David McRaney (1:50)

There it is.

Jordan Ellenberg (1:50)

He has a lot. He was an aphoristic dude.

David McRaney (1:53)

Mathematics is the art of giving the same name to different things. Which is what I'm going to say for the rest of my life is if I understand that in totality like.

Jordan Ellenberg (2:00)

You do, you're going to understand it by the end of this discussion.

David McRaney (2:03)

Okay, I'm looking forward to it. This is.

Podcast Host / Advertiser (2:27)

Welcome to the you are not so smart podcast, episode 328.

David McRaney (2:50)

For me, the bridge of asses segues naturally into a very important question that you spend an incredible amount of time on in the book. And that question is how many holes does a straw have? And I will ask the people for people listening, take a second pause if you have to, and ask yourself, how many holes does the straw have? Ask other people around you this question too and get into it, because believe it or not, you really can get into it. So I will hand off to I will cede the floor to you, sir. How many holes does a straw have?

David McRaney (3:28)

My name is David McRaney. This is the youe Are not so Smart podcast and we will return to that question later in the show about straws and holes and stuff during our interview with Jordan Ellenberg, who is The John D. MacArthur professor of Mathematics at the University of Wisconsin, Madison. His writing has appeared in Slate, the Wall Street Journal, the New York Times, the Washington Post, the Boston Globe, and he is the New York Times best selling author of how not to Be Wrong. And in this episode we will discuss his new book, which is also a New York Times bestseller, the Hidden Geometry of Information, Biology, Strategy, Democracy and Everything Else. In the book he explores how a democracy should choose its representatives, how to stop a pandemic from sweeping the world. How do computers learn to play Go? And why is learning go so much easier for them than learning to read a sentence? Can ancient Greek proportions predict the stock market? What should your kids learn in school if they really want to learn how to think? And all of these questions, he says, are at their core questions about geometry. And to understand what he means by that, stay tuned because right now we're going to start the interview.

David McRaney (4:55)

But let me give me do ask straight away, like who are you and what do you do?

Jordan Ellenberg (4:59)

So I am Jordan Ellenberg and I am a math professor. I was always interested in math my whole life. So I've had in some ways like a very direct career course into being a kid who was like really interested in mathematics to eventually growing up into being a person who's actual job is to think about mathematics on a daily basis. But so, but also somehow, I guess accidentally, I've sort of become a writer, which is something else I was always interested in. And I only came to understand, you know, later in life that those were things that could be combined. I mean, you talked about putting it into words. And of course, you know, if you look at research mathematics, by the way, if you were to, as I'm sure you often do, kind of go down to the library and like thumb through the latest issues of the Annals of Mathematics or the Journal of the American Mathematical Society or what have you, it is mostly words that actually it's not just kind of like strings of computations and the Occasional arcane diagram. When we talk to each other about math, we're mostly sort of saying words and gesturing with our hands. And so it's not sort of so far from the interior life of mathematics to like, do this kind of work.

David McRaney (6:10)

I like that a lot because I feel like. And I'm just going to go ahead and jump into the deep end and then we'll swim back to the other stuff I like.

Jordan Ellenberg (6:17)

All right. I hope we make it okay.

David McRaney (6:18)

We might not, but it's fine. It's just the journey that matters. You were there with me along the way. I like this. I'm on a kick right now. And I've written about this in my own book about how people do and do not update their priors. At some point, I felt like I had needed to explain how brains make sense of anything. And I actually pulled a lot of material that you also talk about, about starting with sensory modalities and geometries, then moving up through propositions and how that's a prop. How a proposition even became a thing that we talked about. And I have on a kick here lately about trying to articulate the ineffable. And it doesn't really matter where it comes from. It just matters that it cross pollinates, because once you have a word for a thing, then you can build other things out of those words.

Podcast Host / Advertiser (7:03)

That's just what.

David McRaney (7:04)

That's just what mathematicians do, basically. They come up with a single doodad to represent another thing.

Jordan Ellenberg (7:09)

Yeah. And it's true. And it's an endless discussion that people have about are we inarticulately recognizing the thing and then articulating it and making a word for it, or are we in some sense actually bringing it into existence by naming it?

Jordan Ellenberg (7:28)

I think in the end, there's no answer to that question. And it's kind of an iterative process where we're kind of.

Jordan Ellenberg (7:34)

Bringing things into existence by naming them and naming the things that exist kind of at once. Like, I don't think there's. I think there's sort of, you know, as with chickens and eggs. Right. There's not really a One is not really prior to the other.

David McRaney (7:47)

You said you didn't much care for geometry and you were in a Hell's.

Podcast Host / Advertiser (7:50)

Angel math team circu.

David McRaney (7:52)

And then you leap very quickly in the book to. If you take a powerful enough dose of psychedelics, you're going to go all the way back down through the mountains and mountains of abstractions to some sort of base layer where you're going to see geometries and topologies. What's going on?

Jordan Ellenberg (8:07)

I mean, I think that what I was trying to convey is that geometry is built into us. I think that's pretty clear and I think sometimes we can lose sight of that because when we do the subject in school, college geometry, it's presented in this, as this very formal and abstract and rigid thing which you might imagine came to us from space, certainly not from ourselves. Now you may think what I'm going to say, I'm going to pivot on you a little bit. You may think I'm going to say, oh, why did we do this terrible thing of presenting this kind of like formal construct instead of this kind of primal in our bodies operation? But I'm not going to say that because I think part of the charm of the subject is that both aspects are there. In other words, like, yeah, geometry is built into our bodies. It's part of the way we perceive the world. We fundamentally are always asking, where are things? Where are they going? What do they look like? Those are all geometric questions. But if we hadn't made that leap into formality, then we would be able to do, then we would be limited, right, by what our bodies can do. We wouldn't be able to do geometry in like 10 dimensional space because we don't live in 10 dimensional space. We'd only be able to do geometry in two and three dimensional space and that would be lacking. So again, you know, Poincare is kind of this recurring figure in the book. He just kind of like pops up everywhere. He talks about it. Like if you go and see like the skeleton of a sponge, you know, under the water, like on the one hand, if there's not that rigid skeleton, the sponge can't really grow. On the other hand, if you see the skeleton after the sponge is gone and you don't recognize that there was some living being there that that is the shape of, then you're not really seeing the point of the skeleton. That's the relationship between the two things. If we just learn the formality and don't understand that that's trying to reflect something that's in us, then I can't blame somebody for being like, what the hell is the point?

