Transcript for #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries

That's a really interesting question. You're going to need to reflect on this question of whether the feeling of producing mathematical output, if you want, is like the process of uttering language of 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, maybe I can't imagine it any other way, right?

Ah, that's a really interesting question. And, you know, one thing 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 Baskara of the Pythagorean theorem. proves which are purely visual, proves 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. There's a famous story and it's a little bit disputed about how accurate this is, but then in Boscore's manuscript, he sort of gives us proof just gives the diagram, and then the entire verbal content of the proof is he just writes under it. Behold. 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 or not try to have a good answer? What do you think?

Wow, you've given me like a lot to say and certainly the experience that you described is so typical with there's two versions of it You know 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 are like who like you are like the rest of it I could take her leave but then in 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 talked me all the time who are like I understood all this stuff where you're trying to figure out what X was, there's some mystery you're trying to solve it, X is the number I figured it out, but then there was this geometry, like what was that? What happened that year? Like 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, right? 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 you know, in the 70s we had, you 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 lift the sound out. and the holes were in this rectangular array of 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, six is and six, eight. It's just like the Dissection Bruce you 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. I like to cast on your podcast. I don't know if that's.

It was fucking unbelievable. Okay. That's the last time.

This story merits it.

exactly. And it's just as you say, um, you know, I knew this 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 didn't 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 isosli's triangle has equal angles at the base like this kind of thing. It didn't vibe with me the way that algebra 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, maybe I was always faded to be an algebraic geometry, which is what I am as a researcher.

Well, it's an absolutely fundamental concept and it starts with the words 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 and 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 a classical building or a painting or something like that. Nowadays, in math. We could use a symmetry to refer to any kind of transformation of an image or a space or an object. So what I talk about in the book is take a figure and stretch it vertically. Make a 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 some shape and it gets kind of, I call this in the book a scrunch. I just made that 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, the dominates contemporary mathematics. Or by contemporary, by the way, I mean like the last like 150 years. We're in a very long time scale 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, 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 all the same side lanes and angles and 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 didn't still work. So this question of like what are the symmetries and which things that you want to study are invariant and those symmetries is absolutely fundamental. This is getting a little abstract, right?

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 and of symmetries, the types of symmetries that we know of, are not sufficient, right? So in other words, like, we're just going to keep on going into the weeds on this.

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, right? It would be very easy to 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 distinguished 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.

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.

Wait, so if you ask people, if you've showed them this morph, if you ask a bunch of people, do they all agree about where the transition is?

Because I was surprised. I think so. Oh my god, okay, we have an empiric.

Well, that's not fair.

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, it's like punk array, we're going to study the three body problem. We're going to study sort of like 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, I guess what, that's like, a problem that punk hooray won a huge prize for like 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 youngest mystery.

Okay, so punk array, he ends up being a major figure in the book, and I know I didn't even really intend for him to be such a big figure, but he's so, he's, he's, he's first and foremost a geometry, 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 flour. Actually, I learned a lot, you know, in math, we're not really trained on our own history. We get a PhD in math, what 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 do we do wrong? And they were like, we gotta get strong in math, like the Germans. We have to be like 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,

Definitely wasn't. I would love to hear more about how it was in the Soviet Union.

I think I'm going to pick up on something you said. I think you would love a book called Dual It Dawn by a mural examiner, which I think some of the things you're responding to when I wrote, I think I first got turned on to by a mural's work, he's a historian of math, and he writes about the story of Ever East Galois, which is a story that's well known to all mathematicians, this kind of like, 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 and is 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 their early Nathan 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 Galaw that way was because Galaw himself was a creature of that era and he romanticized himself. I mean now we know he wrote lots of letters and like he was kind of like I mean in modern terms we would say he was extremely emo like that. Like just we wrote all these letters about his like florid feelings and like 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 Yeah. 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 more eyes of the society around us.

Yes, okay. It's a back to punk array. So he's you know, it's funny. This book is filled with kind of you know, mathematical characters who often are kind of pivish or get into feuds or sort of have like weird enthusiasm because those people are fun to write about and they sort of like say very salty things punk array 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 going to tell me now that the book is out like oh didn't you know about this like incredibly sorted episode 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. 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 and what you might think will be this very simple way. Nothing's going on except gravity. So we're leaving the three body 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 or below 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

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 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. It's the first, well, 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 Puanca Ré was the first to recognize that that phenomenon even even existed.

