correct and ups and downs and all of those kinds of things. That's the interesting part. But the fact that the proof is huge and nobody understands, well, from my outsider's perspective, nobody understands what the heck it is, is not as beautiful as it could have been. I wish it was what Fermat originally said, which is, you know, it's not small enough to fit in the margins of this page, but maybe if he had like a full page or maybe a couple of post-it notes, he would have enough to do the proof. What do you make of, if we could take another of a multitude of tangents, what do you make of Fermat's last theorem? Because the statement, there's a few theorems, there's a few problems that are deemed by the world throughout its history to be exceptionally difficult. And that one in particular is really simple to formulate and really hard to come up with a proof for. And it was like taunted as simple by Fermat himself. Is there something interesting to be said about that x to the n plus y to the n equals z to the n for n of three or greater? Is there a solution to this? And then how do you go about proving that? Like, how would you try to prove that? And what do you learn from the proof that eventually emerged by Andrew Wiles? Yeah, so right, let me just say the background, because I don't know if everybody listening knows the story. So you know, Fermat was an early number theorist, at least an early mathematician, those special adjacent didn't really exist back then. He comes up in the book, actually, in the context of a different theorem of his that has to do with testing whether a number is prime or not. So I write about, he was one of the ones who was salty, and like he would exchange these letters where he and his correspondence would like try to top each other and vex each other with questions and stuff like this. But this particular thing, it's called Fermat's Last Theorem, because it's a note he wrote in his copy of the Disquationis Arithmeticae. Like he wrote, here's an equation, it has no solutions, I can prove it, but the proof is like a little too long to fit in this in the margin of this book. He was just like writing a note to himself. Now let me just say historically, we know that Fermat did not have a proof of this theorem. For a long time, people were like, this mysterious proof that was lost, a very romantic story, right? But Fermat later, he did prove special cases of this theorem and wrote about it, talked to people about the problem. It's very clear from the way that he wrote where he can solve certain examples of this type of equation that he did not know how to do the whole thing. He may have had a deep, simple intuition about how to solve the whole thing that he had at that moment, without ever being able to come up with a complete proof. And that intuition maybe lost the time. – Maybe. But I think we, so, but you're right, that that is unknowable. But I think what we can know is that later, he certainly did not think that he had a proof that he was concealing from people. He thought he didn't know how to prove it, and I also think he didn't know how to prove it. Now, I understand the appeal of saying, wouldn't it be cool if this very simple equation, there was a very simple, clever, wonderful proof that you could do in a page or two. And that would be great. But you know what? There's lots of equations like that that are solved by very clever methods like that, including the special cases that Fermat wrote about, the method of descent, which is very wonderful and important. But in the end, those are nice things that you teach in an undergraduate class, and it is what it is, but they're not big. On the other hand, work on the Fermat problem—that's what we like to call it, because it's not really his theorem, because we don't think he proved it. So, I mean, work on the Fermat problem developed this incredible richness of number theory that we now live in today. And not, by the way, just Wiles, Andrew Wiles being the person who, together with Richard Taylor, finally proved this theorem. But you know how you have this whole moment that people try to prove this theorem, and they fail. And there's a famous false proof by LeMay from the 19th century, where Kummer, in understanding what mistake LeMay had made in this incorrect proof, basically understands something incredible, which is that, you know, a thing we know about numbers is that you can factor them, and you can factor them uniquely. There's only one way to break a number up into primes. Like, if we think of a number like 12, 12 is 2 times 3 times 2. I had to think about it. Right? Or it's 2 times 2 times 3. Of course, you can reorder them. But there's no other way to do it. There's no universe in which 12 is something times 5, or in which there's like four 3s in it. Nope, 12 is like two 2s and a 3. Like, that is what it is. And that's such a fundamental feature of arithmetic that we almost think of it like God's law. You know what I mean? It has to be that way. That's a really powerful idea. It's so cool that every number is uniquely made up of other numbers. And like, made up meaning, like, there's these like basic atoms that form molecules that get built on top of each other. I love it. I mean, when I teach, you know, undergraduate number theory, it's like, it's the first really deep theorem that you prove. What's amazing is, you know, the fact that you can factor a number into primes is much easier. Essentially, Euclid knew it, although he didn't quite put it in that way. The fact that you can do it at all. What's deep is the fact that there's only one way to do it. Or however you sort of chop the number up, you end up with the same set of prime factors. And indeed, what people finally understood at the end of the 19th century is that if you work in number systems slightly more general than the ones we're used to, which it turns out are relevant to Fermat, all of a sudden this stops being true. Things get, I mean, things get more complicated. And now, because you were praising simplicity before, you were like, it's so beautiful, unique factorization. It's so great. Like, so when I tell you that in more general number systems, there is no unique factorization, maybe you're like, that's bad. I'm like, no, that's good, because there's like a whole new world of phenomena to study that you just can't see through the lens of the numbers that we're used to. So I'm, I'm for complication. I'm highly in favor of complication. And every complication is like an opportunity for new things to study. And is that the big, kind of one of the big insights for you from Andrew Wiles' proof? Is there interesting insights about the process that you used to prove that sort of resonates with you as a mathematician? Is there an interesting concept that emerged from it? Is there interesting human aspects to the proof? Whether there's interesting human aspects to the proof itself is an interesting question. Certainly, it has a huge amount of richness. Sort of at its heart is an argument of what's called deformation theory, which was in part created by my PhD advisor, Barry Mazur. Can you speak to what deformation theory is? I can speak to what it's like. Sure. How about that? What does it rhyme with? Right. Well, the reason that Barry called it deformation theory, I think he's the one who gave it the name. I hope I'm not wrong in saying this someday. In your book, you have calling different things by the same name as one of the things in the beautiful map that opens the book. Yes. And this is a perfect example. So, this is another phrase of Poincaré, this incredible generator of slogans and aphorisms. He said, mathematics is the art of calling different things by the same name. That very thing we do, right? When we're like this triangle and this triangle, come on, they're the same triangle. They're just in a different place, right? So, in the same way, it came to be understood that the kinds of objects that you study when you study Fermat's last theorem, and let's not even be too careful about what these objects are. I can tell you there are Galois representations in modular forms, but saying those words is not going to mean so much. But whatever they are, they're things that can be deformed, moved around a little bit. And I think the insight of what Andrew and then Andrew and Richard were able to do was to say something like this. A deformation means moving something just a tiny bit, like an infinitesimal amount. If you really are good at understanding which ways a thing can move in a tiny, tiny, tiny infinitesimal amount in certain directions, maybe you can piece that information together to understand the whole global space in which it can move. And essentially, their argument comes down to showing that two of those big global spaces are actually the same, the fabled R equals T part of their proof, which is at the heart of it. And it involves this very careful principle like that. But that being said, what I just said, it's probably not what you're thinking, because what you're thinking when you think, oh, I have a point in space and I move it around like a little tiny bit, you're using the same kind of thing. You're using your notion of distance that's from calculus. We know what it means for two points in the real line to be close together. So yet another thing that comes up in the book a lot is this fact that the notion of distance is not given to us by God. We could mean a lot of different things by distance. And just in the English language, we do that all the time. We talk about somebody being a close relative. It doesn't mean they live next door to you. It means something else. There's a different notion of distance we have in mind. And there are lots of notions of distances that you could use. In the natural language processing community in AI, there might be some notion of semantic distance or lexical distance between two words. How much do they tend to arise in the same context? That's incredibly important for doing autocomplete and machine translation and stuff like that. And it doesn't have anything to do with, are they next to each other in the dictionary? It's a different kind of distance. Okay, ready? In this kind of number theory, there is a crazy distance called the P-adic distance. I didn't write about this that much in the book because even though I love it, it's a big part of my research life, it gets a little bit into the weeds. But your listeners are going to hear about it now. Please. Where? You know, what a normal person says when they say two numbers are close, they say like, you know, their difference is like a small number, like seven and eight are close because their difference is one and one's pretty small. If we were to be what's called a two-adic number theorist, we'd say, oh, two numbers are close if their difference is a multiple of a large power of two. So like, so like one and 49 are close because their difference is 48 and 48 is a multiple of 16, which is a pretty large power of two. Whereas one and two are pretty far away because the difference between them is one, which is not even a multiple of a power of two at all. It's odd. You want to know what's really far from one? Like one and one 64th, because their difference is a negative power of two, two to the minus six. So those points are quite, quite far away. Two to the power of a large n would be two, if that's the difference between two numbers, then they're close. Yeah. So two to a large power is in this metric a very small number and two to a negative power is a very big number. That's two-adic. Okay. I can't even visualize that. It takes practice. It takes practice. If you've ever heard of the Cantor set, it looks kind of like that. So it is crazy that this is good for anything, right? I mean, this just sounds like a definition that someone would make up to torment you. But what's amazing is there's a general theory of distance where you say any definition you make that satisfies certain axioms deserves to be called a distance. And this... See, I'm sorry to interrupt. My brain, you broke my brain. Awesome. 