David McRaney (10:01)

That's so good. And here's a quote from the book. And for me, this is what would sell it to me and I hope this sells it to everybody else. Ayahuasca. This is you quote. Ayahuasca drinkers have a similar take the drug, reboots the brain and lifts the mind above the Tortured labyrinth it thinks it's stuck in. Mmm, that's good stuff.

Jordan Ellenberg (10:23)

You know, honestly, as a writer, and you must know this, too, the best thing in the world is when there's some line you. You kind of patted yourself on the back for writing, and you're like, yeah, I nailed that one. And then if anybody else ever picks up one of those, you're like, we're friends forever now. I'm gonna send you Christmas cards. Like, now we're friends.

David McRaney (10:39)

Yes, we will. Now drink because of that line, because I feel that, too. Like, whenever I have felt the most love and passion and obsession for mathematical concepts, it came from something intuitive in this way. It plugged into whatever I feel intuitively inspired by. For me, it would be like Conway's Game of Life and Sailor Automaton, stuff like that. That I can get stuck on a rabbit hole of the Internet forever, not really understanding it on a deep level, but. But understanding it at the, like, level of, like, the weather system that. That the molecules are made of, where I'm like, okay, so little doodads adhering to the laws of physics globbing together over time. This will do that. And I start to feel the train of thought that I think Game of Life is supposed to. To inspire in you. And then I see people who have created starting conditions where they create an infinite, fractalized game of Life inside of Game of Life inside of Game of Life, or there's a Turing machine or something in it. And I start to feel whatever it must feel like to actually know what you're reading and doing in geometry. So I really appreciate what you do, even though I feel like I have. Would take me years to get to a level of being an actual novice in this world. So it's really cool that you write about it in a way for people like myself to get a lot out of it.

Jordan Ellenberg (12:03)

I can't resist saying, by the way, I'm about to waste a lot of your time, David, but if you have not already seen the iOS app, golly, I don't know who made this thing. It runs lightning fast on an iPad, and it basically, it has about 100 different lifelike rules that you can explore, or you can just kind of create your own rules and kind of, like, let a rip and sort of see what it does. I guess maybe we should explain this for people who, yeah, please do. Don't know about this kind of wonderful thing. But, you know, the Game of Life is a very simple mathematical construct invented by a fascinating guy called John Conway, who. And he appears in the book again and again, again, a sort of, like, master geometry who kind of pops up. Zelig, like. Like where you least expect it. But one of the things he's most famous for in the world, outside research, mathematics, is this thing called the game of life, where you just, like, mark little squares on a sheet of graph paper. That's how he did it. He did it on paper. Now he'd do it on a computer. And there's a rule that tells you, okay, if these squares are filled in, then at the next turn of the game, this other set of squares are filled in. And then you apply the rule again and again. The configuration of squares changes, and something rather miraculous happens because the rule is very simple. You could write it on about two lines of a sheet of paper, and it produces the most unimaginably baroque. Well, I was going to say patterns, but really, it's the change in the patterns, moving patterns. I really encourage, if you're listening to this, to get this app, golly, which runs really fast, and you can really see this thing go. And it just brings home so vividly to people the way a very short and simple set of rules can produce an astonishing amount of complexity, in a way. I mean, I don't know if Conway thought of it this way, but you can think of it as a metaphor for geometry itself. I mean, the genius of Euclid and his way of organizing the knowledge of his time was to say, hey, you can start from this very simple set of rules about what lines are and what points are and how they behave. And from there, the entire apparatus of everything we know about triangles and circles and parallelograms and what have you, it can all be built up, all this richness from this very small set of initial rules. It's amazing.

David McRaney (14:16)

It's amazing. G O L L Y is the name of the app. I just found it. Yes, I will play with this endlessly. My entry.

Jordan Ellenberg (14:23)

The end of your week, man, I've killed your week.

David McRaney (14:24)

Thanks. The. My entry point of this was this. This concept was a Daniel Dennett book where he mentioned Rules for artificial intelligence. And then he backed up a bit and said, let's talk about Game of Life. And then he was like, you know, this clearly demonstrates how everything around you exists. And I'm like, okay. And it just so happened around that time, xkcd, the comic, created my favorite comic that they've ever done, Bunch of Rocks, where it's a person stuck in an infinite plane with a bunch of rocks, and they use them to create a set of rules. That is Conway's game of life, basically. And eventually it simulates the universe. And, you know, he. But he moves one rock at a time. But on the timescale of the people within the universe, something happened. And, you know, that's the whole way to, like, conceptualize it. I can't get enough of this idea. I can't get enough of the concept that a very simple set of rules then put in motion in frames, so every frame, something changes. Some things in that environment will be better at sticking around than others. And then as you add layers and layers of abstraction to that, as those rules become rules in a game that has rules that are the rules of other games, and up and up you go, you can get to very complex things, and you just start to see this. You start to get a sense, a feeling, an emotional reaction that feels a little bit like, I kind of sort of understand this now. And it feels very similar to that moment when you are on mushrooms or something like that, and you understand everything for three seconds. And then you're like, oh, no, I lost it.

Jordan Ellenberg (15:59)

And by the way, that feeling you're talking about, that feeling of, like, I can just almost touch it, I kind of sort of get it. I can feel that there's something there that I can't quite articulate. See, I feel like the way you're describing it, you're like, that's me, David Graney, and you, Jordan Ellenberg, have mastered it and you know it. No, I want you to know that that experience of I kind of sort of get it, something I can't quite describe, that's like, every day of your life as a research mathematician, like, yes, there's sort of some things you know, but that's not where you want to spend your time, right? You are always, like, at that frontier, and, like, those things that you sort of can't quite talk about, but you can sense that they're there, is, like, that's where you live.

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Podcast Host / Advertiser (17:41)

Okay, that thing.

David McRaney (17:42)

I said I would talk about in the middle of the show. It's not quite the middle of the show, but here's the thing.

Podcast Host / Advertiser (17:46)

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David McRaney (18:31)

To do with it.

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Podcast Host / Advertiser (20:34)

And now we return to our program.

David McRaney (20:41)

My name is David McCraney.

Podcast Host / Advertiser (20:42)

This is the youe Are not so Smart podcast.

David McRaney (20:44)

And what follows is my interview with.

Podcast Host / Advertiser (20:47)

The fantastic and brilliant mathematician Jordan Ellenberg.

David McRaney (20:53)

You say that geometry isn't out there beyond space and time. It's right here with us mixed in with the reasoning of everyday life. Is it beautiful?

Podcast Host / Advertiser (21:01)

Yes.

David McRaney (21:02)

But not bare geometers. See beauty with its work clothes on. Again, you did it.

