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

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

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

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

Quick mention of our sponsors, Secret Sauce, ExpressVPN, Blinkist, and Indeed.

Check them out in the description to support this podcast.

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

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

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

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

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

If the brain is a cake. It is.

Let's just go with me on this. Okay. We'll pause it.

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

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

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

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

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

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

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

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

is like the process of uttering language or producing linguistic output.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

just did something to my brain that had planted this hopeful seed that you can

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

I'm from the 70s. I think you're from a different decade than that. In the 70s,

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

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

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

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

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

six rows of eight holes each, but there's also eight columns of six holes each.

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

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

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

Am I allowed to cuss on your podcast? I don't know if that's...

Oh, fuck yes.

Are we FCC regulated?

Okay. It was fucking unbelievable. Okay. That's the last time.

Get it in there.

This story merits it.

So two different perspectives and the same physical reality.

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

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

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

but I didn't know why that was the case until that moment. And in that moment,

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

building or a painting or something like that. You know, nowadays in math,

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

like playing with objects, we're able to see things that are the same and not

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

that like the change because I mean partially because our perception systems,

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

What's the, who's Poincaré and what's this conjecture that he came up with?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

But there's an extra sense that that excellence will lead to a better world.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

So the French, the Germans and Poincaré. Yes. Okay. So back to Poincaré. So

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

So stable meaning they would sort of like end up in some kind of periodic

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

what we see, right? That's those are the objects we interact with.

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

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

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

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

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

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

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

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

He understood that if you want to study three bodies moving in space,

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

I'd love to dig in a little deeper on that, actually, because I have some

intuitions to work out. Okay, my brain. Well, I'm not a mathematical physicist,

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

But Poincaré conjecture, what is it? The Poincaré conjecture is about curved

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

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

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

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

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

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

right? There's a balloon and that looks one way, and a donut looks another way,

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

It doesn't have to do with intrinsic features of those people's world.

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

in a circle or on a knot in a knot inside the string that forms the knot?

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

but it does not bother me at all. I don't lose one minute of sleep over it.

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

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

live on the surface of a mobius strip? No, I think you can tell. If you live,

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

that you're going to see everywhere around you the back of your own head.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Now, how would you figure out whether it was true or not is another question.

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

you would try to understand what effect the presence of those extra dimensions

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

first of all, to the mug, but to the example you give in the book of the straw,

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

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

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

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

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

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

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

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

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

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

for the way they have. And then people get really heated. People are like,

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

And then people really learn something about each other and people get heated.

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

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

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

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

Yeah.

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

That's like... Is there any truth to that kind of conception?

Yeah, I think that would be somebody who's account... I mean,

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

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

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

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

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

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

like where we've planted our foot. This is what we stand for.

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

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

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

at the bottom. I think a common thing you see when people

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

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

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

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

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

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

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

or we'll say like, okay, where does one hole begin and the other hole end?

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

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

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

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

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

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

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

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

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

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

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

So many people would say there's three holes in a pair of pants. But you know,

for instance, my daughter, when I asked, by the way, talking to kids about this

is super fun. I highly recommend it. What did she say? She said, well,

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

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

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

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

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

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

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

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

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

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

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

So for the straw, I think what a modern mathematician would say is like,

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

To some degree, be able to give inklings of the beauty of the subject.

Yeah, I mean, there's, you know, there's a lot of books that are like,

I don't quite know how to express this well, I'm still laboring to do it, but

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

their skills to do that. I hope there is at any rate.

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

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

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

manum library and in Python, everything is drawn through Python. You have to

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

And you can be like, well, let me sort of train an AI to sort of try to find

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

but it's a question of whether the knot is sliced. It has to do with

something about what kinds of three dimensional surfaces in four dimensions

can be bounded by this knot, but never mind what it means. It's some question.

And it's actually very hard to compute whether a knot is sliced or not.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

I think just like symmetry and the breaking of symmetry is beautiful somehow.

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

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

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

like obviously, we get a huge amount of trouble with that because oftentimes,

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

those are nice things that like, you know, you teach in an undergraduate class,

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

is? I can speak to what it's like. Sure. How about that? What is it rhyme with?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

distance is not given to us by God. We could mean a lot of different things by distance.

And just in the English language, we do that all the time. We talk about somebody being a close

relative. It doesn't mean they live next door to you, right? It means something else. There's a

different notion of distance we have in mind. And there are lots of notions of distances

that you could use in the natural language processing community and AI. There might be some

notion of semantic distance or lexical distance between two words. How much do they tend to

arise in the same context? That's incredibly important for doing autocomplete and machine

translation and stuff like that. And it doesn't have anything to do with, are they next to each

other in the dictionary? It's a different kind of distance. Okay, ready? In this kind of number

theory, there was a crazy distance called the piatic distance. I didn't write about this that

much in the book, because even though I love it, it's a big part of my research life. It gets a

little bit into the weeds, but your listeners are going to hear about it now. Please, you know,

what a normal person says when they say two numbers are close, they say like, you know,

their difference is like a small number, like seven and eight are close because their difference is

one and one's pretty small. If we were to be what's called a two attic number theorist,

we'd say, oh, two numbers are close. If their difference is a multiple of a large power of two.

So like, so like one and 49 are close, because their difference is 48 and 48 is a multiple

of 16, which is a pretty large power of two, whereas whereas one and two are pretty far away,

because the difference between them is one, which is not even a multiple of a power of two at all.

It's odd. You want to know what's really far from one, like one and one 64th,

because their difference is a negative power of two, two to the minus six. So those

points are quite, quite far. Two to the power of a large n would be two.

If that's the difference between two numbers and they're close.

Yeah. So two to a large power is this metric, a very small number and two to a negative power

is a very big number. That's too much. Okay. I can't even visualize that.

It takes practice. It takes practice. If you've ever heard of the Cantor set, it looks kind of like

that. So it is crazy that this is good for anything, right? I mean, this just sounds

like a definition that someone would make up to torment you. But what's amazing is

there's a general theory of distance where you say any definition you make that satisfies

certain axioms deserves to be called a distance and this. See, I'm sorry to interrupt. My brain,

you broke my brain now. Awesome. 10 seconds ago. Because I'm also starting to map for the two out

of case to binary numbers, you know, because we romanticized those. Oh, that's exactly the

right way to think of it. I was trying to mess with numbers. I was trying to see, okay, which ones

are close. And then I'm starting to visualize different binary numbers and how they, which ones

are close to each other. And that's, well, I think there's no, no, it's very similar. That's

exactly the right way to think of it. It's almost like binary numbers written in reverse, right?

Because in a, in a binary expansion, two numbers are close. A number that's small is like 0.000

something. Something that's the decimal and it starts with a lot of zeros. In the two attic

metric, a binary number is very small if it ends with a lot of zeros and then the decimal point.

Gotcha. So it is kind of like binary numbers written backwards is actually, I should have,

that's what I should have said, Lex. That's like very good metaphor.

Oh, okay. But so why is that, why is that interesting? Except for the fact that it's,

it's a beautiful kind of framework, different kind of framework, which you think about distances.