10 seconds ago. Because I'm also starting to map for the two-adic case to binary numbers and, you know, because we romanticize those. Oh, that's exactly the right way to think of it. I was trying to mess with number, you know, trying to see, okay, which ones are close? And then I'm starting to visualize different binary numbers and how they, which ones are close to each other. And, well, I think there's a clear... No, no, it's very similar. That's exactly the way to think of it. It's almost like binary numbers written in reverse. Because in a binary expansion, two numbers are close. A number that's small is like 0.0000 something. Something that's the decimal and it starts with a lot of zeros. In the two-adic metric, a binary number is very small if it ends with a lot of zeros and then the decimal point. Gotcha. So it is kind of like binary numbers written backwards is actually, I should have said, that's what I should have said, Lex. That's a very good metaphor. Okay. But so why is that interesting except for the fact that it's a beautiful kind of framework, different kind of framework, which you think about distances. And you're talking about not just the two-adic, but the generalization of that. Why is that interesting? Yeah, the MEP. And so, because that's the kind of deformation that comes up in Woz's proof. That deformation, we're moving something a little bit, means a little bit in this two-adic sense. Trippy. Okay. No, I mean, it's such a, I mean, I just get excited talking about it. And I just taught this like in the fall semester that... But like reformulating, why is... So you pick a different measure of distance over which you can talk about very tiny changes and then use that to then prove things about the entire thing. Yes. Although, you know, honestly, what I would say, I mean, it's true that we use it to prove things, but I would say we use it to understand things. And then because we understand things better, then we can prove things. But you know, the goal is always the understanding. The goal is not so much to prove things. The goal is not to know what's true or false. I mean, this is the thing I write about in the book Near the End. It's something that is a wonderful, wonderful essay by Bill Thurston, kind of one of the great geometers of our time, who unfortunately passed away a few years ago, called on proof and progress in mathematics. And he writes very wonderfully about how, you know, we're not, it's not a theorem factory where we have a production quota. I mean, the point of mathematics is to help humans understand things. And the way we test that is that we're proving new theorems along the way. That's the benchmark, but that's not the goal. Yeah, but just as a kind of, absolutely. But as a tool, it's kind of interesting to approach a problem by saying, how can I change the distance function? Like what, the nature of distance, because that might start to lead to insights for deeper understanding. Like if I were to try to describe human society by a distance, two people are close if they love each other. And then start to do a full analysis on everybody that lives on earth currently, the 7 billion people. And from that perspective, as opposed to the geographic perspective of distance, and then maybe there could be a bunch of insights about the source of violence, the source of maybe entrepreneurial success or invention or economic success or different systems, communism, capitalism start to, I mean, that's, I guess what economics tries to do, but really saying, okay, let's think outside the box about totally new distance functions that could unlock something profound about the space. Yeah, because think about it. Okay, here's, I mean, now we're going to talk about AI, which you know a lot more about than I do. So just, you know, start laughing uproariously if I say something that's completely wrong. We both know very little relative to what we will know centuries from now. That is a really good, humble way to think about it. I like it. Okay, so let's just go for it. Okay, so I think you'll agree with this, that in some sense, what's good about AI is that we can't test any case in advance. The whole point of AI is to make, or one point of it, I guess, is to make good predictions about cases we haven't yet seen. And in some sense, that's always going to involve some notion of distance, because it's always going to involve somehow taking a case we haven't seen and saying, what cases that we have seen, is it close to, is it like, is it somehow an interpolation between? Now, when we do that, in order to talk about things being like other things, implicitly or explicitly, we're invoking some notion of distance. And boy, we better get it right. Right? If you try to do natural language processing, and your idea of distance between words is how close they are in the dictionary, when you write them in alphabetical order, you are going to get pretty bad translations, right? No, the notion of distance has to come from somewhere else. Yeah, that's essentially what neural networks are doing. That's what word embeddings are doing is coming up with... Yes. In the case of word embeddings, literally, like literally what they are doing is learning a distance function. But those are super complicated distance functions. And it's almost nice to think maybe there's a nice transformation that's simple. Sorry, there's a nice formulation of the distance. Again, with the simple. So you don't... Let me ask you about this. From an understanding perspective, there's the Richard Feynman, maybe attributed to him, but maybe many others, is this idea that if you can't explain something simply that you don't understand it. In how many cases, how often is that true? Do you find there's some profound truth in that? Oh, okay. So you were about to ask, is it true? To which I would say flatly, no. But then you said, you followed that up with, is there some profound truth in it? And I'm like, okay, sure. So there's some truth in it. But it's not true. It's not true. This is your mathematician answer. The truth that is in it is that learning to explain something helps you understand it. But real things are not simple. A few things are, most are not. And I don't, to be honest, I don't, I mean, I don't, we don't really know whether Feynman really said that right or something like that is sort of disputed. But I don't think Feynman could have literally believed that, whether or not he said it. And you know, he was the kind of guy, I didn't know him, but I'm reading his writing, he liked to sort of say stuff, like stuff that sounded good. You know what I mean? So it's, it's totally strikes me as the kind of thing he could have said because he liked the way saying it made him feel. But also knowing that he didn't like literally mean it. Well, I definitely have a lot of friends and I've talked to a lot of physicists and they do derive joy from believing that they can explain stuff simply, or believing it's possible to explain stuff simply, even when the explanation is not actually that simple. Like I've heard people think that the explanation is simple and they do the explanation and I think it is simple, but it's not capturing the phenomena that we're discussing. It's capturing, it somehow maps in their mind, but it's taking as a starting point, as an assumption that there's a deep knowledge and a deep understanding that's actually very complicated. And the simplicity is almost like a poem about the more complicated thing as opposed to a distillation. And I love poems, but a poem is not an explanation. Well, some people might disagree with that, but certainly from a mathematical perspective... No poet would disagree with it. No poet would disagree... You don't think there's some things that can only be described imprecisely? I said explanation. I don't think any poet would say their poem is an explanation. They might say it's a description, they might say it's sort of capturing sort of... Well, some people might say the only truth is like music, right? Not the only truth, but some truth can only be expressed through art. And I mean, that's the whole thing we're talking about, religion and myth. And there's some things that are limited cognitive capabilities, and the tools of mathematics or the tools of physics are just not going to allow us to capture. It's possible consciousness is one of those things. Yes, that is definitely possible. But I would even say, look, I mean, consciousness is a thing about which we're still in the dark as to whether there's an explanation we would understand as an explanation at all. By the way, okay, I gotta give yet one more amazing Poincaré quote, because this guy just never stopped coming up with great quotes. Paul Erdős, another fellow who appears in the book, and by the way, he thinks about this notion of distance of personal affinity, kind of like what you're talking about, that kind of social network and that notion of distance that comes from that. So that's something that Paul Erdős did. Erdős did? Well, he thought about distances in networks. I guess he didn't think about the social network. Oh, that's fascinating. That's how it started, that story of Erdős number. Yeah, okay. It started to distract. But, you know, Erdős was sort of famous for saying, and this is sort of along the lines we were saying, he talked about the book, capital T, capital B, the book. And that's the book where God keeps the right proof of every theorem. So when he saw a proof he really liked, it was like really elegant, really simple, like that's from the book. That's like you found one of the ones that's in the book. He wasn't a religious guy, by the way, he referred to God as the supreme fascist. He was like, but somehow he was like, I don't really believe in God, but I believe in God's book. I mean, it was a yeah. But Poincaré, on the other hand, and by the way, there are other men, Hilda Hudson is one who comes up in this book, she also kind of saw math. She's one of the people who sort of develops the disease model that we now use that we use to sort of track pandemics, this SIR model that sort of originally comes from her work with Ronald Ross, but she was also super, super, super devout. And she also sort of on the other side of the religious coin was like, yeah, math is how we communicate with God. She has a great, all these people are incredibly quotable. She says, you know, math is the truth, the things about mathematics, she's like, they're not the most important of God thoughts, but they're the only ones that we can know