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

Episode Date: June 13, 2021

Jordan Ellenberg is a mathematician and author of Shape and How Not to Be Wrong. Please support this podcast by checking out our sponsors: - Secret Sauce: https://wondery.com/shows/secret-sauce/ - ExpressVPN: https://expressvpn.com/lexpod and use code LexPod to get 3 months free - Blinkist: https://blinkist.com/lex and use code LEX to get 25% off premium - Indeed: https://indeed.com/lex to get $75 credit EPISODE LINKS: Jordan's Website: http://www.jordanellenberg.com Jordan's Twitter: https://twitter.com/JSEllenberg PODCAST INFO: Podcast website: https://lexfridman.com/podcast Apple Podcasts: https://apple.co/2lwqZIr Spotify: https://spoti.fi/2nEwCF8 RSS: https://lexfridman.com/feed/podcast/ YouTube Full Episodes: https://youtube.com/lexfridman YouTube Clips: https://youtube.com/lexclips SUPPORT & CONNECT: - Check out the sponsors above, it's the best way to support this podcast - Support on Patreon: https://www.patreon.com/lexfridman - Twitter: https://twitter.com/lexfridman - Instagram: https://www.instagram.com/lexfridman - LinkedIn: https://www.linkedin.com/in/lexfridman - Facebook: https://www.facebook.com/lexfridman - Medium: https://medium.com/@lexfridman OUTLINE: Here's the timestamps for the episode. On some podcast players you should be able to click the timestamp to jump to that time. (00:00) - Introduction (06:44) - Mathematical thinking (10:21) - Geometry (14:58) - Symmetry (25:29) - Math and science in the Soviet Union (33:09) - Topology (47:57) - Do we live in many more than 4 dimensions? (52:28) - How many holes does a straw have (1:01:53) - 3Blue1Brown (1:07:40) - Will AI ever win a Fields Medal? (1:16:05) - Fermat's last theorem (1:33:23) - Reality cannot be explained simply (1:39:08) - Prime numbers (2:00:37) - John Conway's Game of Life (2:12:29) - Group theory (2:15:45) - Gauge theory (2:23:47) - Grigori Perelman and the Poincare Conjecture (2:33:59) - How to learn math (2:41:08) - Advice for young people (2:43:13) - Meaning of life

The following is a conversation with Jordan Ellberg, a mathematician at University of Wisconsin, and an author who masterfully reveals the beauty and power of mathematics in his 2014 book, How Not to Be Wrong, and His New Book, just released recently called Shape. The Hidden Geometry of Information, Biology, Strategy, Democracy, and Everything Else. Quick mention of our sponsors. Secret sauce, ExpressVPN, Blinkist, and everything else. Quick mention of our sponsors. Secret sauce, express VPN, blinkist, and indeed. Check them out in the description to support this podcast. As a side note, let me say that geometry is what made me fall in love with mathematics

when I was young. It first showed me that something definitive could be stated about this world through intuitive, visual proofs. Somehow that convinced me that math is not just abstract numbers devoid of life, but a part of life, part of this world, part of our search for meaning. As usual, I'll do a few minutes of ads now. I try to make these interesting, but I give you time stamps, so if you skip, please still check out the sponsors by clicking the links in the description.

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Anyway, right now get a free $75 sponsored job credit to upgrade your job post at indeed.com slash Lex, get it at indeed.com slash Lex, terms and conditions apply, offer valid through June 30th, indeed.com slash Lex. This is the Lex Friedman podcast, and here is my conversation with Jordan Allenberg.

If the brain is a cake, it is. Well, let's just go with me on this. Okay. We'll pause it. So for Noam Chomsky, language, the universal grammar, the framework from which language springs is like most of the cake, the delicious chocolate center. And then the rest of cognition that we think of is built on top, extra layers, maybe the icing on the cake, maybe just, maybe consciousness is just like a cherry on top.

Where do you put in this cake mathematical thinking? Is it as fundamental as language? And the Chomsky view, Is it more fundamental in language? Is it echoes of the same kind of abstract framework that he's thinking about in terms of language that they're all really tightly interconnected? That's a really interesting question. You're getting me to reflect on this question

of whether the feeling of producing mathematical output if you want is like the process of, you know, honoring language 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 and propositions. But maybe it's just because that's the way imagine doing mathematics without talking about mathematics and sort of thinking and propositions. But, you know, maybe it's just because that's the way I do mathematics, and maybe I can't imagine it any other way, right? It's a... Well, what about visualizing shapes,

visualizing concepts, to which language is not obviously attachable? Ah, that's a really interesting question. And, you know, one thing that 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 Basca of the Pythagorean Theorem. Proofs which are purely visual.

Proofs where you show that two quantities are the same by taking the same pieces and putting them together one way and making one pieces and putting them together one way and making one shape and putting them together another way and making a different shape. And then observing those two shapes must have the same area because they were built out of the same pieces. You know, there's a famous story and it's a little bit disputed about how accurate this is, but that in Boscow's manuscript, he sort of gives this proof, just gives the diagram, and then the entire verbal content of the proof is he just writes under it. Behold! That's it. There's some dispute about exactly how accurate that is, but so then that'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 together in two different formations to make two different things, is that language or not sure how to answer what do you think? I think it is. I think the process of manipulating the visual elements is the same as the process of manipulating the elements of language. And I think probably the manipulating, the aggregation, the stitching stuff together is the important part. It's not the actual specific elements.

It's more like to me, language is a process, and math is a process. It's not just specific symbols. It's in action. It's ultimately created through action, through change, and so you're constantly evolving ideas. Of course, we kind of attach, there's a certain destination you arrive to that you attach to and you call that a proof, but that's not, that doesn't need to end there. It's just like the end of the chapter and then it goes on and on and on and that kind of way. But I got to ask you about geometry and it's a prominent topic in your

new book shape. So for me, geometry is the thing, just like as you're saying, made me fall in love with mathematics and I was young. So being able to prove something visually just did something to my brain that ads planted this hopeful seed that you can understand the world. Like perfectly, maybe it's an OCD thing, but from a mathematics perspective, like humans are messy, the world is messy, biology is messy, your parents are yelling or making you do stuff, but you know, you can cut through all that BS and truly understand the world through mathematics and nothing like geometry did that for me. For you, you did not immediately fall in love with geometry. So how do you how do you think about geometry? Why is it a special field in mathematics? And how did you fall in love with it? You have. Wow, you've given me like a lot to say. And

certainly the experience that you described is so typical, but 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 or leave. But then at this one moment, it made sense.

This class made sense why I wasn't at all like that. There's other people I I can tell you, because they come and talk to me all the time, who are like, I understood all the stuff where you were trying to figure out what X was, or 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? 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. So, 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, a six by eight array of holes. And I was just kind of 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. Whoa. So eight sixes and six eights,

it's just like the dissection boost 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 cost on your podcast, I don't know if that's, we have to see if I can relate it. Okay, it was like unbelievable to me. I like to cost on your podcast. I don't know if that's we have to see if I can get it.

Okay, it was fucking unbelievable. Okay, that's the last time. Get it in there. This story merits it. So two different perspectives and the same physical reality. Exactly. And it's just as you say, um, you know, I knew the six times eight was the same as eight times six, where I knew my times tables., I knew that that was a fact.

But did I really know it until that moment? That's the question. Right? I knew that I sort of knew that the times table was symmetric, but I didn't know why that was the case until that moment. And in that moment, I could see like, oh, I didn't have to have somebody tell me that. That's information that you can just directly access. That's a really amazing moment.

And as math teachers, that's something that we're really trying to bring to our students. And I was one of those who did not love the kind of Euclidean geometry, ninth grade class of like prove that an isosceles triangle has equal angles at the base. Like this kind of thing, it didn't vibe with me the way that algebra and numbers did. But if you go back to that moment,

from my adult perspective, looking back at what happened with that rectangle, I think that is a very geometric moment. In fact, that moment exactly encapsulates the intertwining of algebra and geometry. This algebraic fact that, well, in the instance, 8 times 6 is equal to 6 times 8, 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 fated to be an algebraic geometry, which is what I am

as a researcher. So that's a kind of transformation, and you talk about symmetry in your book. What the heck is symmetry? What the heck is these kinds of transformation on objects that once you transform them, they seem to be similar? What do you make of it? What's its use in mathematics or maybe broadly in understanding our world? Well, it's an absolutely fundamental concept. And it starts with the word symmetry in the way that we usually use it when we're just like talking English and not talking mathematics, right? Sort of something is, when we say something is symmetrical, we usually means it has what's

called an axis of symmetry. Maybe like the left half of it looks the same as the right half. That would be like a left right axis of symmetry or maybe the top half looks like the bottom half or both, right? Maybe there's sort of a fourfold symmetry where the top looks like the bottom and the left looks like the right. Um, 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, it was actually.

Um, one of my first loves mathematics, what I thought about a lot when I was in college, but the notion of symmetry is actually much more general than the things that we would call symmetry if we were looking at like a classical building or a painting or or or something like that. You know nowadays in in math, we could use a symmetry to refer to any kind of transformation of an image or a space or an object. So what I talk about in the book is take a figure and stretch it vertically. Make it twice as big vertically and make it half as wide. That I would call a symmetry.

It's not a symmetry in the classical sense, but it's a well-defined transformation that has an input and an output, I give you some shape. And it gets kind of, I call this in the book a scotch. It just made that, I had to make up some sort of funny sounding name for it because it doesn't really have a name. And just as you can sort of study which kinds of objects are symmetrical under the operations of switching left and right or switching top and bottom or rotating 40 degrees or what have you, you could study what kinds of things are preserved by this kind of

scrawch symmetry and this kind of more general idea of what a symmetry can be. Let me put it this way. A fundamental mathematical idea. In some sense, I might even say the idea that dominates contemporary mathematics. Or by contemporary, by the way, I mean the last 150 years. We're on 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 like all the same side lengths and angles, and like looks exactly the same, the term of art, if you want it, you would say they're congruent. But one way of saying it is there's a symmetry called translation, which just means move everything three inches to the left. And we want all of our theories to be translation invariant. What that means is that if you prove a theorem about a thing that's over here and then you move it three inches to the left, it would be kind of weird if all of your theorems like didn't still work.

So this question of like, what are the symmetries and which things that you want to study or invariant into those symmetries is absolutely fundamental. Boy, this is getting a little abstract, right? It's not at all abstract. I think this is completely central to everything I think about in terms of artificial intelligence. I don't know if you know about the M-ness dataset with handwritten digits. Yeah. And, you know, I don't smoke much weed or any really, but it certainly feels like it when I look at M-ness and think about the stuff, which is like, what's the difference between one and

two, and why are all the two similar to each other? What kind of transformations are within the category of what makes a thing the same, and what kind of transformations are those that make it different? And symmetry's core to that. In fact, whatever the hell our brain is doing, it's really good at constructing these arbitrary and sometimes novel, which is really important when you look at like the IQ test, or they feel novel, ideas of symmetry of like what like Playing with objects. We're able to see things that are the same and not and Construct almost like little geometric Theories or what makes things the same and not and how to make

Programs do that in AI is a total open question and so I kind of stared and wonder a total open question. And so I kind of stared and wonder how what kind of symmetries are enough to solve the M-nist handwritten recognition problem and write that down. And exactly, and what's so fascinating about the work in that direction from the point of you of a mathematician like me and a Geometer, is that the kind of groups of symmetries, the types of symmetries that we know of are not sufficient. So in other words, we're just gonna keep on going into the weeds on this. The deeper the better. A kind of symmetry that we understand very well is rotation.

So here's what would be easy. If humans, if we recognized a 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 like write a program

that could detect whether something was a rotation of a fixed digit one. Whatever we're doing when you recognize the digit one and distinguish 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. There's so simple, these numbers. I mean, they're really geometric objects. Like we can draw out one, two, three.

It does seem like it should be formalizable. That's why it's so strange. Do you think it's formalizable when something stops being a two and starts being a three, where you can imagine something continuously deforming from being a two to a three? Yeah, but that's, there is a moment. I have myself a written program that literally morphed tos and threes and so on.

And you watch, and there is moments that you notice, depending on the trajectory of that transformation, that morphing, that it is a three, and a two, there's a hard line. Wait, so if you ask people, if you show them this morph, if you ask a bunch of people, do they all agree about where the transition to have question? I was surprised. I think so. Oh my God. OK, we have an empirical problem. But here's the problem. Here's the problem that if I just showed that moment that I agreed on. Well, that's not fair.

No, but say I said, so I want to move with an agreement because that's a fascinating actually question that I want to backtrack from because I just dogmatically said, because I could be very, very wrong. But the morphing really helps that like the change, because I mean, partially because our perception systems, see, this is all probably tied in there. Somehow the change from one to the other, like seeing the video of it allows you to pinpoint the place where it to becomes a 3-much better. If I just showed you one picture, I think you might really, really struggle.

You might call it 7. I think there's something also that we don't often think about, which is it's not just about the static image. It's the transformation of the image, or it's not a static shape. It's the transformation of the shape. There's something in the movement that seems to be not just about our perception system, but fundamental to our cognition, like how we think about stuff. Yeah, and you know, that's part of geometry too.

And in fact, again, another insight of modern geometry is this idea that you know Maybe we would naively think we're gonna study I don't know. Let's you know like punk ore We're gonna study the three-body problem. We're gonna 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 like put it in your computer and see what they do well I guess what that's like a problem that punk ore 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 Mungus mystery you just opened the door and we're gonna walk right in before we return to symmetry What's the who's Pwon punk are and what's what's this conjecture that he came up with? Why is this such a hard problem? Okay, so punk are a he ends up being a major figure in the book and I don't I didn't even really intend for him to be such a big figure but he's so he's um 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 flower. Actually, I learned a lot.