And you're talking about not just the two attic, but the generalization of that.

Yeah, the NEP. And so, so that, because that's the kind of deformation that comes up

in Wiles's, in Wiles's proof that deformation where moving something a little bit means a little

bit in this two attic sense. Okay. No, I mean, it's such, I mean, I just get excited to talk

about it. And I just taught this like in the fall semester that, but it like reformulating, why is,

so you pick a different measure of distance over which you can talk about very tiny changes

and then use that to then prove things about the entire thing. Yes. Although, you know,

honestly, what I would say, I mean, it's true that we use it to prove things, but I would say

we use it to understand things. And then because we understand things better, then we can prove

things. But you know, the goal is always the understanding, the goal is not to, so much

to prove things, the goal is not to know what's true or false. I mean, this is the thing I write

about in the book near the end, and it's something that, so wonderful, wonderful essay by, by Bill

Thurston, kind of one of the great geometers of our time, who unfortunately passed away a few years

ago, um, called on proof and progress in mathematics. And he writes very wonderfully about how,

you know, we're not, it's not a theorem factory where we have a production quota. I mean, the

point of mathematics is to help humans understand things. And the way we test that is that we're

proving new theorems along the way. That's the benchmark, but that's not the goal. Yeah. But

just as a, as a kind of absolutely, but as a tool, it's kind of interesting to approach a problem

by saying, how can I change the distance function? Like what the nature of distance,

because that might start to lead to insights for deeper understanding. Like if I were to try to

describe human society by a distance, two people are close if they love each other, right? And then,

and then start to, uh, and do a full analysis on the everybody that lives on earth currently,

the seven billion people. And from that perspective, as opposed to the geographic perspective of

distance, and then maybe there could be a bunch of insights about the source of, uh, violence,

the source of, uh, maybe entrepreneurial success or invention or economic success or different

systems, communism, capitalism start to, I mean, that's, I guess what economics tries to do. But

really saying, okay, let's think outside the box about totally new distance functions that could

unlock something profound about the space. Yeah. Because think about it. Okay. Here's,

I mean, now we're going to talk about AI, which you know a lot more about than I do. So

just, you know, start laughing uproariously if I say something that's completely wrong.

We both know very little relative to what we will know centuries from now.

That is, that is a really good humble way to think about it. I like it. Okay. So let's just go for it.

Um, okay. So I think you'll agree with this, that in some sense what's good about AI is that

we can't test any case in advance. The whole point of AI is to make, or one point of it,

I guess, is to make good predictions about cases we haven't yet seen. And in some sense,

that's always going to involve some notion of distance because it's always going to involve

somehow taking a case we haven't seen and saying what cases that we have seen, is it close to?

Is it like, is it somehow an interpolation between? Now, when we do that in order to talk about

things being like other things, implicitly or explicitly, we're invoking some notion of distance

and boy, we better get it right. Right? If you try to do natural language processing and your

idea about distance between words is how close they are in the dictionary. When you write them

in alphabetical order, you are going to get pretty bad translations, right? No, the notion of distance

has to come from somewhere else. Yeah, that's essentially when neural networks are doing this.

Well, word embeddings are doing is coming up with. In the case of word embeddings, literally,

like literally what they are doing is learning a distance. But those are super complicated

distance functions. And it's almost nice to think maybe there's a nice transformation that's simple.

Sorry, there's a nice formulation of the distance. Again, with the simple.

Let me ask you about this. From an understanding perspective, there's the Richard Feynman maybe

attributed to him, but maybe many others is this idea that if you can't explain something simply

that you don't understand it. In how many cases, how often is that true? Do you find there's some

profound truth in that? Oh, okay. So you were about to ask, is it true, which I would say

flatly no, but then you said you followed that up with, is there some profound truth in it?

And I'm like, okay, sure. So there's some truth in it. It's not true.

It says your mathematician answer. The truth that is in it is that learning to explain something

helps you understand it. But real things are not simple. A few things are, most are not.

And I don't, to be honest, I don't, I mean, I don't, we don't really know whether Feynman

really said that, right? Or something like that is sort of disputed. But I don't think Feynman

could have literally believed that whether or not he said it. And, you know, he was the kind

of guy, I didn't know him, but I'm reading his writing, he liked to sort of say stuff like stuff

that sounded good. You know what I mean? So it totally strikes me as the kind of thing he could

have said because he liked the way saying it made him feel. But also knowing that he didn't

like literally mean it. Well, I definitely have a lot of friends and I've talked to a lot of

physicists and they do derive joy from believing that they can explain stuff simply or believing

as possible to explain stuff simply, even when the explanation is not actually that simple.

Like I've heard people think that the explanation is simple and they do the explanation. And I

think it is simple, but it's not capturing the phenomena that we're discussing. It's capturing,

it somehow maps in their mind, but it's, it's taking as a starting point as an assumption

that there's a deep knowledge and a deep understanding that's, that's actually very

complicated. And the simplicity is almost like a, almost like a poem about the more complicated

thing as opposed to a distillation. And I love poems, but a poem is not an explanation.

Well, some people might disagree with that, but certainly from a mathematical perspective,

no poet would disagree with it. No poet would disagree. You don't think there's some things

that can only be described in precisely? I said explanation. I don't think any poem would,

I don't think any poet would say their poem is an explanation. They might say it's a description.

They might say it's sort of capturing sort of. Well, some people might say the only truth is like

music, right? That the, the, the, not the only truth, but some truth can only be expressed through

art. And I mean, that's the whole thing we're talking about religion and myth. And there's

some things that are limited cognitive capabilities and the tools of mathematics or the tools of physics

are just not going to allow us to capture. Like it's possible consciousness is one of those things.

Yes, that is definitely possible. But I would even say, look, in consciousness is a thing about

which we're still in the dark as to whether there's an explanation we would, we would understand

as an explanation at all. By the way, okay, I got to give yet one more amazing Poincare quote,

because this guy just never stopped coming up with great quotes that, you know, Paul Erdisch,

another fellow who appears in the book. And by the way, he thinks about this notion of distance of

like personal affinity, kind of like what you're talking about, the kind of social network and

that notion of distance that comes from that. So that's something that Paul Erdisch, Erdisch did.

Well, he thought about distances in networks. I guess he didn't probably, he didn't think about

the social network. That's fascinating. That's how he started that story, where there's no,

but yeah, okay. But you know, Erdisch was sort of famous for saying, and this is sort of one

line you were saying, he talked about the book, capital T, capital B, the book. And that's the

book where God keeps the right proof of every theorem. So when he saw a proof he really liked,

it was like really elegant, really simple, like that's from the book. That's like you found one

of the ones that's in the book. He wasn't a religious guy, by the way, he referred to God

as the supreme fascist. He was like, but somehow he was like, I don't really believe in God,

but I believe in God's book. But Poincaré, on the other hand, and by the way, there are other

men, Hilda Hudson is one who comes up in this book. She also kind of saw math. She's one of the

people who sort of develops the disease model that we now use, that we use to sort of track

pandemics, this SIR model that sort of originally comes from her work with Ronald Ross. But she

was also super, super, super devout. And she also sort of on the other side of the religious coin

was like, yeah, math is how we communicate with God. She has a great, all these people are incredibly

quotable. She says, you know, math is, the truth, the things about mathematics, she's like,

they're not the most important of God's thoughts, but they're the only ones that we can know precisely.