David McRaney (21:10)

That's good stuff. That's how you ease us into Euclid. So that's what I'm going to do. As a segue to go to pull it back from Conway where Euclid eventually gets us, you cleverly bring in Lincoln here. And I remember seeing the movie in which Daniel Day Lewis goes through Euclid for a second and talks about the idea that if two things are equal to the same thing, then those two things are also equal to each other. And this is a proposition that is a self evident truth. And then later on, Lincoln starts dropping some Euclid lines in his speeches. I won't over talk this. You tell me about this. Tell me how you connected Lincoln to Euclid in your book.

Jordan Ellenberg (21:52)

Yeah, so I didn't see this movie. I probably would have saved myself some research time if I had because this was new to me. Actually I learned it writing about this stuff. Sort of Lincoln's fondness for Euclid. And as he told the story to interviewers at the time, he was troubled by the fact this was, by the way, he had been in Congress once, but it was before he was president. So we're talking Lincoln in the 50s, the 1850s. You know, he's going into court day after day and he's being asked to prove things. To demonstrate is the word he uses. And he said he found himself asking, what does this mean? What is a demonstration? What is that? And he realized that he needed to go back to Euclid to understand what a demonstration was. Remember, Lincoln doesn't really have formal schooling, Right. So he's not like, you know, I contrast him in the book with Jefferson, who also loved geometry, but from the point of view of being this Kind of patrician person who had, of course, had all the proper education. Lincoln didn't have that, but Lincoln recognized that he needed to, or wanted to, I should say, go get it. And it's. You know, I love, like, reading just the people around Lincoln talk about him. I probably. I put a lot of quotes in the book, probably too long, because I just love listening to his buddies talk about it. And what, you know, what they say is that, you know, what's special about Lincoln and what does it have to do with Euclid? It wasn't that he was, like, so brilliant. You know what I mean? His friends say, look, lots of people in politics are extremely intelligent and clever. They said, no, what's special about Lincoln is that he had this habit of not wanting to come to a conclusion unless he'd really backed it up, not wanting to say something unless he really felt like he could demonstrate it. It's funny, because I found myself backtracking a bit in the book. I wanted to say, because you could say that's what he got from Euclid, or you could say that's just what he was like. And Euclid was like that, too. And it resonated with him. He found something in Euclid that he was like, oh, this is where I want to be again. Chicken and egg.

David McRaney (23:51)

Yeah. Well, the thing is, this is something in my slot. One of my great heroes is James Burke, and I remember him starting his discussion on the Connection series about how did humans, like, what was the actual monolith that was put forth and caused us to accelerate our evolution, right? And he talks about geometry, is like. It's straight up, like. Like this line and this line and this. You know, if you know these two things about these two things, then now you know a third thing that is not in. In the evidence in front of you. It's in the evidence generated by your brain, yet is a truth. It is a truth that cannot be denied. And the idea that you could extract truth from the universe from incomplete evidence and you could trust that that truth would be. Would follow because of the propositions have been so clearly defined, was the thing that, like, was the Promethean fire, right? So that's. That's in the yearbook, too. You talk about that right away. Using Lincoln as someone who was like, yo, this is crazy, y'. All, Like. Like, this is good stuff. And you write about how in the mind of a. Of a geometer, you don't settle for leaving things half understood. You trace back your steps through using reason, classic reason. And classical logic. And you will, because of that, be able to then work your way forward through steps. There's a back propagation and then an actual propagation that comes about because of being able to think this way. So the gift of this, this, this is something that was never taught to me in any math class ever. But I have since found a better understanding of it later because of people like yourself, that this is how we bootstrapped, how we, you know, ratcheted and extracted from the universe a spark by which we could then, like, gain some momentum and move forward and say, okay, we can build an idea on top of an idea until finally we have very complex ideas, but we can trust that all the way down. It's been rigorously understood, or at least has been shown to be these propositions hold true. And that's why, like, it never made sense to be, like, we had to prove that two plus two equals four. And then it didn't make sense to me ever. That this is very difficult to prove, actually. And, you know, I'm like. Because you make a big point in the book, like, it seems like it should be obvious because we have some sort of possibly biologically created by proteins that are instructed by Genesis, intuition, for certain things that can then be later described using the language of mathematics. But just because it seems obvious doesn't mean that it is, especially not to a mathematician. If you could talk about that at any length. I'm interested in that idea.

Jordan Ellenberg (26:32)

Yeah. And I do. But actually, I'm going to start because now since you opened up the topic, and I like to ask everybody, so I hope it's okay if, like, I view you for one second. What you. What was your experience learning math and especially learning geometry as a kid? Since you brought it up, I'm going to ask, I like to ask everybody.

David McRaney (26:49)

It was connected to nothing but the math itself. So it was, it was. It felt like.

David McRaney (26:56)

It felt like the A. It felt like a task to memorize things, to make A's on tests. And I didn't understand. I had no concept in any of my math classes, even all the way through college math courses, how it connected to anything in the physical, how it connected to any deeper truths philosophically, how, what the history of it was, I had no concept of. Like, where did this start? What was this built on? What is sort of like the, the. What is the genealogy of this idea? The idea that all of this had a philosophical shroud around it, that it was couched in ideas of what does all of this mean? And how does this Help us relate to our very humanity, came way later. It was not any of my math classes. And also the gift of animations and GIFs and things on the Internet allowed me to see things like the way PI. How it actually is related to the natural world or the Fibonacci sequence or anything like that, whose signal has risen above the noise on the Internet. Seeing it visually really changed the way that I understood the thing, and that was completely absent from my entire experience being taught it in schools.

Jordan Ellenberg (28:09)

Yeah, I mean, like, wow, there's a lot there. And I think, you know, one thing that I really try to do, both in this book and the last book actually, is to humanize the practice of mathematics. And what I mean by that, you know, one thing people sometimes mean by that is to say, like, okay, what does this have to do with my daily life? Like, I have, like, a roast beef that's shaped like this. What's its volume? Or whatever? You know, sort of to, like, to make it. To make it relevant. That would be a thing that you might ask. But I mean, something a little bit different. I mean, what you said really resonated with me that you said, like, there was no genealogy. There was no sense of, like, where these ideas came from. Whereas, you know, the truth of the matter is that mathematics is a human activity. And every single formalism we have in mathematics was created by people to solve a problem that they had, and they were less confused after they developed this formalism than they were before. So one of the things I always try to do when I'm writing.

Jordan Ellenberg (29:06)

Is to go back and try to say, I love the word genealogy. Actually, I don't think that word appears in my book. It probably should.

