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

1. 0:00 – 1:01

   Introduction

   1. LFLex Fridman0:00

      The following is a conversation with Jordan Ellenberg, a mathematician at University of Wisconsin and an author who masterfully reveals the beauty and power of mathematics in his 2014 book How Not To Be Wrong, and his new book just released recently called SHAPE: The Hidden Geometry of Information, Biology, Strategy, Democracy, and Everything Else. Quick mention of our sponsors: Secret Sauce, ExpressVPN, Blinkist, and Indeed. Check them out in the description to support this podcast. As a side note, let me say that geometry is what made me fall in love with mathematics when I was young. It first showed me that something definitive could be stated about this world through intuitive, visual proofs. Somehow that convinced me that math is not just abstract numbers devoid of life, but a part of life, part of this world, part of our search for meaning. This is the Lex Fridman podcast, and here is my conversation with Jordan Ellenberg.

2. 1:01 – 4:38

   Mathematical thinking

   1. LFLex Fridman1:01

      If the brain is a cake-

   2. JEJordan Ellenberg1:04

      It is? (laughs)

   3. LFLex Fridman1:05

      Okay. (laughs) Well, let's just, let's just go with me on this, okay?

   4. JEJordan Ellenberg1:07

      Okay. We'll pause it.

   5. LFLex Fridman1:08

      So for Noam Chomsky, um, language, the universal grammar, the framework from which language springs i- 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, um, maybe consciousness is just like a cherry on top. Uh, f- where do you put in this cake, uh, mathematical thinking? Is it as fundamental as language in the Chomsky view? Is it more fundamental than language? Is it echoes of the same kind of abstract framework does he- he's thinking about in terms of language that they're all, like, really tightly interconnected?

   6. JEJordan Ellenberg1:53

      That's a really interesting question. You're getting me to reflect on this question of whether the feeling of producing mathematical output, if you want, is like the process of, you know, uttering language or producing linguistic output. I think it feels something like that, and it's certainly the case... Let me put it this way. It's hard to imagine doing mathematics in a completely non-linguistic way. It's hard to imagine doing mathematics without talking about mathematics and sort of thinking in propositions. But, you know, maybe it's just because that's the way I do mathematics (laughs) and maybe I can't imagine it any other way, right? It's a...

   7. LFLex Fridman2:29

      Well, what about visualizing shapes, visualizing concepts to which language is not obviously attachable?

   8. JEJordan Ellenberg2:38

      Ah, that's a really interesting question. And, you know, one thing it reminds me of is one thing I talk about, uh, in the book, is dissection proofs, these very beautiful proofs of geometric propositions. Um, there's a very famous one by Bhaskara of the, the Pythagorean theorem. Um, proofs which are purely visual, proofs where you show that, uh, two quantities are the same by taking the same pieces and putting them together one way, uh, 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. Um, you know, there's a, there's a famous story, and it's a little bit disputed about how accurate this is, but that in Bhaskara's manuscript he sort of gives this proof, just gives the diagram, and then the, uh, the entire, uh, verbal content of the proof is he just writes under it, "Behold."

   9. LFLex Fridman3:28

      (laughs)

   10. JEJordan Ellenberg3:28

       Like, that's it. (laughs) And it's like... Um, there, there's some dispute about exactly how accurate that is, but- (laughs) So then that's an interesting question. Um, if your proof is a diagram, if your proof is a picture, or even if your proof is like a movie of the same pieces like coming together in two different formations to make two different things, is that language? I'm not sure I have a good answer. What do you think?

   11. LFLex Fridman3:47

       I, 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, more like to me language is a process and math is a process. It's not a, it's not just specific symbols. It's, uh, it's in action. I- i- it's, it's ultimately created through action, through change, and, uh, 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 at the end of the chapter, and then it goes on, and on and on

3. 4:38 – 9:15

   Geometry

   1. LFLex Fridman4:38

      in that kind of way. B- but I gotta ask you about geometry, and it's a prominent topic in your new book SHAPE. So for me, geometry is the thing, just like as you're saying, made me fall in love with mathematics when I was young. So being able to prove something visually just did something to my brain that... It had this... It planted this hopeful seed that you can understand the world, like, perfectly. Uh, maybe it's an OCD thing. But from a mathematics perspective, like, humans are messy, the world is messy, biology's 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 (laughs) fall in love with geometry. So, uh, how do you, uh, how do you think about geometry? Why is it a special field in mathematics?

   2. JEJordan Ellenberg5:36

      (laughs)

   3. LFLex Fridman5:36

      And h- uh, how did you fall in love with it, if you have?

   4. JEJordan Ellenberg5:40

      Wow, you've given me, like, a lot to say.

   5. LFLex Fridman5:41

      (laughs)

   6. JEJordan Ellenberg5:42

      And certainly the experience that you describe is so typical, but there's two versions of it. Um, you know, one thing I say in the book is that geometry is the cilantro of math.

   7. LFLex Fridman5:51

      (laughs)

   8. JEJordan Ellenberg5:51

      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 wasn't it all like that?" There's other people, I can tell you, 'cause 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 a number, I figured it out. But then there was this geometry, like, what was that? What happened that year? Like, I didn't get it. I was, like, lost the whole year and I didn't understand, like, why we even spent the time doing that. So, um, but what everybody agrees on is that it's somehow different.

   9. LFLex Fridman6:21

      Mm-hmm.

   10. JEJordan Ellenberg6:22

       Right? There's something special about it. Um, we're gonna walk around in circles a little bit, but we'll get there. You asked me, um, how I fell in love with math. I have a story about this. Um, when I was a small child, I don't know, maybe like I was six or seven, I don't know. Um, I'm from the '70s. I think you're from a different decade than that. But, you know, in the '70s we had, um, you had a cool wooden box around your stereo. That was the look. Everything was dark wood. Uh, and the box had a bunch of holes in it-

   11. LFLex Fridman6:53

       Mm-hmm.