So she's like, this is the one place where we get to sort of see what God's thinking when we do

mathematics. Again, not a fan of poetry or music. Some people say Hendricks is like,

some people say chapter one of that book is mathematics. And then chapter two is like classic

rock. Right. So like, it's not clear that the, I'm sorry, you just sent me off on a tangent,

just imagining like Erdish at a Hendricks concert, trying to figure out if it was from

the book or not. What I was coming to was just to say, but when Poincaré said about this is he's

like, you know, if like, this is all worked out in the language of the divine and if a divine

being like came down and told it to us, we wouldn't be able to understand it. So it doesn't matter.

So Poincaré was of the view that there were things that were sort of like

inhumanly complex. And that was how they really were. Our job is to figure out the things that

are not like that. They're not like that. All this talk of primes got me hungry for primes.

You, uh, you wrote a blog post, the beauty of bounding gaps, a huge discovery about prime

numbers and what it means for the future of math. Can you tell me about prime numbers? What the

heck are those? What are twin primes? What are prime gaps? What are bounding gaps in primes?

What are all these things? And what, if anything, or what exactly is beautiful about them?

Yeah. So, you know, prime numbers are one of the things that number theorists study the most and

have for millennia. They are numbers which can't be factored. And then you say like, like five,

and then you're like, wait, I can factor five. Five is five times one. Okay, not like that.

That is a factorization. It absolutely is a way of expressing five as a product of two things.

But don't you agree that there's like something trivial about it? It's something you can do to

any number. It doesn't have content the way that if I say that 12 is six times two or 35 is seven

times five, I've really done something to it. I've broken up. So those are the kind of factorizations

that count. And a number that doesn't have a factorization like that is called prime, except

historical side note one, which at some times in mathematical history has been deemed to be a prime,

but currently is not. And I think that's for the best. But I bring it up only because sometimes

people think that, you know, these definitions are kind of, if we think about them hard enough,

we can figure out which definition is true. No, there's just an artifact of mathematics. So

so it's the definition is best for us for our purposes. Well, those edge cases are weird, right?

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

the factorization or as the entirety of the factorization. So the so you somehow get to the

meat of the number by factorizing it. And that seems to get to the core of all of mathematics.

Yeah, you take any number and you factorize it until you can factorize no more. And what you

have left is some big pile of primes. I mean, by definition, when you can't factor anymore,

when you when you're done, when you can't break the numbers up anymore, what's left must be prime,

you know, 12 breaks into two and two and three. So these numbers are the atoms, the building blocks

of all numbers. And there's a lot we know about them. But there's much more that we don't know

them. I'll tell you the first few, there's two, three, five, seven, 11. By the way, they're all

going to be odd from then on, because if they were even, I could factor a two out of them.

But it's not all the odd numbers. Nine isn't prime because it's three times three.

15 isn't prime because it's three times five, but 13 is where we're at two, three, five,

seven, 11, 13, 17, 19, not 21, but 23 is et cetera, et cetera. Okay, so you could go on.

How high could you go if we were just sitting here? By the way, your own brain,

if continuous without interruption, would you be able to go over 100?

I think so. There's always those ones that trip people up. There's a famous one,

the Groten Dieck prime, 57, like sort of Alexander Groten Dieck, the great algebraic

geometry was sort of giving some lecture involving a choice of a prime in general.

And somebody said, like, can't you just choose a prime? And he said, okay, 57,

which is in fact not prime. It's three times 19. Oh, damn. But it was like,

I promise you in some circles, that's a funny story. Okay.

Okay. There's a humor in it. Yes, I would say over 100, I definitely don't remember,

like 107, I think. I'm not sure. Okay. So is there a category of

fake primes that are easily mistaken to be prime? Like 57, I wonder.

Yeah. So I would say 57 and 57 and 51 are definitely like prime offenders. Oh,

I didn't do that on purpose. Oh, well done. I didn't do it on purpose. Anyway,

they're definitely ones that people, or 91 is another classic seven times 13. It really

feels kind of prime, doesn't it? But it is not. Yeah. But there's also, by the way,

but there's also an actual notion of pseudo prime, which is a thing with a formal definition,

which is not a psychological thing. It is a prime which passes a primality test devised by Fermat,

which is a very good test, which if a number fails this test, it's definitely not prime.

And so there was some hope that, oh, maybe if a number passes the test, then it definitely

is prime. That would give a very simple criterion for primality. Unfortunately,

it's only perfect in one direction. So there are numbers, I want to say 341 is the smallest,

which passed the test, but are not prime 341. Is this test easily explainable or no?

Yes, actually. Ready? Let me give you the simplest version of it. You can dress it up a little bit,

but here's the basic idea. I take the number, the mystery number, I raise two to that power.

So let's say your mystery number is six. Are you sorry you asked me? Are you ready?

No, I'm breaking my brain again, but yes. Let's do it. We're going to do a live

demonstration. Let's say your number is six. So I'm going to raise two to the sixth power.

Okay, so if I were working on it, I'd be like, that's two cubes squared. So that's eight times

eight. So that's 64. Now we're going to divide by six, but I don't actually care what the

quotient is, only the remainder. So let's say 64 divided by six is, well, there's a quotient of 10,

but the remainder is four. So you failed because the answer has to be two. For any prime,

let's do it with five, which is prime. Two to the fifth is 32. Divide 32 by five.

And you get six with a remainder of two.

With a remainder of two. For seven, two to the seventh is 128. Divide that by seven.

And let's see. I think that's seven times 14. Is that right? No. Seven times

18 is 126 with a remainder of two, right? 128 is a multiple of seven plus two. So if that

remainder is not two, then that's definitely not prime. Then it's definitely not prime.

And then if it is, it's likely a prime, but not for sure. It's likely a prime, but not for

sure. And there's actually a beautiful geometric proof, which is in the book, actually. That's

like one of the most granular parts of the book, because it's such a beautiful proof I couldn't

not give it. So you draw a lot of like opal and pearl necklaces and spin them. That's kind of the

geometric nature of this proof of Fermat's little theorem. So yeah, so with pseudo primes,

there are primes that are kind of faking it. They pass that test, but there are numbers that

are faking it. They pass that test, but are not actually prime. But the point is,

there are many, many, many theorems about prime numbers. Are there like, there's a bunch of

questions to ask, is there an infinite number of primes? Can we say something about the gap

between primes as the numbers grow larger and larger and larger and so on? Yeah, it's a perfect

example of your desire for simplicity in all things. You know, it would be really simple

if there was only finally many primes. Yes. And then there would be this

finite set of atoms that all numbers would be built up. That's right. That would be very

simple in good and certain ways, but it's completely false. And number theory would be

totally different if that were the case. It's just not true. In fact, this is something else

that Euclid knew. So this is a very, very old fact, like much before, long before we'd had

anything like modern number. The primes are infinite. The primes that there are, right,

the infinite primes. There's an infinite number of primes. So what about the gas between the

primes? Right. So one thing that people recognized and really thought about a lot is that the primes

on average seem to get farther and farther apart as they get bigger and bigger. In other words,

it's less and less common. Like I already told you of the first 10 numbers, two, three, five,

seven, four of them are prime. That's a lot, 40%. If I looked at, you know, 10 digit numbers,

no way would 40% of those be prime. Being prime would be a lot rare in some sense because there's

a lot more things for them to be divisible by. That's one way of thinking of it. It's a lot more

possible for there to be a factorization because there's a lot of things you can try to factor

out of it. As the numbers get bigger and bigger, primality gets rarer and rarer.

