Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries

00:00:00.000 | The following is a conversation with Jordan Ellenberg,

00:00:02.960 | a mathematician at University of Wisconsin,

00:00:05.560 | and an author who masterfully reveals the beauty

00:00:08.920 | and power of mathematics in his 2014 book,

00:00:12.200 | "How Not to Be Wrong," and his new book

00:00:15.000 | just released recently called "Shape,"

00:00:17.560 | the hidden geometry of information, biology, strategy,

00:00:21.160 | democracy, and everything else.

00:00:23.400 | Quick mention of our sponsors,

00:00:25.360 | Secret Sauce, ExpressVPN, Blinkist, and Indeed.

00:00:29.840 | Check them out in the description to support this podcast.

00:00:33.160 | As a side note, let me say that geometry

00:00:35.040 | is what made me fall in love with mathematics

00:00:37.440 | when I was young.

00:00:38.560 | It first showed me that something definitive

00:00:41.080 | could be stated about this world

00:00:42.960 | through intuitive visual proofs.

00:00:45.440 | Somehow that convinced me that math

00:00:47.840 | is not just abstract numbers devoid of life,

00:00:50.600 | but a part of life, part of this world,

00:00:53.400 | part of our search for meaning.

00:00:55.720 | This is the Lex Friedman Podcast,

00:00:57.680 | and here is my conversation with Jordan Ellenberg.

00:01:00.920 | If the brain is a cake--

00:01:03.920 | - It is?

00:01:05.080 | - Well, let's just go with me on this.

00:01:06.960 | - Okay, we'll pause it.

00:01:08.200 | - So for Noam Chomsky, language,

00:01:12.720 | the universal grammar, the framework

00:01:16.040 | from which language springs is like most of the cake,

00:01:20.000 | the delicious chocolate center,

00:01:21.920 | and then the rest of cognition that we think of

00:01:25.080 | is built on top, extra layers,

00:01:27.760 | maybe the icing on the cake,

00:01:29.120 | maybe consciousness is just like a cherry on top.

00:01:33.100 | Where do you put in this cake mathematical thinking?

00:01:37.980 | Is it as fundamental as language in the Chomsky view?

00:01:42.100 | Is it more fundamental than language?

00:01:44.600 | Is it echoes of the same kind of abstract framework

00:01:47.760 | that he's thinking about in terms of language

00:01:49.480 | that they're all really tightly interconnected?

00:01:53.000 | - That's a really interesting question.

00:01:54.880 | You're getting me to reflect on this question

00:01:56.560 | of whether the feeling of producing mathematical output,

00:02:00.800 | if you want, is like the process of uttering language

00:02:04.400 | or producing linguistic output.

00:02:05.960 | I think it feels something like that,

00:02:09.000 | and it's certainly the case.

00:02:10.640 | Let me put it this way.

00:02:11.480 | It's hard to imagine doing mathematics

00:02:14.480 | in a completely non-linguistic way.

00:02:17.400 | It's hard to imagine doing mathematics

00:02:19.680 | without talking about mathematics

00:02:22.040 | and sort of thinking in propositions.

00:02:23.800 | But maybe it's just because that's the way I do mathematics

00:02:26.760 | and maybe I can't imagine it any other way.

00:02:29.560 | - Well, what about visualizing shapes,

00:02:32.720 | visualizing concepts to which language

00:02:35.640 | is not obviously attachable?

00:02:38.200 | - Ah, that's a really interesting question.

00:02:40.080 | And one thing it reminds me of

00:02:41.800 | is one thing I talk about in the book is dissection proofs,

00:02:45.760 | these very beautiful proofs of geometric propositions.

00:02:48.720 | There's a very famous one by Bhaskara

00:02:50.480 | of the Pythagorean theorem.

00:02:53.260 | Proofs which are purely visual,

00:02:56.540 | proofs where you show that two quantities are the same

00:03:00.220 | by taking the same pieces and putting them together one way

00:03:04.340 | and making one shape and putting them together another way

00:03:07.380 | and making a different shape.

00:03:08.460 | And then observing that those two shapes

00:03:09.700 | must have the same area

00:03:10.620 | because they were built out of the same pieces.

00:03:13.100 | There's a famous story,

00:03:16.060 | and it's a little bit disputed about how accurate this is,

00:03:19.260 | but that in Bhaskara's manuscript,

00:03:20.660 | he sort of gives this proof, just gives the diagram,

00:03:22.620 | and then the entire verbal content of the proof

00:03:26.180 | is he just writes under it, "Behold!"

00:03:28.420 | Like that's it.

00:03:29.260 | (laughing)

00:03:31.340 | There's some dispute about exactly how accurate that is.

00:03:33.660 | But so then there's an interesting question.

00:03:36.500 | If your proof is a diagram, if your proof is a picture,

00:03:39.940 | or even if your proof is like a movie of the same pieces

00:03:42.260 | like coming together in two different formations

00:03:43.940 | to make two different things, is that language?

00:03:45.780 | I'm not sure I have a good answer.

00:03:46.620 | What do you think?

00:03:47.700 | - I think it is.

00:03:48.980 | I think the process of manipulating the visual elements

00:03:53.980 | is the same as the process of manipulating

00:03:58.100 | the elements of language.

00:03:59.500 | And I think probably the manipulating, the aggregation,

00:04:02.980 | the stitching stuff together is the important part.

00:04:05.820 | It's not the actual specific elements.

00:04:07.660 | It's more like to me, language is a process

00:04:10.540 | and math is a process.

00:04:11.780 | It's not just specific symbols.

00:04:15.020 | It's in action.

00:04:19.100 | It's ultimately created through action, through change.

00:04:23.700 | And so you're constantly evolving ideas.

00:04:26.180 | Of course, we kind of attach,

00:04:27.860 | there's a certain destination you arrive to

00:04:29.740 | that you attach to and you call that a proof.

00:04:32.080 | But that doesn't need to end there.

00:04:34.580 | It's just at the end of the chapter

00:04:35.980 | and then it goes on and on and on in that kind of way.

00:04:39.140 | But I gotta ask you about geometry

00:04:40.700 | and it's a prominent topic in your new book, "Shape."

00:04:44.860 | - So for me, geometry is the thing,

00:04:48.020 | just like as you're saying,

00:04:49.580 | made me fall in love with mathematics when I was young.

00:04:52.340 | So being able to prove something visually

00:04:55.740 | just did something to my brain that it had this,

00:05:01.000 | it planted this hopeful seed

00:05:02.580 | that you can understand the world like perfectly.

00:05:07.320 | Maybe it's an OCD thing,

00:05:08.580 | but from a mathematics perspective,

00:05:10.460 | like humans are messy, the world is messy,

00:05:12.700 | biology is messy, your parents are yelling

00:05:15.780 | or making you do stuff, but you can cut through all that BS

00:05:19.580 | and truly understand the world through mathematics

00:05:21.980 | and nothing like geometry did that for me.

00:05:25.300 | For you, you did not immediately

00:05:27.620 | fall in love with geometry.

00:05:29.100 | So how do you think about geometry?

00:05:33.900 | Why is it a special field in mathematics?

00:05:36.740 | And how did you fall in love with it if you have?

00:05:39.920 | - Wow, you've given me like a lot to say

00:05:41.780 | and certainly the experience that you describe

00:05:44.340 | is so typical, but there's two versions of it.

00:05:47.480 | One thing I say in the book

00:05:49.260 | is that geometry is the cilantro of math.

00:05:51.740 | People are not neutral about it.

00:05:52.900 | There's people who like you are like,

00:05:55.580 | the rest of it I could take or leave,

00:05:56.900 | but then at this one moment, it made sense.

00:05:59.240 | This class made sense, why wasn't it all like that?

00:06:01.420 | There's other people, I can tell you,

00:06:02.900 | 'cause they come and talk to me all the time,

00:06:04.780 | who are like, I understood all the stuff

00:06:06.780 | where you're trying to figure out what X was,

00:06:08.180 | there's some mystery, you're trying to solve it,

00:06:09.500 | X is a number, I figured it out.

00:06:10.700 | But then there was this geometry, like what was that?

00:06:12.980 | What happened that year?

00:06:13.820 | Like I didn't get it, I was like lost the whole year

00:06:15.540 | and I didn't understand like why we even spent

00:06:17.340 | the time doing that.

00:06:18.300 | But what everybody agrees on is that it's somehow different.

00:06:22.220 | Right, there's something special about it.

00:06:24.260 | We're gonna walk around in circles a little bit,

00:06:27.140 | but we'll get there.

00:06:27.980 | You asked me how I fell in love with math.

00:06:32.500 | I have a story about this.

00:06:33.800 | When I was a small child, I don't know,

00:06:39.060 | maybe like I was six or seven, I don't know.

00:06:42.060 | I'm from the '70s, I think you're from a different decade

00:06:44.180 | than that, but in the '70s, you had a cool wooden box

00:06:48.580 | around your stereo, that was the look,

00:06:50.180 | everything was dark wood.

00:06:51.500 | And the box had a bunch of holes in it

00:06:53.580 | to let the sound out.

00:06:54.620 | And the holes were in this rectangular array,

00:06:58.980 | a six by eight array of holes.

00:07:02.540 | And I was just kind of zoning out in the living room

00:07:04.880 | as kids do, looking at this six by eight rectangular array

00:07:08.700 | of holes.

00:07:09.820 | And if you like, just by kind of focusing in and out,

00:07:12.740 | just by kind of looking at this box,

00:07:14.500 | looking at this rectangle, I was like,

00:07:17.100 | well, there's six rows of eight holes each,

00:07:20.900 | but there's also eight columns of six holes each.

00:07:25.660 | - Whoa.

00:07:26.500 | - So eight sixes and six eights.

00:07:29.060 | It's just like the dissection proofs

00:07:30.100 | we were just talking about.

00:07:30.980 | But it's the same holes.

00:07:32.340 | It's the same 48 holes, that's how many there are,

00:07:34.380 | no matter whether you count them as rows

00:07:36.820 | or count them as columns.

00:07:38.060 | And this was like unbelievable to me.

00:07:40.940 | Am I allowed to cuss on your podcast?

00:07:43.020 | I don't know if that's, are we FCC regulated?

00:07:45.220 | Okay, it was fucking unbelievable.

00:07:47.060 | Okay, that's the last time.

00:07:48.020 | - Get it in there.

00:07:48.860 | - This story merits it.

00:07:49.680 | - So two different perspectives

00:07:51.460 | on the same physical reality.

00:07:54.220 | - Exactly.

00:07:55.260 | And it's just as you say,

00:07:56.820 | I knew the six times eight was the same as eight times six.

00:08:01.460 | I knew my times tables.

00:08:02.620 | I knew that that was a fact.

00:08:04.740 | But did I really know it until that moment?

00:08:06.620 | That's the question.

00:08:08.380 | Right, I knew that,

00:08:09.220 | I sort of knew that the times table was symmetric,

00:08:11.620 | but I didn't know why that was the case until that moment.

00:08:13.940 | And in that moment, I could see like,

00:08:15.140 | oh, I didn't have to have somebody tell me that.

00:08:17.660 | That's information that you can just directly access.

00:08:20.160 | That's a really amazing moment.

00:08:21.540 | And as math teachers,

00:08:22.460 | that's something that we're really trying

00:08:23.540 | to bring to our students.

00:08:25.540 | And I was one of those who did not love

00:08:27.500 | the kind of Euclidean geometry,

00:08:29.300 | ninth grade class of like,

00:08:31.260 | prove that an isosceles triangle

00:08:33.020 | has equal angles at the base,

00:08:34.700 | like this kind of thing.

00:08:35.540 | It didn't vibe with me

00:08:36.360 | the way that algebra and numbers did.

00:08:39.020 | But if you go back to that moment,

00:08:40.720 | from my adult perspective,

00:08:41.860 | looking back at what happened with that rectangle,

00:08:43.620 | I think that is a very geometric moment.

00:08:45.460 | In fact, that moment exactly encapsulates

00:08:49.060 | the intertwining of algebra and geometry.

00:08:53.060 | This algebraic fact that,

00:08:54.900 | well, in the instance,

00:08:55.900 | eight times six is equal to six times eight,

00:08:57.880 | but in general, that whatever two numbers you have,

00:09:00.180 | you multiply them one way,

00:09:01.300 | and it's the same as if you multiply them

00:09:02.780 | in the other order.

00:09:04.780 | It attaches it to this geometric fact

00:09:06.760 | about a rectangle,

00:09:07.600 | which in some sense makes it true.

00:09:09.520 | So, who knows, maybe I was always fated

00:09:11.360 | to be an algebraic geometer,

00:09:12.360 | which is what I am as a researcher.

00:09:15.040 | - So, that's the kind of transformation,

00:09:17.320 | and you talk about symmetry in your book.

00:09:19.360 | What the heck is symmetry?

00:09:22.000 | What the heck is these kinds of transformation

00:09:25.600 | on objects that once you transform them,

00:09:27.800 | they seem to be similar?

00:09:29.440 | What do you make of it?

00:09:30.460 | What's its use in mathematics,

00:09:32.160 | or maybe broadly in understanding our world?

00:09:35.300 | - Well, it's an absolutely fundamental concept,

00:09:37.340 | and it starts with the word symmetry

00:09:39.720 | in the way that we usually use it

00:09:41.180 | when we're just like talking English

00:09:42.600 | and not talking mathematics, right?

00:09:43.760 | Sort of something is,

00:09:44.680 | when we say something is symmetrical,

00:09:46.920 | we usually means it has what's called

00:09:48.640 | an axis of symmetry.

00:09:49.780 | Maybe like the left half of it

00:09:51.200 | looks the same as the right half.

00:09:52.520 | That would be like a left-right axis of symmetry,

00:09:54.960 | or maybe the top half looks like the bottom half,

00:09:57.020 | or both, right?

00:09:57.860 | Maybe there's sort of a fourfold symmetry

00:09:59.280 | where the top looks like the bottom

00:10:00.120 | and the left looks like the right, or more.

00:10:03.600 | And that can take you in a lot of different directions.

00:10:06.340 | The abstract study of what the possible

00:10:08.800 | combinations of symmetries there are,

00:10:10.440 | a subject which is called group theory,

00:10:11.800 | was actually one of my first loves in mathematics,

00:10:14.880 | what I thought about a lot when I was in college.

00:10:17.560 | But the notion of symmetry is actually much more general

00:10:21.280 | than the things that we would call symmetry

00:10:23.060 | if we were looking at like a classical building

00:10:25.440 | or a painting or something like that.

00:10:29.840 | You know, nowadays in math,

00:10:33.260 | we could use a symmetry to refer to

00:10:38.680 | any kind of transformation of an image

00:10:41.540 | or a space or an object.

00:10:43.680 | You know, so what I talk about in the book

00:10:45.840 | is take a figure and stretch it vertically,

00:10:50.560 | make it twice as big vertically,

00:10:53.920 | and make it half as wide.

00:10:58.880 | That I would call a symmetry.

00:11:00.060 | It's not a symmetry in the classical sense,

00:11:03.280 | but it's a well-defined transformation

00:11:05.700 | that has an input and an output.

00:11:07.600 | I give you some shape, and it gets kind of,

00:11:10.920 | I call this in the book a scrunch.

00:11:12.160 | I just had to make up some sort of funny sounding name for it

00:11:14.640 | 'cause it doesn't really have a name.

00:11:18.300 | And just as you can sort of study

00:11:21.700 | which kinds of objects are symmetrical

00:11:23.760 | under the operations of switching left and right

00:11:25.960 | or switching top and bottom

00:11:27.520 | or rotating 40 degrees or what have you,

00:11:29.860 | you could study what kinds of things are preserved

00:11:33.880 | by this kind of scrunch symmetry.

00:11:36.320 | And this kind of more general idea

00:11:39.480 | of what a symmetry can be,

00:11:40.880 | let me put it this way.

00:11:43.680 | A fundamental mathematical idea,

00:11:47.240 | in some sense, I might even say the idea

00:11:48.980 | that dominates contemporary mathematics.

00:11:51.320 | Or by contemporary, by the way,

00:11:52.320 | I mean like the last 150 years.

00:11:54.220 | We're on a very long timescale in math.

00:11:56.180 | I don't mean like yesterday.

00:11:57.020 | I mean like a century or so up till now.

00:12:00.920 | Is this idea that it's a fundamental question

00:12:02.560 | of when do we consider two things to be the same.

00:12:07.100 | That might seem like a complete triviality.

00:12:08.700 | It's not.

00:12:10.000 | For instance, if I have a triangle,

00:12:12.180 | and I have a triangle of the exact same dimensions,

00:12:14.920 | but it's over here, are those the same or different?

00:12:19.300 | Well, you might say, well, look, there's two different things.

00:12:20.940 | This one's over here, this one's over there.

00:12:22.380 | On the other hand, if you prove a theorem about this one,

00:12:25.740 | it's probably still true about this one

00:12:27.660 | if it has like all the same side lanes and angles

00:12:29.800 | and like looks exactly the same.

00:12:31.420 | The term of art, if you want it,

00:12:32.620 | you would say they're congruent.

00:12:34.700 | But one way of saying it is there's a symmetry

00:12:36.780 | called translation, which just means

00:12:38.580 | move everything three inches to the left.

00:12:40.740 | And we want all of our theories

00:12:43.060 | to be translation invariant.

00:12:45.420 | What that means is that if you prove a theorem

00:12:46.900 | about a thing that's over here,

00:12:48.840 | and then you move it three inches to the left,

00:12:51.600 | it would be kind of weird if all of your theorems

00:12:53.420 | like didn't still work.

00:12:55.780 | So this question of like, what are the symmetries

00:12:58.580 | and which things that you want to study

00:12:59.900 | are invariant under those symmetries

00:13:01.420 | is absolutely fundamental.

00:13:02.340 | Boy, this is getting a little abstract, right?

00:13:03.980 | - It's not at all abstract.

00:13:05.120 | I think this is completely central

00:13:08.340 | to everything I think about

00:13:09.740 | in terms of artificial intelligence.

00:13:11.340 | I don't know if you know about the MNIST dataset,

00:13:13.500 | what's handwritten digits.

00:13:14.980 | - Yeah.

00:13:16.240 | - And, you know, I don't smoke much weed or any really,

00:13:21.700 | but it certainly feels like it when I look at MNIST

00:13:24.220 | and think about this stuff, which is like,

00:13:26.380 | what's the difference between one and two?

00:13:28.620 | And why are all the twos similar to each other?

00:13:32.180 | What kind of transformations are within the category

00:13:37.180 | of what makes a thing the same?

00:13:39.260 | And what kind of transformations

00:13:40.720 | are those that make it different?

00:13:42.540 | And symmetry is core to that.

00:13:44.020 | In fact, whatever the hell our brain is doing,

00:13:46.740 | it's really good at constructing these arbitrary

00:13:50.420 | and sometimes novel, which is really important

00:13:53.120 | when you look at like the IQ test,

00:13:55.420 | or they feel novel ideas of symmetry of like,

00:13:59.420 | what like playing with objects,

00:14:02.860 | we're able to see things that are the same and not,

00:14:06.980 | and construct almost like little geometric theories

00:14:11.660 | of what makes things the same and not,

00:14:13.380 | and how to make programs do that in AI

00:14:17.340 | is a total open question.

00:14:19.060 | And so I kind of stared and wonder

00:14:21.820 | how, what kind of symmetries are enough

00:14:25.940 | to solve the MNIST handwritten digit recognition problem

00:14:30.840 | and write that down?

00:14:32.260 | - Exactly, and what's so fascinating

00:14:33.820 | about the work in that direction,

00:14:35.420 | from the point of view of a mathematician like me

00:14:38.260 | and a geometer, is that the kind of groups of symmetries,

00:14:42.500 | the types of symmetries that we know of

00:14:44.540 | are not sufficient, right?

00:14:45.900 | So in other words, like,

00:14:47.540 | we're just gonna keep on going into the weeds on this.

00:14:50.240 | - Let's go.

00:14:51.260 | The deeper, the better.

00:14:52.460 | - You know, a kind of symmetry

00:14:54.420 | that we understand very well is rotation, right?

00:14:56.700 | So here's what would be easy.

00:14:57.900 | If humans, if we recognized a digit as a one,

00:15:01.940 | if it was like literally a rotation

00:15:03.620 | by some number of degrees,

00:15:05.020 | with some fixed one in some typeface,

00:15:08.100 | like Palatino or something,

00:15:10.420 | that would be very easy to understand, right?

00:15:12.060 | It would be very easy to like write a program

00:15:13.900 | that could detect whether something was a rotation

00:15:17.340 | of a fixed digit one.

00:15:20.660 | Whatever we're doing when you recognize the digit one

00:15:22.620 | and distinguish it from the digit two, it's not that.

00:15:25.900 | It's not just incorporating

00:15:27.660 | one of the types of symmetries that we understand.

00:15:32.100 | Now, I would say that I would be shocked

00:15:36.620 | if there was some kind of classical symmetry type formulation

00:15:40.620 | that captured what we're doing

00:15:43.320 | when we tell the difference between a two and a three,

00:15:45.580 | to be honest.

00:15:46.420 | I think what we're doing is actually more complicated

00:15:49.940 | than that, I feel like it must be.

00:15:52.260 | - They're so simple, these numbers.

00:15:53.660 | I mean, they're really geometric objects.

00:15:55.820 | Like we can draw a one, two, three.

00:15:58.580 | It does seem like it should be formalizable.

00:16:01.140 | That's why it's so strange.

00:16:03.180 | - Do you think it's formalizable

00:16:04.180 | when something stops being a two and starts being a three,

00:16:06.820 | where you can imagine something continuously deforming

00:16:08.980 | from being a two to a three?

00:16:11.100 | - Yeah, but that's, there is a moment.

00:16:15.420 | I have myself written programs that literally morph

00:16:18.700 | twos and threes and so on.

00:16:20.740 | And you watch, and there is moments that you notice,

00:16:23.920 | depending on the trajectory of that transformation,

00:16:26.880 | that morphing, that it is a three and a two.

00:16:31.880 | There's a hard line.

00:16:33.580 | - Wait, so if you ask people, if you show them this morph,

00:16:36.340 | if you ask a bunch of people,

00:16:37.340 | do they all agree about where the transition happened?

00:16:39.340 | - That's an interesting question.

00:16:40.180 | - 'Cause I would be surprised.

00:16:41.000 | - I think so.

00:16:41.840 | - Oh my God, okay, we have an empirical dispute.

00:16:42.900 | - But here's the problem.

00:16:44.620 | Here's the problem, that if I just showed that moment

00:16:48.260 | that I agreed on.

00:16:50.940 | - That's not fair.

00:16:51.780 | - No, but say I said, so I wanna move away

00:16:54.500 | from the agreement 'cause that's a fascinating,

00:16:56.460 | actually, question that I wanna backtrack from

00:16:59.460 | because I just dogmatically said,

00:17:03.140 | 'cause I could be very, very wrong.

00:17:04.860 | But the morphing really helps, that the change,

00:17:09.860 | 'cause I mean, partially it's because

00:17:11.340 | our perception systems, see,

00:17:13.340 | it's all probably tied in there.

00:17:15.060 | Somehow the change from one to the other,

00:17:18.020 | like seeing the video of it, allows you to pinpoint

00:17:20.980 | the place where a two becomes a three much better.

00:17:23.660 | If I just showed you one picture,

00:17:25.980 | I think you might really, really struggle.

00:17:30.980 | You might call it a seven.

00:17:32.220 | (laughs)

00:17:34.060 | I think there's something also

00:17:37.380 | that we don't often think about,

00:17:38.940 | which is it's not just about the static image,

00:17:41.580 | it's the transformation of the image,

00:17:43.980 | or it's not a static shape,

00:17:45.540 | it's the transformation of the shape.

00:17:47.580 | There's something in the movement that seems to be

00:17:51.540 | not just about our perception system,

00:17:53.300 | but fundamental to our cognition,

00:17:55.060 | like how we think about stuff.

00:17:57.700 | - Yeah, and that's part of geometry too.

00:18:00.340 | And in fact, again, another insight of modern geometry

00:18:03.220 | is this idea that maybe we would naively think

00:18:06.060 | we're gonna study, I don't know, like Poincaré,

00:18:09.020 | we're gonna study the three-body problem.

00:18:10.420 | We're gonna study three objects in space

00:18:13.460 | moving around subject only to the force

00:18:15.260 | of each other's gravity, which sounds very simple, right?

00:18:17.620 | And if you don't know about this problem,

00:18:18.740 | you're probably like, okay,

00:18:19.580 | so you just put it in your computer and see what they do.

00:18:21.260 | Well, guess what?

00:18:22.100 | That's a problem that Poincaré won a huge prize for,

00:18:25.260 | making the first real progress on in the 1880s,

00:18:27.420 | and we still don't know that much about it 150 years later.

00:18:32.420 | It's a humongous mystery.

00:18:34.860 | - You just open the door, and we're gonna walk right in

00:18:38.060 | before we return to symmetry.

00:18:40.460 | Who's Poincaré, and what's this conjecture

00:18:44.900 | that he came up with?

00:18:45.940 | Why is it such a hard problem?

00:18:48.620 | - Okay, so Poincaré, he ends up being a major figure

00:18:52.180 | in the book, and I didn't even really intend for him

00:18:54.220 | to be such a big figure, but he's first and foremost

00:18:59.220 | a geometer, right?

00:19:00.140 | So he's a mathematician who kind of comes up

00:19:02.620 | in late 19th century France at a time

00:19:07.140 | when French math is really starting to flower.

00:19:09.380 | Actually, I learned a lot.

00:19:10.220 | I mean, in math, we're not really trained

00:19:11.660 | on our own history.

00:19:12.700 | We get a PhD in math, learn about math.

00:19:14.300 | So I learned a lot.

00:19:15.220 | There's this whole kind of moment where France

00:19:18.620 | has just been beaten in the Franco-Prussian War,

00:19:22.080 | and they're like, oh my God, what did we do wrong?

00:19:23.860 | And they were like, we gotta get strong in math

00:19:26.460 | like the Germans.

00:19:27.280 | We have to be more like the Germans,

00:19:28.460 | so this never happens to us again.

00:19:29.880 | So it's very much, it's like the Sputnik moment,

00:19:31.780 | you know, like what happens in America in the '50s and '60s

00:19:34.620 | with the Soviet Union.

00:19:35.460 | This is happening to France,

00:19:36.740 | and they're trying to kind of like

00:19:38.580 | instantly like modernize.

00:19:40.380 | - That's fascinating that the humans and mathematics

00:19:43.100 | are intricately connected to the history of humans.

00:19:46.780 | The Cold War is, I think, fundamental

00:19:50.940 | to the way people saw science and math in the Soviet Union.

00:19:55.180 | I don't know if that was true in the United States,

00:19:56.740 | but certainly it was in the Soviet Union.

00:19:58.540 | - It definitely was, and I would love to hear more

00:19:59.980 | about how it was in the Soviet Union.

00:20:01.580 | - I mean, there was, and we'll talk about the Olympiad.