   12. JEJordan Ellenberg6:53

       ... to let the sound out.

   13. LFLex Fridman6:54

       Yeah.

   14. JEJordan Ellenberg6:55

       Um, and the holes were in this rectangular array, a six by eight array, um, of holes. And I was just kinda like, you know, zoning out in the living room as kids do, looking at this six by eight rectangular array of holes. And if you, like, just by kind of like focusing in and out, just by kinda l- 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."

   15. LFLex Fridman7:25

       Whoa.

   16. JEJordan Ellenberg7:26

       So, eight sixes-

   17. LFLex Fridman7:27

       Mm-hmm.

   18. JEJordan Ellenberg7:28

       ... and six eights. It's just like the dissection booth we were just talking about. But it's the same holes.

   19. LFLex Fridman7:32

       Yeah.

   20. JEJordan Ellenberg7:32

       It's the same 48 holes. That's how many there are. No matter whether you count them as rows or count them as columns. And this was like, unbelievable to me. Am I allowed to cuss on your podcast? I don't know if that's-

   21. LFLex Fridman7:43

       Uh, fuck yes. (laughs)

   22. JEJordan Ellenberg7:44

       Are we FCC regulated? Okay, it was fucking unbelievable. Okay, that's the last time-

   23. LFLex Fridman7:48

       Get it in there.

   24. JEJordan Ellenberg7:48

       This story merits it.

   25. LFLex Fridman7:49

       So, two different perspectives on the same physical reality.

   26. JEJordan Ellenberg7:54

       Exactly. And it's just as you say. Um, you know, I knew that six times eight was the same as eight times six, right? I knew my times tables. Like, I knew that that was a fact. But did I really know it until that moment? That's the question. Right? I knew that, I sort of knew that the times table was symmetric, but I didn't know why that was the case until that moment. And in that moment, I could see like, oh, I didn't have to have somebody tell me that. That's information that you can just directly access. That's a really amazing moment. And as math teachers, that's something that we're really trying to bring to our students. And I was one of those who did not love the kind of Euclidean geometry ninth grade class of like, prove that an isosceles triangle has equal angles at the base. Like, this kind of thing. It didn't vibe with me the way that algebra and numbers did. Um, 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, the intertwining of algebra and geometry. This algebraic fact that, well, in the instance, eight times six is equal to six times eight, but in general, that whatever two numbers you have, you multiply them one way and it's the same as if you multiply them in the other order. Um, it attaches it to this geometric fact about a rectangle, which in some sense makes it true. So, you know, who knows? Maybe I was always fated to be an algebraic geometer, which is what I am as a, as a researcher.

4. 9:15 – 19:46

   Symmetry

   1. JEJordan Ellenberg9:15

   2. LFLex Fridman9:15

      So, that's a kinda transformation, and you talk about symmetry in your book. What the heck is symmetry? W- what the heck is these kinds of transformation on objects that, uh, once you transform them they seem to be similar? Uh, what do you make of it? What's its use in mathematics, or maybe broadly in understanding our world?

   3. JEJordan Ellenberg9:35

      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, was actually, um, one of my first loves in mathematics, what I thought about a lot when I was in college. But the notion of symmetry is actually much more general than the things that we would call symmetry if we were looking at like a classical building or a painting or, uh, or something like that. Um, you know, nowadays in, in math, um, we could use a symmetry to ph- to refer to any kind of transformation of an image or a space or an object. Um, you know, so what I talk about in, in the book is, um, take a figure and stretch it vertically. Make it twice as... Make it twice as big vertically and make it half as wide. Um, 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, um, and it gets kind of... I call this in the book a scronch. I just made up, had to make up some sort of funny sounding name for it-

   4. LFLex Fridman11:14

      (laughs)

   5. JEJordan Ellenberg11:14

      ... 'cause it doesn't really have, um, a name. Um, 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, um, you could study what kinds of things are preserved by this kind of scronched symmetry, and this kind of more general idea of what a symmetry can be.

   6. LFLex Fridman11:40

      Yeah.

   7. JEJordan Ellenberg11:41

      Um, l- let me put it this way. Um, a fundamental mathematical idea, in some sense I might even say the idea that dominates contemporary mathematics. Where by contemporary, by the way, I mean like the last like 150 years. We're on a very long time scale-

   8. LFLex Fridman11:55

      Mm-hmm.

   9. JEJordan Ellenberg11:55

      ... (laughs) in math. I don't mean like yesterday. I mean like A century or so up till now. It's 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. Um, are those the same or different? Well, you might say, like, "Well, look, there's two different things. This one's over here, this one's over there." On the other hand, if you prove a theorem about this one, it's probably still true about this one, if it has, like, all the same side lengths and angles and, like, looks exactly the same. The term of art, if you wanted, 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.

   10. LFLex Fridman12:40

       Mm-hmm.

   11. JEJordan Ellenberg12:40

       And we want all of our theories to be translation invariant. What that means is that if you prove a theorem about a thing, if it'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 wanna study are invariant under those symmetries is absolutely fundamental. Boy, this is getting a little abstract, right?

   12. LFLex Fridman13:04

       It's not at all abstract.

   13. JEJordan Ellenberg13:05

       Okay.

   14. LFLex Fridman13:05

       I think this is, this is completely central to everything I think about in terms of artificial intelligence. I don't know if you know about the MNIST dataset with handwritten digits.

   15. JEJordan Ellenberg13:15

       Yeah.

   16. LFLex Fridman13:16

       And, uh, you know, I don't smoke much weed or any really, but it certainly feels like it when I look at MNIST and think about the stuff, which is like, what's the difference between one and two, and why are all the twos similar to each other? What kind of transformations are within the, the category of what makes a thing the same, and what kind of transformations are those that make it different? And symmetry is core to that. In fact, our, uh, 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. Uh, ideas of symmetry of like, what, like, playing with objects, we're able to see things that are the same and not. And, uh, construct almost like little geometric theories of what makes things the same and not. And how to make, uh, programs do that in AI is a total open question. And so I kinda stare at it and wonder, uh, how, what kind of symmetries are enough to solve the MNIST handwritten digit recognition problem and write that down.

