328 - Shape - Jordan Ellenberg (rebroadcast)

You can go to kittedkit.edu.shop and use the code Smart50, SMART50 at checkout, and you will get half off a set of thinking superpowers in a box. If you want to know more about what I'm talking about, check it out, middle of the show. But we are in the same time zone, which oddly, I can make that connect to some of the ideas in your book where there's some sort of topological truth about the fact that we exist on some sort of similar times in space at the moment, though we are separated geographically. Exactly. It's calling different things by the same name, places on a globe. What, oh my god, do I have a note that says that, that's what math is, calling different things with the same names. Somebody said that. That's what Pancarray said, one of his famous slogans. There it is. He has a lot. He was an aphoristic dude. Mathematics is the art of giving the same name to different things, which is what I'm going to say with my life is if I understand that in totality like you do. You're going to understand it by the end of this discussion. Okay, I'm looking forward to it. Welcome to the You Are Not So Smart Podcast episode 328. For me, the bridge of asses segues naturally into a very important question that you spend at incredible amount of time on 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 a 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 and you sir. How many holes does a straw have? My name is David McRainey. This is the You 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. McOther 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 bestselling 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, Shape, 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 I'm going to start the interview. But let me give me a few asks straight away, like, who are you and what do you do? 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 really interested in mathematics to eventually growing up into being a person whose actual job is to think about mathematics on a daily basis, 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 later in life that those were things that could be combined. I mean, you talked about putting it into words. And of course, if you look at research mathematics by the way, if you were to add some sure you often do, kind of go down to the library and thumb through the latest issues of the annals of mathematics or the journal of the American mathematical society, what have you? It is mostly words 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 it's not sort of so far from the interior life of mathematics to like do this kind of work. That make 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. All right, I hope we make it. Okay, we might not, but it's fine. It's just the journey that matters. The were there with me the long the way. I like this. I'm going to kick right now and I've written about this in my own book about how people do not update their priors. At some point, I had 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 and then moving away, moving up through propositions and how that's a proper, how a proposition even became a thing that we talked about. And I want to 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-polities. Because once you have a word for a thing, then you can build other things out of those words. That's just what that's just what mathematicians do basically. They come up with a single doodad to represent another thing. Yeah, and it's true. And it's an endless discussion that people have about, you know, 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? 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 bringing things into existence by naming them and naming the things that exist kind of at once. Like I don't think there's sort of, you know, as with chickens and eggs, right? There's not really a one that's not really prior to the other. You said you didn't much care for geometry and you were in a Hell's Angel Math team circuit and then you leap really 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? 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 called geometry, it's presented in this as this very formal and abstract and rigid thing, which you might imagine came to us from space, like 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 formal construct instead of this kind of primal in our body's 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, Juan Corre 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. That's so good. And here's a quote from the book. And this is for me, this is what would sell it to me. And I hope this sells it to everybody else. I awast this is you quote, I awastka drinkers have a similar take. The drug reboots the brain and lifts the mind above the tortured labyrinth. It thinks it's stuck in. That's good stuff. You know, honestly, as a writer and you know us know this too, the best thing in the world is when there's some line you kind of padded 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 going to send you Christmas cards like now we're friends. 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 conways game of life and Sailor automaton stuff like that that I can get stuck on a rabbit hole 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's that the molecules are made of or I'm like, okay, so Little do dads 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 tearing 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 wrote you write about it in a way for people like myself to get a lot out of it And 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 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 different life like rules you can explore or you can just kind of create your own rules and kind of like letter rip and sort of see what it does I guess maybe we should explain this for people who are 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 fastening guy called John Conway when he appears in the book again and again Again, that's sort of like master geometry who kind of pops up zealog 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 a V 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 rules is very simple. You could write it on about two lines of issue to paper and it produces the most unimaginably Baroque Well, I was gonna say patterns, but really it's the change in the patterns right moving patterns Mm-hmm. I really encourage you to listen 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 it's amazing. It's in May. It's G-O-L-L-Y is the name of the app. I just just found it. Yes, I will play with this endlessly My into the end of your week man. I've killed your week. 