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The following is a conversation with Jordan Ellenberg, a mathematician at University of

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Wisconsin and an author who masterfully reveals the beauty and power of mathematics in his

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2014 book, How Not to Be Wrong, and his new book, just released recently called Shape,

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The Hidden Geometry of Information, Biology, Strategy, Democracy, and Everything Else.

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As a side note, let me say that geometry is what made me fall in love with mathematics

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when I was young. It first showed me that something definitive could be stated about this

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world through intuitive visual proofs. Somehow, that convinced me that math is not just abstract

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numbers devoid of life, but a part of life, part of this world, part of our search for meaning.

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This is the Lex Friedman podcast, and here is my conversation with Jordan Ellenberg.

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So, for Noam Chomsky, language, the universal grammar, the framework from which language springs

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is like most of the cake, the delicious chocolate center, and then the rest of cognition that we

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think of is built on top, extra layers, maybe the icing on the cake, maybe consciousness is

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just like a cherry on top. Where do you put in this cake mathematical thinking? Is it as fundamental

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as language? In the Chomsky view, is it more fundamental in the language? Is it echoes of

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the same kind of abstract framework that he's thinking about in terms of language that they're

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all really tightly interconnected? That's a really interesting question. You're getting me to reflect

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on this question of whether the feeling of producing mathematical output, if you want,

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I think it feels something like that, and it's certainly the case. Let me put it this way. It's

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hard to imagine doing mathematics in a completely non linguistic way. It's hard to imagine doing

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mathematics without talking about mathematics and thinking and propositions. Maybe it's just

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because that's the way I do mathematics that maybe I can't imagine it any other way.

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

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attachable? That's a really interesting question. One thing it reminds me of is one thing I talk

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about in the book is dissection proofs, these very beautiful proofs of geometric propositions.

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There's a very famous one by Baskara of the Pythagorean theorem. Proofs which are purely

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visual, proofs where you show that two quantities are the same by taking the same pieces and putting

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them together one way and making one shape and putting them together another way and making a

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different shape and then observing that those two shapes must have the same area because they were

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built out of the same pieces. There's a famous story and it's a little bit disputed about how

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accurate this is, but that in Baskara's manuscript he gives this proof, just gives the diagram and

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then the entire verbal content of the proof is he just writes under it, behold. There's some dispute

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about exactly how accurate that is. Then there's an interesting question. If your proof is a diagram,

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if your proof is a picture or even if your proof is a movie of the same pieces coming together

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in two different formations to make two different things, is that language? I'm not sure I have

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a good answer. What do you think? I think it is. I think the process of manipulating the visual

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elements is the same as the process of manipulating the elements of language and I think probably

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the manipulating, the aggregation, the stitching stuff together is the important part. It's not

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the actual specific elements. It's more like, to me, language is a process and math is a process.

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It's not just specific symbols. It's inaction. It's ultimately created through action,

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through change, and so you're constantly evolving ideas. Of course, we kind of attach,

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there's a certain destination you arrive to that you attach to and you call that a proof,

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but that doesn't need to end there. It's just at the end of the chapter and then it goes on

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and on and on in that kind of way. But I got to ask you about geometry and it's a prominent

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topic in your new book shape. For me, geometry is the thing, just like as you're saying,

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made me fall in love with mathematics when I was young. Being able to prove something visually

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understand the world perfectly. Maybe it's an OCD thing, but from a mathematics perspective,

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like humans are messy, the world is messy, biology is messy, your parents are yelling or

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making you do stuff, but you can cut through all that BS and truly understand the world

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through mathematics and nothing like geometry did that for me. For you, you did not immediately

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fall in love with geometry. So how do you think about geometry? Why is it a special field in

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mathematics and how did you fall in love with it if you have? Wow, you've given me a lot to say

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and certainly the experience that you describe is so typical, but there's two versions of it.

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One thing I say in the book is that geometry is the cilantro of math. People are not neutral

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about it. There's people who are like you, the rest of it, I could take or leave, but then at

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this one moment, it made sense. This class made sense. Why wasn't it all like that? There's other

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people I can tell you because they come and talk to me all the time who are like, I understood

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all the stuff where you're trying to figure out what X was. There's some mystery, you're trying

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to solve it. X is the number, I figured it out, but then there was this geometry. What was that?

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What happened that year? I didn't get it. I was lost the whole year and I didn't understand

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why we even spent the time doing that. What everybody agrees on is that it's somehow different.

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There's something special about it. We're going to walk around in circles a little bit,

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but we'll get there. You asked me how I fell in love with math. I have a story about this.

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When I was a small child, I don't know, maybe like I was six or seven, I don't know.

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we had a cool wooden box around your stereo. That was the look. Everything was dark wood.

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The box had a bunch of holes in it to let the sound out. The holes were in this rectangular

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array, a six by eight array of holes. I was just zoning out in the living room as kids do,

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looking at this six by eight rectangular array of holes. If you like just by focusing in and

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out, just by looking at this box, looking at this rectangle, I was like, well, there's

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Eight sixes and six eights. It's just like the dissection proofs we were just talking about,

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but it's the same holes. It's the same 48 holes. That's how many there are, no matter

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whether you count them as rows or count them as columns. This was unbelievable to me.

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Exactly. It's just as you say. I knew the six times eight was the same as eight times six.

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Right? I knew my times table. I knew that that was a fact, but did I really know it

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until that moment? That's the question. I knew that the times table was symmetric,

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I could see like, oh, I didn't have to have somebody tell me that. That's information

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that you can just directly access. That's a really amazing moment. And as math teachers,

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that's something that we're really trying to bring to our students. And I was one of those

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who did not love the kind of Euclidean geometry ninth grade class of like, prove that an isosceles

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triangle has equal angles at the base, like this kind of thing. It didn't vibe with me the way that

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algebra and numbers did. But if you go back to that moment, from my adult perspective,

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looking back at what happened with that rectangle, I think that is a very geometric moment. In fact,

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that moment exactly encapsulates the intertwining of algebra and geometry. This algebraic fact that,

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well, in the instance, eight times six is equal to six times eight. But in general,

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that whatever two numbers you have, you multiply them one way, and it's the same as if you multiply

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them in the other order, it attaches it to this geometric fact about a rectangle, which in some

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sense makes it true. So, you know, who knows, maybe I was always fated to be an algebraic

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geometry, which is what I am as a researcher. So that's the kind of transformation. And you

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talk about symmetry in your book. What the heck is symmetry? What the heck is these kinds of

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transformation on objects that once you transform them, they seem to be similar? What do you make

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of it? What's its use in mathematics or maybe broadly in understanding our world?

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Well, it's an absolutely fundamental concept. And it starts with the word symmetry in the way

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that we usually use it when we're just like talking English and not talking mathematics,

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right? Sort of something is, when we say something is symmetrical, we usually means it has what's

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called an axis of symmetry. Maybe like the left half of it looks the same as the right half,

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that would be like a left, right axis of symmetry, or maybe the top half looks like the bottom half,

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or both, right? Maybe there's sort of a fourfold symmetry where the top looks like the bottom

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and the left looks like the right. Or more, and that can take you in a lot of different

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directions, the abstract study of what the possible combinations of symmetries there are,

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a subject which is called group theory was actually one of my first loves in mathematics

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what I thought about a lot when I was in college. But the notion of symmetry is actually much more

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general than the things that we would call symmetry if we were looking at like a classical

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we could use a symmetry to refer to any kind of transformation of an image or a space or an

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object. You know, so what I talk about in the book is take a figure and stretch it vertically.

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Make it twice as big vertically and make it half as wide. That I would call a symmetry.

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It's not a symmetry in the classical sense, but it's a well defined transformation that

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has an input and an output. I give you some shape and it gets kind of, I call this in the

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book a scrunch. I just made out to have to make up some sort of funny sounding name for it because

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it doesn't really have a name. And just as you can sort of study which kinds of objects are

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symmetrical under the operations of switching left and right or switching top and bottom or

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rotating 40 degrees or what have you, you could study what kinds of things are preserved by

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this kind of scrunch symmetry and this kind of more general idea of what a symmetry can be.

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Let me put it this way. A fundamental mathematical idea, in some sense, I might even say the idea

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that dominates contemporary mathematics. Or by contemporary, by the way, I mean like the

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last like 150 years. We're on a very long time scale in math. I don't mean like yesterday. I mean

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like a century or so up till now. Is this idea that it's a fundamental question of when do we

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consider two things to be the same? That might seem like a complete triviality. It's not. For

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instance, if I have a triangle, and I have a triangle of the exact same dimensions, but it's

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over here, are those the same or different? Well, you might say like, well, look, there's two different

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things. This one's over here. This one's over there. On the other hand, if you prove a theorem

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about this one, it's probably still true about this one. If it has like all the same side lanes

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and angles and like looks exactly the same, the term of art, if you want it, you would say they're

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congruent. But one way of saying it is there's a symmetry called translation, which just means

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move everything three inches to the left. And we want all of our theories to be translation

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invariant. What that means is that if you prove a theorem about a thing that's over here, and then

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you move it three inches to the left, it would be kind of weird if all of your theorems like didn't

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still work. So this question of like, what are the symmetries and which things that you want to

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study are invariant under those symmetries is absolutely fundamental. Boy, this is getting a

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little abstract, right? It's not at all abstract. I think this is completely central to everything

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I think about in terms of artificial intelligence. I don't know if you know about the MNIST dataset

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with handwritten digits. And I don't smoke much weed or any really, but it certainly feels like

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it when I look at MNIST and think about this stuff, which is like, what's the difference between one

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and two? And why are all the twos similar to each other? What kind of transformations

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are within the category of what makes a thing the same? And what kind of transformations

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are those that make a different and symmetries core to that? In fact, whatever the hell our brain

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is doing, it's really good at constructing these arbitrary and sometimes novel, which is really

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important when you look at like the IQ test or they feel novel ideas of symmetry of like what,

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and construct almost like little geometric theories of what makes things the same and not

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and how to make programs do that in AI is a total open question. And so I kind of stared

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and wonder how, what kind of symmetries are enough to solve the MNIST handwritten digit

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recognition problem and write that down? And exactly. And what's so fascinating about the work in that

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direction from the point of view of a mathematician like me in a geometry is that the kind of groups

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and of symmetries, the types of symmetries that we know of are not sufficient, right? So in other

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words, like, we're just going to keep on going into the weeds on this. The deeper the better.

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You know, a kind of symmetry that we understand very well is rotation, right? So here's what

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would be easy if humans, if we recognized a digit as a one, if it was like literally a rotation by

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some number of degrees of some fixed one in some typeface like Palatino or something,

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that would be very easy to understand, right? It would be very easy to like write a program that

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could detect whether something was a rotation of a fixed digit one. Whatever we're doing when

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you recognize the digit one and distinguish it from the digit two, it's not that. It's not just

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incorporating one of the types of symmetries that we understand. Now, I would say that I would be

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shocked if there was some kind of classical symmetry type formulation that captured what we're

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doing when we tell the difference between a two and a three, to be honest. I think what we're

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doing is actually more complicated than that. I feel like it must be. They're so simple, these

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numbers. I mean, they're really geometric objects. Like, we can draw out one, two, three. It does

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seem like it should be formalizable. That's why it's so strange. Do you think it's formalizable

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when something stops being a two and starts being a three, right? You can imagine something

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continuously deforming from being a two to a three. Yeah, but there is a moment. I have myself a

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written program that literally morphed twos and threes and so on. You watch and there's moments

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that you notice depending on the trajectory of that transformation, that morphing, that it is a

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three and a two. There's a hard line. Wait, so if you ask people, if you showed them this morph,

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if you ask a bunch of people, do they all agree about where the transition happened?

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Because I would be surprised. I think so. Oh my God. Okay, we have an empirical view.

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But here's the problem. Here's the problem that if I just showed that moment that I agreed on.

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Well, that's not fair. No, but say I said, so I want to move away from the agreement because

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that's a fascinating actually question that I want to backtrack from because I just

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dogmatically said because I could be very, very wrong. But the morphing really helps

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see this, it's all probably tied in there. Somehow the change from one to the other,

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like seeing the video of it allows you to pinpoint the place where a two becomes a three

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much better. If I just showed you one picture, I think you might really, really struggle. You

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might call a seven. I think there's something also that we don't often think about, which is

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it's not just about the static image. It's the transformation of the image or it's not a static

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shape. It's the transformation of the shape. There's something in the movement that seems to be

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not just about our perception system, but fundamental to our cognition, like how we think

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about stuff. Yeah. That's part of geometry too. And in fact, again, another insight of

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modern geometry is this idea that maybe we would naively think we're going to study,

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I don't know, like PoincarÃ©, we're going to study the three body problem. We're going to study

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sort of like three objects in space moving around subject only to the force of each other's gravity,

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which sounds very simple, right? And if you don't know about this problem, you're probably like,

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okay, so you just put it in your computer and see what they do. Well, guess what? That's a

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problem that PoincarÃ© won a huge prize for, making the first real progress on in the 1880s.

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And we still don't know that much about it 150 years later. I mean, it's the mongest mystery.

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You just opened the door and we're going to walk right in before we return to symmetry.

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And why is it such a hard problem? Okay, so PoincarÃ©, he ends up being a major figure in

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the book. And I didn't even really intend for him to be such a big figure, but he's so, he's,

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he's first and foremost a geometer, right? So he's a mathematician who kind of comes up

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in late 19th century France at a time when French math is really starting to flower.

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Actually, I learned a lot. I mean, you know, in math, we're not really trained on our own history.

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We get a PhD in math and learn about math. So I learned a lot. There's this whole kind of

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moment where France has just been beaten in the Franco, Prussian war. And they're like,

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oh my God, what did we do wrong? And they were like, we got to get strong in math,

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like the Germans, we have to be like more like the Germans. So this never happens to us again.

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So it's very much, it's like the Sputnik moment, you know, like what happens in America in the

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50s and 60s with the Soviet Union, this is happening to France. And they're trying to kind of like,

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instantly like modernize. That's fascinating. The humans and mathematics are intricately connected

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to the history of humans. The Cold War is, I think, fundamental to the way people saw science and

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math in the Soviet Union. I don't know if that was true in the United States, but certainly

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

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Soviet Union. I mean, we'll talk about the Olympiad. I just remember that there was this feeling

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like the world hung in a balance and you could save the world with the tools of science. And

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mathematics was like the superpower that fuels science. And so like people were seen as, you

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know, people in America often idolized athletes. But ultimately, the best athletes in the world,

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they just throw a ball into a basket. So like there's not, what people really enjoy about sports,

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and I love sports, is like excellence at the highest level. But when you take that with mathematics

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and science, people also enjoyed excellence in science and mathematics in the Soviet Union.

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So that created all the usual things you think about with the Olympics, which is like

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extreme competitiveness, right? But it also created this sense that in the modern era in America,

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somebody like Elon Musk, whatever you think of them, like Jeff Bezos, those folks,

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they inspire the possibility that one person or a group of smart people can change the world.

