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?

Well, let’s just go with me on this, okay?

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 than 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 nonlinguistic way.

It’s hard to imagine doing mathematics

without talking about mathematics

and sort of thinking in propositions.

But maybe it’s just because that’s the way I do mathematics,

and maybe I can’t imagine it any other way, right?

Well, what about visualizing shapes,

visualizing concepts to which language

is not obviously attachable?

Ah, that’s a really interesting question.

And 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 sort of gives this proof, just gives the diagram,

and then the entire verbal content of the proof

is he just writes under it, behold.

Like that’s it.

And it’s like, there’s some dispute

about exactly how accurate that is.

But so then there’s an interesting question.

If your proof is a diagram, if your proof is a picture,

or even if your proof is like a movie of the same pieces

like coming together in two different formations

to make two different things, is that language?

I’m not sure I have a good answer.

What do you think?

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 in action.

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’s not, that doesn’t need to end there.

It’s just at the end of the chapter

and then it goes on and on and on in that kind of way.

But I gotta ask you about geometry

and it’s a prominent topic in your new book, Shape.

So for me, geometry is the thing,

just like as you’re saying,

made me fall in love with mathematics when I was young.

So being able to prove something visually

just did something to my brain that it had this,

it planted this hopeful seed

that you can understand the world, like perfectly.

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 like 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 like you are like,

the rest of it I could take or leave,

but then at this one moment, it made sense.

This class made sense, why wasn’t it all like that?

There’s other people, I can tell you,

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 a number, I figured it out.

But then there was this geometry, like what was that?

What happened that year? Like I didn’t get it.

I was like lost the whole year

and I didn’t understand like why we even

spent the time doing that.

So, but what everybody agrees on

is that it’s somehow different, right?

There’s something special about it.

We’re gonna 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.

But in the 70s, we had a cool wooden box

around your stereo.

That was the look, everything was dark wood.

And the box had a bunch of holes in it

to let the sound out.

And the holes were in this rectangular array,

a six by eight array of holes.

And I was just kind of like zoning out

in the living room as kids do,

looking at this six by eight rectangular array of holes.

And if you like, just by kind of like focusing in and out,

just by kind of 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.

Whoa.

So 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.

And this was like unbelievable to me.

Am I allowed to cuss on your podcast?

I don’t know if that’s, 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 in the same physical reality.

Exactly.

And it’s just as you say.

I knew that six times eight was the same as eight times six.

I knew my times tables.

I knew that that was a fact.

But did I really know it until that moment?

That’s the question, right?

I sort of knew that the times table was symmetric,

but I didn’t know why that was the case until that moment.

And in that moment I could see like,

oh, I didn’t have to have somebody tell me that.

That’s information that you can just directly access.

That’s a really amazing moment.

And as math teachers, that’s something

that we’re really trying to bring to our students.

And I was one of those who did not love

the kind of Euclidean geometry ninth grade class

of like prove that an isosceles triangle

has equal angles at the base, like this kind of thing.

It didn’t vibe with me the way that algebra and numbers did.

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, who knows, maybe I was always fated

to be an algebraic geometer,

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.

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

when 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.

Nowadays in math,

we could use a symmetry to refer to

any kind of transformation of an image

or a space or an object.

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 had 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, 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,

what’s handwritten digits.

And you know, I don’t smoke much weed or any really,

but it certainly feels like it when I look at MNIST

and think about 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 it 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 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

and a geometer is that the kind of groups of symmetries,

the types of symmetries that we know of are not sufficient.

So in other words, like we’re just gonna keep on going

into the weeds on this.

The deeper, the better.

A kind of symmetry that we understand very well

is rotation.

So here’s what would be easy.

If humans, if we recognize the digit as a one,

if it was like literally a rotation

by some number of degrees or some fixed one

in some typeface like Palatino or something,

that would be very easy to understand.

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 that’s, there is a moment.

Like I have myself written programs

that literally morph twos and threes and so on.

And you watch, and there is 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 dispute.

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 it’s 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, and 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 gonna study, I don’t know,

like Poincare, we’re gonna study the three body problem.

We’re gonna 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 like put it

in your computer and see what they do.

Well, guess what?

That’s like a problem that Poincare won a huge prize for

like 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 a humongous mystery.

You just opened the door and we’re gonna walk right in

before we return to symmetry.

What’s the, who’s Poincare and what’s this conjecture

that he came up with?

Why is it such a hard problem?

Okay, so Poincare, 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 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, in math, we’re not really trained

on our own history.

We got a PhD in math, learned 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 gotta 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,

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 that 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 it was in the Soviet Union.

It definitely was, and I would love to hear more

about how it was in the Soviet Union.

I mean, there was, and 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 idolize 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,

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 would lead to a better world.

So that created all the usual things you think about

with the Olympics, which is like extreme competitiveness.

But it also created this sense that in the modern era

in America, somebody like Elon Musk, whatever you think

of him, like Jeff Bezos, those folks,

they inspire the possibility that one person

or a group of smart people can change the world.

Like not just be good at what they do,

but actually change the world.

Mathematics was at the core of that.

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 gonna pick up on something you said.