00:20:04.940 | I just remember that there was this feeling

00:20:08.700 | like the world hung in a balance

00:20:14.300 | and you could save the world with the tools of science,

00:20:19.300 | and mathematics was like the superpower that fuels science.

00:20:25.060 | And so like people were seen as, you know,

00:20:30.220 | people in America often idolize athletes,

00:20:32.900 | but ultimately the best athletes in the world,

00:20:35.700 | they just throw a ball into a basket.

00:20:40.020 | So like there's not, what people really enjoy about sports,

00:20:44.260 | I love sports, is like excellence at the highest level.

00:20:48.660 | But when you take that with mathematics and science,

00:20:51.300 | people also enjoyed excellence in science and mathematics

00:20:54.300 | in the Soviet Union, but there's an extra sense

00:20:56.860 | that that excellence will lead to a better world.

00:21:01.380 | So that created all the usual things you think about

00:21:06.380 | with the Olympics,

00:21:08.260 | which is like extreme competitiveness, right?

00:21:12.180 | But it also created this sense that in the modern era

00:21:15.140 | in America, somebody like Elon Musk,

00:21:18.140 | whatever you think of him, like Jeff Bezos, those folks,

00:21:21.460 | they inspire the possibility that one person

00:21:24.500 | or a group of smart people can change the world.

00:21:27.040 | Like not just be good at what they do,

00:21:29.060 | but actually change the world.

00:21:30.660 | Mathematics was at the core of that.

00:21:32.500 | I don't know, there's a romanticism around it too.

00:21:36.040 | Like when you read books about in America,

00:21:39.480 | people romanticize certain things like baseball,

00:21:41.700 | for example, there's like these beautiful poetic writing

00:21:45.680 | about the game of baseball.

00:21:47.420 | The same was the feeling with mathematics and science

00:21:50.620 | in the Soviet Union, and it was in the air.

00:21:53.180 | Everybody was forced to take high level mathematics courses.

00:21:57.260 | Like you took a lot of math, you took a lot of science

00:22:00.480 | and a lot of like really rigorous literature.

00:22:03.260 | Like the level of education in Russia,

00:22:06.580 | this could be true in China, I'm not sure,

00:22:09.180 | in a lot of countries is in whatever that's called,

00:22:14.100 | it's K to 12 in America, but like young people education,

00:22:18.760 | the level they were challenged to learn at is incredible.

00:22:23.340 | It's like America falls far behind, I would say.

00:22:27.980 | America then quickly catches up

00:22:29.900 | and then exceeds everybody else at the like the,

00:22:32.660 | as you start approaching the end of high school to college,

00:22:35.360 | like the university system in the United States

00:22:37.040 | arguably is the best in the world.

00:22:39.280 | But like what we challenge everybody,

00:22:44.180 | it's not just like the A students,

00:22:46.560 | but everybody to learn in the Soviet Union was fascinating.

00:22:50.200 | - I think I'm gonna pick up on something you said.

00:22:52.060 | I think you would love a book called

00:22:53.800 | "Duel at Dawn" by Amir Alexander,

00:22:56.360 | which I think some of the things you're responding to

00:22:58.720 | what I wrote, I think I first got turned on to

00:23:01.020 | by Amir's work, he's a historian of math.

00:23:02.880 | And he writes about the story of Everest Galois,

00:23:06.040 | which is a story that's well known to all mathematicians,

00:23:08.280 | this kind of like very, very romantic figure

00:23:12.880 | who he really sort of like begins the development

00:23:17.360 | of this, well, this theory of groups

00:23:18.980 | that I mentioned earlier,

00:23:20.100 | this general theory of symmetries

00:23:22.360 | and then dies in a duel in his early 20s,

00:23:25.520 | like all this stuff, mostly unpublished.

00:23:28.400 | It's a very, very romantic story that we all learn.

00:23:31.100 | And much of it is true, but Alexander really lays out

00:23:36.880 | just how much the way people thought about math

00:23:39.480 | in those times in the early 19th century

00:23:42.400 | was wound up with, as you say, romanticism.

00:23:44.480 | I mean, that's when the romantic movement takes place.

00:23:47.160 | And he really outlines how people were predisposed

00:23:51.200 | to think about mathematics in that way,

00:23:52.800 | because they thought about poetry that way.

00:23:54.240 | And they thought about music that way.

00:23:55.720 | It was the mood of the era to think about,

00:23:58.280 | we're reaching for the transcendent,

00:23:59.960 | we're sort of reaching for sort of direct contact

00:24:02.040 | with the divine.

00:24:02.880 | And so part of the reason that we think of Galois that way

00:24:06.120 | was because Galois himself was a creature of that era

00:24:08.760 | and he romanticized himself.

00:24:10.660 | I mean, now we know he like wrote lots of letters

00:24:12.720 | and like he was kind of like, I mean, in modern terms,

00:24:14.960 | we would say he was extremely emo.

00:24:16.600 | Like that's, like just, we wrote all these letters

00:24:19.840 | about his like florid feelings and like the fire within him

00:24:22.560 | about the mathematics.

00:24:23.400 | You know, so he, so it's just as you say

00:24:26.320 | that the math history touches human history.

00:24:29.660 | They're never separate because math is made of people.

00:24:32.760 | - Yeah.

00:24:33.600 | - I mean, that's what it's, it's people who do it

00:24:35.580 | and we're human beings doing it.

00:24:36.860 | And we do it within whatever community we're in

00:24:39.160 | and we do it affected by the mores

00:24:42.680 | of the society around us.

00:24:44.120 | - So the French, the Germans and Poincaré.

00:24:47.400 | - Yes, okay, so back to Poincaré.

00:24:48.920 | So he's, you know, it's funny.

00:24:52.560 | This book is filled with kind of, you know,

00:24:54.760 | mathematical characters who often are kind of peevish

00:24:58.920 | or get into feuds or sort of have like weird enthusiasms

00:25:03.040 | 'cause those people are fun to write about

00:25:05.160 | and they sort of like say very salty things.

00:25:07.480 | Poincaré is actually none of this.

00:25:09.560 | As far as I can tell, he was an extremely normal dude.

00:25:13.480 | He didn't get into fights with people

00:25:15.240 | and everybody liked him

00:25:16.320 | and he was like pretty personally modest

00:25:18.080 | and he had very regular habits, you know what I mean?

00:25:20.960 | He did math for like four hours in the morning

00:25:23.760 | and four hours in the evening and that was it.

00:25:25.680 | Like he had his schedule.

00:25:28.240 | I actually, it was like, I still am feeling like

00:25:31.680 | somebody's gonna tell me now that the book is out,

00:25:33.400 | like, oh, didn't you know about this?

00:25:34.760 | Like incredibly sordid episode of this.

00:25:37.040 | As far as I could tell, a completely normal guy.

00:25:39.960 | But he just kind of in many ways creates

00:25:44.960 | the geometric world in which we live

00:25:47.960 | and his first really big success is this prize paper

00:25:52.960 | he writes for this prize offered by the King of Sweden

00:25:56.120 | for the study of the three-body problem.

00:25:59.560 | The study of what we can say about, yeah,

00:26:04.280 | three astronomical objects moving

00:26:07.320 | in what you might think would be this very simple way.

00:26:09.120 | Nothing's going on except gravity.

00:26:11.520 | - So what's the three-body problem?

00:26:13.680 | Why is that a problem?

00:26:15.040 | - So the problem is to understand

00:26:16.840 | when this motion is stable and when it's not.

00:26:20.040 | So stable meaning they would sort of like end up

00:26:21.880 | in some kind of periodic orbit.

00:26:23.680 | Or I guess it would mean, sorry,

00:26:25.440 | stable would mean they never sort of fly off

00:26:26.960 | far apart from each other.

00:26:28.080 | And unstable would mean like eventually they fly apart.

00:26:30.160 | - So understanding two bodies is much easier.

00:26:32.880 | - Yes, exactly.

00:26:33.720 | - When you have the third wheel is always a problem.

00:26:36.560 | - This is what Newton knew.

00:26:37.400 | Two bodies, they sort of orbit each other

00:26:38.760 | in some kind of, either in an ellipse,

00:26:41.320 | which is the stable case.

00:26:42.400 | That's what the planets do that we know.

00:26:45.200 | Or one travels on a hyperbola around the other.

00:26:49.360 | That's the unstable case.

00:26:50.320 | It sort of like zooms in from far away,

00:26:51.920 | sort of like whips around the heavier thing

00:26:54.280 | and like zooms out.

00:26:56.720 | Those are basically the two options.

00:26:58.120 | So it's a very simple and easy to classify story.

00:27:00.840 | With three bodies, just a small switch from two to three,

00:27:04.160 | it's a complete zoo.

00:27:05.200 | It's the first, what we would say now

00:27:07.000 | is it's the first example of what's called chaotic dynamics

00:27:09.920 | where the stable solutions and the unstable solutions,

00:27:13.040 | they're kind of like wound in among each other.

00:27:14.520 | And a very, very, very tiny change

00:27:16.720 | in the initial conditions can make the long-term behavior

00:27:19.400 | of the system completely different.

00:27:21.240 | So Poincaré was the first to recognize

00:27:23.000 | that that phenomenon even existed.

00:27:27.040 | - What about the conjecture that carries his name?

00:27:31.180 | - Right, so he also was one of the pioneers

00:27:36.920 | of taking geometry, which until that point

00:27:41.440 | had been largely the study of two

00:27:44.120 | and three-dimensional objects

00:27:45.240 | 'cause that's like what we see, right?

00:27:47.480 | That's the objects we interact with.

00:27:49.280 | He developed a subject we now called topology.

00:27:53.600 | He called it Analysis Citus.

00:27:55.360 | He was a very well-spoken guy with a lot of slogans,

00:27:57.880 | but that name did not, you can see why that name

00:28:00.320 | did not catch on.

00:28:01.160 | So now it's called topology now.

00:28:03.360 | - Sorry, what was it called before?

00:28:06.320 | - Analysis Citus, which I guess sort of roughly means

00:28:09.400 | like the analysis of location or something like that.

00:28:12.000 | It's a Latin phrase.

00:28:14.200 | Partly because he understood that even to understand stuff

00:28:20.080 | that's going on in our physical world,

00:28:22.440 | you have to study higher-dimensional spaces.

00:28:24.360 | How does this work?

00:28:25.480 | And this is kind of like where my brain went to it

00:28:27.440 | because you were talking about not just where things are,

00:28:29.840 | but what their path is, how they're moving

00:28:31.720 | when we were talking about the path from two to three.

00:28:34.800 | He understood that if you want to study three bodies

00:28:37.600 | moving in space, well, each body, it has a location

00:28:42.600 | where it is, so it has an X coordinate, a Y coordinate,

00:28:45.920 | a Z coordinate, right?

00:28:46.760 | I can specify a point in space by giving you three numbers,

00:28:49.400 | but it also, at each moment, has a velocity.

00:28:52.080 | So it turns out that really to understand what's going on,

00:28:56.480 | you can't think of it as a point, or you could,

00:28:58.880 | but it's better not to think of it as a point

00:29:01.000 | in three-dimensional space that's moving.

00:29:03.240 | It's better to think of it as a point

00:29:04.400 | in six-dimensional space where the coordinates are

00:29:06.520 | where is it and what's its velocity right now.

00:29:09.320 | That's a higher-dimensional space called phase space.

00:29:11.760 | And if you haven't thought about this before,

00:29:13.180 | I admit that it's a little bit mind-bending,

00:29:15.920 | but what he needed then was a geometry

00:29:20.720 | that was flexible enough, not just to talk about

00:29:23.600 | two-dimensional spaces or three-dimensional spaces,

00:29:25.560 | but any dimensional space.

00:29:27.480 | So the sort of famous first line of this paper

00:29:29.320 | where he introduces analysis situs

00:29:30.800 | is no one doubts nowadays that the geometry

00:29:34.280 | of n-dimensional space is an actually existing thing.

00:29:37.760 | I think maybe that had been controversial.

00:29:39.640 | He's saying, "Look, let's face it.

00:29:41.320 | "Just because it's not physical

00:29:42.940 | "doesn't mean it's not there.

00:29:43.980 | "It doesn't mean we shouldn't study it."

00:29:46.080 | - Interesting.

00:29:46.920 | He wasn't jumping to the physical interpretation.

00:29:50.600 | It can be real even if it's not perceivable

00:29:53.640 | to the human cognition.

00:29:55.760 | - I think that's right.

00:29:56.880 | I think, don't get me wrong,

00:29:58.400 | Poincaré never strays far from physics.

00:30:00.300 | He's always motivated by physics,

00:30:02.120 | but the physics drove him to need to think about

00:30:05.720 | spaces of higher dimension,

00:30:07.240 | and so he needed a formalism that was rich enough

00:30:09.420 | to enable him to do that.

00:30:10.480 | And once you do that, that formalism

00:30:12.120 | is also gonna include things that are not physical.

00:30:14.600 | And then you have two choices.

00:30:15.560 | You can be like, "Oh, well, that stuff's trash,"

00:30:17.720 | or, and this is more the mathematician's frame of mind,

00:30:21.280 | if you have a formalistic framework

00:30:23.640 | that seems really good and sort of seems to be

00:30:25.880 | very elegant and work well,

00:30:27.160 | and it includes all the physical stuff,

00:30:29.020 | maybe we should think about all of it.

00:30:30.520 | Like, maybe we should think about it,

00:30:31.360 | thinking, "Oh, maybe there's some gold to be mined there."

00:30:34.600 | And indeed, guess what?

00:30:36.800 | Before long, there's relativity and there's space-time,

00:30:39.080 | and all of a sudden, it's like, "Oh, yeah,

00:30:40.520 | "maybe it's a good idea.

00:30:41.480 | "We already have this geometric apparatus set up

00:30:43.820 | "for how to think about four-dimensional spaces.

00:30:47.280 | "Turns out they're real after all."

00:30:49.240 | This is a story much told in mathematics,

00:30:52.280 | not just in this context, but in many.

00:30:53.680 | - I'd love to dig in a little deeper on that, actually,

00:30:55.640 | 'cause I have some intuitions to work out.

00:31:00.640 | - Okay. - In my brain, but--

00:31:01.660 | - Well, I'm not a mathematical physicist,

00:31:03.580 | so we can work it out together.

00:31:05.580 | - Good, we'll together walk along the path of curiosity.

00:31:10.020 | But Poincaré conjecture, what is it?

00:31:14.340 | - The Poincaré conjecture is about

00:31:16.660 | curved three-dimensional spaces.

00:31:18.900 | So I was on my way there, I promise.

00:31:23.380 | The idea is that we perceive ourselves as living in,

00:31:26.680 | we don't say a three-dimensional space,

00:31:29.120 | we just say three-dimensional space.

00:31:30.520 | You can go up and down, you can go left and right,

00:31:32.240 | you can go forward and back.

00:31:33.200 | There's three dimensions in which we can move.

00:31:35.480 | In Poincaré's theory, there are many possible

00:31:39.800 | three-dimensional spaces.

00:31:41.680 | In the same way that going down one dimension

00:31:45.320 | to sort of capture our intuition a little bit more,

00:31:48.420 | we know there are lots of different

00:31:49.680 | two-dimensional surfaces, right?

00:31:51.080 | There's a balloon, and that looks one way,

00:31:54.080 | and a donut looks another way,

00:31:55.520 | and a Mobius strip looks a third way.

00:31:57.640 | Those are all two-dimensional surfaces

00:31:59.120 | that we can kind of really get a global view of,

00:32:02.360 | because we live in three-dimensional space,

00:32:03.880 | so we can see a two-dimensional surface

00:32:05.480 | sort of sitting in our three-dimensional space.

00:32:07.160 | Well, to see a three-dimensional space whole,

00:32:11.220 | we'd have to kind of have four-dimensional eyes, right?

00:32:13.180 | Which we don't, so we have to use our mathematical eyes,

00:32:14.960 | we have to envision.

00:32:17.400 | The Poincaré conjecture says that there's a very simple way

00:32:22.060 | to determine whether a three-dimensional space

00:32:24.440 | is the standard one, the one that we're used to.

00:32:29.600 | And essentially, it's that it's what's called

00:32:31.840 | fundamental group has nothing interesting in it.

00:32:34.600 | And that I can actually say,

00:32:35.580 | without saying what the fundamental group is,

00:32:36.920 | I can tell you what the criterion is.

00:32:38.960 | This would be good, oh look, I can even use a visual aid.

00:32:40.840 | So for the people watching this on YouTube,

00:32:42.320 | you'll just see this.

00:32:43.160 | For the people on the podcast, you'll have to visualize it.

00:32:46.120 | So Lex has been nice enough to give me a surface

00:32:49.080 | with some interesting topology.

00:32:50.400 | - It's a mug.

00:32:51.240 | - Right here in front of me.

00:32:52.280 | A mug, yes, I might say it's a genus one surface,

00:32:55.120 | but we could also say it's a mug, same thing.

00:32:57.280 | So if I were to draw a little circle on this mug,

00:33:02.960 | oh, which way should I draw it so it's visible?

00:33:04.320 | Like here, okay.

00:33:06.240 | If I draw a little circle on this mug,

00:33:07.480 | imagine this to be a loop of string.

00:33:09.360 | I could pull that loop of string closed

00:33:12.040 | on the surface of the mug, right?

00:33:14.600 | That's definitely something I could do.

00:33:15.840 | I could shrink it, shrink it, shrink it until it's a point.

00:33:18.320 | On the other hand, if I draw a loop

00:33:20.040 | that goes around the handle, I can kind of zhuzh it up here

00:33:23.040 | and I can zhuzh it down there and I can sort of slide it up

00:33:24.840 | and down the handle, but I can't pull it closed, can I?

00:33:27.240 | It's trapped.

00:33:28.800 | Not without breaking the surface of the mug, right?

00:33:30.640 | Not without like going inside.

00:33:32.320 | So the condition of being what's called simply connected,

00:33:37.080 | this is one of Poincare's inventions,

00:33:39.760 | says that any loop of string can be pulled shut.

00:33:42.600 | So it's a feature that the mug simply does not have.

00:33:45.040 | This is a non-simply connected mug

00:33:48.480 | and a simply connected mug would be a cup, right?

00:33:51.040 | You would burn your hand when you drank coffee out of it.

00:33:53.520 | - So you're saying the universe is not a mug?

00:33:56.480 | - Well, I can't speak to the universe,

00:33:59.280 | but what I can say is that regular old space is not a mug.

00:34:04.280 | Regular old space, if you like sort of actually

00:34:07.000 | physically have like a loop of string,

00:34:09.520 | you can pull it shut. - You can always close it.

00:34:10.920 | You can always pull it shut.

00:34:12.600 | - But you know, what if your piece of string

00:34:14.000 | was the size of the universe?

00:34:14.920 | Like what if your piece of string

00:34:16.320 | was like billions of light years long?

00:34:18.080 | Like how do you actually know?

00:34:20.160 | - I mean, that's still an open question

00:34:21.440 | of the shape of the universe.

00:34:22.560 | - Exactly.

00:34:23.800 | - Whether it's, I think there's a lot,

00:34:26.440 | there is ideas of it being a torus.

00:34:28.600 | I mean, there's some trippy ideas

00:34:30.400 | and they're not like weird out there, controversial.

00:34:33.440 | There's legitimate at the center of cosmology debate.

00:34:38.160 | I mean, I think most people think it's flat.

00:34:39.000 | - I think there's somebody who thinks

00:34:39.840 | that there's like some kind of dodecahedral symmetry

00:34:42.160 | or I mean, I remember reading something crazy

00:34:43.600 | about somebody saying that they saw the signature of that

00:34:45.920 | in the cosmic noise or what have you.

00:34:48.520 | I mean.

00:34:49.800 | - To make the flat earthers happy,

00:34:51.380 | I do believe that the current main belief is it's flat.

00:34:56.380 | It's flat-ish or something like that.

00:34:59.800 | The shape of the universe is flat-ish.

00:35:01.960 | I don't know what the heck that means.

00:35:03.120 | I think that has like a very,

00:35:05.600 | I mean, how are you even supposed to think about

00:35:08.680 | the shape of a thing

00:35:11.160 | that doesn't have anything outside of it?

00:35:14.120 | I mean.

00:35:14.960 | - Ah, but that's exactly what topology does.

00:35:16.720 | Topology is what's called an intrinsic theory.

00:35:19.400 | That's what's so great about it.

00:35:20.320 | This question about the mug,

00:35:22.560 | you could answer it without ever leaving the mug, right?

00:35:25.960 | Because it's a question about a loop drawn

00:35:28.920 | on the surface of the mug and what happens

00:35:30.560 | if it never leaves that surface.

00:35:31.760 | So it's like always there.

00:35:33.480 | - See, but that's the difference between the topology

00:35:37.800 | and say if you're like trying to visualize a mug,

00:35:42.480 | that you can't visualize a mug while living inside the mug.

00:35:45.560 | - Well, that's true.

00:35:47.520 | The visualization is harder, but in some sense,

00:35:49.160 | no, you're right, but if the tools of mathematics are there,

00:35:51.960 | I, sorry, I don't wanna fight,

00:35:53.680 | but I was like the tools of mathematics are exactly there

00:35:55.560 | to enable you to think about

00:35:57.000 | what you cannot visualize in this way.

00:35:58.720 | Let me give, let's go,

00:35:59.640 | always to make things easier, go down a dimension.

00:36:02.120 | Let's think about we live on a circle, okay?

00:36:05.800 | You can tell whether you live on a circle or a line segment

00:36:10.800 | because if you live on a circle,

00:36:12.360 | if you walk a long way in one direction,

00:36:13.800 | you find yourself back where you started.

00:36:15.200 | And if you live in a line segment,

00:36:17.280 | you walk for a long enough one direction,

00:36:18.720 | you come to the end of the world.

00:36:20.200 | Or if you live on a line, like a whole line,

00:36:22.880 | an infinite line, then you walk in one direction

00:36:25.880 | for a long time and like,

00:36:27.120 | well, then there's not a sort of terminating algorithm

00:36:28.760 | to figure out whether you live on a line or a circle,

00:36:30.520 | but at least you sort of,

00:36:31.720 | at least you don't discover that you live on a circle.

00:36:35.720 | So all of those are intrinsic things, right?

00:36:37.400 | All of those are things that you can figure out

00:36:39.720 | about your world without leaving your world.

00:36:42.120 | On the other hand, ready?

00:36:43.360 | Now we're gonna go from intrinsic to extrinsic.

00:36:45.240 | Why did I not know we were gonna talk about this,

00:36:46.960 | but why not?

00:36:48.040 | - Why not?

00:36:48.880 | - If you can't tell whether you live in a circle or a knot,

00:36:53.400 | like imagine like a knot floating in three-dimensional space.

00:36:56.960 | The person who lives on that knot, to them it's a circle.

00:36:59.720 | They walk a long way, they come back to where they started.

00:37:01.760 | Now we with our three-dimensional eyes can be like,

00:37:04.320 | oh, this one's just a plain circle

00:37:05.640 | and this one's knotted up.

00:37:06.720 | But that has to do with how they sit

00:37:09.760 | in three-dimensional space.

00:37:10.640 | It doesn't have to do with intrinsic features

00:37:12.160 | of those people's world.

00:37:13.160 | - We can ask you one ape to another.

00:37:14.920 | Does it make you, how does it make you feel

00:37:17.160 | that you don't know if you live in a circle

00:37:19.880 | or on a knot, in a knot,

00:37:22.800 | inside the string that forms the knot?

00:37:26.800 | - I'm gonna be honest-- - I don't even know

00:37:29.560 | how to say that.

00:37:30.400 | - I'm gonna be honest with you.

00:37:31.220 | I don't know if like, I fear you won't like this answer,

00:37:34.600 | but it does not bother me at all.

00:37:37.160 | I don't lose one minute of sleep over it.

00:37:39.400 | - So like, does it bother you that if we look

00:37:41.720 | at like a Mobius strip, that you don't have

00:37:45.160 | an obvious way of knowing whether you are inside

00:37:49.080 | of a cylinder, if you live on a surface of a cylinder

00:37:51.720 | or you live on the surface of a Mobius strip?

00:37:54.000 | - No, I think you can tell.

00:37:57.400 | - If you live-- - Which one?

00:37:59.080 | Because what you do is you like, tell your friend,

00:38:02.440 | hey, stay right here, I'm just gonna go for a walk,

00:38:04.120 | and then you like, walk for a long time in one direction

00:38:06.680 | and then you come back and you see your friend again,

00:38:08.240 | and if your friend is reversed,

00:38:09.360 | then you know you live on a Mobius strip.

00:38:10.720 | - Well, no, because you won't see your friend, right?

00:38:13.800 | - Okay, fair point, fair point on that.

00:38:17.000 | - But you have to believe the stories about,

00:38:19.760 | no, I don't even know.

00:38:21.320 | Would you even know?

00:38:24.160 | Would you really-- - Oh, no, your point is right.

00:38:26.800 | Let me try to think of a better,

00:38:28.120 | let's see if I can do this on the vlog.

00:38:28.960 | - It may not be correct to talk about

00:38:32.480 | cognitive beings living on a Mobius strip

00:38:35.280 | because there's a lot of things taken for granted there,

00:38:37.880 | and we're constantly imagining actual

00:38:40.800 | three-dimensional creatures,

00:38:42.240 | like how it actually feels like

00:38:45.120 | to live on a Mobius strip is tricky to internalize.

00:38:50.120 | - I think that on what's called the real projective plane,

00:38:52.800 | which is kind of even more sort of messed up version

00:38:54.840 | of the Mobius strip, but with very similar features,

00:38:57.480 | this feature of kind of only having one side,

00:39:01.280 | that has the feature that there's a loop of string,

00:39:04.440 | which can't be pulled closed,

00:39:06.720 | but if you loop it around twice along the same path,

00:39:09.640 | that you can pull closed.

00:39:11.280 | That's extremely weird.

00:39:12.920 | - Yeah.

00:39:13.760 | - But that would be a way you could know

00:39:16.200 | without leaving your world that

00:39:17.720 | something very funny is going on.

00:39:20.320 | - You know what's extremely weird?

00:39:21.920 | Maybe we can comment on,

00:39:23.200 | hopefully it's not too much of a tangent,

00:39:24.840 | is I remember thinking about this.

00:39:29.000 | This might be right.

00:39:30.320 | This might be wrong.

00:39:31.760 | But if we now talk about a sphere,

00:39:35.400 | and you're living inside a sphere,

00:39:37.560 | that you're going to see everywhere around you

00:39:42.160 | the back of your own head.

00:39:43.600 | This is very counterintuitive to me to think about,

00:39:51.680 | maybe it's wrong.

00:39:52.520 | But 'cause I was thinking of like Earth,

00:39:55.280 | your 3D thing sitting on a sphere.

00:39:58.120 | But if you're living inside the sphere,

00:40:00.120 | you're going to see, if you look straight,

00:40:03.120 | you're always going to see yourself all the way around.

00:40:06.480 | So everywhere you look,

00:40:07.800 | there's gonna be the back of your own head.