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Like not just be good at what they do, but actually change the world. Mathematics was at

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the core of that. And I don't know, there's a romanticism around it too. Like when you read

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books about in America, people romanticize certain things like baseball, for example,

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there's like these beautiful poetic writing about the game of baseball. The same was the

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feeling with mathematics and science in the Soviet Union. And it was in the air. Everybody was forced

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to take high level mathematics courses. Like you took a lot of math. You took a lot of science

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and a lot of like really rigorous literature. Like the level of education in Russia, this could be

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true in China. I'm not sure. In a lot of countries is in whatever that's called, it's K to 12 in

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America. But like young people education, the level they were challenged to learn at is incredible.

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It's like America falls far behind, I would say. America then quickly catches up and then exceeds

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everybody else. As you start approaching the end of high school to college, like the university

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system in the United States arguably is the best in the world. But like what we challenge

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everybody. It's not just like the good, the A students, but everybody to learn in the Soviet

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Union was fascinating. I think I'm going to pick up on something you said. I think you would love

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a book called Duel at Dawn by Amir Alexander, which I think some of the things you're responding

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to what I wrote, I think I first got turned on to by Amir's work. He's a historian of math,

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and he writes about the story of Everdeast Galois, which is a story that's well known to all

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mathematicians, this kind of like very, very romantic figure who he really sort of like begins

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the development of this, or this theory of groups that I mentioned earlier, this general

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theory of symmetries, and then dies in a duel in his early 20s, like all this stuff

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mostly unpublished. It's a very, very romantic story that we all learn. And much of it is true,

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but Alexander really lays out just how much the way people thought about math in those times in

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the early 19th century was wound up with, as you say, romanticism. I mean, that's when the

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romantic movement takes place. And he really outlines how people were predisposed to think

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about mathematics in that way because they thought about poetry that way, and they thought about

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music that way. It was the mood of the era to think about we're reaching for the transcendent,

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we're sort of reaching for sort of direct contact with the divine. And so part of the reason that

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we think of Galois that way was because Galois himself was a creature of that era, and he

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romanticized himself. I mean, now we know he wrote lots of letters and he was kind of like,

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I mean, in modern times, we would say he was extremely emo. We wrote all these letters about

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his like florid feelings and like the fire within him about the mathematics. So it's just as you say

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that the math history touches human history. They're never separate because math is made of

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people. I mean, that's what it's people who do it and we're human beings doing it and we do it

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within whatever community we're in and we do it affected by the mores of the society around us.

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he's, you know, it's funny, this book is filled with kind of, you know, mathematical characters who

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often are kind of peevish or get into feuds or sort of have like weird enthousiasms because

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those people are fun to write about and they sort of like say very salty things. PoincarÃ©

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is actually none of this. As far as I can tell, he was an extremely normal dude. He didn't get

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into fights with people and everybody liked him and he was like pretty personally modest and he

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had very regular habits, you know what I mean? He did math for like four hours in the morning

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and four hours in the evening and that was it. Like he had his schedule. I actually, it was like,

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I still am feeling like somebody's going to tell me now that the book is out like,

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oh, didn't you know about this like incredibly sorted episode of this? As far as I can tell,

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a completely normal guy, but he just kind of in many ways creates the geometric world

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in which we live and, you know, his first really big success is this prize paper he writes for this

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prize offered by the King of Sweden for the study of the three body problem. The study of what we

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can say about, yeah, three astronomical objects moving in what you might think would be this

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very simple way. Nothing's going on except gravity relating. So what's the three body problem? Why

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is it a problem? So the problem is to understand when this motion is stable and when it's not.

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orbit or I guess it would mean, sorry, stable would mean they never sort of fly off far apart

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from each other and unstable would mean like eventually they fly apart. So understanding

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two bodies is much easier. Yeah, exactly. Two bodies. Newton knew two bodies, they sort of

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orbit each other in some kind of either in an ellipse, which is the stable case, you know,

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that's what planets do that we know. Or one travels on a hyperbola around the other. That's

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the unstable case. It sort of like zooms in from far away, sort of like whips around the

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heavier thing and like zooms out. Those are basically the two options. So it's a very simple

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and easy to classify story. With three bodies, just a small switch from two to three, it's a

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complete zoo. It's the first, what we would say now is it's the first example of what's called

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chaotic dynamics, where the stable solutions and the unstable solutions, they're kind of

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like wound in among each other and a very, very, very tiny change in the initial conditions can

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make the long term behavior of the system completely different. So Poincare was the first

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to recognize that phenomenon even existed. What about the conjecture that carries his name?

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Right. So he also was one of the pioneers of taking geometry, which until that point

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had been largely the study of two and three dimensional objects, because that's like

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He developed the subject we now call topology. He called it analysis C2s. He was a very

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well spoken guy with a lot of slogans, but that name did not, you can see why that name did not

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catch on. So now it's called topology now. Sorry, what was it called before? Analysis C2s,

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which I guess sort of roughly means like the analysis of location or something like that.

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It's a Latin phrase. Partly because he understood that even to understand stuff that's going on

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in our physical world, you have to study higher dimensional spaces. How does this work? And this

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is kind of like where my brain went to it because you were talking about not just where things are,

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but what their path is, how they're moving when we were talking about the path from two to three.

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well, each body, it has a location where it is, so it has an x coordinate, a y coordinate,

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a z coordinate, right? I can specify a point in space by giving you three numbers,

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but it also at each moment has a velocity. So it turns out that really to understand what's

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going on, you can't think of it as a point or you could, but it's better not to think of it as a

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point in three dimensional space that's moving. It's better to think of it as a point in six

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dimensional space where the coordinates are where is it and what's its velocity right now.

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That's a higher dimensional space called phase space. And if you haven't thought about this before,

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I admit that it's a little bit mind bending, but what he needed then was a geometry that was flexible

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enough, not just to talk about two dimensional spaces or three dimensional spaces, but any

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dimensional space. So the sort of famous first line of this paper where he introduces analysis

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situs is no one doubts nowadays that the geometry of n dimensional space is an actually existing

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thing, right? I think that maybe that had been controversial and he's saying like,

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look, let's face it, just because it's not physical, doesn't mean it's not there. It

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doesn't mean we shouldn't study it. Interesting. He wasn't jumping to the physical interpretation.

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It can be real even if it's not perceivable to human cognition. I think that's right. I think,

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don't get me wrong, PoincarÃ© never strays far from physics. He's always motivated by physics,

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but the physics drove him to need to think about spaces of higher dimension. And so he needed a

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formalism that was rich enough to enable him to do that. And once you do that, that formalism is

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also going to include things that are not physical. And then you have two choices. You can be like,

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oh, well, that stuff's trash. Or, and this is more the mathematician's frame of mind.

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If you have a formalistic framework that seems really good and sort of seems to be very elegant

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and work well, and it includes all the physical stuff, maybe we should think about all of it.

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Like maybe we should think about it thinking, maybe there's some gold to be mined there.

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And indeed, guess what? Before long, there's relativity and there's space time. And all

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of a sudden, it's like, oh, yeah, maybe it's a good idea. We already had this geometric apparatus

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set up for how to think about four dimensional spaces. It turns out they're real after all.

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This is a story much told in mathematics, not just in this context, but in many.

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so we can work out together. Good. We'll together walk along the path of curiosity.

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three dimensional spaces. So I was on my way there, I promise. The idea is that we perceive

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ourselves as living in, we don't say a three dimensional space, we just say three dimensional

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space, you know, you can go up and down, you can go left and right, you can go forward and back,

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there's three dimensions in which we can move. In PoincarÃ©'s theory, there are many possible

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three dimensional spaces in the same way that going down one dimension to sort of capture

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our intuition a little bit more. We know there are lots of different two dimensional surfaces,

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and a mobius strip looks a third way. Those are all like two dimensional surfaces that we can

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kind of really get a global view of because we live in three dimensional space, so we can see

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a two dimensional surface sort of sitting in our three dimensional space. Well, to see a three

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dimensional space whole, we'd have to kind of have four dimensional eyes, right, which we don't.

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So we have to use our mathematical eyes, we have to envision. The PoincarÃ© conjecture

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says that there's a very simple way to determine whether a three dimensional space

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is the standard one, the one that we're used to. And essentially, it's that it's what's called

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fundamental group has nothing interesting in it. And that I can actually say without saying what

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the fundamental group is, I can tell you what the criterion is. This would be good. Oh, look,

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I can even use a visual aid. So for the people watching this on YouTube, you'll just see this,

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for the people on the podcast, you'll have to visualize it. So Lex has been nice enough to like

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give me a surface with an interesting topology, some mug right here in front of me, a mug. Yes,

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I might say it's a genus one surface, but we could also say it's a mug, same thing.

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So if I were to draw a little circle on this mug, which way should I draw it? So it's visible,

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like here. Yeah, that's yeah, if I draw a little circle on this mug, imagine this to be a loop of

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string, I could pull that loop of string closed on the surface of the mug, right? That's definitely

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something I could do. I could shrink it, shrink it, shrink it until it's a point. On the other hand,

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if I draw a loop that goes around the handle, I can kind of just it up here and I can just

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it down there and I can sort of slide it up and down the handle, but I can't pull it closed,

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can I? It's trapped. Not without breaking the surface of the mug, right? Not without like going

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inside. So the condition of being what's called simply connected, this is one of punk erase

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inventions, says that any loop of string can be pulled shut. So it's a feature that the mug

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simply does not have. This is a non simply connected mug and a simply connected mug would

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be a cup, right? You would burn your hand when you drank coffee out of it. So you're saying the

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universe is not a mug? Well, I can't speak to the universe, but what I can say is that

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regular old space is not a mug. Regular old space, if you like sort of actually physically have like

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a loop of string. You can always close it. You can pull a shot. You can always pull a shot. But you

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know, what if your piece of string was the size of the universe? What if your piece of string was

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like billions of light years long? Like, how do you actually know? I mean, that's still an open

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question of the shape of the universe. Exactly. Whether it's, I think there's a lot, there is

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ideas of it being a tourist. I mean, there's some trippy ideas and they're not like weird

link |

out there controversial. There's a legitimate at the center of cosmology debate. I mean, I think

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there's some kind of dodecahedral symmetry. I mean, I remember reading something crazy about

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somebody saying that they saw the signature of that in the cosmic noise or what have you. I mean,

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to make the flat earthers happy, I do believe that the current main belief is it's flat.

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It's flat ish or something like that. The shape of the universe is flat ish. I don't know what the

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heck that means. I think that has like a very, I mean, how are you even supposed to think about the

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shape of a thing that doesn't have any thing outside of it? I mean, but that's exactly what

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topology does. Topology is what's called an intrinsic theory. That's what's so great about it.

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This question about the mug, you could answer it without ever leaving the mug, right? Because

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it's a question about a loop drawn on the surface of the mug and what happens if it never leaves

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that surface? So it's like always there. See, but that's the difference between the topology

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and say, if you're like trying to visualize a mug, do you can't visualize a mug while living inside

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the mug? Well, that's true. The visualization is harder, but in some sense, no, you're right,

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but if the tools of mathematics are there, I don't want to fight, but I think the tools of

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mathematics are exactly there to enable you to think about what you cannot visualize in this

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way. Let me give you, always to make things easier, go down a dimension. Let's think about,

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we live on a circle, okay? You can tell whether you live on a circle or a line segment, because

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if you live on a circle, if you walk a long way in one direction, you find yourself back where

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you started. And if you live in a line segment, you walk for a long enough one direction, you

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come to the end of the world. Or if you live on a line, like a whole line, an infinite line,

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then you walk in one direction for a long time. And like, well, then there's not a sort of terminating

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algorithm to figure out whether you live on a line or a circle, but at least you sort of,

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at least you don't discover that you live on a circle. So all of those are intrinsic things,

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right? All of those are things that you can figure out about your world without leaving

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your world. On the other hand, ready, now we're going to go from intrinsic to extrinsic. Why did

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I not know we were going to talk about this, but why not? Why not? If you can't tell whether you

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live in a circle or a knot. Like imagine like a knot floating in three dimensional space. The

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person who lives on that knot, to them it's a circle. They walk a long way, they come back to

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where they started. Now we with our three dimensional eyes can be like, oh, this one's just a plain

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circle and this one's knotted up. But that has to do with how they sit in three dimensional space.

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We can ask you one ape to another. How does it make you feel that you don't know if you live

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I'm going to be honest with you. I don't know if I fear you won't like this answer,

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So does it bother you that if we look at like a mobius strip, that you don't have an obvious way

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of knowing whether you are inside of a cylinder, if you live on a surface of a cylinder or you

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if which one? Because if what you do is you like, tell your friend, hey, stay right here,

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I'm just going to go for a walk. And then you like walk for a long time in one direction.

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And then you come back and you see your friend again. And if your friend is reversed, then

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you know you live on a mobius strip. Well, no, because you won't see your friend, right?

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Okay, fair, fair point, fair point on that. But you have to believe his stories about,

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no, I don't even know. I would, would you even know? Would you really? Oh, no, I know your

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point is right. Let me try to think of a better, let's see if I can do this on the fly. It may not

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be correct to talk about cognitive beings living on a mobius strip because there's a lot of things

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taken for granted there. And we're constantly imagining actual like three dimensional creatures,

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like how it actually feels like to live in a mobius strip is tricky to internalize.

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I think that on what's called the real protective plane, which is kind of even more sort of like

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messed up version of the mobius strip, but with very similar features, this feature of kind of

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like only having one side, that has the feature that there's a loop of string, which can't be

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pulled closed. But if you loop it around twice along the same path, that you can pull closed.

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That's extremely weird. Yeah. But that would be a way you could know without leaving your world that

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something very funny is going on. You know, it's extremely weird. Maybe we can comment on,

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hopefully it's not too much of a tangent is, I remember thinking about this, this might be right,

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this might be wrong. But if you're, if we now talk about a sphere, and you're living inside a sphere,

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That I was, because like, I was, this was very counterintuitive to me to think about,

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maybe it's wrong, but because I was thinking like earth, you know, your 3d thing on sitting on a

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sphere, but if you're living inside the sphere, like you're going to see, if you look straight,

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you're always going to see yourself all the way around. So everywhere you look, there's going

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to be the back of your own head. I think somehow this depends on something of like how the physics

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of light works in this scenario, which I'm sort of finding it hard to bend my. That's true. The

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sea is doing a lot of work, like saying you see something is doing a lot of work. People have

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thought about this. I mean, this metaphor of like, what if we're like little creatures in some sort

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of smaller world, like how could we apprehend what's outside that metaphor just comes back and

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back. And actually, I didn't even realize like how frequent it is. It comes up in the book a lot.