I think you would love a book called

Dual at Dawn by Amir Alexander,

which I think some of the things you’re responding to

and 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 Everest to 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 part of the reason that we think of Gawa that way

was because Gawa 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 terms,

we would say he was extremely emo.

Like we wrote all these letters

about his like florid feelings

and like the fire within him about the mathematics.

And 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 Poincare.

Yes, okay, so back to Poincare.

So he’s, you know, it’s funny.

This book is filled with kind of mathematical characters

who often are kind of peevish or get into feuds

or sort of have like weird enthusiasms

because those people are fun to write about

and they sort of like say very salty things.

Poincare is actually none of this.

As far as I can tell, he was an extremely normal dude

who didn’t get into fights with people

and everybody liked him

and he was like pretty personally modest

and he had very regular habits.

You know what I mean?

He 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 gonna tell me now that the book is out,

like, oh, didn’t you know about this

like incredibly sordid episode?

As far as I could tell, a completely normal guy.

But he just kind of, in many ways,

creates the geometric world in which we live

and 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.

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.

Yes, exactly.

When you have the third wheel is always a problem.

This is what 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 the 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 the 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 longterm behavior of the system

completely different.

So Poincare was the first to recognize

that 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 called topology.

He called it analysis situs.

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 situs, which I guess sort of roughly means

like the analysis of location or something like that.

Like 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 wanna 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 of Cetus

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.

Like it can be real,

even if it’s not perceivable to the human cognition.

I think that’s right.

I think, don’t get me wrong,

Poincare never strays far from physics.

He’s always motivated by physics,

but the physics drove him to need to think about spaces

of higher dimension.

And so he needed a formalism that was rich enough

to enable him to do that.

And once you do that,

that formalism is also gonna include things

that are not physical.

And then you have two choices.

You can be like, oh, well, that stuff’s trash.

Or, and this is more of the mathematicians frame of mind,

if you have a formalistic framework

that like seems really good

and sort of seems to be like very elegant and work well,

and it includes all the physical stuff,

maybe we should think about all of it.

Like maybe we should think about it,

thinking maybe there’s some gold to be mined there.

And indeed, like, you know, guess what?

Like before long there’s relativity and there’s space time.

And like all of a sudden it’s like,

oh yeah, maybe it’s a good idea.

We already had this geometric apparatus like set up

for like how to think about four dimensional spaces,

like turns out they’re real after all.

As I said, you know, this is a story much told

right in mathematics, not just in this context,

but in many.

I’d love to dig in a little deeper on that actually,

cause I have some intuitions to work out.

Okay.

My brain.

Well, I’m not a mathematical physicist,

so we can work them out together.

Good.

We’ll together walk along the path of curiosity,

but Poincare conjecture.

What is it?

The Poincare 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 Poincare’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 Poincare 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 will 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.

It’s a 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, okay.

If I draw a little circle on this mug,

imagine this to be a loop of string.

I could pull that loop of string closed

on the surface of the mug, right?

That’s definitely something I could do.

I could shrink it, shrink it, shrink it until it’s a point.

On the other hand,

if I draw a loop that goes around the handle,

I can kind of zhuzh it up here

and I can zhuzh it down there

and I can sort of slide it up and down the handle,

but I can’t pull it closed, can I?

It’s trapped.

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 Poincare’s 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 pull it shut.

You can always pull it shut.

But what if your piece of string

was the size of the universe?

Like 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.

I think there’s a lot,

there is ideas of it being a torus.

I mean, there’s some trippy ideas

and they’re not like weird out there controversial.

There’s legitimate at the center of a cosmology debate.

I mean, I think most people think it’s flat.

I think there’s some kind of dodecahedral symmetry

or 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,

how are you even supposed to think about the shape

of a thing that doesn’t have any thing outside of it?

I mean.

Ah, 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,

that 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, sorry, 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, let’s go, always to make things easier,

go down to dimension.

Let’s think about we live in a circle, okay?

You can tell whether you live on a circle or a line segment,

because if you live in 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,

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.

Boy, 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.

Does it make you, 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 don’t even know how to say that.

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 like, 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.

Which one?

Because 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 point, fair point on that.

But you have to believe the stories about,

no, I don’t even know, would you even know?

Would you really?

Oh, no, 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 what’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 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,

cause like I was,

this is very counterintuitive to me to think about,

maybe it’s wrong.

But cause I was thinking of like earth,

your 3D thing 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 a lot.

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.

A while ago, yeah.

An adult.

You know, 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 one day he’s in his house

and suddenly there’s like a little circle there

and they’re with him.

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 the three dimensional world,

maybe with the 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 is just a nice way to analyze a universe,

but the reality is, is as exactly we perceive it,

it is three dimensional, or are we just seeing,

are we those flatland creatures

that are just seeing a tiny slice of reality

and the actual reality is many, many, many more dimensions

than the three dimensions we perceive?

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 flatland 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 asked 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 but 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 no 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

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,

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 as some like to call it,

or a five dimensional cube, or a six dimensional cube,

but it is not beyond our cognitive capabilities

to figure out how many corners

a six dimensional cube would have.

That’s what’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?

And you, listener, may try to answer that in your own head.

Yeah, I’m gonna 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 can get very creative.