   17. JEJordan Ellenberg14:32

       Exactly. And what's so fascinating about the work in that direction from the point of view of a, of a mathematician like me and a geometer, um, is that the kind of groups and, of symmetries, the types of symmetries that we know of, um, are not sufficient, right? So, in other words, like, w- we're just gonna keep on going into the weeds on this. Uh...

   18. LFLex Fridman14:50

       Let's go (laughs) . The deeper, the better.

   19. JEJordan Ellenberg14:53

       Uh, you know, a kind of symmetry that we understand very well is rotation.

   20. LFLex Fridman14:56

       Yeah.

   21. JEJordan Ellenberg14:56

       Right? So here's what would be easy. If, if, 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 di- digit one. Um, whatever we're doing when we recognize the digit one and distinguish it from the digit two, it's not that. It's not just incorporating, uh, 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, I think what we're doing is actually more complicated than that. I feel like it must be. Um-

   22. LFLex Fridman15:52

       But they're so simple, these numbers. I mean, they're really geometric objects, like we can draw out one, two, three. Yeah.

   23. JEJordan Ellenberg15:58

       It does seem like it's for- 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?

   24. LFLex Fridman16:11

       Yeah, but that's, there is a moment, like I've, uh, myself have written programs that literally morph twos into threes-

   25. JEJordan Ellenberg16:19

       Yeah.

   26. LFLex Fridman16:19

       ... and so on. And you watch, and there's moments that you notice depending on the trajectory of that transformation, that morphing, that it, uh, it is a three and a two. There's a hard line.

   27. JEJordan Ellenberg16:33

       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 happens?

   28. LFLex Fridman16:39

       That's an interesting question. I think so.

   29. JEJordan Ellenberg16:39

       'Cause I would be surprised.

   30. LFLex Fridman16:40

       I think so.
5. 19:46 – 27:26

   Math and science in the Soviet Union

   1. LFLex Fridman19:46

      The Cold War is, uh, I think fundamental to the way people saw science and math in, uh, the Soviet Union. I don't know if that was true in the United States, but certainly was in the Soviet Union. Uh-

   2. JEJordan Ellenberg19:58

      It definitely was, and I would love to hear more about how it was in the Soviet Union.

   3. LFLex Fridman20:01

      I mean, there is, uh... And we'll talk about the, uh, the Olympiad. I, I just remember that there was this feeling like the world hung in a balance, and you could save the world with the tools of science. And mathematics was, like, the superpower that fuels science. And so, like, people were seen as, uh... Y- 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... Uh, what people really enjoy about sports, and I love sports, is, like, excellence at the highest level. But when you take that with mathematics and science, people also enjoyed excellence in science and mathematics in the Soviet Union, but there's an extra sense that that excellence will lead to a better world. So, that created, uh, all the usual things you, you think about with the Olympics, which is, like, extreme competitiveness, right? But it also created this sense that, uh, in the modern era in America, somebody like Elon Musk, uh, whatever (laughs) y- you think of him, like Jeff Bezos, those folks, they inspire the possibility that one person or a group of smart people can change the world. Like, not just be good at what they do, but actually change the world. Mathematics was at the core of that. Uh, and I- I don't know, there's a romanticism around it too. Like, when you read, uh, books about... In America, y- people romanticize certain things like baseball, for example. There's, like, these beautiful poetic, uh, writing about the game of baseball. The same was the feeling with mathematics and science in the Soviet Union, and it was, it was in the air. Everybody was forced to take high-level mathematics courses. Like, they, you took a lot of math, you took a lot of science, and a lot of, like, really rigorous literature. Like, they, the, the level of education in Russia, this could be true in China, I'm not sure, uh, in, in a lot of countries, is, uh, in, um, whatever that's called. It's K to 12 in, in America, but like young people education. The level they were challenged to- to- to learn at is incredible. It's like America falls far behind, I would say. America then quickly catches up and then exceeds everybody else at the, like the, uh, as you start approaching the end of high school to college. Like, the university system in the United States arguably is the best in the world. But, like, what, what we, uh, challenge everybody. It's not just, like, the good, the A students, but everybody to learn in, in the Soviet Union was fascinating.

   4. JEJordan Ellenberg22:50

      I think I'm gonna pick up on something you said. I think you would love a book called Duel at Dawn by Amir Alexander.

   5. LFLex Fridman22:55

      Mm-hmm.

   6. JEJordan Ellenberg22:56

      Which I- I think some of the things you're responding to in, uh, what I wrote, I think I first got turned onto by Amir's work. He's a historian of math, and he writes about, uh, the story of Évariste Galois, which is a story that's well known to all mathematicians. This kind of, like, very, very romantic figure who... Uh, he really sort of, like, begins the development of this, well, this theory of groups that I mentioned earlier, this general, uh, theory of symmetries, um, 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. Um, and much of it is true, but Alexander really lays out just how much the way people thought about math in those times, in the early 19th century, was wound up with, as you say, romanticism. I mean, that's when the romantic movement takes place. And he really outlines how people were, were predisposed to think about mathematics in that way because they thought about poetry that way, and they thought about music that way. It was the mood of the era to think about we're reaching for the transcendent, we're sort of reaching for sort of direct contact with the divine. And so part of the reason that we think of Galois that way was because Galois himself was a creature of that era, and he romanticized himself.

   7. LFLex Fridman24:10

      Yeah.

   8. JEJordan Ellenberg24:10

      I mean, now, (laughs) now we know he, like, wrote lots of letters and, like, he was kind of like... I mean, in modern terms, we would say he was extremely emo. Like, that's... (laughs)

   9. LFLex Fridman24:17

      (laughs)

   10. JEJordan Ellenberg24:18

       Like, just wrote all these letters about his, like, florid feelings and, like, the fire within him about the mathematics. And, you know, so he... So it- it's just as you say, that the math history touches human history. They're never separate because math is made of people.