00:40:10.120 | - I think somehow this depends on something

00:40:11.840 | of how the physics of light works in this scenario,

00:40:14.000 | which I'm finding it hard to bend my--

00:40:15.760 | - That's true.

00:40:16.600 | The sea is doing a lot of work.

00:40:17.680 | Saying you see something is doing a lot of work.

00:40:20.400 | - People have thought about this,

00:40:21.520 | I mean this metaphor of what if we're little creatures

00:40:25.600 | in some sort of smaller world?

00:40:27.040 | How could we apprehend what's outside?

00:40:28.520 | That metaphor just comes back and back.

00:40:30.400 | And actually I didn't even realize how frequent it is.

00:40:32.960 | It comes up in the book a lot.

00:40:34.400 | I know it from a book called Flatland.

00:40:36.320 | I don't know if you ever read this when you were a kid.

00:40:38.480 | - A while ago, yeah.

00:40:39.320 | - An adult.

00:40:40.480 | This sort of comic novel from the 19th century

00:40:43.600 | about an entire two-dimensional world.

00:40:47.720 | It's narrated by a square, that's the main character.

00:40:50.560 | And the kind of strangeness that befalls him

00:40:54.240 | when one day he's in his house

00:40:55.920 | and suddenly there's a little circle there

00:40:58.520 | and they're with him.

00:41:00.440 | But then the circle starts getting bigger

00:41:03.120 | and bigger and bigger.

00:41:04.360 | And he's like, what the hell is going on?

00:41:06.080 | It's like a horror movie for two-dimensional people.

00:41:08.520 | And of course what's happening

00:41:09.840 | is that a sphere is entering his world.

00:41:12.200 | And as the sphere moves farther and farther into the plane,

00:41:15.800 | it's cross-sectioned, the part of it that he can see.

00:41:18.480 | To him it looks like there's this kind of bizarre being

00:41:21.240 | that's getting larger and larger and larger.

00:41:24.760 | Until it's exactly sort of halfway through.

00:41:27.320 | And then they have this kind of philosophical argument

00:41:29.160 | where the sphere's like, I'm a sphere,

00:41:30.160 | I'm from the third dimension.

00:41:31.040 | The square's like, what are you talking about?

00:41:32.200 | There's no such thing.

00:41:33.360 | And they have this kind of sterile argument

00:41:36.200 | where the square is not able to follow

00:41:39.640 | the mathematical reasoning of the sphere

00:41:40.960 | until the sphere just kind of grabs him

00:41:42.360 | and jerks him out of the plane and pulls him up.

00:41:45.760 | And it's like, now, now do you see?

00:41:47.560 | Now do you see your whole world

00:41:50.240 | that you didn't understand before?

00:41:52.080 | - So do you think that kind of process

00:41:53.800 | is possible for us humans?

00:41:56.560 | So we live in a three-dimensional world,

00:41:58.360 | maybe with a time component four-dimensional.

00:42:01.520 | And then math allows us to go into high dimensions

00:42:06.520 | comfortably and explore the world from those perspectives.

00:42:10.500 | Is it possible that the universe is many more dimensions

00:42:21.360 | than the ones we experience as human beings?

00:42:25.160 | So if you look at the,

00:42:28.800 | especially in physics theories of everything,

00:42:31.960 | physics theories that try to unify general relativity

00:42:35.440 | and quantum field theory,

00:42:37.360 | they seem to go to high dimensions to work stuff out

00:42:42.360 | through the tools of mathematics.

00:42:44.600 | Is it possible, so like the two options are,

00:42:47.680 | one is just a nice way to analyze,

00:42:50.200 | a universe, but the reality is as exactly we perceive it,

00:42:54.680 | it is three-dimensional.

00:42:56.200 | Or are we just seeing, are we those flatland creatures

00:43:00.720 | that are just seeing a tiny slice of reality

00:43:03.720 | and the actual reality is many, many, many more dimensions

00:43:08.720 | than the three dimensions we perceive?

00:43:10.920 | - Oh, I certainly think that's possible.

00:43:12.920 | Now, how would you figure out whether it was true or not

00:43:17.840 | is another question.

00:43:19.920 | And I suppose what you would do,

00:43:22.160 | as with anything else that you can't directly perceive,

00:43:25.040 | is you would try to understand what effect

00:43:30.040 | the presence of those extra dimensions out there

00:43:33.800 | would have on the things we can perceive.

00:43:36.920 | Like what else can you do, right?

00:43:39.320 | And in some sense, if the answer is

00:43:42.260 | they would have no effect,

00:43:44.720 | then maybe it becomes like a little bit of a sterile question

00:43:47.000 | 'cause what question are you even asking, right?

00:43:49.320 | You can kind of posit however many entities that you want.

00:43:53.680 | - Is it possible to intuit how to mess

00:43:56.840 | with the other dimensions while living

00:43:58.800 | in a three-dimensional world?

00:44:00.280 | I mean, that seems like a very challenging thing to do.

00:44:03.000 | The reason flatland could be written

00:44:06.000 | is because it's coming from a three-dimensional writer.

00:44:11.400 | - Yes, but what happens in the book,

00:44:13.840 | I didn't even tell you the whole plot,

00:44:15.200 | what happens is the square is so excited

00:44:17.160 | and so filled with intellectual joy.

00:44:19.920 | By the way, maybe to give the story some context,

00:44:22.160 | you ask is it possible for us humans

00:44:25.120 | to have this experience of being transcendentally jerked

00:44:28.440 | out of our world so we can sort of truly see it from above?

00:44:30.960 | Well, Edwin Abbott, who wrote the book,

00:44:32.720 | certainly thought so because Edwin Abbott was a minister.

00:44:35.900 | So the whole Christian subtext of this book,

00:44:37.840 | I had completely not grasped reading this as a kid,

00:44:41.800 | that it means a very different thing, right?

00:44:43.400 | If sort of a theologian is saying like,

00:44:45.680 | oh, what if a higher being could pull you out

00:44:48.240 | of this earthly world you live in

00:44:50.000 | so that you can sort of see the truth

00:44:51.360 | and really see it from above, as it were.

00:44:54.480 | So that's one of the things that's going on for him.

00:44:56.560 | And it's a testament to his skill as a writer

00:44:58.440 | that his story just works,

00:45:00.000 | whether that's the framework you're coming to it from or not.

00:45:05.000 | But what happens in this book,

00:45:06.800 | and this part now, looking at it through a Christian lens,

00:45:08.960 | it becomes a bit subversive,

00:45:11.160 | is the square is so excited

00:45:13.200 | about what he's learned from the sphere,

00:45:16.760 | and the sphere explains to him what a cube would be.

00:45:18.720 | Oh, it's like you, but three-dimensional,

00:45:20.000 | and the square is very excited.

00:45:21.000 | And the square is like, okay, I get it now.

00:45:23.160 | So now that you explained to me how just by reason

00:45:26.020 | I can figure out what a cube would be like,

00:45:27.360 | like a three-dimensional version of me,

00:45:29.600 | let's figure out what a four-dimensional version

00:45:31.320 | of me would be like.

00:45:32.720 | And then the sphere's like,

00:45:33.960 | what the hell are you talking about?

00:45:34.800 | There's no fourth dimension, that's ridiculous.

00:45:36.480 | Like, there's only three dimensions.

00:45:37.720 | Like, that's how many there are, I can see.

00:45:39.240 | Like, I mean, so it's this sort of comic moment

00:45:40.840 | where the sphere is completely unable to conceptualize

00:45:44.640 | that there could actually be yet another dimension.

00:45:47.900 | So yeah, that takes the religious allegory

00:45:49.880 | to like a very weird place

00:45:50.920 | that I don't really like understand theologically, but.

00:45:53.180 | - That's a nice way to talk about religion and myth

00:45:57.120 | in general as perhaps us trying to struggle,

00:46:01.000 | us meaning human civilization,

00:46:02.440 | trying to struggle with ideas

00:46:04.480 | that are beyond our cognitive capabilities.

00:46:08.640 | - But it's in fact not beyond our capability.

00:46:10.640 | It may be beyond our cognitive capabilities

00:46:13.320 | to visualize a four-dimensional cube,

00:46:16.480 | a tesseract as some like to call it,

00:46:18.360 | or a five-dimensional cube or a six-dimensional cube,

00:46:20.840 | but it is not beyond our cognitive capabilities

00:46:24.000 | to figure out how many corners

00:46:26.680 | a six-dimensional cube would have.

00:46:28.080 | That's what's so cool about us.

00:46:29.420 | Whether we can visualize it or not,

00:46:31.080 | we can still talk about it, we can still reason about it,

00:46:33.440 | we can still figure things out about it.

00:46:36.200 | That's amazing.

00:46:37.280 | - Yeah, if we go back to this,

00:46:40.160 | first of all to the mug,

00:46:41.680 | but to the example you give in the book of the straw,

00:46:45.020 | how many holes does a straw have?

00:46:49.800 | And you, listener, may try to answer that in your own head.

00:46:54.120 | - Yeah, I'm gonna take a drink

00:46:55.120 | while everybody thinks about it

00:46:56.240 | so we can give you a moment.

00:46:57.080 | - A slow sip.

00:46:59.020 | Is it zero, one, or two, or more than that maybe?

00:47:04.020 | Maybe you get very creative.

00:47:06.660 | But it's kind of interesting to dissecting each answer

00:47:11.660 | as you do in the book.

00:47:13.100 | It's quite brilliant, people should definitely check it out.

00:47:15.540 | But if you could try to answer it now,

00:47:18.460 | think about all the options

00:47:20.300 | and why they may or may not be right.

00:47:23.440 | - Yeah, and it's one of these questions

00:47:25.280 | where people on first hearing it think it's a triviality

00:47:28.300 | and they're like, well, the answer is obvious.

00:47:29.780 | And then what happens,

00:47:30.620 | if you ever ask a group of people this,

00:47:31.620 | something wonderfully comic happens,

00:47:33.980 | which is that everyone's like,

00:47:34.820 | well, it's completely obvious.

00:47:36.500 | And then each person realizes that half the person,

00:47:38.980 | the other people in the room

00:47:39.820 | have a different obvious answer

00:47:41.980 | for the way they have.

00:47:42.900 | And then people get really heated.

00:47:44.420 | People are like, I can't believe

00:47:46.100 | that you think it has two holes.

00:47:47.460 | Or like, I can't believe that you think it has one.

00:47:49.660 | And then you really,

00:47:50.740 | people really learn something about each other.

00:47:52.420 | And people get heated.

00:47:54.440 | - I mean, can we go through the possible options here?

00:47:57.740 | Is it zero, one, two, three, 10?

00:48:01.300 | - Sure, so I think most people,

00:48:04.660 | the zero holers are rare.

00:48:06.100 | They would say like, well, look,

00:48:07.740 | you can make a straw by taking a rectangular piece of plastic

00:48:10.180 | and closing it up.

00:48:11.060 | Rectangular piece of plastic doesn't have a hole in it.

00:48:13.760 | I didn't poke a hole in it.

00:48:16.980 | So how can I have a hole?

00:48:18.300 | They'd be like, it's just one thing.

00:48:19.580 | Okay, most people don't see it that way.

00:48:21.940 | That's like--

00:48:22.780 | - Is there any truth to that kind of conception?

00:48:25.860 | - Yeah, I think that would be somebody who's a count.

00:48:28.460 | I mean,

00:48:34.100 | what I would say is you could say the same thing

00:48:37.340 | about a bagel.

00:48:40.460 | You could say, I can make a bagel by taking

00:48:42.220 | a long cylinder of dough, which doesn't have a hole,

00:48:45.060 | and then smushing the ends together.

00:48:47.720 | Now it's a bagel.

00:48:49.060 | So if you're really committed, you can be like,

00:48:50.420 | okay, a bagel doesn't have a hole either.

00:48:51.700 | But who are you if you say a bagel doesn't have a hole?

00:48:54.140 | I mean, I don't know.

00:48:54.980 | - Yeah, so that's almost like

00:48:55.820 | an engineering definition of it.

00:48:58.000 | Okay, fair enough.

00:48:59.020 | So what about the other options?

00:49:02.260 | - So one-hole people would say--

00:49:06.580 | - I like how these are like groups of people,

00:49:09.900 | like we've planted our foot.

00:49:11.700 | This is what we stand for.

00:49:12.940 | There's books written about each belief.

00:49:16.260 | - You know, would say, look, there's a hole,

00:49:17.540 | and it goes all the way through the straw, right?

00:49:18.900 | There's one region of space that's the hole,

00:49:21.920 | and there's one.

00:49:22.760 | And two-hole people would say, well, look,

00:49:24.140 | there's a hole in the top and a hole at the bottom.

00:49:27.340 | I think a common thing you see when people

00:49:31.980 | argue about this, they would take something like this

00:49:35.980 | bottle of water I'm holding, and they'll open it.

00:49:38.820 | And they say, well, how many holes are there in this?

00:49:41.540 | And you say, well, there's one.

00:49:42.580 | There's one hole at the top.

00:49:44.100 | Okay, what if I poke a hole here

00:49:46.380 | so that all the water spills out?

00:49:48.940 | Well, now it's a straw.

00:49:50.300 | So if you're a one-holer, I say to you,

00:49:53.140 | well, how many holes are in it now?

00:49:56.020 | There was one hole in it before,

00:49:57.340 | and I poked a new hole in it.

00:49:59.260 | And then you think there's still one hole,

00:50:01.580 | even though there was one hole and I made one more?

00:50:04.700 | - Clearly not, there's just two holes, yeah.

00:50:07.260 | - And yet, if you're a two-holer,

00:50:09.540 | the one-holer will say, okay, where does one hole begin

00:50:11.740 | and the other hole end?

00:50:12.900 | And in the book, I sort of, in math,

00:50:18.660 | there's two things we do when we're faced

00:50:19.980 | with a problem that's confusing us.

00:50:21.740 | We can make the problem simpler.

00:50:24.540 | That's what we were doing a minute ago

00:50:25.740 | when we were talking about high-dimensional space,

00:50:27.100 | and I was like, let's talk about circles and line segments.

00:50:29.100 | Let's go down a dimension to make it easier.

00:50:31.740 | The other big move we have is to make the problem harder

00:50:35.100 | and try to sort of really face up

00:50:36.700 | to what are the complications.

00:50:37.660 | So what I do in the book is say,

00:50:39.580 | let's stop talking about straws for a minute

00:50:41.260 | and talk about pants.

00:50:42.700 | How many holes are there in a pair of pants?

00:50:46.240 | So I think most people who say there's two holes in a straw

00:50:48.820 | would say there's three holes in a pair of pants.

00:50:52.020 | I guess we're filming only from here.

00:50:54.020 | I could take up, no, I'm not gonna do it.

00:50:56.580 | You'll just have to imagine the pants, sorry.

00:50:59.660 | Lex, if you want to, no, okay, no.

00:51:01.360 | (laughing)

00:51:03.220 | - That's gonna be in the director's cut.

00:51:04.620 | It's a Patreon-only footage.

00:51:06.380 | There you go.

00:51:07.860 | - So many people would say there's three holes

00:51:09.520 | in a pair of pants, but for instance,

00:51:10.940 | my daughter, when I asked this,

00:51:12.000 | by the way, talking to kids about this is super fun.

00:51:14.860 | I highly recommend it.

00:51:16.380 | - What did she say?

00:51:17.900 | - She said, well, yeah, I feel a pair of pants

00:51:21.380 | just has two holes because yes, there's the waist,

00:51:23.660 | but that's just the two leg holes stuck together.

00:51:26.660 | - Whoa, okay.

00:51:28.540 | - Two leg holes, yeah, okay.

00:51:29.820 | - Right, I mean, that really is a good combination.

00:51:30.660 | - 'Cause she's a one-holer for the straw.

00:51:32.380 | - So she's a one-holer for the straw too.

00:51:34.480 | And that really does capture something.

00:51:39.480 | It captures this fact, which is central

00:51:42.900 | to the theory of what's called homology,

00:51:44.460 | which is like a central part of modern topology,

00:51:46.120 | that holes, whatever we may mean by them,

00:51:49.740 | they're somehow things which have an arithmetic to them.

00:51:51.980 | They're things which can be added,

00:51:53.960 | like the waist, like waist equals leg plus leg

00:51:57.220 | is kind of an equation, but it's not an equation

00:51:59.380 | about numbers, it's an equation about some kind of geometric,

00:52:02.940 | some kind of topological thing, which is very strange.

00:52:05.460 | And so, when I come down, like a rabbi,

00:52:09.980 | I like to kind of like come up with these answers

00:52:12.180 | and somehow like dodge the original question

00:52:14.300 | and say like, you're both right, my children.

00:52:15.900 | Okay, so.

00:52:16.740 | So for the straw, I think what a modern mathematician

00:52:23.500 | would say is like, the first version would be to say like,

00:52:28.020 | well, there are two holes,

00:52:29.300 | but they're really both the same hole.

00:52:31.380 | Well, that's not quite right.

00:52:32.300 | A better way to say it is, there's two holes,

00:52:34.780 | but one is the negative of the other.

00:52:37.540 | Now, what can that mean?

00:52:38.740 | One way of thinking about what it means

00:52:41.220 | is that if you sip something like a milkshake

00:52:43.380 | through the straw, no matter what,

00:52:46.460 | the amount of milkshake that's flowing in one end,

00:52:49.980 | that same amount is flowing out the other end.

00:52:53.420 | So they're not independent from each other.

00:52:55.820 | There's some relationship between them.

00:52:57.700 | In the same way that if you somehow could like suck

00:53:01.780 | a milkshake through a pair of pants,

00:53:04.160 | the amount of milkshake, just go with me on this.

00:53:07.980 | - I'm right there with you.

00:53:09.660 | - The amount of milkshake that's coming in

00:53:11.700 | the left leg of the pants,

00:53:13.340 | plus the amount of milkshake that's coming in

00:53:15.100 | the right leg of the pants,

00:53:16.700 | is the same that's coming out the waist of the pants.

00:53:20.780 | - So just so you know, I fasted for 72 hours.

00:53:23.300 | The last three days.

00:53:25.580 | So I just broke the fast

00:53:26.580 | with a little bit of food yesterday.

00:53:27.780 | So this is like, this sounds,

00:53:30.460 | food analogies or metaphors for this podcast

00:53:32.780 | work wonderfully 'cause I can intensely picture it.

00:53:35.740 | - Is that your weekly routine

00:53:36.860 | or just in preparation for talking about geometry

00:53:38.840 | for three hours?

00:53:39.680 | - Exactly, it's just for this.

00:53:41.900 | It's hardship to purify the mind.

00:53:44.140 | No, it's for the first time.

00:53:45.220 | I just wanted to try the experience.

00:53:46.700 | - Oh, wow.

00:53:47.540 | - And just to pause, to do things

00:53:50.820 | that are out of the ordinary,

00:53:52.060 | to pause and to reflect on how grateful I am

00:53:55.980 | to be just alive and be able to do all the cool shit

00:53:59.220 | that I get to do.

00:54:00.060 | - Did you drink water?

00:54:01.340 | - Yes, yes, yes, yes, yes.

00:54:03.020 | Water and salt.

00:54:04.540 | So like electrolytes and all those kinds of things.

00:54:07.180 | But anyway, so the inflow on the top of the pants

00:54:10.620 | equals to the outflow on the bottom of the pants.

00:54:14.620 | - Exactly, so this idea that,

00:54:16.900 | I mean, I think, you know,

00:54:18.980 | Poincaré really had this idea,

00:54:21.340 | this sort of modern idea,

00:54:22.820 | I mean, building on stuff other people did,

00:54:25.020 | Betty is an important one,

00:54:26.700 | of this kind of modern notion of relations between wholes.

00:54:29.900 | But the idea that wholes really had an arithmetic,

00:54:32.260 | the really modern view was really Emmy Noether's idea.

00:54:35.540 | So she kind of comes in and sort of truly puts the subject

00:54:39.380 | on its modern footing that we have now.

00:54:43.300 | So, you know, it's always a challenge.

00:54:45.220 | You know, in the book,

00:54:46.380 | I'm not gonna say I give like a course

00:54:48.620 | so that you read this chapter and then you're like,

00:54:50.300 | oh, it's just like I took like a semester

00:54:51.940 | of algebraic topology.

00:54:53.100 | It's not like this.

00:54:53.940 | And it's always a challenge writing about math

00:54:56.380 | because there are some things

00:54:59.060 | that you can really do on the page and the math is there.

00:55:03.300 | And there's other things which,

00:55:05.260 | it's too much in a book like this

00:55:06.500 | to like do them all the page.

00:55:07.380 | You can only say something about them,

00:55:10.140 | if that makes sense.

00:55:11.140 | So, you know, in the book, I try to do some of both.

00:55:14.780 | I try to,

00:55:17.100 | topics that are,

00:55:18.780 | you can't really compress and really truly say

00:55:22.100 | exactly what they are in this amount of space.

00:55:25.460 | I try to say something interesting about them,

00:55:29.020 | something meaningful about them

00:55:30.180 | so that readers can get the flavor.

00:55:31.780 | And then in other places,

00:55:33.460 | I really try to get up close and personal

00:55:36.660 | and really do the math and have it take place on the page.

00:55:40.260 | - To some degree, be able to give inklings

00:55:44.140 | of the beauty of the subject.

00:55:45.860 | - Yeah, I mean, there's a lot of books that are like,

00:55:48.260 | I don't quite know how to express this well.

00:55:49.740 | I'm still laboring to do it,

00:55:51.020 | but there's a lot of books that are about stuff,

00:55:56.020 | but I want my books to not only be about stuff,

00:56:00.980 | but to actually have some stuff there on the page

00:56:03.580 | in the book for people to interact with directly

00:56:05.580 | and not just sort of hear me talk about distant features,

00:56:07.700 | about distant features of it.

00:56:10.460 | - Right, so not be talking just about ideas,

00:56:13.660 | but actually be expressing the idea.

00:56:16.860 | Is there, you know somebody in the,

00:56:18.700 | maybe you can comment, there's a guy,

00:56:21.420 | his YouTube channel is 3Blue1Brown, Grant Sanderson.

00:56:25.500 | He does that masterfully well.

00:56:27.900 | - Absolutely.

00:56:28.740 | - Of visualizing, of expressing a particular idea

00:56:31.620 | and then talking about it as well, back and forth.

00:56:34.540 | What do you think about Grant?

00:56:37.020 | - It's fantastic.

00:56:37.980 | I mean, the flowering of math YouTube

00:56:40.180 | is like such a wonderful thing

00:56:41.460 | because math teaching,

00:56:44.700 | there's so many different venues

00:56:47.020 | through which we can teach people math.

00:56:48.860 | There's the traditional one, right?

00:56:50.780 | Well, where I'm in a classroom with,

00:56:54.540 | depending on the class, it could be 30 people,

00:56:56.420 | it could be 100 people,

00:56:57.700 | it could, God help me, be 500 people

00:56:59.460 | if it's like the big calculus lecture or whatever it may be.

00:57:01.860 | And there's sort of some,

00:57:02.700 | but there's some set of people of that order of magnitude.

00:57:05.060 | And I'm with them, we have a long time.

00:57:06.460 | I'm with them for a whole semester

00:57:08.340 | and I can ask them to do homework and we talk together.

00:57:10.300 | We have office hours,

00:57:11.140 | if they have one-on-one questions, blah, blah, blah.

00:57:12.500 | It's like a very high level of engagement,

00:57:14.500 | but how many people am I actually hitting at a time?

00:57:17.180 | Like not that many, right?

00:57:19.300 | And you can, and there's kind of an inverse relationship

00:57:22.940 | where the more, the fewer people you're talking to,

00:57:27.740 | the more engagement you can ask for.

00:57:29.300 | The ultimate, of course, is like the mentorship relation

00:57:32.020 | of like a PhD advisor and a graduate student

00:57:35.020 | where you spend a lot of one-on-one time together

00:57:38.020 | for like three to five years.

00:57:41.180 | And the ultimate high level of engagement to one person.

00:57:44.820 | You know, books, this can get to a lot more people

00:57:50.420 | that are ever gonna sit in my classroom

00:57:52.700 | and you spend like however many hours

00:57:56.700 | it takes to read a book.

00:57:58.700 | Somebody like 3Blue1Brown or Numberphile

00:58:01.140 | or people like Vi Hart.

00:58:03.140 | I mean, YouTube, let's face it,

00:58:05.980 | has bigger reach than a book.

00:58:07.900 | Like there's YouTube videos that have many, many,

00:58:09.660 | many more views than like any hardback book

00:58:13.320 | like not written by a Kardashian or an Obama

00:58:15.820 | is gonna sell, right?

00:58:16.660 | So that's, I mean.

00:58:17.580 | And then, you know, those are, you know,

00:58:23.340 | some of them are like longer, 20 minutes long,

00:58:25.540 | some of them are five minutes long,

00:58:26.580 | but they're shorter.

00:58:27.820 | And then even somebody like, look, like Eugenia Chang

00:58:29.740 | is a wonderful category theorist in Chicago.

00:58:31.540 | I mean, she was on, I think, The Daily Show or is it?

00:58:33.620 | I mean, she was on, you know, she has 30 seconds,

00:58:35.820 | but then there's like 30 seconds to sort of say

00:58:37.620 | something about mathematics to like untold

00:58:40.380 | millions of people.

00:58:41.220 | So everywhere along this curve is important.

00:58:43.980 | And one thing I feel like is great right now

00:58:46.580 | is that people are just broadcasting on all the channels

00:58:49.220 | because we each have our skills, right?

00:58:51.900 | Somehow along the way, like I learned how to write books.

00:58:53.820 | I had this kind of weird life as a writer

00:58:55.700 | where I sort of spent a lot of time

00:58:57.100 | like thinking about how to put English words together

00:58:59.620 | into sentences and sentences together into paragraphs,

00:59:01.860 | like at length, which is this kind of like weird

00:59:04.940 | specialized skill.

00:59:06.900 | And that's one thing, but like sort of being able to make

00:59:09.100 | like, you know, winning good looking eye catching videos

00:59:12.980 | is like a totally different skill.

00:59:15.020 | And, you know, probably, you know, somewhere out there,

00:59:16.700 | there's probably sort of some like heavy metal band

00:59:19.500 | that's like teaching math through heavy metal

00:59:21.780 | and like using their skills to do that.

00:59:23.300 | I hope there is at any rate.

00:59:25.020 | - Their music and so on, yeah.

00:59:26.540 | But there is something to the process.

00:59:28.780 | I mean, Grant does this especially well,

00:59:31.700 | which is in order to be able to visualize something,

00:59:36.380 | now he writes programs, so it's programmatic visualization.

00:59:39.540 | So like the things he is basically mostly

00:59:42.900 | through his Manum library in Python,

00:59:46.300 | everything is drawn through Python.

00:59:48.280 | You have to truly understand the topic to be able

00:59:54.340 | to visualize it in that way and not just understand it,

00:59:59.700 | but really kind of think in a very novel way.

01:00:04.380 | It's funny 'cause I've spoken with him a couple of times,

01:00:07.540 | spoken to him a lot offline as well.