precisely. So she's like, this is the one place where we get to sort of see what God's thinking when we do mathematics. Again, not a fan of poetry or music. Some people would say Hendrix is like, some people would say chapter one of that book is mathematics and then chapter two is like classic rock. Right? So like, it's not clear that the... I'm sorry, you just sent me off on a tangent, just imagining like Erdos at a Hendrix concert, like trying to figure out if it was from the book or not. All I, what I was coming to was just to say, but what Poincaré said about this is he's like, you know, if like, this has all worked out in the language of the divine and if a divine being like came down and told it to us, we wouldn't be able to understand it. So it doesn't matter. So Poincaré was of the view that there were things that were sort of like inhumanly complex and that was how they really were. Our job is to figure out the things that are not like that. That are not like that. All this talk of primes got me hungry for primes. You wrote a blog post, the beauty of bounding gaps, a huge discovery about prime numbers and what it means for the future of math. Can you tell me about prime numbers? What the heck are those? What are twin primes? What are prime gaps? What are bounding gaps in primes? What are all these things and what, if anything, or what exactly is beautiful about them? Yeah, so, you know, prime numbers are one of the things that number theorists study the most and have for millennia. They are numbers which can't be factored. And then you say like, like five, and then you're like, wait, I can factor five. Five is five times one. Okay, not like that. That is a factorization. It absolutely is a way of expressing five as a product of two things. But don't you agree there's like something trivial about it? It's something you could do to any number. It doesn't have content the way that if I say that 12 is six times two or 35 is seven times five, I've really done something to it. I've broken up. So those are the kind of factorizations that count. And the number that doesn't have a factorization like that is called prime, except historical side note, one, which at some times in mathematical history has been deemed to be a prime, but currently is not. And I think that's for the best. But I bring it up only because sometimes people think that, you know, these definitions are kind of, if we think about them hard enough, we can figure out which definition is true. No, there's just an artifact of mathematics. So yeah, so it's a question which definition is best for us, for our purposes. Well, those edge cases are weird, right? So it can't be, it doesn't count when you use yourself as a number or one as part of the factorization or as the entirety of the factorization. So you somehow get to the meat of the number by factorizing it. And that seems to get to the core of all of mathematics. Yeah, you take any number and you factorize it until you can factorize no more. And what you have left is some big pile of primes. I mean, by definition, when you can't factor anymore, when you're done, when you can't break the numbers up anymore, what's left must be prime. You know, 12 breaks into two and two and three. So these numbers are the atoms, the building blocks of all numbers. And there's a lot we know about them, but there's much more that we don't know. I'll tell you the first few. There's two, three, five, seven, 11. By the way, they're all going to be odd from then on, because if they were even, I could factor two out of them. But it's not all the odd numbers. Nine isn't prime because it's three times three. 15 isn't prime because it's three times five, but 13 is. Where were we? Two, three, five, seven, 11, 13, 17, 19. Not 21, but 23 is, et cetera, et cetera. Okay, so you could go on. How high could you go if we were just sitting here? By the way, your own brain. Oh. Continuous, without interruption. Would you be able to go over 100? I think so. There's always those ones that trip people up. There's a famous one, the Grotendieck prime, 57. Alexander Grotendieck, the great algebraic geometer, was giving some lecture involving a choice of a prime in general. And somebody said, can't you just choose a prime? And he said, okay, 57, which is in fact not prime. It's three times 19. Oh, damn. But it was like, I promise you in some circles, it's a funny story. Okay. It's not that. But there's a humor in it. Yes, I would say over 100, I definitely don't remember. Like 107, I think. I'm not sure. Okay. So is there a category of like fake primes that are easily mistaken to be prime? Like 57, I wonder. Yeah, so I would say 57 and 51 are definitely like prime offenders. Oh, I didn't do that on purpose. Oh, well done. Didn't do it on purpose. Anyway, they're definitely ones that people, or 91 is another classic seven times 13. It really feels kind of prime, doesn't it? But it is not. Yeah. So there's also, by the way, but there's also an actual notion of pseudo prime, which is a thing with a formal definition, which is not a psychological thing. It is a prime which passes a primality test devised by Fermat, which is a very good test, which if a number fails this test, it's definitely not prime. And so there was some hope that, oh, maybe if a number passes the test, then it definitely is prime. That would give a very simple criterion for primality. Unfortunately, it's only perfect in one direction. So there are numbers, I want to say 341 is the smallest, which pass the test, but are not prime 341. Is this test easily explainable or no? Yes, actually. Ready? Let me give you the simplest version of it. You can dress it up a little bit, but here's the basic idea. I take the number, the mystery number, I raise two to that power. So let's say your mystery number is six. Are you sorry you asked me? Are you ready to follow? No, you're breaking my brain again, but yes. Okay, let's do it. We're going to do a live demonstration. Let's say your number is six. So I'm going to raise two to the sixth power. Okay, so if I were working, I'd be like, that's two cubed squared. That's eight times eight. So that's 64. Now we're going to divide by six, but I don't actually care what the quotient is, only the remainder. So let's see, 64 divided by six is, well, there's a quotient of 10, but the remainder is four. So you failed because the answer has to be two. For any prime, let's do it with five, which is prime. Two to the fifth is 32, divide 32 by five, and you get six with a remainder of two. With a remainder of two. The remainder of two. For seven, two to the seventh is 128, divide that by seven, and let's see, I think that's seven times 14. Is that right? No. Seven times 18 is 126 with a remainder of two, right? 128 is a multiple of seven plus two. So if that remainder is not two, then that's definitely not prime. Then it's definitely not prime. And then if it is, it's likely a prime, but not for sure. It's likely a prime, but not for sure. And there's actually a beautiful geometric proof, which is in the book, actually. That's one of the most granular parts of the book, because it's such a beautiful proof, I could not give it. So you draw a lot of opal and pearl necklaces and spin them. That's the geometric nature of this proof of Fermat's little theorem. So yeah, so with pseudo primes, there are primes that are faking it, they pass that test, but there are numbers that are faking it, they pass that test, but are not actually prime. But the point is, there are many, many, many theorems about prime numbers. Are there, like there's a bunch of questions to ask. Is there an infinite number of primes? Can we say something about the gap between primes as the numbers grow larger and larger and larger and so on? Yeah, it's a perfect example of your desire for simplicity in all things. You know what would be really simple? If there was only finitely many primes, and then there would be this finite set of atoms that all numbers would be built up. That would be very simple and good in certain ways, but it's completely false. And number theory would be totally different if that were the case. It's just not true. In fact, this is something else that Euclid knew. So this is a very, very old fact, like much before, long before we had anything like modern number theory. Their primes are infinite. Their primes that there are, that write the infinite numbers. There's an infinite number of primes. So what about the gaps between the primes? Right. So one thing that people recognized and really thought about a lot is that the primes, on average, seem to get farther and farther apart as they get bigger and bigger. In other words, it's less and less common. Like I already told you of the first 10 numbers, 2, 3, 5, 7, 4 of them are prime. That's a lot, 40%. If I looked at, you know, 10 digit numbers, no way would 40% of those be prime. Being prime would be a lot rarer, in some sense, because there's a lot more things for them to be divisible by. That's one way of thinking of it. It's a lot more possible for there to be a factorization because there's a lot of things you can try to factor out of it. As the numbers get bigger and bigger, primality gets rarer and rarer. And the extent to which that's the case, that's pretty well understood. But then you can ask more fine-grained questions. And here is one. A twin prime is a pair of primes that are two apart, like 3 and 5, or like 11 and 13, or like 17 and 19. And one thing we still don't know is, are there infinitely many of those? We know on average they get farther and farther apart, but that doesn't mean there couldn't be like occasional folks that come close together. And indeed, we think that there are. And one interesting question, I mean, this is... Because I think you might say, well, how could one possibly have a right to have an opinion about something like that? We don't have any way of describing a process that makes primes. Sure, you can look at your computer and see a lot of them, but the fact that there's a lot, why is that evidence that there's infinitely many? Maybe I can go on my computer and find 10 million. Well, 10 million is pretty far from infinity, right? So how is that evidence? There's a lot of things. There's a lot more than 10 million atoms. That doesn't mean there's infinitely many atoms in the universe, right? I mean, on most people's physical theories, there's probably not, as I understand it. Okay, so why would we think this? The answer is that it turns out to be incredibly productive and enlightening to think about primes as if they were random numbers, as if they were randomly distributed according to a certain law. Now, they're not. They're not random. There's no chance involved. It's completely deterministic whether a number is prime or not. And yet, it just turns out to be phenomenally useful in mathematics to say, even if something is governed by a deterministic law, let's just pretend it wasn't. Let's just pretend that they were produced by some random process and see if the behavior is roughly the same. And if it's not, maybe change the random process. Maybe make the randomness a little bit different and tweak it and see if you can find a random process that matches the behavior we see. And then maybe you predict that other behaviors of the system are like that of the