I mean, you know, in math, we're not really trained on our own history. We get a PhD in math, learn about math. So I learned a lot. There's this whole kind of moment where France has just been beaten in the Franco-Prussian war. And they're like, oh my god, what did we do wrong? And they were like, we 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 instantly like modernize. That this fascinating, the humans and mathematics are intricately connected to the history of humans. The Cold War is, I think, fundamental to the way people saw science and math in the Soviet

Union. I don't know if that was true in the United States, but certainly wasn't the Soviet Union. It definitely wasn't. I would love to hear more about how it was in the Soviet Union. I mean, there is a, and we'll talk about the Olympiad. I just remember that there was this feeling, like the world hung in a balance, and you could save the world with the tools of science. with the tools of science and mathematics was like the superpower that fuels science and so like

People were seen as you know people in America often idolize athletes, but ultimately the best athletes in the world They just throw a ball into a basket. So like there's not what people really enjoy about sports, and I love sports, is like excellence at the highest level. But when you take that with mathematics and science, people also enjoyed excellence in science and mathematics and Soviet Union. But there's an extra sense that that excellence will lead to a better world. that excellence will lead to a better world. So that created all the usual things you think about with the Olympics, which is like extreme competitiveness, right? But it also created this sense that in the modern era in America, somebody like Elon Musk, what are you?

You think of them like Jeff Bezos, those folks, they inspire the possibility that one person or a group of smart people can change the world. Like, not just be good at what they do, but actually change the world. Mathematics is at the core of that. I don't know. There's a romanticism around it too. Like when you read books about in America, people romanticize certain things like baseball, for example.

There's like these beautiful poetic writing about the game of baseball. The same was the feeling with mathematics and science and the Soviet Union, and it was in the air. Everybody was forced to take high level mathematics courses. Like, you took a lot of math. You took a lot of science and a lot of like really rigorous literature. Like, the lot of science and a lot of like really rigorous literature.

Like the level of education in Russia, this could be true in China. I'm not sure in a lot of countries is in whatever that's called. It's K-12 in America, but like young people education, the level they were challenged to learn at is incredible. It's like America falls far behind, I would say. America then quickly catches up and then exceeds everybody else. As you start approaching the end of high school to college, like the University of

the United States arguably is the best in the world. But like what we challenge everybody, it's not just like the good, the world. But like, what we challenge everybody, it's not just like the good, the A-students, but everybody to learn in the Soviet Union was fascinating. I think I'm gonna pick up on something you said. I think you would love a book called, Dual It Dawn by a mere Alexander, which I think some of the things you're responding to,

and what I wrote, I think I first got turned on to by a mere's work, he's a historian of math, and he writes about the story of Ever East Galwa, 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 in his early 20s, like all this stuff mostly unpublished. It's a very, very romantic story that we all learn.

And much of it is true, but Alexander really lays out just how much the way people thought about math in those times in the early 18th 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 were sort of reaching for sort of direct contact with the divine.

And so part of the reason that we think of Gala that way was because Gala himself was a creature of that era and he romanticized himself. I mean, now, now he wrote lots of letters and like, he was kind of like, I mean, in modern times, we would say he was extremely emo. Like that's, like, we wrote all these letters about his like, florid feelings and like,

the fire within him about the mathematics. And you 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 in whatever community we're in and we do it affected by the mores of this is of this idea around us. So the French, the Germans and Pancarré, yes, okay, so back to Pancarré. 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's, because those people are fun get into feuds or sort of have like weird enthusiasm. Um, because those people are fun to write about and they sort of like say very salty things.

Pancare 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 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 of this episode of it, as far as I could tell, a completely normal guy, but he just kind of, in many ways, creates the geometric world in which we live. And, you know, his first really big success is this prize paper he writes for this prize offered by the King of Sweden for the study of the three-body problem. The study of what we can say about, yeah, three astronomical objects moving and what you might think would be this very simple way. Nothing's going on except gravity relating to the three-body problem. Why is that a problem?

So the problem is to understand when this motion is stable and when it's not. So stable meaning they would sort of like end up in some kind of periodic orbid, or I guess it would mean, sorry, stable would mean they never sort of fly off far apart from each other and unstable would mean like eventually they fly apart. So understanding two bodies is much easier. Yeah, exactly. When you have a third, two bodies stay.

Third wheel is all the problem. This is what Newton knew. Two bodies, they sort of orbit each other and some kind of either in an ellipse, which is the stable case. You know, that's what the planets do that we know. Or one travels on a hyperbola around the other. That's the unstable case. It sort of like zooms in from far away, sort of like whips around heavier thing and like zooms out. Those are basically the two options. So it's a very simple and

easy to classify a story. With three bodies, just a small switch from two to three, it's a complete zoo. It's the first example of what's called chaotic dynamics, where the stable solutions and the unstable solutions, they're kind of like wounding 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 Poinca Ray was the first to recognize that that phenomenon even existed. What about the conjecture that carries his name? Right. So he also was one of the pioneers of taking geometry, which until that point had been largely the study of two and three dimensional objects because that's like what we see, right? That's those are the

objects we interact with. He developed a 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 and not, you can see what that name did not catch on. So now it's called topology now. Sorry, what was it called before? Analysis Cetus, which I guess sort of roughly means like the analysis of location or something like that. It's a Latin phrase. Partly because he understood that even to understand stuff that's going on in our physical world, you have to study

higher dimensional spaces. How does this work? And this is kind of like where my brain went to it because you were talking about not just where things are, but what their path is, how they're moving when we were talking about the path from two to three. He understood that if you want to study three bodies moving in space, well each body, it has a location where it is, so it has an x-coordinate,

a y-coordinate, a z-coordinate, right? I can specify a point in space by giving you three numbers, but it also, at each moment, has a velocity. So, it turns out that really to understand what's going on, you can't think of it as a point, or you could, but it's better not to think of it as a point in three-dimensional space that's moving. It's better to think of it as a point in six-dimensional space where the coordinates are, where is it, and what's its velocity right now? That's a higher-dimensional space called phase space. And if you haven't thought about this before, I admit that it's a little bit mind-bending.

But what he needed then was a geometry that was flexible enough, not just to talk about two dimensional spaces or three dimensional spaces, but any dimensional space. The sort of famous first line of this paper where he introduces an LCC to is no end doubts, nowadays that the geometry of end dimensional space is an actually existing thing.

Right, I think that had been controversial. And he's saying, like, look, let's face it, just because it's not physical, it doesn't mean it's not there. It doesn't mean we shouldn't study. It's interesting. He wasn't jumping to the physical interpretation. Like, it can be real even if it's not perceivable to human cognition.

I think that's right. I think, don't get me wrong. Juan Carre never strays far from physics. He's always motivated by physics. But the physics drove him to need to think about spaces of higher dimension and so he needed a formalism That was rich enough to enable him to do that and once you do that that formalism is also gonna 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 than half the dishes frame of mind if you have a

or but I think this is more than a fthetitions frame of mind. If you have a formalistic framework that seems really good and seems to be very elegant and work well and it includes all the physical stuff, maybe we should think about all of it. Maybe we should think about it. Maybe there's some gold to be mine there. And indeed, guess what? Before long there's relativity and there's space time and all of a sudden it's like, oh yeah, maybe it's a good idea. We already had this geometric apparatus like set up for like how

to think about four dimensional spaces like turns out there real after all as I said you know this is a story much told right in mathematics not just in this context but in many. I'd love to dig in a little deeper on that actually because I have some intuitions to work out. Okay my brain. Well I'm not a mathematical physicist, so we can work it out together. Good. We'll together walk along the path of curiosity. But, Poincare, conjecture, what is it?

Poincare conjecture is 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 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 Poincarei's theory, there are many possible, three-dimensional spaces. In the same way that going down one dimension to sort of capture our intuition a little bit more, we know there are lots of different two dimensional surfaces, right? There's a balloon and that looks one way and a donut looks another way and a mobius strip looks a third way. Those are all like two dimensional surfaces that we can kind of really get a global view of because we live in three-dimensional space.

So we can see a two-dimensional surface sort of sitting in our three-dimensional space. Well, to see a three-dimensional space whole, we'd have to kind of have four-dimensional eyes, right, which we don't. So we have to use our mathematical eyes, we have to envision. The Poincare conjecture says that there's a very simple way to determine whether a three-dimensional space is the standard one, the one that we're used to. And essentially, it's that it's what's called Fundamental Group has nothing interesting in it. And that I can actually say without saying

what the Fundamental Group is, I can tell you what the criterion is. This would be good, oh look, I can even use a visual aid. So for the people watching this on YouTube, you'll see this for the people on the podcast, you'll have to visualize it. So Lex has been nice enough to like give me a surface with an interesting topology. It's a mug right here in front of me. A mug, yes, I might say it's a genus one surface, but we could also say it's a mug, same thing. So if I were to draw a little

circle on this mug, oh, which way should I draw it so it's visible? Like here, okay. That's good. 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 judge it up here and I can judge it down there and I can sort of slide it up and down the handle, but I can't pull it close, if I draw a loop that goes around the handle, I can kind of just sit up here and I can just sit 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 without going inside. So the condition of being what's called simply connected, this is one of punk-or-age inventions, says that any loop of string can be pulled shut.

So it's a feature that the mug simply does not have. This is a non-simply connected mug and a simply connected mug would be a cup. You would burn your hand when you drank coffee out of it. So you're saying the universe is not a mug? Well, I can't speak to the universe, but what I can say is that regular old space is not a mug. Regular old space, if you like, sort of actually physically have a loop of string, you can always pull a shot. You can always pull a shot.

But, you know, what if your piece of string was the size of the universe? What if your piece of string was billions of light years long? How do you actually know? I mean, that's still an open question of the shape of the universe. Exactly. Whether it's, I think there's a lot, there is ideas of it being a tourist. I mean, there's, there's some trippy ideas and they're not like weird out there controversial. There's a legitimate at the center of cosmology debate. I mean, I think the somebody who thinks

that there's like some kind of decahedral symmetry. I mean, I remember reading something crazy about somebody saying that they saw this signature of that in the cosmic noise or what have you. I mean, to make the flat earthers happy, I do believe that the current main belief is, it's flat, it's flat-ish or something like that. The shape of the universe is flat-ish. I don't know what the heck that means. I think that has like a very... How are you even supposed to think about the shape of a thing that doesn't have any thing outside of it? I mean, that's exactly what topology does. Topology is what's called an intrinsic theory. That's what's so great about it. This question about the mug, you could answer it without ever leaving the mug, right? Because it's a question about a loop drawn

on the surface of the mug and what happens if it never leaves that surface. So it's like always there. See, but the difference between the topology and say if you're like trying to visualize a mug, you can't visualize a mug while living inside the mug. Well, that's true. The visualization is harder, but in some sense,

no, you're right, but the tools of mathematics are there. I, I, I, I, I don't want to fight, but I think the tools of mathematics are exactly there to enable you to think about what you cannot visualize in this way. Let me give, let's go, always to make things easier, go down or to mention. Let's think about we live on a circle, okay? You can tell whether you live on a circle or a line segment because if you live on a circle, if you walk

a long way in one direction, you find yourself back where you started and if you live in a line segment, you walk for long enough one direction, you come to the end of the world or if you live on a line, like a whole line, an infinite line, then you walk in one direction for a long time, and like, well, then there's not a sort of terminating algorithm to figure out whether you live on a line or a circle, but at least you sort of, at least you don't discover

that you live on a circle. So all of those are intrinsic things, right? All of those are things that you can figure out about your world without leaving your world. On the other hand, ready now we're gonna go from intrinsic to extrinsic. Why, did I not know we were gonna talk about this? But why not?

Why not? If you can't tell whether you live in a circle or a knot, like imagine like a knot floating in three-dimensional space. The person who lives on that knot, to them, it's a circle. They walk a long way, they come back to where they started. Now, we with our three-dimensional eyes can be like, oh, this one's just a plain circle and this one's knotted up.

But that'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. We can ask you one ape to another. Does it make you, how does it make you feel that you don't know, feel living a circle or on a knot, in a not,

in, inside the string that forms the not. I'm gonna 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. So like, does it bother you that if we look at like a mobius strip, that you don't have an obvious way of knowing whether you are inside of a cylinder, if you live on a surface of a cylinder, or you live on the surface of a mobius strip? No, I think you can tell. If you live, which one? Because what you do is you like,

tell your friend, hey, stay right here, I'm just gonna go for a walk, and then you 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 mobious trip. Well, no, because you won't see your friend, right? Okay, fair point, fair point on that.

But you have to believe a story is about, no, I don't even know. Would you even know? Would you really? Oh, no, I don't even know. I would, would you even know? Would you really, oh, no, your point is right. Let me try to think of a better, let's see if I can do this on the floor. It may not be correct to talk about cognitive beings living on a mobius trip, because there's a lot of things

taken for granted there, and we're constantly imagining actual like three-dimensional creatures. Like how it actually feels like to live in a moment We're out of there and we're constantly imagining actual three-dimensional creatures. How it actually feels like to live in a mobile strip is tricky, to internalize. I think that on what's called the real protective plane, which is even more messed up version of the maybe a strip, but with very similar features, this feature of only having one side, 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. But that would be a way you could know without leaving your world that something very funny is going on. You know it's extremely weird. Maybe we can comment on. Hopefully it's not too much of a tangent. I remember thinking about this.