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I know it from a book called Flatland. I don't know if you ever read this when you were a kid

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or an adult, you know, this, this sort of comic novel from the 19th century about an entire

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two dimensional world. It's narrated by a square. That's the main character. And

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the kind of strangeness that befalls him when, you know, one day he's in his house and suddenly

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there's like a little circle there and there with him. And then the circle, but then the circle

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like starts getting bigger and bigger and bigger. And he's like, what the hell is going on? It's

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like a horror movie like for two dimensional people. And of course, what's happening is that a

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sphere is entering his world. And as the sphere kind of like moves farther and farther into the

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plane, it's cross section, the part of it that he can see to him. It looks like there's like this

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kind of bizarre being that's like getting larger and larger and larger until it's exactly sort of

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halfway through. And then they have this kind of like philosophical argument where the sphere is

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like, I'm a sphere. I'm from the third dimension. The square is like, what are you talking about?

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There's no such thing. And they have this kind of like sterile argument where the square is not

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able to kind of like follow the mathematical reasoning of the sphere until the sphere just

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kind of grabs him and like jerks him out of the plane and pulls him up. And it's like, now,

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like now do you see, like now do you see your whole world that you didn't understand before?

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So do you think that kind of process is possible for us humans? So we live in a three dimensional

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world, maybe with a time component, four dimensional. And then math allows us to go

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high into high dimensions comfortably and explore the world from those perspectives.

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Like, is it possible that the universe is many more dimensions than the ones we experience as human

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beings? So if you look at the, you know, especially in physics theories of everything,

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physics theories that try to unify general relativity and quantum field theory, they seem to

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go to high dimensions to work stuff out through the tools of mathematics. Is it possible, so like

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the two options are one, it's just a nice way to analyze the universe, but the reality is as

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exactly we perceive it, it is three dimensional. Or are we just seeing, are we those flat land

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creatures? They're just seeing a tiny slice of reality. And the actual reality is many, many,

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many more dimensions than the three dimensions we perceive. Oh, I certainly think that's possible.

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And I suppose what you would do as with anything else that you can't directly perceive is

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out there would have on the things we can perceive. Like what else can you do, right?

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And in some sense, if the answer is they would have no effect, then maybe it becomes like a

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little bit of a sterile question because what question are you even asking, right? You can kind

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of posit however many entities that you want. Is it possible to intuit how to mess with the other

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dimensions while living in a three dimensional world? I mean, that seems like a very challenging

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thing to do. The reason flat land could be written is because it's coming from a three dimensional

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writer. Yes, but what happens in the book, I didn't even tell you the whole plot,

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what happens is the square is so excited and so filled with intellectual joy. By the way,

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maybe to give the story some context, you ask like, is it possible for us humans to have this

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experience of being transcendentally jerked out of our world so we can sort of truly see it from

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above? Well, Edwin Abbott, who wrote the book, certainly thought so because Edwin Abbott was

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a minister. So the whole Christian subtext of this book, I had completely not grasped

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reading this as a kid, that it means a very different thing, right? If sort of a theologian

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is saying like, oh, what if a higher being could like pull you out of this earthly world you live

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in so that you can sort of see the truth and like really see it from above as it were. So that's

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one of the things that's going on for him. And it's a testament to his skill as a writer that

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his story just works, whether that's the framework you're coming to it from or not.

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But what happens in this book and this part now looking at it through a Christian lens,

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it becomes a bit subversive is the square is so excited about what he's learned from the sphere

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and the sphere explains to him like what a cube would be. Oh, it's like you have a three

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dimensional and the square is very excited and the square is like, okay, I get it now. So like,

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now that you explained to me how just by reason, I can figure out what a cube would be like,

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like a three dimensional version of me, like, let's figure out what a four dimensional version

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of me would be like. And then the sphere is like, what the hell are you talking about? There's a

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fourth dimension, that's ridiculous. Like, there's three dimensions, like that's how many there are,

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I can see. Like, I mean, it's this sort of comic moment where the sphere is completely unable to

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conceptualize that there could actually be yet another dimension. So yeah, that takes the

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religious allegory to like a very weird place that I don't really like understand theologically,

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but that's a nice way to talk about religion and myth. In general, as perhaps us trying to

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struggle with us, meaning human civilization, trying to struggle with ideas that are beyond

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our cognitive capabilities. But it's in fact not beyond our capability, it may be beyond

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our cognitive capabilities to visualize a four dimensional cube, a Tesseract is something like

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to call it, or a five dimensional cube or a six dimensional cube, but it is not beyond our cognitive

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capabilities to figure out how many corners a six dimensional cube would have. That's what's so

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cool about us, whether we can visualize it or not, we can still talk about it, we can still reason

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about it, we can still figure things out about it. That's amazing. Yeah, if we go back to this,

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how many holes does a straw have? A new listener may try to answer that in your own head.

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Yeah, I'm going to take a drink while everybody thinks about it so we can give you a moment.

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A slow sip. Is it zero, one, or two, or more than that maybe? Maybe you get very creative,

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but it's kind of interesting to dissecting each answer as you do in the book is quite brilliant.

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People should definitely check it out, but if you could try to answer it now, think about all the

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options and why they may or may not be right. Yeah, it's one of these questions where people

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on first hearing it think it's a triviality and they're like, well, the answer is obvious.

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And then what happens if you ever ask a group of people this, something wonderfully comic happens,

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which is that everyone's like, well, it's completely obvious. And then each person

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realizes that half the person, the other people in the room have a different obvious answer,

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I can't believe that you think it has two holes or like, I can't believe that you think it has one.

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

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Sure. So I think most people, the zero holders are rare. They would say like, well, look,

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you can make a straw by taking a rectangular piece of plastic and closing it up.

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Rectangular piece of plastic doesn't have a hole in it. I didn't poke a hole in it when I...

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So how can I have a hole? It's like, it's just one thing. Okay. Most people don't see it that way.

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what I would say is you could say the same thing about a bagel. You could say I can make a bagel

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by taking like a long cylinder of dough, which doesn't have a hole, and then smushing the ends

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together. Now it's a bagel. So if you're really committed, you can be like, okay, a bagel doesn't

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have a hole either. But like, who are you if you say a bagel doesn't have a hole? I mean, I don't know.

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

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other options? So you know, one whole people would say... I like how these are like groups of people,

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There's books written about each belief. You know, would say, look, there's like a hole and it goes

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all the way through the straw, right? There's one region of space, that's the hole, and there's one.

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And two whole people would say like, well, look, there's a hole in the top and the hole

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argue about this, they would take something like this bottle of water I'm holding,

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and go open it. And they say, well, how many holes are there in this? And you say like, well,

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there's one, there's one hole at the top. Okay, what if I like poke a hole here so that all the

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water spills out? Well, now it's a straw. So if you're a one hole, or I say to you like, well,

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how many holes are in it now? There was one hole in it before, and I poked a new hole in it.

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And then you think there's still one hole, even though there was one hole, and I made one more?

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Clearly not. There's just two holes. Yeah. And yet, if you're a two hole, or the one hole,

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And in the book, I sort of, in math, there's two things we do when we're faced with a problem

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that's confusing us. We can make the problem simpler. That's what we were doing a minute ago

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when we were talking about high dimensional space. And I was like, let's talk about like

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circles and line segments. Let's go down a dimension to make it easier. The other big

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move we have is to make the problem harder, and try to sort of really like face up to

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what are the complications. So what I do in the book is say like, let's stop talking about straws,

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remove it and talk about pants. How many holes are there in a pair of pants? So I think most

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people who say there's two holes in a straw would say there's three holes in a pair of pants.

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I guess like, I mean, I guess we're filming only from here. I could take up. No, I'm not going

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to do it. You'll just have to imagine the pants. Sorry. Yeah. Lex, if you want to, no, okay, no.

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That's going to be in the director's, that's the Patreon only footage. There you go.

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yeah, I feel a pair of pants like just has two holes because yes, there's the waist,

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but that's just the two leg holes stuck together. Whoa. Okay. Two leg holes. Yeah. Okay. I mean,

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that really is a one color for the straw. So she's a one holder for the straw too. And

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and that really does capture something. It captures this fact, which is central to the

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theory of what's called homology, which is like a central part of modern topology that

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holds whatever we may mean by them. There are somehow things which have an arithmetic to them.

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There are things which can be added. Like the waist, like waist equals leg plus leg is kind

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of an equation, but it's not an equation about numbers. It's an equation about some kind of

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geometric, some kind of topological thing, which is very strange. And so, you know,

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when I come down, you know, like a rabbi, I like to kind of like come up with these answers

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and somehow like dodge the original question and say, like, you're both right, my children. Okay.

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the first version would be to say like, well, there are two holes, but they're really both

link |

the same hole. Well, that's not quite right. A better way to say it is there's two holes,

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but one is the negative of the other. Now, what can that mean? One way of thinking about what it

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means is that if you sip something like a milkshake through the straw, no matter what,

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the amount of milkshake that's flowing in one end, that same amount is flowing out

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the other end. So they're not independent from each other. There's some relationship

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between them in the same way that if you somehow could like suck a milkshake through a pair of

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pants, the amount of milkshake, just go with me on this Bob experiment. I'm right there with you.

link |

The amount of milkshake that's coming in the left leg of the pants, plus the amount of milkshake

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that's coming in the right leg of the pants is the same that's coming out the waste of the pants.

link |

So just so you know, I fasted for 72 hours yesterday, the last three days. So I just broke

link |

the fast with a little bit of food yesterday. So this is like, this sounds, food analogies

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or metaphors for this podcast work wonderfully, because I can intensely picture it.

link |

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

link |

Exactly, it's just for this. It's hardship to purify the mind. No, for the first time,

link |

I just wanted to try the experience and just to pause, to do things that are out of the

link |

ordinary, to pause and to reflect on how grateful I am to be just alive and be able to do all the

link |

cool shit that I get to do. So did you drink water? Yes, yes, yes, yes, yes. Water and salt,

link |

so like electrolytes and all those kinds of things. But anyway, so the inflow on the top

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of the pants equals to the outflow on the bottom of the pants. Exactly. So this idea that,

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I mean, I think, you know, Poincare really have this idea, this sort of modern idea. I mean,

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building on stuff that other people did, Betty is an important one of this kind of modern notion

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of relations between holes. But the idea that holes really had an arithmetic, the really modern view

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was really Emy Nurtur's idea. So she kind of comes in and sort of truly puts the subject

link |

on its modern footing that we have now. So, you know, it's always a challenge, you know,

link |

in the book, I'm not going to say I give like a course so that you read this chapter and then

link |

you're like, oh, it's just like I took like a semester of algebraic apology. It's not like

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this. And it's always, you know, it's always a challenge writing about math because there are

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some things that you can really do on the page and the math is there. And there's other things,

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which it's too much in a book like this to like do them all the page, you can only say something

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about them, if that makes sense. So, you know, in the book, I try to do some of both. I try to do,

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I try to topics that are, you can't really compress and really truly say exactly what they are

link |

in this amount of space. I try to say something interesting about them, something meaningful

link |

about them so that readers can get the flavor. And then in other places, I really try to get

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up close and personal and really do the math and have it take place on the page.

link |

there's a lot of books that are about stuff. But I want my books to not only be about stuff,

link |

but to actually have some stuff there on the page in the book for people to interact with

link |

directly and not just sort of hear me talk about distant features about, distant features of it.

link |

Right. So, not be talking just about ideas, but the actually be expressing the idea.

link |

You know somebody in the, maybe you can comment, there's a guy, his YouTube channel is 3Blue1Brown,

link |

Grant Sanderson, he does that masterfully well. Absolutely. Of visualizing, of expressing a

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particular idea and then talking about it as well back and forth. What do you think about Grant?

link |

It's fantastic. I mean, the flowering of math YouTube is like such a wonderful thing because,

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you know, math teaching, there's so many different venues through which we can teach

link |

people math. There's the traditional one, right? Well, where I'm in a classroom with, you know,

link |

depending on the class, it could be 30 people, it could be 100 people, it could God help me be a

link |

500 people, if it's like the big calculus lecture or whatever it may be. And there's sort of some,

link |

but there's some set of people of that order of magnitude. And I'm with them for, we have a long

link |

time, I'm with them for a whole semester. And I can ask them to do homework and we talk together,

link |

we have office hours that they have one on one questions, a lot of that's like a very high level

link |

of engagement. But how many people am I actually hitting at a time? Like not that many, right?

link |

And you can, and there's kind of an inverse relationship where the fewer people you're

link |

talking to, the more engagement you can ask for. The ultimate, of course, is like the mentorship

link |

relation of like a PhD advisor and a graduate student where you spend a lot of one on one time

link |

together for like, you know, three to five years. And the ultimate high level of engagement to one

link |

person. You know, books, I can, this can get to a lot more people that are ever going to sit in my

link |

classroom and you spend like, however many hours it takes to read a book, somebody like

link |

Three Blue One Brown or Numberphile or people like Vi Hart, I mean, YouTube, let's face it,

link |

has bigger reach than a book. Like there's YouTube videos that have many, many, many more views than

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like, you know, any hardback book like not written by a Kardashian or an Obama is going to sell,

link |

right? So that's, I mean, and then, you know, those are, you know, some of them are like

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longer, 20 minutes long, some of them are five minutes long, but they're, you know, they're

link |

shorter. And then even so, look, look, like Virginia Chang is a wonderful category theorist

link |

in Chicago. I mean, she was on, I think the Daily Show or is it, I mean, she was on, you know,

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she has 30 seconds, but then there's like 30 seconds to sort of say something about mathematics

link |

to like untold millions of people. So everywhere along this curve is important. One thing I feel

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like is great right now is that people are just broadcasting on all the channels because we each

link |

have our skills, right? Somehow along the way, like I learned how to write books, I had this kind

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of weird life as a writer where I sort of spent a lot of time like thinking about how to put

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English words together into sentences and sentences together into paragraphs, like

link |

at length, which is this kind of like weird specialized skill. And that's one thing, but

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like sort of being able to make like, you know, winning good looking eye catching videos is like

link |

a totally different skill. And you know, probably, you know, somewhere out there, there's probably

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sort of some like heavy metal band that's like teaching math through heavy metal and like using

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Their music and so on. Yeah. But there is something to the process. I mean, Grant does this

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especially well, which is in order to be able to visualize something that he writes programs.

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So it's programmatic visualization. So like the things he is basically mostly through his

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you have to truly understand the topic to be able to to visualize it in that way,

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and not just understand it, but really kind of thinking a very novel way. It's funny because

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I've spoken with him a couple of times, spoken to him a lot offline as well. He really doesn't

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think he's doing anything new, meaning like he sees himself as very different from maybe like a

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researcher. But it feels to me like he's creating something totally new, like that act of understanding

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visualizing is as powerful or has the same kind of inkling of power as does the process of proving

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something. It just it doesn't have that clear destination, but it's pulling out an insight

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and creating multiple sets of perspective that arrive at that insight. And to be honest, it's

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something that I think we haven't quite figured out how to value inside academic mathematics in

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the same way. And this is a bit older that I think we haven't quite figured out how to value

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the development of computational infrastructure. You know, we all have computers as our partners now

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and people build computers that sort of assist and participate in our mathematics. They build

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those systems and that's a kind of mathematics too, but not in the traditional form of proving

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theorems and writing papers. But I think it's coming. Look, I mean, I think, you know, for example,

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the Institute for Computational Experimental Mathematics at Brown, which is like a, you know,

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it's a NSF funded math institute, very much part of sort of traditional math academia,

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they did an entire theme semester about visualizing mathematics, looking to the same kind

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of thing that they would do for like an up and coming research topic, like that's pretty cool.