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 those are the objects we interact with. He developed this subject, we now called topology. He called it analysis C2. He was a very well-spoken guy with a lot of slogans, but that named, you can see where that named, and that catch on. So now it's called topology now. Sorry, what was it called before? Analysis C2s. Okay, I guess sort of roughly means like the analysis of location or something like that. Like, it's a Latin phrase. Partly because you 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, three bodies moving in space, well, each, each body, it has a location where it is. So it has an x coordinate, a y coordinate is the coordinate, right? I can specify a point in space by giving you three numbers. But it also, and 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 a 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 introduces analyses status is, is no-end-outs nowadays that the geometry of end-dimensional space is an actual existing thing. 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.

I think that's right. I think don't give me wrong. Juanca Reyn never strives 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's stuff's trash. Or, but I think, and this is more of the mathematicians frame of mind. If you have a Formalistic framework that seems really good and sort of seems to be like very elegant and workwell and it includes all the physical stuff. Maybe we should think about all of that. Maybe we should think about it. Maybe there's some gold to be mine there. 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 like turns out they're real after all. This is a story much told right in mathematics, not just in this context, but in many.

Well, I'm not a mathematical physicist, so we can work out together.

The punk or a conjecture. It's about curved three-dimensional spaces. So I wasn't 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 Poincare'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 it does not look another way, and it maybe a 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 punk or a 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 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 have to visualize it. So Lex has been nice enough to like give me a surface was an interesting topology. Some 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. Oh, which way should I draw it so it's visible? Like here. Okay. That's yeah. 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 just it up here and I can just it down there and I can sort of slide it up and down the handle. But I can't pull it close. Can I? It's trapped. Not without breaking the surface of the mug, right? Not with like going inside. So the condition of being what's called simply connected. This is what I'm going to mention 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, but you'd rank off you 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 kind of push it. But you know, what if your piece of string was the size of the universe? Like what if your point, your piece of string was like billions of light years long? Like how do you actually know?

I mean, I think there's somebody who thinks that there's like some kind of do-decker-headroll symmetry. I remember reading something crazy about somebody saying that they saw the signature of that in the cosmic noise or what have you.

But that's exactly what the apology does, the apology 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.

Well, that's true. The visualization is harder, but in some sense, no, you're right, but at the tools that mathematics are there. Sorry, I don't want to fight, but I think the tools that mathematics are exactly there to enable you to think about what you cannot visualize in this way. Let me give it, let's always make things easier. Go down or 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 along 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 and 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 not floating in three-dimensional space. The person who lives on that not, to them, it's a circle. Yeah. 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 not it up, but that's a that's a 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.

I don't know how to stay. 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. It does. I don't lose one minute of sleep over it.

No, I think you can tell. If you live, it's 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 mobile strip. Well, no, because you won't see your friend, right? Okay, fair point fair point on that.

Would you really know your point is right? Let me try to think of a better. If I could do this.

That has the feature that there's a loop of string, which can't be pulled close. 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 the way you could know without leaving your world that something very funny is going on.

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.

People have thought about this. I mean, 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. We're all good. Yeah. And adult. You know, 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 there with him. But then the circle starts getting bigger and bigger and bigger and is like what the hell is going on? It's like a horror movie like for conventional 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 section, the part of it that you can see to him it looks like there's like this kind of bizarre being, it'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 this fear is like I'm a sphere I'm from the third dimension the squares 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 did you think that kind of processes

Oh, I certainly think that's possible. Now, how would you figure out whether it was true or not has another question? 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 pause it however many entities that

Yes, but, 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 this story some context, you ask, like, is it possible for us humans to have this experience of being transcendental, transcendental, jerked out of our world? So he means sort of 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 sort of a theologian is saying like, oh, what if a higher being could like pull you out of this earthly world you live in so that you can sort of see the truth and like 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 skills, 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 from the square is like, okay, I get it now, so like, 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, like, let's figure out what a four dimensional version of me would be like. And then this year is like, what the hell are you talking about? There's no fourth dimension. That's ridiculous. Like the three dimensions. Like that's how many there are. I can see. I mean, 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 it's in fact not beyond our capabilities. And maybe beyond our cognitive capabilities to visualize a four dimensional cube, a test or act as something like the 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, that'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 there, 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. 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 know, you really like people really learn something about each other. And people get heated.

Sure. So I think, you know, most people The zero-holeers are rare. They would say, like, well, look, you can make a straw by taking a rectangular piece of plastic and closing it up. A rectangular piece of plastic doesn't have a hole in it. I didn't poke a hole in it when I knew. So how can I have a hole? It's just one thing. Okay. Most people don't see it that way. Is there any truth to that kind of concept? Yeah, I think that would be somebody who's a count. 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 shmushing 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?