random process. And so that's kind of like, it's funny, because I think when you talk to people about the twin prime conjecture, people think you're saying, wow, there's like some deep structure there that like makes those primes be like close together again and again. And no, it's the opposite of deep structure. What we say when we say we believe the twin prime conjecture is that we believe the primes are like sort of strewn around pretty randomly. And if they were, then by chance you would expect there to be infinitely many twin primes. And we're saying, yeah, we expect them to behave just like they would if they were random dirt. You know, the fascinating parallel here is I just got a chance to talk to Sam Harris, and he uses the prime numbers as an example. Often, I don't know if you're familiar with who Sam is. He uses that as an example of there being no free will. Wait, where does he get this? Well, he just uses as an example of it might seem like this is a random number generator, but it's all like formally defined. So if we keep getting more and more primes, then like that might feel like a new discovery, and that might feel like a new experience, but it's not. It was always written in the cards. But it's funny that you say that, because a lot of people think of like randomness, the fundamental randomness within the nature of reality might be the source of something that we experience as free will. And you're saying it's like useful to look at prime numbers as a random process in order to prove stuff about them. But fundamentally, of course, it's not a random process. Well, not in order to prove some stuff about them so much as to figure out what we expect to be true and then try to prove that. Because here's what you don't want to do, try really hard to prove something that's false. That makes it really hard to prove the thing if it's false. So you certainly want to have some heuristic ways of making good guesses about what's true. So yeah, here's what I would say. You're going to be imaginary Sam Harris now. You're talking about prime numbers, and you're like, but prime numbers are completely deterministic. And I'm saying, well, let's treat them like a random process. And then you say, but you're just saying something that's not true. They're not a random process, they're deterministic. And I'm like, okay, great. You hold to your insistence that it's not a random process. Meanwhile, I'm generating insight about the primes that you're not because I'm willing to sort of pretend that there's something that they're not in order to understand what's going on. Yeah, so it doesn't matter what the reality is. What matters is what framework of thought results in the maximum number of insights. Yeah, because I feel, look, I'm sorry, but I feel like you have more insights about people if you think of them as like, beings that have wants and needs and desires and do stuff on purpose. Even if that's not true, you still understand better what's going on by treating them in that way. Don't you find, look, you work on machine learning, don't you find yourself sort of talking about what the machine is trying to do? In a certain instance, do you not find yourself drawn to that language? Well, I- It knows this, it's trying to do that, it's learning that. I'm certainly drawn to that language to the point where I receive quite a bit of criticisms for it. Because I, you know, like- Oh, I'm on your side, man. Yeah, so especially in robotics, I don't know why, but robotics people don't like to name their robots, or they certainly don't like to gender their robots, because the moment you gender a robot, you start to anthropomorphize. If you say he or she, you start to, in your mind, construct like a life story in your mind. You can't help it. You create like a humorous story to this person, you start to- This person, this robot, you start to project your own- But I think that's what we do to each other. I think that's actually really useful for the engineering process, especially for human-robot interaction. And yes, for machine learning systems, for helping you build an intuition about a particular problem. It's almost like asking this question, you know, when a machine learning system fails in a particular edge case, asking, like, what were you thinking about? Like, asking, like, almost like when you're talking about, to a child who just did something bad, you want to understand, like, what was, how did they see the world? Maybe there's a totally new- Maybe you're the one that's thinking about the world incorrectly. And yeah, that anthropomorphization process, I think, is ultimately good for insight. And the same as, I agree with you, I tend to believe about free will as well. Let me ask you a ridiculous question, if it's okay. Of course. I've just recently, most people go on, like, rabbit hole, like, YouTube things, and I went on a rabbit hole off and do of Wikipedia. And I found a page on finitism, ultra finitism, and intuitionism, or- I forget what it's called. Yeah, intuitionism. Intuitionism. That seemed pretty- that seemed pretty interesting. I have it on my to-do list to actually, like, look into, like, is there people who, like, formally- like, real mathematicians are trying to argue for this. But the belief there, I think, let's say finitism, that infinity is fake. Meaning, infinity may be, like, a useful hack for certain, like, a useful tool in mathematics, but it really gets us into trouble, because there's no infinity in the real world. Maybe I'm sort of not expressing that fully correctly, but basically saying, like, there's things that are- once you add into mathematics things that are not provably within the physical world, you're starting to inject- to corrupt your framework of reason. What do you think about that? I mean, I think- okay, so first of all, I'm not an expert, and I couldn't even tell you what the difference is between those three terms, finitism, ultra finitism, and intuitionism, although I know they're related, I tend to associate them with the Netherlands in the 1930s. Okay, I'll tell you, can I just quickly comment, because I read the Wikipedia page? The difference in ultra- That's, like, the ultimate sentence of the modern age. Can I just comment, because I read the Wikipedia page? That sums up our moment. Bro, I'm basically an expert. Ultra- ultra finitism- so, finitism says that the only infinity you're allowed to have is that the natural numbers are infinite. So, like, those numbers are infinite. So, like, one, two, three, four, five, the integers are infinite. The ultra-finitism says, nope, even that infinity is fake. That's- I'll bet ultra-finitism came second. I'll bet it's, like, when there's, like, a hardcore scene, and then one guy's, like, oh, now there's a lot of people in this scene. I have to find a way to be more hardcore than the hardcore people. It's all back to the emo talk. Yeah. Okay, so is there any- are you ever- because I'm often uncomfortable with infinity, like, psychologically. I, you know, I have trouble when that sneaks in there. It's because it works so damn well, I get a little suspicious, because it could be almost like a crutch or an oversimplification that's missing something profound about reality. Well, so, first of all, okay, if you say, like, is there, like, a serious way of doing mathematics that doesn't really treat infinity as a real thing, or maybe it's kind of agnostic and it's like, I'm not really going to make a firm statement about whether it's a real thing or not. Yeah, that's called most of the history of mathematics, right? So it's only after Cantor, right, that we really are sort of, okay, we're going to, like, have a notion of, like, the cardinality of an infinite set and, like, do something that you might call, like, the modern theory of infinity. That said, obviously, everybody was drawn to this notion. And no, not everybody was comfortable with it. Look, I mean, this is what happens with Newton, right? I mean, so Newton understands that to talk about tangents and to talk about instantaneous velocity, he has to do something that we would now call taking a limit, right? The fabled dy over dx, if you sort of go back to your calculus class, for those who've taken calculus, remember this mysterious thing. And, you know, what is it? What is it? Well, he'd say, like, well, it's like you sort of divide the length of this line segment by the length of this other line segment, and then you make them a little shorter, and you divide again, and then you make them a little shorter, and you divide again, and then you just keep on doing that until they're infinitely short, and then you divide them again. These quantities that are, like, they're not zero, but they're also smaller than any actual number, these infinitesimals. Well, people were queasy about it, and they weren't wrong to be queasy about it, right? From a modern perspective, it was not really well-formed. There's this very famous critique of Newton by Bishop Berkeley, where he says, like, what these things you define, like, you know, they're not zero, but they're smaller than any number. Are they the ghosts of departed quantities? That was this, like, ultra-par. And on the one hand, he was right. It wasn't really rigorous by modern standards. On the other hand, like, Newton was out there doing calculus, and other people were not, right? So I think a sort of intuitionist view, for instance, I would say would express serious doubt. And by the way, it's not just infinity. It's not it's like saying, I think we would express serious doubt that, like, the real numbers exist. Now, most people are comfortable with the real numbers. Well, computer scientists with floating point number, I mean, floating point arithmetic. That's a great point, actually. I think, in some sense, this flavor of doing math, saying, we shouldn't talk about things that we cannot specify in a finite amount of time, there's something very computational in flavor about that. And it's probably not a coincidence that it becomes popular in the 30s and 40s, which is also like, kind of like the dawn of ideas about formal computation, right? You probably know the timeline better than I do. Sorry, what becomes popular? These ideas that maybe we should be doing math in this more restrictive way where even a thing that, you know, because look, the origin of all this is like, number represents a magnitude, like the length of a line. So, I mean, the idea that there's a continuum, there's like, is pretty old, but that, you know, just because something is old doesn't mean we can't reject it if we want to. Well, a lot of the fundamental ideas in computer science, when you talk about the complexity of problems, to Turing himself, they rely on infinity as well. The ideas that kind of challenge that, the whole space of machine learning, I would say, challenges that. It's almost like the engineering approach to things, like the floating point arithmetic. The other one that, back to John Conway, that challenges this idea, I mean, maybe to tie in the ideas of deformation theory and limits to infinity, is this idea of cellular automata. With John Conway looking at the game of life, Stephen Wolfe from his work, that I've been a big fan of for a while, of cellular automata. I was wondering if you have ever encountered these kinds of