This might be right. This might be wrong. But if you're, if we now talk about a sphere and you're living inside a sphere that you're going to see everywhere around you, the back of your own head. That I was, because like, I was, this was very counterintuitive to me to think about, maybe it's wrong. But, because I was thinking in like earth, you know, your 3D thing on, sitting on a sphere. But if you're living inside the sphere, like you're going to see, if you look straight, you're always going to see yourself all the way around.

So everywhere you look, this can be the back of your own head. I think somehow this depends on something of like how the physics of light works in this scenario, which I'm sort of planning and hard to bend my. I mean, that's true. The C's doing a lot of, like saying you see something's doing a lot of work. 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

where I'll go, yeah. And I don't, you know, this sort of, this sort of comic novel from the 19th century about an entire two-dimensional world, it's narrated by a square, that's the main character, and the kind of strangeness that befalls him when, you know, one day he's in his house and suddenly there's like a little circle there and there with him. And then the circle, the circle, like it starts getting bigger and bigger and bigger. And he's like, what the hell is going on?

It's like a horror movie for people. And of course, what's happening is that a sphere is entering his world. And as the sphere 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 this bizarre being. It's getting larger and larger and larger until it's exactly halfway through. Then they have this philosophical argument with a sphere.

I'm a sphere, I'm from the third dimension, the square is like, what are you talking about? There's no such thing. They have this sterile argument where the square is not able to follow the mathematical reasoning of the sphere until the sphere just grabs him and jerks him out of the plane and pulls him up and it's like now like now do you see like now do you see your whole world that you didn't understand before so do you think that kind of process is possible for us humans so we live in the three-dimensional world maybe with the time component for dimensional dimensional world, maybe with the time component for dimensional. And then math allows us to go into high dimensions comfortably and explore the world from those perspectives. Like, is it possible that the universe is many more dimensions than the ones we experience as human beings.

So if you look at the, especially in physics theories of everything, physics theories that try to unify general relativity and quantum field theory, they seem to go to high dimensions to work stuff out through the tools and mathematics. Is it possible? So like the two options are one, it's just a nice way to analyze a universe, but the reality is as exactly we perceive it, it is three dimensional. Or are we just seeing, are we those flat land creatures? They're just seeing a tiny slice of reality.

And the actual reality is many, many, many more dimensions than the three dimensions we perceive. Oh, I certainly think that's possible. Now, how would you figure out whether it was true or not is another question? And I suppose what you would do, as with anything else that you can't directly perceive, is you would try to understand what effect the presence of those extra dimensions out there would have on the things we can perceive.

Like what else can you do, right? And in some sense, if the answer is they would have no effect, then maybe it becomes like a little bit of a sterile question, because what question are you even asking, right? You can kind of posit however many entities that you want. Is it possible to intuit how to mess with the other dimensions while living in a three-dimensional world? I mean, that seems like a very challenging thing to do. We, the reason flatland could be written is because it's coming from a three-dimensional writer. 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 the stories from context, you ask, like, is it possible for us humans to have this experience of being transcendentally jerked out of our world so we can truly see it from above? Well, Edwin Abbott, who wrote the book, certainly thought so

because Edwin Abbott was a minister. So the whole Christian subtext of this book, I had completely not grasped reading this as a kid, that it means a very different thing, right? If a theologian is saying, like, oh, what if a higher being could pull you out of this earthly world you live in so that you can sort of see the truth and really see it from above as it were.

So that's one of the things that's going on for him. And it's a testament to his skill as a writer that his story just works, whether that's the framework you're coming to it from, or not. But what happens in this book, and this part now looking out 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 like what a cube would be. Oh, it's like you have a three-dimensional, and the square is very excited, and the square is like, okay, I get it now, so like, now that you explain to me how just by reason I can figure out what a cube would be. Oh, it's like a cube at three dimensional and the square is very excited and 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 the sphere is like, what the hell are you talking about? There's no fourth dimension in that particular. It's only the three dimensions, like that's how many there are, I can see.

Like 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 understand, be a lot. But- That's a nice way to talk about religion and myth

in general as perhaps us trying to struggle, us meaning human civilization, trying to struggle, us meaning human civilization, trying to struggle with ideas that are beyond our cognitive capabilities. But it's in fact not beyond our capabilities. It may be beyond our cognitive capabilities to visualize a four-dimensional cube, a Tesseract, something like to call it, or a five-dimensional cube, or a six-dimensional cube. But it is not beyond our cognitive capabilities to figure out how many corners a six dimensional

cube would have. That's what's so cool about us, whether we can visualize it or not, we can still talk about it, we can still reason about it, we can still figure things out about it. That's amazing. Yeah. If we go back to this, first of all, to the mug, but to the example you give in the book of the straw, how many holes does a straw have? And you listener may try to answer that in your own head.

Yeah, I'm going to take a drink while everybody thinks about it. So you give you a little slow sip. Is it zero, one, or two 2 or more than that maybe maybe you get very creative but it's kind of interesting to dissecting each answer as you do in the book. It's quite brilliant people should definitely check it out but if you could try to answer it now like think about all the options and why they may or may not be right. Yeah, it's one of it's one of these questions where people on first hearing it think it's a triviality and they're like, well, the answer is obvious.

And then what happens if you ever ask a group of people this, something wonderfully common happens, which is that everyone's like, well, it's completely obvious. And then each person realizes that half the person that other people in the room have a different obvious answer. Yeah. By the way, they have, and other people in the room have a different obvious answer. For the way that they have. And then people get really heated. People are like, I can't believe that you think it has two holes.

Or like, I can't believe that you think it has one. And then, you know, you really, like, people really learn something about each other. And people get heated. I mean, can we go through the possible options here? Is it zero, one, two, three, ten? Sure. So I think, you know, most people, the zero holders 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. That's like, is there any truth to that kind of conception? Yeah, I think that would be somebody who's account. I mean, what I would say is you could say the same thing, about a bagel. You could say I can make a bagel by taking like a long cylinder of dough, which doesn't have a hole, and then smushing the ends together.

Now it's a bagel. So if you're really committed, you can be like, okay, a bagel doesn't have a hole either, but like, who are you if you say a bagel doesn't have a hole? I mean, I don't know. Yeah, so that's almost like an engineering definition of it. Okay, fair enough. So what about the other options?

So, you know, one whole people would say Like how these are like groups of people like where we've planted our foot. Yes This book's written about each Believe you know, what say look there's like a hole and it goes all the way through the straw right? There's one reason of space. That's the hole. And there's one. And two whole people would say, like, well, look, there's a hole at the top and the hole at the bottom.

I think a common thing you see when people argue about this, they would take something like this bottle of water I'm holding, and be able to open it. And they'd say, well, how many holes are there in this? And you say, like, well, there's one, there's one hole at the top. Okay, what if I like poke a hole here so that all the water spills out?

Well, now it's a straw. So if you're a one-holeer, I say to you, like, well, how many holes are in it now? There was one hole in it before, and I poked a new hole in it. And then you think there's still one hole, even though there was one hole in it. I made you think there's still one hole, even though there was one hole on it, made one more.

Clearly not, there's just two holes. Yeah. And yet, if you're a two hole, the one hole will say, like, okay, where does one hole begin in the other hole end? Yeah. Like, what's it like, and in the book, I sort of, you know, in math, there's two things we do when we're faced

with a problem that's confusing us. We can make the problem simpler. That's what we were doing a minute ago when you were talking about high dimensional space and I was like, let's talk about like circles and line segments, let's go down to 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. Not, I'm not gonna do it.

You just have to imagine the path. Sorry. Lex, if you want, no, okay, no. That's go. Uh, that's going to be in the director's cut. It's a Patreon only footage. There you go. So many people would say there's three holes in a pair of pants. But, you know, for instance, my daughter when I asked, is by the way, talking to kids about this, is super fun. I highly recommend it.

Um, what did she say? She said, well, yeah, I feel a pair of pants, like just has two holes, because yes, there's the waist, but that's just the two leg holes stuck together. Whoa, okay. Two leg holes, yeah, okay. Right, I mean, that really is one color. She's a one-holar for the straw. So she's a one-holar for the straw too.

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 have an arithmetic to them. There are things which can be added. Like the waste, like waste equals leg plus leg is kind of an equation, but it's not an equation about numbers. It's an equation about some kind of geometric, some kind of topological thing, which is very strange. And so, you know, when I come down, you know, like a rabbi, I like to kind of like come up with these answers, as somehow like dodge the original question and say, like, you're both right, my children.

Okay, so yeah. So for the straw, I think what a modern mathematician would say is like, the first version would be to say, like, well, there are two holes, but they're really both the same hole. Well, that's not quite right. A better way to say it is, there's two holes, but one is the negative of the other. Now, what can that mean? One way of thinking about what it means is that if you sip something like a milkshake through the straw, no matter what, the amount of milkshake that's flowing in one end,

that same amount is flowing out the other end. So they're not independent from each other. There's some relationship between them. In the same way that if you somehow could like suck a milkshake through a pair of pants, the amount of milkshake, just go with me on this topic experiment. The amount of milkshake that's coming in the left leg of the pants plus the amount of milkshake that's coming in the right leg of the pants is the same that's coming out the waist of the pants. So just so you know, I fasted for 72 hours

the last three days. So I just broke the fast with a little bit of food yesterday. So this is like, this sounds, food analogies are metaphors for this podcast work wonderfully because I can intensely picture it. Is that your weekly routine or just in preparation for talking about geometry for three hours? Exactly, just for this.

It's hardship to purify the mind. No, it's for the first time I just wanted to try the experience. Oh wow and just to To pause to do things that are out of the ordinary to pause and to Reflect on how grateful I am to be just alive and be able to do all the cool shit that I get to do so did you drink water? Yeah, yes Yes Water and salt so like electrolytes and all those kinds of things and be able to do all the cool shit that I get to do so. Did you drink water? Yeah, yes, yes, yes, yes. Water and salt, so electrolytes and all those kinds of things. But anyway, so the inflow on the top of the pants

equals to the outflow on the bottom of the pants. Exactly, so this idea that, I mean, I think, Pongka Ray really had this idea, this sort of modern idea. I mean, building on, you know, Poincaré really have these idea, this sort of modern idea. I mean, building on stuff other people did, Betty is an important one of this kind of modern notion of relations between holes, but the idea that holes really had an arithmetic, the really modern view was really, I mean, notars idea. So she kind of comes in and sort of truly puts the subject on its modern footing that we have now.

So you know, it's always a challenge in the book. I'm not going to say I give a course so that you read this chapter and then you're like, oh, it's just like I took a semester of algebraic apology. It's not like this. And it's always a challenge writing about math because there are some things that you can really do on the page and the math is there. And there's other things which it's too much in a book like this to 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. To some degree be able to give inklinks

of the beauty of the subject. 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 of it.

Right. So not be talking just about ideas, but to actually be expressing the idea. You know somebody in the maybe you can comment, there a guy his YouTube channel is three blue one brown grand sanerson he does that masterfully well absolutely of a visualizing of expressing a particular idea and then talking about it as well back and forth what do you what do you think about grant it's fantastic I mean the flowering of math youtube is like such a wonderful thing because Fantastic. I mean the flowering of math YouTube is like such a wonderful thing because You know math teaching There's so many different venues through which we can teach people math. There's the traditional one, right?

Well, where I'm in a classroom With you're depending on the class. It could be 30 people. It could be a hundred 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 some set of people of that order of magnitude. And we have a long time, I'm with them for a whole semester, and I can ask them to do homework and we talk together, we have office hours, if they have one-on-one questions to add up to that. That's a very high level of engagement,

but how many people am I actually hitting at a time? Like not that many, right? And there's kind of an inverse relationship where the more, the fewer people you're talking to, the more engagement you can ask for. The ultimate, of course, is like the mentorship relation of like a PhD advisor and a graduate student where you spend a lot of one on one time together for like, 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 than are ever going to sit in my classroom and you spend like, however many hours it takes to read a book. Somebody like 3 blue one brown or number file or people people like Vi Heart. I mean, YouTube, let's face it, has bigger reach than a book. There's YouTube videos that have many, many, many more views than any hardbacked book, not written by a Kardashian or an Obama, it's going to sell. And then those are some of them are longer, 20 minutes 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 so, look, look, like, Jeannie Achen, there's a wonderful category 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 to sort of say something about mathematics to, like, untold millions of people. So everywhere along this curve is important. One thing I feel like is great right now is that people are just broadcasting on all the channels because we each have our skills, right? Somehow along the way, like I learned how to write books. I had this kind of weird life as a writer where I sort of spent a lot of time like thinking

about how to put English words together into sentences and sentences together into paragraphs, like at length, which is this kind of, like, weird specialized skill. And that's one thing, but, like, sort of, being able to make, like, you know, winning, good-looking, eye-catching videos is, like, a totally different skill. And, you know, probably, you know, somewhere out there, there's probably, sort of, some, like, heavy metal band that's, like, teaching math through heavy metal and, like, using their skills to do that. I hope there is. And they're music and so on. Yeah. But there is something to the process. I mean, Grant does this, especially well, which is in order to be able to visualize something. Now, he writes programs. So it's programmatic visualization. So like

the the thing he is basically mostly through his madam library and Python, everything is drawn through Python. You have to truly understand the topic to be able to visualize it in that way. And not just understand it, but really kind of thinking a very novel way. It's funny because I've spoken with them a couple times, I've spoken on the Malat offline as well. He really doesn't think he's doing anything new, meaning like he sees himself as very different from maybe like a researcher. But it feels to me like he's creating something totally new like that act of understanding visualizing is as powerful or has

the same kind of inkling of power as does the process of proving something. You know, it just it doesn't have that clear destination, but it's it's pulling out an insight and creating multiple sets of perspective that arrive at that insight. And to be honest, it's something that I think we haven't quite figured out how to value inside economic 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-participated 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. I mean, I think, for example, the Institute for Computational Experimental Mathematics at Brown, which is like a, you know, it's an NSF-funded math institute very much part of sort of 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. Yeah, I'm hoping to see more and more of that from like MIT faculty, from faculty, from all the the top universities in the world. Let me ask you this weird question about the field model, which is the Nobel Prize in mathematics. Do you think since we're talking about computers, there will one day come a time when a computer and AI system will win a field model? No.