01:00:09.860 | He really doesn't think he's doing anything new,

01:00:14.100 | meaning like he sees himself as very different

01:00:17.380 | from maybe like a researcher,

01:00:19.260 | but it feels to me like he's creating something

01:00:25.460 | totally new, like that act of understanding

01:00:27.980 | and visualizing is as powerful or has the same kind

01:00:32.220 | of inkling of power as does the process

01:00:34.660 | of proving something.

01:00:35.920 | It doesn't have that clear destination,

01:00:39.940 | but it's pulling out an insight and creating multiple sets

01:00:43.980 | of perspective that arrive at that insight.

01:00:47.140 | - And to be honest, it's something that I think

01:00:49.220 | we haven't quite figured out how to value

01:00:53.340 | inside academic mathematics in the same way,

01:00:55.380 | and this is a bit older, that I think we haven't quite

01:00:57.300 | figured out how to value the development

01:00:59.460 | of computational infrastructure.

01:01:01.020 | We all have computers as our partners now,

01:01:02.900 | and people build computers that sort of assist

01:01:07.820 | and participate in our mathematics.

01:01:09.260 | They build those systems, and that's a kind

01:01:11.660 | of mathematics too, but not in the traditional form

01:01:13.980 | of proving theorems and writing papers.

01:01:16.380 | But I think it's coming.

01:01:17.500 | I mean, I think, for example, the Institute

01:01:21.300 | for Computational Experimental Mathematics at Brown,

01:01:23.900 | which is like a, you know, it's a NSF-funded math institute,

01:01:27.780 | very much part of sort of traditional math academia.

01:01:29.820 | They did an entire theme semester

01:01:31.700 | about visualizing mathematics, like the same kind

01:01:33.900 | of thing that they would do for like an up-and-coming

01:01:36.980 | research topic, like that's pretty cool.

01:01:38.500 | So I think there really is buy-in

01:01:40.220 | from the mathematics community to recognize

01:01:43.940 | that this kind of stuff is important

01:01:45.380 | and counts as part of mathematics,

01:01:47.500 | like part of what we're actually here to do.

01:01:50.540 | - Yeah, I'm hoping to see more and more of that

01:01:51.980 | from like MIT faculty, from faculty

01:01:54.380 | from all the top universities in the world.

01:01:57.500 | Let me ask you this weird question about the Fields Medal,

01:01:59.980 | which is the Nobel Prize in mathematics.

01:02:02.920 | Do you think, since we're talking about computers,

01:02:05.500 | there will one day come a time when a computer,

01:02:10.500 | an AI system, will win a Fields Medal?

01:02:14.080 | - No.

01:02:16.860 | (Lex laughs)

01:02:18.420 | - That's what a human would say.

01:02:19.700 | Why not?

01:02:20.540 | (Lex laughs)

01:02:21.360 | - Is that like, that's like my cap shot,

01:02:23.380 | that's like the proof that I'm a human,

01:02:24.500 | is I deny that I'm not?

01:02:25.340 | (Lex laughs)

01:02:26.820 | - What is, how does he want me to answer?

01:02:28.940 | Is there something interesting to be said about that?

01:02:31.980 | - Yeah, I mean, I am tremendously interested

01:02:34.660 | in what AI can do in pure mathematics.

01:02:37.860 | I mean, it's, of course, it's a parochial interest, right?

01:02:40.540 | You're like, why am I not interested in like how it can

01:02:42.100 | like help feed the world or help solve like real-world

01:02:44.460 | problems, I'm like, can it do more math?

01:02:46.180 | Like, what can I do?

01:02:47.580 | We all have our interests, right?

01:02:49.620 | But I think it is a really interesting conceptual question.

01:02:53.720 | And here too, I think it's important to be kind of

01:02:58.720 | historical, because it's certainly true that there's lots

01:03:01.800 | of things that we used to call research in mathematics

01:03:04.920 | that we would now call computation.

01:03:07.400 | Tasks that we've now offloaded to machines, like, you know,

01:03:10.720 | in 1890, somebody could be like, here's my PhD thesis,

01:03:13.780 | I computed all the invariants of this polynomial ring

01:03:18.200 | under the action of some finite group,

01:03:19.900 | doesn't matter what those words mean,

01:03:21.380 | just it's like some thing that in 1890 would take a person

01:03:25.040 | a year to do and would be a valuable thing

01:03:27.200 | that you might wanna know.

01:03:28.040 | And it's still a valuable thing that you might wanna know,

01:03:29.900 | but now you type a few lines of code in Macaulay or Sage

01:03:34.700 | or Magma, and you just have it.

01:03:37.720 | So we don't think of that as math anymore,

01:03:40.240 | even though it's the same thing.

01:03:41.640 | - What's Macaulay, Sage and Magma?

01:03:43.360 | - Oh, those are computer algebra programs.

01:03:45.020 | So those are like sort of bespoke systems

01:03:46.860 | that lots of mathematicians use.

01:03:48.100 | - Is that similar to Maple and Mathematica?

01:03:49.580 | - Yeah, oh yeah, so it's similar to Maple and Mathematica,

01:03:51.580 | yeah, but a little more specialized, but yeah.

01:03:54.700 | - It's programs that work with symbols

01:03:56.620 | and allow you to do, can you do proofs?

01:03:58.140 | Can you do kind of little leaps and proofs?

01:04:01.060 | - They're not really built for that,

01:04:02.420 | and that's a whole other story.

01:04:04.780 | - But these tools are part of the process

01:04:06.380 | of mathematics now.

01:04:07.300 | - Right, they are now, for most mathematicians,

01:04:09.440 | I would say, part of the process of mathematics.

01:04:11.620 | And so, you know, there's a story I tell in the book

01:04:14.740 | which I'm fascinated by, which is, you know,

01:04:17.740 | so far, attempts to get AIs to prove interesting theorems

01:04:22.740 | have not done so well.

01:04:27.260 | It doesn't mean they can't.

01:04:28.100 | There's actually a paper I just saw,

01:04:29.700 | which has a very nice use of a neural net

01:04:32.420 | to find counter examples to conjecture.

01:04:34.580 | Somebody said like, well, maybe this is always that.

01:04:37.180 | And you can be like, well, let me sort of train an AI

01:04:39.260 | to sort of try to find things where that's not true,

01:04:43.140 | and it actually succeeded.

01:04:43.980 | Now, in this case, if you look at the things that it found,

01:04:47.880 | you say like, okay, I mean,

01:04:51.360 | these are not famous conjectures, okay?

01:04:53.600 | So like, somebody wrote this down, maybe this is so.

01:04:56.200 | Looking at what the AI came up with, you're like,

01:05:00.880 | you know, I'll bet if like five grad students

01:05:03.720 | had thought about that problem,

01:05:04.680 | they wouldn't have come up with that.

01:05:05.520 | I mean, when you see it, you're like,

01:05:06.800 | okay, that is one of the things you might try

01:05:08.400 | if you sort of like put some work into it.

01:05:10.480 | Still, it's pretty awesome.

01:05:12.600 | But the story I tell in the book,

01:05:14.800 | which I'm fascinated by is there is,

01:05:19.800 | okay, we're gonna go back to knots.

01:05:21.480 | - It's cool.

01:05:22.320 | - There's a knot called the Conway knot.

01:05:23.920 | After John Conway, who maybe we'll talk about

01:05:25.400 | a very interesting character also.

01:05:26.400 | - Yeah, it's a small tangent.

01:05:28.160 | Somebody I was supposed to talk to,

01:05:29.480 | and unfortunately he passed away,

01:05:30.760 | and he's somebody I find as an incredible mathematician,

01:05:35.240 | incredible human being.

01:05:36.240 | - Oh, and I am sorry that you didn't get a chance

01:05:38.320 | because having had the chance to talk to him a lot

01:05:40.320 | when I was a postdoc, yeah, you missed out.

01:05:44.140 | There was no way to sugarcoat it.

01:05:45.160 | I'm sorry that you didn't get that chance.

01:05:46.640 | - Yeah, it is what it is.

01:05:47.920 | So, knots.

01:05:50.120 | - Yeah, so there was a question,

01:05:51.520 | and again, it doesn't matter the technicalities

01:05:53.860 | of the question, but it's a question

01:05:55.000 | of whether the knot is sliced.

01:05:56.320 | It has to do with something about what kinds

01:05:59.960 | of three-dimensional surfaces in four dimensions

01:06:02.340 | can be bounded by this knot.

01:06:03.480 | But never mind what it means.

01:06:04.360 | It's some question, and it's actually very hard

01:06:07.720 | to compute whether a knot is sliced or not.

01:06:12.880 | And in particular, the question of the Conway knot,

01:06:16.900 | whether it was sliced or not, was particularly vexed.

01:06:21.320 | Until it was solved just a few years ago

01:06:24.620 | by Lisa Piccirillo, who actually,

01:06:26.260 | now that I think of it, was here in Austin.

01:06:27.700 | I believe she was a grad student at UT Austin at the time.

01:06:29.980 | I didn't even realize there was an Austin connection

01:06:31.540 | to this story until I started telling it.

01:06:34.100 | In fact, I think she's now at MIT,

01:06:35.780 | so she's basically following you around.

01:06:38.140 | If I remember correctly. - The reverse.

01:06:39.820 | - There's a lot of really interesting richness

01:06:41.340 | to this story.

01:06:42.660 | One thing about it is her paper was rather,

01:06:45.580 | was very short.

01:06:46.860 | It was very short and simple.

01:06:48.100 | Nine pages, of which two were pictures.

01:06:50.060 | Very short for a paper solving a major conjecture.

01:06:54.580 | And it really makes you think about what we mean

01:06:55.860 | by difficulty in mathematics.

01:06:57.380 | Do you say, oh, actually the problem wasn't difficult

01:06:59.420 | because you could solve it so simply?

01:07:00.820 | Or do you say, well no, evidently it was difficult

01:07:03.260 | because the world's top topologist worked on it

01:07:05.980 | for 20 years and nobody could solve it,

01:07:07.340 | so therefore it is difficult.

01:07:08.540 | Or is it that we need some new category of things

01:07:11.460 | about which it's difficult to figure out

01:07:13.500 | that they're not difficult?

01:07:15.700 | - I mean, this is the computer science formulation,

01:07:18.640 | but the journey to arrive at the simple answer

01:07:23.640 | may be difficult, but once you have the answer,

01:07:26.640 | it will then appear simple.

01:07:28.720 | And I mean, there might be a large category,

01:07:30.620 | I hope there's a large set of such solutions,

01:07:37.380 | because once we stand at the end of the scientific process

01:07:42.380 | that we're at the very beginning of,

01:07:46.100 | or at least it feels like,

01:07:47.540 | I hope there's just simple answers to everything.

01:07:50.140 | That we'll look and it'll be simple laws

01:07:53.580 | that govern the universe,

01:07:55.100 | simple explanation of what is consciousness,

01:07:58.020 | of what is love, is mortality fundamental to life,

01:08:02.340 | what's the meaning of life,

01:08:05.460 | are humans special or we're just another

01:08:08.020 | sort of reflection of all that is beautiful

01:08:12.500 | in the universe in terms of life forms,

01:08:15.220 | all of it is life and just has different,

01:08:18.380 | when taken from a different perspective,

01:08:19.900 | is all life can seem more valuable or not,

01:08:22.460 | but really it's all part of the same thing.

01:08:24.180 | All those will have a nice two equations,

01:08:26.520 | maybe one equation.

01:08:28.100 | - Why do you think you want those questions

01:08:30.800 | to have simple answers?

01:08:32.740 | - I think just like symmetry and the breaking of symmetry

01:08:36.780 | is beautiful somehow,

01:08:38.160 | there's something beautiful about simplicity.

01:08:41.420 | I think it--

01:08:42.260 | - So it's aesthetic.

01:08:43.420 | - It's aesthetic, yeah.

01:08:44.580 | But it's aesthetic in the way that happiness is an aesthetic.

01:08:49.620 | Why is that so joyful that a simple explanation

01:08:55.380 | that governs a large number of cases is really appealing?

01:09:01.900 | Even when it's not,

01:09:04.380 | like obviously we get a huge amount of trouble with that

01:09:07.320 | because oftentimes it doesn't have to be connected

01:09:11.420 | with reality or even that explanation

01:09:13.460 | could be exceptionally harmful.

01:09:15.560 | Most of like the world's history

01:09:18.140 | that was governed by hate and violence

01:09:20.980 | had a very simple explanation at the core

01:09:23.940 | that was used to cause the violence and the hatred.

01:09:26.940 | So like we get into trouble with that,

01:09:28.740 | but why is that so appealing?

01:09:30.380 | And in its nice forms in mathematics,

01:09:33.820 | like you look at the Einstein papers,

01:09:36.580 | why are those so beautiful?

01:09:38.020 | And why is the Andrew Wiles'

01:09:39.860 | proof of the Fermat's last theorem not quite so beautiful?

01:09:43.240 | Like what's beautiful about that story

01:09:45.620 | is the human struggle of like the human story

01:09:48.940 | of perseverance, of the drama,

01:09:51.700 | of not knowing if the proof is correct

01:09:53.900 | and ups and downs and all of those kinds of things.

01:09:56.060 | That's the interesting part.

01:09:57.220 | But the fact that the proof is huge and nobody understand,

01:09:59.380 | well, from my outsider's perspective,

01:10:01.220 | nobody understands what the heck it is,

01:10:03.180 | is not as beautiful as it could have been.

01:10:06.700 | I wish it was what Fermat originally said,

01:10:09.220 | which is, you know, it's not small enough

01:10:14.220 | to fit in the margins of this page,

01:10:17.200 | but maybe if he had like a full page

01:10:19.300 | or maybe a couple of post-it notes,

01:10:20.820 | he would have enough to do the proof.

01:10:22.940 | What do you make of,

01:10:23.860 | if we could take another of a multitude of tangents,

01:10:27.740 | what do you make of Fermat's last theorem?

01:10:29.260 | Because the statement, there's a few theorems,

01:10:31.660 | there's a few problems that are deemed by the world

01:10:35.540 | throughout its history to be exceptionally difficult.

01:10:37.780 | And that one in particular is really simple to formulate

01:10:42.380 | and really hard to come up with a proof for.

01:10:46.260 | And it was like taunted as simple by Fermat himself.

01:10:51.260 | Is there something interesting to be said

01:10:52.860 | about that X to the N plus Y to the N

01:10:56.780 | equals Z to the N for N of three or greater.

01:11:00.140 | Is there a solution to this?

01:11:02.540 | And then how do you go about proving that?

01:11:04.300 | Like, how would you try to prove that?

01:11:08.140 | And what do you learn from the proof

01:11:09.940 | that eventually emerged by Andrew Wiles?

01:11:12.100 | - Yeah, so right, let me just say the background,

01:11:14.380 | 'cause I don't know if everybody listening knows the story.

01:11:17.020 | So, you know, Fermat was an early number theorist,

01:11:21.940 | at least sort of an early mathematician.

01:11:23.100 | Those special adjacent didn't really exist back then.

01:11:27.060 | He comes up in the book actually in the context

01:11:29.420 | of a different theorem of his that has to do with testing

01:11:32.620 | whether a number is prime or not.

01:11:34.620 | So I write about, he was one of the ones who was salty

01:11:37.420 | and like he would exchange these letters

01:11:39.460 | where he and his correspondence would like

01:11:41.140 | try to top each other and vex each other with questions

01:11:44.100 | and stuff like this.

01:11:44.940 | But this particular thing,

01:11:46.020 | it's called Fermat's last theorem

01:11:49.300 | because it's a note he wrote

01:11:54.100 | in his copy of the "Disquisitonis Arithmeticae."

01:11:57.380 | Like he wrote, "Here's an equation.

01:11:59.860 | "It has no solutions.

01:12:00.780 | "I can prove it, but the proof's like a little too long

01:12:03.260 | "to fit in the margin of this book."

01:12:05.540 | He was just like writing a note to himself.

01:12:07.140 | Now, let me just say historically,

01:12:08.580 | we know that Fermat did not have a proof of this theorem.

01:12:11.620 | For a long time, people were like,

01:12:14.420 | this mysterious proof that was lost,

01:12:16.020 | a very romantic story, right?

01:12:17.260 | But Fermat later, he did prove special cases of this theorem

01:12:23.260 | and wrote about it, talked to people about the problem.

01:12:27.380 | It's very clear from the way that he wrote

01:12:29.100 | where he can solve certain examples

01:12:30.740 | of this type of equation

01:12:32.140 | that he did not know how to do the whole thing.

01:12:35.740 | - He may have had a deep, simple intuition

01:12:39.940 | about how to solve the whole thing

01:12:41.820 | that he had at that moment

01:12:43.820 | without ever being able to come up with a complete proof.

01:12:47.060 | And that intuition maybe lost the time.

01:12:49.100 | - Maybe.

01:12:51.900 | But I think we, but you're right, that that is unknowable.

01:12:54.540 | But I think what we can know is that later,

01:12:56.980 | he certainly did not think that he had a proof

01:12:59.140 | that he was concealing from people.

01:13:00.660 | He thought he didn't know how to prove it,

01:13:04.380 | and I also think he didn't know how to prove it.

01:13:06.420 | Now, I understand the appeal of saying,

01:13:10.260 | wouldn't it be cool if this very simple equation,

01:13:12.580 | there was a very simple, clever, wonderful proof

01:13:16.060 | that you could do in a page or two.

01:13:17.420 | And that would be great, but you know what?

01:13:19.020 | There's lots of equations like that

01:13:20.380 | that are solved by very clever methods like that,

01:13:22.220 | including the special cases that Fermat wrote about,

01:13:24.380 | the method of descent, which is very wonderful.

01:13:26.900 | But in the end, those are nice things

01:13:31.740 | that you teach in an undergraduate class,

01:13:33.940 | and it is what it is, but they're not big.

01:13:37.020 | On the other hand, work on the Fermat problem,

01:13:41.620 | that's what we like to call it,

01:13:42.460 | because it's not really his theorem,

01:13:44.100 | because we don't think he proved it.

01:13:46.020 | I mean, work on the Fermat problem

01:13:49.220 | and develop this incredible richness of number theory

01:13:52.380 | that we now live in today.

01:13:55.060 | And not, by the way, just Wiles, Andrew Wiles being

01:13:58.340 | the person who, together with Richard Taylor,

01:13:59.700 | finally proved this theorem.

01:14:01.780 | But you have this whole moment

01:14:03.260 | that people try to prove this theorem and they fail,

01:14:06.560 | and there's a famous false proof by LeMay

01:14:08.780 | from the 19th century where Kummer,

01:14:11.460 | in understanding what mistake LeMay had made

01:14:14.500 | in this incorrect proof,

01:14:16.320 | basically understands something incredible,

01:14:18.340 | which is that a thing we know about numbers

01:14:20.940 | is that you can factor them,

01:14:24.500 | and you can factor them uniquely.

01:14:26.940 | There's only one way to break a number up into primes.

01:14:30.300 | Like if we think of a number like 12,

01:14:32.260 | 12 is two times three times two.

01:14:35.500 | I had to think about it.

01:14:36.700 | Or it's two times two times three,

01:14:39.700 | of course you can reorder them.

01:14:41.580 | But there's no other way to do it.

01:14:43.220 | There's no universe in which 12 is something times five,

01:14:46.140 | or in which there's like four threes in it.

01:14:47.520 | Nope, 12 is like two twos and a three.

01:14:49.160 | Like that is what it is.

01:14:50.700 | And that's such a fundamental feature of arithmetic

01:14:54.820 | that we almost think of it like God's law.

01:14:56.540 | You know what I mean?

01:14:57.380 | It has to be that way.

01:14:58.200 | - That's a really powerful idea.

01:14:59.660 | It's so cool that every number

01:15:02.540 | is uniquely made up of other numbers.

01:15:04.740 | And like made up meaning,

01:15:08.380 | like there's these like basic atoms that form molecules

01:15:12.820 | that get built on top of each other.

01:15:15.420 | - I love it.

01:15:16.240 | I teach undergraduate number theory.

01:15:18.100 | It's like, it's the first really deep theorem

01:15:22.240 | that you prove.

01:15:23.600 | What's amazing is the fact that you can factor

01:15:26.520 | a number into primes is much easier.

01:15:29.020 | Essentially Euclid knew it,

01:15:30.380 | although he didn't quite put it in that way.

01:15:33.740 | The fact that you can do it at all.

01:15:34.900 | What's deep is the fact that there's only one way to do it.

01:15:38.860 | Or however you sort of chop the number up,

01:15:40.680 | you end up with the same set of prime factors.

01:15:44.860 | And indeed what people finally understood

01:15:48.300 | at the end of the 19th century is that

01:15:51.960 | if you work in number systems slightly more general

01:15:54.680 | than the ones we're used to,

01:15:56.180 | which it turns out are relevant to Fermat,

01:16:00.340 | all of a sudden this stops being true.

01:16:03.120 | Things get, I mean, things get more complicated.

01:16:08.000 | And now because you were praising simplicity before,

01:16:10.040 | you were like, it's so beautiful, unique factorization.

01:16:12.920 | It's so great.

01:16:13.760 | It's like, it's when I tell you that

01:16:15.020 | in more general number systems

01:16:16.660 | there is no unique factorization,

01:16:18.340 | maybe you're like, that's bad.

01:16:19.340 | I'm like, no, that's good.

01:16:20.220 | Because there's like a whole new world of phenomena

01:16:22.540 | to study that you just can't see through the lens

01:16:25.100 | of the numbers that we're used to.

01:16:26.980 | So I'm for complication.

01:16:29.940 | I'm highly in favor of complication.

01:16:32.620 | Every complication is like an opportunity

01:16:34.620 | for new things to study.

01:16:35.900 | - And is that the big,

01:16:37.320 | kind of one of the big insights for you

01:16:40.540 | from Andrew Wiles' proof?

01:16:42.900 | Is there interesting insights about the process

01:16:46.280 | that you used to prove that sort of resonates

01:16:49.580 | with you as a mathematician?

01:16:51.360 | Is there an interesting concept that emerged from it?

01:16:54.400 | Is there interesting human aspects to the proof?

01:16:57.140 | - Whether there's interesting human aspects

01:16:59.860 | to the proof itself is an interesting question.

01:17:02.640 | Certainly it has a huge amount of richness.

01:17:05.520 | Sort of at its heart is an argument

01:17:07.680 | of what's called deformation theory.

01:17:12.420 | Which was in part created by my PhD advisor, Barry Mazur.

01:17:17.420 | - Can you speak to what deformation theory is?

01:17:20.160 | - I can speak to what it's like.

01:17:21.920 | - Sure. - How about that?

01:17:22.920 | - What does it rhyme with?

01:17:24.640 | - Right, well, the reason that Barry called it

01:17:27.320 | deformation theory, I think he's the one

01:17:29.460 | who gave it the name.

01:17:30.860 | I hope I'm not wrong in saying some day.

01:17:32.320 | - In your book, you have calling different things

01:17:35.120 | by the same name as one of the things

01:17:37.860 | in the beautiful map that opens the book.

01:17:40.400 | - Yes, and this is a perfect example.

01:17:42.040 | So this is another phrase of Poincaré,

01:17:44.080 | this incredible generator of slogans and aphorisms.

01:17:46.800 | He said, "Mathematics is the art of calling

01:17:48.320 | "different things by the same name."

01:17:49.880 | That very thing we do, right?

01:17:52.520 | When we're like this triangle and this triangle,

01:17:54.000 | come on, they're the same triangle,

01:17:55.040 | they're just in a different place, right?

01:17:56.520 | So in the same way, it came to be understood

01:18:00.440 | that the kinds of objects that you study

01:18:04.160 | when you study Fermat's last theorem,

01:18:10.200 | and let's not even be too careful

01:18:12.120 | about what these objects are.

01:18:13.720 | I can tell you they are Galois representations

01:18:15.760 | in modular forms, but saying those words

01:18:18.400 | is not gonna mean so much.

01:18:19.700 | But whatever they are, they're things

01:18:21.680 | that can be deformed, moved around a little bit.

01:18:25.900 | And I think the insight of what Andrew,

01:18:28.440 | and then Andrew and Richard were able to do,

01:18:31.280 | was to say something like this.

01:18:33.680 | A deformation means moving something just a tiny bit,

01:18:36.680 | like an infinitesimal amount.

01:18:39.400 | If you really are good at understanding

01:18:41.440 | which ways a thing can move in a tiny, tiny, tiny,

01:18:44.680 | infinitesimal amount in certain directions,

01:18:46.760 | maybe you can piece that information together

01:18:49.240 | to understand the whole global space

01:18:51.200 | in which it can move.

01:18:52.520 | And essentially, their argument comes down

01:18:54.400 | to showing that two of those big global spaces

01:18:57.320 | are actually the same, the fabled R equals T

01:19:00.080 | part of their proof, which is at the heart of it.

01:19:05.240 | And it involves this very careful principle like that.

01:19:09.560 | But that being said, what I just said,

01:19:11.880 | it's probably not what you're thinking,

01:19:14.640 | because what you're thinking when you think,

01:19:16.280 | oh, I have a point in space and I move it around

01:19:18.500 | like a little tiny bit,

01:19:19.700 | you're using your notion of distance

01:19:26.720 | that's from calculus.

01:19:28.320 | We know what it means for two points in the real line

01:19:30.180 | to be close together.

01:19:32.960 | So yet another thing that comes up in the book a lot

01:19:37.080 | is this fact that the notion of distance

01:19:41.200 | is not given to us by God.

01:19:42.600 | We could mean a lot of different things by distance.

01:19:44.640 | And just in the English language, we do that all the time.

01:19:46.520 | We talk about somebody being a close relative.

01:19:49.020 | It doesn't mean they live next door to you, right?

01:19:51.040 | It means something else.

01:19:52.800 | There's a different notion of distance we have in mind.

01:19:54.840 | And there are lots of notions of distances

01:19:57.520 | that you could use.

01:19:58.840 | In the natural language processing community in AI,

01:20:01.560 | there might be some notion of semantic distance

01:20:04.120 | or lexical distance between two words.

01:20:06.340 | How much do they tend to arise in the same context?

01:20:08.720 | That's incredibly important for doing auto-complete

01:20:13.440 | and machine translation and stuff like that.

01:20:15.440 | And it doesn't have anything to do with,

01:20:16.400 | are they next to each other in the dictionary?

01:20:17.960 | It's a different kind of distance.

01:20:19.260 | Okay, ready?

01:20:20.100 | In this kind of number theory,

01:20:21.840 | there was a crazy distance called the P-adic distance.

01:20:25.120 | I didn't write about this that much in the book,

01:20:26.760 | because even though I love it

01:20:27.580 | and it's a big part of my research life,

01:20:28.640 | it gets a little bit into the weeds,

01:20:29.760 | but your listeners are gonna hear about it now.

01:20:32.520 | - Please.

01:20:33.360 | - Where, you know, what a normal person says

01:20:35.920 | when they say two numbers are close,

01:20:37.740 | they say like, you know,

01:20:38.920 | their difference is like a small number,

01:20:40.260 | like seven and eight are close,

01:20:41.720 | because their difference is one and one's pretty small.