objects, you ever looked at them as a mathematician, where you have very simple rules of tiny little objects that when taken as a whole, create incredible complexities, but are very difficult to analyze, very difficult to make sense of, even though the one individual object, one part, it's like what we were saying about Andrew Wiles, you can look at the deformation of a small piece to tell you about the whole. It feels like with cellular automata, or any kind of complex systems, it's often very difficult to say something about the whole thing, even when you can precisely describe the operation of the local neighborhoods. Yeah, I mean, I love that subject. I haven't really done research in it myself, I've played around with it. I'll send you a fun blog post I wrote where I made some cool texture patterns from cellular automata. And those are really always compelling, it's like you create simple rules and they create some beautiful textures, it doesn't make any sense. Actually, did you see there was a great paper, I don't know if you saw this, like a machine learning paper. I don't know if you saw the one I'm talking about, where they were learning the textures, like let's try to reverse engineer and learn a cellular automaton that can produce a texture that looks like this from the images. Very cool. And as you say, the thing you said is I feel the same way when I read machine learning papers, that what's especially interesting is the cases where it doesn't work. What does it do when it doesn't do the thing that you tried to train it to do? That's extremely interesting. Yeah, that was a cool paper. So yeah, so let's start with a game of life. Let's start with John Conway. So Conway... So yeah, so let's start with John Conway again. I don't know, from my outsider's perspective, there's not many mathematicians that stand out throughout the history of the 20th century. He's one of them. I feel like he's not sufficiently recognized. I think he's pretty recognized. Okay, well... I mean, he was a full professor at Princeton for most of his life, he was sort of in certainly the pinnacle of... Yeah, but I found myself every time I talk about Conway and how excited I am about him, I have to constantly explain to people who he is. And that's always a sad sign to me. But that's probably true for a lot of mathematicians. I was about to say, I feel like you have a very elevated idea of how famous... This is what happens when you grow up in the Soviet Union, or you think the mathematicians are like very, very famous. Yeah, but I'm not actually so convinced at a tiny tangent that that shouldn't be so. I mean, it's not obvious to me that that's one of the... Like, if I were to analyze American society, that perhaps elevating mathematical and scientific thinking to a little bit higher level would benefit the society. Well, both in discovering the beauty of what it is to be human, and for actually creating cool technology, better iPhones. But anyway, John Conway. Yeah, and Conway is such a perfect example of somebody whose humanity was, and his personality was like wound up with his mathematics, right? It's what's not sometimes I think people who are outside the field think of mathematics as this kind of like, cold thing that you do separate from your existence as a human being. No way your personality is in there, just as it would be in like a novel you wrote, or a painting you painted, or just like the way you walk down the street, like it's in there, it's you doing it. And Conway was certainly a singular personality. I think anybody would say that he was playful, like everything was a game to him. Now, what you might think I'm going to say, and it's true, is that he sort of was very playful in his way of doing mathematics. But it's also true, it went both ways. He also sort of made mathematics out of games. He like looked at, he was a constant inventor of games with like crazy names. And then he was sort of analyzed those games mathematically, to the point that he and then later collaborating with Knuth, like, you know, created this number system, the serial numbers, in which actually each number is a game. There's a wonderful book about this called, I mean, there are his own books, and then there's like a book that he wrote with Burlekemp and Guy called Winning Ways, which is such a rich source of ideas. And he too kind of has his own crazy number system, in which, by the way, there are these infinitesimals, the ghosts of departed quantities, they're in there. They're in there, now, not as ghosts, but as like certain kind of two player games. So, you know, he was a guy, so I knew him when I was a postdoc. And I knew him at Princeton, and our research overlapped in some ways. Now it was on stuff that he had worked on many years before, the stuff I was working on kind of connected with stuff in group theory, which somehow keeps seems to keep coming up. And so I often would like sort of ask him a question, I would sort of come upon him in the common room, and I would ask him a question about something. And just anytime you turned him on, you know what I mean? You sort of asked a question, it was just like turning a knob and winding him up, and he would just go and you would get a response that was like, so rich and went so many places and taught you so much, and usually had nothing to do with your question. Usually your question was just a prompt to him, you couldn't count on actually getting the question. He had those brilliant curious minds, even at that age. Yeah, it was definitely a huge loss. But on his Game of Life, which was I think he developed in the 70s, as almost like a side thing, a fun little experiment. Yeah, the Game of Life is this, it's a very simple algorithm. It's not a very complicated algorithm. It's a very simple algorithm. It's not really a game per se, in the sense of the kinds of games that he liked, where people played against each other. And but essentially, it's a game that you play with marking little squares on a sheet of graph paper. And in the 70s, I think he was like, literally doing it with like a pen on graph paper, you have some configuration of squares, some of the squares in the graphic are filled in, and some of the squares are not. And then there's a rule, a single rule that tells you at the next stage, which squares are filled in, and which squares are not. Sometimes an empty square gets filled in, that's called birth. Sometimes a square that's filled in gets erased, that's called death. And there's rules for which squares are born and which squares die. It's the rule is very simple, you can write it on one line. And then the great miracle is that you can start from some very innocent looking, little small set of boxes and get these results of incredible richness. And of course, nowadays, you don't do it on paper. Nowadays, you do it on a computer. There's actually a great iPad app called Gali, which I really like that has like, Conway's original rule and like, gosh, like hundreds of other variants. And it's lightning fast. So you can just be like, I want to see 10,000 generations of this rule play out like faster than your eye can even follow. And it's like, amazing. So I highly recommend it if this is at all intriguing to you getting Gali on your iOS device. And you can do this kind of process, which I really enjoy doing, which is almost from like putting a Darwin hat on or a biologist hat on and doing analysis of a higher level of abstraction, like the organisms that spring up, because there's different kinds of organisms, like you can think of them as species and they interact with each other. They can, there's gliders, they shoot different, there's like, things that can travel around, there's things that can, glider guns, that can generate those gliders. They're, you can use the same kind of language as you would about describing a biological system. So it's a wonderful laboratory. And it's kind of a rebuke to someone who doesn't think that like very, very rich, complex structure can come from very simple underlying laws. Like, it definitely can. Now, here's what's interesting. If you just pick like some random rule, you wouldn't get interesting complexity. I think that's one of the most interesting things of these, one of the most interesting features of this whole subject, that the rules have to be tuned just right, like a sort of typical rule set doesn't generate any kind of interesting behavior. But some do. And I don't think we have a clear way of understanding which do and which don't. I don't, maybe Stephen thinks he does. I don't know. No, no, it's a giant mystery. What Stephen Wolfram did is, now there's a whole interesting aspect to the fact that he's a little bit of an outcast in the mathematics and physics community, because he's so focused on a particular, his particular work. I think if you put ego aside, which I think, unfairly, some people are not able to look beyond, I think his work is actually quite brilliant. But what he did is exactly this process of Darwin-like exploration. He's taking these very simple ideas and writing a thousand page book on them, meaning like, let's play around with this thing. Let's see. And can we figure anything out? Spoiler alert, no, we can't. In fact, he does a challenge. I think it's like rule 30 challenge, which is quite interesting, just simply for machine learning people, for mathematics people, is can you predict the middle column? For his, it's a 1D cellular automata. Can you, generally speaking, can you predict anything about how a particular rule will evolve, just in the future? Very simple. Just looking at one particular part of the world, just zooming in on that part, you know, 100 steps ahead, can you predict something? And the challenge is to do that kind of prediction so far as nobody's come up with an answer. But the point is, we can't, we don't have tools, or maybe it's impossible, or, I mean, he has these kind of laws of irreducibility that he refers to, but it's poetry. It's like, we can't prove these things. It seems like we can't. That's the basic, it almost sounds like ancient mathematics or something like that, where you're like, the gods will not allow us to predict the cellular automata. But that's fascinating that we can't. I'm not sure what to make of it. And there's power to calling this particular set of rules Game of Life, as Conway did, because, not exactly sure, but I think he had a sense that there's some core ideas here that are fundamental to life, to complex systems, to the way life emerged on Earth. I'm not sure I think Conway thought that. It's something that, I mean, Conway always had a rather ambivalent relationship with the Game of Life, because I think he saw it as, it was certainly the thing he was most famous for in the outside world. And I think that his view, which is correct, is that he had done things that were much deeper mathematically than that. And I think it always aggrieved him a bit that he was like the Game of Life guy, when he proved all these wonderful theorems and created all these wonderful games, created the serial numbers. He was a very tireless guy who just did an incredibly variegated array of stuff. So he was exactly the kind of person who you would never want to reduce to one achievement, you know what I mean? Let me ask about group theory. You mentioned