Jordan Ellenberg (29:13)

To say, like, not just what is the idea? But, like, what was somebody trying to do when they created this idea? Because that's where I mean, every idea comes from. Somebody having a problem and then finding a way. And then finding a way to solve it.

David McRaney (29:27)

Yeah.

Jordan Ellenberg (29:29)

By the way, even in math education, for professional mathematicians, even when you go get a PhD, we don't learn the history of the subject. I learned this stuff to write the books. I often don't know what people are trying to do.

David McRaney (29:42)

Look, as a science journalist, usually one of the tricks of science journalism is to go into the history of an idea and to use that as the narrative that helps leaven the bread. Right. So you say, this is where this comes from, and this person was ridiculous, and this was a maniac. And then you can kind of. And I learned all that from watching James Burke. And it Even in the history of. Of I mainly write about social science, but. Which is nothing but maniacs doing really weird stuff until somebody comes along and says, you really. Have you thought about maybe this is unethical? And so. And the. But even like in the history of ideas I learned that just recently that the story had been told about Socrates forever was way more fun. Whenever you find out that the people of his era didn't really believe in original ideas. They thought that if you came up with an idea that it was a God or a godlike entity came along and put it in your head but for some purpose. And Socrates comes along with these very strange ideas and they're like, hey, where are you getting this from? Because nobody else is getting those. And he's like, I get them from another dimension. I get. There's another set of.

Jordan Ellenberg (30:56)

Wow, that's like Flatland. Yeah, I didn't even know that I would have put that in if I'd known that.

David McRaney (31:00)

There's another. Socrates said that there was an alternate dimension where there were Daemonia who would give him ideas, and they weren't giving them to anybody else. And the reaction in Athens was, okay, we probably don't want ideas coming from other dimensions, so will you drink this, please?

David McRaney (31:20)

Wow.

Jordan Ellenberg (31:20)

Yeah, I could have used that. I'm sure Edwin Abbott knew that. I didn't know that. But I'm sure Edwin Abbott, who wrote Flatland, knew that and consciously had it in mind as he sort of told his story of like the sphere invading Flatland from three dimensional space, bringing with it these kind of dangerous ideas about the third dimension.

David McRaney (31:35)

Yes, it's perfect. That's. I mean, I feel like your book does this so well. It does it from every possible direction. It says this is the history of the idea, this is the humanity of the idea. This is the reason that your humanity connects this idea. And also here's some drawings and some crazy shit that you've never heard of before that will show you how weird my life is as a mathematician. It's really good. I really like the way that you presented it and I've seen a couple of authors attempt it, but I think you stuck the landing. Maybe best of of all.

Jordan Ellenberg (32:08)

Oh, thank you. You know, I think this idea of like another thing that just reminded me to connect this idea of talking about the history and genealogy with education is that in some sense what's required of you to write about where these ideas came from and what's required of you in the classroom as a teacher, which is what you know I am most of the time when I'm not writing books. Both require you to imagine your way into the state of not understanding the thing. And when you're a professional mathematician, something that I've known very well for 25 years, there is an act of imagination required to put myself into the mind and into the cognitive situation of my students who are just learning it for the first time. And if you can't do that, it's hard to be a good teacher. But that's exactly what you're doing. When you do the history, you're saying, let me go. Let's roll back the clock to the moment at which literally nobody on earth really clearly and articulately understood this concept.

David McRaney (33:11)

Piaget, the psychologist who's famous for the experiments showing that kids don't understand how much water is in a glass, if it's tall.

David McRaney (33:21)

He was doing all that to try to demonstrate how brains update as they are revealed, the magnitude of their ignorance in one domain to the next. And you write in the book the ultimate reason this is a quote, the ultimate reason for teaching kids to write a proof is not that the world is full of proofs, it's that the world is full of non proofs and grownups need to know the difference. It's hard to settle for a non proof once you really familiarize yourself with the genuine article. That's great. I love everything about it. It relates back to the Piaget thing. That gets on my mind a lot right now. And you also say that what Lincoln took from Euclid was integrity. There's the principle that one does not say a thing unless one is justified, fair and square, that one has the right to say it. And geometry in that sense is a form of honesty. And for me, that like tunnels all the way down to. Yeah, that's why math is great, right? Because you're, you're really digging into the bedrock of what is going on exactly out there right now. What is all this? And oddly enough, on a piece of paper you can find things that not just like the other line of the triangle that you didn't see at this point, with the. The amount of work that's been done that you're like getting notions of a very intense alternate dimensional concepts that perhaps will turn out to be observed with some sort of tool or instrument one day. And to me it feels like, hey, that's some demonia from another dimension whispering in our ears saying, take a look at this stuff that you haven't seen yet. I don't know. I love how all of it fits together for me personally. Yeah.

Jordan Ellenberg (34:58)

And I think it's funny you say that and it makes me reflect. I don't want to oversell in the following sense. There are plenty of professional mathematicians out there, like well versed in all manner of advanced geometry. And I'm sure I'm sometimes one of them who believe all kinds of crazy stuff which is like, not boring. So, you know, maybe the way I'd say it like this, I'm always tempted to oversell, especially when crafting an aphorism. But I don't want to say it makes you invulnerable, but maybe I'd say it gives you a tool. It's like you probably have friends who know a lot about meditation, right?

David McRaney (35:37)

Sure.

Jordan Ellenberg (35:38)

And are they serene all the time? Like. No, but they have a tool. Right? There is like a tool they have access to that has value to them. And maybe it's a little more like that because I wish I could tell you that the world of mathematicians was a world in which we just constantly went around being utterly honest with each other and with ourselves.

David McRaney (35:59)

There's a thing where I think Neil Grasse Tyson borrowed this from a comedian. But the idea, you show a dog a card trick and nothing happens there. So if you try to show.

David McRaney (36:13)

Euclidean geometry to an ant, nothing happens. And I think he took the. That farther was like, I can imagine a similar thing happening a couple of ratcheted degrees up to us from some super sentient life form. I like to think that human beings could eventually figure out everything. But there may be a cognitive restraint in that regard. We may be rationally bounded, as they say in psychology. I don't know. But I do know it excites me to no end. The idea that we can force our way into a higher understanding of the natural universe through this trick we figured out this language we figured out to describe the world. And geometry is like the essence of it. And you, you, that's the, the point. You drill home quite a bit in the book. The. You talk about Euclidean geometry. Two, two things equal, the same thing are thereby equal to each other. And then that seems obvious, but then you have to like prove it somehow. You have to show what are the atoms and molecules of that idea. And then you pivot into non Euclidean geometry and you use this phrase that I've never heard before and I'd like to hear you wax poetic about it. And that is the bridge of asses.