   11. LFLex Fridman24:32

       Yeah.

   12. JEJordan Ellenberg24:33

       I mean, that's what it's- 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, uh, the morays of the soc- of the society around us.

   13. LFLex Fridman24:44

       So, the French, the Germans and Poincaré.

   14. JEJordan Ellenberg24:47

       Yes, okay. So back to Poincaré. So-

   15. LFLex Fridman24:49

       Poincaré.

   16. JEJordan Ellenberg24:49

       ... (laughs) um, he's, you know, it's funny, this book is filled with kind of, you know, mathematical characters who often are kind of peevish or get into feuds or sort of have like weird enthusiasms, um, 'cause those people are fun to write about and they sort of like say very salty things. Poincaré is actually none of this. As far as I can tell, he was an extremely normal dude (laughs) who 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, 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. Um, I actually was like... I, I still am feeling like somebody's gonna tell me now that the book is out, like, "Oh, didn't you know about this, like, incredibly sorted episode of his life?" (laughs)

   17. LFLex Fridman25:37

       Yeah.

   18. JEJordan Ellenberg25:37

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

   19. LFLex Fridman26:12

       So, what's the three body problem? Why is that a problem?

   20. JEJordan Ellenberg26:15

       So, so the problem is to understand, um, when this motion is stable and when it's not. So stable meaning they would sort of like end up in some kind of periodic orbit. Or, 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.

   21. LFLex Fridman26:30

       So understanding two bodies is much easier.

   22. JEJordan Ellenberg26:32

       Yeah, exactly.

   23. LFLex Fridman26:33

       When you add a third, uh, (laughs) -

   24. JEJordan Ellenberg26:34

       Two bodies, they-

   25. LFLex Fridman26:35

       ... third wheel is always a problem.

   26. JEJordan Ellenberg26:35

       This is what Newton knew. Two bodies, they sort of orbit each other in some kind of a, uh, either in an ellipse, which is the stable case, you know, that's what the planets do that we know, um, or, uh, one travels on a hyperbola around the other, that's the unstable case. It sort of like zooms in from far away, sort of like whips around the heavier thing and like zooms out. Um, those are basically the two options. So it's a very simple and easy to classify story. With three bodies, just a small switch from two to three, uh, it's a complete zoo. It's the first exam- what we would say now is it's the first example of what's called chaotic dynamics.

   27. LFLex Fridman27:09

       Mm-hmm.

   28. JEJordan Ellenberg27:10

       Where the stable solutions and the unstable solutions, they're kind of like wound in among each other, and a very, very, very tiny change in the initial conditions can make the long-term behavior of the system completely different. So Poincaré was the first to recognize that that phenomenon even, uh, even existed.

6. 27:26 – 42:15

   Topology

   1. JEJordan Ellenberg27:27

   2. LFLex Fridman27:27

      What about the, uh, conjecture that carries his name?

   3. JEJordan Ellenberg27:31

      Right. So he also, um, was one of the pioneers of taking geometry, um, which until that point had been largely the study of two and three dimensional objects 'cause that's like what we see, right? (laughs) That's- those are the objects we interact with. Um, he developed that subject we now call topology. He called it analysis situs. He was a very well-spoken guy with a lot of slogans, but that name did not r- you can see why that name did not catch on. So now-

   4. LFLex Fridman28:01

      Mm-hmm.

   5. JEJordan Ellenberg28:01

      ... it's- it's called topology now. Um-

   6. LFLex Fridman28:05

      Sorry, what was it called before?

   7. JEJordan Ellenberg28:06

      Analysis situs.

   8. LFLex Fridman28:08

      Okay.

   9. JEJordan Ellenberg28:08

      Which I guess sort of roughly means like the analysis of location or something like that, like, um-

   10. LFLex Fridman28:12

       Huh.

   11. JEJordan Ellenberg28:12

       It's a, it's a Latin phrase. Um, 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, 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.

   12. LFLex Fridman28:33

       Mm-hmm.

   13. JEJordan Ellenberg28:34

       Um, he understood that if you want to study three dem- three bodies moving in space, well each, uh, 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.

   14. LFLex Fridman29:09

       Mm-hmm.

   15. JEJordan Ellenberg29:09

       That's a higher dimensional space called phase space i- and if you haven't thought about this before, I admit that it's a little bit mind-bending. But what he needed then was a geometry that was flexible enough not just to talk about two-dimensional spaces or three-dimensional spaces, but any dimensional space. So the sort of famous first line of this paper where he introduces analysis situs is, is no one doubts nowadays that the geometry of n-dimensional space is an actually existing thing, right? I think that ha- maybe that had been controversial, and he's saying like, "Look, let's face it, just because it's not physical doesn't mean it's not there, doesn't mean we shouldn't study it."

   16. LFLex Fridman29:45

       Interesting. He wasn't jumping to the physic- uh, the physical interpretation. Like it does- it can be real even if it's not perceivable to human cognition.

   17. JEJordan Ellenberg29:55

       I think, I think that's right. I think... Don't get me wrong, Poincaré never strays far from physics. He's always motivated by physics. But the physics drove him to need to think about spaces of higher dimension, and so he needed a formalism that was rich enough to enable him to do that, and once you do that, that formalism is also gonna include things that are not physical. And then you have two choices. You can be like, "Oh, well, that stuff's trash." Or, but I th- and this is more the mathematician's frame of mind, if you have a (laughs) formalistic framework that like seems really good and sort of seems to be like very elegant and work well and it includes all the physical stuff, maybe we should think about all of it. (laughs)

   18. LFLex Fridman30:30

       (laughs)

   19. JEJordan Ellenberg30:30

       Like maybe we should think about it, you know, maybe there's some gold to be mined there. Um, and indeed, like, you know, guess what? Like before long there's relativity and there's space time and like all of a sudden it's like, oh yeah, maybe it's a good idea. We already have this geometric apparatus like set up for like how to think about four-dimensional (laughs) -... spaces, like, turns out they're real after all. And so, you know, this is a- a- a story much told, right, in mathematics. Not just in this context, but in many.