So you know, one whole people would say, look, these are like groups of people.

You know, what's they look? There's like a hole. It goes all the way through the straw, right? It's one region of space. That's the hole. Yeah. And there's one. And two whole 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, a bottle of water, I'm holding. I'm gonna 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 goes 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 one hole at 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 on it. And then you think there's still one hole, even though there was one hole on it. And then you think there's still one hole, even though there was one hole on it.

And then you think there still one hole, even though there was one hole on it. And then you think there still one hole, even though there was one hole on it. And then you think there still one hole, even though there was one hole on it. And then you think there still one hole, even though there was one hole on it. And then you think there still one hole on it. And then you think there still one hole, even though there was one hole on it. And then you think there still one hole on it. And then you think there still one hole on it. And then you think there still one hole We can make the problem simpler. That's what we were doing a minute ago and you were talking about high dimensional space. And I was like, let's talk about like circles and lines, I can slowly go down and dimension and 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 from it 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. No, I'm not going to do it. I don't know. You just have to imagine the pair. Sorry. Yeah. Like, if you want, no, okay, no.

So many people would say there's three holes in the 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.

I mean, that's one color for the straw. This is a one hole for the straw to 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 holds whatever we may mean by them. There are somehow things which haven't 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 what 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 for the for the straw, I think what a modern mathematician would say is like the first version would be to say like well, they're 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 I 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.

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. be the waste of the pants.

Is that your weekly routine or just in preparation for talking about geometry for three hours? Exactly.

Exactly, so this idea that I mean, I think, you know, Ponca Ray really had these ideas, this sort of modern idea. I mean, building on stuff that other people did, Betty is an important one of this kind of modern notion of relations between holes. But the idea that holes really had in arithmetic, the really modern view was really, I mean, there are two ideas. So she kind of comes in and sort of truly puts the subject honest modern footing that we have that we have now so you know it's always a challenge you know in the book I'm not gonna 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 biology it's not like this and it's always a you know 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 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.

Yeah, I mean, 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, just different distant features of it.

It's fantastic. I mean, the flowering of math YouTube is 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're depending on the class. It could be 30 people. It could be 100 people. It could God help me be a 500 people if it's like the big calculus lecture or whatever it may be. And there's sort of something, but there's some set of people of that order of magnitude. And I'm with them for 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 that they have one on one questions, blah, blah, blah. That'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, 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, you know, three to five years. And the ultimate high level of engagement to one person. You know, books, I can, this can get to a lot more people that are ever going to sit in my classroom and you spend like, however many hours it takes to read a book somebody like three blue one brown or number file or um people like by heart 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 no mama is gonna 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 though he looked like Eugenia Chang, there's a wonderful category of theorist in Chicago. I mean, I 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 that sort of say something about mathematics to like untold. millions of people so everywhere along this curve is important one thing I feel like it's 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 is a writer where I sort of spent a lot of time 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. Yeah. Right.

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 are system 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 a traditional math academia. They did an entire theme semester about visualizing mathematics, looking to 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.

No.

Why not? Is that like, that's like my capture.

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 interested in like how it can like help feed the world? or else I'm like, there's a problem that I'm like, can I 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 and 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 invariance of this polynomial ring under the action of some finite group. Doesn't matter what those words mean, just it's like something that an 1890 would take a person a year to do and would be a valuable thing to 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 Macauley 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.

Oh, those are computer algebra programs. So those are like sort of bespoke systems that lots of mathematicians use as similar to Maple and yeah. Oh, yeah. So it's similar to Maple and Mathematica. Yeah. But a little more specialized. But yeah.

They're not really built for that, and that's a whole other story.

Right. They are now for most mathematicians, I would say, part of the process of mathematics. And so, you know, the story I told in the book, which I'm fascinated by, which is, you know, so far attempts to get AI's to prove interesting theorems have not done so well. It doesn't mean they can. It's actually a paper I just saw, which is a very nice use of a neural net defined 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 contractors. Yes, okay. So like somebody wrote this down, maybe this is so. Looking at what the AI came up with, you're like, you know, a bed of like, Five grad students had thought about that problem. 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, okay, we're gonna go back to knots. It's cool. There's a knot called the Conway knot. After John Conway, maybe we'll talk about it very interesting character also.

Oh, when I am sorry that you didn't get a chance because having had the chance talked a lot, when I was a postdoc, yeah, you missed out. There's no way to sugarcoat it. I'm sorry that you didn't get that chance.