Jordan Ellenberg (37:26)

Yeah. You know, the bridge of asses is the pons, us and aurum as we like to say if we're in polite company or just if we think we're on. People who don't, don't know that it means the same thing.

Jordan Ellenberg (37:36)

It's a famous proof that appears pretty early in Euclid in the first book, but in some sense it's one of the first really difficult proofs. So it's called that because it's something that we know people need to be led across the first time I see it, if you want to get technical about it, which I will for one moment. It's the statement that an isosceles triangle, a triangle with two sides the same, also has two angles the same. And I like writing about it in the book because it's a perfect example of something which, on the one hand, to really explain to yourself formally how you know that's true is a bit complicated. And yet there's also something that speaks directly to the intuition, because what you feel is that if you have a triangle and two sides are the same, that you can just flip it over and it doesn't change. It's the same viewed, like, from your left, through your left eye, and through your right eye. And that's an example of something that undoubtedly Euclid understood, but it wasn't really in his toolkit. He really resists writing about symmetry because it somehow wasn't something that he built into his system. But in fact, I mean, the perception of symmetry, that's definitely something that's built in. And I would say to a modern geometer, that notion is completely fundamental, like what counts as a symmetry. This comes back to Poincare's maxim about calling two things by the same name. Is a triangle, when you flip it over, is it the same triangle or a different one? There's no right answer to that question. It sort of depends which things you care about.

David McRaney (39:33)

Well, for me, the bridge of asses segues naturally into a very important question that you spend an incredible amount of time on in the book. And that question is, how many holes does a straw have? And I will ask the people. For people listening, take a second pause if you have to, and ask yourself, how many holes does the straw have? Ask other people around you this question, too, and get into it, because, believe it or not, you really can get into it. So I will hand off to. I will see the floor to you, sir. How many holes does a straw have?

Jordan Ellenberg (40:07)

Yeah. Let me preface it by saying that people really do get into it uniformly. I almost feel like if I were doing events in person for this book, which under the circumstances I'm not doing. I almost feel like I'd be tempted to just start with that and say, take 10 minutes and argue about that before I even talk about anything else. And then you'll really know what this book is about before I say one word about the book. Because people really get into it. It's one of those things where people think the answer is obvious, and then they're absolutely stunned to find out that the person standing next to them also thinks the answer is obvious, but it's not the same answer.

Jordan Ellenberg (40:44)

And of course, you know, it leads you into asking, wait, what is a hole? What does that even mean? So just to say, I mean, there's sort of three answers that people give. Some people will say, well, there's no holes in it because a straw, you can make it out of a rectangle of plastic, right? Attach the ends together. Now it's a straw. A square doesn't have a hole in it. I didn't punch a hole in it. So still no hole. That's one view. It's a minority view. You know, people who say there's one hole will say, there's almost no argument. They'll just say, like, well, look at it. There's the hole. It goes all the way through. And people who say two holes will say, well, look, there's like a hole in the top and there's a hole in the bottom, right?

David McRaney (41:27)

And you talk about, like, there's an argument to be made where, like, how many holes does a vase have? And they're like, it's got one. So if I poke a hole in the bottom, how many does it have? And the thing is, some people will say, same hole, and others will say, no, I have added a hole. But to use language to describe it, you necessarily must say, I poke a hole in the bottom. So therefore, I have added a hole to the thing. And now we don't know if we're playing a language game or we're playing a mathematical game. And I love it.

Jordan Ellenberg (41:56)

But this is scenario where I'm a big believer that the mere fact that everybody who hears about this actually feels moved to argue about it. They feel, on some level, personally attacked if somebody doesn't agree with them about the answer. You can see people get really head up about this. And I love. I cannot even tell you how many of these videos I watched on the Internet while I was prepping this chapter. As a math teacher, you love that because we look at it and we say, the reason that people are getting so exercised about this issue is that they're recognizing that there is an actual mathematical issue here. They may not call it that, they may not use those words to say it, but it exactly speaks to my contention that that sort of math sense is in us all. And we kind of like react when something touches that nerve. We're like, wow, what's going on?

David McRaney (42:47)

I feel the same way about the. Is a hot dog a sandwich? Because it. Cause what's. Because what's happening there is. You're having to consider categorical thinking. And what are words abstractions for? And what are we trying to articulate? And what do these things define? And what are the agreed upon terms and what is the proposition? All of these things which usually are in the domain of philosophy and then at some point in the domain of psychology, and then on even all the way up to politics at some point. These are, at their heart, can be mapped onto logic and reason, principles that are mathematical in nature. And you really illustrate that in the book with this. How many holes does a straw have issued?

Jordan Ellenberg (43:26)

Yeah, absolutely. And by the way. But when you say they can be mapped on to things that are logical and mathematical in nature, that is absolutely true. But at the same time, what I would not say. And it's always a danger because in math we kind of have an imperial tendency that we have to constantly resist. What I would not say is that those things can be reduced to purely mathematical questions. So, I mean, what's nice about the holes in the straw? And then I go on from there, by the way, to talk about how many holes are there in a pair of pants?

David McRaney (43:52)

This, this.

Jordan Ellenberg (43:53)

Then it gets harder still.

David McRaney (43:54)

This got me. I was like, I really wanted to actually be in the depths of a psychedelic freak out. Because when you said how many holes do pants have? I'm like, oh, no.

David McRaney (44:07)

Now I have to think, right, because.

Jordan Ellenberg (44:09)

It'S a challenge for the one holers. Right? Because the people who are very, very sure there's only one hole in a straw, those people will still talk about the two leg holes of their pants.

Podcast Host / Advertiser (44:18)

Mm.

David McRaney (44:18)

What's the topology of the inner range of a pair of pants? I have to follow on the side of. We got two holes here. But you say in the book, a straw has two holes, but they're the same hole. Which is the way that. The way you rise above or way beneath language in some way. That is very satisfying.

Jordan Ellenberg (44:38)

Well, exactly. I mean, and somehow it almost has to be that way. If there's two very compelling answers that both seem to sort of strike a chord and feel Right. Mathematically, it's very rare that the right conclusion is one is right and one is wrong. Usually it's, we just have to sort of understand the right vantage from which they're both correct. Because if there weren't, if that vantage didn't exist, then probably people wouldn't feel so strongly in favor of, of both of those.

David McRaney (45:04)

There's a diagram where you can see a so three dimensional object that from one perspective looks like a sphere and from the other perspective it looks like a cube. But it's just topologically complicated and depends on what vantage you take. And I find that fascinating and wonderful.

