Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries #190

Transcript

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

00:00:02 a mathematician at University of Wisconsin

00:00:05 and an author who masterfully reveals the beauty

00:00:08 and power of mathematics in his 2014 book,

00:00:12 How Not To Be Wrong, and his new book,

00:00:15 just released recently, called Shape,

00:00:17 The Hidden Geometry of Information, Biology,

00:00:20 Strategy, Democracy, and Everything Else.

00:00:23 Quick mention of our sponsors,

00:00:25 Secret Sauce, ExpressVPN, Blinkist, and Indeed.

00:00:29 Check them out in the description to support this podcast.

00:00:33 As a side note, let me say that geometry

00:00:35 is what made me fall in love with mathematics

00:00:37 when I was young.

00:00:38 It first showed me that something definitive

00:00:41 could be stated about this world

00:00:42 through intuitive visual proofs.

00:00:45 Somehow, that convinced me that math

00:00:47 is not just abstract numbers devoid of life,

00:00:50 but a part of life, part of this world,

00:00:53 part of our search for meaning.

00:00:55 This is the Lex Friedman podcast,

00:00:57 and here is my conversation with Jordan Ellenberg.

00:01:01 If the brain is a cake.

00:01:03 It is?

00:01:05 Well, let’s just go with me on this, okay?

00:01:07 Okay, we’ll pause it.

00:01:08 So for Noam Chomsky, language,

00:01:12 the universal grammar, the framework

00:01:16 from which language springs is like most of the cake,

00:01:19 the delicious chocolate center,

00:01:21 and then the rest of cognition that we think of

00:01:25 is built on top, extra layers,

00:01:27 maybe the icing on the cake,

00:01:28 maybe consciousness is just like a cherry on top.

00:01:34 Where do you put in this cake mathematical thinking?

00:01:37 Is it as fundamental as language?

00:01:40 In the Chomsky view, is it more fundamental than language?

00:01:44 Is it echoes of the same kind of abstract framework

00:01:47 that he’s thinking about in terms of language

00:01:49 that they’re all really tightly interconnected?

00:01:52 That’s a really interesting question.

00:01:54 You’re getting me to reflect on this question

00:01:56 of whether the feeling of producing mathematical output,

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

00:02:04 or producing linguistic output.

00:02:07 I think it feels something like that,

00:02:09 and it’s certainly the case.

00:02:10 Let me put it this way.

00:02:11 It’s hard to imagine doing mathematics

00:02:14 in a completely nonlinguistic way.

00:02:17 It’s hard to imagine doing mathematics

00:02:19 without talking about mathematics

00:02:22 and sort of thinking in propositions.

00:02:23 But maybe it’s just because that’s the way I do mathematics,

00:02:26 and maybe I can’t imagine it any other way, right?

00:02:29 Well, what about visualizing shapes,

00:02:32 visualizing concepts to which language

00:02:35 is not obviously attachable?

00:02:38 Ah, that’s a really interesting question.

00:02:40 And one thing it reminds me of is one thing I talk about

00:02:43 in the book is dissection proofs,

00:02:45 these very beautiful proofs of geometric propositions.

00:02:48 There’s a very famous one by Baskara

00:02:50 of the Pythagorean theorem, proofs which are purely visual,

00:02:56 proofs where you show that two quantities are the same

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

00:03:04 and making one shape and putting them together another way

00:03:07 and making a different shape,

00:03:08 and then observing that those two shapes

00:03:09 must have the same area

00:03:10 because they were built out of the same pieces.

00:03:14 There’s a famous story,

00:03:16 and it’s a little bit disputed about how accurate this is,

00:03:19 but that in Baskara’s manuscript,

00:03:20 he sort of gives this proof, just gives the diagram,

00:03:22 and then the entire verbal content of the proof

00:03:26 is he just writes under it, behold.

00:03:28 Like that’s it.

00:03:29 And it’s like, there’s some dispute

00:03:32 about exactly how accurate that is.

00:03:33 But so then there’s an interesting question.

00:03:36 If your proof is a diagram, if your proof is a picture,

00:03:39 or even if your proof is like a movie of the same pieces

00:03:42 like coming together in two different formations

00:03:43 to make two different things, is that language?

00:03:45 I’m not sure I have a good answer.

00:03:46 What do you think?

00:03:47 I think it is. I think the process

00:03:51 of manipulating the visual elements

00:03:55 is the same as the process

00:03:56 of manipulating the elements of language.

00:03:59 And I think probably the manipulating, the aggregation,

00:04:02 the stitching stuff together is the important part.

00:04:05 It’s not the actual specific elements.

00:04:07 It’s more like, to me, language is a process

00:04:10 and math is a process.

00:04:11 It’s not just specific symbols.

00:04:15 It’s in action.

00:04:19 It’s ultimately created through action, through change.

00:04:23 And so you’re constantly evolving ideas.