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 slight. So it has to do with, um, something about what kinds of three-dimensional services and four-dimensions can be bounded by this not. But never mind what it means. It's some question, and it's actually very hard to compute whether or not is slice or not. In particular, the question of the conway not, whether it was slice or not, was particularly vexed. Until it was solved, just a few years ago, by Lisa Picker-Rillow, 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 it was an Austin connection to the story that 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 very short, it was very short and simple. Nine pages of which two were pictures. very short for 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, well, no, evidently it was difficult because like the world's top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top top

Why do you think you want those questions to have simple answers?

Yeah, so, right, let me just say the background because I don't know if everybody listening is, no, no is the story. So, you know, Fermat? was an early number theorist. I was going to have an early mathematician that specialization 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 try to top each other and vex each other with questions and stuff like this. But this particular thing. It's called for Mazlas theorem, because it's a note he wrote in his copy of the Disquishionist arithmetic. 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. But he is just like writing a note to himself. Now, let me just say historically, we know that Vermont did not have a proof of this theorem. For a long time, people were like, this mysterious proof that was lost in a very romantic story, right? Fair mouth 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.

Maybe, but you're right that that is a noble, 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. I understand the appeal of saying, like, wouldn't it be cool if it's very simple equation? There was like 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 are from our road about the method of descent, which is like very wonderful and important. But in the end, those are nice things that, like, you know, you teach an undergraduate class. And it is what it is, but they're not big. Um, on the other hand, work on the Fermat problem. It's what we like to call it because it's not really his theorem because we don't think you proved this. So, I mean, work on the Fermat problem developed this like incredible richness of number theory that we now live in today. Like, and not, by the way, just wild 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 Lamay from the 19th century where Comer in understanding what mistake Lamay had made in this incorrect proof basically understand 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. If we think of a number like 12, 12 is two times three times two. I had to think about it. Or it's two times two times three, 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 five, or in which there's like four, three's in it. Nope, 12 is like two, two's in a three. 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.

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 and never into primes is much easier. Essentially, you could know it all though. He didn't quite put it in that 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 irrelevant to Forma, all of a sudden this stops being true. Things get I mean, things get more complicated and now because you're 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 of 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 four complications. I'm a highly in favor of complication. And every complication is like an opportunity for new things to study.

whether this is interesting human aspects to the proof itself as 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 Mayzer.

I can speak to what it's like.

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 the Sunday.

Yes, and this is a perfect example. So this is another phrase of punk array, this incredible generator of slogans and aphorisms. He said mathematics is the art of calling different things by the same name. That very thing, 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, when you study from our last theorem, and let's not even be too careful about what these objects are. I can tell you there are gaural representations in modular forms, but saying those words. is not going to mean so much. But whatever they are, there are 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. Deformation means moving something just a tiny bit that can infinitesisable amount. If you really are good at understanding which ways a thing can move in a tiny, 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. They're 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 your notion of distance, from calculus. We know what it means for like two points in the real line to be close together. get 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 being listening which we do that all the time we talk about somebody being a close relative. It doesn't mean they live next door to you, right? 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, you know, in the natural language processing community in AI, there might be some notion of semantic distance or electrical distance between two words. How much do they tend to arise in the same context? That's incredibly important for, you know, doing auto-complete 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, right? It's a different kind of distance. Okay, ready? In this kind of number theory, there was a crazy distance called the Piatic 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. 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 attic number theorist, we'd say, oh, two numbers are close. If their difference is a multiple of a large power of two. 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 wanted them what's really far from one, like one and 164th. Because their difference is a negative power of two. Two to the minus six, so those points are quite, quite far.

Yeah, so two to a large power is a very small number and two to a negative power is a very big number.

It takes practice. It takes practice. If you've ever heard of the canter 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 the satisfy 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

I was exactly the right way to think of it.

I think there's very similar, that's exactly where we would think of it. It's almost like binary numbers written in reverse, because in a binary expansion, two numbers are closed. A number that's small is like 0.00 something. Something that's the decimal and it starts with a lot of zeros. In the two-addict metric, a binary number is very small. If it ends with a lot of zeros, and then the decimal point, got you. So it is kind of like binary numbers written backwards is actually, oh, I should have said, that's what I should have said, Lex. That's like very good metaphor.

Yeah, anything. And so because that's the kind of deformation that comes up. in wilds is is in wilds is proof that definition we're moving something a little bit means a little bit in this to add exactly 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 it like reformulating wise

Yes, although 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 to, 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, so wonderful, wonderful essay by, 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 pointed 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, because think about it. Okay, here's it. I mean, now we're gonna talk about AI, which you know a lot more about than I do. So just, you know, start laughing up royally, if I say something that's completely wrong.