   20. LFLex Fridman30:53

       I'd love to dig in a little deeper on that actually, 'cause I have some, uh, intuitions to work out (laughs) -

   21. JEJordan Ellenberg31:00

       Okay.

   22. LFLex Fridman31:01

       ... in- (laughs) in my brain. But-

   23. JEJordan Ellenberg31:02

       Well, I'm not a mathematical physicist, so we can, like work them out together. (laughs)

   24. LFLex Fridman31:05

       I- I, good. We'll, uh, we'll- we'll- we'll together walk along the path of curiosity. But Poincaré, uh, conjecture, what is it?

   25. JEJordan Ellenberg31:14

       The Poincaré conjecture is about curved three-dimensional spaces. So, I was on my way there, I promise. Um, the idea is that we perceive ourselves as living in ... we don't say a three-dimensional space, we just say three-dimensional space. You know, you can go up and down, you can go left and right, you can go forward and back. There's three dimensions in which we can move. In Poincaré's theory, there are many possible three-dimensional spaces. In the same way that going down one dimension to sort of capture our intuition a little bit more, we know there are lots of different two-dimensional surfaces, right? There's a balloon, and that looks one way, and a donut looks another way, and a Mobius strip looks a third way. Those are all, like two-dimensional surfaces that we can kind of really, uh, 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 hole, 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. Um, the Poincaré conjecture, uh, says that there's a very simple way to determine whether a three-dimensional space, um, is the standard one, the one that we're used to. Um, and essentially it's that it's what's called fundamental group has nothing interesting in it. And that- 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 be able to see this. For the people, uh, on the podcast, you'll have to visualize it. So Lex has been nice enough to like give me a surface with some interesting topology.

   26. LFLex Fridman32:50

       It's a mug.

   27. JEJordan Ellenberg32:51

       Right here in front of me. A mug, yes. I might say it's a genus-1 surface, but we could also say it's a mug. Same thing. Um, so if I were to draw a little circle on this mug, uh, which way should I draw it so it's visible? Like here? Okay.

   28. LFLex Fridman33:05

       Yeah, that's... yeah.

   29. JEJordan Ellenberg33:06

       If I draw a little circle on this mug, imagine this to be a loop of string. I could pull that loop of string closed on the surface of the mug, right? That's definitely something I could do. I could shrink it, shrink it, shrink it until it's a point. On the other hand, if I draw a loop that goes around the handle, I can kind of zhuzh it up here and I can zhuzh it down there and I can sort of slide it up and down the handle, but I can't pull it closed, can I? It's trapped.

   30. LFLex Fridman33:27

       Mm-hmm.
7. 42:15 – 46:45

   Do we live in many more than 4 dimensions?

   1. LFLex Fridman42:15

      ... Is it possible (laughs) that the universe is, uh, many more dimensions than the ones we experience as human beings? So, if you look at, um, the, you know, especially in physics, theories of everything, uh, physics theories that try to unify general relativity and quantum field theory, they seem to go to high dimensions to work stuff out through the tools of mathematics. Is it possible... So, like, the two options are one, it's just a nice way to analyze the universe, but the reality is, is as exactly we perceive it, it is three dimensional. Or... are we just seeing, are we those Flatland creatures that are just seeing a tiny slice of reality, and the actual reality is many, many, many more dimensions than the three dimensions we perceive?

   2. JEJordan Ellenberg43:10

      Oh, I certainly think that's possible. Um, now, how would you figure out whether it was true or not-

   3. LFLex Fridman43:17

      Yeah.

   4. JEJordan Ellenberg43:17

      ... is another question. Um, and I suppose what you would do, as with anything else that you can't directly perceive, is, um, 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.

   5. LFLex Fridman43:53

      Well, is it, is it possible to intuit how to mess with the other dimensions while living in a three-dimensional world? I mean, that- that seems like a very challenging thing to do. We, w- the reason Flatland could be written is because it's coming from a three-dimensional writer.

   6. JEJordan Ellenberg44:11

      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 story some context, you asked, like, is it possible for us humans to have this experience of being transcend- transcendentally jerked out of our world-

   7. LFLex Fridman44:29

      Mm-hmm.

   8. JEJordan Ellenberg44:29

      ... so we can sort of truly see it from above. Well, Edwin Abbott, who wrote the book, certainly thought so, because Edwin Abbott was a minister. So the whole Christian subtext to this book I had completely not grasped reading this as a kid, that it means a very different thing, right, if sort of a theologian is- is saying, like, "Oh, what if a higher being could, like, pull you out of this earthly world you live in so that you can sort of see the truth and, like, really see it, uh, 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.

   9. LFLex Fridman45:04

      Mm-hmm.

   10. JEJordan Ellenberg45:04

       Um, but what happens in this book, and this part now looking at it through a Christian lens it becomes a bit subversive (laughs) , 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 but three-dimensional," and the square is very excited. Then 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?"

   11. LFLex Fridman45:34

       (laughs)

   12. JEJordan Ellenberg45:34

       "There's no fourth dimension. That's ridiculous. Like, there's only, there's 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, uh, conceptualize that there could actually be (laughs) yet another dimension. So yeah, that takes the religious allegory to, like, a very weird place that I don't really, like, understand theologically, but-

   13. LFLex Fridman45:53

       Well, that's a nice way to talk about religion and myth in general, as perhaps us trying to struggle, us meaning human civilization, trying to struggle with ideas that are beyond our cognitive capabilities.

   14. JEJordan Ellenberg46:08

       But it's in fact not beyond our capability. It may be beyond our cognitive capabilities to visualize a four-dimensional cube, a tesseract some like to call it, or a five-dimensional cube, or a six-dimensional cube, but it is not beyond our cognitive capabilities to figure out how many corners a six-dimensional cube would have. That's what 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.