Jordan Ellenberg (45:19)

Right. And imagine it would be a sterile argument to be like, well, which is it? Is it actually round or is it actually square? Right. The question is really like, why do I have this ability to perceive it as round and also to perceive it as square? Like, what is its nature that causes it to be that way? That's the right question.

David McRaney (45:39)

Yes. And combining, it's combining the perspectives and then like realizing that alone, I can't do this on a piece of paper, I might be able to describe it. But somehow combining your perspective and my perspective does come to some sort of higher order view of the thing. And I think that's really cool.

Jordan Ellenberg (45:55)

And this is what, you know, and I write about. It's a little bit hard to do in audio without pictures, but this is what the great geometer Amy Noether like was able to understand that any notion of whole that was like flexible enough to actually reason about.

Jordan Ellenberg (46:11)

Had to have this property that holes could be added and subtracted to each other. They form what's called a group that you can do arithmetic with them. So that in particular it makes sense to say that, you know, just as like 3 and negative 3 are two different numbers and yet they have a relation with each other. They're not completely independent from each other. You know, that was what in kind of nurture's theory, the two holes in a straw. I start by saying there's two holes and they're the same, but really then I go back and revise that once we understand it a little bit better and say there are two holes, but one is the negative of the other. That's really the right way to think of it. And right, you feel that way because after all, in any given moment the milkshake is either flowing one way through the straw or the other. And if it's flowing in one hole, it's flowing out the other. There is some kind of relation of oppositeness between the Two, you need to.

David McRaney (46:58)

Get with your publisher and have a paper straw.

David McRaney (47:03)

Come with the book. Like, maybe it's, like, glued to the spine. That would be. That would be killer.

David McRaney (47:22)

Talk a lot about mosquitoes at some point in service of a greater idea. But you lead into it by talking about something called the Scraunch plane. What's a Scraunch plane?

Jordan Ellenberg (47:32)

Oh, so this is where I really try to sort of express something about this, like, just how general this notion of symmetry can be. Cause I think people have, you know, the word symmetry is an English word. People know what it means for a building or someone's face to be symmetrical or something like this. And they imagine, you know, flipping something from left to right or flipping something upside down as a possible way that something could be symmetric. But the notion is, like, much more general than that. You know, for instance, you could think of expanding something by a factor of 10 as being a certain kind of symmetry. You could say, what kind of things look the same? If you expand them by a factor of 10, that's a little bit hard to imagine, but it is a kind of symmetry. And just as.

David McRaney (48:21)

You can.

Jordan Ellenberg (48:22)

You can imagine somebody saying, I have a shape, and you have a shape that's exactly the same, but twice as big. You might. Under some circumstances, you might call those the same shape. If all you care about are the angles between things and the relative sizes of things, somebody else who actually cares how big something is might not consider them the same. The scranch. I just try to get really a little bit exotic and talk about these kinds of funny transformations where you may say, huh, these things really don't seem the same to me. But I could imagine myself into a world where I consider this kind of one directional stretching, which just I called a scrunch, because it just seemed like a funny word. And I like words that sound funny. It's not a technical term. You don't like Kant?

David McRaney (49:06)

No. You just did what Kant did. You made a word.

Jordan Ellenberg (49:09)

Wait, what words did Kant make up?

David McRaney (49:11)

He made up the word angst.

Jordan Ellenberg (49:12)

Yes, he told me that.

David McRaney (49:13)

Which I think is great. Like, it's not that it didn't exist. It's not. Nobody had that feeling before. But he was like, it really makes it. If I turn this into a brick, I can then build things out of this brick instead of. And then that gives me the ability to have a much more complicated abstraction. So, yeah, scrotch away. Yeah.

Jordan Ellenberg (49:32)

So what's. You know, so you can set it up as this kind of purely abstract exercise and Saying, like, what if you allow things to kind of get stretchy in one direction and change what you think of as their shape, you know, which things would stay the same and which things would be different. And on the one hand, that seems like a possibly sterile abstract exercise. On the other hand, what it turns out is that in the history of physics you eventually have to accept that these kinds of symmetries are the ones that space time actually has. So this is what happens when you start to really wrestle with how relativity works. And you know, this phenomenon of the Lorenz contraction that things do at relativistic speeds, they undergo a transformation that we sort of slowly and naive beings think of as changing their shape, right? We call it a contraction. We say, oh, the train gets kind of like smushy as it gets near the speed of light, right? But from a sort of more geometric viewpoint, we would just say, no, what the actual symmetries of space are, are not what we thought. And that thing's not changing its shape, it's just undergoing some kind of four dimensional scranch. But so a la poincure, thinking of things correctly means thinking of two things as the same if they're related to each other by that kind of relativistic symmetry. So it's really an interesting story, and I'm no physicist by any means, but.

Jordan Ellenberg (50:57)

I think Poincare understood what kind of formal geometry was required and what kind of symmetries would go well with what was being understood about the speed of light. And the speed of light is the limit. But I think he wasn't quite willing to accept that like space was actually like that.

David McRaney (51:17)

I love it.

Jordan Ellenberg (51:18)

He understood exactly the right geometric formalism. But it took Einstein to really say, like, no, that's not just like a formal thing that you're working out what the symmetries are. That's actually how space is. It's just not like we thought.

David McRaney (51:29)

See, that's the genealogy of the idea. That's what makes it so fantastic to me. The.

David McRaney (51:35)

Something excites me about. There's a story from art history with the creation of perspective, which there's a long story behind how that happened. But you look at art before perspective, the heads are odd and the bodies aren't in the right place and everybody's the same size, no matter where they are on the landscape. And there was a urge to put things larger if they were more important. It really there wasn't an attempt to make things seem as what we would say today, photorealistic and work of art. But Then once, but.

Jordan Ellenberg (52:11)

But of course people were seeing the same things that we see.

David McRaney (52:13)

That's what's nuts. You know, they're like, I'm looking at the street and then I paint this thing that looks nothing like the street. But once the geometry of perspective was understood by artists, and there's a great story behind how that happened, but we don't have time to get into it. But it evolves. Putting a mirror in front of a church and then painting through the mirror to make the thing match up perfectly, basically tracing on the real world, you could take the lines from that and extend them out infinitely as best you could imagine it. And all of a sudden you're like, oh, perspective and vanishing points. And once that's introduced into art, all of a sudden there's this moment where art looks like the real world from that point forward until you want to play with that and get crazy again.

David McRaney (52:56)

But the same, that thinking tool of perspective and vanishing points, the same sort of thing happens all throughout mathematics and geometry where, oh, what a great thinking tool. Thanks for making it. I will now use it to construct this thing with it. And that's how you get Einstein in the course of the genealogy of the history of ideas. I think that is so cool.