01:20:44.420 | If we were to be what's called a two-adic number theorist,

01:20:48.600 | we'd say, oh, two numbers are close

01:20:50.920 | if their difference is a multiple of a large power of two.

01:20:56.600 | So like one and 49 are close

01:21:00.920 | because their difference is 48,

01:21:03.000 | and 48 is a multiple of 16,

01:21:04.800 | which is a pretty large power of two.

01:21:06.680 | Whereas one and two are pretty far away

01:21:09.720 | because the difference between them is one,

01:21:12.480 | which is not even a multiple of a power of two at all.

01:21:14.240 | It's odd.

01:21:15.600 | You wanna know what's really far from one?

01:21:17.680 | Like one and 1/64,

01:21:20.560 | because their difference is a negative power of two,

01:21:24.720 | two to the minus six.

01:21:25.640 | So those points are quite, quite far away.

01:21:28.400 | - Two to the power of a large n would be two,

01:21:32.260 | if that's the difference between two numbers

01:21:35.600 | and they're close.

01:21:37.120 | - Yeah, so two to a large power is,

01:21:39.200 | in this metric, a very small number,

01:21:41.640 | and two to a negative power is a very big number.

01:21:44.800 | - That's two-adic, okay.

01:21:46.760 | I can't even visualize that.

01:21:48.680 | - It takes practice.

01:21:49.680 | It takes practice.

01:21:50.520 | If you've ever heard of the Cantor set,

01:21:51.800 | it looks kind of like that.

01:21:54.080 | So it is crazy that this is good for anything, right?

01:21:57.280 | I mean, this just sounds like a definition

01:21:58.840 | that someone would make up to torment you.

01:22:00.600 | But what's amazing is there's a general theory of distance

01:22:05.520 | where you say any definition you make

01:22:08.360 | that satisfies certain axioms

01:22:09.840 | deserves to be called a distance, and this--

01:22:12.000 | - See, I'm sorry to interrupt.

01:22:13.840 | My brain, you broke my brain.

01:22:15.440 | - Awesome.

01:22:16.520 | - 10 seconds ago.

01:22:18.080 | 'Cause I'm also starting to map,

01:22:20.380 | for the two-adic case, to binary numbers,

01:22:23.840 | 'cause we romanticize those.

01:22:25.280 | So I was trying to--

01:22:26.120 | - Oh, that's exactly the right way to think of it.

01:22:27.280 | - I was trying to mess with number,

01:22:28.800 | and I was trying to see, okay, which ones are close?

01:22:31.840 | And then I'm starting to visualize

01:22:33.000 | different binary numbers and how they,

01:22:35.620 | which ones are close to each other.

01:22:37.320 | And I'm not sure, well, I think there's a--

01:22:39.520 | - No, no, it's very similar.

01:22:40.560 | That's exactly the right way to think of it.

01:22:41.960 | It's almost like binary numbers written in reverse.

01:22:44.520 | - Right.

01:22:45.360 | - Because in a binary expansion, two numbers are close.

01:22:47.360 | A number that's small is like .0000 something.

01:22:50.840 | Something that's the decimal,

01:22:51.680 | and it starts with a lot of zeros.

01:22:53.200 | In the two-adic metric, a binary number is very small

01:22:56.840 | if it ends with a lot of zeros,

01:22:59.120 | and then the decimal point.

01:23:01.680 | - Gotcha.

01:23:02.520 | - So it is kind of like binary numbers written backwards

01:23:04.040 | is actually, that's what I should have said, Lex.

01:23:06.480 | (Lex laughing)

01:23:07.400 | That's a very good metaphor.

01:23:08.800 | - No, you said, okay, but so why is that interesting,

01:23:12.040 | except for the fact that it's a beautiful kind of framework,

01:23:17.040 | different kind of framework

01:23:19.640 | of which to think about distances.

01:23:21.000 | And you're talking about not just the two-adic,

01:23:23.120 | but the generalization of that.

01:23:24.320 | Why is that interesting? - Yeah, the NEP.

01:23:26.320 | Because that's the kind of deformation that comes up

01:23:29.400 | in Wiles' proof, that deformation,

01:23:32.760 | where moving something a little bit

01:23:34.400 | means a little bit in this two-adic sense.

01:23:36.960 | - Trippy, okay.

01:23:38.160 | - No, I mean, I just get excited talking about it,

01:23:40.160 | and I just taught this in the fall semester.

01:23:42.800 | - But reformulating, why is...

01:23:49.520 | So you pick a different measure of distance

01:23:52.920 | over which you can talk about very tiny changes,

01:23:57.080 | and then use that to then prove things

01:23:59.760 | about the entire thing.

01:24:01.480 | - Yes, although, honestly, what I would say,

01:24:05.240 | I mean, it's true that we use it to prove things,

01:24:08.400 | but I would say we use it to understand things.

01:24:10.680 | And then because we understand things better,

01:24:12.600 | then we can prove things.

01:24:13.680 | But the goal is always the understanding.

01:24:15.400 | The goal is not so much to prove things.

01:24:17.960 | The goal is not to know what's true or false.

01:24:19.880 | I mean, this is the thing I write about

01:24:20.920 | in the book "Near the End."

01:24:21.760 | I mean, it's something that, it's a wonderful,

01:24:23.520 | wonderful essay by Bill Thurston,

01:24:26.600 | kind of one of the great geometers of our time,

01:24:28.200 | who unfortunately passed away a few years ago,

01:24:30.800 | called on proof and progress in mathematics.

01:24:34.040 | And he writes very wonderfully about how,

01:24:36.120 | you know, it's not a theorem factory

01:24:39.040 | where we have a production quota.

01:24:40.600 | I mean, the point of mathematics

01:24:41.760 | is to help humans understand things.

01:24:44.440 | And the way we test that is that we're proving new theorems

01:24:47.280 | along the way, that's the benchmark,

01:24:48.520 | but that's not the goal.

01:24:50.040 | - Yeah, but just as a kind of, absolutely,

01:24:52.440 | but as a tool, it's kind of interesting

01:24:54.920 | to approach a problem by saying,

01:24:57.440 | how can I change the distance function?

01:24:59.780 | Like what, the nature of distance,

01:25:03.720 | because that might start to lead to insights

01:25:07.080 | for deeper understanding.

01:25:08.400 | Like if I were to try to describe human society

01:25:12.600 | by distance, two people are close if they love each other.

01:25:17.160 | - Right.

01:25:18.000 | - And then start to do a full analysis

01:25:21.040 | on everybody that lives on earth currently,

01:25:23.840 | the 7 billion people.

01:25:25.300 | And from that perspective,

01:25:27.680 | as opposed to the geographic perspective of distance,

01:25:30.840 | and then maybe there could be a bunch of insights

01:25:33.000 | about the source of violence,

01:25:35.580 | the source of maybe entrepreneurial success or invention

01:25:40.080 | or economic success or different systems,

01:25:42.960 | communism, capitalism, start to,

01:25:44.680 | I mean, that's, I guess, what economics tries to do,

01:25:47.480 | but really saying, okay, let's think outside the box

01:25:50.520 | about totally new distance functions

01:25:52.840 | that could unlock something profound about the space.

01:25:57.240 | - Yeah, because think about it, okay, here's,

01:25:59.620 | I mean, now we're gonna talk about AI,

01:26:01.200 | which you know a lot more about than I do,

01:26:02.980 | so just start laughing uproariously

01:26:05.840 | if I say something that's completely wrong.

01:26:07.080 | - We both know very little relative

01:26:09.840 | to what we will know centuries from now.

01:26:12.600 | - That is a really good, humble way to think about it.

01:26:15.720 | I like it, okay, so let's just go for it.

01:26:18.360 | Okay, so I think you'll agree with this,

01:26:20.520 | that in some sense, what's good about AI

01:26:23.040 | is that we can't test any case in advance.

01:26:26.360 | The whole point of AI is to make,

01:26:27.840 | or one point of it, I guess, is to make good predictions

01:26:30.560 | about cases we haven't yet seen.

01:26:32.640 | And in some sense, that's always gonna involve

01:26:34.840 | some notion of distance,

01:26:35.960 | because it's always gonna involve

01:26:37.840 | somehow taking a case we haven't seen

01:26:40.040 | and saying what cases that we have seen is it close to,

01:26:43.840 | is it like, is it somehow an interpolation between.

01:26:46.820 | Now, when we do that, in order to talk about

01:26:49.960 | things being like other things,

01:26:52.040 | implicitly or explicitly,

01:26:53.480 | we're invoking some notion of distance,

01:26:55.600 | and boy, we better get it right.

01:26:57.640 | If you try to do natural language processing

01:26:59.200 | and your idea of distance between words

01:27:01.200 | is how close they are in the dictionary,

01:27:03.160 | when you write them in alphabetical order,

01:27:04.440 | you are gonna get pretty bad translations, right?

01:27:08.160 | No, the notion of distance has to come from somewhere else.

01:27:11.520 | - Yeah, that's essentially

01:27:13.000 | what neural networks are doing,

01:27:14.120 | that's what word embeddings are doing,

01:27:15.520 | is coming up with-- - Yes.

01:27:17.320 | In the case of word embeddings, literally,

01:27:18.760 | like literally what they are doing

01:27:20.080 | is learning a distance-- - But those are

01:27:21.320 | super complicated distance functions,

01:27:23.560 | and it's almost nice to think

01:27:26.200 | maybe there's a nice transformation that's simple.

01:27:31.200 | Sorry, there's a nice formulation of the distance.

01:27:34.480 | - Again with the simple.

01:27:36.560 | So you don't, let me ask you about this.

01:27:39.740 | From an understanding perspective,

01:27:43.400 | there's the Richard Feynman, maybe attributed to him,

01:27:45.600 | but maybe many others,

01:27:47.320 | is this idea that if you can't explain something simply,

01:27:52.440 | that you don't understand it.

01:27:54.080 | In how many cases, how often is that true?

01:28:00.680 | Do you find there's some profound truth in that?

01:28:05.560 | Oh, okay, so you were about to ask, is it true,

01:28:07.680 | to which I would say flatly no,

01:28:09.320 | but then you followed that up with,

01:28:11.240 | is there some profound truth in it?

01:28:13.200 | And I'm like, okay, sure, so there's some truth in it.

01:28:15.400 | - But it's not true. - It's just not.

01:28:17.200 | (laughing)

01:28:19.440 | - This is your mathematician answer.

01:28:22.760 | - The truth that is in it is that learning

01:28:25.760 | to explain something helps you understand it.

01:28:30.000 | But real things are not simple.

01:28:33.460 | A few things are, most are not.

01:28:36.660 | And I don't, to be honest, I don't, I mean, I don't,

01:28:39.500 | we don't really know whether Feynman really said that,

01:28:41.140 | right, or something like that is sort of disputed,

01:28:43.100 | but I don't think Feynman could have literally believed that,

01:28:46.220 | whether or not he said it.

01:28:47.260 | And you know, he was the kind of guy,

01:28:49.040 | I didn't know him, but I'm reading his writing,

01:28:51.420 | he liked to sort of say stuff, like stuff that sounded good.

01:28:55.060 | You know what I mean?

01:28:55.900 | So it totally strikes me as the kind of thing

01:28:57.660 | he could have said because he liked the way saying it

01:29:00.220 | made him feel.

01:29:02.260 | But also knowing that he didn't like literally mean it.

01:29:04.540 | - Well, I definitely have a lot of friends

01:29:07.820 | and I've talked to a lot of physicists,

01:29:09.600 | and they do derive joy from believing

01:29:12.780 | that they can explain stuff simply.

01:29:14.580 | Or believing it's possible to explain stuff simply,

01:29:17.860 | even when the explanation's not actually that simple.

01:29:20.220 | Like I've heard people think that the explanation's simple

01:29:23.980 | and they do the explanation, and I think it is simple,

01:29:27.620 | but it's not capturing the phenomena that we're discussing.

01:29:30.620 | It's capturing, it somehow maps in their mind,

01:29:33.100 | but it's taking as a starting point,

01:29:36.020 | as an assumption that there's a deep knowledge

01:29:38.220 | and a deep understanding that's actually very complicated.

01:29:41.820 | And the simplicity is almost like a poem

01:29:45.260 | about the more complicated thing,

01:29:46.860 | as opposed to a distillation.

01:29:48.740 | - And I love poems, but a poem is not an explanation.

01:29:51.740 | (laughing)

01:29:53.500 | - Well, some people might disagree with that,

01:29:55.580 | but certainly from a mathematical perspective--

01:29:57.500 | - No poet would disagree with it.

01:29:59.620 | - No poet would disagree.

01:30:01.220 | You don't think there's some things

01:30:02.740 | that can only be described imprecisely?

01:30:06.540 | - As an explanation, I don't think any poet

01:30:09.140 | would say their poem is an explanation.

01:30:10.540 | They might say it's a description,

01:30:11.820 | they might say it's sort of capturing sort of--

01:30:14.440 | - Well, some people might say the only truth is like music.

01:30:17.700 | Not the only truth, but some truth

01:30:21.700 | can only be expressed through art.

01:30:23.580 | And I mean, that's the whole thing we're talking about,

01:30:26.900 | religion and myth.

01:30:28.220 | There's some things that are limited cognitive capabilities

01:30:32.380 | and the tools of mathematics or the tools of physics

01:30:35.220 | are just not going to allow us to capture.

01:30:37.380 | Like, it's possible consciousness is one of those things.

01:30:40.280 | - Yes, that is definitely possible.

01:30:44.620 | But I would even say, look, I mean,

01:30:46.420 | consciousness is a thing about which we're still in the dark

01:30:48.460 | as to whether there's an explanation

01:30:50.420 | we would understand as an explanation at all.

01:30:53.620 | By the way, okay, I gotta give yet one more

01:30:55.420 | amazing Poincaré quote, 'cause this guy

01:30:56.780 | has never stopped coming up with great quotes.

01:30:59.080 | Paul Erdős, another fellow who appears in the book,

01:31:02.820 | and by the way, he thinks about this notion of distance

01:31:05.500 | of personal affinity, kind of like what you're talking about,

01:31:08.540 | that kind of social network and that notion of distance

01:31:11.260 | that comes from that, so that's something that Paul Erdős--

01:31:13.260 | - Erdős did?

01:31:14.340 | - Well, he thought about distances and networks.

01:31:16.020 | I guess he didn't think about the social network--

01:31:17.780 | - Oh, that's fascinating.

01:31:18.620 | That's how it started, that story of Erdős number.

01:31:20.060 | Yeah, okay, sorry to distract.

01:31:21.820 | - But Erdős was sort of famous for saying,

01:31:25.100 | and this is sort of along the lines of what we were saying,

01:31:26.820 | he talked about the book, capital T, capital B, the book,

01:31:31.340 | and that's the book where God keeps the right proof

01:31:33.420 | of every theorem, so when he saw a proof he really liked,

01:31:36.340 | it was really elegant, really simple,

01:31:37.980 | like that's from the book, that's like you found

01:31:39.820 | one of the ones that's in the book.

01:31:42.000 | He wasn't a religious guy, by the way.

01:31:44.700 | He referred to God as the supreme fascist.

01:31:47.020 | But somehow he was like, I don't really believe in God,

01:31:49.720 | but I believe in God's book.

01:31:53.300 | But Poincaré, on the other hand,

01:31:54.980 | and by the way, there are other,

01:31:57.020 | Hilda Hudson is one who comes up in this book.

01:31:58.700 | She also kind of saw math.

01:32:00.800 | She's one of the people who sort of develops

01:32:03.940 | the disease model that we now use,

01:32:06.900 | that we use to sort of track pandemics,

01:32:08.380 | this SIR model that sort of originally comes

01:32:10.380 | from her work with Ronald Ross.

01:32:11.940 | But she was also super, super, super devout,

01:32:14.500 | and she also, sort of on the other side

01:32:17.380 | of the religious coin, was like, yeah,

01:32:18.580 | math is how we communicate with God.

01:32:20.700 | She has a great, all these people are incredibly quotable.

01:32:22.560 | She says, you know, math is,

01:32:24.700 | the truth, the things about mathematics,

01:32:26.620 | she's like, they're not the most important of God thoughts,

01:32:29.460 | but they're the only ones that we can know precisely.

01:32:32.660 | So she's like, this is the one place

01:32:34.020 | where we get to sort of see what God's thinking

01:32:35.460 | when we do mathematics.

01:32:37.340 | - Again, not a fan of poetry or music.

01:32:39.140 | Some people say Hendrix is like,

01:32:41.020 | some people say chapter one of that book is mathematics,

01:32:44.300 | and then chapter two is like classic rock.

01:32:46.840 | Right, so like, it's not clear that the--

01:32:51.340 | I'm sorry, you just sent me off on a tangent,

01:32:52.720 | just imagining like Erdős at a Hendrix concert,

01:32:55.040 | trying to figure out if it was from the book or not.

01:32:58.460 | What I was coming to was just to say,

01:33:00.960 | but what Poincaré said about this is he's like,

01:33:03.320 | if this has all worked out in the language of the divine,

01:33:08.440 | and if a divine being came down and told it to us,

01:33:12.440 | we wouldn't be able to understand it, so it doesn't matter.

01:33:15.000 | So Poincaré was of the view that there were things

01:33:17.360 | that were sort of inhumanly complex,

01:33:19.320 | and that was how they really were.

01:33:21.000 | Our job is to figure out the things that are not like that.

01:33:23.740 | - That are not like that.

01:33:25.560 | All this talk of primes got me hungry for primes.

01:33:28.460 | You wrote a blog post, "The Beauty of Bounding Gaps,"

01:33:32.520 | a huge discovery about prime numbers

01:33:35.240 | and what it means for the future of math.

01:33:37.280 | Can you tell me about prime numbers?

01:33:40.780 | What the heck are those?

01:33:41.820 | What are twin primes?

01:33:42.780 | What are prime gaps?

01:33:43.700 | What are bounding gaps in primes?

01:33:46.720 | What are all these things?

01:33:47.820 | And what, if anything,

01:33:49.780 | or what exactly is beautiful about them?

01:33:52.100 | - Yeah, so prime numbers are one of the things

01:33:57.100 | that number theorists study the most and have for millennia.

01:34:01.180 | They are numbers which can't be factored.

01:34:06.220 | And then you say, like, five.

01:34:08.140 | And then you're like, wait, I can factor five.

01:34:09.780 | Five is five times one.

01:34:11.820 | Okay, not like that.

01:34:13.500 | That is a factorization.

01:34:14.540 | It absolutely is a way of expressing five

01:34:16.700 | as a product of two things.

01:34:18.380 | But don't you agree that there's

01:34:19.540 | something trivial about it?

01:34:20.900 | It's something you could do to any number.

01:34:22.300 | It doesn't have content the way that if I say

01:34:24.340 | that 12 is six times two or 35 is seven times five,

01:34:27.640 | I've really done something to it.

01:34:28.960 | I've broken up.

01:34:29.800 | So those are the kind of factorizations that count.

01:34:31.740 | And a number that doesn't have a factorization like that

01:34:34.460 | is called prime, except, historical side note,

01:34:38.100 | one, which at some times in mathematical history

01:34:42.440 | has been deemed to be a prime, but currently is not.

01:34:46.020 | And I think that's for the best.

01:34:47.140 | But I bring it up only because sometimes people think

01:34:49.020 | that these definitions are kind of,

01:34:52.220 | if we think about them hard enough,

01:34:53.540 | we can figure out which definition is true.

01:34:55.700 | - No, there's just an artifact of mathematics.

01:34:58.820 | - Yeah, we do.

01:35:00.060 | - So it's--

01:35:00.900 | - It's a question of which definition is best for us,

01:35:03.460 | for our purposes.

01:35:04.300 | - Well, those edge cases are weird, right?

01:35:06.020 | So it can't be, it doesn't count when you use yourself

01:35:11.020 | as a number or one as part of the factorization,

01:35:15.020 | or as the entirety of the factorization.

01:35:17.300 | So you somehow get to the meat of the number

01:35:22.900 | by factorizing it.

01:35:24.180 | And that seems to get to the core of all of mathematics.

01:35:27.420 | - Yeah, you take any number and you factorize it

01:35:29.920 | until you can factorize no more.

01:35:31.420 | And what you have left is some big pile of primes.

01:35:33.900 | I mean, by definition, when you can't factor anymore,

01:35:36.380 | when you're done, when you can't break the numbers up

01:35:39.020 | anymore, what's left must be prime.

01:35:40.900 | You know, 12 breaks into two and two and three.

01:35:45.300 | So these numbers are the atoms,

01:35:46.860 | the building blocks of all numbers.

01:35:49.380 | And there's a lot we know about them,

01:35:52.180 | but there's much more that we don't know them.

01:35:53.420 | I'll tell you the first few.

01:35:54.340 | There's two, three, five, seven, 11.

01:35:58.160 | By the way, they're all gonna be odd from then on,

01:36:00.780 | because if they were even, I could factor two out of them.

01:36:03.060 | But it's not all the odd numbers.

01:36:04.300 | Nine isn't prime, 'cause it's three times three.

01:36:06.460 | 15 isn't prime, 'cause it's three times five, but 13 is.

01:36:08.860 | Where were we?

01:36:09.700 | Two, three, five, seven, 11, 13, 17, 19.

01:36:13.820 | Not 21, but 23 is, et cetera, et cetera.

01:36:15.900 | Okay, so you could go on.

01:36:17.060 | - How high could you go if we were just sitting here?

01:36:19.580 | By the way, your own brain.

01:36:20.980 | Continuous, without interruption.

01:36:23.940 | Would you be able to go over 100?

01:36:25.940 | - I think so.

01:36:27.080 | There's always those ones that trip people up.

01:36:29.060 | There's a famous one, the Grotendieck prime, 57.

01:36:31.740 | Like sort of Alexander Grotendieck,

01:36:33.340 | the great algebraic geometer, was sort of giving

01:36:35.700 | some lecture involving a choice of a prime in general,

01:36:38.660 | and somebody said, like, can't you just choose a prime?

01:36:41.460 | And he said, okay, 57, which is in fact not prime.

01:36:43.500 | It's three times 19.

01:36:45.780 | - Oh, damn.

01:36:46.620 | - But it was like, I promise you in some circles

01:36:49.260 | it's a funny story, okay.

01:36:50.500 | - There's a humor in it.

01:36:55.780 | - Yes, I would say over 100 I definitely don't remember.

01:36:59.220 | Like 107, I think.

01:37:01.420 | I'm not sure.

01:37:02.260 | - So is there a category of fake primes

01:37:08.500 | that are easily mistaken to be prime?

01:37:12.900 | Like 57, I wonder.

01:37:14.700 | - Yeah, so I would say 57 and 51

01:37:19.700 | are definitely like prime offenders.

01:37:21.860 | Oh, I didn't do that on purpose.

01:37:23.020 | - Oh, well done.

01:37:24.300 | - Didn't do it on purpose.

01:37:25.340 | Anyway, there are definitely ones that people,

01:37:28.180 | or 91 is another classic, seven times 13.

01:37:30.660 | It really feels kind of prime, doesn't it?

01:37:32.900 | But it is not.

01:37:33.740 | But there's also, by the way,

01:37:36.900 | but there's also an actual notion of pseudo prime,

01:37:39.580 | which is a thing with a formal definition,

01:37:41.460 | which is not a psychological thing.

01:37:43.380 | It is a prime which passes a primality test

01:37:47.540 | devised by Fermat, which is a very good test,

01:37:50.380 | which if a number fails this test,

01:37:52.540 | it's definitely not prime.

01:37:54.580 | And so there was some hope that,

01:37:55.620 | oh, maybe if a number passes the test,

01:37:57.300 | then it definitely is prime.

01:37:58.420 | That would give a very simple criterion for primality.

01:38:00.660 | Unfortunately, it's only perfect in one direction.

01:38:03.980 | So there are numbers, I wanna say 341 is the smallest,

01:38:09.540 | which pass the test, but are not prime, 341.

01:38:12.380 | - Is this test easily explainable or no?

01:38:14.740 | - Yes, actually.

01:38:15.820 | Ready, let me give you the simplest version of it.

01:38:18.260 | You can dress it up a little bit,

01:38:19.340 | but here's the basic idea.

01:38:20.640 | I take the number, the mystery number.

01:38:25.140 | I raise two to that power.

01:38:27.300 | So let's say your mystery number is six.

01:38:32.620 | Are you sorry you asked me?

01:38:33.860 | Are you ready to throw it?

01:38:34.700 | - No, you're breaking my brain again, but yes.

01:38:37.100 | - Let's do it.

01:38:38.180 | We're gonna do a live demonstration.

01:38:40.020 | Let's say your number is six.

01:38:43.300 | So I'm gonna raise two to the sixth power.

01:38:45.940 | Okay, so if I were working, I'd be like,

01:38:47.260 | that's two cubed squared, so that's eight times eight.

01:38:49.740 | So that's 64.

01:38:51.620 | Now we're gonna divide by six,

01:38:53.480 | but I don't actually care what the quotient is,

01:38:54.940 | only the remainder.

01:38:56.060 | So let's see, 64 divided by six is,

01:39:01.380 | well, there's a quotient of 10, but the remainder is four.

01:39:05.420 | So you failed because the answer has to be two.

01:39:08.620 | For any prime, let's do it with five, which is prime.

01:39:13.220 | Two to the fifth is 32.

01:39:15.540 | Divide 32 by five, and you get six

01:39:20.260 | with a remainder of two.

01:39:21.460 | - With a remainder of two, yeah.

01:39:24.220 | - For seven, two to the seventh is 128.

01:39:26.700 | Divide that by seven, and let's see.

01:39:29.440 | I think that's seven times 14.

01:39:31.580 | Is that right?

01:39:32.420 | No.

01:39:34.700 | Seven times 18 is 126, with a remainder of two, right?

01:39:39.700 | 128 is a multiple of seven plus two.

01:39:43.340 | So if that remainder is not two--

01:39:46.480 | - Then that's definitely not prime.

01:39:47.320 | - Then it's definitely not prime.

01:39:49.460 | - And then if it is, it's likely a prime, but not for sure.

01:39:53.300 | - It's likely a prime, but not for sure.

01:39:54.620 | And there's actually a beautiful geometric proof,

01:39:56.220 | which is in the book, actually.

01:39:57.200 | That's one of the most granular parts of the book,

01:39:58.700 | 'cause it's such a beautiful proof, I could not give it.

01:40:00.420 | So you draw a lot of opal and pearl necklaces,

01:40:05.380 | and spin them.

01:40:06.220 | That's kind of the geometric nature

01:40:07.420 | of this proof of Fermat's little theorem.

01:40:09.980 | So yeah, so with pseudoprimes,

01:40:13.620 | there are primes that are kind of faking it.

01:40:14.740 | They pass that test, but there are numbers

01:40:16.540 | that are faking it that pass that test,

01:40:17.940 | but are not actually prime.

01:40:19.300 | But the point is,

01:40:21.820 | there are many, many, many theorems about prime numbers.

01:40:30.060 | - Are there, like there's a bunch of questions to ask.