it a few times. What is group theory? What is an adjacent idea from group theory that you find beautiful? Well, so I would say group theory sort of starts as the general theory of symmetry, is that, you know, people looked at different kinds of things and said, like, as we said, like, oh, it could have maybe all there is is symmetry from left to right, like a human being, right? That's roughly bilateral, bilaterally symmetric, as we say. So, so there's two symmetries. And then you're like, well, wait, didn't I say there's just one there's just left to right? Well, we always count the symmetry of doing nothing. We always count the symmetry that's like, there's flip and don't flip. Those are the two configurations that you can be in. So there's two. You know, something like a rectangle is bilaterally symmetric, you can flip it left to right, but you can also flip it top to bottom. So there's actually four symmetries. There's do nothing, flip it left to right, and flip it top to bottom or do both of those things. A square, there's even more, because now you can rotate it, you can rotate it by 90 degrees. So you can't do that. That's not a symmetry of the rectangle. If you try to rotate it 90 degrees, you get a rectangle oriented in a different way. So a person has two symmetries, a rectangle for a square, eight different kinds of shapes have different numbers of symmetries. And the real observation is that that's just not like a set of things. They can be combined, you do one symmetry, then you do another. The result of that is some third symmetry. So a group really abstracts away this notion of saying it's just some collection of transformations you can do to a thing where you combine any two of them to get a third. So you know, a place where this comes up in computer science is in sorting, because the ways of permuting a set, the ways of taking sort of some set of things you have on the table and putting them in a different order, shuffling a deck of cards, for instance, those are the symmetries of the deck. And there's a lot of them. There's not two, there's not four, there's not eight, think about how many different orders a deck of card can be in each one of those is the result of applying a symmetry to the original deck. So a shuffle is a symmetry, right? You're reordering the cards. If I shuffle and then you shuffle, the result is some other kind of thing you might call a double shuffle, which is a more complicated symmetry. So group theory is kind of the study of the general abstract world that encompasses all these kinds of things. But then of course, lots of things that are way more complicated than that. Like infinite groups of symmetries, for instance. So they can be infinite, huh? Oh yeah. Okay. Well, okay, ready? Think about the symmetries of the line. You're like, okay, I can reflect it left to right, you know, around the origin. Okay, but I could also reflect it left to right, grabbing somewhere else, like at one or two or pi or anywhere. Or I could just slide it some distance. That's a symmetry. Slide it five units over. So there's clearly infinitely many symmetries of the line. That's an example of an infinite group of symmetries. Is it possible to say something that kind of captivates, keeps being brought up by physicists, which is gauge theory, gauge symmetry, as one of the more complicated type of symmetries? Is there an easy explanation of what the heck it is? Is that something that comes up on your mind at all? Well, I'm not a mathematical physicist, but I can say this. It is certainly true that it has been a very useful notion in physics to try to say, like, what are the symmetry groups of the world? Like, what are the symmetries under which things don't change, right? So we just, I think we talked a little bit earlier about, it should be a basic principle that a theorem that's true here is also true over there. Yes. And same for a physical law, right? I mean, if gravity is like this over here, it should also be like this over there. Okay. What that's saying is we think translation in space should be a symmetry. All the laws of physics should be unchanged if the symmetry we have in mind is a very simple one like translation. And so then there becomes a question, like, what are the symmetries of the actual world with its physical laws? And one way of thinking, this is an oversimplification, but like one way of thinking of this big shift from before Einstein to after is that we just changed our idea about what the fundamental group of symmetries were. So that things like the Lorenz contraction, things like these bizarre relativistic phenomena, or Lorenz would have said, oh, to make this work, we need a thing to change its shape if it's moving nearly the speed of light. Well, under the new framework, it's much better. No, it wasn't changing its shape. You were just wrong about what counted as a symmetry. Now that we have this new group, the so-called Lorenz group, now that we understand what the symmetries really are, we see it was just an illusion that the thing was changing its shape. Yeah, so you can then describe the sameness of things under this weirdness that is general relativity, for example. Yeah, yeah, still, I wish there was a simpler explanation of like exact, I mean, gauge symmetries, pretty simple general concept about rulers being deformed. I've actually just personally been on a search, not a very rigorous or aggressive search, but for something I personally enjoy, which is taking complicated concepts and finding the sort of minimal example that I can play around with, especially programmatically. That's great. I mean, this is what we try to train our students to do, right? I mean, in class, this is exactly what, this is like best pedagogical practice. I do hope there's simple explanation, especially like I've, in my sort of drunk random walk, drunk walk, whatever that's called, sometimes stumble into the world of topology. And like quickly, like, you know, when you like go into a party and you realize this is not the right party for me. So whenever I go into topology, it's like so much math everywhere. I don't even know what, it feels like, this is me like being a hater, is I think there's way too much math. Like there are two, the cool kids who just want to have like everything is expressed through math because they're actually afraid to express stuff simply through language. That's my hater formulation of topology. But at the same time, I'm sure that's very necessary to do sort of rigorous discussion, but I feel like- But don't you think that's what gauge symmetry is like? I mean, it's not a field I know well, but it certainly seems like- Yes, it is like that. But my problem with topology, okay, and even like differential geometry is like, you're talking about beautiful things. Like if they could be visualized, it's open question if everything could be visualized, but you're talking about things that could be visually stunning, I think. But they are hidden underneath all of that math. Like if you look at the papers that are written in topology, if you look at all the discussions on Stack Exchange, they're all math dense, math heavy. And the only kind of visual things that emerge every once in a while is like something like a Mobius strip. Every once in a while, some kind of simple visualizations. Well, there's the vibration, there's the hop vibration, or all of those kinds of things that somebody, some grad student from like 20 years ago wrote a program in Fortran to visualize it, and that's it. And it just, you know, it makes me sad because those are visual disciplines, just like computer vision is a visual discipline. So you can provide a lot of visual examples. I wish topology was more excited and in love with visualizing some of the ideas. I mean, you could say that, but I would say for me, a picture of the hop vibration does nothing for me. Whereas like when you're like, oh, it's like about the Quaternions, it's like a subgroup of the Quaternions, and I'm like, oh, so now I see what's going on. Like, why didn't you just say that? Why were you like showing me this stupid picture instead of telling me what you were talking about? Oh, yeah, yeah. I'm just saying, no, but it goes back to what you were saying about teaching that like people are different in what they'll respond to. So I think there's no, I mean, I'm very opposed to the idea that there's one right way to explain things. I think there's a huge variation in like, you know, our brains like have all these like weird, like hooks and loops. And it's like very hard to know like what's going to latch on and it's not going to be the same thing for everybody. So I think monoculture is bad, right? I think that's and I think we're agreeing on that point that like, it's good that there's like a lot of different ways in and a lot of different ways to describe these ideas, because different people are going to find different things illuminating. But that said, I think there's a lot to be discovered when you force little like silos of brilliant people to kind of find a middle ground or like, aggregate or come together in a way. So there's like people that do love visual things. I mean, there's a lot of disciplines, especially in computer science that they're obsessed with visualizing, visualizing data, visualizing neural networks. I mean, neural networks themselves are fundamentally visual. There's a lot of work in computer vision that's very visual. And then coming together with some folks that were like deeply rigorous and are like totally lost in multi-dimensional space where it's hard to even bring them back down to 3D. They're very comfortable in this multi-dimensional space. So forcing them to kind of work together to communicate, because it's not just about public communication of ideas. It's also, I feel like when you're forced to do that public communication, like you did with your book, I think deep, profound ideas can be discovered that's like applicable for research and for science. Like there's something about that simplification, or not simplification, but distillation or condensation or whatever the hell you call it, compression of ideas that somehow actually stimulates creativity. And I'd be excited to see more of that in the mathematics community. Can you- Look, let me make a crazy metaphor. Maybe it's a little bit like the relation between prose and poetry, right? I mean, you might say like, why do we need anything more than prose? You're trying to convey some information. So you just like say it. Well, poetry does something, right? It's sort of, you might think of it as a kind of compression. Of course, not all poetry is compressed. Like not all, some of it is quite baggy, but like you are kind of, often it's compressed, right? A lyric poem is often sort of like a compression of what would take a long time and be complicated to explain in prose into sort of a different mode that is going to hit in a different way. We talked about Poincaré conjecture. There's a guy, he's Russian, Grigori Pearlman. He proved Poincaré's conjecture. If you can comment on the proof itself, if that stands out to you as something interesting, or the human story of it, which is he turned down the Fields Medal for the proof. Is there something you find inspiring or insightful about the proof itself, or about