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So I think there really is buy in from the mathematics community to recognize that this

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kind of stuff is important and counts as part of mathematics, like part of what we're actually

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here to do. Yeah, I'm hoping to see more and more of that from like MIT faculty, from faculty,

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from all the top universities in the world. Let me ask you this weird question about the

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Fields Medal, which is the Nobel Prize in mathematics. Do you think, since we're talking

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about computers, there will one day come a time when a computer and AI system will win a Fields

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Medal? No, of course, that's what a human would say. Why not? Is that like your, that's like

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my cap shot? That's like the proof that I'm a human? Yeah. What does, how does he want me to

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answer? Is there something interesting to be said about that? Yeah, I mean, I am tremendously

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interested in what AI can do in pure mathematics. I mean, it's, of course, it's a parochial

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interest, right? You're like, why am I not interested in like how it can like help feed

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the world or help solve like various problems? I'm like, can it do more math? What can I do?

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We all have our interests, right? But I think it is a really interesting conceptual question.

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And here too, I think it's important to be kind of historical because it's certainly true that

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there's lots of things that we used to call research in mathematics that we would now call

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computation, tasks that we've now offloaded to machines. Like, you know, in 1890, somebody

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could be like, here's my PhD thesis. I computed all the invariance of this polynomial ring under

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the action of some finite group. Doesn't matter what those words mean, just it's like something

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that in 1890 would take a person a year to do and would be a valuable thing that you might want to

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know. And it's still a valuable thing that you might want to know. But now you type a few lines

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of code in Macaulay or Sage or Magma and you just have it. So we don't think of that as

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math anymore, even though it's the same thing. What's Macaulay, Sage and Magma? Oh, those are

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

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use. That's similar to Maple. Yeah. Oh, yeah. So it's similar to Maple in Mathematica. Yeah.

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But a little more specialized, but yeah. It's programs that work with symbols and allow you

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to do, can you do proofs? Can you do kind of little leaps and proofs? They're not really

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built for that. And that's a whole other story. But these tools are part of the process of

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mathematics now. Right. They are now, for most mathematicians, I would say, part of the process

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of mathematics. And so there's a story I tell in the book, which I'm fascinated by, which is,

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so far, attempts to get AIs to prove interesting theorems have not done so well. It doesn't mean

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they can. There's actually a paper I just saw, which has a very nice use of a neural net defined

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counter examples to conjecture. Somebody said like, well, maybe this is always that. Yeah.

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things where that's not true. And it actually succeeded. Now, in this case, if you look at

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the things that it found, you say like, okay, I mean, these are not famous conjectures. Yes.

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Okay. So like somebody wrote this down, maybe this is so. Looking at what the AI came up with,

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you're like, you know, I bet if like five grad students had thought about that problem, they

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wouldn't have thought about that. I mean, when you see it, you're like, okay, that is one of the

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things you might try if you sort of like put some work into it. Still, it's pretty awesome.

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But the story I tell in the book, which I'm fascinated by is there is, there's a, okay,

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we're going to go back to knots. It's cool. There's a knot called the Conway knot.

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After John Conway, who maybe we'll talk about a very interesting character, awesome.

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Yeah. It's a small tangent. Somebody I was supposed to talk to, and unfortunately,

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he passed away. And he's somebody I find is an incredible mathematician, incredible human being.

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Oh, and I am sorry that you didn't get a chance because having had the chance to talk to him

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a lot when I was, you know, when I was a postdoc. Yeah, you missed out. There's no way to sugar

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code it. I'm sorry that you didn't get that chance. Yeah, it is what it is. So knots.

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Yeah. So there was a question. And again, it doesn't matter the technicalities of the question,

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And in particular, the question of the Conway knot, whether it was sliced or not,

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was particularly vexed until it was solved just a few years ago by Lisa Piccarillo,

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who actually, now that I think of it, was here in Austin. I believe she was a grad student

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at UT Austin at the time. I didn't even realize there was an Austin connection to this story

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until I started telling it. She is, in fact, I think she's now at MIT. So she's basically following

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you around. If I remember correctly, there's a lot of really interesting richness to this story.

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One thing about it is her paper was very short and simple, nine pages of which two were pictures.

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Very short for a paper solving a major conjecture. And it really makes you think about what we mean

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by difficulty in mathematics. Do you say, oh, actually, the problem wasn't difficult because

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you could solve it so simply? Or do you say, well, no, evidently it was difficult because

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the world's top depositors worked on it for 20 years and nobody could solve it. So therefore,

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it is difficult. Or is it that we need some new category of things about which it's

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difficult to figure out that they're not difficult? I mean, this is the computer science

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formulation, but the journey to arrive at the simple answer may be difficult. But once you

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have the answer, it will then appear simple. And I mean, there might be a large category.

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I hope there's a large set of such solutions. Because once we stand at the end of the scientific

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process that we're at the very beginning of, or at least it feels like, I hope there's just

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simple answers to everything that will look and it'll be simple laws that govern the universe,

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simple explanation of what is consciousness, of what is love, is mortality fundamental to life,

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what's the meaning of life, are human special or we're just another sort of reflection of

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and all that is beautiful in the universe in terms of like life forms, all of it is life and just

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has different, when taken from a different perspective is all life can seem more valuable

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or not. But really, it's all part of the same thing. All those will have a nice like two equations,

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maybe one equation. Why do you think you want those questions to have simple answers?

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There's something beautiful about simplicity. I think it's aesthetic. It's aesthetic. Yeah.

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Or but it's aesthetic in the way that happiness is an aesthetic. Why is that so joyful that a

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simple explanation that governs a large number of cases is really appealing? Even when it's not,

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it doesn't have to be connected with reality or even that explanation could be exceptionally

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harmful. Most of like the world's history that was governed by hate and violence had a very

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simple explanation at the core that was used to cause the violence and the hatred. So like we

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get into trouble with that. But why is that so appealing? And in its nice forms in mathematics,

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like you look at the Einstein papers, why are those so beautiful? And why is the Andrew Wiles

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proof of the Fermat's last theorem not quite so beautiful? Like what's beautiful about that story

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is the human struggle of like the human story of perseverance, of the drama of not knowing if

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the proof is correct and ups and downs and all of those kinds of things. That's the interesting part.

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But the fact that the proof is huge and nobody understood, well, from my outsider's perspective,

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nobody understands what the heck it is, is not as beautiful as it could have been. I wish it was

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what Fermat originally said, which is, you know, it's not small enough to fit in the margins of

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this page. But maybe if he had like a full page or maybe a couple of post notes, he would have

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enough to do the proof. What do you make of, if we could take another of a multitude of tangents,

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what do you make of Fermat's last theorem? Because the statement, there's a few theorems,

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there's a few problems that are deemed by the world throughout its history to be exceptionally

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difficult. And that one in particular is really simple to formulate and really hard to come up

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with a proof for. And it was like taunted as simple by Fermat himself. Is there something

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interesting to be said about that x to the n plus y to the n equals z to the n for n of three or

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greater? Is there a solution to this? And then how do you go about proving that? Like how would you

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try to prove that? And what do you learn from the proof that eventually emerged by Andrew Wiles?

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Yeah, so let me just say the background, because I don't know if everybody listening

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knows the story. So you know, Fermat was an early number theorist, always sort of an early mathematician,

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those special adjacent didn't really exist back then. He comes up in the book actually in the

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context of a different theorem of his that has to do with testing whether a number is prime

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or not. So I write about he was one of the ones who was salty and like he would exchange these

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letters where he and his correspondents would like try to top each other and vex each other with

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questions and stuff like this. But this particular thing, it's called Fermat's last theorem because

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it's a note he wrote in his in his copy of the Discretionist arithmetic guy, like he wrote,

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here's an equation, it has no solutions, I can prove it, but the proof is like a little too long

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to fit in this in the margin of this book, he was just like writing a note to himself.

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Now, let me just say historically, we know that Fermat did not have a proof of this theorem.

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For a long time, people like, you know, people were like, this mysterious proof that was lost,

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a very romantic story, right? But Fermat later, he did prove special cases of this theorem,

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and wrote about it, talked to people about the problem. It's very clear from the way that he

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wrote where he can solve certain examples of this type of equation that he did not know how to do

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the whole thing. He may have had a deep, simple intuition about how to solve the whole thing

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that he had at that moment without ever being able to come up with a complete proof, and that

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intuition may be lost in time. Maybe, but I think we, so, but you're right, that is unknowable,

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but I think what we can know is that later, he certainly did not think that he had a proof that

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he was concealing from people. He thought he didn't know how to prove it, and I also think he

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didn't know how to prove it. Now, I understand the appeal of saying like, wouldn't it be cool

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if this very simple equation, there was like a very simple, clever, wonderful proof that you

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could do in a page or two, and that would be great. But you know what, there's lots of equations

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like that that are solved by very clever methods like that, including the special cases that Fermat

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wrote about, the method of descent, which is like very wonderful and important. But in the end,

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and it is what it is, but they're not big. On the other hand, work on the Fermat problem,

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that's what we like to call it because it's not really his theorem because we don't think you

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proved it. So, I mean, work on the Fermat problem, develop this like incredible richness of number

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theory that we now live in today, like, and not by the way, just while Andrew Wiles being

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the person who together with Richard Taylor finally proved this theorem. But you know how

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you have this whole moment that people try to prove this theorem and they fail, and there's a

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famous false proof by LeMay from the 19th century, where Kummer, in understanding what mistake LeMay

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had made in this incorrect proof, basically understands something incredible, which is that,

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you know, a thing we know about numbers is that you can factor them and you can factor them

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uniquely. There's only one way to break a number up into primes. Like, if we think of a number like

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12, 12 is 2 times 3 times 2. I had to think about it, right? Or it's 2 times 2 times 3,

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of course, you can reorder them. But there's no other way to do it. There's no universe in which

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12 is something times 5, or in which there's like four threes in it. Nope, 12 is like two twos in a

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three. Like, that is what it is. And that's such a fundamental feature of arithmetic that we almost

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think of it like God's law. You know what I mean? It has to be that way. That's a really powerful

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idea. It's so cool that every number is uniquely made up of other numbers. And like made up meaning

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like there's these like basic atoms that form molecules that get built on top of each other.

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I love it. I mean, when I teach, you know, undergraduate number theory, it's like,

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it's the first really deep theorem that you prove. What's amazing is, you know, the fact that you can

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factor a number into primes is much easier. Essentially Euclid knew it all, though he didn't

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quite put it in that way. The fact that you can do it at all. What's deep is the fact that

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there's only one way to do it that or however you sort of chop the number up, you end up with the

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same set of prime factors. And indeed, what people finally understood at the end of the 19th century

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is that if you work in number systems slightly more general than the ones we're used to,

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which it turns out irrelevant to Fermat, all of a sudden, this stops being true.

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Things get, I mean, things get more complicated. And now, because you were praising simplicity

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before, you were like, it's so beautiful, unique factorization. It's so great. Like it's when I

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tell you that in more general number systems, there is no unique factorization. Maybe you're

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like, that's bad. I'm like, no, that's good, because there's like a whole new world of phenomena to

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study that you just can't see through the lens of the numbers that we're used to. So I'm, I'm

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for complication. I'm highly in favor of complication, because every complication is like an opportunity

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for new things to study. And is that the big kind of one of the big insights for you from

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Andrew Wiles's proof? Is there interesting insights about the process they used to prove

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that sort of resonates with you as a mathematician? Is there interesting concept that emerged from

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it? Is there interesting human aspects to the proof? Whether there's interesting human aspects

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to the proof itself is an interesting question. Certainly, it has a huge amount of richness.

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Sort of at its heart is an argument of, of what's called deformation theory, which was,

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in part, created by my PhD advisor, Barry Mazer. Can you speak to what deformation theory

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Right. Well, the reason that Barry called it deformation theory, I think he's the one who

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gave it the name. I hope I'm not wrong in saying Sunday. In your book, you have calling different

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things by the same name as one of the things in the beautiful map that opens the book.

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Yes. And this is a perfect example. So this is another phrase of PoincarÃ©, this like incredible

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generator of slogans and aphorisms. He said, mathematics is the art of calling different

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things by the same name. That very thing, that very thing we do, right? When we're like this

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triangle and this triangle, come on, they're the same triangle. They're just in a different place,

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right? So in the same way, it came to be understood that the kinds of objects that you study

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when you study, when you study Fermat's last theorem, and let's not even be too careful about

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what these objects are. I can tell you, they're gaol representations and modular forms, but saying

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those words is not going to mean so much. But whatever they are, they're things that can be

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deformed, moved around a little bit. And I think the insight of what Andrew and then Andrew and

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Richard were able to do was to say something like this. A deformation means moving something just a

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tiny bit, like an infinitesimal amount. If you really are good at understanding which ways

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a thing can move in a tiny, tiny, tiny infinitesimal amount in certain directions, maybe you can

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piece that information together to understand the whole global space in which it can move.

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And essentially, their argument comes down to showing that two of those big global spaces

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are actually the same, the fabled r equals t, part of their proof, which is at the heart of it.

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And it involves this very careful principle like that. But that being said, what I just said,

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it's probably not what you're thinking, because what you're thinking, when you think, oh, I have a

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point in space and I move it around like a little tiny bit, you're using your notion of distance

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that's from calculus. We know what it means for like two points on the real line to be close

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together. So yet another thing that comes up in the book a lot is this fact that the notion of

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distance is not given to us by God. We could mean a lot of different things by distance.

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And just in the English language, we do that all the time. We talk about somebody being a close

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relative. It doesn't mean they live next door to you, right? It means something else. There's a

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different notion of distance we have in mind. And there are lots of notions of distances

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that you could use in the natural language processing community and AI. There might be some

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notion of semantic distance or lexical distance between two words. How much do they tend to

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arise in the same context? That's incredibly important for doing autocomplete and machine

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translation and stuff like that. And it doesn't have anything to do with, are they next to each

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other in the dictionary? It's a different kind of distance. Okay, ready? In this kind of number

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theory, there was a crazy distance called the piatic distance. I didn't write about this that

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much in the book, because even though I love it, it's a big part of my research life. It gets a

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little bit into the weeds, but your listeners are going to hear about it now. Please, you know,

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what a normal person says when they say two numbers are close, they say like, you know,

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their difference is like a small number, like seven and eight are close because their difference is

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one and one's pretty small. If we were to be what's called a two attic number theorist,

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we'd say, oh, two numbers are close. If their difference is a multiple of a large power of two.

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So like, so like one and 49 are close, because their difference is 48 and 48 is a multiple

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of 16, which is a pretty large power of two, whereas whereas one and two are pretty far away,

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because the difference between them is one, which is not even a multiple of a power of two at all.