   15. LFLex Fridman46:37

       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,

8. 46:45 – 56:11

   How many holes does a straw have

   1. LFLex Fridman46:45

      uh, how many holes does a straw have? And you, listener, may, uh, try to answer that in your own head. (laughs)

   2. JEJordan Ellenberg46:54

      Yeah, I'm gonna take a drink while everybody thinks about it so we can give you one -

   3. LFLex Fridman46:57

      (laughs) A slow sip. Is it, uh, zero, one, or two, or more- more than that maybe? Maybe you can get very creative. But, uh, it's kind of interesting to, uh, each, uh, dissecting each answer as you do in the book, uh, is 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.

   4. JEJordan Ellenberg47:23

      Yeah, and 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 in- wonderfully comic happens, which is that everyone's like, "Well, it's completely obvious," and then each person realizes that half the person, the other people in the room have a different obvious answer-

   5. LFLex Fridman47:41

      Yeah.

   6. JEJordan Ellenberg47:41

      ... for the way they have, and then people get really heated. People are like, "I can't believe that you think it has two holes," or like, "I can't believe that you think it has one." And then, you know, you really, like, people really learn something about each other, and people get heated.

   7. LFLex Fridman47:54

      I mean, can we go through the possible options here? Is it zero, one, two, three, 10?

   8. JEJordan Ellenberg48:00

      (laughs) Sure. So I think, you know, most people... The zero holers are rare. They would say like, "Well look, uh, 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."

   9. LFLex Fridman48:13

      Mm-hmm.

   10. JEJordan Ellenberg48:13

       Uh, "I didn't poke a hole in it when I-" (laughs)

   11. LFLex Fridman48:16

       Yeah.

   12. JEJordan Ellenberg48:17

       "... so how can I have a hole?" They're like, "It's just one thing." Okay, most people don't see it that way. That's like a, um...

   13. LFLex Fridman48:23

       Is there any truth to that kind of conception?

   14. JEJordan Ellenberg48:25

       Yeah, I think that would be somebody whose account... I mean, what I would say is you could say the same thing, um, 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 smooshing the ends together. Now it's a bagel." So if you're really committed you could 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?

   15. LFLex Fridman48:54

       (laughs)

   16. JEJordan Ellenberg48:54

       I mean, I don't know.

   17. LFLex Fridman48:54

       Yeah, so that's almost like an engineering definition of it. Okay, fair enough. So what's- what about the other options?

   18. JEJordan Ellenberg49:01

       Um, so you know, one hole people...... would say, um-

   19. LFLex Fridman49:07

       I like how these are, like, groups of people. Like, we're, we've planted our foot-

   20. JEJordan Ellenberg49:11

       Yes, team one-hole.

   21. LFLex Fridman49:13

       This is what we stand for. There's books written about each belief.

   22. JEJordan Ellenberg49:16

       You know, would say, "Look, there's a hole and it goes all the way through the straw," right? There's, it's one region of space that's the hole.

   23. LFLex Fridman49:21

       Yeah.

   24. JEJordan Ellenberg49:22

       And there's one. And two-hole people would say like, "Well, look, there's a hole in the top and a hole at the bottom." Um, I think a common thing you see when people, um, argue about this, they would take something like this, uh, bottle of water I'm holding. Maybe I'll open it. And they say, "Well, how many holes are there in this?" And you say, like, "Well, there's one. There's one hole at the top." Okay, what if I, like, poke a hole here so that all the water spills out?

   25. LFLex Fridman49:48

       Yeah.

   26. JEJordan Ellenberg49:48

       Well, now it's a straw.

   27. LFLex Fridman49:50

       Yeah.

   28. JEJordan Ellenberg49:51

       So if you're a one-holer, I say to you, like, "Well, how many holes are in it now?" There was a, there was one hole in it before and I poked a new hole in it.

   29. LFLex Fridman49:58

       Mm-hmm.

   30. JEJordan Ellenberg49:59

       And then you think there's still one hole even though there was-
9. 56:11 – 1:01:57

   3Blue1Brown

   1. LFLex Fridman56:11

      So, not be talking just about ideas, but the, actually be expressing the idea. Is there, you know somebody in the, maybe you can comment, there's a guy, uh, his YouTube channel is 3Blue1Brown, Grant Sanderson. He does that masterfully well.

   2. JEJordan Ellenberg56:27

      Absolutely.

   3. LFLex Fridman56:28

      Of, uh, visualizing, of expressing a particular idea and then talking about it as well, back and forth. Uh, what do you th- what do you think about Grant?

   4. JEJordan Ellenberg56:37

      It's, it's fantastic. I mean, the flowering of math YouTube is, like, such a wonderful thing because, you know, math teaching, there's so many different venues through which we can teach people math. Th- there's the traditional one, right? What, where I'm in a classroom with, you know, depending on the class it could be 30 people, it could be 100 people, it could, God help me, be 500 people if it's like a big calculus lecture or whatever it may be. And there's sort of some, but there's some set of people of that order of magnitude, and I'm with them for, we have a long time. I'm with them for a whole semester and I can ask them to do homework and we talk together, we have office hours if they have one-on-one questions, blah, blah, blah. That's like a very high level of engagement, but how many people am I actually hitting at a time? Like, not that many.

   5. LFLex Fridman57:18

      Mm-hmm.

   6. JEJordan Ellenberg57:18

      Right? Um, and you can... And there's kind of an inverse relationship where the more enga- the, the fewer people you're talking to, the more engagement you can ask for. The ultimate, of course, is like the mentorship relation of like a PhD advisor and a graduate student where you spend a lot of one-on-one time together for like, you know, three to five years. And then, like, the ultimate high level of engagement to one person.

   7. LFLex Fridman57:44

      Mm-hmm.