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 just so happened around that time XK CD the comic created my favorite comic that they've ever done a 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 time scale 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 abstracts into that as Though there's 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 a an emotional reaction that feels a little bit like Hmm, I kind of sort of understand this now and it feels very similar to that moment when you're when you are on mushrooms Or something like that and you understand everything for three seconds and then you're like oh no lost it And by the way that 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 grainy and you Jordan Elmer of master did 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 that's where you live Okay, that thing 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 so curiosity is this unusually common trait of people who listen to this podcast you may have noticed that about yourself And if you're the kind of person who wants to understand how minds work and sometimes don't work which is Clearly who you are because you listen to the show you are probably super interested in critical thinking If you are the kind of person who is right now listening to this podcast Then you might also be curious to find out about the higher order thinking skills course that I am co-presenting at the executive thinking academy the executive thinking academy. It's about executive thinking like the executive centers of your brain But also executive thinking too. That's what you want to do with it It's a four-week course to level up your strategic creative critical and executive thinking skills But it's a bit different because first it's not a passive exercise and watching a video and then filling out some multiple choice questions instead You will be actively participating in hands-on activities using templates and frameworks That you can use well beyond the course itself It's a genuinely interactive experience that will help you to think in new ways You also get the full set of kitted thinking tools with more than 200 beautifully designed physical cards In these fancy magnetic boxes that you can use to plan and facilitate workshops Elevate brain storming sessions supercharge strategy planning and much more these cards They they have digital versions they have yard codes on them They have a whole like thing that you can use on a website to make them cool The course is incredible and it shows you a bunch of ways to use those cards at your workplace or anywhere else And you'll have the option to learn collaboratively with a small group of like-minded peers So that you're holding each other accountable and encouraging each other to push your thinking boundaries Plus you'll just get access to this one course you get 12 months of membership to the executive thinking academy itself And that includes webinars in Q&A sessions with global thought leaders with authors with academics It's a whole lot of stuff and you get 50% off if you use the code smart 50 at checkout when you visit kitted.shop half off Smart 50 Kitted.shop if you are curious to learn more and to join me For next months higher order thinking skills course Head over there right now click on the link in the show notes and lock in your place And now we return to our program My name is David McRainey. This is the you are not so smart podcast and what follows is my interview with the fantastic and brilliant mathematician Jordan Ellen Burke You say that a geometry isn't out there beyond space and time is right here with us mixed in with the reasoning of everyday life Is it beautiful? Yes, but not bare Geometers see beauty with its work clothes on again you did it Oh That's good stuff. That's how you ease us into Euclid So that's what I'm going to do as a as a segue to go to pull it back From Conway where we 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 that they those two things are also equal to each other and this is a Proposition that is a self-evident truth and later on Lincoln starts dropping some Euclid lines and is in his And his speeches I won't overtalk this you tell me about this, tell me how you connected Lincoln to Euclid in your book. Yeah, so I didn't see this movie. I probably would have saved myself some research time if I have because this was new to me actually. I learned a 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, you know, 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 can trust him in the book with Jefferson who also loved geometry, but for 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 put a lot of quotes in the book, probably too long, because I just love listening to his buddies talk about him. 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 like, look, lots of people in politics are like 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. That's what it's funny because I find 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 he resonated with him. He like found something in Euclid that he was like, oh, this is where I want to be. Again, chicken and egg. Yeah. Well, the thing is, you know, this, this is something in my thought, my 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 calls this to accelerate our evolution, right? And he talks about geometry. He's like, it's straight up, 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 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 your book too. You talk about that right away, using Lincoln as someone who was like, yo, this is crazy. Y'all like, this is good stuff. And you write about how in the mind of a, of a geometry, 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 is something that was never taught to me in any math class ever. But I have since, we found a better understanding of it later because of people like yourself. This is how we bootstrapped, you know, how we, you know, ratched it 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 a very complex ideas. But we can trust that all the way down, it's been rigorously understood or at least it's been shown to be these propositions hold true. And that's why, like it never made this be like, we had to prove that two plus two equals four. And then it didn't mix this me ever that this is very difficult to prove actually. And you know, then 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 extracted by genes 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, and especially not to a mathematician. If you can talk about that in any length, I'm interested in that idea. Yeah, and I do, but actually, I'm going to start because now, you since you opened up the topic, and I like to ask everybody. So I hope it's okay if I have you for one second. 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. It was connected to nothing, but the math itself. So it was it felt like 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 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 is all 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 gifts and things on the internet allowed me to see things like the way pie actually what it's how it actually is related to the natural world or the Fibonacci sequence or anything like that that wrote the who's signal has risen about 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 in schools. 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 but 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 it's sort of 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 when 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 tried to do when I'm writing 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. 