00:04:26 Of course, we kind of attach,

00:04:27 there’s a certain destination you arrive to

00:04:29 that you attach to and you call that a proof,

00:04:32 but that’s not, that doesn’t need to end there.

00:04:34 It’s just at the end of the chapter

00:04:36 and then it goes on and on and on in that kind of way.

00:04:39 But I gotta ask you about geometry

00:04:40 and it’s a prominent topic in your new book, Shape.

00:04:44 So for me, geometry is the thing,

00:04:48 just like as you’re saying,

00:04:49 made me fall in love with mathematics when I was young.

00:04:53 So being able to prove something visually

00:04:56 just did something to my brain that it had this,

00:05:01 it planted this hopeful seed

00:05:02 that you can understand the world, like perfectly.

00:05:07 Maybe it’s an OCD thing,

00:05:08 but from a mathematics perspective,

00:05:10 like humans are messy, the world is messy, biology is messy.

00:05:14 Your parents are yelling or making you do stuff,

00:05:17 but you can cut through all that BS

00:05:19 and truly understand the world through mathematics

00:05:22 and nothing like geometry did that for me.

00:05:25 For you, you did not immediately fall in love

00:05:28 with geometry, so how do you think about geometry?

00:05:33 Why is it a special field in mathematics?

00:05:36 And how did you fall in love with it if you have?

00:05:39 Wow, you’ve given me like a lot to say.

00:05:41 And certainly the experience that you describe

00:05:44 is so typical, but there’s two versions of it.

00:05:48 One thing I say in the book

00:05:49 is that geometry is the cilantro of math.

00:05:51 People are not neutral about it.

00:05:52 There’s people who like you are like,

00:05:55 the rest of it I could take or leave,

00:05:56 but then at this one moment, it made sense.

00:05:59 This class made sense, why wasn’t it all like that?

00:06:01 There’s other people, I can tell you,

00:06:02 because they come and talk to me all the time,

00:06:04 who are like, I understood all the stuff

00:06:06 where you’re trying to figure out what X was,

00:06:08 there’s some mystery you’re trying to solve it,

00:06:09 X is a number, I figured it out.

00:06:10 But then there was this geometry, like what was that?

00:06:12 What happened that year? Like I didn’t get it.

00:06:14 I was like lost the whole year

00:06:15 and I didn’t understand like why we even

00:06:17 spent the time doing that.

00:06:18 So, but what everybody agrees on

00:06:20 is that it’s somehow different, right?

00:06:22 There’s something special about it.

00:06:25 We’re gonna walk around in circles a little bit,

00:06:27 but we’ll get there.

00:06:27 You asked me how I fell in love with math.

00:06:32 I have a story about this.

00:06:36 When I was a small child, I don’t know,

00:06:39 maybe like I was six or seven, I don’t know.

00:06:42 I’m from the 70s.

00:06:42 I think you’re from a different decade than that.

00:06:44 But in the 70s, we had a cool wooden box

00:06:48 around your stereo.

00:06:49 That was the look, everything was dark wood.

00:06:51 And the box had a bunch of holes in it

00:06:53 to let the sound out.

00:06:56 And the holes were in this rectangular array,

00:06:58 a six by eight array of holes.

00:07:02 And I was just kind of like zoning out

00:07:04 in the living room as kids do,

00:07:06 looking at this six by eight rectangular array of holes.

00:07:09 And if you like, just by kind of like focusing in and out,

00:07:12 just by kind of looking at this box,

00:07:14 looking at this rectangle, I was like,

00:07:17 well, there’s six rows of eight holes each,

00:07:21 but there’s also eight columns of six holes each.

00:07:25 Whoa.

00:07:26 So eight sixes and six eights.

00:07:29 It’s just like the dissection proofs

00:07:30 we were just talking about, but it’s the same holes.

00:07:32 It’s the same 48 holes.

00:07:33 That’s how many there are,

00:07:34 no matter whether you count them as rows

00:07:36 or count them as columns.

00:07:38 And this was like unbelievable to me.

00:07:41 Am I allowed to cuss on your podcast?

00:07:43 I don’t know if that’s, are we FCC regulated?

00:07:45 Okay, it was fucking unbelievable.

00:07:47 Okay, that’s the last time.

00:07:48 Get it in there.

00:07:48 This story merits it.

00:07:49 So two different perspectives in the same physical reality.

00:07:54 Exactly.

00:07:55 And it’s just as you say.

00:07:58 I knew that six times eight was the same as eight times six.

00:08:01 I knew my times tables.

00:08:02 I knew that that was a fact.

00:08:04 But did I really know it until that moment?

00:08:06 That’s the question, right?

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

00:08:11 but I didn’t know why that was the case until that moment.

00:08:13 And in that moment I could see like,

00:08:15 oh, I didn’t have to have somebody tell me that.

00:08:17 That’s information that you can just directly access.

00:08:20 That’s a really amazing moment.

00:08:21 And as math teachers, that’s something

00:08:22 that we’re really trying to bring to our students.