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because their difference is a negative power of two, two to the minus six. So those

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Yeah. So two to a large power is this metric, a very small number and two to a negative power

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It takes practice. It takes practice. If you've ever heard of the Cantor set, it looks kind of like

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that. So it is crazy that this is good for anything, right? I mean, this just sounds

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like a definition that someone would make up to torment you. But what's amazing is

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there's a general theory of distance where you say any definition you make that satisfies

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certain axioms deserves to be called a distance and this. See, I'm sorry to interrupt. My brain,

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you broke my brain now. Awesome. 10 seconds ago. Because I'm also starting to map for the two out

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of case to binary numbers, you know, because we romanticized those. Oh, that's exactly the

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right way to think of it. I was trying to mess with numbers. I was trying to see, okay, which ones

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are close. And then I'm starting to visualize different binary numbers and how they, which ones

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are close to each other. And that's, well, I think there's no, no, it's very similar. That's

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exactly the right way to think of it. It's almost like binary numbers written in reverse, right?

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Because in a, in a binary expansion, two numbers are close. A number that's small is like 0.000

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something. Something that's the decimal and it starts with a lot of zeros. In the two attic

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metric, a binary number is very small if it ends with a lot of zeros and then the decimal point.

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Gotcha. So it is kind of like binary numbers written backwards is actually, I should have,

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Oh, okay. But so why is that, why is that interesting? Except for the fact that it's,

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it's a beautiful kind of framework, different kind of framework, which you think about distances.

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And you're talking about not just the two attic, but the generalization of that.

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Yeah, the NEP. And so, so that, because that's the kind of deformation that comes up

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in Wiles's, in Wiles's proof that deformation where moving something a little bit means a little

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bit in this two attic sense. Okay. No, I mean, it's such, I mean, I just get excited to talk

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about it. And I just taught this like in the fall semester that, but it like reformulating, why is,

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so you pick a different measure of distance over which you can talk about very tiny changes

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and then use that to then prove things about the entire thing. Yes. Although, you know,

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honestly, what I would say, I mean, it's true that we use it to prove things, but I would say

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we use it to understand things. And then because we understand things better, then we can prove

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things. But you know, the goal is always the understanding, the goal is not to, so much

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to prove things, the goal is not to know what's true or false. I mean, this is the thing I write

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about in the book near the end, and it's something that, so wonderful, wonderful essay by, by Bill

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Thurston, kind of one of the great geometers of our time, who unfortunately passed away a few years

link |

ago, um, called on proof and progress in mathematics. And he writes very wonderfully about how,

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you know, we're not, it's not a theorem factory where we have a production quota. I mean, the

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point of mathematics is to help humans understand things. And the way we test that is that we're

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proving new theorems along the way. That's the benchmark, but that's not the goal. Yeah. But

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just as a, as a kind of absolutely, but as a tool, it's kind of interesting to approach a problem

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by saying, how can I change the distance function? Like what the nature of distance,

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because that might start to lead to insights for deeper understanding. Like if I were to try to

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describe human society by a distance, two people are close if they love each other, right? And then,

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and then start to, uh, and do a full analysis on the everybody that lives on earth currently,

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the seven billion people. And from that perspective, as opposed to the geographic perspective of

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distance, and then maybe there could be a bunch of insights about the source of, uh, violence,

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the source of, uh, maybe entrepreneurial success or invention or economic success or different

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systems, communism, capitalism start to, I mean, that's, I guess what economics tries to do. But

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really saying, okay, let's think outside the box about totally new distance functions that could

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unlock something profound about the space. Yeah. Because think about it. Okay. Here's,

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I mean, now we're going to talk about AI, which you know a lot more about than I do. So

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just, you know, start laughing uproariously if I say something that's completely wrong.

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That is, that is a really good humble way to think about it. I like it. Okay. So let's just go for it.

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Um, okay. So I think you'll agree with this, that in some sense what's good about AI is that

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we can't test any case in advance. The whole point of AI is to make, or one point of it,

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I guess, is to make good predictions about cases we haven't yet seen. And in some sense,

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that's always going to involve some notion of distance because it's always going to involve

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somehow taking a case we haven't seen and saying what cases that we have seen, is it close to?

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Is it like, is it somehow an interpolation between? Now, when we do that in order to talk about

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things being like other things, implicitly or explicitly, we're invoking some notion of distance

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and boy, we better get it right. Right? If you try to do natural language processing and your

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idea about distance between words is how close they are in the dictionary. When you write them

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in alphabetical order, you are going to get pretty bad translations, right? No, the notion of distance

link |

has to come from somewhere else. Yeah, that's essentially when neural networks are doing this.

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Well, word embeddings are doing is coming up with. In the case of word embeddings, literally,

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like literally what they are doing is learning a distance. But those are super complicated

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distance functions. And it's almost nice to think maybe there's a nice transformation that's simple.

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Let me ask you about this. From an understanding perspective, there's the Richard Feynman maybe

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attributed to him, but maybe many others is this idea that if you can't explain something simply

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that you don't understand it. In how many cases, how often is that true? Do you find there's some

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profound truth in that? Oh, okay. So you were about to ask, is it true, which I would say

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flatly no, but then you said you followed that up with, is there some profound truth in it?

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It says your mathematician answer. The truth that is in it is that learning to explain something

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helps you understand it. But real things are not simple. A few things are, most are not.

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And I don't, to be honest, I don't, I mean, I don't, we don't really know whether Feynman

link |

really said that, right? Or something like that is sort of disputed. But I don't think Feynman

link |

could have literally believed that whether or not he said it. And, you know, he was the kind

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of guy, I didn't know him, but I'm reading his writing, he liked to sort of say stuff like stuff

link |

that sounded good. You know what I mean? So it totally strikes me as the kind of thing he could

link |

have said because he liked the way saying it made him feel. But also knowing that he didn't

link |

like literally mean it. Well, I definitely have a lot of friends and I've talked to a lot of

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physicists and they do derive joy from believing that they can explain stuff simply or believing

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as possible to explain stuff simply, even when the explanation is not actually that simple.

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Like I've heard people think that the explanation is simple and they do the explanation. And I

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think it is simple, but it's not capturing the phenomena that we're discussing. It's capturing,

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it somehow maps in their mind, but it's, it's taking as a starting point as an assumption

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that there's a deep knowledge and a deep understanding that's, that's actually very

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complicated. And the simplicity is almost like a, almost like a poem about the more complicated

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thing as opposed to a distillation. And I love poems, but a poem is not an explanation.

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Well, some people might disagree with that, but certainly from a mathematical perspective,

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no poet would disagree with it. No poet would disagree. You don't think there's some things

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that can only be described in precisely? I said explanation. I don't think any poem would,

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I don't think any poet would say their poem is an explanation. They might say it's a description.

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They might say it's sort of capturing sort of. Well, some people might say the only truth is like

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music, right? That the, the, the, not the only truth, but some truth can only be expressed through

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art. And I mean, that's the whole thing we're talking about religion and myth. And there's

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some things that are limited cognitive capabilities and the tools of mathematics or the tools of physics

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are just not going to allow us to capture. Like it's possible consciousness is one of those things.

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Yes, that is definitely possible. But I would even say, look, in consciousness is a thing about

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which we're still in the dark as to whether there's an explanation we would, we would understand

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as an explanation at all. By the way, okay, I got to give yet one more amazing Poincare quote,

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because this guy just never stopped coming up with great quotes that, you know, Paul Erdisch,

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another fellow who appears in the book. And by the way, he thinks about this notion of distance of

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like personal affinity, kind of like what you're talking about, the kind of social network and

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that notion of distance that comes from that. So that's something that Paul Erdisch, Erdisch did.

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Well, he thought about distances in networks. I guess he didn't probably, he didn't think about

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the social network. That's fascinating. That's how he started that story, where there's no,

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but yeah, okay. But you know, Erdisch was sort of famous for saying, and this is sort of one

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line you were saying, he talked about the book, capital T, capital B, the book. And that's the

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book where God keeps the right proof of every theorem. So when he saw a proof he really liked,

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it was like really elegant, really simple, like that's from the book. That's like you found one

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of the ones that's in the book. He wasn't a religious guy, by the way, he referred to God

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as the supreme fascist. He was like, but somehow he was like, I don't really believe in God,

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but I believe in God's book. But PoincarÃ©, on the other hand, and by the way, there are other

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men, Hilda Hudson is one who comes up in this book. She also kind of saw math. She's one of the

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people who sort of develops the disease model that we now use, that we use to sort of track

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pandemics, this SIR model that sort of originally comes from her work with Ronald Ross. But she

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was also super, super, super devout. And she also sort of on the other side of the religious coin

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was like, yeah, math is how we communicate with God. She has a great, all these people are incredibly

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quotable. She says, you know, math is, the truth, the things about mathematics, she's like,

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they're not the most important of God's thoughts, but they're the only ones that we can know precisely.

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So she's like, this is the one place where we get to sort of see what God's thinking when we do

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mathematics. Again, not a fan of poetry or music. Some people say Hendricks is like,

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some people say chapter one of that book is mathematics. And then chapter two is like classic

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rock. Right. So like, it's not clear that the, I'm sorry, you just sent me off on a tangent,

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just imagining like Erdish at a Hendricks concert, trying to figure out if it was from

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the book or not. What I was coming to was just to say, but when PoincarÃ© said about this is he's

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like, you know, if like, this is all worked out in the language of the divine and if a divine

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being like came down and told it to us, we wouldn't be able to understand it. So it doesn't matter.

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inhumanly complex. And that was how they really were. Our job is to figure out the things that

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are not like that. They're not like that. All this talk of primes got me hungry for primes.

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You, uh, you wrote a blog post, the beauty of bounding gaps, a huge discovery about prime

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numbers and what it means for the future of math. Can you tell me about prime numbers? What the

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heck are those? What are twin primes? What are prime gaps? What are bounding gaps in primes?

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What are all these things? And what, if anything, or what exactly is beautiful about them?

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Yeah. So, you know, prime numbers are one of the things that number theorists study the most and

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have for millennia. They are numbers which can't be factored. And then you say like, like five,

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and then you're like, wait, I can factor five. Five is five times one. Okay, not like that.

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That is a factorization. It absolutely is a way of expressing five as a product of two things.

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But don't you agree that there's like something trivial about it? It's something you can do to

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any number. It doesn't have content the way that if I say that 12 is six times two or 35 is seven

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times five, I've really done something to it. I've broken up. So those are the kind of factorizations

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that count. And a number that doesn't have a factorization like that is called prime, except

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historical side note one, which at some times in mathematical history has been deemed to be a prime,

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but currently is not. And I think that's for the best. But I bring it up only because sometimes

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people think that, you know, these definitions are kind of, if we think about them hard enough,

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we can figure out which definition is true. No, there's just an artifact of mathematics. So

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so it's the definition is best for us for our purposes. Well, those edge cases are weird, right?

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So so it can't you can't be it doesn't count when you use yourself as a number or one as part of

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the factorization or as the entirety of the factorization. So the so you somehow get to the

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meat of the number by factorizing it. And that seems to get to the core of all of mathematics.

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Yeah, you take any number and you factorize it until you can factorize no more. And what you

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have left is some big pile of primes. I mean, by definition, when you can't factor anymore,

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when you when you're done, when you can't break the numbers up anymore, what's left must be prime,

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you know, 12 breaks into two and two and three. So these numbers are the atoms, the building blocks

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of all numbers. And there's a lot we know about them. But there's much more that we don't know

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them. I'll tell you the first few, there's two, three, five, seven, 11. By the way, they're all

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going to be odd from then on, because if they were even, I could factor a two out of them.

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But it's not all the odd numbers. Nine isn't prime because it's three times three.

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15 isn't prime because it's three times five, but 13 is where we're at two, three, five,

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seven, 11, 13, 17, 19, not 21, but 23 is et cetera, et cetera. Okay, so you could go on.

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I think so. There's always those ones that trip people up. There's a famous one,

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the Groten Dieck prime, 57, like sort of Alexander Groten Dieck, the great algebraic

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geometry was sort of giving some lecture involving a choice of a prime in general.

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Okay. There's a humor in it. Yes, I would say over 100, I definitely don't remember,

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they're definitely ones that people, or 91 is another classic seven times 13. It really

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feels kind of prime, doesn't it? But it is not. Yeah. But there's also, by the way,

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but there's also an actual notion of pseudo prime, which is a thing with a formal definition,

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which is not a psychological thing. It is a prime which passes a primality test devised by Fermat,

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which is a very good test, which if a number fails this test, it's definitely not prime.

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And so there was some hope that, oh, maybe if a number passes the test, then it definitely

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it's only perfect in one direction. So there are numbers, I want to say 341 is the smallest,

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which passed the test, but are not prime 341. Is this test easily explainable or no?

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Yes, actually. Ready? Let me give you the simplest version of it. You can dress it up a little bit,

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but here's the basic idea. I take the number, the mystery number, I raise two to that power.

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So let's say your mystery number is six. Are you sorry you asked me? Are you ready?

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demonstration. Let's say your number is six. So I'm going to raise two to the sixth power.

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Okay, so if I were working on it, I'd be like, that's two cubes squared. So that's eight times

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eight. So that's 64. Now we're going to divide by six, but I don't actually care what the

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quotient is, only the remainder. So let's say 64 divided by six is, well, there's a quotient of 10,

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but the remainder is four. So you failed because the answer has to be two. For any prime,

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let's do it with five, which is prime. Two to the fifth is 32. Divide 32 by five.

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With a remainder of two. For seven, two to the seventh is 128. Divide that by seven.

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18 is 126 with a remainder of two, right? 128 is a multiple of seven plus two. So if that

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remainder is not two, then that's definitely not prime. Then it's definitely not prime.

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And then if it is, it's likely a prime, but not for sure. It's likely a prime, but not for

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sure. And there's actually a beautiful geometric proof, which is in the book, actually. That's

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like one of the most granular parts of the book, because it's such a beautiful proof I couldn't

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not give it. So you draw a lot of like opal and pearl necklaces and spin them. That's kind of the

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geometric nature of this proof of Fermat's little theorem. So yeah, so with pseudo primes,

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there are primes that are kind of faking it. They pass that test, but there are numbers that

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are faking it. They pass that test, but are not actually prime. But the point is,

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there are many, many, many theorems about prime numbers. Are there like, there's a bunch of

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questions to ask, is there an infinite number of primes? Can we say something about the gap

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between primes as the numbers grow larger and larger and larger and so on? Yeah, it's a perfect

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example of your desire for simplicity in all things. You know, it would be really simple

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finite set of atoms that all numbers would be built up. That's right. That would be very

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simple in good and certain ways, but it's completely false. And number theory would be

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totally different if that were the case. It's just not true. In fact, this is something else

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that Euclid knew. So this is a very, very old fact, like much before, long before we'd had

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anything like modern number. The primes are infinite. The primes that there are, right,

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the infinite primes. There's an infinite number of primes. So what about the gas between the

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primes? Right. So one thing that people recognized and really thought about a lot is that the primes

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on average seem to get farther and farther apart as they get bigger and bigger. In other words,

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it's less and less common. Like I already told you of the first 10 numbers, two, three, five,

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seven, four of them are prime. That's a lot, 40%. If I looked at, you know, 10 digit numbers,

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no way would 40% of those be prime. Being prime would be a lot rare in some sense because there's

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a lot more things for them to be divisible by. That's one way of thinking of it. It's a lot more

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possible for there to be a factorization because there's a lot of things you can try to factor

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out of it. As the numbers get bigger and bigger, primality gets rarer and rarer.