But it’s kind of interesting to each,

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, and 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

that 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, you know, you really,

like 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, you know, 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.

A rectangular piece of plastic doesn’t have a hole in it.

I didn’t poke a hole in it when I,

so how can I have a hole?

They’d be like, it’s just one thing.

Okay, most people don’t see it that way.

That’s like a…

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 schmushing 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 we’ve planted our foot, this is what we stand for.

There’s books written about each belief.

You know, I would say, look, there’s like a hole

and it goes all the way through the straw, right?

It’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 a 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 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.

Yeah.

So if you’re a one holder, 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, this is two holes.

Yeah.

And yet if you’re a two holder, the one holder will say like,

okay, where does one hole begin and the other hole end?

Yeah.

And in the book, I sort of, you know, 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 like 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, you know, what I do in the book is say like,

let’s stop talking about straws for a minute

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, I mean, I guess we’re filming only from here.

I could take up, no, I’m not gonna do it.

You’ll just have to imagine the pants, sorry.

Yeah.

Lex, if you want to, no, okay, no.

That’s gonna be in the director’s cut.

That’s that 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 good combination.

So she’s a one holder for the straw.

So she’s a one holder for the straw too.

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 holes, whatever we may mean by them,

they’re somehow things which have an arithmetic to them.

They’re 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.

Yeah.

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 thought 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 waist of the pants.

So just so you know, I fasted for 72 hours

the last three days.

So I just broke the fast with a little bit of food yesterday.

So 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, this is just for this.

It’s hardship to purify the mind.

No, it’s for the first time,

I just wanted to try the experience.

Oh, wow.

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 had this idea,

this sort of modern idea.

I mean, building on stuff 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 Emmy Noether’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 gonna 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 anthropology.

It’s not like this and 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 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 of it.

Right, so not be talking just about ideas,

but the actually be expressing the idea.

Is there, you know, somebody in the,

maybe you can comment, there’s a guy,

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 math teaching, there’s so many different venues

through which we can teach people math.

There’s the traditional one, right?

Where I’m in a classroom with, depending on the class,

it could be 30 people, it could be a hundred 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, we have a long time.

I’m with them for a whole semester

and I can ask them to do homework and we talk together.

We have office hours, if they have one on one questions,

a lot of, it’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 more, 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 three to five years.

And the ultimate high level of engagement to one person.

Books, this can get to a lot more people

that are ever gonna 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 any hardback book

like not written by a Kardashian or an Obama

is gonna sell, right?

So that’s, I mean,

and then those are, some of them are like longer,

20 minutes long, some of them are five minutes long,

but they’re shorter.

And then even some of you look like Eugenia Chang

who’s a wonderful category theorist in Chicago.

I mean, she was on, I think the Daily Show or is it,

I mean, she was on, 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.

And one thing I feel like is great right now

is that people are just broadcasting on all the channels

because we each have our skills, right?

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 winning, good looking, eye catching videos

is like a totally different skill.

And probably 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,

now he writes programs, so it’s programmatic visualization.

So like the things he is basically mostly

through his Manum library and Python,

everything is drawn through Python.

You have to truly understand the topic

to be able to visualize it in that way

and not just understand it,

but really kind of think in 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 and visualizing

is as powerful or has the same kind of inkling of power

as does the process of proving something.

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.

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, for example,

the Institute for Computational Experimental Mathematics

at Brown, which is like, it’s a NSF funded math institute,

very much part of sort of traditional math academia.

They did an entire theme semester

about visualizing mathematics,

looking at the same kind of thing that they would do

for like an up and coming research topic.

Like that’s pretty cool.

So I think there really is buy in

from 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,

an AI system will win a Fields Medal?

No.

Of course, that’s what a human would say.

Why not?

Is that like, that’s like my captcha?

That’s like the proof that I’m a human?

Is that like the lie that I know?

Yeah.

What is, 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 interested in like,

how it can like help feed the world

or help solve like real social problems?

I’m like, can it do more math?

Like, 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 invariants of this polynomial ring

under the action of some finite group.

Doesn’t matter what those words mean,

just it’s like some thing that in 1890

would take a person a year to do

and would be a valuable thing that you might wanna know.

And it’s still a valuable thing that you might wanna 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 and…

Yeah, oh yeah, so it’s similar to Maple and 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, you know, there’s a story I tell in the book,

which I’m fascinated by, which is, you know,

so far, attempts to get AIs

to prove interesting theorems have not done so well.

It doesn’t mean they can.

There’s actually a paper I just saw,

which has a very nice use of a neural net

to find counter examples to conjecture.

Somebody said like, well, maybe this is always that.

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.

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 come up with 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, okay, we’re gonna go back to knots.

There’s a knot called the Conway knot.

After John Conway, maybe we’ll talk about

a very interesting character also.

Yeah, it’s a small tangent.

Somebody I was supposed to talk to

and unfortunately he passed away

and he’s somebody I find as 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 a postdoc, yeah, you missed out.

There’s no way to sugarcoat 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 slice.

It has to do with something about what kinds

of three dimensional surfaces and four dimensions

can be bounded by this knot.

But nevermind what it means, it’s some question.

And it’s actually very hard to compute

whether a knot is slice or not.