   8. JEJordan Ellenberg57:45

      Um, you know, books, I can, this can get to a lot more people than are ever gonna sit in my classroom and you spend, like, uh, however many hours it takes to read a book. Uh, somebody like 3Blue1Brown or Numberphile or, um, people like Vi Hart. I mean, YouTube, let's face it, has bigger reach than a book. Like, there's YouTube videos that have many, many, many more views than, like, you know, any hardback book, like, not written by a Kardashian or an Obama is gonna sell, right? So that's...

   9. LFLex Fridman58:17

      Yeah.

   10. JEJordan Ellenberg58:17

       I mean, um, and y- and, and then, you know, those are, you know, some of them are, like, longer, 20 minutes long, some of them are five minutes long, but they're, you know, they're shorter. And even so, you look, look, like, Eugenia Cheng is 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.

   11. LFLex Fridman58:34

       Mm-hmm.

   12. JEJordan Ellenberg58:35

       She has 30 seconds, but then there's like 30 seconds to sort of say something about math- mathematics to, like, untold millions of people. So everywhere along this curve is im- is important. And one thing I feel like is great right now is that people are just broadcasting on all the channels, because we each have our skills, right?

   13. LFLex Fridman58:51

       Mm-hmm.

   14. JEJordan Ellenberg58:51

       Somehow along the way, like, I learned how to write books. I had this sort of 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. (laughs) And that's one thing, but, like, sort of being able to make, like, you know, winning, good-looking, eye-catching videos is, like, a totally different skill. And, you know, probably, you know, somewhere out there there's probably sort of some, like, heavy metal band that's, like, teaching math through heavy metal and, like, using their skills to do that. I hope there is at any rate. (laughs)

   15. LFLex Fridman59:24

       (laughs) Through 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 things he is, is basically mostly through his, uh, Manim Library in, in Python, e- everything is drawn through Python. You have to, um, you have to truly understand the topic to be able to, to visualize it in that way, and not just understand it but really kinda think in a very novel way. It's funny 'cause I- I- I've spoken with him a couple of times, I've spoken with him a lot offline as well. He really doesn't think he's doing anything new, meaning, like, he sees himself as very different from maybe, like, a researcher. But it feels to me like he's creating something totally new. Like, that act of understanding and visualizing is as powerful or has the same kind of inkling of power as does the process of proving something. You know, it, 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.

   16. JEJordan Ellenberg1:00:47

       And to be honest, it's something that I think we haven't quite figured out how to value-

   17. LFLex Fridman1:00:53

       Yeah.

   18. JEJordan Ellenberg1:00:53

       ... inside academic mathematics in the same way, and this is a bit older, that I think we haven't quite figured out how to value the development of computational infrastructure. You know, we all have computers as our partners now and people build computers that sort of assist and participate in our mathematics. They build those systems and that's a kind of mathematics too, but not in the traditional form of proving theorems and writing papers. But I think it's coming. Look, I mean, I think...You know, for example, the Institute for, uh, Computational and Experimental Mathematics at Brown, which is like a v- you know, a... It's a NSF-funded math institute, very much part of sort of traditional math academia. They did an entire theme semester about visualizing mathematics, looking at 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, uh, 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.

   19. LFLex Fridman1:01:50

       Yeah, I'm hoping to see more and more of that from, like, MIT faculty, from faculty from all the, eh, the top universities in the world.

10. 1:01:57 – 1:10:22

    Will AI ever win a Fields Medal?

    1. LFLex Fridman1:01:57

       Let me ask you this weird question about the Fields Medal, which is the Nobel Prize in mathematics. Do you think, since we're talking about computers, there will one day come a time when, uh, a computer, an AI system, will win a Fields Medal?

    2. JEJordan Ellenberg1:02:16

       No. (laughs)

    3. LFLex Fridman1:02:18

       Of cour- that's what a human would say, so why not?

    4. JEJordan Ellenberg1:02:20

       (laughs)

    5. LFLex Fridman1:02:20

       (laughs)

    6. JEJordan Ellenberg1:02:21

       Is, is that, like, th- y- that that's like my capture? That's like the proof that I'm a human, is I deny that I know?

    7. LFLex Fridman1:02:25

       (laughs) Yeah. What does (laughs) , how does he want me to answer? Is there something interesting to be said about that?

    8. JEJordan Ellenberg1:02:32

       Uh, yeah. I mean, I am tremendously interested in what AI can do in pure mathematics. I mean, it's of, of course, it's a parochial interest, right? You're like, "Why am I interested in, like, how it can, like, help feed the world (laughs) or help solve, like, real such problems?"

    9. LFLex Fridman1:02:44

       Yeah.

    10. JEJordan Ellenberg1:02:44

        I'm like, "Can it do more math? Like, but what can, (laughs) what can I do?" We all have our interests, right? Um.

    11. LFLex Fridman1:02:49

        Yes.

    12. JEJordan Ellenberg1:02:49

        But I think it is a really interesting conceptual question. And here too, I think, um, it's important to be kind of historical, because it's certainly true that there's lots of things that we used to call research in mathematics that we would now call computation.

    13. LFLex Fridman1:03:07

        Yeah.

    14. JEJordan Ellenberg1:03:07

        Tasks that we've now offloaded to machines. Like, you know, in 1890, somebody could be like, "Here's my PhD thesis. I computed all the invariants of this polynomial ring under the action of some finite group." It doesn't matter what those words mean, just it's like something that in 1890 would take a person a year to do, and would be a valuable thing that you might want to know. And it's still a valuable thing that you might want to know, but now you type a few lines of code in Macaulay or Sage, uh, or Magma, and you just have it. So we don't think of that as math anymore, even though it's the same thing.

    15. LFLex Fridman1:03:41

        What's Macaulay, Sage, and Magma?

    16. JEJordan Ellenberg1:03:43

        Oh, those are computer algebra programs. So those are, like, sort of bespoke systems that lots of mathematicians use.