And the extent to which that's the case, that's pretty well understood. But then you can ask more

fine grained questions. And here is one. A twin prime is a pair of primes that are two apart,

like three and five, or like 11 and 13, or like 17 and 19. And one thing we still don't know is,

are there infinitely many of those? We know on average, they get farther and farther apart,

but that doesn't mean there couldn't be like occasional folks that come close together.

And indeed, we think that there are. And one interesting question, I mean, this is,

because I think you might say like, well, why, how could one possibly have a right to have an

opinion about something like that? Like what, you know, we don't have any way of describing a

process that makes primes like, sure, you can like look at your computer and see a lot of them.

But the fact that there's a lot, why is that evidence that there's infinitely many, right?

Maybe I can go on the computer and find 10 million. Well, 10 million is pretty far from

infinity, right? So how is that, how is that evidence? There's a lot of things. There's like

a lot more than 10 million atoms. That doesn't mean there's infinitely many atoms in the universe,

right? I mean, on most people's physical theories, there's probably not, as I understand it.

Okay. So why would we think this? The answer is that we've, that it turns out to be like

incredibly productive and enlightening to think about primes as if they were random numbers,

as if they were randomly distributed according to a certain law. Now they're not,

they're not random. There's no chance involved. It's completely deterministic,

whether a number is prime or not. And yet it just turns out to be phenomenally useful in

mathematics to say, even if something is governed by a deterministic law, let's just pretend it

wasn't, let's just pretend that they were produced by some random process and see if the behavior is

roughly the same. And if it's not, maybe change the random process, maybe make the randomness a

little bit different and tweak it and see if you can find a random process that matches the behavior

we see. And then maybe you predict that other behaviors of the system are like that of the

random process. And so that's kind of like, it's funny because I think when you talk to people

about the twin prime conjecture, people think you're saying, wow, there's like some deep structure

there that like makes those primes be like close together again and again. And no, it's the opposite

of deep structure. What we say when we say we believe the twin prime conjecture is that we

believe the primes are like sort of strewn around pretty randomly. And if they were, then by chance,

you would expect there to be infinitely many twin primes. And we're saying, yeah, we expect them

to behave just like they would if they were random dirt. You know, the fascinating parallel here is

I just got a chance to talk to Sam Harris. And he uses the prime numbers as an example.

Often, I don't know if you're familiar with who Sam is. He uses that as an example of there being

no free will. Wait, where does he get this? Well, he just uses as an example of it might seem like

this is a random number generator, but it's all like formally defined. So if we keep getting more

and more primes, then like that might feel like a new discovery and that might feel like a new

experience, but it's not. It was always written in the cards. But it's funny that you say that

because a lot of people think of like randomness, the fundamental randomness within the nature of

reality might be the source of something that we experience as free will. And you're saying it's

like useful to look at prime numbers as a random process in order to prove stuff about them.

But fundamentally, of course, it's not a random process. Well, not in order to prove some stuff

about them so much as to figure out what we expect to be true and then try to prove that.

Because here's what you don't want to do. Try really hard to prove something that's false.

That makes it really hard to prove the thing if it's false. So you certainly want to have some

heuristic ways of guessing, making good guesses about what's true. So yeah, here's what I would

say. You're going to be imaginary Sam Harris now. You are talking about prime numbers and you are like,

but prime numbers are completely deterministic. And I'm saying, well, let's treat them like

a random process. And then you say, but you're just saying something that's not true. They're

not a random process. They're deterministic. And I'm like, okay, great, you hold to your

insistence that it's not a random process. Meanwhile, I'm generating insight about the

primes that you're not because I'm willing to sort of pretend that there's something that

they're not in order to understand what's going on. Yeah, so it doesn't matter what the reality is.

What matters is what framework of thought results in the maximum number of insights.

Yeah, because I feel, look, I'm sorry, but I feel like you have more insights about people if you

think of them as like beings that have wants and needs and desires and do stuff on purpose.

Even if that's not true, you still understand better what's going on by treating them in that

way. Don't you find, look, when you work on machine learning, don't you find yourself sort

of talking about what the machine is trying to do in a certain instance? Do you not find

yourself drawn to that language? It knows this. It's trying to do that. It's learning that.

I'm certainly drawn to that language to the point where I receive quite a bit of criticisms for it

because I, you know, like, oh, I'm on your side, man. So especially in robotics, I don't know why,

but robotics people don't like to name their robots. They certainly don't like to gender

their robots because the moment you gender a robot, you start to anthropomorphize.

If you say he or she, you start to, you know, in your mind construct like a,

like a life story in your mind. You can't help it. It's like you create like a humorous story to

this person. You start to, this person, this robot, you start to project your own. But I think

that's what we do to each other. And I think that's actually really useful for the engineering

process, especially for human robot interaction. And yes, for machine learning systems,

for helping you build an intuition about a particular problem. It's almost like asking this

question, you know, when a machine learning system fails in a particular edge case, asking

like, what were you thinking about? Like, like asking like almost like when you're talking about

to a child who just did something bad, you want to understand like what was,

how did they see the world? Maybe there's a totally new, maybe you're the one that's

thinking about the world incorrectly. And yeah, that anthropomorphization process,

I think is ultimately good for insight. And the same as I agree with you, I tend to believe

about free will as well. Let me ask you a ridiculous question if it's okay.

Of course. I've just recently, most people go on like rabbit hole, like YouTube things. And

I went on a rabbit hole often do of Wikipedia. And I found a page on finitism, ultra finitism,

and intuitionism, or I forget what it's called. Yeah, intuitionism. Intuitionism.

That seemed pretty, pretty interesting. I have a my to do list actually like looking to, like,

is there people who like formally attract like real mathematicians are trying to argue for this.

But the belief there, I think, let's say finitism that infinity is, is fake.

Meaning, infinity might be like a useful hack for certain, like a useful tool in mathematics,

but it really gets us into trouble. Because there's no infinity in the real world. Maybe I'm sort of

not expressing that fully correctly, but basically saying like there's things there.

And once you add into mathematics, things that are not provably within the physical world,

you're starting to inject, to corrupt your framework of reason. What do you think about that?

I mean, I think, okay, so first of all, I'm not an expert. And I couldn't even tell you what the

difference is between those three terms finitism, ultra finitism, and intuitionism, although I

know what they're related, I tend to associate them with the Netherlands in the 1930s.