And in particular, the question of the Conway knot,

whether it was slice 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.

In fact, I think she’s now at MIT,

so she’s basically following you around.

If I remember correctly.

The reverse.

There’s a lot of really interesting richness to this story.

One thing about it is her paper was rather,

was very short, it was very short and simple.

Nine pages of which two were pictures.

Very short for like a paper solving a major conjecture.

And it really makes you think about what we mean

by difficulty in mathematics.

Like, do you say, oh, actually the problem wasn’t difficult

because you could solve it so simply?

Or do you say like, well, no, evidently it was difficult

because like the world’s top topologists,

many, you know, worked on it for 20 years

and nobody could solve it, so therefore it is difficult.

Or is it that we need sort of some new category

of things that about which it’s difficult

to figure out that they’re not difficult?

I mean, this is the computer science formulation,

but the sort of 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, you know, 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 we’ll look and it’ll be simple laws

that govern the universe,

simple explanation of what is consciousness,

what is love, is mortality fundamental to life,

what’s the meaning of life, are humans special

or we’re just another sort of reflection

of 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, but.

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, what is that?

So it’s aesthetic.

It’s aesthetic, yeah.

Or, but it’s aesthetic in the way

that happiness is an aesthetic.

Like, 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 has,

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 this 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 understands, 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,

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 it 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 right, so to give,

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,

at least 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 copy of the Disquisitiones Arithmetic I.

Like he wrote, here’s an equation, it has no solutions.

I can prove it, but the proof’s like a little too long

to fit in 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 were like this mysterious proof

that was lost, a very romantic story, right?

But a fair amount 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 maybe lost the time.

Maybe, 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 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 he proved it.

So, I mean, work on the Fermat problem

developed this like incredible richness of number theory

that we now live in today.

Like, and not, by the way,

just Wiles, Andrew Wiles being the person

who, together with Richard Taylor,

finally proved this theorem.

But you know how you have this whole moment

that people try to prove this theorem

and they fail,

and there’s a famous false proof by 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 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 two times three times two.

I had to think about it.

Or it’s two times two times three,

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 five,

or in which there’s like four threes in it.

Nope, 12 is like two twos and a three.

Like that is what it is.

And that’s such a fundamental feature of arithmetic

that we almost think of it like God’s law.

You know what I mean?

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 undergraduate number theory,

it’s like, it’s the first really deep theorem

that you prove.

What’s amazing is the fact

that you can factor a number into primes is much easier.

Essentially Euclid knew it,

although 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

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 are relevant 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, so 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 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

that you used to prove that sort of resonates

with you as a mathematician?

Is there an 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 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.

How about that?

What does 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 it’s a name.

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 Poincare,

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 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 Fermat’s Last Theorem,

and let’s not even be too careful

about what these objects are.

I can tell you there are gaol representations

in modular forms, but saying those words

is not gonna 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 like machine translation and stuff like that.

And it doesn’t have anything to do with

are they next to each other in the dictionary, right?

It’s a different kind of distance.

Okay, ready?

In this kind of number theory,

there was a crazy distance called the peatic distance.

I didn’t write about this that much in the book

because even though I love it

and it’s a big part of my research life,

it gets a little bit into the weeds,

but your listeners are gonna hear about it now.

Please.

What a normal person says

when they say two numbers are close,

they say like 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 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 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.

That’s odd.

You wanna know what’s really far from one?

Like one and 1 64th

because their difference is a negative power of two,

two to the minus six.

So those points are quite, quite far away.

Two to the power of a large N would be two,

if that’s the difference between two numbers

then they’re close.

Yeah, so two to a large power is in this metric

a very small number

and two to a negative power is a very big number.

That’s two attic.

Okay, I can’t even visualize that.

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

to satisfy certain axioms deserves to be called a distance

and this.

See, I’m sorry to interrupt.

My brain, you broke my brain.

Awesome.

10 seconds ago.

Cause I’m also starting to map for the two attic case

to binary numbers.

And you know, cause we romanticize those.

So I was trying to.

Oh, that’s exactly the right way to think of it.

I was trying to mess with number,

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 I’m not sure.

Well, I think there’s a.

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.

Because in a binary expansion, two numbers are close.

A number that’s small is like 0.0000 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 said,

that’s what I should have said, Lex.

That’s a very good metaphor.

Okay, but so why is that interesting

except for the fact that it’s a beautiful kind of framework,

different kind of framework

of which to think about distances.

And you’re talking about not just the two attic,

but the generalization of that.

Why is that interesting?

Yeah, the NEP.

And so that, because that’s the kind of deformation

that comes up in Wiles’s proof,

that deformation where moving something a little bit

means a little bit in this two attic sense.

Trippy, okay.

No, I mean, it’s such a,

I mean, I just get excited talking 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, 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 the goal is always the understanding.

The goal is not so much to prove things.

The goal is not to know what’s true or false.

I mean, this is something I write about

in the book, Near the End.

And it’s something that,

it’s a wonderful, wonderful essay by Bill Thurston,

kind of one of the great geometers of our time,

who unfortunately passed away a few years ago,

called on proof and progress in mathematics.