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. Yeah, but you know it's even it by the way even in 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 like write the books. I haven't done what we're trying to do. Look as a as a 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 like and even in the history of of I mainly read about social science but which is nothing but maniacs. Yeah, doing really weird stuff until somebody comes along and says you really have you thought about maybe this is unethical and the and the but even like in the history of ideas I learned that just recently that a story I'd ever I'd been told about Socrates forever was way more fun whenever you whenever you find out that the the people of his era didn't really believe in original ideas they thought that if you if you came up with an idea that it was you know a god or a godlike entity came along and put it in your head before 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 that there's another there another set of Wow that's like flatland. You know I wouldn't identify knowing that. There's another Socrates said that there was an alternate dimension where there were demonia 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 Wow yeah I could have used that I'm sure Edwin Abbott knew that I didn't know that but I'm sure Edwin Abbott he wrote flatland knew that and consciously how to didn't 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 yes it's perfect and 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 it's 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 all. Oh, thank you. 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 is 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, right? And when you're a professional mathematician, something that I've known very well for 25 years, there is an active 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 to clock to the moment at which literally nobody on earth really clearly and particularly understood this concept. He is a psychologist who's famous for the experiments showing that kids don't understand how much water is in a glass if it's tall. 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 is 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-proves 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 P.J. thing that I get on my mind a lot right now. And you also say that what Lincoln took from you, Cliff, was integrity. There's the principle that one does not say a thing unless one has 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 tunnels all the way down to, yeah, that's why math is great, because 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 amount of work that's been done, 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. Yeah, and I think, you know, 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 work. So, you know, maybe the way I'd say it like this, I don't, I'm always tempted to oversell, especially when crafting an aphorism. But I don't want to say it makes to you invulnerable, but maybe I'd say it gives you a tool. It's like you probably have friends who know a lot about like meditation, right? Sure. 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 like constantly went around being like utterly honest with each other and with ourselves. There's a thing where I think Neil Grass Tyson borrowed this from a comedian, but the idea you know, you show a dog or a card trick and you know, nothing happens there. So like if you try to show you clitian geometry to an ant, nothing happens. And I think he took the that for farther was like I can imagine a similar thing happening, a couple of ratcheted degrees up to us from some supercentiate 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 know in 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 that's the point you drill home quite a bit in the book. You talk about you clitian geometry two things equal the same thing or thereby equal to each other and then that seems obvious, but then you have to like prove it somehow. You have to show the what are the atoms and molecules of that idea. And then you pivot into non-ucleidian geometry and you use this phrase that I've never heard before and I like to hear you wax poetic about it. And that is the bridge of asses. Yeah, the bridge of asses is the ponds us in Oramas. We like to say if we're in polite company or just if we think we're on people who don't know that it means the same thing. It's a famous proof that it 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 and I sosseles triangle, 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 on two sides of 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 geometry, that notion is completely fundamental like what counts as a symmetry. This comes back to Poincareus Maxim about calling to things by the same name. Is it 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. Well, for me, the bridge of asses segues naturally into a very important question that you spend at incredible amount of time on the book. So I will hand off to, I will see the floor to you, sir. Yeah, let me preface it by saying that people really do get into it uniformly. I almost feel like, you know, if I were doing events in person for this book, which under the circumstances I'm not doing, I almost feel like I'll 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. 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 the 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? And 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 why I'm you. It's a minority of you. You know, people who say, there's one hole will say there's almost no argument. I thought, let's say, like, well, look at it. There's there's the hole that goes all the way through. And people who say two holes will say, well, look, there's like a hole on the top and there's a hole in the bottom. Right. And you talk about like there's an argument to be made where like how many holes does a vase have? Either 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 that 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 But this is narrower. I'm a big believer that the mere fact that everybody who hears about this actually feels Move 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 like 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 Exercise about this issue is that they're recognizing that there is an actual mathematical issue here They might 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 we kind of like react When something touches that nerve. We're like wow. What's going on? I feel the same way about the Is a hot dog a sandwich because it because what's Because what's happening there is you're having to consider a categorical thinking and what are words abstractions for and what is it what are we trying to articulate and what is one of these things define and what is the what are they 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 these are at their heart can be mapped on to logic and reason principles that are Mathematical in nature and you really illustrate that in the book with this how many holes this is straw-haff issue 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 when I would not say and it's always 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? This this that gets harder still 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 