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 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 the 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 yeah right if you try to do natural language processing and your idea about of distance between words is how close they are in the dictionary when you write them in alphabetical order you were going to get pretty bad translation right no the notion of distance has to come from somewhere else yeah that that's essentially when your own networks are doing this will order meetings are doing is yes coming up with uh... In the case of word about it is literally, literally what they are doing is learning a distance.

Again with this simple.

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? Okay. Sure. So there's some truth in it. But it's not true.

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, 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 bringing me to his writing. He liked this sort of say stuff, like stuff that sounded good. You know what I mean? So 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 literally mean it.

And I love poems. What a poem is not an explanation.

No poet would disagree with it.

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.

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, we would understand it as an explanation at all. By the way, okay, I gotta give, yeah, one more amazing point correct, because this guy just never stopped coming up with great quotes that, um, you know Paul Airdish another fellow who appears in the book and by the way he thinks about this notion of distance of like personal affinity kind of like you were talking about the kind of social network and that notion of distance that comes from that so that's something that Paul Airdish did well he thought about distances in that work because I guess he didn't probably he didn't think about the social work.

It's hard to distract. But you know Airdish was sort of famous for saying and this is sort of one lens we're 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, there was a, I'm, yeah. but punk array in the other hand. And by the way, there are other methods. Hilda Hudson is one who comes up in this book. She also kind of saw a math. She's one of the people who sort of develops. The disease model that we now use, that we use to sort of attract pandemics, this SIR model, that sort of originally comes from her work with Ronald Ross. But she was also super, super devout. And she also sort of from 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 portable. She says, you know, math isn't 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.

All right, so like it's not clear that the, I'm sorry you just sent me off on a tangent just imagining like airdish at a Hendrix concert. Trying to figure out if it was from the book or not. What I was coming to you was just a sip, but one point, all right, I said about this is he's like, you know, if like, this is all worked out in the language of the divine and if a divine being like came down and told to do us, we wouldn't be able to understand it, so it doesn't matter. So Punk array was at the view that there were things that were sort of like in humanly complex, and that was how they really were. Our job is to figure out the things that are not like that. They're not like that.

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. I mean, you're like, wait, I can't factor five. Five is five times one. Okay, not like that. That is the factorization. It absolutely is a way of expressing five as a product of two things. But don't you agree that there's something trivial about it? It's something you can 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 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 times 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.

So what's the definition is best for us?

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 them. 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 the non because if they were even, I get fact right two out of them. But it's not all the odd numbers. Nine isn't prying because it's three times three. Fifteen isn't prying because it's three times five, but thirteen is where we're ready. Two, three, five, seven, eleven, thirteen, seventeen, nineteen. Not twenty one, but twenty three is, et cetera, et cetera. Okay, so you could go on.

I think so. There's always those ones that trip people up. There's a famous one, the Grotindique Prime 57, like sort of Alexander Grotindique, the great algebraic geometry, sort of giving some lecture involving a choice of a prime in general on somebody said, like, can't you just choose a prime and you 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. Um, that's one of them. But I'm

Yeah, so I would say 57 and 507 and 501 are definitely like prime offenders. Oh, I didn't do that on purpose. Oh, well done didn't do it on purpose. Anyway, there definitely ones that people or 91 is another classic seven times 13. It really feels kind of primed, doesn't it, but it is not Yeah. But 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 to devise 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. Now we 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 passed the test, but are not prime, 341.

Yes, actually. Um, ready. Let me give you the simplest version of it. You can dress it up a little bit, but here's the basic idea. Uh, I take the number, the mystery number. I raised two to that power. So let's say your mystery number is six. Are you sorry you asked me?

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 going to be like that's two cube squares 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 say 64 divided by 6 is, there's a quotient of 10, but the remainder is 4. So you failed because the answer has to be 2. For any prime, let's do a 5, which is prime. 2 to the fifth is 32 divided 32 by 5, and you get 6 with the remainder of 2. The remainder of two here seven two to the seventh is one twenty eight divide that by seven and let's see I think that seven times fourteen is that right now seven times eighteen is one twenty six with a remainder of two right one twenty eight is a multiple seven plus two so if that remainder is not two And then that's definitely not prime.

It's likely a prime but not for sure. And it's actually a beautiful geometric proof which is in the book actually. That's like one of the most granular parts of the book because it's such a beautiful proof I couldn't not give it. So you draw a lot of like. Opal and pearl necklaces and spin them. That's kind of the geometric nature of the of this proof of firm as little theorem. So yeah, so with pseudo-primes, there are primes that are kind of faking that they pass that test, but there are numbers that are faking it that pass that test, but not actually prime. But the point is, there are many, many, many theorems about prime numbers.

