Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries
Guests: Jordan Ellenberg
Full transcript · Read the full conversation in order, with its original context.
Contents20 chapters
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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,
§02Transcript · 08:15–16:33
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.
§03Transcript · 16:33–25:31
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
§04Transcript · 25:31–33:06
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.
§05Transcript · 33:06–40:33
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.
§06Transcript · 40:33–48:47
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.
§07Transcript · 48:47–56:40
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
§08Transcript · 56:40–1:05:06
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,
§09Transcript · 1:05:06–1:13:34
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,
§10Transcript · 1:13:34–1:21:41
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
§11Transcript · 1:21:41–1:29:33
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.
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,
§12Transcript · 1:29:33–1:36:46
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.
§13Transcript · 1:36:46–1:44:49
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.
§14Transcript · 1:44:50–1:52:54
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,
§15Transcript · 1:52:54–2:01:22
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
§16Transcript · 2:01:22–2:09:24
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
§17Transcript · 2:09:24–2:18:00
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.
Every once in a while, some kind of simple visualizations.
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
§18Transcript · 2:18:00–2:26:42
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.
§19Transcript · 2:26:42–2:35:00
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,
§20Transcript · 2:35:01–2:42:02
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.