Okay, I'll tell you, can I just quickly comment because I read the Wikipedia page,

the difference in ultra... That's like the ultimate sentence of the modern age.

Can I just comment because I read the Wikipedia page? That sums up our moment.

Bro, I'm basically an expert. Ultra finitism, so finitism says that the only infinity you're

allowed to have is that the natural numbers are infinite. So like those numbers are infinite.

So like one, two, three, four, five, the integers are infinite. The ultra finitism says, nope,

even that infinity is fake. I'll bet ultra finitism came second. I'll bet it's like when

there's like a hardcore scene and then one guy's like, oh, now there's a lot of people in the

scene. I have to find a way to be more hardcore than the hardcore people. All back to the emo talk.

Yeah. Okay, so is there any... Are you ever... Because I'm often uncomfortable with infinity.

Like psychologically. I have trouble when that sneaks in there. It's because it works so damn

well. I get a little suspicious because it could be almost like a crutch or an oversimplification

that's missing something profound about reality. Well, so first of all, okay, if you say like,

is there like a serious way of doing mathematics that doesn't really treat infinity as a real thing

or maybe it's kind of agnostic and is like, I'm not really going to make a firm statement about

whether it's a real thing or not. Yeah, that's called most of the history of mathematics, right?

So it's only after Cantor, right, that we really are sort of, okay, we're going to like

have a notion of like the cardinality of an infinite set and like

do something that you might call like the modern theory of infinity. That said, obviously,

everybody was drawn to this notion and no, not everybody was comfortable with it. Look,

I mean, this is what happens with Newton, right? I mean, so Newton understands that to talk about

tangents and to talk about instantaneous velocity, he has to do something that we would now call

taking a limit, right? The fabled dy over dx. If you sort of go back to your calculus class,

for those who've taken calculus, remember this mysterious thing and you know, what is it? What

is it? Well, he'd say like, well, it's like you sort of divide the length of this line segment

by the length of this other line segment and then you make them a little shorter and you

divide again and then you make them a little shorter and you divide again and then you just

keep on doing that until they're like infinitely short and then you divide them again. These quantities

that are like, they're not zero, but they're also smaller than any actual number, these infinitesimals.

Well, people were queasy about it and they weren't wrong to be queasy about it, right? From a

modern perspective, it was not really well formed. There's this very famous critique of Newton by

Bishop Berkeley where he says like, what these things you define like, you know, they're not zero,

but they're smaller than any number. Are they the ghosts of departed quantities? That was

this like ultra line of Newton. And on the one hand, he was right. It wasn't really rigorous

by modern standards. On the other hand, like Newton was out there doing calculus and other

people were not writing. It works. It works. I think a sort of intuitionist view, for instance,

I would say would express serious doubt. And by the way, it's not just infinity. It's like

saying, I think we would express serious doubt that like, the real numbers exist.

Now, most people are comfortable with the real numbers.

Well, computer scientists with floating point number, I mean,

floating point arithmetic.

It's a great point, actually. I think in some sense, this flavor of doing math saying,

we shouldn't talk about things that we cannot specify in a finite amount of time. There's

something very computational in flavor about that. And it's probably not a coincidence that it

becomes popular in the 30s and 40s, which is also like kind of like the dawn of ideas about

formal computation, right? You probably know the timeline better than I do. Sorry, what becomes

popular? These ideas that maybe we should be doing math in this more restrictive way where

even a thing that, you know, because look, the origin of all this is like,

you know, number represents a magnitude, like the length of a line. Like so, I mean, the idea that

there's a continuum, there's sort of like, there's like,

is pretty old, but that, you know, just because something is old doesn't mean we can't reject

it if we want to. Well, a lot of the fundamental ideas in computer science, when you talk about

the complexity of problems to Turing himself, they rely on infinity as well.

Well, the ideas that kind of challenge that, the whole space of machine learning, I would say,

challenges that. It's almost like the engineering approach to things, like the floating point

arithmetic. The other one that, back to John Conway, that challenges this idea, I mean,

maybe to tie in the ideas of deformation theory and limits to infinity is this idea of cellular

automata. With John Conway looking at the game of life, Steven Wolfram's work, that I've been a

big fan of for a while of cellular automata. I was wondering if you have, if you have ever

encountered these kinds of objects, you ever looked at them as a mathematician, where you have

very simple rules of tiny little objects that when taken as a whole create incredible complexities,

but are very difficult to analyze, very difficult to make sense of, even though the one individual

object, one part, it's like what we were saying about Andrew Wiles, you can look at the deformation

of a small piece to tell you about the whole. It feels like with cellular automata or any kind of

complex systems, it's often very difficult to say something about the whole thing, even when you

can precisely describe the operation of the local neighborhoods. Yeah, I mean, I love that subject.

I haven't really done research in it myself. I've played around with it. I'll send you a fun blog post

I wrote where I made some cool texture patterns from cellular automata that I... And those are

really always compelling. It's like you create simple rules and they create some beautiful

textures. It doesn't make any sense. Actually, did you see there was a great paper? I don't know if

you saw this, like a machine learning paper. Yes. I don't know if you saw the one I'm talking about,

where they were learning the texture as like, let's try to reverse engineer and learn a cellular

automaton that can reduce texture that looks like this from the images. Very cool. And as you say,

the thing you said is I feel the same way when I read machine learning paper is that

what's especially interesting is the cases where it doesn't work. What does it do when it

doesn't do the thing that you tried to train it to do? That's extremely interesting. Yeah,

that was a cool paper. So yeah, so let's start with the game of life. Let's start with...

Or let's start with John Conway. So Conway...

Yeah, so let's start with John Conway again. Just I don't know, from my outsider's perspective,

there's not many mathematicians that stand out throughout the history of the 20th century.

And he's one of them. I feel like he's not sufficiently recognized. I think he's pretty

recognized. Okay. Well, I mean, he was a full professor of Princeton for most of his life.

He was sort of certainly the pinnacle of... Yeah, but I found myself every time I talk about Conway

and how excited I am about him. I have to constantly explain to people who he is.

And that's always a sad sign to me. But that's probably true for a lot of mathematicians.

I was about to say, I feel like you have a very elevated idea of how famous... This is

what happens when you grow up in the Soviet Union or you think the mathematicians are very, very

famous. Yeah, but I'm not actually so convinced at a tiny tangent that that shouldn't be so.