Jordan Ellenberg (53:14)

One thing I don't know though, and I wonder if you know, is at that moment, at the moment of that shift in representational art, when people saw the new paintings, were they like, oh, this looks way better. This is awesome. Or are they like, the old paintings look good. Those look like paintings. This looks like some weird uncanny thing that makes me uncomfortable that I don't like. I mean, I actually have no idea which it is. Was there instant uptake?

David McRaney (53:37)

There was when I only know about the Bruno Shelley or Brunoncelli who did the Battistrie in Italy, who's the guy who's like, famously, I invented perspective. Although I'm sure that he was co invented or co discovered. But he and the apparently the Greeks had also some at some places in the ancient world had figured it out too. But then it was lost. And we don't know how they figured it out. But for that one particular dude who did that one particular thing, he had lines around the street to come look at that painting. Because you could look at the painting and then look at the actual building and look at the painting and look at the actual building. And apparently it freaked people the floor out. And they were like, oh, whoa. It was like avatar or jaws 3D or something. It was a tick. It really caused quite the stir.

Jordan Ellenberg (54:28)

I Bet there was people at the time who were like the vinyl collectors, but of painting who were like, I like the hisses and pops. Like, I don't like it to look exactly like the church. I like this sort of more human touch.

David McRaney (54:37)

Yeah, I like it when the horse is as big as the person. I mean, that's exactly so something that I also have gotten onto a kick about and went down a rabbit hole on the Internet about. And I was so happy when this appeared in the book was arbore iation. The concept that. I think you see this a lot in. Woo Ah. Music in the background, stock art kind of stuff that floats around the Internet where people independently. I'm not making fun of that at all. I think it's great. Will independently say, oh, look, an eye looks a lot like a galaxy or trees seem to be very similar to rivers, which seem to be very similar to circulatory systems and lightning bolt patterns and that sort of thing. I love that there's not only is that something that scientists already know about and there's already a word for it, and a deep study of it that has led to incredible bonkers mathematical insights. Tell me everything there is to know about it.

Jordan Ellenberg (55:40)

Yeah, well, you know, when I started writing this book, I think, you know, originally I was like, well, maybe it'll be organized around like, each chapter about a different geometry. And that ended up not to be a real workable concept because I'm just too disorganized that I like, I start researching one thing and I want to write about some other thing. And it never fits into the neat categories I imagine at the beginning. I'm sure it's the same for you because I know what kind of stuff you write and I'm sure it also, like things escape their neat categories that you might have imagined on day one.

David McRaney (56:10)

The book that was eventually finished does not appear anything like the one that it started as. So that's a great part of writing books. Yeah. Go ahead.

Jordan Ellenberg (56:20)

Let's praise patient and understanding editors who are happy with not getting the book that we told that we were going to write.

David McRaney (56:26)

Sure, my book proposal and my final product are not close, but I will credit my editor, Nikki Papadopoulos, who took a chapter out of the middle and said, if you start with this chapter, the whole book makes sense. And she was completely right. So that's. That's. Yes, praise to editors.

Jordan Ellenberg (56:43)

But, yeah, one thing that did survive from the original concept was that, you know, the geometry of the tree, which is so fundamental and appears in so many places like from literal trees themselves to family trees. So I mean, the very metaphor we use suggests that there's some geographic, sorry, geometric similarity, taxonomies of all kind.

David McRaney (57:02)

Like a tree of life.

David McRaney (57:07)

The building up of a small idea to a giant one or the other way around. Like you see this pattern all over.

Jordan Ellenberg (57:13)

Yeah, yeah. I mean, I didn't even. That was one of the things that I didn't get to do. Like Darwin and the tree of life, this famous diagram that's in his book, not mentioned in my book, I just didn't get to it. But the thing that I write about the most is the tree of a game. Because it turns out that the geometry of a tree exactly describes what happens when you play a game with specified rules. And it reminded me of something that you said near the beginning of our talk together, that it's not just that you're learning the rules of a game, you're learning that a game has rules and what kind of rules they are. Like maybe that you take turns and that maybe that the board doesn't spontaneously change by chance, like.

Jordan Ellenberg (57:58)

During your play. Well, there's a certain class of games that are like that two player games, like no chance having a definite ending, which all of them are described by a tree. And what that means is that even though games like Checkers or Nim, which is a sort of like very small scale game, or Tic Tac Toe or Connect four or I forgot which names I already said, or Chess or Go, those are all obviously games with different rules, but they have the same kind of rules, which means that the method by which they can be analyzed is in the end exactly the same. That's why, for instance, a modern machine learning engine.

Jordan Ellenberg (58:46)

Essentially doesn't care which game it's learning. It's not built to learn Go, it's built to learn how to play well, a game with any set of rules of that certain kind. And so I think it's something that people tend not to appreciate, which is that a game of that form, a game like chess, or a game like Tic Tac Toe, which probably feel to you pretty different.

Jordan Ellenberg (59:12)

They're really different only in size.

Jordan Ellenberg (59:15)

There is an answer to the question of whether a perfect chess player.

Jordan Ellenberg (59:23)

If two perfect chess players matched up, would the first player to go always win or would the second player always win, or would it always end in a draw? There's an actual answer to that question. We don't know it, but there's an answer to that question the same way there's an answer to the Question of what is the product of these two 500 digit numbers. It's just that the chess problem is much harder and we don't have the.

David McRaney (59:44)

Answer and we don't know it because there's just so many variables in chess. Is that why?

Jordan Ellenberg (59:49)

Exactly. But it's a matter of time, right? If we sort of have like an infinitely fast, infinitely large computer, or not even infinitely, just like much bigger than one that could exist in the physical universe, it is a finite problem that is in principle solvable. Of course, when you say in principle, you're hiding a lot. But it's not a different kind of problem than multiplying two numbers together. It's just a different scale.