And he writes very wonderfully about how,

we’re not, it’s not a theorem factory

where you 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 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 start to do a full analysis

on the everybody that lives on earth currently,

the 7 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 violence,

the source of 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 gonna talk about AI,

which you know a lot more about than I do.

So just 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 a really good humble way to think about it.

I like it.

Okay, so let’s just go for it.

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 gonna involve

some notion of distance,

because it’s always gonna 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.

If you try to do natural language processing

and your idea of distance between words

is how close they are in the dictionary,

when you write them in alphabetical order,

you are gonna get pretty bad translations, right?

No, the notion of distance has to come from somewhere else.

Yeah, that’s essentially what neural networks are doing,

that’s what word embeddings are doing is coming up with.

In the case of word embeddings, literally,

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.

So you don’t, 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?

To 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. But it’s not true.

It’s just not.

That’s such a 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 to be honest, 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 he was the kind of guy, I didn’t know him,

but I’ve been reading his writing,

he liked to sort of say stuff, like stuff that sounded good.

You know what I mean?

So it’s 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 it’s 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’s somehow maps in their mind,

but it’s taking as a starting point,

as an assumption that there’s a deep knowledge

and a deep understanding that’s actually very complicated.

And the simplicity is 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 imprecisely?

As an explanation.

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.

Not the only truth,

but some truths 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.

And.

Yes, that is definitely possible.

But I would even say,

look, I mean, consciousness is a thing about

which we’re still in the dark

as to whether there’s an explanation

we would understand it 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,

Paul Erdős, 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 Erdős.

Erdős did?

Well, he thought about distances and networks.

I guess he didn’t probably,

he didn’t think about the social network.

Oh, that’s fascinating.

And that’s how it started that story of Erdős number.

Yeah, okay.

It’s hard to distract.

But you know, Erdős was sort of famous for saying,

and this is sort of long lines we’re 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.

I mean, it was,

but Poincare on the other hand,

and by the way, there were other managers.

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 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 will say Hendrix 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 Erdos at a Hendrix concert,

like trying to figure out if it was from the book or not.

What I was coming to was just to say,

but what 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.

That are not like that.

All this talk of primes got me hungry for primes.

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 and 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, 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 there’s like something trivial about it?

It’s something you could 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 it’s a question of which definition is best for us,

for our purposes.

Well, those edge cases are weird, right?

So it 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 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’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,

or there’s much more that we don’t know about them.

I’ll tell you the first few.

There’s two, three, five, seven, 11.

By the way, they’re all gonna be odd from then on

because if they were even, I could factor out

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 were we?

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 Grotendeek prime 57,

like sort of Alexander Grotendeek,

the great algebraic geometer was sort of giving

some lecture involving a choice of a prime in general.

And somebody said, 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

it’s a funny story.

But 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, like, I mean.

So is there a category of like fake primes

that are easily mistaken to be prime?

Like 57, I wonder.

Yeah, so I would say 57 and 51 are definitely

like prime offenders.

Oh, I didn’t do that on purpose.

Oh, well done.

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 pass 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, you’re 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 see, 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, yeah.

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 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 that pass that test,

but are not actually prime.

But the point is, there are many, many,

many theorems about prime numbers.

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 what would be really simple?

If there was only finitely many primes

and then there would be this finite set of atoms

that all numbers would be built up.

That would be very simple and good in 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’ve had anything

like modern number theory.

The primes are infinite.

The primes that there are, right.

There’s an infinite number of primes.

So what about the gaps 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 10 digit numbers,

no way would 40% of those be prime.

Being prime would be a lot rarer.

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 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,

well, how could one possibly have a right

to have an opinion about something like that?

We don’t have any way of describing a process

that makes primes.

Sure, you can 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 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 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.

There 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

at 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.

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.

Like you are talking about prime numbers

and you are like,

but prime numbers are completely deterministic.

And I’m saying like,

well, but 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?

Well, 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,

in your mind, construct like a life story.

In your mind, you can’t help it.

There’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 is, 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

finiteism, ultra finiteism and intuitionism

or into, I forget what it’s called.

Yeah, intuitionism.

Intuitionism.

That seemed pretty, pretty interesting.

I have it on my to do list actually like look into

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 finiteism

that infinity 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

that 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, finiteism, ultra finiteism

and intuitionism, although I know they’re related

and 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 finiteism.

So, finiteism 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 finiteism says, nope, even that infinity is fake.

I’ll bet ultra finiteism 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.

It’s all back to the emo, Doc.

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 it’s like, I’m not really gonna make a firm statement

about whether it’s a real thing or not.

Yeah, that’s called most of the history of mathematics.

So it’s only after Cantor that we really are sort of,

okay, we’re gonna 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.

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 have taken calculus

and 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 burn 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, right?

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.

That’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,

because look, the origin of all this is like,

number represents a magnitude, like the length of a line.

So I mean, the idea that there’s a continuum,

there’s sort of like, it’s pretty old,

but 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 an infinity as 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,

Stephen Wolfram’s work,

that I’ve been a big fan of for a while, 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 on 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, but.

And those are really always compelling

is 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 like learning the texture

as like let’s try to like reverse engineer

and like 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.

Like what does it do when it doesn’t do the thing

that you tried to train it to do?

That’s extremely interesting.