a like oh No, I Now I have to think of it's a challenge for the one-holeers 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. Mm-hmm What's the topology of the 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 sad 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 Uh, that's a favor of of both of those There's a diagram where you can see a so three-dimensional object that from one perspective looks like a a sphere and from the other perspective it looks like a cube But it's it's just topologically complicated and depends on what advantage you take and uh, I find that fascinating and wonderful 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 that's The question is really like why do I have this ability to perceive it as round and also to perceive it? Square like what is its nature that causes it to be that way? Mm-hmm. That's the right question. Yes, and combining it's combining the perspectives and then like realizing that Alone I can't do this Well, to 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 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 uh, Geometer Emmy Noether Like was able to understand that any notion of whole that was like flexible enough to actually reason about Um 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 three and negative three are two different numbers and yet they have a relation with each other Mm-hmm. They're not completely independent from each other You know, that was what uh in kind of Nurtre'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 I throw 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 get with your publisher and have a paper straw Come with the book like maybe it's like glue to the spine That would be that would be killer Talk a lot about mosquitoes at some point in service of a greater idea But you lead into about talking about something called the scotch plane Um, what's a scotch plane? 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 because 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 um, you know flipping something from left to right or flipping something upside down as a possible way There's something could be symmetric, but the notion is like much more general than that um You know for instance you could think of Expanding something by a factor of 10 as being a certain kind of symmetry Uh, 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 um But it is a kind of symmetry and just as um And you can 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 answer some circumstances you might call those the same shape right if all you care about 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 so The scotch I just try to get really a little bit exotic and talk about these kinds of funny transformations where you use where you may say Huh these things really don't seem the same to me, but I can imagine myself into a world where I consider this kind of One directional stretching which just I call to scotch because it just seemed like a funny word And I like words that sound funny. It's not a technical term. You don't know you like hot. No, you just did what condom you made a word Wait, what words did Khan make up? He made up the word yes 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 uh, yeah scrunch away Yes, so what's you know so You can set it up with 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 uh this phenomenon of the Lorenz contraction. Mm-hmm 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 like gets near the speed of light, right? But From a sort of more geometric viewpoint, we would just say no 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 scrunch, but a la puanca ray 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 it's really an interesting story and I know physicists by any means, but um I think puanca ray understood What kind of formal geometry was required and what kind of symmetries would like go well with What was being understood about the speed of light and the speed of light is a limit But I think he wasn't quite willing to accept it like space was actually like that I love it. 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 that's see that's that's the genealogy of the idea That's what makes it so fantastic to me the I Something excites me about The there's a story from art history with you know the the creation of a perspective which there's a long story behind how that happened but the you know 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 an urge to put things larger if they were more important. It really wasn't an attempt to make things seem as what we would say today photorealistic and in the work of art. But then once. But of course, people were seeing the same things that we see. That's what we're 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 it was. I don't have time to get into it, but it involves 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. And. But the same. That thinking tool of perspective and vanishing points, the same sort of thing that 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's so cool. One thing I don't know the one I wonder if you know is at that moment, at the moment of that shift in representational art, when people saw the new paintings where they like, "Oh, this looks way better. This is awesome." Or 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 what it is. Was it instant. Was there instant uptake? There was. When I only know about the Bruner Shelley, or Bruner Shelley, who did the Bittistry in Italy, who is the guy who's like famously. I invented perspective, although I'm sure that he was co-invented or co-discovered. But he. And 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 then look at the painting and look at the actual building. And apparently it freaked people the fuck out. And they were like, "Oh, whoa." It was like a avatar, or Jaws 3D or something. It really caused quite the stir. I bet there was people at the time who were like the vinyl collectors, but the painting, who were like, "I like the Hisses and Pops." I don't like it to look exactly like the church, but I like this sort of more human touch. Yeah, I like it when the horse is as big as the person. 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 Arbor Eation, the concept that I think you see this a lot in a woo-aw music in the background, stock art kind of stuffed, it floats around the internet. Where's people independently? I'm not making fun of that at all. I think it's great. We'll 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's 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. 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 did not be a real workable concept because I'm just too disorganized. And I like, I started research of one thing and I want to write about some other thing. It never fits into the unique 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 unique categories that you might have imagined on day one. 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. Let's, let's, let's praise patience and understanding editors who are happy with not getting the book that we told that we were going to write. For sure. As in my book proposal, my final product are not close. And, 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, that's, yes, praise to editors. 