Yeah, it's a perfect example of your desire for simplicity in all things. You know, would be really simple. If there was only finally 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 three would be totally different if that were the case. It's just not true. In fact, this is something else that you could news. This is a very, very old fact, like much before long before we had anything like modern numbers. The prime's are infinite. The prime's that there are that right.

Right. So one thing that people recognized and really thought about a lot is that the prime's on average seem to get farther and farther apart as they get bigger and bigger. In other words, it's less and less common. I already told you of the first 10 numbers, two, three, five, seven, four 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 rare. 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, a 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 three and five 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, why, how could one possibly have a right to have an opinion about something like that? like what you know we don't have any way of describing a process that makes primes like sure you can like 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 right maybe I can go on the computer and find 10 million what 10 million is pretty far from infinity right so how is that how is that evidence there's a lot of things there's like 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 random. There's no chance involved. There'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 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 random this a little bit different and tweak it, and see if you can find the 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 at 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 conductor is that we believe the primes are like sort of screwed 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 to behave just like they would if they were random dirt.

Well, not an order to prove 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 guessing making a guess 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 are like prime numbers are completely deterministic. And I'm saying like, 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, OK, great. You hold to your insistence on a random process. Meanwhile, I'm generating insight about the problems 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, 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 a look what you work on machine learning? Don't you find yourself sort of talking about what the machine is, what the machine is trying to do? in a certain instance, do you not find yourself drawn to that language? Well, it knows this, it's trying to do that, it's learning that.

I mean, I think, okay, so first of all, I'm not an expert. I couldn't even tell you what the difference is between those three terms, finitism, ultra-finitism and intuitionism, although I know it, they're related and I tend to associate them with the Netherlands in the 1930s.

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.

A bit ultra-friendly doesn't came second. I bet it's like when there's like a hard core 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 hard core than the hard core people.

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 canter, 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 DIY of a DX, if you sort of go back to your calculus class with those who've taken calculus and 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 like 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 infinite testimonials. 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. I 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 smart line of Newton. And on the one hand, he was right. It wasn't really rigorous, I'm understand it. On the other hand, like Newton was out there doing calculus. Out of the people were not, right?

New words. I think I think a sort of intuition is few, for instance, I would say would express serious doubt. And it's not about the way it's not just infinity. 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.

That's a great point, actually. 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 and flavor about that. And it's probably not a coincidence that it becomes popular in the 30s and 40s, which is also kind of like the dawn of ideas about formal computation, right? You probably know the timeline better than I do.

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, you know, number represents a magnitude, like the length of a line. Like, so I mean, the idea that there's a continuum. There's like, there's like, is pretty old, but that, you know, just because nothing is old doesn't mean we can't reject it if we want to.

Yeah, I mean, I love this. I've been really done researching it myself. I've played around with it. I'll send you a fun blog post. I wrote what I made some cool texture patterns from Sawyer on top of that. Um, but, um,

It doesn't mean you see that it's a great paper. I don't know if you saw this. Like a machine learning paper. Yes. I don't know if you saw the one I'm talking about where they were like learning the texture is like let's try to like reverse engineer and like learn a cellular automaton that can reduce 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 remissioning learning paper is that what's especially interesting is the case is where it doesn't work. Like, what does it do when it doesn't do the thing that you tried to train it? Yeah, to do. That's extremely interesting. Yeah, that was a cool paper. So yeah, so let's start with the game of life. Let's start with John Conway. So Conway.

I think he's pretty recognized.

Well, I mean, I'm super, he was a full professor of Princeton for most of his life. He was sort of certainly the pinnacle of.

I was about to tell you, I feel like you have a very elevated idea of how famous, but this is what happens when you grow up in the Soviet Union or you think the mathematicians are like very, very famous.

Yeah, and Conway is such a perfect example of somebody whose humanity was, and his personality was like a wound up with his mathematics, right? It's what sometimes I think people who are outside the field think of mathematics is this kind of like, Cold thing that you do separate from your existence is 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 walked 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 may think I'm gonna say and it's true is that he's sort of was very playful in his way of doing mathematics, but it's also true and went both ways. He also sort of made mathematics out of games. He looked at, it 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 call. I mean, there are his own books, and then there's like a book that he wrote with Burley Camping 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 infinitesimal's, the ghosts of departed quantities. They're in there, 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 a Princeton, and our research overlapped in some ways. Now it was on stuff that he had worked on many years before, and the stuff I was working on, kind of connected with stuff in group theory, which somehow keeps him to keep coming up. And so I often, but like sort of ask more questions, I would sort of come upon them in the common room and I would ask more questions about something. And just anytime you turned him on, you know what I mean, you sort of asked the question, it was just like turning a novel and winding him up and he would just go and you would get a response that was like, So rich and was so many places and taught you so much. And usually, I'd nothing to do with your question. Yeah. Usually, your question was just a prompt to him. You couldn't count on actually keeping the question.