That's what a human would say. Why not? Is that like, that's like my cap shot? That's like the proof that I'm Why not? Is that like my cap shot? That's like the proof that I'm a human. Is like the pilot of that. Yeah. How does he want me to answer? Is there something interesting to be said about that? Yeah, I am tremendously interested in what AI can do in pure mathematics.

I mean, it's of course, it's a perocal interest, right? You're like, why am I not interested in like how I can like help feed the world? I'm like, there's problems. 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. It doesn't matter what those words mean, Just it's like some thing that an 1890 would take a person a year to do and would be a valuable thing that you might want to know. And it's still a valuable thing that you might want to know. But now you type a few lines of code in McCauley or Sage or magma and you just have it. So we don't think of that as math anymore, even though it's the same thing. What's McCauley's sage in Mangal?

Oh, those are computer algebra programs. So those are like sort of bespoke systems that lots of mathematicians use. That's similar to Maple and... Yeah, oh yeah, so it's similar to Maple and Mathematica, yeah. But a little more specialized, but yeah. It's programs that work with symbols and allow you to do...

Can you do proofs? Can you do kind of little leaps and proofs? They're not really built for that one. That's a whole other story. But these tools are part of the process of mathematics now. into proofs, can you do kind of little leaps and proofs? They're not really built for that one. That's a whole other story. But these tools are part of the process of mathematics now. Right. They are now for most mathematicians, I would say, part of the process of mathematics.

And so, there's a story I tell in the book, which I'm fascinated by, which is, 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 has a very nice use of a neural net defined counter examples to conjecture. Somebody said, maybe this is always that. You can be like, well, let me train an AI to AI to sort of try to find things where that's not true. And it actually succeeded.

Now, in this case, if you look at the things that it found, you say like, okay, I mean, these are not famous conjectures. Yes. Okay. So like, somebody worked this down, maybe this is so. Looking at what the AI came up with, you're like, you know, I bet if like five grad students had thought about that problem, they wouldn't have thought. I mean, when you see it, you're like,

okay, that is one of the things you might try if you sort of like put some work into it. Still, it's pretty awesome. But the story I tell in the book, which I'm fascinated by, is there is, okay, we're gonna go back to knots. It's cool. There's a knot called the Conway knot. After John Conway, who maybe will talk about a very interesting character also.

Yeah, there's a small tangent. Somebody I was supposed to talk to and unfortunately he passed away and he's somebody I find and there's an incredible mathematician, incredible human beings. Oh, and I am sorry that you didn't get a chance because having had the chance to talk to him a lot

when I was a postdoc, yeah, you missed out. There's no way to sugarcoat it. I'm sorry that you didn't get a chance because having had the chance to talk to him a lot when I was, you know, when I was a postdoc, um, yeah, you missed out. There's no way to sugar code it. I'm sorry that you didn't get that chance. Yeah. It is what it is. So, knots. Yeah. So there was a question.

And again, it doesn't matter the technicalities of the question, but it's a question of whether the knot is sliced. It has to do with, um, something about what kinds of three-dimensional surfaces and four dimensions can be bounded by this knot. But never mind what it means. It's some question, and it's actually very hard to compute whether a knot is slice or not.

And in particular, the question of the con way knot, whether it was slice or not, was particularly vexed. Until it was solved just a few years ago by Lisa Piccarillo, 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 this story until I started telling it. In fact, I think she's now at MIT, so she's basically following you around. If I remember correctly, I think you're 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 in which two were pictures. Very short for paper solving a major conjecture. And it really makes you think about what we mean by difficulty in mathematics. Like do you say, oh, actually the problem wasn't difficult because you could solve it so simply, or do you say, like, well, no,

evidently it was difficult because like the world's top topologist, many, you know, worked on it wasn't difficult because you could solve it so simply. Or do you say, well, no, evidently, it was difficult because the world's top depologist worked on it for 20 years and nobody could solve it. So therefore, it is difficult. Or is it that we need some new category of things about which it's difficult to figure out that they're not difficult? I mean, this is the computer science formulation, but the journey to arrive at the simple answer may be difficult, but once you have the answer, it will then appear simple. I mean, there might be a large set of such solutions, because once we stand at the end of the scientific process

that we're at the very beginning of, or at least it feels like, I hope there's just simple answers to everything that will look and it'll be simple laws that govern the universe, simply an explanation of what is consciousness, of what is love, is mortality, fundamental to life,

what's the meaning of life. Are humans special or we're just another sort of reflection of all that is beautiful in the universe in terms of like life forms, all of it is life and has different, when taken from a different perspective as all life can seem more valuable or not, but really it's all part of the same thing. When taking from a different perspective is all life can seem more valuable or not, but really it's all part of the same thing. All those will have a nice like two equations, maybe one equation. Why do you think you want those questions to have simple answers?

I think just like symmetry and the breaking of symmetry is beautiful somehow. There's something beautiful about simplicity. I think it's aesthetic, what is that? It's aesthetic, yeah. But it's aesthetic in the way that happiness is an aesthetic. Why is that so joyful, that a simple explanation that governs a large number of cases is really appealing? Even when it's not, like obviously we get

a huge amount of trouble with that because oftentimes it doesn't have to be connected with the reality or even that explanation can be exceptionally harmful. Most of the world's history that was governed by hate and violence had a very simple explanation at the court That was used to cause the violence and the hatred so like we get into trouble with that But why is that so appealing and in this nice forms in mathematics like you look at the Einstein papers

Why are those so beautiful and why is the angel wiles proof of the pharmaceuticals last theorem? Not quite so beautiful like what's beautiful about that story is the human struggle of like the human story of perseverance of the drama of not knowing if the proof is correct and ups and downs and all of those kinds of things that's the interesting part but the fact that the proof is huge nobody understands well from my outsider's perspective nobody understands what the heck it is is not as beautiful as it could have been. I wish it was what from my originally said, which is, you know, it's not small enough to fit in the margins

of this page, but maybe if he had like a full page or maybe like a couple of posted notes, he would have enough to do the proof. What do you make of, if we could take another of a multitude of tangents? What do you make of Fermat's last theorem? Because the statement, there's a few theorems, there's a few problems that are deemed by the world throughout its history to be exceptionally difficult. That one in particular is really simple to formulate and really hard to come up with a proof for.

And it was like taunted as simple by, for my own self. Is there something interesting to be said about that X to the N plus Y to the N equals Z to the N for N of three or greater? Is there a solution to this? and then how do you go about proving that? Like, how would you try to prove that and would you learn from the proof that eventually emerged by Andrew Walsh? Yeah, so let me just say the background because I don't know if everybody listening, no, no is the story. So, you know, Fermat was an early number theory. It's only sort of an early mathematician.

Those special adjacent didn't really exist back then. He comes up in the book actually in the context of a different theorem of his that has to do with testing whether a number is prime or not. So I write about, he was one of the ones who was salty. And like he would exchange these letters where he and his correspondence would try to top each other and vex each other with questions and stuff like this. But this particular thing, it's called Fermat's last theorem because it's a note he wrote

in his copy of the Disquistionist, Arithmetic Eilich. He wrote, here's an equation, it has no solutions. I can prove it, but it proof's like a little too long to fit in this, in the margin of this book. He was just like writing a note to himself. Now, let me just say historically, we know that Vermont did not have a proof of this theorem. For a long time, people were like,

this mysterious proof that was lost the very romantic story, right? But, fair amount later, he did prove special cases of this theorem and wrote about it, talked to people about the problem. It's very clear from the way that he wrote where he can solve certain examples of this type of equation that he did not know how to do the whole thing. He may have had a deep, simple intuition about how to solve the whole thing that he had at that moment

without ever being able to come up with a complete proof. And that intuition may be lost at time. Maybe. But you're right, that is unknowable. But I think what we can know is that later he certainly did not think that he had a proof that he was concealing from people. Yes. He thought he didn't know how to prove it, and I also think he didn't know how to prove

it. Now, I understand the appeal of saying, like, wouldn't it be cool if there's a very simple equation? There was a very simple, clever, wonderful proof that you could do in a page or two. And that would be great. But you know what? There's lots of equations like that that are solved by very clever methods like that, including the special cases that Fermat wrote about, the method of descent, which is like very wonderful

and important. But in the end, those are nice things that you teach in an undergraduate class, and it is what it is, but they're not big. On the other hand, work on the Fermat problem, it's what we like to call it, because it's not really his theorem because we don't think he proved that.

So I mean, work on the Firm-O-Problem developed this like incredible richness of number theory that we now live in today, like and not by the way, just while 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 La Mée from the 19th century where Kumar, in understanding

what mistake La Mée had made in this incorrect proof, basically understands something incredible, which is that, you know, a thing we know about numbers is that you can factor them and you can factor them uniquely. There's only one way to break a number up into primes. Like, if we think of a number like 12, 12 is 2 times 3 times 2. I had to think about it. Right? Or it's 2 times 2 times 3. Of course, you can reorder them. But there's no other way to do it. There's no universe in which 12 is something comes five or in which there's like four threes in it. Nope. 12 is like two twos in a three. Like that is what it is. And that's such a fundamental feature of arithmetic that we almost think of it like God's law. You know what I mean? It has to be that way. It's just that's a really powerful idea.

It's it's so cool that every number is uniquely made up of other numbers. And like made up meaning like there's these like basic atoms that form molecules that get built on top of each other. I love it. I mean, when I teach, you know, undergraduate number theory, it's like, it's the first really deep theorem that you prove. What's amazing is, you know, the fact that you can factor a number into primes is much easier. Essentially, you could knew it all the way, didn't quite put it in that way.

The fact that you can do it at all, what's deep is the fact that there's only one way to do it, or however you sort of chop the number up, you end up with the same set of prime factors. And indeed, what people finally understood at the end of the 19th century is that if you work in number systems slightly more general than the ones we're used to, which it turns out irrelevant to Firmah, all of a sudden this stops being true. Things get, I mean, things get more complicated and now because you were praising simplicity before, you were like, it's so beautiful, unique factorization. It's so great.

So when I tell you that in more general number systems, there is no unique factorization, maybe you're like, that's bad. I'm like, no, that's good because there's like a whole new world of phenomena to study that you just can't see through the lens of the numbers that we're used to. So I'm for complication. I'm a highly in favor of complication.

Every complication is like an opportunity for new things to study. And is that the big kind of one of the big insights for you from Angel Wiles' proof. Is there interesting insights about the process, the use to prove that sort of resonates with you as a mathematician? Is there interesting concept that emerged from it? Is there interesting human aspects to the proof? Whether there's interesting human aspects to the proof itself is an interesting question.

Certainly, it has a huge amount of richness. So it added its heart is an argument of what's called deformation theory, which was, in part, created by my PhD advisor, Barry Meiser. Can you speak to what deformation theory is? I can speak to what it's like. Sure. How about that? What is it rhyme with? Right.

Well, the reason that Barry called it, deformation theory, I think he's the one who gave it the name. I hope I'm not wrong in saying the same thing. In your book, you have calling different things by the same name as one of the things in the beautiful map that opens the book. Yes, and this is a perfect example. So this is another phrase of Pancarré, 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, 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 Fermat's last theorem, and let's not even be too careful about what these objects are. I can tell you there are gauw 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, a deformation means moving something just a tiny bit, like an infinitesimal amount.

If you really are good at understanding which ways a thing can move in the tiny, tiny, tiny, infinite, testable amount in certain directions, maybe you can piece that information together to understand the whole global space in which it can move. And essentially, their argument comes down to showing that two of those big global spaces are actually the same, the fabled R equals T part of their proof, which is at the heart of it. And it involves this very careful principle like that. But that being said, what I just said, it's probably not what you're thinking

because what you're thinking, when you think, oh, I have a point in space and I move it around like a little tiny bit, you're using your notion of distance that's from calculus. We know what it means for like two points in the real line to be close together. So, 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 in the English language, we do that all the time. We talk about somebody being a close relative. It doesn't mean they live next door to you, 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 lexical 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 like

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 periodic distance. I didn't write about this that much in the book because even though I love it, it's a big part of my research life, it gets a little bit into the weeds, but your listeners are going to hear about it now. Please. Where? You know, what a normal person says, when they say two numbers are close, they say like, you know, their difference is like a small number. Like seven and eight are close because their difference is one and one's pretty small.

If we were to be what's called a two 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 in 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 in multiple of a power of two at all.