I mean, it's not obvious to me that that's one of the... If I were to analyze American society that

perhaps elevating mathematical and scientific thinking to a little bit higher level would

benefit the society, well, both in discovering the beauty of what it is to be human and for

actually creating cool technology, better iPhones. But anyway, John Conway. Yeah, and Conway is such

a perfect example of somebody whose humanity was and his personality was like a wound up with his

mathematics. So it's not... Sometimes I think people who are outside the field think of mathematics

as this kind of cold thing that you do separate from your existence as a human being. No way,

your personality is in there just as it would be in a novel you wrote or a painting you painted or

just the way you walk down the street. It's in there, it's you doing it. And Conway was certainly

a singular personality. I think anybody would say that he was playful, like everything was

a game to him. Now, what you may think I'm going to say, and it's true is that he sort of was very

playful in his way of doing mathematics, but it's also true. It went both ways. He also sort of made

mathematics out of games. He looked at... He was a constant inventor of games with like crazy names

and then he was sort of analyzed those games mathematically to the point that he and then

later collaborating with Knuth like created this number system, the serial numbers, in which actually

each number is a game. There's a wonderful book about this called... I mean, there are his own

books and then there's like a book that he wrote with Berlecamp and Guy called Winning Ways, which is

such a rich source of ideas. And he too kind of has his own crazy number system in which, by the

way, there are these infinitesimals, the ghosts of departed quantities. They're in there now not as

ghosts, but as like certain kind of two player games. So, you know, he was a guy. So I knew him

when I was a postdoc and I knew him at Princeton and our research overlapped in some ways. Now,

it was on stuff that he had worked on many years before. The stuff I was working on kind of connected

with stuff in group theory, which somehow seems to keep coming up. And so I often would like to

sort of ask him a question. I would sort of come upon him in the common room and I would ask him a

question about something. And just anytime you turned him on, you know what I mean? You sort of

asked a question. It was just like turning a knob and winding him up and he would just go and you

would get a response that was like so rich and went so many places and taught you so much. And

usually had nothing to do with your question. Yeah. Usually your question was just a prompt.

Damn, you couldn't count on actually getting the question. Yeah, there's brilliant curious minds

that even at that age, yeah, it was definitely a huge loss. But on his game of life, which was I

think he developed in the 70s as almost like a side thing, a fun little experiment. Yeah, the game

of life is this. It's a very simple algorithm. It's not really a game per se in the sense of the

kinds of games that he liked, where people played against each other. But essentially, it's a game

that you play with marking little squares on the sheet of graph paper. And in the 70s, I think he

was like literally doing it with like a pen on graph paper, you have some configuration of squares,

some of the squares in the graph paper are filled in, some are not. And then there's a rule, a single

rule that tells you at the next stage, which squares are filled in and which squares are not.

Sometimes an empty square gets filled in, that's called birth, sometimes a square that's filled

in gets erased, that's called death. And there's rules for which squares are born, which squares die.

It's the rule is very simple, you can write it on one line. And then the great miracle is that you

can start from some very innocent looking little small set of boxes and get these results of

incredible richness. And of course, nowadays, you don't do it on paper. Nowadays, you do it on a

computer. There's actually a great iPad app called Golly, which I really like that has like

Conway's original rule and like, gosh, like hundreds of other variants. And it's a lightning

fast. So you can just be like, I want to see 10,000 generations of this rule play out like faster

than your eye can even follow. And it's like amazing. So I highly recommend it if this is

at all intriguing to you getting golly on your iOS device. And you can do this kind of process,

which I really enjoy doing, which is almost from like putting a Darwin hat on or a biologist hat

on and doing analysis of a higher level of abstraction, like the organisms that spring up,

because there's different kinds of organisms, like you can think of them as species, and they

interact with each other. They can, there's gliders, they shoot different, there's like

things that can travel around. There's things that can glider guns that can generate those gliders.

And you can use the same kind of language as you would about describing a biological system.

So it's a wonderful laboratory. And it's kind of a rebuke to someone who doesn't think that like

very, very rich complex structure can come from very simple underlying laws. Like

it definitely can't. Now, here's what's interesting. If you just picked like some random rule,

you wouldn't get interesting complexity. I think that's one of the most interesting

things of these, one of these most interesting features of this whole subject that the rules

have to be tuned just right. Like a sort of typical rule set doesn't generate any kind of

interesting behavior. But some do. I don't think we have a clear way of understanding

which do and which don't. I don't know. Maybe Steven thinks he does. I don't know, but...

No, no, it's a giant mystery. What Steven Wolfram did is... Now, there's a whole interesting aspect

to the fact that he's a little bit of an outcast in the mathematics and physics community,

because he's so focused on a particular, his particular work. I think if you put ego aside,

which I think unfairly, some people are not able to look beyond. I think his work is

actually quite brilliant. But what he did is exactly this process of Darwin like exploration

is taking these very simple ideas and writing a thousand page book on them, meaning like,

let's play around with this thing. Let's see. And can we figure anything out? Spoiler alert? No,

we can't. In fact, he does a challenge. I think it's like a rule 30 challenge, which is quite

interesting. Just simply for machine learning people, for mathematics people, is can you predict

the middle column? For his, it's a 1D cellular automata. Can you predict, generally speaking,

can you predict anything about how a particular rule will evolve just in the future? Very simple.

Just look at one particular part of the world. Just zooming in on that part,

100 steps ahead. Can you predict something? And the challenge is to do that kind of prediction

so far as nobody's come up with an answer. But the point is, we can't. We don't have tools,

or maybe it's impossible, or, I mean, he has these kind of laws of irreducibility. They hear

first of his poetry. It's like, we can't prove these things. It seems like we can't. That's the

basic... It almost sounds like ancient mathematics or something like that, where you're like, the

gods will not allow us to predict the cellular automata. But that's fascinating that we can't.

I'm not sure what to make of it. And there's power to calling this particular set of rules,

game of life, as Conway did. Because I'm not exactly sure, but I think he had a sense that

there's some core ideas here that are fundamental to life, to complex systems, to the way life

emerged on Earth. I'm not sure I think Conway thought that. It's something that... I mean,

Conway always had a rather ambivalent relationship with the game of life, because I think he saw it

as... It was certainly the thing he was most famous for in the outside world. And I think that he,

his view, which is correct, is that he had done things that were much deeper mathematically than

that. And I think it always aggrieved him a bit that he was the game of life guy when he proved

all these wonderful theorems and created all these wonderful games, created the serial numbers.

He was a very tireless guy who just did an incredibly variegated array of stuff. So he was

exactly the kind of person who you would never want to reduce to one achievement. You know what

I mean? Let me ask you about group theory. You mentioned it a few times. What is group theory?

What is an idea from group theory that you find beautiful? Well, so I would say group theory

sort of starts as the general theory of symmetry is that people looked at different kinds of things

and said, like, as we said, like, oh, it could have maybe all there is the symmetry from left to

right, like a human being, right? Or that's roughly bilaterally symmetric, as we say. So

there's two symmetries. And then you're like, well, wait, didn't I say there's just one? There's

just left to right? Well, we always count the symmetry of doing nothing. We always count the

symmetry that's like, there's flip and don't flip. Those are the two configurations that you can be

in. So there's two. You know, something like a rectangle is bilaterally symmetric, you can flip

it left to right, but you can also flip it top to bottom. So there's actually four symmetries.

There's do nothing, flip it left to right, and flip it top to bottom or do both of those things.