the man? Yeah, I mean, one thing I really like about the proof, and partly that's because it's sort of a thing that happens again and again in this book. I mean, I'm writing about geometry and the way it sort of appears in all these kind of real world problems. But it happens so often that the geometry you think you're studying is somehow not enough. You have to go one level higher in abstraction and study a higher level of geometry. And the way that plays out is that, you know, Poincaré asks a question about a certain kind of three-dimensional object. Is it the usual three-dimensional space that we know, or is it some kind of exotic thing? And so, of course, this sounds like it's a question about the geometry of the three-dimensional space. But no, but no, Perelman understands. And by the way, in a tradition that involves Richard Hamilton and many other people, like most really important mathematical advances, this doesn't happen alone. It doesn't happen in a vacuum. It happens as the culmination of a program that involves many people. Same with Wiles, by the way. I mean, we talked about Wiles, and I want to emphasize that starting all the way back with Kummer, who I mentioned in the 19th century, but Gerhard Frey and Maser and Ken Ribbitt and like many other people are involved in building the other pieces of the arch before you put the keystone in. We stand on the shoulders of giants. Yes. So, what is this idea? The idea is that, well, of course, the geometry of the three-dimensional object itself is relevant, but the real geometry you have to understand is the geometry of the space of all three-dimensional geometries. Whoa. You're going up a higher level. Because when you do that, you can say, now let's trace out a path in that space. There's a mechanism called Ricci flow. And again, we're outside my research area. So, for all the geometric analysts and differential geometers out there listening to this, I'm doing my best and I'm roughly saying it. So, the Ricci flow allows you to say like, okay, let's start from some mystery three-dimensional space, which Poincaré would conjecture is essentially the same thing as our familiar three-dimensional space, but we don't know that. And now you let it flow. You sort of like let it move in its natural path according to some almost physical process and ask where it winds up. And what you find is that it always winds up. You've continuously deformed it. There's that word deformation again. And what you can prove is that the process doesn't stop until you get to the usual three-dimensional space. And since you can get from the mystery thing to the standard space by this process of continually changing and never kind of having any sharp transitions, then the original shape must have been the same as the standard shape. That's the nature of the proof. Now, of course, it's incredibly technical. I think, as I understand it, I think the hard part is proving that the favorite word of AI people, you don't get any singularities along the way. But of course, in this context, singularity just means acquiring a sharp kink. It just means becoming non-smooth at some point. So. Just saying something interesting about formal about the smooth trajectory through this weird space. Yeah. But yeah, so what I like about it is that it's just one of many examples of where it's not about the geometry you think it's about. It's about the geometry of all geometries, so to speak. And it's only by kind of like being jerked out of Flatland, right? Same idea. It's only by sort of seeing the whole thing globally at once that you can really make progress on understanding like the one thing you thought you were looking at. It's a romantic question, but what do you think about him turning down the Fields Medal? Is that just our Nobel Prizes and Fields Medals just the cherry on top of the cake and really math itself, the process of curiosity of pulling at the string of the mystery before us? That's the cake. And then the awards are just icing and clearly I've been fasting and I'm hungry. But do you think it's tragic or just a little curiosity that he turned down the medal? Well, it's interesting because on the one hand, I think it's absolutely true that right in some kind of like vast spiritual sense, like awards are not important, like not important the way that sort of like understanding the universe is important. On the other hand, most people who are offered that prize accept it. So there's something unusual about his choice there. I wouldn't say I see it as tragic. I mean, maybe if I don't really feel like I have a clear picture of why he chose not to take it. I mean, he's not alone in doing things like this. People sometimes turn down prizes for ideological reasons. Probably more often in mathematics. I mean, I think I'm right in saying that Peter Schultz turned down sort of some big monetary prize because he just, you know, I mean, I think at some point you have plenty of money. And maybe you think it sends the wrong message about what the point of doing mathematics is. I do find that there's most people accept, you know, most people give it a prize. Most people take it. I mean, people like to be appreciated. Like I said, we're people. Not that different from most other people. But the important reminder that that turning down the prize serves for me is not that there's anything wrong with the prize. And there's something wonderful about the prize, I think. The Nobel Prize is trickier because so many Nobel Prizes are given. First of all, the Nobel Prize often forgets many, many of the important people throughout history. Second of all, there's like these weird rules to it. It's only three people and some projects have a huge number of people. And it's like this, it, I don't know. It doesn't kind of highlight the way science has done on some of these projects in the best possible way. But in general, the prizes are great. But what this kind of teaches me and reminds me is sometimes in your life, there'll be moments when the thing that you would really like to do, society would really like you to do, is the thing that goes against something you believe in, whatever that is, some kind of principle, and standing your ground in the face of that. It's something, I believe most people will have a few moments like that in their life. Maybe one moment like that. And you have to do it. That's what integrity is. So like, it doesn't have to make sense to the rest of the world, but to stand on that, like, to say no, it's interesting. Because I think- But do you know that he turned down the prize in service of some principle? Because I don't know that. Well, yes, that seems to be the inkling, but he has never made it super clear. But the inkling is that he had some problems with the whole process of mathematics that includes awards, like this hierarchies and reputations and all those kinds of things, and individualism that's fundamental to American culture. He probably, because he visited the United States quite a bit, that he probably, you know, it's like all about experiences. And he may have had, you know, some parts of academia, some pockets of academia can be less than inspiring, perhaps sometimes, because of the individual egos involved. Not academia, people in general, smart people with egos. And if they, if you interact with a certain kinds of people, you can become cynical too easily. I'm one of those people that I've been really fortunate to interact with incredible people at MIT and academia in general, but I've met some assholes. And I tend to just kind of, when I run into difficult folks, I just kind of smile and send them all my love and just kind of go around. But for others, those experiences can be sticky. Like they can become cynical about the world when folks like that exist. So it's, he may have become a little bit cynical about the process of science. Well, you know, it's a good opportunity. Let's posit that that's his reasoning, because I truly don't know. It's an interesting opportunity to go back to almost the very first thing we talked about, the idea of the Mathematical Olympiad, because of course, that is, so the International Mathematical Olympiad is like a competition for high school students solving math problems. And in some sense, it's absolutely false to the reality of mathematics, because just as you say, it is a contest where you win prizes. The aim is to sort of be faster than other people. And you're working on sort of canned problems that someone already knows the answer to, like not problems that are unknown. So, you know, in my own life, I think when I was in high school, I was like very motivated by those competitions. And like I went to the Math Olympiad and... You won it. Well, there's something I have to explain to people, because it says, I think it says on Wikipedia that I won a gold medal. And in the real Olympics, they only give one gold medal at each event. I just have to emphasize that the International Math Olympiad is not like that. The gold medals are awarded to the top 112th of all participants. So sorry to bust the legend or anything like that. You're an exceptional performer in terms of achieving high scores on the problems, and they're very difficult. So you've achieved a high level of performance on the... In this very specialized skill. And by the way, it was a very Cold War activity. You know, in 1987, the first year I went, it was in Havana. Americans couldn't go to Havana back then. It was a very complicated process to get there. And they took the whole American team on a field trip to the Museum of American Imperialism in Havana, so we could see what America was all about. How would you recommend a person learn math? So somebody who's young, or somebody my age, or somebody older, who've taken a bunch of math, but wants to rediscover the beauty of math, and maybe integrate it into their work more so than the research space, and so on. Is there something you could say about the process of incorporating mathematical thinking into your life? I mean, the thing is, it's in part a journey of self-knowledge. You have to know what's gonna work for you, and that's gonna be different for different people. So there are totally people who, at any stage of life, just start reading math textbooks. That is a thing that you can do, and it works for some people and not for others. For others, a gateway is, I always recommend the books of Martin Gardner, another person we haven't talked about, but who also, like Conway, embodies that spirit of play. He wrote a column in Scientific American for decades called Mathematical Recreations, and there's such joy in it and such fun. And the columns are collected into books, and the books are old now, but for each generation of people who discover them, they're completely fresh. And they give a totally different way into the subject than reading a formal textbook, which for some people would be the right thing to do. And working contest-style problems, too, those are bound to books, especially Russian and Bulgarian problems. There's book after book of problems from those contexts. That's gonna motivate some people. For some people, it's gonna be like watching well-produced videos, a totally different format. I feel like I'm not answering your question. I'm sort of saying there's no one answer. And it's a