It's odd. You want to know it's really far from one? Like one and one sixty-fourth. Because their difference is a negative power of 2, 2 to the minus 6. So those points are quite, quite far ahead. 2 to the power of a large n would be 2, if that's the difference between 2 numbers and they're close. Yeah, so 2 to a large power is this metric, a very small number, and two to a negative power

is a very big number. That's too adequate. Okay. I can't even visualize that. It takes practice. It takes practice. If you've ever heard of the cantor set, it looks kind of like that. So it is crazy that this is good for anything, right?

I mean, this just sounds like a definition that someone would make up to torment you. But what's amazing is, there's a general theory of distance where you say any definition you make the satisfy certain axioms deserves to be called a distance and this. See, I'm sorry to interrupt. My brain, you broke my brain now. Awesome. 10 seconds ago. Because I'm also starting to map for the two out of case to binary numbers. Because we romanticized those. Exactly the right way to think of it.

I was trying to mess with number, trying to see which ones are close, and then I'm starting to visualize different binary numbers and which ones are close to each other. Well, I think there's a key in it. No, it's very similar. That's exactly the way we would think of it. It's almost like binary numbers written in reverse. Right.

Because in a binary expansion, two numbers are close. A number that's small is like 0.00000 something. Something that's the decimal and it starts with a lot of zeros. In the two attic 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, I should have said, that's what I should have said, Lex.

That's like very good metaphor. Oh, you said. Okay, but so why is that interesting except for the fact that it's a beautiful kind of framework, different kind of framework, which you think about distances. And you're talking about not just the two-added, but the generalization of that. What's the other thing? Yeah, the NEP.

Because that's the kind of deformation that comes up in Wiles's proof, that deformation, we're moving something a little bit, means a little bit in this two-added therapy. Okay. No, I mean, I can just get excited to talk about it, and I just taught this in the fall semester. But reformulating, why is...

So you pick a different measure of distance over which you can talk about very tiny changes and then use that to then prove things about the entire thing. use that to then prove things about the entire thing. Yeah, so though, 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 the goal is always the understanding.

The goal is not 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, and it's something that's a wonderful, wonderful essay by Bill Thurston, kind of one of the great geometers of our time who unfortunately passed away a few years ago, called on proof and progress in mathematics. And he writes very wonderfully about how, you know, we're not, it's not a theorem factory where we have a production quota. I mean, the point of mathematics is to help humans understand things.

And the way we test that is that we're proving new theorems along the way. That's the benchmark, but that's not the goal. Yeah, but just as a kind of, absolutely, but as a tool, it's kind of interesting to approach a problem by saying, how can I change the distance function? Like what the nature of distance? Because that might start to lead to insights for deep understanding. Like if I were to try to describe human society by a distance to people close if they love each other, right?

And then start to do a full analysis on the everybody that lives on Earth currently, the 7 billion people. And from that perspective, as opposed to the geographic perspective of distance. And then maybe there could be a bunch of insights about the source of violence, the source of maybe entrepreneurial success or invention or economic success or different systems, communism, capitalism, start to, I mean, that's, I guess what economics tries to do, but really saying, okay, let's think outside the box about totally new distance functions that can unlock something profound about the space. Yeah, because think about it, okay, here's, I mean, now we're going to talk about AI,

which you know a lot more about than I do. So just, you know, start laughing up royally if I say something that's completely wrong. We both know very little relative to what we will know centuries from now. That is a really good humble way to think about it. I like it. Okay, so let's just go for it. Okay, so I think you'll agree with this that in some sense what's good about AI is that We can't test any case in advance the whole point of AI is to make or one point of it

I guess is to make good predictions about cases we haven't yet seen and in some sense that's always gonna involve some notion of distance Because it's always gonna 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. If you try to do natural language processing and your idea of distance between words is how close they are in the dictionary, when you write them in alphabetical order, you are gonna get pretty bad translations, right? No, the notion of distance has to come from somewhere else. Yeah, that's essentially what neural networks are doing,

this word embeddings are doing is coming up with... In the case of word embeddings, literally, like literally what they are doing is learning a distance. But those are super complicated distance functions and it's almost nice to think maybe there's a nice transformation that's simple. Sorry, there's a nice formulation of the distance again with the simple. So you don't, let me ask you about this. From an understanding perspective, there's the Richard Feynman, maybe attributed to him,

but maybe many others is this idea that if you can't explain something simply that you don't understand it. In how many cases, how often is that true? Do you find there's some profound truth in that? Oh, okay, so you were about to ask, is it true to which I would say flatly no, but then you said, you followed that up with is there some profound truth in it?

And I'm like, okay, sure. So there's some truth in it. It's not true. It's not true. This is your mathematician answer. The truth that is in it is that learning to explain something helps you understand it. But real things are not simple. A few things are, most are not.

And to be honest, I don't, I mean, I don't, we don't really know whether Feynman really said that right or something like that is sort of disputed, but I don't think Feynman could have literally believed that, whether or not he said it. And, you know, he was the kind of guy, I didn't know him, but I'm reading his writing. He liked to sort of say stuff, like stuff that sounded good. You know what I mean?

So it totally strikes me as the kind of thing he could have said because he liked the way saying it made him feel. But also knowing that he didn't like literally mean it. Well, I definitely have, I have a lot of friends and I've talked to a lot of physicists and they do derive joy from believing that they can explain stuff simply or believing as possible to explain stuff simply. Even when the explanation is not actually that simple, like I've heard,

I've heard people think that the explanation is simple and they do the explanation and I think it is simple But it's not capturing the phenomena that we're discussing. It's capturing it somehow maps in their mind But it's it's taking as a starting point as an assumption that there's a deep knowledge and a deep understanding that's That's actually very complicated and the simplicity is almost like a poem about the more complicated thing as opposed to our distillation. And I love poems, but a poem is not an explanation. Well, some people might disagree with that, but certainly from a mathematical perspective. No poet would disagree with it.

No poet would disagree. You don't think there's some things that can only be described imprecisely? I said explanation. I don't think any poem, I don't think any poet would say their poem is an explanation. They might say it's a description.

They might say it's sort of capturing sort of. Well, some people might say the only truth is like music, all right, that not the only truth, but some truth can only be expressed through art. And, I mean, that's the whole thing. We're talking about religion and myth and there's some things that are limited cognitive capabilities and the tools of mathematics or the tools of physics are just not going to allow us to capture. Like, it's possible consciousness is one of those things. Yes, that is definitely possible.

But I would even say, look, I'm consciousness is a thing about which we're still in the dark as to whether there's an explanation we would understand as an explanation at all. By the way, okay, I gotta give yet one more amazing point of rake quote, because this guy just never stopped coming up with great quotes that, you know, Paul Erdisch, another fellow who appears in the book. And by the way, he thinks about this notion of distance of personal affinity, kind of

like what you're talking about, the kind of social network and that notion of distance that comes from that. So that's something that Paul Erdisch did. Well, he thought about distances in networks. I guess he didn't think about the social network. That's fascinating. That's how it started that story, Vertish, number. Yeah, okay.

But Erdisch was sort of famous for saying, okay. But it's right. But you know, Eridish was sort of famous for saying, and this is sort of one line you were saying, he talked about the book, Capital T, Capital B, the book, and that's the book where God keeps the right proof of every theorem. So when he saw a proof, he really liked it.

It was like really elegant, really simple. Like that's from the book. That's like you found one of the ones that's in the book. He wasn't a religious guy by the way. He referred to God as the supreme fascist. He was like, but somehow he was like, I don't really believe in God, but I believe in God's book. I mean, it was like, but Poincarein the other hand. And by the way, there are other managers held a 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 track pandemics, this SIR model, that sort of originally comes from her work with Ronald Ross, but she was also super, super, super devout, and she also sort of 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 cordable. She says, you know, math is the truth, the things about mathematics, she was like, yeah, math is how we communicate with God. She has a great, all these people are incredibly quotable. She says, you know, math is the truth, the things about mathematics. She's like, they're not the most important of God's thoughts, but they're the only ones that we can know precisely. So she's like, this is the one place where we get to sort of see what God's thinking when we do mathematics.

Again, not a fan of poetry or music. Some people say Hendrix is like, some, some people say chapter one of that book is mathematics, and then chapter two is like classic rock. Okay. All right, so like, it's not clear that the, I'm sorry, you just sent me off on a tangent, just imagining like,

Erdish at a Hendrix concert, trying to figure out if it was from the book or not. All of what I was coming to with, just to say, but one point, all right, instead about, is he was 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 it to us,

we wouldn't be able to understand it, so it doesn't matter. So, point, Karay was of the view that there were things that were sort of like inhumanly complex, and that was how they really were. Our job is to figure out the things that are not like that. They're not like that. All this talk of primes got me hungry for primes. You are blog posts, the beauty of bounty gaps, a huge discovery about prime numbers and

what it means for the future of math. Can you tell me about prime numbers, what the heck are those? What are twin primes? What are prime gaps? What are bound to gaps in primes, what are all these things? And what if anything, or what exactly is beautiful about them? Yeah, so, you know, prime numbers are one of the things that a number theorists study the most and have for millennia, they are numbers which can't be factored.

And then you say like five. And then you're like, wait, I can factor five. Five is five times one. Okay, not like that. That is a factorization. It absolutely is a way of expressing five as a product of two things. But don't you agree that there's like 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 6 times 2 or 35 is 7 times 5, I've really done something to it. I've broken up. So those are the kind of factorizations that count. And the number that doesn't have a factorization like that is called prime, except historical side note.

One, which at some times in mathematical history has been deemed to be a prime, but currently is not, and I think that's for the best. But I bring it up only because a lot of people think that these definitions are kind of, if we think about them hard enough, we can figure out which definition is true. No, there's just an artifact in mathematics. So, what definition is best for us, for our purposes? Well those edge cases are weird, right? So it can't be, it doesn't count when you use yourself as a number or one as part of the factorization or as the entirety of the factorization.

So you somehow get to the meat of the number by factorizing it, and that seems to get to the core of all of mathematics. Yeah, you take any number and you factorize it until you can factorize no more, and what you have left is some big pile of primes. I mean, by definition, when you can't factor anymore, when you're done, when you can't break the numbers of 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 gonna be odd from the nod, because if they were even, I could fact, write two out of them, but it's not all the odd numbers. Nine isn't prime because it's three times three. Fifteen isn't prime because it's three times five.

But 13 is where we're rate 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. How high could you go if we were just sitting here, by the way, your own brain? If I continue this without interruption, would you be able to go over 100? I think so. There's always those ones that trip people up. There's a famous one, the Groten Deak Prime, 57, like sort of Alexander Groten Deak, the great algebraic geometry, was sort of giving some lecture involving a choice of a prime in general, and somebody said, like, can't you just choose a prime?

And he said, okay, 57, which is in fact not prime. It's three times 19. Oh, damn. But it was like, I promise you in some circles, that's a funny story. Okay. That's one of the, but there's a humor in it. Yes, I would say over 100, I definitely don't remember.

Like, 107, I think. I'm not sure. Okay, like, so so is there category of like fake primes that that are easily mistaken to be prime like 57 I wonder. Yeah, so I would say 57 and 57 and 51 are definitely like prime offenders. Oh, I didn't do that on purpose. Oh, well done. They 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 prime, doesn't it, but it is not. Yeah. So there's also, by the way, but there's also an actual notion of pseudo prime, which is a thing with a formal definition, which is not a psychological thing.

It is a prime which passes a primality test, devised by Fermat, which is a very good test, which if a number fails this test, it's definitely not prime. And so there was some hope that, oh, maybe if a number passes the test, then it definitely is prime. That would give a very simple criterion for primality. Unfortunately, it's only perfect in one direction. So there are numbers. I want to say 341 is the smallest, which past the test, but are not prime 341. Is this test easily

explainable or no? Yes, actually. Ready, let me give you the simplest version of it. You can dress it up a little bit, but here's the basic idea. I take the number, the mystery number. I raise two to that power. So let's say your mystery number is six. Are you sorry you asked me? Are you ready to thought of it?

No, I'm breaking my brain again. But yes, let's do it. We're going to do a live demonstration. Let's say your number is six. So I'm gonna raise two to the sixth power. Okay, so if I were working on it, I'd be like, that's two cubes squared, so that's eight times eight, so that's 64.

Now we're gonna divide by six, but I don't actually care what the quotient is, only the remainder. So let's say 64 divided by six is, well, 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 5th is 32, divide 32 by 5,

and you get 6 with a remainder of two. With a remainder of two here. For seven, two to the seventh is 128, divide that by seven, and let's see, I think that's seven times 14, is that right? No. Seven times 18 is 126 with a remainder of two, right? 128 is a multiple of seven plus two. So if that remainder is not two,

then that's definitely not prime. That is definitely not prime. And then if it is, it's likely a prime, but not for sure. It's likely a prime for not for sure. And there'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 could 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 this proof of Fermat's little theorem. So yeah, so as 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 are not actually prime. But the point is, that test but not actually prime. But the point is, there are many, many, many theorems about prime numbers. Are there, like, there's a bunch of questions to ask, is there an infinite number of primes? Can we say something about the gap between primes as the numbers go larger and larger and larger and so on? Yeah, it's a perfect example of your desire for simplicity in all things. You know what would be really simple?