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. But essentially, it's a game that you play with marking little squares on the Shindu graph paper. In the 70s, I think he was literally doing it with a pen on graph paper. You have some configuration of squares, some of the squares on the graph paper are filled in, some 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 a race, that's called death, and there's rules for which squares are born, which squares die. The rules 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. get these results of incredible richness and of course now it is you don't do it on paper nowadays you do in a computer it's actually a great iPad app called galley which I really like that has like Convoy'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 that if this is at all intriguing to you getting golly on your IOS device and you can do this kind of process which I really enjoy doing which is almost like putting a Darwin head on or a biologist head on and doing analysis of

So it's a wonderful laboratory and it's kind of a rebuked 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 picked like some random rule You wouldn't get interesting complexity. I think that's one of the most interesting things of these 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. Yeah, but some do. 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.

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 and mathematically than that. And I think it always like agreed to him a bit, but he was like the game of life guy. When he proved all these wonderful theorems and like, I mean, I created all these wonderful games, like, created the sort of numbers. I mean, he did, I mean, He was a very tireless guy who just like did like an incredibly, very agitated array of stuff. So he was exactly the kind of person who you would never want to like reduce the like one achievement. You know what I mean?

Well, so I would say group theories sort of starts as the general theory of symmetries 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 the symmetry from left to right. Like a human being, right? That's roughly bilaterally symmetric, as we say. So there's two symmetries. I mean, 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 at 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 sciences and is insorting because the ways of permuting a set. The ways of taking sort of some set of things you have in 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 the deck of card can be. And 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 it a double shop bowl, 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. Oh, yeah. Okay. Well, okay, ready. Think about the symmetries of the line. You're like, okay, I can reflect it. left to write, you know, around the origin. Okay, but I could also reflect it left to write, grabbing somewhere else, like at one or two or pie 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.

Well, I'm not a mathematical physicist, but I can say this. It is certainly true that it's been a very useful notion in physics to try to say, like, What are the symmetry groups like 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 I theorem this 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 and 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 is an oversimplification but like one way of thinking of this big shift from, uh, before Einstein to after is that we just changed our idea of what the fundamental group of symmetries were. So that things like the Lorenz contraction, things like these bizarre relativistic phenomenon or Lorenz would have said, oh, to make this work, we need a thing to, um, to change its shape. If it's moving, Yeah, nearly a speed of light. Well, under the new frame of framework, it's much better to like, oh, no, it wasn't changing in state. You were just wrong about what counted as a symmetry. Now that we have this new group that's so called the Rends group. Now that we understand what the symmetries really are, we see it was just an illusion that the thing was changing in state.

That's, I mean, that 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 mean, it's not a field I know well, but it certainly seems like. Yes, it is like that.

I mean, you could say that, but I would say for me a picture of the hot vibration does nothing for me. When you're like, oh, it's like about the quaternions. It's like a subgroup of the quaternions. I'm like, oh, so now I see what's going on. Why didn't you just say that? Why are you like showing me this stupid picture instead of telling me what you were talking about?

I'm just saying nobody goes back to what you're saying about teaching that like people are different and 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. 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 gonna find different things illuminating.

Can you look, let me make a crazy metaphor, maybe it's a little bit like the relation between pros and poetry, right? I mean, if you might say, why do we need anything more than pros? You're trying to convey some information, so you just say it. Well, poetry does something, right? You might think of it as a kind of compression. Of course, not all poetry is compressed, not awesome. 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 and prose into sort of a different mode that is going to hit in a different way.

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 in 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, punk array 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. Paramount 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 Comer, who I mentioned in the 19th century, but Gerhard Fry and Mazer and Ken Ribbit, and like many other people, are involved in building the other pieces of the arch before you put the keystone in.

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. 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 yes there's a mechanism called regi flow and again we're outside my research area so for all the geometric analysts and differential geometers out there listening to this if I please I'm doing my best and I'm roughly saying so this the regi flow allows you to say like okay let's start from some mystery three-dimensional space which punk array 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 let it move and it's 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.