David McRaney (60:23)

Assimilation and accommodation. The idea of how we make sense of the world by increasing levels of complexity. They, they talk a great deal in that domain of psychology about cognitive psychology about games. Because a game is, and you describe it perfectly in the book. You, the first thing is to try to understand the geometry of the board and then the rules of the game and then what is the geometry of the gameplay. And eventually that's the meta game. And you see, you hear that all the time in competitive video games where people are trying to figure out the meta. They begin with something like Call of Duty and they understand, okay, this is the game world and how I interact in it. And that's the physics of it. Okay, here are the rules of what I can do to interact in that world. And now there is a better way to do those things, which is the meta game to play if I want to win. And you write in the book, there is, it is unknowable to find the perfect way to play. Just like. Because Call of Duty has more variables than chess. If you're, you know, so strangely strange to say that, but it is true. And this also maps onto. This is how brains prefer to make sense of things. That's why the storytelling is such an essential part of conveying ideas, because it maps onto assimilation and accommodation and what it's like to figure out a game. And that's why artificial intelligence is start out by trying to learn games because that's learning the universe of a very. Learning the rules of a very tiny universe. I'm too excited about this. I feel it. The storytelling is the same thing because a good story always starts. If you can think of all the best movies. First shot is the establishing shot. That's the geometry of the board. Then we see the characters and how they can interact with each other and what they do. That's the, that's the rules of the game, and then the plot is the geometry of the tree. That's the metagame that's being played by the story and how it plays out unknowable to the viewer because you don't know. You haven't seen the movie till its completion yet. So.

Jordan Ellenberg (62:22)

I wish you all could see me nodding vigorously. It doesn't come through in podcasts form, but that's what I do to show approval.

David McRaney (62:28)

You talk about strategies for not being wrong, and you have this nice little almost like it feels like a cone of some kind, where you say, if a decision you have to make is exactly identical with one you've made before, make the decision you now consider in retrospect the right one. Otherwise flip a coin.

David McRaney (62:48)

Is that a rule to live by? Tell me what you think of that.

Jordan Ellenberg (62:51)

No, I mean, that's sort of meant to be, in some sense a cartoon tune of a way that might superficially seem to be a good rule for living by, but it's actually completely useless because it's a way of retrospectively being completely right about everything. On every data point you've tested and observed, it's completely right. But faced with even the slightest amount of novelty, you're like, I'm sorry, this is not identical with the situation that I've been in before. Thus I'm completely lost. So, I mean, in some sense I don't think. Here's a point I think I didn't make in the book. But now the problem with talking about your book is you see all the things you should have said. Really what you are doing is saying it's not so much I want a situation that's identical with a situation I've seen before. I want to be able to measure if a situation is like in some way is similar to a situation I've encountered before, and then maybe do the same thing that worked in that situation. But the moment you say that, the moment you introduce some notion of similarity on situations, you're being geometric. You're sort of saying there's some space of all possible situations in which some things are close to each other and some things are far away. So this is an incredible conceptual leap from just when are two things exactly the same versus when are they close enough to being the same that my decision in one case is a reliable guide to what to do in the other case. That's a drastic conceptual leap. I mean, this is kind of a philosophical chestnut at this point, right? But you yourself are not identical with the you of a year ago Or a month ago or a minute ago. And yet, obviously, it's conceptually useful to sort of assert some kind of identity between those various renditions of you. Right. And to sort of be like, what worked for one me is like, you know, that's probably the way to go if I'm going to guess what's going to work for the present me. It would be, like, absurd to reject that geometry.

David McRaney (64:38)

It's great because you talk about Kasparov in the book, playing these chess games at a level as high as anybody's going to get, and still, you know, he could be defeated by a computer. But he talked about two humans playing each other. He marveled at when things surprised him. And you have this great quote. This is another one of your incredible quotes. I love this quote. Human chess is not. We're talking about the tree metaphors we used earlier. Human chess is not a tree. It's a battle that takes place in a tree. I'm going to write that I have a blackboard of quotes that I have elf of my kitchen. I'm putting that one quote in there because I feel like that says a lot about a lot of things. It's okay to be a human being playing human games in a world that likely so has this mathematical substrate that is, if we had a computer powerful enough, could demonstrate it.

Jordan Ellenberg (65:27)

Yeah. And people ask a lot, you know, with a sort of tone of worry in their voice, like, well, what happens to chess? Like, what happens to go when the machines are better at the game than human beings? And one of the things, you know, much more than either of those games in the book, I write about checkers because I love the story of checkers. It's, like, so fascinating, and it's, like, farther along, right? I mean, this sort of era of human domination in checkers ended earlier because I've been obsessed with the story of Marion Tinsley, like, the kind of greatest human checkers player who ever lived or ever will live now for years. And I was so happy to get a chance to write about it. But what I think is so interesting is that people did not stop playing checkers even though the game is solved, even though we now know mathematically that two perfect checkers players will always play to a draw, just like tic tac toe. And you might think, like, well, then I guess people would just stop. Nope, people still play. There are still human championships of checkers. And I think what that shows you is that people play because the point of a game is not to win. The point of a game is to play the game.

David McRaney (66:48)

Geometry was at one time seen as dangerous as this thing that represents a source of authority. Because the things you can show in something like the Pythagorean theorem is true, and it's not true because of any individual who discovered it. It's true no matter how it had ever been discovered. And that is a separate authority from anything that you chose before then to be the authority on what is, it is not. So what is and is not reality, what is and is not true. And you relate that to flatlanders and all sorts of things. So with that as my tee up to you, take us out of here with some deep thoughts concerning what it's like to discover things that are true beyond the authorities.

Jordan Ellenberg (67:32)

I. And it's, you know, that's kind of where I end the book with these two poems of Rita Dove. You know, I knew was a very famous and distinguished poet, but I didn't know was like sort of a childhood math fan who wrote a couple of poems like about learning math. I mean, who knew and writes about this kind of electric moment of acquiring knowledge. As you say, it's sort of like not because somebody told you, not because it's written in the book, but because the knowledge is just there, available to you to build up from first principles by yourself, incontrovertible once you understand it. That is a grasping of power, if understood correctly. And you said geometry used to be seen as dangerous. I think it should still be seen as dangerous. Let's show some respect.

Jordan Ellenberg (68:25)

I think it's still true that hopefully it's a way for. It's a very powerful moment for a child or an adult to understand that they can make knowledge by themselves and the authority is themselves in their own insight. There's a lot lacking in our math classrooms. Just like talking to people about their education. I think there's plenty of times that people are still finding that in school. People are, you know, people who say. Because there are plenty of people who say, like, geometry is the only thing I liked. That was the thing where I sort of. I really liked the way it clicked together. They're describing that feeling of making their own knowledge, which is an amazing thing, which ideally. Right. Like all school would do. It's hard to meet ideals. But, you know, I think I see writing a book like this as continuous with my teaching, you know, what I do in the classroom. And so we try to bring that to our students in the classroom, Whether it's a third grader learning math for the first time or a college student and I hope in a small way I'm able to bring that to some people with my books.

Jordan Ellenberg (69:43)

Foreign.

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David McRaney (70:01)

A little bit more head to you.

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Jordan Ellenberg (71:57)

Sam.

David McRaney (72:40)

What's the topology of the inner range of a pair of pants?

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