If there was only finally many primes. Yes. And then there would be this finite set of atoms that all numbers would be built up. That's right. That would be very simple in good and 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. At primes there are. The primes that there are right behind the numbers. There's an A-friend number of primes. So what about the gaps between the primes? Right. So one thing that people recognized and really thought about a lot is that the primes on average Seem to get farther and farther apart as they get bigger and bigger in other words It's less and less common like I already told you of the first 10 numbers 2 3 5 7 4 them a 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 lot rarer. In some sense, because there's a lot more things for them to be divisible by. That's one way of thinking of it. It's a lot more possible for there to be a factorization because there's a lot of things you can try to factor out of it. As the numbers get bigger and bigger,

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, we don't have any way of describing a process that makes primes like, sure, you can look at your computer and see a lot of them, but the fact that there's a lot, why is that evidence that there's infinitely many, right?

Maybe I can go on the computer and find 10 million, well, 10 million is pretty far from infinity, right? So how is that evidence? There's a lot of things. There's a lot more than 10 million atoms, that doesn't mean there's infinitely many atoms far from infinity, right? So how is that evidence? There's a lot of things. There's a lot more than 10 million atoms. That doesn't mean there's infinitely many atoms in the universe, right? I mean, on most people's physical theories, there's probably not, as I understand it. Okay, so why would we think this?

The answer is that it turns out to be incredibly productive and enlightening to think about primes as if they were random numbers, as if they were randomly distributed according to a certain law. Now they're not. They're not random. There's no chance involved. They're completely deterministic whether a number is prime or not. And yet it just turns out to be phenomenally useful in mathematics to say, even if something

is governed by a deterministic law, let's just pretend it wasn't. Let's just pretend that they were produced by some random process and see if the behavior is roughly the same. And if it's not, maybe change the random process, maybe make the randomness a little bit different and tweak it, and see if you can find a random process that matches the behavior we see. And then maybe you predict that other behaviors of the system are like that of the random process.

And so that's kind of like, it's funny because I think when you talk to people about the twin prime conjecture, people think you're saying, wow, there's like some deep structure there that like makes those primes be like close together again and again. And no, it's the opposite of deep structure. What we say, when we say we believe the twin prime conjecture, is that we believe the primes are like sort of strewn around pretty randomly. And if they were then by chance, you would expect there to be infinitely many twin primes. And we're

saying, yeah, we expect them to behave just like they would if they were random dirt. You know, the fascinating parallel here is as you got a chance to talk to Sam Harris. And he uses the prime numbers as an example, often, I don't know if you're familiar with who Sam is, he uses the prime numbers as an example often, I don't know if you're familiar with who Sam is, he uses that as an example of there being no free will. Wait, where does he get this? Well, he just uses as an example of,

it might seem like this is a random number generator, but it's all like formally defined. So if we keep getting more and more primes, then like that might feel like a new discovery and that might feel like a new experience, but it's not. It was always written in the cards. But it's funny that you say that because a lot of people think of like randomness, the fundamental randomness within the nature of reality might be the source of something that we experience as free will. And you're saying it's like useful to look at prime numbers as a random process in order

to prove stuff about them, but fundamentally of course it's not a random process. Well, not in order to prove 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 wanna have some heuristic ways

of guessing, making guesses about what's true. So yeah, here's what I would say. You're gonna be imaginary Sam Harris now. Like, you are talking about prime numbers and you are like, but prime numbers are completely deterministic. I'm saying, well, but let's treat them like a random process. You say, but you're just saying something that's not true. They're not a random process.

They're deterministic. I'm like, okay, great. You hold to your insistence that it's not a random process. Meanwhile, I'm generating insight about the primes that you're not because I'm willing to pretend that there's something that they're not in order to understand what's going on. Yeah, so it doesn't matter what the reality is, what matters is what's, what framework of thought results in the maximum number of insights. Yeah, because I feel, look, I'm sorry, but I feel like you have more insights about people

if you think of them as like beings that have wants and needs and desires and do stuff on purpose, even if that's not true, you still understand better what's going on by treating them in that way. Don't you find, look, what you work on machine learning? Don't you find yourself sort of talking about what the machine is trying to do in a certain instance? Do you not find yourself drawn to that language? Well, it knows this, it's trying to do that.

It's learning that. I'm certainly drawn to that language to the point where I received quite a bit of criticisms for it because I you know Like oh, I'm like your side man. So especially in robotics. I don't know why but robotics people don't like to Name their robots or they they certainly don't like to gender their robots because the moment you gender a robot You start to anthropomorphize. If you say he or she, you start to, in your mind, construct like a life story in your mind, you can't help it. You create a humorous story to this person. You start to connect this person. This robot, you start to project your own

and, but I think that's what we do to each other. I think that's actually really useful for the engineering process, especially for human robot interaction's actually really useful for the engineering process, especially for human-robial interaction. And yes, for machine learning systems, for helping you build an intuition about a particular problem. And it's almost like asking this question, you know, when a machine learning system fails in a particular edge case,

asking like, what were you thinking about? Like asking like almost like when you're talking about to a child who just does something bad, you, you, you, you want to understand like, what was um, how did they see the world? Maybe there's a totally new, maybe you're the one that's thinking about the world or incorrectly. And uh, yeah, that at the pro-morphization process, I think is ultimately good for insight. And the same is I agree with you.

I tend to believe about free will as well. Let me ask you a ridiculous question. It's okay. Of course. I've just recently, most people go on like rabbit hole, like YouTube things. And I went on a rabbit hole often due of Wikipedia. And I found a page on finitism, ultra finitism and intuitionism. Or I forget what it's called. Yeah, intuitionism, intuitionism. That seemed pretty interesting. I have a much to do list actually, like looking to like, is there people who like formally attract, like real mathematicians are trying to argue for this?

But the belief there, I think, let's say, finitism, that infinity is fake, meaning, um, infinity might be like a useful hack for certain, like a useful tool in mathematics, but it really gets us into trouble because there's no infinity in the real world. Maybe I'm not expressing that fully correctly, but basically saying, there's things that are, once you add into mathematics, things that are not probably within the physical world, you're starting to inject, to corrupt your framework of reason. What do you think about that?

I mean, I think, okay, so first of all, I'm not an expert and I couldn't even tell you what the difference is between those three terms, finite, ultrafinatism and intuitionism, although I know that they're related and I tend to associate them with the Netherlands in the 1930s. Okay, I'll tell tell you can I just quickly comment because I read the Wikipedia page the difference in ultra that like the ultimate sentence of the modern age Can I just comment because I read the Wikipedia page that sums up our moment Bro I'm basically an expert

ultra-finatism So, finalism says that the only infinity you're allowed to have is that the natural numbers are infinite. So, like, those numbers are infinite. So, like, one, two, three, four, five, the integers are infinite. The ultrafinatism says, nope, even that infinity is fake. That's what I bet ultra-finatism came second. I bet it's like when there's like a hardcore scene and then one guy is like,

oh, now there's a lot of people in the scene, I have to find a way to be more hardcore than the hardcore people. It's all back to the emo talk. Yeah. Okay, so is there any, are you ever, because I'm often uncomfortable with the infinity, like psychologically, I have trouble when that sneaks in there.

It works so damn well, I get a little suspicious because it could be almost like a crutch or an oversimplification that's missing something profound about reality. Well, so first of all, okay, if you say like, is there like a serious way of doing mathematics that doesn't really treat infinity as a real thing, or maybe it's kind of agnostic, and it's like, I'm not really gonna make a firm statement about whether it's a real thing or not, yeah, that's called most of the history of mathematics.

Right, so it's only after Cantor, right, that we really are sort of, okay, we're gonna like have a notion of like the cardinality of an infinite set and like do something that you might call like the modern theory of infinity. That said, obviously everybody was drawn to this notion and no, not everybody was comfortable with it. Look, I mean, this is what happens with Newton, right? I mean, so Newton understands that to talk about tangents and to talk about instantaneous

velocity, he has to do something that we would now call taking a limit, right? The fabled dy over dx, if you sort of go back to your calculus class, 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 testimals. 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-profile of Newton. They're not zero, but they're smaller than any number. Are they the ghosts of departed quantities? That was just like ultra-profile of new And on the one hand, he was right It wasn't really rigorous. I'm on his standards on the other hand like Newton was out there doing calculus Another people were not right. It works. It works I think I think a sort of intuitionist view for instance I would say would express serious

I think a sort of intuition is few, for instance, I would say would express serious down. And 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. Well, computer scientists with floating point number, I mean, the floating point arithmetic. That's a great point, actually. I think in some sense, this flavor of doing math saying, we shouldn't talk about things that we cannot specify in a finite amount of time. There's something very computational in flavor about that. And it's probably not a coincidence that it becomes popular in the 30s and 40s, which is also like kind of like the dawn of ideas about formal computation, right?

You probably know the timeline better than I do. Sorry, what becomes popular? These ideas that maybe we should be doing math in this more restrictive way, where even a thing that, you know, because look, the origin of all this is like, you know, number represents a magnitude, like the length of a line. Like so, I mean, the know, number represents a magnitude like the length of a line. Like so,

I mean, the idea that there's a continuum. There's sort of like, it's like, um, is pretty old, but that, you know, just because something is old doesn't mean we can't reject it if we want to. Well, a lot of the fundamental ideas in computer science, when you talk about the complexity of problems of, of, of, and to touring himself, they rely on an infinity as well. The ideas that kind of challenge that, the whole space and machine learning, I would say, challenges that, it's almost like the engineering approach to things, like the floating point of arithmetic. The other one that, back to John Conway,

that challenges this idea. I mean, maybe to tie in the ideas of deformation theory and and limits to infinity, it's this idea of cellular automata with John Conway looking at the game of life, Stephen Wolfram's work that I've been a big fan of for a while, or cellular time. I was wondering if you have ever encountered these kinds of objects. You have ever looked at them as a mathematician where you have very simple rules of tiny little objects that when taking as a whole create incredible complexities, but are very difficult to analyze, very difficult to make sense of even though the one individual object, one part, it's like what we were saying about angel wiles, like you can look at the deformation of a small piece to tell you about the whole.

It feels like we'll sell your automata or any kind of complex systems. It's often very difficult to say something about the whole thing, even when you can precisely describe the operation of the local neighborhoods. Yeah, I mean, I love that subject. I haven't really done research in it myself. I've played around with it. I'll send you a fun blog post I wrote where I made some cool texture patterns from Celia

or Tomatata. But, and those are really always compelling. It's like, you create simple rules and they create some beautiful textures. It doesn't make any sense. I'm a software entrepreneur that I, but, and those are really always compelling, is like you create simple rules and they create some beautiful textures. It doesn't mean you sound so. Actually, do you see there was 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 was talking about, where they were learning the texture is like, let's try to like reverse engineer and like learn a software entrepreneur in the reduced texture that looks like this. From the images, very cool.

And as you say, the thing you said is, I feel the same way when I read Michigan learning paper is that what's especially interesting is the cases 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, yeah, that was a cool paper. So yeah, so let's start with the game of life. Let's start with, or let's start with John Conway. So Conway, so yeah, so let's start with John game of life, let's start with, or let's start with John Conway. So Conway, so yeah, so let's start with John Conway again. Just, I don't know, from my outsider's perspective, there's not many mathematicians that stand out throughout the history of the 20th century. And he's one of them.

I feel like he's not sufficiently recognized. I think he's pretty recognized. Okay, well, I mean, he was a full professor of Princeton for most of his life. He was sort of in certainly the pinnacle of. Yeah, but I found myself every time I talk about Conway and how excited I am about him. I have to constantly explain to people who he is. And that's that's always a sad sign to me. But that's probably true for a lot of mathematics.

I was about to say, I feel like you have a very elevated idea of how famous about it. This is what happens when you grow up in the Soviet Union, or you think the mathematicians are very, very famous. Yeah, but I'm not actually so convinced at a tiny tangent that that shouldn't be so. I mean, there's, it's not obvious to me that that's one of the, like, if I were to analyze American society that perhaps elevating mathematical scientific thinking to a little bit higher level would benefit the society. Well, both in discovering the beauty of what it is to be human and for actually creating

cool technology, better iPhones. But anyway, John Conway. Yeah, Conway is such a perfect example of somebody whose humanity was, and his personality was like a wound up with his mathematics, right? So it's not, sometimes I think people who are outside the field think of mathematics as this kind of like cold thing that you do separate from your existence as a human being. No way, your personality is in there, just as it would be in like a novel you wrote or a painting you painted or just like the way you walk down the street. Like it's in there, it's you doing it.

And Conway was certainly a singular personality. I think anybody would say that he was playful, like everything was a game to him. And now what you might think I'm gonna say and it's true is that he sort of was very playful in his way of doing mathematics, but it's also true. It went both ways. He also sort of made very playful in his way of doing mathematics. But it's also true. It went both ways.

He also sort of made mathematics out of games. He looked at, he was a constant inventor of games with crazy names. And then he was sort of analyzed those games mathematically to the point that he and then later collaborating with Knuth, created this number system, the serial numbers, in which actually each number is a game. There's a wonderful book about this called, I mean, there are his own books and then there's like a book that he wrote with Berla Camp and Guy called Winningways, which is such a rich source of ideas. And he too kind of has his own crazy number system in which by the way, there are these infinitesimals the ghosts of departed quantities. They're in there now, not as ghosts, but as like certain kind of two player games. So, you know, he was a guy. So I knew him when I was a postdoc, and I knew him at Princeton, and our research overlapped in some ways.

Now it was on stuff that he had worked on many years before, the stuff I was working on connected with stuff in group theory, which somehow keeps coming up. And so I often would ask him a question. I would come upon him in the common room and ask him a question about something. And just any time you turned him on, you know what I mean?