Yeah, 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.

So 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 at Princeton

for most of his life.

He was sort of certainly at 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 like very, very famous.

Yeah, but I’m not actually so convinced at a tiny tangent

that that shouldn’t be so.

I mean, there’s, it’s not obvious to me

that that’s one of the,

like 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 wound up

with his mathematics, right?

And so it’s not, sometimes I think people

who are outside the field think of mathematics

as this kind of like 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 like a novel you wrote

or a painting you painted

or just like the way you walk down the street.

Like 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 might think I’m gonna 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 like looked at, he was a constant inventor of games

or like crazy names.

And then he would sort of analyze 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 Berlekamp 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, 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 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 the 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.

Usually your question was just a prompt to him.

You couldn’t count on actually getting the question answered.

Yeah, those brilliant, curious minds 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.

His 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 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

and which squares die.

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 in 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 wanna 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.

Cause 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.

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.

Now, here’s what’s interesting.

If you just pick 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.

And I don’t think we have a clear way of understanding

which do and which don’t.

Maybe Steven thinks he does, I don’t know.

No, no, it’s a giant mystery where 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.

He’s 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 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, generally speaking,

can you predict anything about

how a particular rule will evolve just in the future?

Very simple.

Just looking 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 like, we can’t.

We don’t have tools or maybe it’s impossible or,

I mean, he has these kind of laws of irreducibility

that he refers to, but it’s 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 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 emerge 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.

I mean, 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 symmetries,

that people looked at different kinds of things

and said, as we said, oh, it could have,

maybe all there is is symmetry from left to right,

like a human being, right?

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.

And then 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 four, 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

the 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 it 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 can be 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 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, this isn’t 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 phenomenon

or Lorenz would have said, oh, to make this work,

we need a thing to change its shape

if it’s moving nearly the speed of light.

Well, under the new framework, it’s much better.

You say, 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, gauge symmetries,

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 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?

It’s, so whenever I go into topology,

it’s like so much math everywhere.

I don’t even know what, it feels like this is me

like being a hater, 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.

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 can 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, you know, it’s 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 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, no, but it goes back

to what you were saying about teaching

that like people are different in what they’ll respond to.

So I think there’s no, I mean, I’m very opposed

to the idea that there’s a 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 gonna latch on

and it’s not gonna 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 gonna find different things illuminating.

But that said, I think there’s a lot to be discovered

when you force little like silos of brilliant people

to kind of find a 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 they’re 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, 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 all, 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 gonna hit in a different way.

We talked about Poincare conjecture.

There’s a guy, he’s Russian, Grigori Perlman.

He proved Poincare’s conjecture.

If you can comment on the proof itself,

if that stands out to you as something interesting

or the human story of it,

which is he turned down the field’s metal 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 Poincare 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, Perelman 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 wanna emphasize

that starting all the way back with Kummer,

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’ve 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 of geometries.

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 being jerked out of flatland.

Same idea.

It’s only by sort of seeing the whole thing globally at once

that you can really make progress on understanding

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?

Is that just, are Nobel Prizes and Fields Medals

just the 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 awards 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 down 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,

like 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 sometimes turn down prizes for ideological reasons,

but probably more often in mathematics.

I mean, I think I’m right in saying that

Peter Schultz turned down sort of some big monetary prize

because he just, you know, I mean, I think he,

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 give it a prize.

Most people take it.

I mean, people like to be appreciated,

but like I said, we’re people.

Not that different from most other people.

But the important reminder that that turning down

a 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

that it’s only three people

and some projects have a huge number of people.

And it’s like this, it, 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 standing your ground in the face of that

is 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 like, it doesn’t have to make sense

to the rest of the world, but to stand on that,

like 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 a 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 and…

You won it twice and got, I mean…

Well, there’s something I have to explain to people

because it says, I think it says on Wikipedia

that I won a gold medal.

And in the real Olympics,

they only give one gold medal in each event.

I just have to emphasize

that the International Math Olympiad is not like that.

The gold medals are awarded

to the top 112th of all participants.

So sorry to bust the legend or anything like that.

Well, you’re an exceptional performer

in terms of achieving high scores on the problems

and they’re very difficult.

So you’ve achieved a high level of performance on the…

In this very specialized skill.

And by the way, it was a very Cold War activity.

You know, in 1987, the first year I went,

it was in Havana.

Americans couldn’t go to Havana back then.

It was a very complicated process to get there.

And they took the whole American team on a field trip

to the Museum of American Imperialism in Havana

so we could see what America was all about.

How would you recommend a person learn math?

So somebody who’s young or somebody my age

or somebody older who’ve taken a bunch of math

but wants to rediscover the beauty of math

and maybe integrate it into their work

more solid in the research space and so on.

Is there something you could say about the process of…

Incorporating mathematical thinking into your life?

I mean, the thing is,

it’s in part a journey of self knowledge.

You have to know what’s gonna work for you

and that’s gonna be different for different people.

So there are totally people who at any stage of life

just start reading math textbooks.

That is a thing that you can do

and it works for some people and not for others.

For others, a gateway is, I always recommend

the books of Martin Gardner,

another sort of person we haven’t talked about

but who also, like Conway, embodies that spirit of play.