01:40:32.140 | Is there an infinite number of primes?

01:40:34.700 | Can we say something about the gap between primes

01:40:37.500 | as the numbers grow larger and larger and larger and so on?

01:40:41.020 | - Yeah, it's a perfect example of your desire

01:40:43.220 | for simplicity in all things.

01:40:44.660 | You know what would be really simple?

01:40:46.300 | If there was only finitely many primes,

01:40:48.820 | and then there would be this finite set of atoms

01:40:51.560 | that all numbers would be built up on.

01:40:53.880 | That would be very simple and good in certain ways,

01:40:56.900 | but it's completely false.

01:40:58.900 | And number theory would be totally different

01:41:00.220 | if that were the case.

01:41:01.040 | It's just not true.

01:41:01.880 | In fact, this is something else that Euclid knew.

01:41:04.700 | So this is a very, very old fact,

01:41:07.540 | like much before, long before we had anything

01:41:10.340 | like modern number theory.

01:41:11.180 | - The primes are infinite.

01:41:12.140 | - The primes that there are, right, the infinite primes.

01:41:14.020 | - There's an infinite number of primes.

01:41:15.460 | So what about the gaps between the primes?

01:41:17.740 | - Right, so one thing that people recognized

01:41:20.460 | and really thought about a lot is that the primes,

01:41:22.220 | on average, seem to get farther and farther apart

01:41:25.840 | as they get bigger and bigger.

01:41:27.020 | In other words, it's less and less common.

01:41:29.100 | Like I already told you of the first 10 numbers,

01:41:31.100 | two, three, five, seven, four of them are prime.

01:41:32.900 | That's a lot, 40%.

01:41:34.660 | If I looked at 10-digit numbers,

01:41:38.500 | no way would 40% of those be prime.

01:41:40.580 | Being prime would be a lot rarer,

01:41:41.980 | in some sense because there's a lot more things

01:41:43.900 | for them to be divisible by.

01:41:45.820 | That's one way of thinking of it.

01:41:47.100 | It's a lot more possible for there to be a factorization

01:41:49.380 | because there's a lot of things

01:41:50.340 | you can try to factor out of it.

01:41:52.100 | As the numbers get bigger and bigger,

01:41:53.380 | primality gets rarer and rarer.

01:41:55.820 | The extent to which that's the case,

01:42:00.260 | that's pretty well understood.

01:42:01.700 | But then you can ask more fine-grained questions.

01:42:03.980 | Here is one.

01:42:04.860 | A twin prime is a pair of primes that are two apart,

01:42:11.740 | like three and five, or like 11 and 13,

01:42:14.940 | or like 17 and 19.

01:42:17.340 | One thing we still don't know is

01:42:19.160 | are there infinitely many of those?

01:42:21.960 | We know on average they get farther and farther apart,

01:42:24.100 | but that doesn't mean there couldn't be occasional

01:42:26.500 | folks that come close together.

01:42:30.180 | And indeed, we think that there are.

01:42:32.940 | And one interesting question,

01:42:35.220 | 'cause I think you might say,

01:42:39.420 | how could one possibly have a right

01:42:41.020 | to have an opinion about something like that?

01:42:44.020 | We don't have any way of describing

01:42:45.660 | a process that makes primes.

01:42:48.420 | Sure, you can look at your computer and see a lot of them,

01:42:52.300 | but the fact that there's a lot,

01:42:53.860 | why is that evidence that there's infinitely many?

01:42:55.900 | Maybe I can go on my computer and find 10 million.

01:42:57.620 | Well, 10 million is pretty far from infinity,

01:42:59.900 | so how is that evidence?

01:43:01.580 | There's a lot of things.

01:43:02.480 | There's a lot more than 10 million atoms.

01:43:04.140 | That doesn't mean there's infinitely many atoms

01:43:05.460 | in the universe.

01:43:06.300 | I mean, on most people's physical theories,

01:43:07.700 | there's probably not, as I understand it.

01:43:10.140 | Okay, so why would we think this?

01:43:13.180 | The answer is that it turns out to be

01:43:16.660 | incredibly productive and enlightening

01:43:19.700 | to think about primes as if they were random numbers,

01:43:23.260 | as if they were randomly distributed

01:43:24.940 | according to a certain law.

01:43:26.100 | Now, they're not.

01:43:27.060 | They're not random.

01:43:27.900 | There's no chance involved.

01:43:28.900 | It's completely deterministic

01:43:30.180 | whether a number is prime or not,

01:43:31.660 | and yet it just turns out to be phenomenally useful

01:43:35.460 | in mathematics to say,

01:43:38.140 | even if something is governed by a deterministic law,

01:43:41.780 | let's just pretend it wasn't.

01:43:43.140 | Let's just pretend that they were produced

01:43:44.480 | by some random process and see if the behavior

01:43:46.600 | is roughly the same,

01:43:47.980 | and if it's not, maybe change the random process.

01:43:49.660 | Maybe make the randomness a little bit different

01:43:51.100 | and tweak it and see if you can find a random process

01:43:53.820 | that matches the behavior we see,

01:43:55.380 | and then maybe you predict that other behaviors

01:43:58.460 | of the system are like that of the random process.

01:44:02.900 | And so that's kind of like, it's funny,

01:44:04.060 | because I think when you talk to people

01:44:05.260 | about the twin prime conjecture,

01:44:07.420 | people think you're saying,

01:44:09.940 | wow, there's like some deep structure there

01:44:12.400 | that like makes those primes be like close together

01:44:15.180 | again and again, and no,

01:44:16.540 | it's the opposite of deep structure.

01:44:18.260 | What we say when we say we believe

01:44:19.780 | the twin prime conjecture is that we believe

01:44:21.500 | the primes are like sort of strewn around pretty randomly,

01:44:24.540 | and if they were, then by chance you would expect

01:44:26.660 | there to be infinitely many twin primes,

01:44:28.180 | and we're saying, yeah, we expect them to behave

01:44:29.620 | just like they would if they were random dirt.

01:44:32.500 | - You know, the fascinating parallel here

01:44:34.860 | is I just got a chance to talk to Sam Harris,

01:44:38.380 | and he uses the prime numbers as an example often.

01:44:42.220 | I don't know if you're familiar with who Sam is.

01:44:45.460 | He uses that as an example of there being no free will.

01:44:50.280 | Wait, where does he get this?

01:44:52.360 | - Well, he just uses as an example of it might seem

01:44:55.680 | like this is a random number generator,

01:44:58.440 | but it's all like formally defined.

01:45:01.760 | So if we keep getting more and more primes,

01:45:05.080 | then like that might feel like a new discovery,

01:45:09.160 | and that might feel like a new experience, but it's not.

01:45:12.120 | It was always written in the cards.

01:45:14.320 | But it's funny that you say that

01:45:15.680 | because a lot of people think of like randomness.

01:45:19.200 | The fundamental randomness within the nature of reality

01:45:23.440 | might be the source of something

01:45:25.960 | that we experience as free will.

01:45:27.840 | And you're saying it's like useful to look at prime numbers

01:45:30.240 | as a random process in order to prove stuff about them,

01:45:35.240 | but fundamentally, of course, it's not a random process.

01:45:38.860 | - Well, not in order to prove stuff about them

01:45:40.960 | so much as to figure out what we expect to be true

01:45:43.780 | and then try to prove that.

01:45:44.620 | 'Cause here's what you don't wanna do,

01:45:45.960 | try really hard to prove something that's false.

01:45:48.380 | That makes it really hard to prove the thing if it's false.

01:45:51.120 | So you certainly wanna have some heuristic ways

01:45:53.040 | of guessing, making good guesses about what's true.

01:45:55.160 | So yeah, here's what I would say.

01:45:56.720 | You're gonna be imaginary Sam Harris now.

01:45:58.720 | You are talking about prime numbers,

01:46:01.000 | and you are like,

01:46:01.840 | "But prime numbers are completely deterministic."

01:46:04.040 | And I'm saying like,

01:46:04.880 | "Well, but let's treat them like a random process."

01:46:07.000 | And then you say,

01:46:08.200 | "But you're just saying something that's not true.

01:46:09.560 | "They're not a random process, they're deterministic."

01:46:10.960 | And I'm like, "Okay, great, you hold to your insistence

01:46:13.080 | "that it's not a random process.

01:46:13.920 | "Meanwhile, I'm generating insight about the primes

01:46:15.820 | "that you're not because I'm willing to sort of pretend

01:46:17.700 | "that there's something that they're not

01:46:18.600 | "in order to understand what's going on."

01:46:20.460 | - Yeah, so it doesn't matter what the reality is.

01:46:22.960 | What matters is what framework of thought

01:46:27.960 | results in the maximum number of insights.

01:46:30.800 | - Yeah, 'cause I feel, look, I'm sorry,

01:46:32.440 | but I feel like you have more insights about people

01:46:34.220 | if you think of them as like beings

01:46:37.360 | that have wants and needs and desires

01:46:39.120 | and do stuff on purpose.

01:46:40.920 | Even if that's not true,

01:46:41.860 | you still understand better what's going on

01:46:43.540 | by treating them in that way.

01:46:44.640 | Don't you find, look, when you work on machine learning,

01:46:46.520 | don't you find yourself sort of talking about

01:46:48.540 | what the machine is trying to do in a certain instance?

01:46:52.800 | Do you not find yourself drawn to that language?

01:46:54.960 | - Well, I-- - Oh, it knows this,

01:46:56.480 | it's trying to do that, it's learning that.

01:46:59.000 | - I'm certainly drawn to that language

01:47:00.960 | to the point where I receive quite a bit of criticisms

01:47:03.400 | for it 'cause I, you know, like--

01:47:05.400 | - Oh, I'm on your side, man.

01:47:07.040 | - So especially in robotics, I don't know why,

01:47:09.720 | but robotics people don't like to name their robots.

01:47:14.260 | Or they certainly don't like to gender their robots

01:47:17.000 | because the moment you gender a robot,

01:47:18.760 | you start to anthropomorphize.

01:47:20.600 | If you say he or she, you start to,

01:47:23.440 | in your mind, construct like a life story in your mind.

01:47:27.800 | You can't help it.

01:47:29.040 | There's like, you create like a humorous story

01:47:31.520 | to this person, you start to, this person, this robot,

01:47:35.600 | you start to project your own,

01:47:37.320 | but I think that's what we do to each other.

01:47:38.800 | I think that's actually really useful

01:47:40.480 | for the engineering process,

01:47:42.640 | especially for human-robot interaction,

01:47:44.580 | and yes, for machine learning systems,

01:47:46.620 | for helping you build an intuition

01:47:48.020 | about a particular problem.

01:47:49.900 | It's almost like asking this question,

01:47:51.920 | you know, when a machine learning system fails

01:47:55.940 | in a particular edge case, asking like,

01:47:57.860 | "What were you thinking about?"

01:47:59.820 | Like, asking like almost like when you're talking about

01:48:02.980 | to a child who just did something bad,

01:48:06.620 | you wanna understand like, what was,

01:48:09.400 | how did they see the world?

01:48:12.060 | Maybe there's a totally new,

01:48:13.040 | maybe you're the one that's thinking

01:48:14.520 | about the world incorrectly.

01:48:16.840 | And yeah, that anthropomorphization process,

01:48:19.880 | I think is ultimately good for insight,

01:48:21.400 | and the same as I agree with you,

01:48:23.640 | I tend to believe about free will as well.

01:48:26.680 | Let me ask you a ridiculous question, if it's okay.

01:48:28.900 | - Of course.

01:48:30.240 | - I've just recently, most people go on like rabbit hole,

01:48:34.400 | like YouTube things, and I went on a rabbit hole

01:48:37.360 | often do of Wikipedia,

01:48:39.800 | and I found a page on finitism,

01:48:44.800 | ultra finitism and intuitionism,

01:48:49.080 | or I forget what it's called.

01:48:51.160 | - Yeah, intuitionism.

01:48:52.120 | - Intuitionism.

01:48:53.740 | That seemed pretty interesting.

01:48:55.600 | I have it on my to-do list to actually like look into,

01:48:58.420 | like, is there people who like formally,

01:49:00.800 | like real mathematicians are trying to argue for this.

01:49:03.600 | But the belief there, I think, let's say finitism,

01:49:07.480 | that infinity is fake.

01:49:10.160 | Meaning infinity may be like a useful hack for certain,

01:49:16.880 | like a useful tool in mathematics,

01:49:18.900 | but it really gets us into trouble,

01:49:22.500 | because there's no infinity in the real world.

01:49:26.680 | Maybe I'm sort of not expressing that fully correctly,

01:49:31.020 | but basically saying like there's things there,

01:49:33.420 | once you add into mathematics,

01:49:37.080 | things that are not provably within the physical world,

01:49:41.080 | you're starting to inject, to corrupt your framework

01:49:45.840 | of reason.

01:49:46.680 | What do you think about that?

01:49:49.240 | - I mean, I think, okay, so first of all,

01:49:50.560 | I'm not an expert, and I couldn't even tell you

01:49:54.040 | what the difference is between those three terms,

01:49:56.960 | finitism, ultra finitism, and intuitionism.

01:49:59.000 | Although I know they're related,

01:50:00.000 | I tend to associate them with the Netherlands in the 1930s.

01:50:02.680 | - Okay, I'll tell you, can I just quickly comment,

01:50:04.920 | because I read the Wikipedia page?

01:50:06.880 | The difference in ultra-

01:50:07.920 | - That's like the ultimate sentence of the modern age.

01:50:10.520 | Can I just comment, because I read the Wikipedia page.

01:50:12.680 | That sums up our moment.

01:50:14.680 | - Bro, I'm basically an expert.

01:50:16.520 | Ultra finitism, so finitism says that the only infinity

01:50:22.600 | you're allowed to have is that the natural numbers

01:50:25.240 | are infinite.

01:50:26.160 | So like those numbers are infinite.

01:50:29.280 | So like one, two, three, four, five,

01:50:32.240 | the integers are infinite.

01:50:35.480 | The ultra finitism says, nope, even that infinity is fake.

01:50:39.600 | That's-

01:50:41.520 | - I'll bet ultra finitism came second.

01:50:43.160 | I'll bet it's like when there's like a hardcore scene,

01:50:44.800 | and then one guy's like, oh, now there's a lot of people

01:50:47.240 | in this scene, I have to find a way to be more hardcore

01:50:49.200 | than the hardcore people.

01:50:50.240 | - It's all back to the emo talk, yeah.

01:50:52.480 | Okay, so is there any, are you ever,

01:50:54.800 | 'cause I'm often uncomfortable with infinity,

01:50:58.080 | like psychologically.

01:50:59.440 | I have trouble when that sneaks in there.

01:51:04.600 | It's 'cause it works so damn well,

01:51:06.640 | I get a little suspicious,

01:51:09.360 | because it could be almost like a crutch

01:51:12.560 | or an oversimplification that's missing something profound

01:51:15.520 | about reality.

01:51:16.440 | - Well, so first of all, okay, if you say like,

01:51:20.720 | is there like a serious way of doing mathematics

01:51:24.920 | that doesn't really treat infinity as a real thing,

01:51:29.300 | or maybe it's kind of agnostic, and it's like,

01:51:30.880 | I'm not really gonna make a firm statement

01:51:32.620 | about whether it's a real thing or not.

01:51:33.920 | Yeah, that's called most of the history of mathematics.

01:51:36.360 | Right, so it's only after Cantor, right,

01:51:38.360 | that we really are sort of, okay, we're gonna like,

01:51:43.120 | have a notion of like the cardinality of an infinite set

01:51:45.600 | and like do something that you might call

01:51:49.040 | like the modern theory of infinity.

01:51:51.280 | That said, obviously, everybody was drawn to this notion,

01:51:54.040 | and no, not everybody was comfortable with it.

01:51:55.720 | Look, I mean, this is what happens with Newton, right?

01:51:57.640 | I mean, so Newton understands that to talk about tangents

01:52:01.320 | and to talk about instantaneous velocity,

01:52:03.440 | he has to do something that we would now call

01:52:06.560 | taking a limit, right?

01:52:08.660 | The fabled dy over dx, if you sort of go back

01:52:11.220 | to your calculus class, for those who've taken calculus,

01:52:13.080 | remember this mysterious thing.

01:52:14.800 | And you know, what is it?

01:52:17.320 | What is it?

01:52:18.160 | Well, he'd say like, well, it's like you sort of

01:52:20.360 | divide the length of this line segment

01:52:24.040 | by the length of this other line segment,

01:52:25.240 | and then you make them a little shorter,

01:52:26.280 | and you divide again, and then you make them

01:52:27.520 | a little shorter, and you divide again,

01:52:28.740 | and then you just keep on doing that

01:52:29.720 | until they're like infinitely short,

01:52:30.760 | and then you divide them again.

01:52:32.520 | These quantities that are like, they're not zero,

01:52:36.360 | but they're also smaller than any actual number,

01:52:41.360 | these infinitesimals.

01:52:43.440 | Well, people were queasy about it,

01:52:46.360 | and they weren't wrong to be queasy about it, right?

01:52:48.200 | From a modern perspective, it was not really well formed.

01:52:50.080 | There's this very famous critique of Newton

01:52:52.280 | by Bishop Berkeley, where he says like,

01:52:54.480 | what, these things you define, like, you know,

01:52:57.840 | they're not zero, but they're smaller than any number.

01:53:00.280 | Are they the ghosts of departed quantities?

01:53:02.360 | (laughing)

01:53:03.600 | That was this like, ultra-par.

01:53:04.840 | - That's a good line.

01:53:05.680 | - Of Newton.

01:53:06.840 | And on the one hand, he was right.

01:53:10.040 | It wasn't really rigorous by modern standards.

01:53:11.720 | On the other hand, like, Newton was out there

01:53:12.960 | doing calculus, and other people were not, right?

01:53:15.360 | - It works, it works.

01:53:17.400 | - I think a sort of intuitionist view, for instance,

01:53:20.640 | I would say would express serious doubt.

01:53:23.640 | And it's not just infinity.

01:53:25.920 | It's like saying, I think we would express serious doubt

01:53:28.080 | that like, the real numbers exist.

01:53:30.320 | Now, most people are comfortable with the real numbers.

01:53:36.280 | - Well, computer scientists with floating point number,

01:53:39.160 | I mean, floating point arithmetic.

01:53:42.680 | - That's a great point, actually.

01:53:44.680 | I think, in some sense, this flavor of doing math,

01:53:48.360 | saying we shouldn't talk about things

01:53:51.200 | that we cannot specify in a finite amount of time,

01:53:53.600 | there's something very computational in flavor about that.

01:53:55.960 | And it's probably not a coincidence

01:53:57.560 | that it becomes popular in the '30s and '40s,

01:54:01.720 | which is also kind of like the dawn of ideas

01:54:04.960 | about formal computation, right?

01:54:06.160 | You probably know the timeline better than I do.

01:54:07.920 | - Sorry, what becomes popular?

01:54:09.600 | - These ideas that maybe we should be doing math

01:54:12.160 | in this more restrictive way, where even a thing that,

01:54:16.120 | because look, the origin of all this is like,

01:54:18.520 | a number represents a magnitude, like the length of a line.

01:54:22.640 | So, I mean, the idea that there's a continuum,

01:54:26.080 | there's sort of like, is pretty old,

01:54:30.600 | but just 'cause the thing is old

01:54:31.920 | doesn't mean we can't reject it if we want to.

01:54:34.200 | - Well, a lot of the fundamental ideas in computer science,

01:54:36.600 | when you talk about the complexity of problems,

01:54:40.060 | to Turing himself, they rely on infinity as well.

01:54:45.080 | The ideas that kind of challenge that,

01:54:47.600 | the whole space of machine learning,

01:54:48.800 | I would say, challenges that.

01:54:51.000 | It's almost like the engineering approach to things,

01:54:53.020 | like the floating point arithmetic.

01:54:54.640 | The other one that, back to John Conway,

01:54:57.360 | that challenges this idea, I mean,

01:55:01.160 | maybe to tie in the ideas of deformation theory

01:55:04.280 | and limits to infinity, is this idea of cellular automata.

01:55:09.280 | With John Conway looking at "The Game of Life,"

01:55:17.360 | Stephen Wolfram's work, that I've been a big fan of

01:55:20.840 | for a while, of cellular automata.

01:55:22.560 | I was wondering if you have ever encountered

01:55:24.880 | these kinds of objects,

01:55:26.920 | you ever looked at them as a mathematician,

01:55:29.320 | where you have very simple rules of tiny little objects,

01:55:34.320 | that when taken as a whole, create incredible complexities,

01:55:38.000 | but are very difficult to analyze,

01:55:39.800 | very difficult to make sense of,

01:55:41.960 | even though the one individual object, one part,

01:55:45.120 | it's like what we were saying about Andrew Wiles,

01:55:46.960 | like you can look at the deformation of a small piece

01:55:49.800 | to tell you about the whole.

01:55:51.360 | It feels like with cellular automata,

01:55:54.440 | or any kind of complex systems,

01:55:56.400 | it's often very difficult to say something

01:55:59.800 | about the whole thing,

01:56:01.640 | even when you can precisely describe the operation

01:56:05.120 | of the local neighborhoods.

01:56:09.400 | - Yeah, I mean, I love that subject.

01:56:10.960 | I haven't really done research in it myself.

01:56:12.640 | I've played around with it.

01:56:13.520 | I'll send you a fun blog post I wrote,

01:56:15.040 | where I made some cool texture patterns

01:56:17.360 | from cellular automata that I, but--

01:56:21.000 | - And those are really always compelling,

01:56:22.480 | is like you create simple rules,

01:56:24.120 | and they create some beautiful textures.

01:56:25.800 | It doesn't make any sense.

01:56:26.640 | - Actually, did you see there was a great paper,

01:56:28.000 | I don't know if you saw this,

01:56:29.000 | like a machine learning paper.

01:56:30.640 | - Yes, yes.

01:56:31.480 | - I don't know if you saw the one I'm talking about,

01:56:32.320 | where they were learning the textures,

01:56:33.400 | like let's try to reverse engineer,

01:56:35.640 | and learn a cellular automaton that can produce texture

01:56:38.200 | that looks like this, from the images.

01:56:40.320 | Very cool, and as you say, the thing you said is,

01:56:43.800 | I feel the same way when I read machine learning papers,

01:56:46.040 | that what's especially interesting

01:56:47.640 | is the cases where it doesn't work.

01:56:49.520 | Like what does it do when it doesn't do

01:56:50.920 | the thing that you tried to train it to do?

01:56:53.360 | That's extremely interesting.

01:56:54.480 | Yeah, yeah, that was a cool paper.

01:56:56.120 | So yeah, so let's start with the game of life.

01:56:58.360 | Let's start with, or let's start with John Conway.

01:57:02.320 | So Conway, so yeah, so let's start with John Conway again.

01:57:06.080 | Just, I don't know, from my outsider's perspective,

01:57:08.620 | there's not many mathematicians that stand out

01:57:11.520 | throughout the history of the 20th century.

01:57:13.800 | He's one of them.

01:57:15.080 | I feel like he's not sufficiently recognized.

01:57:18.200 | - I think he's pretty recognized.

01:57:20.120 | - Okay, well.

01:57:21.080 | - I mean, he was a full professor at Princeton

01:57:24.320 | for most of his life.

01:57:25.160 | He was sort of in, certainly at the pinnacle of.

01:57:27.040 | - Yeah, but I found myself, every time I talk about Conway

01:57:30.120 | and how excited I am about him,

01:57:32.180 | I have to constantly explain to people who he is.

01:57:36.620 | And that's always a sad sign to me.

01:57:39.520 | But that's probably true for a lot of mathematicians.

01:57:41.480 | - I was about to say, I feel like you have

01:57:43.160 | a very elevated idea of how famous,

01:57:44.920 | this is what happens when you grow up in the Soviet Union,

01:57:46.680 | or you think the mathematicians are very, very famous.

01:57:49.840 | - Yeah, but I'm not actually so convinced

01:57:51.800 | at a tiny tangent that that shouldn't be so.

01:57:54.600 | I mean, there's, it's not obvious to me

01:57:57.600 | that that's one of the, like, if I were to analyze

01:58:00.600 | American society, that perhaps elevating mathematical

01:58:04.080 | and scientific thinking to a little bit higher level

01:58:07.000 | would benefit the society.

01:58:08.700 | Well, both in discovering the beauty of what it is

01:58:11.260 | to be human and for actually creating cool technology,

01:58:14.960 | better iPhones.

01:58:16.200 | But anyway, John Conway.

01:58:18.120 | - Yeah, and Conway is such a perfect example

01:58:20.000 | of somebody whose humanity was, and his personality

01:58:23.040 | was like wound up with his mathematics, right?

01:58:25.040 | It's what's not, sometimes I think people

01:58:26.760 | who are outside the field think of mathematics

01:58:28.640 | as this kind of like cold thing that you do

01:58:31.220 | separate from your existence as a human being.

01:58:33.080 | No way, your personality is in there,

01:58:34.760 | just as it would be in like a novel you wrote

01:58:37.120 | or a painting you painted, or just like the way

01:58:39.080 | you walk down the street.

01:58:40.080 | Like, it's in there, it's you doing it.

01:58:41.760 | And Conway was certainly a singular personality.

01:58:46.240 | I think anybody would say that he was playful,

01:58:50.960 | like everything was a game to him.

01:58:54.240 | Now, what you might think I'm gonna say,

01:58:56.600 | and it's true, is that he sort of was very playful

01:58:59.280 | in his way of doing mathematics.

01:59:01.800 | But it's also true, it went both ways.

01:59:03.700 | He also sort of made mathematics out of games.

01:59:06.240 | He like looked at, he was a constant inventor of games

01:59:08.880 | with like crazy names, and then he would sort of

01:59:11.240 | analyze those games mathematically.

01:59:15.240 | To the point that he, and then later collaborating

01:59:17.240 | with Knuth, like, you know, created this number system,

01:59:20.680 | the serial numbers, in which actually each number

01:59:23.720 | is a game.

01:59:25.200 | There's a wonderful book about this called,

01:59:26.640 | I mean, there are his own books, and then there's

01:59:27.920 | like a book that he wrote with Burle Camp and Guy

01:59:29.760 | called Winning Ways, which is such a rich source of ideas.

01:59:34.760 | And he too kind of has his own crazy number system,

01:59:41.740 | in which, by the way, there are these infinitesimals,

01:59:44.240 | the ghosts of departed quantities, they're in there.

01:59:46.680 | Now, not as ghosts, but as like certain kind

01:59:48.840 | of two-player games.

01:59:50.020 | So, you know, he was a guy, so I knew him when I was

01:59:57.960 | a post-doc, and I knew him at Princeton,

02:00:01.280 | and our research overlapped in some ways.

02:00:03.620 | Now, it was on stuff that he had worked on many years

02:00:05.480 | before, the stuff I was working on kind of connected

02:00:07.400 | with stuff in group theory, which somehow seems

02:00:09.560 | to keep coming up.

02:00:13.880 | And so I often would like sort of ask him a question,

02:00:16.000 | I would sort of come upon him in the common room,

02:00:17.660 | and I would ask him a question about something.

