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OUTLINE:
0:00 - Introduction
1:01 - Mathematical thinking
4:38 - Geometry
9:15 - Symmetry
19:46 - Math and science in the Soviet Union
27:26 - Topology
42:15 - Do we live in many more than 4 dimensions?
46:45 - How many holes does a straw have
56:11 - 3Blue1Brown
1:01:57 - Will AI ever win a Fields Medal?
1:10:22 - Fermat's last theorem
1:27:41 - Reality cannot be explained simply
1:33:25 - Prime numbers
1:54:54 - John Conway's Game of Life
2:06:46 - Group theory
2:10:03 - Gauge theory
2:18:05 - Grigori Perelman and the Poincare Conjecture
2:28:17 - How to learn math
2:35:26 - Advice for young people
2:37:31 - Meaning of life

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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.

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.

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 just, 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 echoes of the same kind of abstract framework that he's thinking about in terms of language that they're all like really tightly interconnected?

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, you know, uttering language or producing linguistic output.

I think it feels something like that, and it's certainly the case.

Let me put it this way, it's hard to imagine doing mathematics in a completely non-linguistic way.

It's hard to imagine doing mathematics without talking about mathematics and 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 1 thing it reminds me of is 1 thing I talk about in the book is dissection proofs, these very beautiful proofs of geometric propositions.

There's a very famous 1 by Bhaskara of the Pythagorean theorem.

Proofs where you show that 2 quantities are the same by taking the same pieces and putting them together 1 way, and making 1 shape, and putting them together another way, and making a different shape, and then observing those 2 shapes must have the same area because they were built out of the same pieces.

You know, there's a famous story, and it's a little bit disputed about how accurate this is but that in Bhaskara'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.

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 2 different formations to make 2 different things, is that language?

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 more like to me, language is a process and math is a process.

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 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.

Why is it a special field in mathematics?

And how did you fall in love with it if you have?

And certainly the experience that you describe is so typical, but there's 2 versions of it.

1 thing I say in the book is that geometry is the cilantro of math.

There's people who are like, who like you, are like, the rest of it I could take or leave, but then at this 1 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 were trying to figure out what X was.

There's some mystery you're trying to solve it.

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

I was like lost the whole year and I didn't understand why we even spent the time doing that.

But what everybody agrees on is that it's somehow different, right?

We're going to walk around in circles a little bit, but we'll get there.

You asked me how I fell in love with math.

When I was a small child, I don't know, maybe like I was 6 or 7, I don't know.

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

But in the 70s, you 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 6 by 8 array of holes.

And I was just kind of like, you know, zoning out in the living room as kids do, looking at this 6 by 8 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 6 rows of 8 holes each, but there's also 8 columns of 6 holes each.

So 8 sixes and 6 eights, it's just like the dissection proofs we were just talking about.

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.

I don't know if that's, are we FCC regulated?

So 2 different perspectives on the same physical reality.

and it's just as you say, you know, I knew that 6 times 8 was the same as 8 times 6, right?

I knew my times tables, like I knew that that was a fact.

But did I really know it until that moment?

Right, I knew that, I sort of knew that the times table was symmetric, but I didn't know why that was the case until that moment.

And in that moment, I could see like, oh, I didn't have to have somebody tell me that.

That's information that you can just directly access.

And as math teachers, that's something that we're really trying to bring to our students.

And I was 1 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, 8 times 6 is equal to 6 times 8, but in general, that whatever 2 numbers you have, you multiply them 1 way, and it's the same as if you multiply them in the other order.

It attaches it to this geometric fact about a rectangle which in some sense makes it true.

So, you know, who knows, maybe I was always fated to be an algebraic 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 these kinds of transformation on objects that once you transform them, they seem to be similar?

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

Well, it's an absolutely fundamental concept and it starts with the word symmetry in the way that we usually use it when we're just like talking English and not talking mathematics, right?

Sort of something is when we say something is symmetrical, we usually means it has what's called an axis of symmetry, maybe like the left half of it looks the same as the right half, that would be like a left right axis of symmetry, or maybe the top half looks like the bottom half, or both, right?

Maybe there's sort of a 4 fold 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 1 of my first loves in mathematics, what I thought about a lot when I was in college.

But the notion of symmetry is actually much more general than the things that we would call symmetry if we were looking at like a classical building or a painting or something like that.

You know, nowadays in math, we could use asymmetry to refer to any kind of transformation of an image or a space or an object.

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

Make it twice as big vertically and make it half as wide.

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.

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 timescale.

In math, I don't mean like yesterday, I mean, like, a century or so, up till now, is this idea that's a fundamental question of when do we consider 2 things to be the same.

That might seem like a complete triviality.

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 2 different things.

This one's over here, this one's over there.

On the other hand, if you prove a theorem about this 1, it's probably still true about this 1 if it has all the same side lanes and angles and looks exactly the same.

The term of art, if you want it, you would say they're congruent.

But 1 way of saying it is there's a symmetry called translation, which just means move everything 3 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 3 inches to the left, it would be kind of weird if all of your theorems didn't still work.