He wrote a column in Scientific American for decades

called Mathematical Recreations

and there’s such joy in it and such fun.

And these books, the columns are collected into books

and the books are old now

but for each generation of people who discover them,

they’re completely fresh.

And they give a totally different way into the subject

than reading a formal textbook,

which for some people would be the right thing to do.

And working contest style problems too,

those are bound to books,

especially like Russian and Bulgarian problems.

There’s book after book problems from those contexts.

That’s gonna motivate some people.

For some people, it’s gonna be like watching

well produced videos, like a totally different format.

Like I feel like I’m not answering your question.

I’m sort of saying there’s no one answer

and it’s a journey where you figure out

what resonates with you.

For some people, it’s the self discovery

is trying to figure out why is it that I wanna know?

Okay, I’ll tell you a story.

Once when I was in grad school,

I was very frustrated with my lack of knowledge

of a lot of things as we all are

because no matter how much we know,

we don’t know much more and going to grad school

means just coming face to face

with the incredible overflowing vault of your ignorance.

So I told Joe Harris, who was an algebraic geometer,

a professor in my department,

I was like, I really feel like I don’t know enough

and I should just take a year of leave

and just read EGA, the holy textbook,

Elements de Géométrie Algebraique,

the Elements of Algebraic Geometry.

I’m just gonna, I feel like I don’t know enough

so I’m just gonna sit and read this like 1500 page

many volume book.

And he was like, and Professor Harris was like,

that’s a really stupid idea.

And I was like, why is that a stupid idea?

Then I would know more algebraic geometry.

He’s like, because you’re not actually gonna do it.

Like you learn.

I mean, he knew me well enough to say like,

you’re gonna learn because you’re gonna be working

on a problem and then there’s gonna be a fact from EGA

that you need in order to solve your problem

that you wanna solve and that’s how you’re gonna learn it.

You’re not gonna learn it without a problem

to bring you into it.

And so for a lot of people, I think if you’re like,

I’m trying to understand machine learning

and I’m like, I can see that there’s sort of

some mathematical technology that I don’t have,

I think you like let that problem

that you actually care about drive your learning.

I mean, one thing I’ve learned from advising students,

math is really hard.

In fact, anything that you do right is hard.

And because it’s hard, like you might sort of have some idea

that somebody else gives you, oh, I should learn X, Y and Z.

Well, if you don’t actually care, you’re not gonna do it.

You might feel like you should,

maybe somebody told you you should,

but I think you have to hook it to something

that you actually care about.

So for a lot of people, that’s the way in.

You have an engineering problem you’re trying to handle,

you have a physics problem you’re trying to handle,

you have a machine learning problem you’re trying to handle.

Let that not a kind of abstract idea

of what the curriculum is, drive your mathematical learning.

And also just as a brief comment that math is hard,

there’s a sense to which hard is a feature, not a bug,

in the sense that, again,

maybe this is my own learning preference,

but I think it’s a value to fall in love with the process

of doing something hard, overcoming it,

and becoming a better person because of it.

Like I hate running, I hate exercise,

to bring it down to like the simplest hard.

And I enjoy the part once it’s done,

the person I feel like in the rest of the day

once I’ve accomplished it, the actual process,

especially the process of getting started in the initial,

like it really, I don’t feel like doing it.

And I really have, the way I feel about running

is the way I feel about really anything difficult

in the intellectual space, especially in mathematics,

but also just something that requires

like holding a bunch of concepts in your mind

with some uncertainty, like where the terminology

or the notation is not very clear.

And so you have to kind of hold all those things together

and like keep pushing forward through the frustration

of really like obviously not understanding certain like

parts of the picture, like your giant missing parts

of the picture and still not giving up.

It’s the same way I feel about running.

And there’s something about falling in love

with the feeling of after you went through the journey

of not having a complete picture,

at the end having a complete picture,

and then you get to appreciate the beauty

and just remembering that it sucked for a long time

and how great it felt when you figured it out,

at least at the basic.

That’s not sort of research thinking,

because with research, you probably also have to

enjoy the dead ends with learning math

from a textbook or from video.

There’s a nice.

I don’t think you have to enjoy the dead ends,

but I think you have to accept the dead ends.

Let’s put it that way.

Well, yeah, enjoy the suffering of it.

So the way I think about it, I do, there’s an.

I don’t enjoy the suffering.

It pisses me off.

You have to accept that it’s part of the process.

It’s interesting.

There’s a lot of ways to kind of deal with that dead end.

There’s a guy who’s the ultra marathon runner,

Navy SEAL, David Goggins, who kind of,

I mean, there’s a certain philosophy of like,

most people would quit here.

And so if most people would quit here and I don’t,

I’ll have an opportunity to discover something beautiful

that others haven’t yet.

And so like any feeling that really sucks,

it’s like, okay, most people would just like,

go do something smarter.

And if I stick with this,

I will discover a new garden of fruit trees that I can pick.

Okay, you say that, but like,

what about the guy who like wins

the Nathan’s hot dog eating contest every year?

Like when he eats his 35th hot dog,

he like correctly says like,

okay, most people would stop here.

Are you like lauding that he’s like,

no, I’m gonna eat the 35th hot dog.