02:00:19.080 | And just, anytime you turned him on, you know what I mean?

02:00:23.760 | You sort of asked a question, it was just like turning

02:00:26.860 | a knob and winding him up, and he would just go,

02:00:29.160 | and you would get a response that was like,

02:00:31.720 | so rich and went so many places, and taught you so much.

02:00:37.360 | And usually had nothing to do with your question.

02:00:39.360 | - Yeah.

02:00:40.200 | - Usually your question was just a prompt to him.

02:00:43.060 | You couldn't count on actually getting the question

02:00:44.560 | answered.

02:00:45.400 | - He had those brilliant, curious minds, even at that age.

02:00:47.400 | Yeah, it was definitely a huge loss.

02:00:50.360 | But on his Game of Life, which was, I think he developed

02:00:55.800 | in the '70s, as almost like a side thing,

02:00:58.800 | a fun little experiment.

02:00:59.640 | - Yeah, the Game of Life is this,

02:01:01.400 | it's a very simple algorithm.

02:01:05.160 | It's not really a game, per se, in the sense of the kinds

02:01:08.600 | of games that he liked, where people played against

02:01:10.400 | each other, but essentially it's a game that you play

02:01:15.400 | with marking little squares on a sheet of graph paper.

02:01:20.380 | And in the '70s, I think he was literally doing it

02:01:22.340 | with a pen on graph paper.

02:01:24.220 | You have some configuration of squares, some of the squares

02:01:26.660 | on the graph paper are filled in, some are not.

02:01:28.900 | And then there's a rule, a single rule, that tells you

02:01:32.340 | at the next stage, which squares are filled in,

02:01:36.460 | and which squares are not.

02:01:38.100 | Sometimes an empty square gets filled in,

02:01:39.740 | that's called birth.

02:01:40.580 | Sometimes a square that's filled in gets erased,

02:01:43.020 | that's called death.

02:01:43.940 | And there's rules for which squares are born

02:01:45.860 | and which squares die.

02:01:47.100 | The rule is very simple, you can write it on one line.

02:01:53.660 | And then the great miracle is that you can start

02:01:56.220 | from some very innocent-looking little small set of boxes

02:02:00.580 | and get these results of incredible richness.

02:02:04.180 | And of course, nowadays you don't do it on paper.

02:02:05.700 | Nowadays you do it on a computer.

02:02:07.020 | There's actually a great iPad app called Golly,

02:02:09.340 | which I really like, that has Conway's original rule

02:02:12.820 | and gosh, hundreds of other variants.

02:02:15.600 | And it's lightning fast, so you can just be like,

02:02:17.580 | I wanna see 10,000 generations of this rule play out

02:02:21.620 | faster than your eye can even follow.

02:02:23.020 | And it's amazing, so I highly recommend it

02:02:25.140 | if this is at all intriguing to you,

02:02:26.380 | getting Golly on your iOS device.

02:02:29.460 | - And you can do this kind of process,

02:02:30.780 | which I really enjoy doing, which is almost

02:02:33.660 | like putting a Darwin hat on or a biologist hat on

02:02:37.220 | and doing analysis of a higher level of abstraction,

02:02:41.500 | like the organisms that spring up.

02:02:43.540 | 'Cause there's different kinds of organisms.

02:02:45.140 | Like you can think of them as species

02:02:46.880 | and they interact with each other.

02:02:48.620 | They can, there's gliders, they shoot,

02:02:51.020 | there's like things that can travel around,

02:02:54.320 | there's things that can, glider guns,

02:02:56.720 | that can generate those gliders.

02:02:58.480 | - Exactly, right, these can--

02:03:00.140 | - You can use the same kind of language as you would

02:03:02.420 | about describing a biological system.

02:03:04.620 | - So it's a wonderful laboratory

02:03:06.260 | and it's kind of a rebuke to someone

02:03:07.940 | who doesn't think that very, very rich,

02:03:10.980 | complex structure can come from very simple

02:03:15.340 | underlying laws, like it definitely can.

02:03:18.940 | Now, here's what's interesting.

02:03:20.620 | If you just picked some random rule,

02:03:23.680 | you wouldn't get interesting complexity.

02:03:26.220 | I think that's one of the most interesting things

02:03:28.380 | of these, one of the most interesting features

02:03:31.460 | of this whole subject, that the rules

02:03:32.740 | have to be tuned just right.

02:03:34.140 | Like a sort of typical rule set doesn't generate

02:03:36.700 | any kind of interesting behavior.

02:03:38.780 | But some do, and I don't think we have a clear way

02:03:43.780 | of understanding which do and which don't.

02:03:45.420 | Maybe Stephen thinks he does, I don't know.

02:03:47.340 | - No, no, it's a giant mystery.

02:03:48.860 | What Stephen Wolfram did is,

02:03:51.820 | now there's a whole interesting aspect

02:03:55.540 | to the fact that he's a little bit of an outcast

02:03:57.620 | in the mathematics and physics community

02:03:59.900 | because he's so focused on a particular,

02:04:02.660 | his particular work, I think if you put ego aside,

02:04:05.780 | which I think, unfairly, some people are not able

02:04:09.100 | to look beyond, I think his work is actually quite brilliant.

02:04:11.880 | But what he did is exactly this process

02:04:13.840 | of Darwin-like exploration, is taking these very simple

02:04:16.940 | ideas and writing a thousand page book on them,

02:04:19.880 | meaning like, let's play around with this thing, let's see.

02:04:23.500 | And can we figure anything out?

02:04:25.460 | Spoiler alert, no, we can't.

02:04:27.320 | In fact, he does a challenge, I think it's like

02:04:31.940 | a rule 30 challenge, which is quite interesting,

02:04:34.140 | just simply for machine learning people,

02:04:36.380 | for mathematics people, is can you predict

02:04:40.460 | the middle column?

02:04:41.820 | For his, it's a 1D cellular automata.

02:04:45.980 | Can you, generally speaking, can you predict anything

02:04:49.940 | about how a particular rule will evolve,

02:04:53.100 | just in the future?

02:04:54.380 | Very simply, just looking at one particular part

02:04:58.020 | of the world, just zooming in on that part,

02:05:01.380 | you know, 100 steps ahead, can you predict something?

02:05:04.740 | And the challenge is to do that kind of prediction,

02:05:08.800 | so far as nobody's come up with an answer,

02:05:10.340 | but the point is, we can't, we don't have tools,

02:05:14.600 | or maybe it's impossible, or, I mean,

02:05:17.140 | he has these kind of laws of irreducibility,

02:05:19.980 | they hear first, but it's poetry,

02:05:21.500 | it's like we can't prove these things.

02:05:22.880 | It seems like we can't, that's the basic,

02:05:25.240 | it almost sounds like ancient mathematics

02:05:28.500 | or something like that, where you're like,

02:05:30.060 | the gods will not allow us to predict

02:05:32.660 | the cellular automata, but that's fascinating

02:05:36.460 | that we can't, I'm not sure what to make of it,

02:05:39.100 | and there's power to calling this particular set

02:05:42.540 | of rules game of life, as Conway did,

02:05:45.460 | because I'm not exactly sure, but I think he had a sense

02:05:49.500 | that there's some core ideas here that are fundamental

02:05:53.220 | to life, to complex systems, to the way life

02:05:56.540 | emerged on Earth.

02:05:58.520 | I'm not sure I think Conway thought that.

02:06:01.760 | It's something that, I mean, Conway always had

02:06:03.240 | a rather ambivalent relationship with the game of life,

02:06:05.920 | because I think he saw it as, it was certainly

02:06:10.920 | the thing he was most famous for in the outside world,

02:06:14.720 | and I think that he, his view, which is correct,

02:06:18.680 | is that he had done things that were much deeper

02:06:20.360 | mathematically than that, you know what I mean?

02:06:22.120 | And I think it always aggrieved him a bit

02:06:24.320 | that he was like the game of life guy,

02:06:26.240 | when he proved all these wonderful theorems,

02:06:28.640 | and created all these wonderful games,

02:06:32.120 | created the theorem of numbers.

02:06:33.620 | He was a very tireless guy who just did

02:06:37.680 | an incredibly variegated array of stuff,

02:06:40.840 | so he was exactly the kind of person who you would

02:06:43.640 | never want to reduce to one achievement,

02:06:45.640 | you know what I mean?

02:06:46.920 | - Let me ask about group theory.

02:06:50.400 | You mentioned it a few times.

02:06:51.800 | What is group theory?

02:06:53.440 | What is an idea from group theory that you find beautiful?

02:06:57.420 | - Well, so I would say group theory sort of starts

02:07:01.960 | as the general theory of symmetry,

02:07:04.480 | is that people looked at different kinds of things

02:07:08.240 | and said, as we said, oh, it could have,

02:07:12.920 | maybe all there is is symmetry from left to right,

02:07:16.400 | like a human being, right?

02:07:17.720 | That's roughly bilaterally symmetric, as we say.

02:07:22.660 | So there's two symmetries, and then you're like,

02:07:24.240 | well, wait, didn't I say there's just one,

02:07:25.460 | there's just left to right?

02:07:26.700 | Well, we always count the symmetry of doing nothing.

02:07:30.100 | We always count the symmetry that's like,

02:07:31.860 | there's flip and don't flip.

02:07:33.100 | Those are the two configurations that you can be in.

02:07:35.220 | So there's two.

02:07:36.060 | You know, something like a rectangle

02:07:40.240 | is bilaterally symmetric.

02:07:41.560 | You can flip it left to right,

02:07:42.600 | but you can also flip it top to bottom.

02:07:44.540 | So there's actually four symmetries.

02:07:47.680 | There's do nothing, flip it left to right,

02:07:50.320 | and flip it top to bottom, or do both of those things.

02:07:53.020 | A square, there's even more,

02:07:59.700 | because now you can rotate it.

02:08:01.700 | You can rotate it by 90 degrees.

02:08:03.080 | So you can't do that, that's not a symmetry of the rectangle.

02:08:04.920 | If you try to rotate it 90 degrees,

02:08:06.180 | you get a rectangle oriented in a different way.

02:08:08.880 | So a person has two symmetries, a rectangle four,

02:08:13.160 | a square eight, different kinds of shapes

02:08:15.420 | have different numbers of symmetries.

02:08:17.520 | And the real observation is that

02:08:19.940 | that's just not like a set of things.

02:08:21.940 | They can be combined.

02:08:25.060 | You do one symmetry, then you do another.

02:08:27.700 | The result of that is some third symmetry.

02:08:31.020 | So a group really abstracts away this notion of saying,

02:08:34.840 | it's just some collection of transformations

02:08:41.180 | you can do to a thing where you combine

02:08:42.980 | any two of them to get a third.

02:08:44.380 | And so, you know, a place where this comes up

02:08:45.620 | in computer science is in sorting,

02:08:48.260 | because the ways of permuting a set,

02:08:50.500 | the ways of taking sort of some set of things

02:08:52.340 | you have on the table and putting them

02:08:53.260 | in a different order,

02:08:54.260 | shuffling a deck of cards, for instance,

02:08:56.100 | those are the symmetries of the deck.

02:08:57.580 | And there's a lot of them.

02:08:58.420 | There's not two, there's not four, there's not eight.

02:09:00.140 | Think about how many different orders

02:09:01.580 | a deck of card can be in.

02:09:02.620 | Each one of those is the result of applying a symmetry

02:09:05.980 | to the original deck.

02:09:07.660 | So a shuffle is a symmetry, right?

02:09:09.060 | You're reordering the cards.

02:09:10.620 | If I shuffle and then you shuffle,

02:09:12.940 | the result is some other kind of thing

02:09:16.020 | you might call a double shuffle,

02:09:17.780 | which is a more complicated symmetry.

02:09:19.980 | So group theory is kind of the study

02:09:22.180 | of the general abstract world that encompasses

02:09:25.980 | all these kinds of things.

02:09:27.020 | But then of course, like lots of things

02:09:29.380 | that are way more complicated than that.

02:09:31.780 | Like infinite groups of symmetries, for instance.

02:09:33.540 | - So they can be infinite, huh?

02:09:35.100 | - Oh yeah.

02:09:35.940 | - Okay.

02:09:36.780 | - Well, okay, ready?

02:09:37.620 | Think about the symmetries of the line.

02:09:41.180 | You're like, okay, I can reflect it left to right,

02:09:45.020 | you know, around the origin.

02:09:46.820 | Okay, but I could also reflect it left to right,

02:09:49.580 | grabbing somewhere else, like at one or two

02:09:52.180 | or pi or anywhere.

02:09:54.620 | Or I could just slide it some distance.

02:09:56.420 | That's a symmetry.

02:09:57.340 | Slide it five units over.

02:09:58.540 | So there's clearly infinitely many symmetries of the line.

02:10:01.220 | That's an example of an infinite group of symmetries.

02:10:03.500 | - Is it possible to say something that kind of captivates,

02:10:06.940 | keeps being brought up by physicists,

02:10:09.420 | which is gauge theory, gauge symmetry,

02:10:11.740 | as one of the more complicated type of symmetries?

02:10:14.900 | Is there an easy explanation of what the heck it is?

02:10:18.380 | Is that something that comes up on your mind at all?

02:10:21.860 | - Well, I'm not a mathematical physicist,

02:10:23.340 | but I can say this.

02:10:24.380 | It is certainly true that it has been a very useful notion

02:10:29.380 | in physics to try to say, like,

02:10:31.860 | what are the symmetry groups of the world?

02:10:34.580 | Like, what are the symmetries

02:10:35.660 | under which things don't change, right?

02:10:36.980 | So we just, I think we talked a little bit earlier

02:10:39.220 | about it should be a basic principle

02:10:40.700 | that a theorem that's true here is also true over there.

02:10:44.180 | And same for a physical law, right?

02:10:45.660 | I mean, if gravity is like this over here,

02:10:47.660 | it should also be like this over there.

02:10:49.140 | Okay, what that's saying is we think translation in space

02:10:52.620 | should be a symmetry.

02:10:53.980 | All the laws of physics should be unchanged

02:10:56.500 | if the symmetry we have in mind

02:10:57.820 | is a very simple one like translation.

02:10:59.660 | And so then there becomes a question,

02:11:03.780 | like what are the symmetries of the actual world

02:11:07.940 | with its physical laws?

02:11:09.780 | And one way of thinking, this is an oversimplification,

02:11:12.860 | but like one way of thinking of this big shift

02:11:15.940 | from before Einstein to after

02:11:22.380 | is that we just changed our idea

02:11:25.260 | about what the fundamental group of symmetries were.

02:11:29.740 | So that things like the Lorenz contraction,

02:11:31.780 | things like these bizarre relativistic phenomena

02:11:34.300 | where Lorenz would have said, oh, to make this work,

02:11:37.660 | we need a thing to change its shape.

02:11:42.660 | If it's moving nearly the speed of light.

02:11:47.460 | Well, under the new framework, it's much better.

02:11:50.260 | He's like, oh no, it wasn't changing its shape.

02:11:51.700 | You were just wrong about what counted as a symmetry.

02:11:54.420 | Now that we have this new group, the so-called Lorenz group,

02:11:57.380 | now that we understand what the symmetries really are,

02:11:59.220 | we see it was just an illusion

02:12:00.300 | that the thing was changing its shape.

02:12:02.940 | - Yeah, so you can then describe the sameness of things

02:12:05.780 | under this weirdness that is general relativity,

02:12:08.820 | for example.

02:12:10.940 | - Yeah, yeah, still.

02:12:14.420 | I wish there was a simpler explanation of exact,

02:12:16.940 | I mean, gauge symmetries,

02:12:19.820 | pretty simple general concept

02:12:22.380 | about rulers being deformed.

02:12:24.360 | It's just that I've actually just personally

02:12:29.820 | been on a search, not a very rigorous or aggressive search,

02:12:34.740 | but for something I personally enjoy,

02:12:38.000 | which is taking complicated concepts

02:12:41.000 | and finding the minimal example

02:12:44.800 | that I can play around with, especially programmatically.

02:12:47.600 | - That's great.

02:12:48.440 | This is what we try to train our students to do, right?

02:12:50.220 | I mean, in class, this is exactly what,

02:12:52.620 | this is best pedagogical practice.

02:12:54.600 | - I do hope there's simple explanation,

02:12:57.380 | especially I've, in my drunk, random walk,

02:13:02.380 | drunk walk, whatever that's called,

02:13:06.620 | sometimes stumble into the world of topology

02:13:09.120 | and quickly, you know when you go into a party

02:13:13.140 | and you realize this is not the right party for me?

02:13:16.000 | So whenever I go into topology,

02:13:18.780 | it's like so much math everywhere.

02:13:22.220 | I don't even know what,

02:13:23.300 | it feels like, this is me being a hater,

02:13:26.300 | I think there's way too much math.

02:13:27.620 | Like there are two, the cool kids who just wanna have,

02:13:30.980 | like everything is expressed through math

02:13:33.140 | because they're actually afraid to express stuff

02:13:35.060 | simply through language.

02:13:37.140 | That's my hater formulation of topology.

02:13:39.620 | But at the same time, I'm sure that's very necessary

02:13:41.620 | to do sort of rigorous discussion.

02:13:43.260 | But I feel like--

02:13:44.660 | - But don't you think that's what gauge symmetry is like?

02:13:46.300 | I mean, it's not a field I know well,

02:13:47.500 | but it certainly seems like--

02:13:48.540 | - Yes, it is like that.

02:13:49.820 | But my problem with topology, okay,

02:13:52.500 | and even like differential geometry,

02:13:56.620 | is like you're talking about beautiful things.

02:14:00.980 | Like if they could be visualized,

02:14:02.940 | it's open question if everything could be visualized,

02:14:05.620 | but you're talking about things

02:14:06.780 | that could be visually stunning, I think.

02:14:09.860 | But they are hidden underneath all of that math.

02:14:14.400 | Like if you look at the papers that are written in topology,

02:14:19.220 | if you look at all the discussions on Stack Exchange,

02:14:21.580 | they're all math dense, math heavy.

02:14:23.940 | And the only kind of visual things

02:14:27.220 | that emerge every once in a while

02:14:29.540 | is like something like a Mobius strip.

02:14:32.700 | Every once in a while,

02:14:33.540 | some kind of simple visualizations.

02:14:38.540 | Well, there's the vibration,

02:14:40.460 | there's the hop vibration,

02:14:41.780 | or all of those kinds of things

02:14:43.140 | that some grad student from like 20 years ago

02:14:46.660 | wrote a program in Fortran to visualize it, and that's it.

02:14:50.180 | And it just, you know, it makes me sad

02:14:52.620 | because those are visual disciplines,

02:14:55.020 | just like computer vision is a visual discipline.

02:14:57.900 | So you can provide a lot of visual examples.

02:14:59.660 | I wish topology was more excited

02:15:03.340 | and in love with visualizing some of the ideas.

02:15:07.180 | - I mean, you could say that,

02:15:08.060 | but I would say for me,

02:15:09.020 | a picture of the hop vibration does nothing for me.

02:15:11.900 | Whereas like when you're like,

02:15:13.140 | oh, it's like about the quaternions,

02:15:14.700 | it's like a subgroup of the quaternions,

02:15:16.060 | and I'm like, oh, so now I see what's going on.

02:15:17.860 | Like, why didn't you just say that?

02:15:18.860 | Why were you like showing me this stupid picture

02:15:20.540 | instead of telling me what you were talking about?

02:15:22.420 | - Oh, yeah, yeah.

02:15:24.980 | - I'm just saying, no,

02:15:25.820 | but it goes back to what you were saying about teaching,

02:15:27.340 | that like people are different in what they'll respond to.

02:15:29.740 | So I think there's no, I mean,

02:15:31.020 | I'm very opposed to the idea

02:15:32.460 | that there's one right way to explain things.

02:15:34.380 | I think there's a huge variation in like, you know,

02:15:37.220 | our brains like have all these like weird,

02:15:39.020 | like hooks and loops,

02:15:40.860 | and it's like very hard to know like what's gonna latch on,

02:15:43.260 | and it's not gonna be the same thing for everybody.

02:15:46.100 | So I think monoculture is bad, right?

02:15:49.460 | I think that's, and I think we're agreeing on that point,

02:15:51.540 | that like, it's good that there's like

02:15:53.300 | a lot of different ways in

02:15:54.580 | and a lot of different ways to describe these ideas

02:15:56.460 | because different people are gonna find

02:15:58.100 | different things illuminating.

02:15:59.740 | - But that said, I think there's a lot to be discovered

02:16:04.420 | when you force little like silos of brilliant people

02:16:09.420 | to kind of find a middle ground,

02:16:15.260 | or like aggregate or come together in a way.

02:16:20.220 | So there's like people that do love visual things.

02:16:23.560 | I mean, there's a lot of disciplines,

02:16:25.740 | especially in computer science,

02:16:27.020 | that are obsessed with visualizing, visualizing data,

02:16:30.140 | visualizing neural networks.

02:16:31.500 | I mean, neural networks themselves are fundamentally visual.

02:16:34.100 | There's a lot of work in computer vision that's very visual.

02:16:36.820 | And then coming together with some folks

02:16:39.140 | that were like deeply rigorous

02:16:41.140 | and are like totally lost in multi-dimensional space

02:16:43.620 | where it's hard to even bring them back down to 3D.

02:16:46.220 | They're very comfortable in this multi-dimensional space.

02:16:50.300 | So forcing them to kind of work together to communicate,

02:16:53.500 | because it's not just about public communication of ideas.

02:16:57.300 | It's also, I feel like when you're forced

02:16:59.180 | to do that public communication,

02:17:00.580 | like you did with your book,

02:17:02.100 | I think deep, profound ideas can be discovered

02:17:05.780 | that's like applicable for research and for science.

02:17:08.740 | Like there's something about that simplification,

02:17:10.980 | not simplification, but distillation or condensation

02:17:15.380 | or whatever the hell you call it,

02:17:17.020 | compression of ideas that somehow

02:17:19.860 | actually stimulates creativity.

02:17:22.160 | And I'd be excited to see more of that

02:17:25.220 | in the mathematics community.

02:17:27.820 | Can you--

02:17:28.660 | - Let me make a crazy metaphor.

02:17:29.500 | Maybe it's a little bit like the relation

02:17:31.140 | between prose and poetry, right?

02:17:32.620 | I mean, you might say,

02:17:33.700 | "Why do we need anything more than prose?

02:17:34.980 | "You're trying to convey some information."

02:17:36.460 | So you just say it.

02:17:38.500 | Well, poetry does something, right?

02:17:40.500 | It's sort of, you might think of it as a kind of compression.

02:17:43.340 | Of course, not all poetry is compressed.

02:17:44.940 | Like not all, some of it is quite baggy.

02:17:47.660 | But like you are kind of, often it's compressed, right?

02:17:52.660 | A lyric poem is often sort of like a compression

02:17:55.600 | of what would take a long time

02:17:57.740 | and be complicated to explain in prose

02:18:00.300 | into sort of a different mode

02:18:03.300 | that is gonna hit in a different way.

02:18:05.360 | - We talked about Poincaré conjecture.

02:18:08.980 | There's a guy, he's Russian, Grigori Perlman.

02:18:14.640 | He proved Poincaré's conjecture.

02:18:16.600 | If you can comment on the proof itself,

02:18:19.180 | if that stands out to you as something interesting,

02:18:21.560 | or the human story of it,

02:18:23.160 | which is he turned down the Fields Medal for the proof.

02:18:26.960 | Is there something you find inspiring or insightful

02:18:32.760 | about the proof itself or about the man?

02:18:36.160 | - Yeah, I mean, one thing I really like about the proof,

02:18:40.600 | and partly that's because it's sort of a thing

02:18:42.920 | that happens again and again in this book.

02:18:45.080 | I mean, I'm writing about geometry

02:18:46.480 | and the way it sort of appears

02:18:47.740 | in all these kind of real world problems.

02:18:50.200 | But it happens so often that the geometry

02:18:52.720 | you think you're studying is somehow not enough.

02:18:56.920 | You have to go one level higher in abstraction

02:18:59.240 | and study a higher level of geometry.

02:19:01.600 | And the way that plays out is that,

02:19:04.080 | Poincaré asks a question about a certain kind

02:19:06.380 | of three-dimensional object.

02:19:07.840 | Is it the usual three-dimensional space that we know,

02:19:10.320 | or is it some kind of exotic thing?

02:19:13.060 | And so of course, this sounds like it's a question

02:19:15.120 | about the geometry of the three-dimensional space.

02:19:17.600 | But no, Perelman understands.

02:19:20.200 | And by the way, in a tradition that involves

02:19:21.920 | Richard Hamilton and many other people,

02:19:23.560 | like most really important mathematical advances,

02:19:26.320 | this doesn't happen alone.

02:19:27.400 | It doesn't happen in a vacuum.

02:19:28.480 | It happens as the culmination of a program

02:19:30.160 | that involves many people.

02:19:31.280 | Same with Wiles, by the way.

02:19:32.400 | I mean, we talked about Wiles,

02:19:33.480 | and I want to emphasize that starting all the way back

02:19:36.200 | with Kummer, who I mentioned in the 19th century,

02:19:38.160 | but Gerhard Frey and Maser and Ken Ribbit

02:19:42.200 | and like many other people are involved

02:19:45.200 | in building the other pieces of the arch

02:19:47.200 | before you put the keystone in.

02:19:48.280 | - We stand on the shoulders of giants.

02:19:50.360 | - Yes.

02:19:51.180 | So what is this idea?

02:19:56.080 | The idea is that, well, of course,

02:19:57.420 | the geometry of the three-dimensional object itself

02:19:59.900 | is relevant, but the real geometry,

02:20:01.640 | you have to understand, is the geometry of the space

02:20:04.720 | of all three-dimensional geometries.

02:20:07.400 | Whoa.

02:20:08.680 | You're going up a higher level.

02:20:10.520 | Because when you do that, you can say,

02:20:12.000 | now let's trace out a path in that space.

02:20:17.000 | There's a mechanism called Ricci flow.

02:20:19.800 | And again, we're outside my research area,

02:20:21.080 | so for all the geometric analysts

02:20:23.320 | and differential geometers out there listening to this,

02:20:25.760 | if I, please, I'm doing my best and I'm roughly saying it.

02:20:29.440 | So the Ricci flow allows you to say like,

02:20:32.200 | okay, let's start from some mystery three-dimensional space,

02:20:35.360 | which Poincaré would conjecture is essentially

02:20:37.720 | the same thing as our familiar three-dimensional space,

02:20:39.480 | but we don't know that.

02:20:41.200 | And now you let it flow.

02:20:44.120 | You sort of like let it move in its natural path,

02:20:47.460 | according to some almost physical process,

02:20:50.080 | and ask where it winds up.

02:20:51.400 | And what you find is that it always winds up.

02:20:54.320 | You've continuously deformed it.

02:20:55.680 | There's that word deformation again.

02:20:58.280 | And what you can prove is that the process doesn't stop

02:21:00.160 | until you get to the usual three-dimensional space.