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And the extent to which that's the case, that's pretty well understood. But then you can ask more

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fine grained questions. And here is one. A twin prime is a pair of primes that are two apart,

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like three and five, or like 11 and 13, or like 17 and 19. And one thing we still don't know is,

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are there infinitely many of those? We know on average, they get farther and farther apart,

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but that doesn't mean there couldn't be like occasional folks that come close together.

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And indeed, we think that there are. And one interesting question, I mean, this is,

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because I think you might say like, well, why, how could one possibly have a right to have an

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opinion about something like that? Like what, you know, we don't have any way of describing a

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process that makes primes like, sure, you can like look at your computer and see a lot of them.

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But the fact that there's a lot, why is that evidence that there's infinitely many, right?

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Maybe I can go on the computer and find 10 million. Well, 10 million is pretty far from

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infinity, right? So how is that, how is that evidence? There's a lot of things. There's like

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a lot more than 10 million atoms. That doesn't mean there's infinitely many atoms in the universe,

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right? I mean, on most people's physical theories, there's probably not, as I understand it.

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Okay. So why would we think this? The answer is that we've, that it turns out to be like

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incredibly productive and enlightening to think about primes as if they were random numbers,

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as if they were randomly distributed according to a certain law. Now they're not,

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whether a number is prime or not. And yet it just turns out to be phenomenally useful in

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mathematics to say, even if something is governed by a deterministic law, let's just pretend it

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wasn't, let's just pretend that they were produced by some random process and see if the behavior is

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roughly the same. And if it's not, maybe change the random process, maybe make the randomness a

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little bit different and tweak it and see if you can find a random process that matches the behavior

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we see. And then maybe you predict that other behaviors of the system are like that of the

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random process. And so that's kind of like, it's funny because I think when you talk to people

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about the twin prime conjecture, people think you're saying, wow, there's like some deep structure

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there that like makes those primes be like close together again and again. And no, it's the opposite

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of deep structure. What we say when we say we believe the twin prime conjecture is that we

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believe the primes are like sort of strewn around pretty randomly. And if they were, then by chance,

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you would expect there to be infinitely many twin primes. And we're saying, yeah, we expect them

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to behave just like they would if they were random dirt. You know, the fascinating parallel here is

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I just got a chance to talk to Sam Harris. And he uses the prime numbers as an example.

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Often, I don't know if you're familiar with who Sam is. He uses that as an example of there being

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no free will. Wait, where does he get this? Well, he just uses as an example of it might seem like

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this is a random number generator, but it's all like formally defined. So if we keep getting more

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and more primes, then like that might feel like a new discovery and that might feel like a new

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experience, but it's not. It was always written in the cards. But it's funny that you say that

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because a lot of people think of like randomness, the fundamental randomness within the nature of

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reality might be the source of something that we experience as free will. And you're saying it's

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like useful to look at prime numbers as a random process in order to prove stuff about them.

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But fundamentally, of course, it's not a random process. Well, not in order to prove some stuff

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about them so much as to figure out what we expect to be true and then try to prove that.

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Because here's what you don't want to do. Try really hard to prove something that's false.

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That makes it really hard to prove the thing if it's false. So you certainly want to have some

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heuristic ways of guessing, making good guesses about what's true. So yeah, here's what I would

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say. You're going to be imaginary Sam Harris now. You are talking about prime numbers and you are like,

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but prime numbers are completely deterministic. And I'm saying, well, let's treat them like

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a random process. And then you say, but you're just saying something that's not true. They're

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not a random process. They're deterministic. And I'm like, okay, great, you hold to your

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insistence that it's not a random process. Meanwhile, I'm generating insight about the

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primes that you're not because I'm willing to sort of pretend that there's something that

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they're not in order to understand what's going on. Yeah, so it doesn't matter what the reality is.

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What matters is what framework of thought results in the maximum number of insights.

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Yeah, because I feel, look, I'm sorry, but I feel like you have more insights about people if you

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think of them as like beings that have wants and needs and desires and do stuff on purpose.

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Even if that's not true, you still understand better what's going on by treating them in that

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way. Don't you find, look, when you work on machine learning, don't you find yourself sort

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of talking about what the machine is trying to do in a certain instance? Do you not find

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yourself drawn to that language? It knows this. It's trying to do that. It's learning that.

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I'm certainly drawn to that language to the point where I receive quite a bit of criticisms for it

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because I, you know, like, oh, I'm on your side, man. So especially in robotics, I don't know why,

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but robotics people don't like to name their robots. They certainly don't like to gender

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their robots because the moment you gender a robot, you start to anthropomorphize.

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like a life story in your mind. You can't help it. It's like you create like a humorous story to

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this person. You start to, this person, this robot, you start to project your own. But I think

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that's what we do to each other. And I think that's actually really useful for the engineering

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process, especially for human robot interaction. And yes, for machine learning systems,

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for helping you build an intuition about a particular problem. It's almost like asking this

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question, you know, when a machine learning system fails in a particular edge case, asking

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like, what were you thinking about? Like, like asking like almost like when you're talking about

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how did they see the world? Maybe there's a totally new, maybe you're the one that's

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thinking about the world incorrectly. And yeah, that anthropomorphization process,

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I think is ultimately good for insight. And the same as I agree with you, I tend to believe

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Of course. I've just recently, most people go on like rabbit hole, like YouTube things. And

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I went on a rabbit hole often do of Wikipedia. And I found a page on finitism, ultra finitism,

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and intuitionism, or I forget what it's called. Yeah, intuitionism. Intuitionism.

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That seemed pretty, pretty interesting. I have a my to do list actually like looking to, like,

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is there people who like formally attract like real mathematicians are trying to argue for this.

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Meaning, infinity might be like a useful hack for certain, like a useful tool in mathematics,

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but it really gets us into trouble. Because there's no infinity in the real world. Maybe I'm sort of

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not expressing that fully correctly, but basically saying like there's things there.

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And once you add into mathematics, things that are not provably within the physical world,

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you're starting to inject, to corrupt your framework of reason. What do you think about that?

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I mean, I think, okay, so first of all, I'm not an expert. And I couldn't even tell you what the

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difference is between those three terms finitism, ultra finitism, and intuitionism, although I

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know what they're related, I tend to associate them with the Netherlands in the 1930s.

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Okay, I'll tell you, can I just quickly comment because I read the Wikipedia page,

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Bro, I'm basically an expert. Ultra finitism, so finitism says that the only infinity you're

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allowed to have is that the natural numbers are infinite. So like those numbers are infinite.

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So like one, two, three, four, five, the integers are infinite. The ultra finitism says, nope,

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even that infinity is fake. I'll bet ultra finitism came second. I'll bet it's like when

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there's like a hardcore scene and then one guy's like, oh, now there's a lot of people in the

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scene. I have to find a way to be more hardcore than the hardcore people. All back to the emo talk.

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Yeah. Okay, so is there any... Are you ever... Because I'm often uncomfortable with infinity.

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Like psychologically. I have trouble when that sneaks in there. It's because it works so damn

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well. I get a little suspicious because it could be almost like a crutch or an oversimplification

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that's missing something profound about reality. Well, so first of all, okay, if you say like,

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is there like a serious way of doing mathematics that doesn't really treat infinity as a real thing

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or maybe it's kind of agnostic and is like, I'm not really going to make a firm statement about

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whether it's a real thing or not. Yeah, that's called most of the history of mathematics, right?

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So it's only after Cantor, right, that we really are sort of, okay, we're going to like

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do something that you might call like the modern theory of infinity. That said, obviously,

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everybody was drawn to this notion and no, not everybody was comfortable with it. Look,

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I mean, this is what happens with Newton, right? I mean, so Newton understands that to talk about

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tangents and to talk about instantaneous velocity, he has to do something that we would now call

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taking a limit, right? The fabled dy over dx. If you sort of go back to your calculus class,

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for those who've taken calculus, remember this mysterious thing and you know, what is it? What

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is it? Well, he'd say like, well, it's like you sort of divide the length of this line segment

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by the length of this other line segment and then you make them a little shorter and you

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divide again and then you make them a little shorter and you divide again and then you just

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keep on doing that until they're like infinitely short and then you divide them again. These quantities

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that are like, they're not zero, but they're also smaller than any actual number, these infinitesimals.

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Well, people were queasy about it and they weren't wrong to be queasy about it, right? From a

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modern perspective, it was not really well formed. There's this very famous critique of Newton by

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Bishop Berkeley where he says like, what these things you define like, you know, they're not zero,

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but they're smaller than any number. Are they the ghosts of departed quantities? That was

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this like ultra line of Newton. And on the one hand, he was right. It wasn't really rigorous

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by modern standards. On the other hand, like Newton was out there doing calculus and other

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people were not writing. It works. It works. I think a sort of intuitionist view, for instance,

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I would say would express serious doubt. And by the way, it's not just infinity. It's like

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saying, I think we would express serious doubt that like, the real numbers exist.

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It's a great point, actually. I think in some sense, this flavor of doing math saying,

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we shouldn't talk about things that we cannot specify in a finite amount of time. There's

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something very computational in flavor about that. And it's probably not a coincidence that it

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becomes popular in the 30s and 40s, which is also like kind of like the dawn of ideas about

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formal computation, right? You probably know the timeline better than I do. Sorry, what becomes

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popular? These ideas that maybe we should be doing math in this more restrictive way where

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you know, number represents a magnitude, like the length of a line. Like so, I mean, the idea that

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is pretty old, but that, you know, just because something is old doesn't mean we can't reject

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it if we want to. Well, a lot of the fundamental ideas in computer science, when you talk about

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Well, the ideas that kind of challenge that, the whole space of machine learning, I would say,

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challenges that. It's almost like the engineering approach to things, like the floating point

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arithmetic. The other one that, back to John Conway, that challenges this idea, I mean,

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maybe to tie in the ideas of deformation theory and limits to infinity is this idea of cellular

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automata. With John Conway looking at the game of life, Steven Wolfram's work, that I've been a

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big fan of for a while of cellular automata. I was wondering if you have, if you have ever

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encountered these kinds of objects, you ever looked at them as a mathematician, where you have

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very simple rules of tiny little objects that when taken as a whole create incredible complexities,

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but are very difficult to analyze, very difficult to make sense of, even though the one individual

link |

object, one part, it's like what we were saying about Andrew Wiles, you can look at the deformation

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of a small piece to tell you about the whole. It feels like with cellular automata or any kind of

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complex systems, it's often very difficult to say something about the whole thing, even when you

link |

can precisely describe the operation of the local neighborhoods. Yeah, I mean, I love that subject.

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I haven't really done research in it myself. I've played around with it. I'll send you a fun blog post

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I wrote where I made some cool texture patterns from cellular automata that I... And those are

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really always compelling. It's like you create simple rules and they create some beautiful

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textures. It doesn't make any sense. Actually, did you see there was a great paper? I don't know if

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you saw this, like a machine learning paper. Yes. I don't know if you saw the one I'm talking about,

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where they were learning the texture as like, let's try to reverse engineer and learn a cellular

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automaton that can reduce texture that looks like this from the images. Very cool. And as you say,

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the thing you said is I feel the same way when I read machine learning paper is that

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what's especially interesting is the cases where it doesn't work. What does it do when it

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doesn't do the thing that you tried to train it to do? That's extremely interesting. Yeah,

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that was a cool paper. So yeah, so let's start with the game of life. Let's start with...

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Yeah, so let's start with John Conway again. Just I don't know, from my outsider's perspective,

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there's not many mathematicians that stand out throughout the history of the 20th century.

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And he's one of them. I feel like he's not sufficiently recognized. I think he's pretty

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recognized. Okay. Well, I mean, he was a full professor of Princeton for most of his life.

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He was sort of certainly the pinnacle of... Yeah, but I found myself every time I talk about Conway

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and how excited I am about him. I have to constantly explain to people who he is.

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And that's always a sad sign to me. But that's probably true for a lot of mathematicians.

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I was about to say, I feel like you have a very elevated idea of how famous... This is

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what happens when you grow up in the Soviet Union or you think the mathematicians are very, very

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famous. Yeah, but I'm not actually so convinced at a tiny tangent that that shouldn't be so.

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I mean, it's not obvious to me that that's one of the... If I were to analyze American society that

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perhaps elevating mathematical and scientific thinking to a little bit higher level would

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benefit the society, well, both in discovering the beauty of what it is to be human and for

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actually creating cool technology, better iPhones. But anyway, John Conway. Yeah, and Conway is such

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a perfect example of somebody whose humanity was and his personality was like a wound up with his

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mathematics. So it's not... Sometimes I think people who are outside the field think of mathematics

link |

as this kind of cold thing that you do separate from your existence as a human being. No way,

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your personality is in there just as it would be in a novel you wrote or a painting you painted or

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just the way you walk down the street. It's in there, it's you doing it. And Conway was certainly

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a singular personality. I think anybody would say that he was playful, like everything was

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a game to him. Now, what you may think I'm going to say, and it's true is that he sort of was very

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playful in his way of doing mathematics, but it's also true. It went both ways. He also sort of made

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mathematics out of games. He looked at... He was a constant inventor of games with like crazy names

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and then he was sort of analyzed those games mathematically to the point that he and then

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later collaborating with Knuth like created this number system, the serial numbers, in which actually

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each number is a game. There's a wonderful book about this called... I mean, there are his own

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books and then there's like a book that he wrote with Berlecamp and Guy called Winning Ways, which is

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such a rich source of ideas. And he too kind of has his own crazy number system in which, by the

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way, there are these infinitesimals, the ghosts of departed quantities. They're in there now not as

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ghosts, but as like certain kind of two player games. So, you know, he was a guy. So I knew him

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when I was a postdoc and I knew him at Princeton and our research overlapped in some ways. Now,

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it was on stuff that he had worked on many years before. The stuff I was working on kind of connected

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with stuff in group theory, which somehow seems to keep coming up. And so I often would like to

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sort of ask him a question. I would sort of come upon him in the common room and I would ask him a

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question about something. And just anytime you turned him on, you know what I mean? You sort of

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asked a question. It was just like turning a knob and winding him up and he would just go and you

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would get a response that was like so rich and went so many places and taught you so much. And

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usually had nothing to do with your question. Yeah. Usually your question was just a prompt.