You sort of asked the question, it was just like turning a knob and winding him up and he would just go and you would get a response that was like so rich and with so many places and taught you so much. And usually had nothing to do with your question. Yeah, usually your question was just a prompt to him. You couldn't count on actually getting the question.

Yeah, that's brilliant, curious minds. At that age, it was definitely a huge loss. But on his game of life, which was I think he developed in the 70s, is almost like a side thing. I found the word experiment. Yeah, the game of life is this, 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 it's people played against each other. And, but essentially, it's a game that you play with marking little squares under sheet of graph paper. And 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 in 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 erased, that's called death, And there's rules for which squares are born and which squares die. It's, the rule is very simple. You can write it on one line.

And then the great miracle is that you can start from some very innocent looking little small set of boxes and get these results of incredible richness. And of course, nowadays you don't do it on paper. Nowadays you do it on a computer. It's actually a great iPad app called Gali, which I really like that has like Conway's original rule and like, gosh, like hundreds of other variants. And it's a lightning fast. So you can just be like, I want to see 10,000 generations of this rule play out like faster than your eye can even follow and it's like amazing.

So I highly recommend it if this is at all intriguing to you getting golly on your Iost device. And you can do this kind of process which I really enjoy doing which is almost from like putting a Darwin head on or a biologist head on and doing analysis of a higher level of abstraction like the organisms that spring up, because there's different kinds of organisms, like you can think of them as species, and they interact with each other.

They can, there's gliders, they shoot different, there's like things that can travel around, there's things that can glide or guns, they can generate those gliders. They're, and you can use the same kind of language as you would about describing a biological system. So it's a wonderful laboratory and it's kind of a rebuke to someone who doesn't think that like very, very rich complex structure can come from very simple underlying laws like it definitely can't. Now here's what's interesting. If you just

picked like some random rule you wouldn't get interesting complexity. I, 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, one 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. But some do.

And I don't think we have a clear way of understanding which do and which do. I don't maybe even think see that. I don't think we have a clear way of understanding which do in which don't. I don't maybe even think see that. I don't know. No, no, it's a giant mystery. Stephen Wolfram did is, now there's a whole interesting aspect to the fact that he's a little bit of an alcatastin, the mathematics and physics community, because he's so focused on a particular work. I think if you put ego aside, which I think unfairly some people are not able to look beyond. I think his work is actually quite brilliant, but what he did is exactly this process of Darwin-like exploration is taking these very simple ideas and writing a thousand-page book on them,

meaning like, let's play around with this thing, let's see. And can we figure anything out? Spoiler alert? No, we can't. In fact, he does a challenge. I think it's like a rule 30 challenge, which is quite interesting, just simply for machine learning people, for mathematics people, is can you predict the middle column for his? It's a it's a one D cellular tomat. Can you, generally speaking, can you predict anything about how a particular rule will evolve just in the future? Very simple. Just look at one particular part of the world, just zooming in on that part.

You know, hundreds steps ahead, can you predict something? And the challenge is to do that kind of prediction so far as nobody's come up with an answer. But the point is, like, we can't, we don't have tools or maybe it's impossible. Or I mean, he has these kind of laws of your disability. They hear first to boost poetry. It's like, we can't prove these things. It seems like we can't. That's the basic, it almost sounds like ancient mathematics or something like that, where you,

like the gods will not allow us to predict a cellular automata, but that's fascinating that we can't. I'm not sure what to make of it. And there's power to calling this particular set of rules game of life as Conway did because Not exactly sure, but I think he had a sense that there's some core ideas here that are fundamental to life to complex systems to the way life emerged on earth

I'm not sure I think Conway thought that it's's something that, I mean, Conway always had to rather ambivalent relationship with the game of life because I think he saw it as, it was certainly the thing he was most famous for in the outside world. And I think that his view, which is correct, is that he had done things that were much deeper mathematically than that. And I think it always like a grieved him a bit, but he was like the game of life guy. When, you know, he proved all these wonderful theorems and like,

did I mean, created all these wonderful games, like created the surreal numbers, like, I mean, he did, I mean, he was a very tireless guy who like just like did like an incredibly variegated array of stuff. So he was exactly the kind of person who you would never want to like reduce to like one achievement, you know what I mean? Let me ask you about group theory. You mentioned a few times.

What is group theory? What is an idea from group theory that you find beautiful? Well, so I would say group theory sort of starts as the general theory of symmetry is that you know, people looked at different kinds of things and said like, as we said, like, oh, we could have maybe all there is a symmetry from left to right. Like a human being, right? Or that's roughly bilaterally symmetric as we say.

So there's two symmetries. And then you're like, well, wait, didn't I say there's just one? There's just left to right. Well, we always count the symmetry of doing nothing. We always count the symmetry that's like, there's flip and don't flip. Those are the two configurations that you can be in. So there's two. You know, something like a rectangle is bilaterally symmetric.

You can flip it left to right, but you can also flip it top to bottom. So there's actually four symmetries. There's do nothing, flip it left to right, and flip it top to bottom, or do both of those things. A square, there's even more, because now you can rotate it. You can rotate it by nine degrees. So you can't do that. That's not a symmetry at the rectangle.

If you try to rotate it at 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 is in sorting 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 a deck of card can be in each one of those is the result of applying a symmetry To the original deck. So a shuffle is a symmetry, right? You're re-ordering the cards. If if I shuffle and then you shuffle the result is some Other kind of thing you might call a double a double shuffle, which is a more complicated

Symmetry so group theory is kind of the study of the general abstract world that encompasses all of these kinds of things. But then of course, like lots of things that are way more complicated than that. Like infinite groups of symmetries, for instance. So that would be interesting, huh? Oh yeah. OK.

Well, OK. Ready? Think about the symmetries of the line. You're like, OK, I can reflect it left to right, around the origin. Okay, but I could also reflect it left to right, 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. Is it possible to say something that kind of captivates keeps being brought up by physicists, which is gauge theory, gauge symmetry, as one of the more complicated type of symmetries? Is there, is there an easy explanation of what the heck it is? Is that something that comes up on your mind at all?

Well, I'm not a mathematical physicist, but I can say this, it is certainly true that it'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 the things don't change, right? So we just, I think we talked a little bit earlier

about it should be a basic principle that a theorem that's true here is also true over there. And same for a physical law, right? I mean, if gravity is like this over here, it should also be like this over there. Okay, what that's saying is we think translation in space should be a symmetry. All the laws of physics should be unchanged. If the symmetry we have in mind is a very simple one, like translation. And so then there becomes a question like,

what are the symmetries of the actual world with its physical laws? And one way of thinking is an oversimplification, but like one way of thinking of this big shift from before Einstein to after is that we just changed our idea about what the fundamental group of symmetries were, so that things like the Lorenz contraction, things like these bizarre, relativistic phenomenon, or Lorenz would have said, oh, to make this work, we need a thing to change its shape. If it's moving nearly a speed of light, well, under the new frame of framework, it's much better. You're like, no, it wasn't changing its shape.

You were just wrong about what counted as a symmetry. Now that we have this new group, the so-called the Rens group, now that we understand what the symmetry's really are, we see it was just an illusion that the thing was changing in shape. Yeah, so you can then describe the sameness of things under this weirdness that is general relativity, for example. Yeah, still, I wish there was a simpler explanation of like, it, you know, gauge symmetries are pretty simple general concept about rulers being deformed. It is just that I, I've actually just personally been on a search, not a very rigorous

or aggressive search, but for something I personally enjoy, which is taking complicated concepts and finding the sort of minimal example that I can play around with, especially programmatically. That's great. I mean, this is what we try to train our students to do, right? I mean, in class, this is exactly what this is like best pedagogical practice. I do hope there's simple explanation, especially like I've in my sort of drunk random walk, drunk walk, whatever it is that's called, sometimes stumbling to the world of topology.

And like quickly, like, you know when you like go into a party and you realize this is not the right party for me. So whenever I go into topology, it's like so much math everywhere. I don't even know what, it feels like this is me like being a hater, I think there's way too much math. Like there are two cool kids who just wanna have, like everything is expressed to math

because they're actually afraid to express stuff simply through language. That's my hater formulation of topology. But at the same time, I'm sure that's very necessary to do sort of rigorous discussion. But I feel like, but don't you think that's what gauge geometry is like?

I mean, it's not a field I know well, but it certainly seems like. Yes, it is like that. But my problem with topology, okay? And even different geometry is like, you're talking about beautiful things. Like, if they could be visualized, it's open question if everything could be visualized, but you're talking about things that

could be visually stunning, I think. But they are hidden underneath all of that math. Like, if you look at the papers that are written in the anthropology, if you look at all the discussions on stack exchange, they're all math dense, math heavy. And the only kind of visual things that emerge every once in a while is like something like a mobius strip. Every once in a while, some kind of simple visualizations. Well, there's the vibration, there's the hop vibration or all those kinds of things that somebody, some grad student from like 20 years ago, wrote a program in

Fortran to visualize it. And that's it. And it just, you know, it makes me sad because those are visual disciplines. Just like computer vision is a visual discipline. So you can provide a lot of visual examples. I wish topology was more excited and in love with visualizing some of the ideas. I mean, you could say that, but I would say for me, a picture of the hot vibration does nothing for me. Whereas like when you're like,

oh, it's like about the quaternions, it's like a subgroup of the quaternions. And I'm like, oh, so now I see what's going on. Like why didn't you just say that? Why were you like showing me this stupid picture instead of telling me what you were talking about? Oh, yeah, yeah. I'm just saying, nobody goes back

to what we were saying about teaching that people are different in what they'll respond to. So I think there's no, I mean, I'm very opposed to the idea that there's one right way to explain things. I think there's a huge variation in like, you know, our brains have all these weird hooks and loops and it's like very hard to know like what's gonna latch on

and it's not gonna be the same thing for everybody. So, that's... I think monoculture is bad, right? I think that's, and I think we're agreeing on that point that like it's good that there's like a lot of different ways in and a lot of different ways to describe these ideas because different people are gonna find different things illuminating.

But that said, I think there's a lot to be discovered when you force little silos of brilliant people to kinda find a middle ground or aggregate or come together in a way. So there's people that do love visual things. I mean, there's a lot of disciplines, especially in computer science, that are obsessed with visualizing data,

visualizing neural networks. I mean, neural networks in themselves are fundamentally visual. There's a lot of working computer vision. That's very visual. And then coming together with some folks that were like deeply rigorous and are like totally lost in multidimensional space where it's hard to even bring them back down to 3D. They're very comfortable in this

multidimensional space so forcing them to kind of work together to communicate because it's not just about public communication of ideas. It's also I feel like when you're forced to do that public communication like you did with your book, I think deep profound ideas can be discovered. That's like applicable for research and for science. Like there's something about that simplification, not simplification, but distillation or condensation or whatever the hell you call it, compression of ideas that somehow actually stimulates creativity. And I'd be excited to see more of that in the mathematics community. Can you... Let me make a crazy metaphor. Maybe it's a little bit like the relation between pros and poetry.

You might say, like, 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, like 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 in prose into sort of a different mode that is going to hit in a different way. We talked about punk-array conjecture. There's a guy, he's Russian, Gidegoi Perlman. He proved punk-array conjecture. If you can comment on the proof itself, if that stands out to you, something interesting, or the human story of it,

which is, you turn down the field's metal for the proof. Is there something you find inspiring or insightful about the proof itself or about the man? Yeah, I mean, one thing I really like about the proof, and partly that's because it's sort of a thing that happens again and again in this book. I mean, I'm writing about geometry in the way it sort of appears in all these kind of real world problems. And, but it happens so often that the geometry you think you're studying is somehow not enough.

You have to go one level higher in abstraction and study a higher level of geometry. And the way that plays out is that, you know, Poincaré asks a question about a certain kind of three-dimensional object. Is it the usual three-dimensional space that we know or is it some kind of exotic thing? And so of course, this sounds like it's a question

about the geometry of the three-dimensional space. But no, parallel man 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 the vacuum. It happens as the culmination of a program that involves many people. Same with Wiles, by the way.

I mean, we talked about Wiles, and I want to emphasize that starting all the way back with Kummer, who I mentioned in the 19th century, but Gerhard Fry and Mazer and Ken Ribbitt, and like many other people are involved in building the other pieces of the arch before you put the keystone in. We stand on the shoulders of jazz. 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 three-dimensional object itself is relevant, but the real geometry you have to understand is the geometry of the space of all three-dimensional geometries. Whoa. You're going up a higher level, because when you do that, you can say, now let's trace out a path in that space. There's a mechanism called Regi-flow. Again, we're outside my research area, so we're 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 it.

So the Regi flow allows you to say, like, okay, let's start from some mystery three dimensional space, which Pwonco Ray 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 in its natural path according to some almost physical process and ask where

it winds up. And what you find is that it always winds up. You've continuously deformed it. There's that word deformation again. And what you can prove is that the process doesn't stop until you get to the usual three-dimensional space. And since you can get from the mystery process doesn't stop until you get to the usual three-dimensional space. And since you can get from the mystery thing to the standard space by this process of continually changing and never kind of having any sharp transitions, then the original

shape must have been the same as the standard shape. That's the nature of the proof. Now, of course, it's incredibly technical. I think, as I understand it, I think the hard part is proving that the favorite word of AI people, you don't get any singularities along the way. But of course, in this context, singularity just means acquiring a sharp kink.