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 the geometry of all geometries. So to speak, and it's only by, and kind of like, you've kind of like being jerked out of flat land, right? Same idea. It's only by sort of seeing the whole thing globally or once that you can really make progress on understanding the one thing you thought you were looking at.

Well, it's interesting because on the one hand, I think it's absolutely true that right in some kind of vast spiritual sense, like awards are not important, like not important the way that's sort of like understanding the universe is important. On the other hand, most people who are offered that prize accepted, you know, so there's something unusual about his... 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, it's not, 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 Schultzer like turned down sort of some big monetary prize because he just, you know, I mean, I think he at some point you have plenty of money. And maybe you think it's in the wrong message about what the point of doing mathematics is. Um, I do find that there's most people except, you know, most people are giving a prize, most people take it. I mean, people like to be appreciated, but we're like I said, where people, yes, not that different from most other people.

It's interesting, because I think... But do you know that you turned down the prize in services, some principle?

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 the international mathematical Olympiad. It's a good competition for high school students solving math problems. And in some sense, it's absolutely false to the reality of mathematics, because it 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 very motivated by those competitions. And like, I went to the mathland be at, and you won it. I wasn't, I mean, well, there's something I had 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 in each event. I just have to emphasize that the International Math Olympia 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.

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, when in 1987 the first year I went, it was in Havana. Americans couldn't go to have had a fact that 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 a van so we could see what America was all about.

I mean, the thing is it's in part a journey of self-knowledge. You have to know what's going to work for you and that's going to be different from 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, you know, I always recommend like the books of Martin Gardner or another sort of person we haven't talked about, but he also like Conway and Bodies, 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 these books, the cons are collecting the 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 you know working contest style problems to those are bound to books like especially like Russian and Bulgarian problems right there's book after book problems from those context that's going to motivate some people for some people it's going to be like watching well produced videos like a totally different format like I feel like I'm not answering your question I'm sort of saying there's no one answer and like it's a journey where you figure out what resonates with you for some people it's 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 what's more, and going to grad school means just coming face to face with like the incredible overflowing vault of your ignorance, right? So I told Joe Harris, who was an algebraic jameter, a 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 Text book, and I'm all on to jam three algebraic, like the elements of algebraic geometry. I'm just I feel like I don't know enough. So I just going to sit and like read this like 1500 page many volume. Yeah, book. Um, and he was like, Professor Harris, like, that's a really stupid idea. And I was like, why is that a stupid idea? Then I would know more algebra geometry. So because you're not actually going to do it like you learn. I mean, he knew me well enough to say, like, you're going to learn because you're going to be working on a problem and then there's going to be a fact from HGA, 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 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. Um, 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 machine learning problem you're trying to handle. Let that not kind of abstract the idea of what the curriculum is. Drive your mathematical learning.

And I see you up to enjoy the dead ends, but I think you have to accept the dead ends. Let me let's put it that way.

It pisses me off. It's not that it's part of the process.

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 lotting that he's like, no, I'm gonna eat the 30th hot dog.

And you know, I have kids, so this is actually a live issue for me, right? I actually, it's not a father's room, and I actually do have to give advice to two young people all the time. It'll 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, but I say 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 really want to do. I always 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 to make the high self-esteem choice, make the choice sort of take yourself out of it and like If you didn't have the, you can probably figure out what the version of you feels completely confident would do. And do that and see what happens. And I think that's often like pretty good advice.

Right, but it doesn't mean I would never say to somebody, you should just go have high self esteem. Yeah. You shouldn't have doubts. Now 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.

I mean, look the way I feel is that, you know, mathematics, as we've discussed, like it underlies the way we think about constructing learning machines and underlies physics, it can be you. I mean, 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. Like ask a rabbi. No, I mean, yeah, you know, I wrote a lot in the last book, how not to be wrong. Yeah, 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 wage or I mean he's probably among all mathematicians he's the one who's best known for this can you actually like apply mathematics to kind of these transcendent questions. But it'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 and he almost kind of says, the opposite if you ask blaze Pascal like why do you believe in god it's he'd be like oh cuz I met god you know you had this kind of like psychedelic experience is like a mystical experience where as he tells it he just like directly in counter god it's like okay because 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 can 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, God's there, he'll tell you, you know, you don't eat all of this.

Yes, this is what I was trying to keep my head all at once. Well, I was writing and I probably should have drawn this picture earlier on the process. Maybe it would have made my organization easier. I actually drew it only at the end.

Right on the edge, right on the edge, not on the center.

Thank you, Lex. We went to some amazing places today. This is really fun.