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Damn, you couldn't count on actually getting the question. Yeah, there's brilliant curious minds

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that even at that age, yeah, it was definitely a huge loss. But on his game of life, which was I

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think he developed in the 70s as almost like a side thing, a fun little experiment. Yeah, the game

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of life is this. It's a very simple algorithm. It's not really a game per se in the sense of the

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kinds of games that he liked, where people played against each other. But essentially, it's a game

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that you play with marking little squares on the sheet of graph paper. And in the 70s, I think he

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was like literally doing it with like a pen on graph paper, you have some configuration of squares,

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some of the squares in the graph paper are filled in, some are not. And then there's a rule, a single

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rule that tells you at the next stage, which squares are filled in and which squares are not.

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Sometimes an empty square gets filled in, that's called birth, sometimes a square that's filled

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in gets erased, that's called death. And there's rules for which squares are born, which squares die.

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It's the rule is very simple, you can write it on one line. And then the great miracle is that you

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can start from some very innocent looking little small set of boxes and get these results of

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incredible richness. And of course, nowadays, you don't do it on paper. Nowadays, you do it on a

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computer. There's actually a great iPad app called Golly, which I really like that has like

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Conway's original rule and like, gosh, like hundreds of other variants. And it's a lightning

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fast. So you can just be like, I want to see 10,000 generations of this rule play out like faster

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than your eye can even follow. And it's like amazing. So I highly recommend it if this is

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at all intriguing to you getting golly on your iOS device. And you can do this kind of process,

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which I really enjoy doing, which is almost from like putting a Darwin hat on or a biologist hat

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on and doing analysis of a higher level of abstraction, like the organisms that spring up,

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because there's different kinds of organisms, like you can think of them as species, and they

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interact with each other. They can, there's gliders, they shoot different, there's like

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things that can travel around. There's things that can glider guns that can generate those gliders.

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And you can use the same kind of language as you would about describing a biological system.

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So it's a wonderful laboratory. And it's kind of a rebuke to someone who doesn't think that like

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very, very rich complex structure can come from very simple underlying laws. Like

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it definitely can't. Now, here's what's interesting. If you just picked like some random rule,

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you wouldn't get interesting complexity. I think that's one of the most interesting

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things of these, one of these most interesting features of this whole subject that the rules

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have to be tuned just right. Like a sort of typical rule set doesn't generate any kind of

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interesting behavior. But some do. I don't think we have a clear way of understanding

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which do and which don't. I don't know. Maybe Steven thinks he does. I don't know, but...

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No, no, it's a giant mystery. What Steven Wolfram did is... Now, there's a whole interesting aspect

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to the fact that he's a little bit of an outcast in the mathematics and physics community,

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because he's so focused on a particular, his particular work. I think if you put ego aside,

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which I think unfairly, some people are not able to look beyond. I think his work is

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actually quite brilliant. But what he did is exactly this process of Darwin like exploration

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is taking these very simple ideas and writing a thousand page book on them, meaning like,

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let's play around with this thing. Let's see. And can we figure anything out? Spoiler alert? No,

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we can't. In fact, he does a challenge. I think it's like a rule 30 challenge, which is quite

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interesting. Just simply for machine learning people, for mathematics people, is can you predict

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the middle column? For his, it's a 1D cellular automata. Can you predict, generally speaking,

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can you predict anything about how a particular rule will evolve just in the future? Very simple.

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100 steps ahead. Can you predict something? And the challenge is to do that kind of prediction

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so far as nobody's come up with an answer. But the point is, we can't. We don't have tools,

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or maybe it's impossible, or, I mean, he has these kind of laws of irreducibility. They hear

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first of his poetry. It's like, we can't prove these things. It seems like we can't. That's the

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basic... It almost sounds like ancient mathematics or something like that, where you're like, the

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gods will not allow us to predict the cellular automata. But that's fascinating that we can't.

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I'm not sure what to make of it. And there's power to calling this particular set of rules,

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game of life, as Conway did. Because I'm not exactly sure, but I think he had a sense that

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there's some core ideas here that are fundamental to life, to complex systems, to the way life

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emerged on Earth. I'm not sure I think Conway thought that. It's something that... I mean,

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Conway always had a rather ambivalent relationship with the game of life, because I think he saw it

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as... It was certainly the thing he was most famous for in the outside world. And I think that he,

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his view, which is correct, is that he had done things that were much deeper mathematically than

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that. And I think it always aggrieved him a bit that he was the game of life guy when he proved

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all these wonderful theorems and created all these wonderful games, created the serial numbers.

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He was a very tireless guy who just did an incredibly variegated array of stuff. So he was

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exactly the kind of person who you would never want to reduce to one achievement. You know what

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I mean? Let me ask you about group theory. You mentioned it a few times. What is group theory?

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What is an idea from group theory that you find beautiful? Well, so I would say group theory

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sort of starts as the general theory of symmetry is that people looked at different kinds of things

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and said, like, as we said, like, oh, it could have maybe all there is the symmetry from left to

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right, like a human being, right? Or that's roughly bilaterally symmetric, as we say. So

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there's two symmetries. And then you're like, well, wait, didn't I say there's just one? There's

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just left to right? Well, we always count the symmetry of doing nothing. We always count the

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symmetry that's like, there's flip and don't flip. Those are the two configurations that you can be

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in. So there's two. You know, something like a rectangle is bilaterally symmetric, you can flip

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it left to right, but you can also flip it top to bottom. So there's actually four symmetries.

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There's do nothing, flip it left to right, and flip it top to bottom or do both of those things.

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A square, there's even more because now you can rotate it. You can rotate it by 90 degrees. So

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you can't do that. That's not a symmetry of the rectangle. If you try to rotate it 90 degrees,

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you get a rectangle oriented in a different way. So a person has two symmetries, a rectangle for

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a square, eight, different kinds of shapes have different numbers of symmetries. And the real

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observation is that that's just not like a set of things. They can be combined. You do one symmetry,

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then you do another. The result of that is some third symmetry. So a group really abstracts away

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this notion of saying it's just some collection of transformations you can do to a thing where

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you combine any two of them to get a third. So, you know, a place where this comes up in computer

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science is in sorting because the ways of permuting a set, the ways of taking sort of some set of

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things you have on the table and putting them in a different order, shuffling a deck of cards,

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for instance, those are the symmetries of the deck. And there's a lot of them. There's not two,

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there's not four, there's not eight. Think about how many different orders a deck of card can be in.

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Each one of those is the result of applying a symmetry to the original deck. So a shuffle is

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a symmetry, right? You're reordering the cards. If I shuffle and then you shuffle, the result is some

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other kind of thing you might call a double shuffle, which is a more complicated symmetry.

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So group theory is kind of the study of the general abstract world that encompasses all

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these kinds of things. But then, of course, like lots of things that are way more complicated than

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that. Like infinite groups of symmetries, for instance. So they give you infinite, huh? Oh,

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yeah. Okay. Well, okay, ready? Think about the symmetries of the line. You're like, okay, I can

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reflect it left to right, you know, around the origin. Okay, but I could also reflect it left

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to right grabbing somewhere else, like at one or two or pi or anywhere. Or I could just slide it

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some distance. That's a symmetry. Slide it five units over. So there's clearly infinitely many

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Is it possible to say something that kind of captivates, keeps being brought up by physicists,

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which is gauge theory, gauge symmetry, as one of the more complicated type of symmetries? Is there

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an easy explanation of what the heck it is? Is that something that comes up on your mind at all?

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Well, I'm not a mathematical physicist, but I can say this. It is certainly true

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that it has been a very useful notion in physics to try to say, like, what are the symmetry groups,

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like, of the world? Like, what are the symmetries under which things don't change, right? So we

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just, I think we talked a little bit earlier about, it should be a basic principle that a theorem

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that's true here is also true over there. And same for a physical law, right? I mean,

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All the laws of physics should be unchanged if the symmetry we have in mind is a very simple one,

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like translation. And so then there becomes a question, like, what are the symmetries

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of the actual world with its physical laws? And one way of thinking is an oversimplification,

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but like one way of thinking of this big shift from before Einstein to after is that we just

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changed our idea about what the fundamental group of symmetries were. So that things like

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the Lorenz contraction, things like these bizarre relativistic phenomena or Lorenz would

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have said, oh, to make this work, we need a thing to change its shape if it's moving

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nearly a speed of light. Well, under the new framework, it's much better. You're like, oh,

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no, it wasn't changing its shape. You were just wrong about what counted as a symmetry.

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Now that we have this new group, the so called Lorenz group, now that we understand what the

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symmetries really are, we see it was just an illusion that the thing was changing its shape.

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Yeah, so you can then describe the sameness of things under this weirdness that is

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general relativity, for example. Yeah. Yeah, still, I wish there was a simpler explanation

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of like exact, I mean, you know, gauge symmetries is a pretty simple general concept about rulers

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being deformed. I've actually just personally been on a search, not a very rigorous or aggressive

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search, but for something I personally enjoy, which is taking complicated concepts and finding

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the sort of minimal example that I can play around with, especially programmatically.

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That's great. I mean, this is what we try to train our students to do, right? I mean, in class,

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I do hope there's simple explanation, especially like I've in my sort of drunk random walk, drunk

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walk, whatever it is that's called, sometimes stumble into the world of topology. And like

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quickly, like, you know, when you go into a party and you realize this is not the right party for

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me, whenever I go into topology is like so much math everywhere. I don't even know what it feels

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like this is me like being a hater is I think there's way too much math. Like there are two,

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the cool kids who just want to have like everything is expressed through math, because they're

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actually afraid to express stuff simply through language. That's my hater formulation of topology.

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But at the same time, I'm sure that's very necessary to do sort of rigorous discussion.

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But I feel like... But don't you think that's what gauge symmetry is like? I mean,

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

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But my problem with topology, okay, and even like differential geometry is like,

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you're talking about beautiful things. Like if they could be visualized, it's open question if

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everything could be visualized, but you're talking about things that could be visually stunning,

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I think. But they are hidden underneath all of that math. Like if you look at the papers that

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are written in topology, if you look at all the discussions on stack exchange, they're all math

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dense, math heavy. And the only kind of visual things that emerge every once in a while is like

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simple visualizations. Well, there's the vibration, there's the hop vibration or

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all those kinds of things that somebody, some grad student from like 20 years ago wrote a

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program in Fortran to visualize it. And that's it. And it's just... It makes me sad because

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those are visual disciplines. Just like computer vision is a visual discipline. So you can provide

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a lot of visual examples. I wish topology was more excited and in love with visualizing some of the

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ideas. I mean, you could say that, but I would say for me, a picture of the hop vibration does

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nothing for me. Whereas like when you're, it's like, oh, it's like about the quaternions. It's

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like a subgroup of the quaternions. And I'm like, oh, so now I see what's going on. Like, why didn't

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you just say that? Why were you like showing me this stupid picture instead of telling me what

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you were talking about? Oh, yeah. Yeah. I'm just saying, nobody goes back to what we were saying

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about teaching that like people are different and what they'll respond to. So I think there's no...

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I mean, I'm very opposed to the idea that there's one right way to explain things. I think there's

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like a huge variation in like, you know, our brains like have all these like weird like

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hooks and loops. And it's like very hard to know like what's going to latch on. And it's not going

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to be the same thing for everybody. So I think monoculture is bad, right? I think that's...

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And I think we're agreeing on that point that like it's good that there's like a lot of different

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ways in and a lot of different ways to describe these ideas because different people are going to

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find different things illuminating. But that said, I think there's a lot to be discovered when you

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force little silos of brilliant people to kind of find the middle ground or like

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aggregate or come together in a way. So there's like people that do love visual things.

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I mean, there's a lot of disciplines, especially in computer science that are obsessed with

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visualizing, visualizing data, visualizing neural networks. I mean, neural networks themselves

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are fundamentally visual. There's a lot of work in computer vision that's very visual.

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And then coming together with some folks that were like deeply rigorous and are like totally

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lost in multi dimensional space where it's hard to even bring them back down to 3D.

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They're very comfortable in this multi dimensional space. So forcing them to kind of work together

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to communicate because it's not just about public communication of ideas. It's also,

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I feel like when you're forced to do that public communication like you did with your book,

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I think deep profound ideas can be discovered. That's like applicable for research and for

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science. Like there's something about that simplification or not simplification, but

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distillation or condensation or whatever the hell you call it compression of ideas that

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somehow actually stimulates creativity. And I'd be excited to see more of that in the

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mathematics community. Can you? Let me make a crazy metaphor. Maybe it's a little bit like

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why do we need anything more than prose? You're trying to convey some information. So you just

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like say it. Well, poetry does something, right? It's sort of, you might think of it as a kind

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of compression. Of course, not all poetry is compressed, like not awesome. Some of it is quite

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baggy, but like you are kind of often it's compressed, right? A lyric poem is often sort of

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like a compression of what would take a long time and be complicated to explain in prose into sort

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of a different mode that is going to hit in a different way. We talked about PoincarÃ© conjecture

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here. There's a guy, he's Russian, Grigori Perlman. He proved PoincarÃ© conjecture. If you

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can comment on the proof itself, if that stands out to you, something interesting or the human

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story of it, which is he turned down the Fields Medal for the proof. Is there something you find

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inspiring or insightful about the proof itself or about the man? Yeah. I mean, one thing I really

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like about the proof, and partly that's because it's sort of a thing that happens again and again

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in this book. I mean, I'm writing about geometry and the way it sort of appears in all these kind

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of real world problems. But it happens so often that the geometry you think you're studying

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is somehow not enough. You have to go one level higher in abstraction and study a higher level

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of geometry. And the way that plays out is that PoincarÃ© asks a question about a certain kind

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of three dimensional object. Is it the usual three dimensional space that we know, or is it some kind

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of exotic thing? And so, of course, this sounds like it's a question about the geometry of the

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three dimensional space. But no, Perlman understands. And by the way, in a tradition that involves

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Richard Hamilton and many other people, like most really important mathematical advances,

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this doesn't happen alone. It doesn't happen in a vacuum. It happens as the culmination of a program

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that involves many people. Same with Wiles, by the way. I mean, we talked about Wiles, and I

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want to emphasize that starting all the way back with Coomer, who I mentioned in the 19th century,

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but Gerhard Frey and Mazer and Ken Ribbit and like many other people are involved in building

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the other pieces of the arch before you put the keystone in. We stand on the shoulders of giants.

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Yes. So, what is this idea? The idea is that, well, of course, the geometry of the three

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dimensional object itself is relevant. But the real geometry you have to understand is

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the geometry of the space of all three dimensional geometries. Whoa. You're going up a higher level.