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 tree. So I mean, the very metaphor we use suggests that there's some geographic, sorry, geometric similarity. Textonies of all kinds, like a tree of life, the building up of a small idea to a giant one, or the other way around. Like you see this pattern all over, yeah. Yeah. I mean, I didn't even, I was one of the things that I didn't get to do like, you know, 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, you know, one, 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 you know, remind me of something that you said, you know, near the beginning of our talk together, that you not only it's not just that you're learning the rules of a game, you're learning that a game has rules and one kind of rules they are. Like maybe that you take turns and that maybe that, you know, the board doesn't spontaneously change by chance, like during your, during your play. Well, there's a certain class of games that are like that, you know, 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 nimm, which is a sort of like very small-scale game, or Tic Tac Toe, or Connect 4, or I've forgotten what I, 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. And that's why, if you know, for instance, a modern machine learning engine 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, again, with any set of rules of a certain kind. And so, you know, 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 CXX Toe, which probably feel to you pretty different, they're really different only in size. Like, there is an answer to the question of whether a perfect chess player, you know, if two perfect chess players matched up, would the first player to go always win, or would the second player always to go always win, or would 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 like, what is the product of these two 500 digit numbers? It's just that the chess problem is much harder and we don't know the answer. And we don't know it because there are just so many variables in chess, is that what? 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. A simulation and accommodation, the idea of how we make sense of the world by increasing levels of complexity, 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, the first thing is the try to understand the geometry of the board, and then the rules of the game, and then what is the geometry of the game play, and eventually that's the meta game. And 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're right in the book, there is, it is unknowable to find the perfect way to play, just like, Call of Duty has more variables than chess, if you're, you know, so strangely, just strange to say that, but it is true. And this also maps onto, this is how it 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's the same thing, 'cause a good story, if you 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 rules of the game. And then the plot is the geometry of the tree. That's the meta-game that's being played by the story and how it plays out, unknowable to the viewer because you haven't seen the movie until it's completion yet. So, mm. I wish you all could see me nodding vigorously. It doesn't come through in podcast form, but that's what I do to show approval. 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. (laughs) Is that a rule to live by? Tell me what you think of that. - No, I mean, that's sort of meant to be in some sense a cartoon 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 have 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 of talking about your book 'cause you see all the things you should have said. You know, really what you are doing is saying, it's not so much that I want a situation is 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, right? 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 or 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. And I mean, this is kind of a philosophical chestnut at this point, right? But like 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. - It's great because you talk about Cass Prove 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're using 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, alpha 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. - Yeah. And people ask a lot, you know, with a sort of tone of worry on their voice, like, well, what happens to chess? Like what happens to go and 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 and checkers ended earlier because I've been obsessed with this story of Mary and Tinsley, like the kind of greatest human checkers player who ever lived or ever will live now for years. And it 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. (upbeat music) (upbeat music) 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 it is not reality, what is it is not true. And you relate that to flat letters 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. And it's, that's kind of the where I end the book. I would these two poems of Rita Dove, you know, I knew it was a very famous and distinguished poet, but I didn't know it was like sort of a childhood math man who wrote a couple of poems 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 in Contravertible once you understand it. That is a grasping of power, if understood correctly. And you know what you said, geometry used to be seen as dangerous? I think it should still be seen as dangerous. Let's show some respect. I think it's still true that hopefully it's a way for 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 grade or 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. That is it for this episode of The You Are Not So Smart Podcast. For links to everything we talked about, and a little bit more, head to YouAreNotSoSmart.com or check the show notes right there inside your podcast player. You can find my book, How Minds Change, wherever they put books on shelves and ship them in trucks, details, or at DavidMcRainey.com. And there's links to all of that in the show notes as well. On that website, you can find a round table video with a group of persuasion experts who are featured in the book. You can read a sample chapter, download a discussion guide, send it for the newsletter, read reviews, and hire me to come give a lecture wherever you are at. And I'll give you workshops on how to change people's minds and how minds change. For all the past episodes of This Podcast, go to Spotify, Apple, podcasts, Amazon Music, Audible, or YouAreNotSoSmart.com. Follow me on Twitter and threads and Instagram and Blue Sky. And everything else, @DavidMcRainey, follow the show @NoughtSmartBlog. We're also on Facebook, slash YouAreNotSoSmart. And if you'd like to support this one person operation, like really keep it going, 'cause I don't have any editors, staff, nothing is. Just me. Go to patreon.com/YouAreNotSoSmartPitchInitAnyAmount. And it will help a whole lot. You'll also get the show ad free after that. And if you pitch in higher amounts, I'll send you a post or a t-shirt, signed books, all sorts of things. The opening music that is clashed by Karev and Pallas. 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