I am, I am.

In the long arc of history, that man is onto something.

Which brings up this question.

What advice would you give to young people today,

thinking about their career, about their life,

whether it’s in mathematics, poetry,

or hot dog eating contest?

And you know, I have kids,

so this is actually a live issue for me, right?

I actually, it’s not a thought experiment.

I actually do have to give advice

to two young people all the time.

They don’t listen, but I still give it.

You know, one thing I often say to students,

I don’t think I’ve actually said this to my kids yet,

but I say it to students a lot is,

you know, you come to these decision points

and everybody is beset by self doubt, right?

It’s like, not sure like what they’re capable of,

like not sure what they really wanna do.

I always, I sort of tell people like,

often when you have a decision to make,

one of the choices is the high self esteem choice.

And I always tell them, make the high self esteem choice.

Make the choice, sort of take yourself out of it

and like, if you didn’t have those,

you can probably figure out what the version of you

that feels completely confident would do.

And do that and see what happens.

And I think that’s often like pretty good advice.

That’s interesting.

Sort of like, you know, like with Sims,

you can create characters.

Create a character of yourself

that lacks all the self doubt.

Right, but it doesn’t mean,

I would never say to somebody,

you should just go have high self esteem.

You shouldn’t have doubts.

No, you probably should have doubts.

It’s okay to have them.

But sometimes it’s good to act in the way

that the person who didn’t have them would act.

That’s a really nice way to put it.

Yeah, that’s like from a third person perspective,

take the part of your brain that wants to do big things.

What would they do?

That’s not afraid to do those things.

What would they do?

Yeah, that’s really nice.

That’s actually a really nice way to formulate it.

That’s very practical advice.

You should give it to your kids.

Do you think there’s meaning to any of it

from a mathematical perspective, this life?

If I were to ask you,

we talked about primes, talked about proving stuff.

Can we say, and then the book that God has,

that mathematics allows us to arrive

at something about in that book.

There’s certainly a chapter

on the meaning of life in that book.

Do you think we humans can get to it?

And maybe if you were to write cliff notes,

what do you suspect those cliff notes would say?

I mean, look, the way I feel is that mathematics,

as we’ve discussed, it underlies the way we think

about constructing learning machines.

It underlies physics.

It can be used.

I mean, it does all this stuff.

And also you want the meaning of life?

I mean, it’s like, we already did a lot for you.

Like, ask a rabbi.

No, I mean, I wrote a lot in the last book,

How Not to Be Wrong.

I wrote a lot about Pascal, a fascinating guy who is

a sort of very serious religious mystic,

as well as being an amazing mathematician.

And he’s well known for Pascal’s wager.

I mean, he’s probably among all mathematicians.

He’s the one who’s best known for this.

Can you actually like apply mathematics

to kind of these transcendent questions?

But what’s interesting when I really read Pascal

about what he wrote about this,

I started to see that people often think,

oh, this is him saying, I’m gonna use mathematics

to sort of show you why you should believe in God.

You know, mathematics has the answer to this question.

But he really doesn’t say that.

He almost kind of says the opposite.

If you ask Blaise Pascal, like, why do you believe in God?

He’d be like, oh, cause I met God.

You know, he had this kind of like psychedelic experience.

It’s like a mystical experience where as he tells it,

he just like directly encountered God.

It’s like, okay, I guess there’s a God, I met him last night.

So that’s it.

That’s why he believed.

It didn’t have to do with any kind.

You know, the mathematical argument was like

about certain reasons for behaving in a certain way.

But he basically said, like, look,

like math doesn’t tell you that God’s there or not.

Like, if God’s there, he’ll tell you.

You know, you don’t even.

I love this.

So you have mathematics, you have, what do you have?

Like a way to explore the mind, let’s say psychedelics.

You have like incredible technology.

You also have love and friendship.

And like, what the hell do you want to know

what the meaning of it all is?

Just enjoy it.

I don’t think there’s a better way to end it, Jordan.

This was a fascinating conversation.

I really love the way you explore math in your writing.

The willingness to be specific and clear

and actually explore difficult ideas,

but at the same time stepping outside

and figuring out beautiful stuff.

And I love the chart at the opening of your new book

that shows the chaos, the mess that is your mind.

Yes, this is what I was trying to keep in my head

all at once while I was writing.

And I probably should have drawn this picture

earlier in the process.

Maybe it would have made my organization easier.

I actually drew it only at the end.

And many of the things we talked about are on this map.

The connections are yet to be fully dissected, investigated.

And yes, God is in the picture.

Right on the edge, right on the edge, not in the center.

Thank you so much for talking to me.

It is a huge honor that you would waste

your valuable time with me.

Thank you, Lex.

We went to some amazing places today.

This was really fun.

Thanks for listening to this conversation

with Jordan Ellenberg.

And thank you to Secret Sauce, ExpressVPN, Blinkist,

and Indeed.

Check them out in the description to support this podcast.

And now let me leave you with some words from Jordan

in his book, How Not To Be Wrong.

Knowing mathematics is like wearing a pair of X ray specs

that reveal hidden structures underneath the messy

and chaotic surface of the world.

Thank you for listening and hope to see you next time.