02:21:02.080 | And since you can get from the mystery thing

02:21:04.640 | to the standard space by this process of continually changing

02:21:08.400 | and never kind of having any sharp transitions,

02:21:13.000 | then the original shape must have been the same

02:21:16.320 | as the standard shape.

02:21:17.520 | That's the nature of the proof.

02:21:18.800 | Now, of course, it's incredibly technical.

02:21:20.480 | I think, as I understand it,

02:21:21.520 | I think the hard part is proving

02:21:23.400 | that the favorite word of AI people,

02:21:25.680 | you don't get any singularities along the way.

02:21:27.920 | But of course, in this context,

02:21:30.520 | singularity just means acquiring a sharp kink.

02:21:34.400 | It just means becoming non-smooth at some point.

02:21:37.040 | So just saying something interesting about formal,

02:21:41.000 | about the smooth trajectory through this weird space.

02:21:44.400 | - Yeah, but yeah, so what I like about it

02:21:46.720 | is that it's just one of many examples

02:21:48.280 | of where it's not about the geometry you think it's about.

02:21:51.640 | It's about the geometry of all geometries, so to speak.

02:21:55.960 | And it's only by kind of like being jerked out of Flatland,

02:21:59.360 | right, same idea.

02:22:00.200 | It's only by sort of seeing the whole thing globally at once

02:22:04.120 | that you can really make progress on understanding

02:22:05.840 | the one thing you thought you were looking at.

02:22:08.440 | - It's a romantic question,

02:22:09.520 | but what do you think about him

02:22:11.160 | turning down the Fields Medal?

02:22:13.000 | Is that just, are Nobel Prizes and Fields Medals

02:22:17.040 | just the cherry on top of the cake

02:22:20.000 | and really math itself,

02:22:21.960 | the process of curiosity,

02:22:25.240 | of pulling at the string of the mystery before us,

02:22:28.480 | that's the cake?

02:22:29.560 | And then the awards are just icing.

02:22:33.800 | - Man, clearly I've been fasting and I'm hungry,

02:22:37.200 | but do you think it's tragic

02:22:42.200 | or just a little curiosity that he turned down the medal?

02:22:46.360 | - Well, it's interesting because on the one hand,

02:22:48.400 | I think it's absolutely true that right,

02:22:50.760 | in some kind of like vast spiritual sense,

02:22:55.440 | like awards are not important,

02:22:57.320 | like not important the way that sort of like

02:22:59.240 | understanding the universe is important.

02:23:02.480 | On the other hand,

02:23:03.320 | most people who are offered that prize accept it.

02:23:05.440 | It's, so there's something unusual about his choice there.

02:23:10.440 | I wouldn't say I see it as tragic.

02:23:14.400 | I mean, maybe if I don't really feel like

02:23:16.200 | I have a clear picture of why he chose not to take it.

02:23:19.280 | I mean, it's not,

02:23:20.120 | he's not alone in doing things like this.

02:23:22.080 | People sometimes turn down prizes for ideological reasons,

02:23:25.280 | probably more often in mathematics.

02:23:28.000 | I mean, I think I'm right in saying that Peter Schultz

02:23:30.720 | like turned down sort of some big monetary prize

02:23:33.880 | 'cause he just, you know, I mean, I think he,

02:23:36.600 | at some point you have plenty of money

02:23:39.200 | and maybe you think it sends the wrong message

02:23:41.280 | about what the point of doing mathematics is.

02:23:44.360 | - I do find that there's--

02:23:46.520 | - But most people accept.

02:23:47.480 | You know, most people give it a prize,

02:23:48.800 | most people take it.

02:23:49.620 | I mean, people like to be appreciated,

02:23:50.840 | but like I said, we're people.

02:23:52.480 | - Yes.

02:23:53.320 | - Not that different from most other people.

02:23:54.640 | - But the important reminder that that turning down

02:23:57.840 | the prize serves for me is not that there's anything wrong

02:24:01.480 | with the prize and there's something wonderful

02:24:03.600 | about the prize, I think.

02:24:04.960 | The Nobel prize is trickier

02:24:07.640 | because so many Nobel prizes are given.

02:24:10.360 | First of all, the Nobel prize often forgets

02:24:12.320 | many of the important people throughout history.

02:24:15.640 | Second of all, there's like these weird rules to it

02:24:18.960 | that's only three people and some projects

02:24:21.120 | have a huge number of people and it's like this,

02:24:24.080 | it, I don't know, it doesn't kind of highlight

02:24:29.080 | the way science has done on some of these projects

02:24:32.280 | in the best possible way.

02:24:33.660 | But in general, the prizes are great.

02:24:34.960 | But what this kind of teaches me and reminds me

02:24:37.360 | is sometimes in your life, there'll be moments

02:24:39.920 | when the thing that you would really like to do,

02:24:46.200 | society would really like you to do

02:24:49.620 | is the thing that goes against something you believe in,

02:24:53.920 | whatever that is, some kind of principle,

02:24:56.040 | and stand your ground in the face of that.

02:24:59.840 | It's something, I believe most people

02:25:02.680 | will have a few moments like that in their life,

02:25:05.120 | maybe one moment like that.

02:25:06.440 | And you have to do it, that's what integrity is.

02:25:09.100 | So it doesn't have to make sense to the rest of the world,

02:25:11.160 | but to stand on that, to say no, it's interesting.

02:25:15.520 | 'Cause I think-- - But do you know

02:25:16.480 | that he turned down the prize in service of some principle?

02:25:20.040 | 'Cause I don't know that.

02:25:20.960 | - Well, yes, that seems to be the inkling,

02:25:22.720 | but he has never made it super clear.

02:25:24.560 | But the inkling is that he had some problems

02:25:26.960 | with the whole process of mathematics

02:25:28.720 | that includes awards, like this hierarchies

02:25:32.000 | and the reputations and all those kinds of things,

02:25:34.520 | and individualism that's fundamental to American culture.

02:25:37.680 | He probably, 'cause he visited the United States quite a bit,

02:25:41.200 | that he probably, it's all about experiences.

02:25:46.200 | And he may have had some parts of academia,

02:25:51.520 | some pockets of academia can be less than inspiring,

02:25:54.760 | perhaps sometimes, because of the individual egos involved.

02:25:57.640 | Not academia, people in general, smart people with egos.

02:26:01.200 | And if you interact with a certain kinds of people,

02:26:05.640 | you can become cynical too easily.

02:26:07.480 | I'm one of those people that I've been really fortunate

02:26:10.760 | to interact with incredible people at MIT

02:26:12.840 | and academia in general, but I've met some assholes.

02:26:15.560 | And I tend to just kind of,

02:26:17.080 | when I run into difficult folks,

02:26:19.200 | I just kind of smile and send them all my love

02:26:21.360 | and just kind of go around.

02:26:23.080 | But for others, those experiences can be sticky.

02:26:26.720 | Like they can become cynical about the world

02:26:29.800 | when folks like that exist.

02:26:31.640 | So he may have become a little bit cynical

02:26:35.480 | about the process of science.

02:26:37.200 | - Well, you know, it's a good opportunity.

02:26:38.600 | Let's posit that that's his reasoning,

02:26:40.200 | 'cause I truly don't know.

02:26:42.360 | It's an interesting opportunity to go back

02:26:43.800 | to almost the very first thing we talked about,

02:26:46.320 | the idea of the Mathematical Olympiad.

02:26:48.360 | Because of course, that is,

02:26:50.520 | so the International Mathematical Olympiad

02:26:52.120 | is like a competition for high school students

02:26:54.640 | solving math problems.

02:26:55.880 | And in some sense, it's absolutely false

02:26:59.240 | to the reality of mathematics.

02:27:00.400 | Because just as you say, it is a contest

02:27:03.760 | where you win prizes.

02:27:05.240 | The aim is to sort of be faster than other people.

02:27:10.160 | And you're working on sort of canned problems

02:27:13.880 | that someone already knows the answer to,

02:27:15.720 | like not problems that are unknown.

02:27:18.520 | So, you know, in my own life,

02:27:20.600 | I think when I was in high school,

02:27:21.960 | I was like very motivated by those competitions.

02:27:24.320 | And like I went to the Math Olympiad.

02:27:26.160 | - You won it twice and got, I mean.

02:27:28.600 | - Well, there's something I have to explain to people,

02:27:30.200 | because it says, I think it says on Wikipedia

02:27:32.240 | that I won a gold medal.

02:27:33.480 | And in the real Olympics,

02:27:35.280 | they only give one gold medal in each event.

02:27:37.440 | I just have to emphasize

02:27:38.520 | that the International Math Olympiad is not like that.

02:27:40.840 | The gold medals are awarded

02:27:42.240 | to the top 1/12 of all participants.

02:27:45.040 | So sorry to bust the legend or anything like that.

02:27:47.320 | - Well, you're an exceptional performer

02:27:48.840 | in terms of achieving high scores on the problems,

02:27:51.840 | and they're very difficult.

02:27:53.240 | So you've achieved a high level of performance on the--

02:27:56.280 | - In this very specialized skill.

02:27:57.880 | And by the way, it was a very Cold War activity.

02:28:00.520 | You know, in 1987, the first year I went, it was in Havana.

02:28:04.640 | Americans couldn't go to Havana back then.

02:28:06.120 | It was a very complicated process to get there.

02:28:08.720 | And they took the whole American team

02:28:10.120 | on a field trip to the Museum of American Imperialism

02:28:13.600 | in Havana so we could see what America was all about.

02:28:17.580 | - How would you recommend a person learn math?

02:28:22.580 | So somebody who's young or somebody my age

02:28:26.380 | or somebody older who've taken a bunch of math

02:28:29.740 | but wants to rediscover the beauty of math

02:28:32.100 | and maybe integrate it into their work more so

02:28:34.700 | than the research space and so on.

02:28:38.540 | Is there something you could say about the process

02:28:40.660 | of incorporating mathematical thinking into your life?

02:28:46.220 | - I mean, the thing is it's in part

02:28:48.340 | a journey of self-knowledge.

02:28:49.700 | You have to know what's gonna work for you

02:28:52.540 | and that's gonna be different for different people.

02:28:54.780 | So there are totally people who at any stage of life

02:28:58.100 | just start reading math textbooks.

02:29:00.820 | That is a thing that you can do

02:29:02.460 | and it works for some people and not for others.

02:29:05.420 | For others, a gateway is, you know,

02:29:07.220 | I always recommend like the books of Martin Gardner,

02:29:09.620 | another sort of person we haven't talked about,

02:29:11.420 | but who also, like Conway, embodies that spirit

02:29:14.340 | of play, he wrote a column in Scientific American

02:29:17.380 | for decades called "Mathematical Recreations"

02:29:19.820 | and there's such joy in it and such fun.

02:29:22.700 | And these books, the columns are collected into books

02:29:25.380 | and the books are old now, but for each generation

02:29:27.500 | of people who discover them, they're completely fresh.

02:29:30.140 | And they give a totally different way into the subject

02:29:32.540 | than reading a formal textbook,

02:29:35.140 | which for some people would be the right thing to do.

02:29:38.660 | And, you know, working contest style problems too,

02:29:40.900 | those are bound to books, like especially like writing

02:29:43.100 | to books, like especially like Russian

02:29:44.580 | and Bulgarian problems, right?

02:29:45.660 | There's book after book of problems from those contexts.

02:29:47.780 | That's gonna motivate some people.

02:29:50.060 | For some people, it's gonna be like watching

02:29:51.580 | well-produced videos, like a totally different format.

02:29:54.300 | Like I feel like I'm not answering your question.

02:29:56.060 | I'm sort of saying there's no one answer

02:29:57.780 | and like it's a journey where you figure out

02:30:00.340 | what resonates with you.

02:30:01.900 | For some people, it's the self-discovery

02:30:04.300 | is trying to figure out why is it that I wanna know.

02:30:06.780 | Okay, I'll tell you a story.

02:30:07.620 | Once when I was in grad school, I was very frustrated

02:30:10.540 | with my like lack of knowledge of a lot of things.

02:30:12.500 | As we all are, because no matter how much we know,

02:30:14.100 | we don't know much more and going to grad school

02:30:15.820 | means just coming face to face with like the incredible

02:30:18.780 | overflowing vault of your ignorance, right?

02:30:20.340 | So I told Joe Harris, who was an algebraic geometer

02:30:23.740 | a professor in my department, I was like,

02:30:26.540 | I really feel like I don't know enough

02:30:27.620 | and I should just like take a year of leave

02:30:29.340 | and just like read EGA, the holy textbook,

02:30:32.620 | (speaking in foreign language)

02:30:34.460 | elements of algebraic geometry.

02:30:35.740 | This like, I'm just gonna, I feel like I don't know enough.

02:30:38.700 | So I was gonna sit and like read this like 1500 page,

02:30:42.060 | many volume book.

02:30:44.060 | And he was like, Professor Harris was like,

02:30:48.300 | that's a really stupid idea.

02:30:49.460 | And I was like, why is that a stupid idea?

02:30:50.820 | Then I would know more algebraic geometry.

02:30:52.660 | He's like, because you're not actually gonna do it.

02:30:53.940 | Like you learn.

02:30:55.780 | I mean, he knew me well enough to say like,

02:30:57.140 | you're gonna learn because you're gonna be working

02:30:58.860 | on a problem and then there's gonna be a fact from EGA

02:31:01.060 | you need in order to solve your problem

02:31:03.020 | that you wanna solve and that's how you're gonna learn it.

02:31:05.300 | You're not gonna learn it without a problem

02:31:06.820 | to bring you into it.

02:31:07.980 | And so for a lot of people, I think if you're like,

02:31:10.660 | I'm trying to understand machine learning

02:31:12.420 | and I'm like, I can see that there's sort of

02:31:14.420 | some mathematical technology that I don't have.

02:31:19.420 | I think you like let that problem

02:31:22.580 | that you actually care about drive your learning.

02:31:26.020 | I mean, one thing I've learned from advising students,

02:31:27.860 | you know, math is really hard.

02:31:31.140 | In fact, anything that you do right is hard.

02:31:35.860 | And because it's hard, like,

02:31:40.140 | you might sort of have some idea that somebody else gives you

02:31:42.260 | oh, I should learn X, Y, and Z.

02:31:44.460 | Well, if you don't actually care, you're not gonna do it.

02:31:46.380 | You might feel like you should,

02:31:47.420 | maybe somebody told you you should,

02:31:48.860 | but I think you have to hook it to something

02:31:51.740 | that you actually care about.

02:31:52.700 | So for a lot of people, that's the way in.

02:31:54.500 | You have an engineering problem you're trying to handle.

02:31:57.180 | You have a physics problem you're trying to handle.

02:31:59.500 | You have a machine learning problem you're trying to handle.

02:32:02.020 | Let that, not a kind of abstract idea

02:32:04.980 | of what the curriculum is, drive your mathematical learning.

02:32:08.340 | - And also just as a brief comment, that math is hard.

02:32:12.180 | There's a sense to which hard is a feature, not a bug.

02:32:15.220 | In the sense that, again,

02:32:16.980 | maybe this is my own learning preference,

02:32:19.740 | but I think it's a value to fall in love

02:32:23.660 | with the process of doing something hard,

02:32:26.020 | overcoming it, and becoming a better person

02:32:29.220 | because like, I hate running.

02:32:31.060 | I hate exercise to bring it down to like the simplest hard.

02:32:35.740 | And I enjoy the part once it's done,

02:32:39.780 | the person I feel like for the rest of the day

02:32:42.020 | once I've accomplished it.

02:32:43.100 | The actual process, especially the process

02:32:45.180 | of getting started in the initial,

02:32:47.540 | like it really, I don't feel like doing it.

02:32:49.580 | And I really have, the way I feel about running

02:32:51.660 | is the way I feel about really anything difficult

02:32:55.140 | in the intellectual space, especially in mathematics,

02:32:58.500 | but also just something that requires

02:33:01.860 | like holding a bunch of concepts in your mind

02:33:04.860 | with some uncertainty, like where the terminology

02:33:08.260 | or the notation is not very clear.

02:33:10.260 | And so you have to kind of hold all those things together

02:33:13.340 | and like keep pushing forward through the frustration

02:33:16.060 | of really like obviously not understanding certain,

02:33:19.660 | like parts of the picture,

02:33:21.660 | like your giant missing parts of the picture,

02:33:24.380 | and still not giving up.

02:33:26.580 | It's the same way I feel about running.

02:33:29.020 | And there's something about falling in love

02:33:32.820 | with the feeling of after you went through the journey

02:33:36.140 | of not having a complete picture,

02:33:38.180 | at the end, having a complete picture,

02:33:40.620 | and then you get to appreciate the beauty

02:33:42.460 | and just remembering that it sucked for a long time

02:33:46.020 | and how great it felt when you figured it out,

02:33:48.780 | at least at the basic.

02:33:49.940 | That's not sort of research thinking,

02:33:52.020 | 'cause with research, you probably also have to

02:33:55.220 | enjoy the dead ends.

02:33:57.260 | With learning math from a textbook or from video,

02:34:02.540 | there's a nice--

02:34:03.380 | - I don't think you have to enjoy the dead ends,

02:34:04.580 | but I think you have to accept the dead ends.

02:34:06.340 | Let's put it that way.

02:34:07.960 | - Well, yeah, enjoy the suffering of it.

02:34:11.180 | The way I think about it, I do, there's an--

02:34:17.060 | - I don't enjoy the suffering, it pisses me off,

02:34:19.060 | but I accept that it's part of the process.

02:34:21.220 | - It's interesting, there's a lot of ways

02:34:22.420 | to kind of deal with that dead end.

02:34:24.540 | There's a guy who's an ultra marathon runner,

02:34:26.420 | Navy SEAL, David Goggins, who kind of,

02:34:30.060 | I mean, there's a certain philosophy of like,

02:34:32.800 | most people would quit here.

02:34:36.000 | And so if most people would quit here, and I don't,

02:34:42.420 | I'll have an opportunity to discover something beautiful

02:34:45.140 | that others haven't yet.

02:34:46.400 | So like, any feeling that really sucks,

02:34:52.900 | it's like, okay, most people would just like

02:34:56.940 | go do something smarter.

02:34:58.620 | And if I stick with this, I will discover a new garden

02:35:03.100 | of fruit trees that I can pick.

02:35:06.100 | - Okay, you say that, but like, what about the guy

02:35:08.340 | who like wins the Nathan's hot dog eating contest every year?

02:35:11.260 | Like when he eats his 35th hot dog, he like correctly says,

02:35:13.980 | like, okay, most people would stop here.

02:35:16.020 | Are you like lauding that he's like,

02:35:18.420 | no, I'm gonna eat the 36th hot dog?

02:35:19.980 | - I am, I am, I am.

02:35:21.580 | In the long arc of history, that man is onto something.

02:35:26.260 | Which brings up this question.

02:35:28.380 | What advice would you give to young people today,

02:35:30.940 | thinking about their career, about their life,

02:35:33.980 | whether it's in mathematics, poetry,

02:35:37.260 | or hot dog eating contest?

02:35:39.340 | (laughing)

02:35:40.620 | - And you know, I have kids, so this is actually

02:35:42.740 | a live issue for me, right?

02:35:43.860 | I actually, it's not a thought experiment.

02:35:45.700 | I actually do have to give advice to young people

02:35:47.900 | all the time.

02:35:48.740 | They don't listen, but I still give it.

02:35:50.540 | You know, one thing I often say to students,

02:35:55.340 | I don't think I've actually said this to my kids yet,

02:35:56.740 | but I say it to students a lot, is,

02:35:58.980 | you know, you come to these decision points,

02:36:02.020 | and everybody is beset by self-doubt, right?

02:36:06.620 | It's like, not sure what they're capable of,

02:36:09.740 | like, not sure what they really wanna do.

02:36:14.740 | I always, I sort of tell people, like,

02:36:16.540 | often when you have a decision to make,

02:36:18.460 | one of the choices is the high self-esteem choice.

02:36:22.740 | And I always tell them, make the high self-esteem choice.

02:36:24.620 | Make the choice, sort of take yourself out of it,

02:36:26.780 | and like, if you didn't have those,

02:36:29.620 | you can probably figure out what the version of you

02:36:32.180 | that feels completely confident would do,

02:36:34.180 | and do that, and see what happens.

02:36:36.500 | And I think that's often, like, pretty good advice.

02:36:40.100 | - That's interesting, sort of like,

02:36:41.860 | you know, like with Sims, you can create characters.

02:36:44.980 | Like, create a character of yourself

02:36:47.820 | that lacks all of the self-doubt.

02:36:50.260 | - Right, but it doesn't mean, I would never say to somebody,

02:36:52.940 | you should just go have high self-esteem.

02:36:56.220 | You shouldn't have doubts.

02:36:57.180 | No, you probably should have doubts.

02:36:58.220 | It's okay to have them, but sometimes it's good to act

02:37:01.340 | in the way that the person who didn't have them would act.

02:37:04.260 | - That's a really nice way to put it.

02:37:08.460 | Yeah, that's like, from a third-person perspective,

02:37:13.060 | take the part of your brain that wants to do big things.

02:37:16.540 | What would they do?

02:37:18.180 | That's not afraid to do those things.

02:37:20.060 | What would they do?

02:37:21.500 | Yeah, that's really nice.

02:37:24.420 | That's actually a really nice way to formulate it.

02:37:26.340 | That's very practical advice.

02:37:27.540 | You should give it to your kids.

02:37:29.100 | Do you think there's meaning to any of it,

02:37:32.660 | from a mathematical perspective, this life?

02:37:35.780 | If I were to ask you, we're talking about primes,

02:37:40.900 | talking about proving stuff.

02:37:42.480 | Can we say, and then the book that God has,

02:37:47.340 | that mathematics allows us to arrive at something about,

02:37:50.540 | in that book, there's certainly a chapter

02:37:52.820 | on the meaning of life in that book.

02:37:54.980 | Do you think we humans can get to it?

02:37:57.380 | And maybe, if you were to write Cliff Notes,

02:37:59.500 | what do you suspect those Cliff Notes would say?

02:38:01.500 | - I mean, look, the way I feel is that,

02:38:03.340 | you know, mathematics, as we've discussed,

02:38:05.900 | like it underlies the way we think

02:38:07.580 | about constructing learning machines.

02:38:09.220 | It underlies physics.

02:38:10.560 | It can be, I mean, it does all this stuff.

02:38:14.600 | And also, you want the meaning of life?

02:38:17.180 | I mean, it's like, we already did a lot for you.

02:38:18.820 | Like, ask a rabbi.

02:38:20.440 | (laughing)

02:38:22.600 | No, I mean, I think, you know, I wrote a lot

02:38:24.680 | in the last book, How Not to Be Wrong.

02:38:27.720 | I wrote a lot about Pascal, a fascinating guy,

02:38:30.660 | who is a sort of very serious religious mystic,

02:38:35.120 | as well as being an amazing mathematician.

02:38:37.200 | And he's well-known for Pascal's Wager.

02:38:38.860 | I mean, he's probably, among all mathematicians,

02:38:40.240 | he's the one who's best known for this,

02:38:42.320 | can you actually apply mathematics

02:38:44.120 | to kind of these transcendent questions?

02:38:49.800 | But what's interesting, when I really read Pascal

02:38:53.040 | about what he wrote about this,

02:38:54.560 | you know, I started to see that people often think,

02:38:56.320 | oh, this is him saying, I'm gonna use mathematics

02:38:58.960 | to sort of show you why you should believe in God.

02:39:03.340 | You know, to really, that's,

02:39:04.480 | mathematics has the answer to this question.

02:39:07.240 | But he really doesn't say that.

02:39:08.920 | He almost kind of says the opposite.

02:39:11.880 | If you ask Blaise Pascal, like, why do you believe in God?

02:39:15.160 | He'd be like, oh, 'cause I met God.

02:39:16.600 | You know, he had this kind of like psychedelic experience,

02:39:20.160 | this like mystical experience, where, as he tells it,

02:39:23.400 | he just like directly encountered God.

02:39:24.960 | It's like, okay, I guess there's a God.

02:39:26.060 | I met him last night, so that's it.

02:39:27.960 | That's why he believed.

02:39:29.120 | It didn't have to do with any kind of,

02:39:30.320 | you know, the mathematical argument was like

02:39:32.520 | about certain reasons for behaving in a certain way.

02:39:36.760 | But he basically said, like, look,

02:39:38.320 | like math doesn't tell you that God's there or not.

02:39:41.080 | Like, if God's there, he'll tell you, you know?

02:39:44.360 | You don't even-- - I love this.

02:39:45.720 | So you have mathematics, you have, what do you have?

02:39:50.440 | Like, ways to explore the mind, let's say psychedelics.

02:39:53.760 | You have like incredible technology.

02:39:56.600 | You also have love and friendship.

02:39:59.640 | And like, what the hell do you wanna know

02:40:01.800 | what the meaning of it all is?

02:40:02.880 | Just enjoy it. (laughs)

02:40:04.920 | I don't think there's a better way to end it, Jordan.

02:40:07.000 | This was a fascinating conversation.

02:40:08.520 | I really love the way you explore math in your writing,

02:40:14.120 | the willingness to be specific and clear

02:40:18.440 | and actually explore difficult ideas,

02:40:21.200 | but at the same time stepping outside

02:40:23.120 | and figuring out beautiful stuff.

02:40:25.080 | And I love the chart at the opening of your new book

02:40:30.080 | that shows the chaos, the mess that is your mind.

02:40:33.280 | - Yes, this is what I was trying to keep in my head

02:40:35.560 | all at once while I was writing.

02:40:38.040 | And I probably should have drawn this picture earlier

02:40:40.680 | in the process.

02:40:41.520 | Maybe it would have made my organization easier.

02:40:43.080 | I actually drew it only at the end.

02:40:45.400 | - And many of the things we talked about are on this map.

02:40:48.600 | The connections are yet to be fully dissected

02:40:51.800 | and investigated.

02:40:52.640 | And yes, God is in the picture.

02:40:56.720 | - Right on the edge, right on the edge, not in the center.

02:40:59.480 | - Thank you so much for talking to me.

02:41:01.660 | It is a huge honor that you would waste

02:41:03.440 | your valuable time with me.

02:41:04.840 | - Thank you, Lex.

02:41:06.680 | We went to some amazing places today.

02:41:07.800 | This was really fun.

02:41:09.640 | - Thanks for listening to this conversation

02:41:11.200 | with Jordan Ellenberg.

02:41:12.360 | And thank you to Secret Sauce, ExpressVPN, Blinkist,

02:41:16.640 | and Indeed.

02:41:18.000 | Check them out in the description to support this podcast.

02:41:21.360 | And now let me leave you with some words from Jordan

02:41:24.120 | in his book, "How Not to Be Wrong."

02:41:26.640 | "Knowing mathematics is like wearing a pair of X-ray specs

02:41:30.600 | that reveal hidden structures underneath the messy

02:41:33.500 | and chaotic surface of the world."

02:41:35.780 | Thank you for listening and hope to see you next time.

02:41:38.520 | (upbeat music)

02:41:41.100 | (upbeat music)

02:41:43.680 | [BLANK_AUDIO]

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

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