journey where you figure out what resonates with you. For some people, the self-discovery is trying to figure out why is it that I want to know? Okay, I'll tell you a story. Once when I was in grad school, I was very frustrated with my lack of knowledge of a lot of things, as we all are, because no matter how much we know, we don't know much more. And going to grad school means just coming face-to-face with the incredible overflowing vault of your ignorance, right? So I told Joe Harris, who was an algebraic geometer of professor in my department, I was like, I really feel like I don't know enough. And I should just like take a year of leave and just like read EGA, the holy textbook, the elements of algebraic geometry. I feel like I don't know enough. So I was gonna sit and like read this like 1500 page, many volume book. And he was like, and Professor Harris was like, that's a really stupid idea. And I was like, why is that a stupid idea? Then I would know more algebraic geometry, because you're not actually going to do it like you learn. I mean, he knew me well enough to say like, you're gonna learn because you're gonna be working on a problem. And then there's gonna be a fact from EGA you need in order to solve your problem that you want to solve. And that's how you're going to learn it. You're not going to learn it without a problem to bring you into it. And so for a lot of people, I think if you're like, I'm trying to understand machine learning, and I'm like, I can see that there's sort of some mathematical technology that I don't have, I think you like let that problem that you actually care about drive your learning. I mean, one thing I've learned from advising students, you know, math is really hard. In fact, anything that you do right, is hard. And because it's hard, like, you might sort of have some idea that somebody else gives you, oh, I should learn x, y and z. Well, if you don't actually care, you're not going to do it, you might feel like you should, maybe somebody told you you should, but I think you have to hook it to something that you actually care about. So for a lot of people, that's the way in, you have an engineering problem you're trying to handle, you have a physics problem you're trying to handle, you have a machine learning problem you're trying to handle, let that not a kind of abstract idea of what the curriculum is, drive your mathematical learning. And also just as a brief comment, that math is hard, there's a sense to which hard is a feature, not a bug, in the sense that, again, maybe this is my own learning preference, but I think it's a value to fall in love with the process of doing something hard, overcoming it, and becoming a better person because of it. Like, I hate running, I hate exercise, to bring it down to like the simplest hard, and I enjoy the part once it's done. The person I feel like for the rest of the day once I've accomplished it, the actual process, especially the process of getting started in the initial, like it really, I don't feel like doing it. And I really have, the way I feel about running is the way I feel about really anything difficult in the intellectual space, especially mathematics, but also just something that requires like holding a bunch of concepts in your mind with some uncertainty, like where the terminology or the notation is not very clear, and so you have to kind of hold all those things together and like keep pushing forward through the frustration of really like obviously not understanding certain parts of the picture, like your giant missing parts of the picture, and still not giving up. It's the same way I feel about running. And there's something about falling in love with the feeling of after you went through the journey of not having a complete picture, at the end, having a complete picture, and then you get to appreciate the beauty, and just remembering that it sucked for a long time, and how great it felt when you figured it out, at least at the basic. That's not sort of research thinking, because with research, you probably also have to enjoy the dead ends. With learning math from a textbook or from a video, there's a nice- I don't think you have to enjoy the dead ends, but I think you have to accept the dead ends. Let's put it that way. Well, yeah, enjoy the suffering of it. So, the way I think about it, I do, there's an- I don't enjoy the suffering. It pisses me off, but I accept that it's part of the process. It's interesting. There's a lot of ways to kind of deal with that dead end. There's a guy who's an ultra marathon runner, Navy SEAL, David Goggins, who kind of, I mean, there's a certain philosophy of like, most people would quit here. And so, if most people would quit here, and I don't, I'll have an opportunity to discover something beautiful that others haven't yet. So, like, any feeling that really sucks, it's like, okay, most people would just like, go do something smarter. If I stick with this, I will discover a new garden of fruit trees that I can pick. Okay, you say that, but like, what about the guy who like, wins the Nathan's hot dog eating contest every year? Like, when he eats his 35th hot dog, he like, correctly says like, okay, most people would stop here. Like, are you like, lauding that he's like, no, I'm going to eat the 36th hot dog? I am, I am. In the long arc of history, that man is onto something. Which brings up this question, what advice would you give to young people today, thinking about their career, about their life, whether it's in mathematics, poetry, or hot dog eating contest? And you know, I have kids, so this is actually a live issue for me, right? I actually, it's not a thought experiment. I actually do have to give advice to young people all the time. They don't listen, but I still give it. You know, one thing I often say to students, I don't think I've actually said this to my kids yet, but I say it to students a lot is, you know, you come to these decision points, and everybody is beset by self-doubt, right? It's like, not sure like, what they're capable of, like, not sure what they're, what they really want to do. I always, I sort of tell people, like, often when you have a decision to make, one of the choices is the high self-esteem choice. And I always tell them, make the high self-esteem choice. Make the choice, sort of take yourself out of it, and like, if you didn't have those, you can probably figure out what the version of you that feels completely confident would do. And do that, and see what happens. And I think that's often, like, pretty good advice. That's interesting. Sort of like, you know, like with Sims, you can create characters. Like, create a character of yourself that lacks all of the self-doubt. Right, but it doesn't mean, I would never say to somebody, you should just go have high self-esteem. You shouldn't have doubts. No, you probably should have doubts. It's okay to have them, but sometimes it's good to act in the way that the person who didn't have them would act. That's a really nice way to put it. Yeah, that's like, from a third person perspective, take the part of your brain that wants to do big things. What would they do? That's not afraid to do those things. What would they do? Yeah, that's really nice. That's actually a really nice way to formulate it. That's very practical advice. You should give it to your kids. Do you think there's meaning to any of it from a mathematical perspective, this life? If I were to ask you, we talked about primes, talked about proving stuff. Can we say, and then the book that God has, that mathematics allows us to arrive at something about in that book, there's certainly a chapter on the meaning of life in that book. Do you think we humans can get to it? And maybe if you were to write Cliff Notes, what do you suspect those Cliff Notes would say? I mean, look, the way I feel is that mathematics, as we've discussed, it underlies the way we think about constructing learning machines. It underlies physics. It does all this stuff. And also, you want the meaning of life? I mean, it's like, we already did a lot for you. Ask a rabbi. No, I mean, I wrote a lot in the last book, How Not to Be Wrong. I wrote a lot about Pascal, a fascinating guy, who is a sort of very serious religious mystic, as well as being an amazing mathematician. And he's well known for Pascal's Wager. I mean, he's probably among all mathematicians, he's the one who's best known for this. Can you actually apply mathematics to these transcendent questions? But what's interesting, when I really read Pascal, about what he wrote about this, you know, I started to see that people often think, oh, this is him saying, I'm going to use mathematics to sort of show you why you should believe in God. You know, to really, that's this, mathematics has the answer to this question. But he really doesn't say that. He almost kind of says the opposite. If you ask Blaise Pascal, like, why do you believe in God? He'd be like, oh, because I met God. You know, he had this kind of like, psychedelic experience, this like a mystical experience, where, as he tells it, he just like directly encountered God. It's like, okay, I guess there's a God, I met him last night. So that's, that's it. That's why he believed. It didn't have to do with any kind of, you know, the mathematical argument was like, about certain reasons for behaving in a certain way. But he basically said, like, look, like, math doesn't tell you that God's there or not. Like, if God's there, he'll tell you, you know, you don't even, I love this. So you have, you have mathematics, you have, what do you, what do you have, like, ways to explore the mind, let's say psychedelics, you have like incredible technology, you also have love and friendship and like, what, what the hell do you want to know what the meaning of it all is? Just enjoy it. I don't think there's a better way to end it, Jordan. This was a fascinating conversation. I really love the way you explore math in your writing, the willingness to be specific and clear and actually explore difficult ideas, but at the same time, stepping outside and figuring out beautiful stuff. And I love the chart at the opening of your new book that shows the chaos, the mess that is your mind. Yes, this is what I was trying to keep in my head all at once while I was writing. And I probably should have drawn this picture earlier in the process. Maybe it would have made my organization easier. I actually drew it only at the end. And many of the things we talked about are on this map. The connections are yet to be fully dissected and investigated. And yes, God is in the picture. Right on the edge, right on the edge, not in the center. Thank you so much for talking. It is a huge honor that you would waste your valuable time with me. Thank you, Lex. We went to some amazing places today. This was really fun. Thanks for listening to this conversation with Jordan Ellenberg. And thank you to Secret Sauce, ExpressVPN, Blinkist, and Indeed. Check them out in the description to support this podcast. And now let me leave you with some words from Jordan in his book, How Not to Be Wrong. Knowing mathematics is like wearing a pair of x-ray specs that reveal hidden structures underneath the messy and chaotic surface of the world. Thank you for listening and hope to see you next time.