    17. LFLex Fridman1:03:48

        Is that similar to Maple and Mathem-

    18. JEJordan Ellenberg1:03:49

        Yeah. Oh, yeah. So it's similar to Maple and Mathematica, yeah.

    19. LFLex Fridman1:03:51

        Okay.

    20. JEJordan Ellenberg1:03:52

        But a li- a little more specialized, but yeah.

    21. LFLex Fridman1:03:54

        It's programs that work with symbols and allow you to do... Can you do proofs? Can you do kind, kind of little, little leaps and proofs and...

    22. JEJordan Ellenberg1:04:01

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

    23. LFLex Fridman1:04:04

        But these tools are part of the process of mathematics now?

    24. JEJordan Ellenberg1:04:07

        Right. They are now, for most mathematicians, I would say, part of the process of mathematics. And so, um, you know, there's a story I tell in the book which I'm fascinated by, which is I... You know, so far, attempts to get AIs to prove interesting theorems have not done so well. It doesn't mean they can. There's actually a paper I just saw which has a very nice use of a neural net to find counterexamples to conjectures. Somebody said like, "Well, maybe this is always that."

    25. LFLex Fridman1:04:36

        Yeah.

    26. JEJordan Ellenberg1:04:37

        And you can be like, "Well, let me sort of train an AI to sort of try to find things where that's not true," and it actually succeeded. Now, in this case, if you look at the things that it found, you say like... Okay, I mean, these are not famous conjectures. (laughs)

    27. LFLex Fridman1:04:52

        Yes.

    28. JEJordan Ellenberg1:04:53

        Okay? So like, "Somebody wrote this down. Maybe this is so." Um, 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 would have come up with that."

    29. LFLex Fridman1:05:05

        Yeah, they would have solved it. Gotcha.

    30. JEJordan Ellenberg1:05:05

        There, I mean, when, 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."
11. 1:10:22 – 1:27:41

    Fermat's last theorem

    1. LFLex Fridman1:10:22

       the proof. What do you make of, if we could take a- an- another of a multitude of tangents, what do you make of Fermat's Last Theorem? Because the statement, there's a few theorems, there's a few problems that are deemed by the world throughout its history to be exceptionally difficult, and that one in particular is, uh, really simple to formulate and, uh, really hard to come up with a proof for. And it was like taunted as simple, uh, by Fermat himself. Is there something interesting to be said about that x to the n plus y to the n equals z to the n for n of three or greater, is there a solution to this, and then how do you go about proving that? Like how would you, uh, try to prove that and what did you learn from the proof that eventually emerged by Andrew Wiles?

    2. JEJordan Ellenberg1:11:12

       Yeah. So right, so to give, let me just say the background 'cause I don't know if everybody listening is-

    3. LFLex Fridman1:11:15

       Yes. They'll be wonderful.

    4. JEJordan Ellenberg1:11:16

       ... knows the story. So you know Fermat, uh, was an early number theorist. Well he's sort of an early mathematician. Those special adjacent didn't really exist back then. Um, he comes up in the book actually in the context of, um, a different theorem of his that has to do with testing whether a number is prime or not. So I write about, he was one of the ones who was salty and like he would exchange these letters where he and his correspondence would like try to top each other and vex each other with questions and stuff like this. But this particular thing, um, it's called Fermat's Last Theorem because it's a note he wrote, uh, in, in his, uh, in his copy of the Discussiones Arithmeticae. Like he wrote, "Here's an equation. It has no solutions. I can prove it but the 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 Fermat did not have a proof for this theorem. For a long time people like, you know, people were like, this mysterious proof that was lost, a very romantic story, right? But, uh, Fermat later, he did prove special cases of this theorem and wrote about it, talked to people about the problem. Uh, and 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.

    5. LFLex Fridman1:12:35

       He may have had a deep simple intuition about this, 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 to time.

    6. JEJordan Ellenberg1:12:50

       Maybe. But I think we, and so 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. He-

    7. LFLex Fridman1:13:01

       Yes.

    8. JEJordan Ellenberg1:13:01

       ... uh, 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 this very simple equation there was like a very simple, clever, wonderful proof that you could do in a page or two?" And that would be great but you know what? There's lots of equations like that that are solved by very clever methods like that, including the special cases that Fermat wrote about, the method of dissent which is like very wonderful and imp- but in the end-Those are nice things that, like, you know, you teach in an undergraduate class, um, and it is what it is, but they're not big.

    9. LFLex Fridman1:13:37

       Mm-hmm.

    10. JEJordan Ellenberg1:13:38

        Um, on the other hand, work on the Fermat problem, that's what we like to call it, because it's not really his theorem because we don't think he proved it, (laughs) so... I mean, work on the, the Fermat problem developed this, like, incredible richness of number theory that we now live in today. Like, and not, by the way, just Wiles, Andrew Wiles being the person who together with Richard Taylor finally proved this theorem. But you know how you have this whole moment that people try to prove this theorem and they fail, and there's a famous false proof by Lame from the 19th century, where Kummer, in understanding what mistake Lame had made in this incorrect proof, basically understands something incredible, which is that, you know, a thing we know about numbers is that, um, 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 two times three times two. I had to think about it. Uh, (laughs) right? It's, or it's two times two times three. Of course, you can reorder them.

    11. LFLex Fridman1:14:41

        Right.

    12. JEJordan Ellenberg1:14:41

        But there's no other way to do it. There's no universe in which 12 is something times five, or in which there's, like, four threes in it. Nope, 12 is, like, two twos and 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?

    13. LFLex Fridman1:14:56

        Mm-hmm.

    14. JEJordan Ellenberg1:14:56

        It has to be that way.

    15. LFLex Fridman1:14:58

        So that's a really powerful idea. It's, it's so cool that every number is uniquely made up of other numbers. A- and, like, made up meaning, like, there's these, like, basic atoms that form molecules, that, that for- that get built on top of each other.

Episode duration: 2:41:47