So this question of 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?

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 1 and 2?

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?

In fact, whatever the hell our brain is doing, it's really good at constructing these arbitrary and sometimes novel, which is really important when you look at like the IQ test or they feel novel ideas of symmetry of like what, like playing with objects, we're able to see things that are the same and not and construct almost like little geometric theories of what makes things the same and not and how to make programs do that in AI is a total open question.

And so I kind of stared and wonder how, what kind of symmetries are enough to solve the MNIST, handwritten digit recognition problem and write that down.

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

So in other words, like, we're just gonna keep on going into the weeds on this.

You know, a kind of symmetry that we understand very well is rotation.

If humans, if we recognized a digit as a 1, if it was like literally a rotation by some number of degrees with some fixed 1 in some typeface like Palatino or something, that would be very easy to understand, right, it would be very easy to like write a program that could detect whether something was a rotation of a fixed digit 1.

Whatever we're doing when you recognize the digit 1 and distinguish it from the digit 2, it's not that.

It's not just incorporating 1 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 2 and a 3, 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 123, 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 2 and starts being a 3?

Where you can imagine something continuously deforming from being a 2 to a 3.

Yeah, but that's, there is a moment, like I've 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 3 and a 2.

Wait, so if you ask people, if you show them this morph, if you ask a bunch of people, do they all agree about where the transition happened?

Oh my God, okay, we have an empirical dispute.

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

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, cause I mean, partially it's because our perception systems see this, it's all probably tied in there.

Somehow the change from 1 to the other, like seeing the video of it, allows you to pinpoint the place where a 2 becomes a 3 much better.

If I just showed you 1 picture, I think you might really, really struggle.

Like 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.

And in fact, again, another insight of modern geometry is this idea that, you know, maybe we would naively think we're going to study, I don't know, let's, you know, like Poincaré, we're going to study the 3 body problem.

We're going to study sort of like 3 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.

That's like a problem that Poincaré 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.

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

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

Okay, so Poincaré, he ends up being a major figure in the book, and I didn't even really intend for him to be such a big figure, but he's so, he's first and foremost a geometer, right?

So he's a mathematician who kind of comes up in late 19th century France At a time when French math is really starting to flower.

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

We got a PhD in math What 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 more like the Germans so this never happens to us again.

So It's very much, it's like the Sputnik moment, you know, like what happens in America in the 50s and 60s with the Soviet Union.

This is happening to France, and they're trying to kind of like instantly modernize.

That's fascinating that the humans and mathematics are intricately connected to the history of humans.

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's a, 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 will lead to a better world.

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

But it also created this sense that in the modern era in America, somebody like Elon Musk, whatever you think of them, like Jeff Bezos, those folks, they inspire the possibility that 1 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.

And I don't know, there's a romanticism around it too.

Like when you read books about in America, people romanticize certain things like baseball, for example, there's like these beautiful poetic writing about the game of baseball.

The same was the feeling with mathematics and science in the Soviet Union.

Everybody was forced to take high level mathematics courses.

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 at the like the, 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 what we challenge everybody, it's not just like 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 Duel at Dawn by Amir Alexander, which I think some of the things you're responding to what I wrote, I think I first got turned on to by Amir's work.

And he writes about the story of Everest Galwa, 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, well this theory of groups that I mentioned earlier, this general theory of symmetries, and then dies in a duel in his early 20s, like all this stuff, mostly unpublished.

It's a very, very romantic story that we all learn.

And much of it is true, but Alexander really lays out just how much the way people thought about math in those times in the early 19th century was wound up with, as you say, romanticism.

I mean, that's when the romantic movement takes place.

And he really outlines how people were predisposed to think about mathematics in that way because they thought about poetry that way and they thought about music that way.

It was the mood of the era to think about we're reaching for the transcendent, we're sort of reaching for sort of direct contact with the divine.

And so part of the reason that we think of Gawa that way was because Gawa himself was a creature of that era and he romanticized himself.

I mean, now we know he like wrote lots of letters and like he was kind of like, I mean, in modern terms we would say he was extremely emo.

Like that's, like just, we wrote all these letters about his like florid feelings and like the fire within him about the mathematics.

And you know, so he, so it's just as you say that the math history touches human history.

They're never separate because math is made of people.

I mean, that's what, it's people who do it and we're human beings doing it and we do it within whatever community we're in and we do it affected by the mores of the society around us.

So The French, the Germans, and Poincaré.

So, he's, you know, it's funny, this book is filled with kind of, you know, mathematical characters who often are kind of peevish or get into feuds or sort of have like weird enthusiasms, Because those people are fun to write about.

And they sort of like say very salty things.

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

And he was like, pretty personally modest.

He did math for like, 4 hours in the morning and 4 hours in the evening and that was it like he had his schedule.

I actually it was like I still am feeling like somebody's gonna tell me now that the book is out, like, oh, didn't you know about this 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, 3 astronomical objects moving in what you might think would be this very simple way.

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.

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Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries | Lex Fridman Podcast #190