A square, there's even more because now you can rotate it. You can rotate it by 90 degrees. So

you can't do that. That's not a symmetry of the rectangle. If you try to rotate it 90 degrees,

you get a rectangle oriented in a different way. So a person has two symmetries, a rectangle for

a square, eight, different kinds of shapes have different numbers of symmetries. And the real

observation is that that's just not like a set of things. They can be combined. You do one symmetry,

then you do another. The result of that is some third symmetry. So a group really abstracts away

this notion of saying it's just some collection of transformations you can do to a thing where

you combine any two of them to get a third. So, you know, a place where this comes up in computer

science is in sorting because the ways of permuting a set, the ways of taking sort of some set of

things you have on the table and putting them in a different order, shuffling a deck of cards,

for instance, those are the symmetries of the deck. And there's a lot of them. There's not two,

there's not four, there's not eight. Think about how many different orders a deck of card can be in.

Each one of those is the result of applying a symmetry to the original deck. So a shuffle is

a symmetry, right? You're reordering the cards. If I shuffle and then you shuffle, the result is some

other kind of thing you might call a double shuffle, which is a more complicated symmetry.

So group theory is kind of the study of the general abstract world that encompasses all

these kinds of things. But then, of course, like lots of things that are way more complicated than

that. Like infinite groups of symmetries, for instance. So they give you infinite, huh? Oh,

yeah. Okay. Well, okay, ready? Think about the symmetries of the line. You're like, okay, I can

reflect it left to right, you know, around the origin. Okay, but I could also reflect it left

to right grabbing somewhere else, like at one or two or pi or anywhere. Or I could just slide it

some distance. That's a symmetry. Slide it five units over. So there's clearly infinitely many

symmetries of the line. That's an example of an infinite group of symmetries.

Is it possible to say something that kind of captivates, keeps being brought up by physicists,

which is gauge theory, gauge symmetry, as one of the more complicated type of symmetries? Is there

an easy explanation of what the heck it is? Is that something that comes up on your mind at all?

Well, I'm not a mathematical physicist, but I can say this. It is certainly true

that it has been a very useful notion in physics to try to say, like, what are the symmetry groups,

like, of the world? Like, what are the symmetries under which things don't change, right? So we

just, I think we talked a little bit earlier about, it should be a basic principle that a theorem

that's true here is also true over there. And same for a physical law, right? I mean,

if gravity is like this over here, it should also be like this over there.

Okay. What that's saying is we think translation in space should be a symmetry.

All the laws of physics should be unchanged if the symmetry we have in mind is a very simple one,

like translation. And so then there becomes a question, like, what are the symmetries

of the actual world with its physical laws? And one way of thinking is an oversimplification,

but like one way of thinking of this big shift from before Einstein to after is that we just

changed our idea about what the fundamental group of symmetries were. So that things like

the Lorenz contraction, things like these bizarre relativistic phenomena or Lorenz would

have said, oh, to make this work, we need a thing to change its shape if it's moving

nearly a speed of light. Well, under the new framework, it's much better. You're like, oh,

no, it wasn't changing its shape. You were just wrong about what counted as a symmetry.

Now that we have this new group, the so called Lorenz group, now that we understand what the

symmetries really are, we see it was just an illusion that the thing was changing its shape.

Yeah, so you can then describe the sameness of things under this weirdness that is

general relativity, for example. Yeah. Yeah, still, I wish there was a simpler explanation

of like exact, I mean, you know, gauge symmetries is a pretty simple general concept about rulers

being deformed. I've actually just personally been on a search, not a very rigorous or aggressive

search, but for something I personally enjoy, which is taking complicated concepts and finding

the sort of minimal example that I can play around with, especially programmatically.

That's great. I mean, this is what we try to train our students to do, right? I mean, in class,

this is exactly what this is like best pedagogical practice.

I do hope there's simple explanation, especially like I've in my sort of drunk random walk, drunk

walk, whatever it is that's called, sometimes stumble into the world of topology. And like

quickly, like, you know, when you go into a party and you realize this is not the right party for

me, whenever I go into topology is like so much math everywhere. I don't even know what it feels

like this is me like being a hater is I think there's way too much math. Like there are two,

the cool kids who just want to have like everything is expressed through math, because they're

actually afraid to express stuff simply through language. That's my hater formulation of topology.

But at the same time, I'm sure that's very necessary to do sort of rigorous discussion.

But I feel like... But don't you think that's what gauge symmetry is like? I mean,

it's not a field I know well, but it certainly seems like... Yes, it is like that. Okay.

But my problem with topology, okay, and even like differential geometry is like,

you're talking about beautiful things. Like if they could be visualized, it's open question if

everything could be visualized, but you're talking about things that could be visually stunning,

I think. But they are hidden underneath all of that math. Like if you look at the papers that

are written in topology, if you look at all the discussions on stack exchange, they're all math

dense, math heavy. And the only kind of visual things that emerge every once in a while is like

something like a mobius strip. Every once in a while, some kind of

simple visualizations. Well, there's the vibration, there's the hop vibration or

all those kinds of things that somebody, some grad student from like 20 years ago wrote a

program in Fortran to visualize it. And that's it. And it's just... It makes me sad because

those are visual disciplines. Just like computer vision is a visual discipline. So you can provide

a lot of visual examples. I wish topology was more excited and in love with visualizing some of the

ideas. I mean, you could say that, but I would say for me, a picture of the hop vibration does

nothing for me. Whereas like when you're, it's like, oh, it's like about the quaternions. It's

like a subgroup of the quaternions. And I'm like, oh, so now I see what's going on. Like, why didn't

you just say that? Why were you like showing me this stupid picture instead of telling me what

you were talking about? Oh, yeah. Yeah. I'm just saying, nobody goes back to what we were saying

about teaching that like people are different and what they'll respond to. So I think there's no...

I mean, I'm very opposed to the idea that there's one right way to explain things. I think there's

like a huge variation in like, you know, our brains like have all these like weird like

hooks and loops. And it's like very hard to know like what's going to latch on. And it's not going

to be the same thing for everybody. So I think monoculture is bad, right? I think that's...

And I think we're agreeing on that point that like it's good that there's like a lot of different

ways in and a lot of different ways to describe these ideas because different people are going to

find different things illuminating. But that said, I think there's a lot to be discovered when you

force little silos of brilliant people to kind of find the middle ground or like

aggregate or come together in a way. So there's like people that do love visual things.

I mean, there's a lot of disciplines, especially in computer science that are obsessed with

visualizing, visualizing data, visualizing neural networks. I mean, neural networks themselves

are fundamentally visual. There's a lot of work in computer vision that's very visual.

And then coming together with some folks that were like deeply rigorous and are like totally

lost in multi dimensional space where it's hard to even bring them back down to 3D.