It just means becoming non-smooth at some point. So, just saying something interesting about formal about the smooth trajectory through this weird space of boundaries. Yeah, but that's 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 kind of like being jerked out of flatland, right? Same idea. It's only by sort of seeing the whole thing globally at once that you can really make progress on understanding

like the one thing you thought you were looking at. It's a romantic question, but what do you think about him turning down the field metal? Is that just our Nobel prizes in field metals, just the cherry on top of the cake and really math itself, the process of curiosity, of pulling at the string of the mystery before us.

That's the cake and then the words are just icing and clearly I've been fasting and I'm hungry. But do you think it's tragic or just a little curiosity that he turned on the metal? Well, it's interesting because on the one hand, I think it's absolutely true that right in some kind of like vast spiritual sense, like awards are not important, looking not important the way that sort of like understanding the universe is important. On the other hand, most people who are offered that prize accepted, you know, it's it is 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 have sometimes turn down prizes for ideological reasons. Probably more often in mathematics. I mean, I think I'm right in saying that Peter Schultz like turned down sort of some big monetary prize because he just, you know what I mean, I think he, at some point you have plenty of money. And maybe you think it sends the wrong message about what the point of doing mathematics is. I do find that there's most people accept. Most people give it a prize,

most people take it. I mean, people like to be appreciated, but like I said, we're people. Not that different from most other people. But the important reminder that that turning down the prize serves for me is not that there's anything wrong with the prize and there's something wonderful about the prize, I think. The Nobel Prize is trickier, because so many Nobel prizes are given. First of all, the Nobel Prize often forgets many, many of the important people throughout history. Second of all, there's like these weird rules to it.

There's only three people and some projects have a huge number of people and it's like this. It, I don't know, it doesn't kind of highlight the way science is done on some of these projects in the best possible way. But in general, the prizes are great. What this kind of teaches me and reminds me is sometimes in your life, there'll be moments when the thing that you you would really like to do society would really like you to do is the thing that goes against something you believe in whatever that is some kind of principle and stand your ground

in the face of that is, I believe most people will have a few moments like that in their life, maybe one moment like that, and you have to do it, that's what integrity is. So it doesn't have to make sense to the rest of the world but to stand on that, like to say no. It's interesting, because I think- But do you know that he turned down the prize

in the service of some principle? Because I know that. Well, yes, that seems to be the inkling, but he has never made it super clear. But the inkling is that he had some problems with the whole process of mathematics that includes awards, like this hierarchies and the reputations and all those kinds of things

and individualism that's fundamental to American culture. He probably, because he visited the United States quite a bit, that he probably, you know, it's, it's like all about experiences. And he may have had, you know, some parts of academia, some pockets of academia can be less than inspiring, perhaps sometimes, because the individual ego is involved, not academia, people in general, smart people with egos.

And if they, if you interact with a certain kinds of people, you can become cynical too easily. I'm one of those people that I've been really fortunate to interact with incredible people at MIT and academia in general, but I've met some assholes. And I tend to just kind of when I run into difficult folks, I just kind of smile and send them all my love and just kind of go around. But for others, those experiences can be sticky. They can become cynical about the world when folks like that exist. So he may have become a little bit cynical about the process of science. Well, you know, it's a good opportunity. Let's posit that that's his reason because I truly don't know.

It's an interesting opportunity to go back to almost the very first thing we talked about the idea of the mathematical Olympiad because of course that is So the international mathematical and be add is a competition for high school students solving math problems and In some sense, it's absolutely false to the reality of mathematics. Because just as you say, it is a contest where you win prizes. The aim is to sort of be faster than other people. And you're working on sort of canned problems that someone already knows the answer to, like not problems that are unknown.

So, you know, in my own life, I think when I was in high school, I was like very motivated by those competitions. And I went to the Math Olympiad and you won it. I always and got, I mean, well, there's something I have to explain to people because it says, I think it says on Wikipedia that I won a gold medal and in the real Olympics, they only give one gold medal in each event. I just have to emphasize that the international Math Olympiad is not like that. The gold medals are awarded to the top 112th of all participants. So sorry to bust the legend or anything.

Well, you had exceptional performance in terms of achieving high scores and the problems and they're very difficult. So you've achieved a high level performance on the in this very specialized scale. And by the way, it was very it was a very cold war activity. You know, when I in 1987, the first year I went, it was in Havana. Americans couldn't go to Havana back then. It was a very complicated process to get there. And they took the whole American team on a field trip to the Museum of American Imperialism in Havana, so we could see what America was all about. How would you recommend a person learn math?

So somebody who's young or somebody my age or somebody older who've taken a bunch of math but wants to rediscover the beauty of math and maybe integrate it into their work more solid in the resource space and so on. Is there something you could say about the process of incorporating mathematical thinking into your life? I mean, the thing is, it's in part a journey of self-knowledge.

You have to know what's going to work for you, and that's going to be different for different people. So there are totally people who, at any stage of life, just start reading math textbooks. That is a thing that you can do, and it works for some people and not for others. For others, a gateway is, you know, I always recommend like the books of Martin Gardner, another sort of person we haven't talked about,

but who also like Conway embodies that spirit of play. He wrote a column in Scientific American for decades, called Mathematical Recreations, and there's such joy in it and such fun. And these books, the columns are collected into books and the books are old now, but for each generation of people who discover them, they're completely fresh. And they give a totally different way into the subject than reading a formal textbook, which for some people would be the right thing to do.

And working contest style problems too, those are bound to books, like especially like Russian and Bulgarian problems, right? There's book after book problems from those contacts. That's gonna motivate some people. For some people, it's gonna 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 wanna 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 the incredible, overflowing vault of your ignorance, right? So I told Joe Harris, who was an algebraic geometry professor in my department, I was like, I really feel like I don't know enough and I should just like take a year of leave

and just like read EGA, the Holy Textbook, and I'm all into geometry, I was like the elements of algebraic geometry. It's like, I'm just, I feel like I don't know enough so I was just gonna sit and like read this like, 1500 page, many volume book. And he was like, and the professor here was like, that's a really stupid idea.

And I was like, why is that a stupid idea? Then I would know more algebraic geometry. It's like, because you're not actually gonna do it, like you learn, I mean, he knew me well enough to say, like, you're gonna learn because you're gonna be working on a problem, and then there's gonna be a fact from HGA, you need in order to solve your problem that you wanna to solve and that's how you're gonna learn it. You're not gonna learn it without a problem to bring you into it. And so for a lot of people,

I think if you're like, I'm trying to understand machine learning and I'm like, I can see that there's sort of some mathematical technology that I don't have. I think you like let that problem that you actually care about drive your learning. I mean, one thing I've learned from advising students, math is really hard. In fact, anything that you do right is hard. And because it's hard, 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 might have some idea that somebody else gives you, oh, I should learn x, y, and z.

Well, if you don't actually care, you're not going to do it. You might feel like you should. Maybe somebody told you you should. But I think you have to hook it to something that you actually care about. So for a lot of people, that's the way in. You have an engineering problem you're trying to handle. You have a physics problem you're trying to handle. You have a machine learning problem you're trying to handle. You have a machine learning problem, you're trying to handle. Let that not a kind of abstract idea

of what the curriculum is, drive your mathematical learning. And also just a brief comment that math is hard, there's a sense to which hard is a feature, not a bug. In a sense that, again, this is maybe my own learning preference, but I think it's a value to fall in love with the process of doing something hard, overcoming it, and becoming a better person because I hate running. I hate exercise to bring it down to the simplest hard.

I enjoy the part once it's done. The person I feel like in the rest of the day once I've accomplished it. The actual process, especially the process of getting started in the initial, I don't feel like doing it. I really feel about running as the way I feel about anything difficult in the intellectual space, especially in mathematics, but also just something that requires like holding a bunch of concepts in your mind with some uncertainty, like where this the terminology or the notation is not very clear. And so you have to kind of hold all those things together and like keep pushing forward

to the frustration of really like obviously not understanding certain like parts of the picture, like you're giant missing parts of the picture, like you're missing parts of the picture, and still not giving up. It's the same way I feel about running. And there's something about falling in love with the feeling of after you went to the journey of not having a complete picture. At the end, having a complete picture,

and then you get to appreciate the beauty, and just remembering that it sucked for a long time picture at the end, having a complete picture. And then you get to appreciate the beauty. And just remembering that it sucked for a long time and how great it felt when you figured it out, at least at the basic. That's not sort of research thinking because with research, you probably also have to enjoy the dead ends with learning math from a textbook or from video, there's a nice. I think you have to enjoy the dead ends, but I think you have to accept the dead ends. Let me just let's put it that way. Well, yeah, enjoy the suffering of it.

The way I think about it, I do, there's an effort. I don't enjoy this suffering. It pisses me off by the exact, it's part of the process. It's interesting. There's a lot of ways to kind of deal with that dead end. There's a guy who's the ultra marathon runner, Navy SEAL, David Goggins, who kind of, I mean, there's a certain philosophy of like most people would quit here. And so if most people would quit here, and I don't, I'll have an opportunity to discover something beautiful that others haven't yet. And so if most people would quit here, and I don't,

I'll have an opportunity to discover something beautiful that others haven't yet. So like, anything, any feeling that really sucks, it's like, okay, most people would just like, go do something smarter. And if I stick with this, I will discover a new garden of fruit trees that I can pick. Okay, you say that, but like, what about the guy who like wins the Nathan's hot dog eating

contest every year? Like, when he eats his 35th hot dog, he like, correctly says, like, okay, most people would stop here. Like, are you like lauding that he's like, no, I'm gonna eat the 30th hot dog. I am. I am. In the long, in the long he's like, no, I'm gonna eat the 30s. I am. I am. Okay. In the long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long, long you know, I have kids, so this is actually a live issue for me, right? I actually, it's not a thought of it, but I actually do have to give advice to two young people all the time. They don't listen, but I still give it. You know, one thing I often say to students, I don't think I've actually said this to my kids yet, but I say to students a lot is, you know, you come to these decision points and everybody is beset

myself out, right? It's like not sure like what they're capable of, like not sure what they really want to do. I always, I sort of tell people like often when you have a decision to make one of the choices is the high self-esteem choice. And I always make the high self-esteem choice, make the choice, sort of take yourself out of it and like if you didn't have those,

you can probably figure out what the version of you feels completely confident would do. And do that and see what happens. And I think that's often like pretty good advice. That's interesting, sort of like, you know, like with Sims, you can create characters. I could create a character of yourself that lacks all of the self-doubt. Right, but it doesn't mean I would never say to somebody, you should just go have high self-esteem. 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. That's a really nice way to put it. Yeah, that's a, that's a like, from a third person perspective, take the part of your brain that wants to do big things. What would they do? That's not afraid to do those things. What would they do?

Yeah, that's, that's really nice. That's actually a really do those things. What would they do? Yeah. That's really nice. That's actually a really nice way to formulate it. It's very practical advice. You should give it to your kids. Do you think there's meaning to any of it from a mathematical perspective? This life. If I were to ask you, we talked about primes, talking about proving stuff. Can we say, and then the

book that God has that mathematics allows us to arrive at something about in that book? There's certainly a chapter on the meaning of life in that book. Do you think we humans can get to it? And maybe if you were to write Clifnose, what do you suspect those Clifnose would say? 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 are, it's a lot for you. Like ask a rabbi. No, I mean, I wrote a lot in the last book, not to be wrong. I wrote a lot about Pascal, a fascinating guy who is a very serious religious mystic as well as being an amazing mathematician. And he's well known for Pascal's wager.

I mean, he's probably among all mathematicians, he's the one who's best known for this. Can you actually apply mathematics to kind of these transcendent questions? But what's interesting when I really read Pascal about what he wrote about this, you know, I started to see that people often think, oh, this is him saying, I'm gonna use mathematics to sort of show you why you should believe in God. You know, to really, that's,

this mathematics has the answer to this question. But he really doesn't say that. He almost kind of says the opposite. If you ask Blaze Pascal, like, why do you believe in God? It's, he'd be like, oh, because I met God. You know, he had this kind of like psychedelic experience, it's like a mystical experience where, as he tells it,

he just like directly encountered God. And it's like, okay, I guess there's a God. I met him last night. So that's it. That's why I believe it didn't have to do with any kind. You know, the mathematical argument was like about certain reasons for behaving in a certain way. But he basically said like, look, like math doesn't tell you

that God's there or not. Like, if God's there, he'll tell you, you know, you know, I love this. So you have, you have mathematics, you have what do you, what do you have like a waste explore the mind? Let's say psychedelics. You have like incredible technology. You also have love and friendship and like what, what the hell do you want to know what the meaning of it all is just enjoy it?

I don't think there's a better way to end it. Jordan this was a fascinating conversation. I really love the way you explore math in your writing. The willingness to be specific and clear and actually explore difficult ideas, but at the same time stepping outside and figuring out beautiful stuff. And I love the chart at the opening of your new book that shows the chaos, the mess that is your mind. Yes, this is what I was trying to keep in my head all at once. 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. And many of the things we talked about are on this map. The connections are yet to be fully dissected and investigated. And yes, God is in the picture. Right on the edge, right on the edge, not in the center.

Thank you so much for talking to me as a huge honor that you would waste your valuable time with me. Thank you, Lex. We went to some amazing places today. This is really fun. Thanks for listening to this conversation with Jordan Allenberg. And thank you to Secret Sauce, ExpressVPN, Blinkist, and indeed. Check them out in the description to support this podcast. And now let me leave you with some words from Jordan in his book, How Not to Be Wrong. Knowing mathematics is like wearing a pair of X-ray specs that reveal hidden structures underneath the messy and chaotic surface of the world.

Thank you for listening and hope to see you next time. you