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Because when you do that, you can say, now let's trace out a path in that space. There's a mechanism

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called Ricci flow. And again, we're outside my research area. So for all the geometric analysts

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and differential geometers out there listening to this, if I please, I'm doing my best and I'm

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roughly saying it. So the Ricci flow allows you to say like, okay, let's start from some

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mystery three dimensional space, which Poincare would conjecture is essentially the same thing

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as our familiar three dimensional space, but we don't know that. And now you let it flow,

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you sort of like let it move in its natural path according to some almost physical process

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and ask where it winds up. And what you find is that it always winds up. You've continuously

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deformed it. There's that word deformation again. And what you can prove is that the process doesn't

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stop until you get to the usual three dimensional space. And since you can get from the mystery thing

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having any sharp transitions, then the original shape must have been the same as the standard

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shape. That's the nature of the proof. Now, of course, it's incredibly technical. I think

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as I understand it, I think the hard part is proving that the favorite word of AI people,

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you don't get any singularities along the way. But of course, in this context, singularity just

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means acquiring a sharp kink. It just means becoming non smooth at some point. So

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just saying something interesting about formal about the smooth trajectory through this weird

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space. But yeah, so what I like about it is that it's just one of many examples of where

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it's not about the geometry you think it's about. It's about the geometry of all geometries,

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so to speak. And it's only by kind of like kind of like being jerked out of Flatland,

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right? Same idea. It's only by sort of seeing the whole thing globally at once that you can

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really make progress on understanding like the one thing you thought you were looking at.

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It's a romantic question. But what do you think about him turning down the Fields Medal?

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Well, is that just our Nobel Prizes and Fields Medal is just a cherry on top of the cake and

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really math itself, the process of curiosity of pulling at the string of the mystery before us.

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That's the cake. And then the words are just icing and clearly I've been fasting and I'm hungry.

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But do you think it's tragic or just a little curiosity that he turned on the medal?

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Well, it's interesting because on the one hand, I think it's absolutely true that right in some kind

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of like vast spiritual sense, like awards are not important, not important the way that sort of

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like understanding the universe is important. On the other hand, most people who are offered that

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prize accept it. So there's something unusual about his choice there. I wouldn't say I see it as

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tragic. I mean, maybe if I don't really feel like I have a clear picture of why he chose not to take

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it. I mean, he's not alone in doing things like this. People have sometimes turned down prizes

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for ideological reasons, probably more often in mathematics. I mean, I think I'm right in saying

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that Peter Schultz had like turned down sort of some big monetary prize because he just, you know,

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I mean, I think at some point you have plenty of money. And maybe you think it sends the wrong

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message about what the point of doing mathematics is. I do find that there's most people accept.

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You know, most people have given a prize. Most people take it. I mean, people like to be appreciated.

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But the important reminder that that turning down the prize serves for me is not that there's

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anything wrong with the prize. And there's something wonderful about the prize, I think.

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The Nobel Prize is trickier because so many Nobel Prizes are given. First of all, the Nobel

link |

Prize often forgets many, many of the important people throughout history. Second of all,

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there's like these weird rules to it. There's only three people and some projects have a huge

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number of people. I don't know. It doesn't kind of highlight the way science is done

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on some of these projects in the best possible way. But in general, the prizes are great.

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But what this kind of teaches me and reminds me is sometimes in your life, there'll be moments

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when the thing that you would really like to do, society would really like you to do,

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is the thing that goes against something you believe in, whatever that is, some kind of

link |

principle, and stand your ground in the face of that. It's something, I believe most people

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will have a few moments like that in their life, maybe one moment like that. And you have to do

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it. That's what integrity is. So it doesn't have to make sense to the rest of the world,

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But do you know that he turned down the prize in service of some principle? Because I don't know

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that. Well, yes, that seems to be the inkling, but he has never made it super clear. But the

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inkling is that he had some problems with the whole process of mathematics that includes awards,

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like this hierarchies and the reputations and all those kinds of things and individualism that's

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fundamental to American culture. He probably, because he visited the United States quite a bit,

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that he probably... It's all about experiences. And he may have had some parts of academia,

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some pockets of academia can be less than inspiring, perhaps sometimes, because of the

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individual egos involved, not academia, people in general, smart people with egos. And if you

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interact with certain kinds of people, you can become cynical too easily. I'm one of those people

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that I've been really fortunate to interact with incredible people at MIT and academia in general,

link |

but I've met some assholes. And I tend to just kind of... When I run into difficult folks,

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I just kind of smile and send them all my love and just kind of go around. But for others,

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those experiences can be sticky. Like they can become cynical about the world when folks like

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that exist. So he may have become a little bit cynical about the process of science.

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Well, you know, it's a good opportunity. Let's posit that that's his reasoning because I truly

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don't know. It's an interesting opportunity to go back to almost the very first thing we talked

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about, the idea of the mathematical Olympiad. Because of course, that is... So the international

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mathematical Olympiad is like a competition for high school students solving math problems. And

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in some sense, it's absolutely false to the reality of mathematics because just as you say,

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it is a contest where you win prizes. The aim is to sort of be faster than other people.

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And you're working on sort of canned problems that someone already knows the answer to,

link |

like not problems that are unknown. So, you know, in my own life, I think when I was in

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high school, I was like very motivated by those competitions. And like I went to the math Olympiad

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and... You won it. I won. I mean... Well, there's something I have to explain to people because

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it says... I think it says on the Wikipedia that I won a gold medal. And in the real Olympics,

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they only give one gold medal in each event. I just have to emphasize that the international

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math Olympiad is not like that. The gold medals are awarded to the top one 12th of all participants.

link |

performer in terms of achieving high scores on the problems and they're very difficult.

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So you've achieved a high level of performance on the... In this very specialized skill. And by

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the way, it was a very cold war activity. You know, in 1987, the first year I went,

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it was in Havana. Americans couldn't go to Havana back then. It was a very complicated

link |

process to get there. And they took the whole American team on a field trip to the Museum

link |

How would you recommend a person learn math? So somebody who's young or somebody my age or

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somebody older who've taken a bunch of math but wants to rediscover the beauty of math

link |

I mean, the thing is, it's in part a journey of self knowledge. You have to know

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what's going to work for you and that's going to be different for different people. So there are

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totally people who at any stage of life just start reading math textbooks. That is a thing

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that you can do and it works for some people and not for others. For others, a gateway is,

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you know, I always recommend like the books of Martin Gardner, another sort of person we

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haven't talked about, but who also like Conway embodies that spirit of play. He wrote a column

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in Scientific American for decades called Mathematical Recreations and there's such joy in it and such

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fun. And these books, the columns are collected into books and the books are old now, but for

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each generation of people who discover them, they're completely fresh and they give a totally

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different way into the subject than reading a formal textbook, which for some people would be the right

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thing to do. And, you know, working contest style problems too, those are bound into books,

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like especially like Russian and Bulgarian problems, right? There's book after book

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problems from those contexts. That's going to motivate some people. For some people,

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it's going to be like watching well produced videos, like a totally different format. Like,

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I feel like I'm not answering your question. I'm sort of saying there's no one answer and like,

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Okay, I'll tell you a story. Once when I was in grad school, I was very frustrated with my like

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lack of knowledge of a lot of things as we all are, because no matter how much we know, we don't

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know much more and going to grad school means just coming face to face with like the incredible

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overflowing fault of your ignorance, right? So I told Joe Harris, who was an algebraic

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geometer, a professor in my department, I was like, I really feel like I don't know enough. And I

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should just like take a year of leave and just like read EGA, the holy textbook, elements of

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algebraic geometry. This like, I'm just gonna, I feel like I don't know enough. So I just gonna

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sit and like read this like 1500 page, many volume book. And he was like, and the professor

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Harris, like, that's a really stupid idea. And I was like, why is that a stupid idea? Then I would

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know more algebraic geometry, because you're not actually going to do it like you learn.

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I mean, he knew me well enough to say like, you're going to learn because you're going to be working

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on a problem. And then there's going to be a fact from EGA you need in order to solve your problem

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that you want to solve. And that's how you're going to learn it. You're not going to learn it

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without a problem to bring you into it. And so for a lot of people, I think if you're like,

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I'm trying to understand machine learning, and I'm like, I can see that there's sort of some

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mathematical technology that I don't have. I think you like, let that problem that you actually care

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about drive your learning. I mean, one thing I've learned from advising students, you know,

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And because it's hard, like, you might sort of have some idea that somebody else gives you, oh,

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I should learn X, Y, and Z. Well, if you don't actually care, you're not going to do it. You

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might feel like you should. Maybe somebody told you you should. But I think you have to hook it

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to something that you actually care about. So for a lot of people, that's the way in. You have an

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engineering problem you're trying to handle. You have a physics problem you're trying to handle.

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You have a machine learning problem you're trying to handle. Let that not a kind of abstract idea

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of what the curriculum is, drive your mathematical learning. And also just as a brief comment,

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that math is hard. There's a sense to which hard is a feature, not a bug. In the sense that, again,

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this maybe this is my own learning preference. But I think it's a value to fall in love with

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the process of doing something hard, overcoming it, and becoming a better person. Because like,

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I hate running. I hate exercise to bring it down to like the simplest hard. And I enjoy the part

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once it's done. The person I feel like for the rest in the rest of the day, once I've accomplished

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it, the actual process, especially the process of getting started in the initial, like it really,

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I don't feel like doing it. And I really have the way I feel about running is the way I feel about

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really anything difficult in the intellectual space, especially mathematics, but also

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just something that requires like holding a bunch of concepts in your mind with some uncertainty,

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like where the terminology or the notation is not very clear. And so you have to kind of hold all

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those things together and like keep pushing forward through the frustration of really,

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like, obviously not understanding certain like parts of the picture, like your giant missing

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parts of the picture, and still not giving up. It's the same way I feel about running. And there's

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something about falling in love with the feeling of after you went through the journey of not having

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a complete picture at the end, having a complete picture, and then you get to appreciate the beauty

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and just remembering that it sucked for a long time. And how great it felt when you figured

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it out, at least at the basic, that's not sort of research thinking. Because with research,

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you probably also have to enjoy the dead ends with learning math from a textbook or from a video.

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There's a nice... I think you have to enjoy the dead ends, but I think you have to accept the

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dead ends. That means that let's put it that way. Well, yeah, enjoy the suffering of it. So

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the way I think about it, I do... I don't enjoy the suffering. It pisses me off,

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by the way. You've got to accept that it's part of the process. It's interesting. There's a lot of

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ways to kind of deal with that dead end. There's a guy who's ultramarathon runner, Navy Seal, David

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Goggins, who kind of... I mean, there's a certain philosophy of most people would quit here.

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And so if most people would quit here, and I don't, I'll have an opportunity to discover

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something beautiful that others haven't yet. So any feeling that really sucks, it's like, okay,

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Okay, you say that, but what about the guy who wins the Nathan's hotdog eating contest every year?

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Like when he eats his 35th hotdog, he correctly says, okay, most people would stop here.

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I am. In the long arc of history, that man is onto something, which brings up this question.

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What advice would you give to young people today, thinking about their career, about their life,

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You know, one thing I often say to students, I don't think I've actually said this to

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my kids yet, but I say to students a lot is, you know, you come to these decision points,

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and everybody is beset by self doubt, right? It's like, not sure like what they're capable of,

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like not sure what they're, what they really want to do. I always, I sort of tell people,

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like often when you have a decision to make, one of the choices is the high self esteem choice.

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And I always make the high self esteem choice, make the choice, sort of take yourself out of it,

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and like if you didn't have those, you can probably figure out what the version of you

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feels completely confident would do and do that and see what happens. And I think that's often

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like pretty good advice. That's interesting. Sort of like, you know, like with Sims,

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you can create characters, like create a character of yourself that lacks all the self doubt.

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Right, but it doesn't mean, I would never say to somebody, you should just go have

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high self esteem. Yeah, you shouldn't have doubts. No, you probably should have doubts.

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It's okay to have them. But sometimes it's good to act in the way that the person who didn't have

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them would act. That's a really nice way to put it. Yeah, that's a, that's a like from a third person

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perspective. Take the part of your brain that wants to do big things. What would they do?

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That's not afraid to do those things. What would they do? Yeah, that's, that's really nice. That's

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actually a really nice way to formulate it. It's very practical advice. You should go to your kids.

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Do you think there's meaning to any of it from a mathematical perspective, this life?

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Can we say, and then the book that God has, that mathematics allows us to arrive at something about

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in that book, there's certainly a chapter on the meaning of life in that book. Do you think we humans

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can get to it? And maybe if you were to write cliff notes, what do you suspect those cliff notes

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would say? I mean, look, the way I feel is that, you know, mathematics, as we've discussed, like it

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underlies the way we think about constructing learning machines and underlies physics. It

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can be you. I mean, it does all this stuff. And also you want the meaning of life. I mean,

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it's like, we already did a lot for you. Like ask a rabbi. No, I mean, I wrote a lot in the

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last book, how not to be wrong. Yeah, I wrote a lot about Pascal, a fascinating guy who is a sort

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of very serious religious mystic as well as being an amazing mathematician. And he's well

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known for Pascal's wager. I mean, he's probably among all mathematicians, he's the ones who's

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best known for this. Can you actually like apply mathematics to kind of these transcendent

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questions? But what's interesting when I really read Pascal about what he wrote about this,

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you know, I started to see that people often think, oh, this is him saying, I'm going to use

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mathematics to sort of show you why you should believe in God, you know, to really that's this

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is mathematics has the answer to this question. But he really doesn't say that he almost kind of

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says the opposite. If you ask blaze Pascal, like why do you believe in God? It's he'd be like,

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oh, because I met God, you know, he had this kind of like psychedelic experience, this is like

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a mystical experience where as he tells it, he just like directly encountered God. It's like,

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okay, I guess there's a God, I met him last night. So that's that's it. That's why he believed it

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didn't have to do with any kind of you know, the mathematical argument was like, about certain

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reasons for behaving in a certain way. But he basically said like, look, like math doesn't tell

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you that God's there or not like, if God's there, he'll tell you, you know, you know, I love this.

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So you have you have mathematics, you have, what do you what do you have, like a waste of

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sport of the mind, let's say psychedelics, you have like incredible technology, you also have

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love and friendship and like, what, what the hell do you want to know what the meaning of it all is?

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Just enjoy it. I don't think there's a better way to end it. Jordan, this was a fascinating

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conversation. I really love the way you explore math in your writing, the, the willingness to be

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specific and clear and actually explore difficult ideas, but at the same time, stepping outside

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and figuring out beautiful stuff. And I love the chart at the opening of your new book that shows

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the chaos, the mess that is your mind. Yes, this is what I was trying to keep in my head all at once

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while I was writing. And I probably should have drawn this picture earlier on the process. Maybe

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it would have made my organization easier. I actually drew it only at the end. And many of the

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things we talked about are on this map, the connections are yet to be fully dissected and

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dissected and investigated. And yes, God is in the picture. Right on the edge, right on the edge,

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not in the center. Thank you so much for talking. It is a huge honor that you would waste your valuable

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

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Thanks for listening to this conversation with Jordan Ellenberg. And thank you to

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Secret Sauce, ExpressVPN, Blinkist and Indeed. Check them out in the description to support

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this podcast. And now let me leave you with some words from Jordan in his book, How Not to Be Wrong.

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Knowing mathematics is like wearing a pair of xray specs that reveal hidden structures

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underneath the messy and chaotic surface of the world. Thank you for listening and hope to see you