They're very comfortable in this multi dimensional space. So forcing them to kind of work together

to communicate because it's not just about public communication of ideas. It's also,

I feel like when you're forced to do that public communication like you did with your book,

I think deep profound ideas can be discovered. That's like applicable for research and for

science. Like there's something about that simplification or not simplification, but

distillation or condensation or whatever the hell you call it compression of ideas that

somehow actually stimulates creativity. And I'd be excited to see more of that in the

mathematics community. Can you? Let me make a crazy metaphor. Maybe it's a little bit like

the relation between prose and poetry, right? I mean, if you might say like,

why do we need anything more than prose? You're trying to convey some information. So you just

like say it. Well, poetry does something, right? It's sort of, you might think of it as a kind

of compression. Of course, not all poetry is compressed, like not awesome. Some of it is quite

baggy, but like you are kind of often it's compressed, right? A lyric poem is often sort of

like a compression of what would take a long time and be complicated to explain in prose into sort

of a different mode that is going to hit in a different way. We talked about Poincaré conjecture

here. There's a guy, he's Russian, Grigori Perlman. He proved Poincaré conjecture. If you

can comment on the proof itself, if that stands out to you, something interesting or the human

story of it, which is he turned down the Fields Medal for the proof. Is there something you find

inspiring or insightful about the proof itself or about the man? Yeah. I mean, one thing I really

like about the proof, and partly that's because it's sort of a thing that happens again and again

in this book. I mean, I'm writing about geometry and the way it sort of appears in all these kind

of real world problems. But it happens so often that the geometry you think you're studying

is somehow not enough. You have to go one level higher in abstraction and study a higher level

of geometry. And the way that plays out is that Poincaré asks a question about a certain kind

of three dimensional object. Is it the usual three dimensional space that we know, or is it some kind

of exotic thing? And so, of course, this sounds like it's a question about the geometry of the

three dimensional space. But no, Perlman understands. And by the way, in a tradition that involves

Richard Hamilton and many other people, like most really important mathematical advances,

this doesn't happen alone. It doesn't happen in a vacuum. It happens as the culmination of a program

that involves many people. Same with Wiles, by the way. I mean, we talked about Wiles, and I

want to emphasize that starting all the way back with Coomer, who I mentioned in the 19th century,

but Gerhard Frey and Mazer and Ken Ribbit and like many other people are involved in building

the other pieces of the arch before you put the keystone in. We stand on the shoulders of giants.

Yes. So, what is this idea? The idea is that, well, of course, the geometry of the three

dimensional object itself is relevant. But the real geometry you have to understand is

the geometry of the space of all three dimensional geometries. Whoa. You're going up a higher level.

Because when you do that, you can say, now let's trace out a path in that space. There's a mechanism

called Ricci flow. And again, we're outside my research area. So for all the geometric analysts

and differential geometers out there listening to this, if I please, I'm doing my best and I'm

roughly saying it. So the Ricci flow allows you to say like, okay, let's start from some

mystery three dimensional space, which Poincare would conjecture is essentially the same thing

as our familiar three dimensional space, but we don't know that. And now you let it flow,

you sort of like let it move in its natural path according to some almost physical process

and ask where it winds up. And what you find is that it always winds up. You've continuously

deformed it. There's that word deformation again. And what you can prove is that the process doesn't

stop until you get to the usual three dimensional space. And since you can get from the mystery thing

to the standard space by this process of continually changing and never kind of

having any sharp transitions, then the original shape must have been the same as the standard

shape. That's the nature of the proof. Now, of course, it's incredibly technical. I think

as I understand it, I think the hard part is proving that the favorite word of AI people,

you don't get any singularities along the way. But of course, in this context, singularity just

means acquiring a sharp kink. It just means becoming non smooth at some point. So

just saying something interesting about formal about the smooth trajectory through this weird

space. But yeah, so what I like about it is that it's just one of many examples of where

it's not about the geometry you think it's about. It's about the geometry of all geometries,

so to speak. And it's only by kind of like kind of like being jerked out of Flatland,

right? Same idea. It's only by sort of seeing the whole thing globally at once that you can

really make progress on understanding like the one thing you thought you were looking at.

It's a romantic question. But what do you think about him turning down the Fields Medal?

Well, is that just our Nobel Prizes and Fields Medal is just a cherry on top of the cake and

really math itself, the process of curiosity of pulling at the string of the mystery before us.

That's the cake. And then the words are just icing and clearly I've been fasting and I'm hungry.

But do you think it's tragic or just a little curiosity that he turned on the medal?

Well, it's interesting because on the one hand, I think it's absolutely true that right in some kind

of like vast spiritual sense, like awards are not important, not important the way that sort of

like understanding the universe is important. On the other hand, most people who are offered that

prize accept it. So there's something unusual about his choice there. I wouldn't say I see it as

tragic. I mean, maybe if I don't really feel like I have a clear picture of why he chose not to take

it. I mean, he's not alone in doing things like this. People have sometimes turned down prizes

for ideological reasons, probably more often in mathematics. I mean, I think I'm right in saying

that Peter Schultz had like turned down sort of some big monetary prize because he just, you know,

I mean, I think at some point you have plenty of money. And maybe you think it sends the wrong

message about what the point of doing mathematics is. I do find that there's most people accept.

You know, most people have given a prize. Most people take it. I mean, people like to be appreciated.

Like I said, we're people. Yes. Not that different from most other people.

But the important reminder that that turning down the prize serves for me is not that there's

anything wrong with the prize. And there's something wonderful about the prize, I think.

The Nobel Prize is trickier because so many Nobel Prizes are given. First of all, the Nobel

Prize often forgets many, many of the important people throughout history. Second of all,

there's like these weird rules to it. There's only three people and some projects have a huge

number of people. I don't know. It doesn't kind of highlight the way science is done

on some of these projects in the best possible way. But in general, the prizes are great.

But what this kind of teaches me and reminds me is sometimes in your life, there'll be moments

when the thing that you would really like to do, society would really like you to do,

is the thing that goes against something you believe in, whatever that is, some kind of

principle, and stand your ground in the face of that. It's something, I believe most people

will have a few moments like that in their life, maybe one moment like that. And you have to do

it. That's what integrity is. So it doesn't have to make sense to the rest of the world,

but to stand on that, to say no. It's interesting because I think...

But do you know that he turned down the prize in service of some principle? Because I don't know

that. Well, yes, that seems to be the inkling, but he has never made it super clear. But the

inkling is that he had some problems with the whole process of mathematics that includes awards,

like this hierarchies and the reputations and all those kinds of things and individualism that's

fundamental to American culture. He probably, because he visited the United States quite a bit,

that he probably... It's all about experiences. And he may have had some parts of academia,

some pockets of academia can be less than inspiring, perhaps sometimes, because of the

individual egos involved, not academia, people in general, smart people with egos. And if you

interact with certain kinds of people, you can become cynical too easily. I'm one of those people

that I've been really fortunate to interact with incredible people at MIT and academia in general,

but I've met some assholes. And I tend to just kind of... When I run into difficult folks,

I just kind of smile and send them all my love and just kind of go around. But for others,

those experiences can be sticky. Like they can become cynical about the world when folks like

that exist. So he may have become a little bit cynical about the process of science.

Well, you know, it's a good opportunity. Let's posit that that's his reasoning because I truly

don't know. It's an interesting opportunity to go back to almost the very first thing we talked

about, the idea of the mathematical Olympiad. Because of course, that is... So the international

mathematical Olympiad is like a competition for high school students solving math problems. And

in some sense, it's absolutely false to the reality of mathematics because just as you say,

it is a contest where you win prizes. The aim is to sort of be faster than other people.

And you're working on sort of canned problems that someone already knows the answer to,

like not problems that are unknown. So, you know, in my own life, I think when I was in

high school, I was like very motivated by those competitions. And like I went to the math Olympiad