00:08:25 And I was one of those who did not love

00:08:27 the kind of Euclidean geometry ninth grade class

00:08:30 of like prove that an isosceles triangle

00:08:33 has equal angles at the base, like this kind of thing.

00:08:35 It didn’t vibe with me the way that algebra and numbers did.

00:08:39 But if you go back to that moment,

00:08:40 from my adult perspective,

00:08:41 looking back at what happened with that rectangle,

00:08:43 I think that is a very geometric moment.

00:08:45 In fact, that moment exactly encapsulates

00:08:49 the intertwining of algebra and geometry.

00:08:53 This algebraic fact that, well, in the instance,

00:08:55 eight times six is equal to six times eight.

00:08:57 But in general, that whatever two numbers you have,

00:09:00 you multiply them one way.

00:09:01 And it’s the same as if you multiply them

00:09:02 in the other order.

00:09:03 It attaches it to this geometric fact about a rectangle,

00:09:07 which in some sense makes it true.

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

00:09:11 to be an algebraic geometer,

00:09:12 which is what I am as a researcher.

00:09:15 So that’s the kind of transformation.

00:09:17 And you talk about symmetry in your book.

00:09:20 What the heck is symmetry?

00:09:22 What the heck is these kinds of transformation on objects

00:09:26 that once you transform them, they seem to be similar?

00:09:29 What do you make of it?

00:09:30 What’s its use in mathematics

00:09:32 or maybe broadly in understanding our world?

00:09:35 Well, it’s an absolutely fundamental concept.

00:09:37 And it starts with the word symmetry

00:09:39 in the way that we usually use it

00:09:41 when we’re just like talking English

00:09:42 and not talking mathematics, right?

00:09:43 Sort of something is, when we say something is symmetrical,

00:09:46 we usually means it has what’s called an axis of symmetry.

00:09:49 Maybe like the left half of it

00:09:51 looks the same as the right half.

00:09:52 That would be like a left, right axis of symmetry.

00:09:55 Or maybe the top half looks like the bottom half or both.

00:09:57 Maybe there’s sort of a fourfold symmetry

00:09:59 where the top looks like the bottom

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

00:10:03 And that can take you in a lot of different directions.

00:10:06 The abstract study of what the possible combinations

00:10:09 of symmetries there are,

00:10:10 a subject which is called group theory

00:10:11 was actually one of my first loves in mathematics

00:10:14 when I thought about a lot when I was in college.

00:10:17 But the notion of symmetry is actually much more general

00:10:21 than the things that we would call symmetry

00:10:23 if we were looking at like a classical building

00:10:25 or a painting or something like that.

00:10:30 Nowadays in math,

00:10:35 we could use a symmetry to refer to

00:10:38 any kind of transformation of an image

00:10:41 or a space or an object.

00:10:43 So what I talk about in the book is

00:10:48 take a figure and stretch it vertically,

00:10:50 make it twice as big vertically

00:10:53 and make it half as wide.

00:10:58 That I would call a symmetry.

00:11:00 It’s not a symmetry in the classical sense,

00:11:03 but it’s a well defined transformation

00:11:05 that has an input and an output.

00:11:07 I give you some shape and it gets kind of,

00:11:10 I call this in the book a scrunch.

00:11:12 I just had to make up some sort of funny sounding name

00:11:14 for it because it doesn’t really have a name.

00:11:20 And just as you can sort of study

00:11:21 which kinds of objects are symmetrical

00:11:23 under the operations of switching left and right

00:11:26 or switching top and bottom

00:11:27 or rotating 40 degrees or what have you,

00:11:31 you could study what kinds of things are preserved

00:11:33 by this kind of scrunch symmetry.

00:11:36 And this kind of more general idea

00:11:39 of what a symmetry can be.

00:11:42 Let me put it this way.

00:11:44 A fundamental mathematical idea,

00:11:47 in some sense, I might even say the idea

00:11:49 that dominates contemporary mathematics.

00:11:51 Or by contemporary, by the way,

00:11:52 I mean like the last like 150 years.

00:11:54 We’re on a very long time scale in math.

00:11:56 I don’t mean like yesterday.

00:11:57 I mean like a century or so up till now.

00:12:00 Is this idea that it’s a fundamental question

00:12:02 of when do we consider two things to be the same?

00:12:07 That might seem like a complete triviality.

00:12:08 It’s not.

00:12:10 For instance, if I have a triangle

00:12:13 and I have a triangle of the exact same dimensions,

00:12:14 but it’s over here, are those the same or different?

00:12:19 Well, you might say, well, look,

00:12:20 there’s two different things.

00:12:21 This one’s over here, this one’s over there.

00:12:22 On the other hand, if you prove a theorem about this one,

00:12:25 it’s probably still true about this one

00:12:27 if it has like all the same side lanes and angles

00:12:29 and like looks exactly the same.

00:12:31 The term of art, if you want it,

00:12:32 you would say they’re congruent.

00:12:34 But one way of saying it is there’s a symmetry

00:12:36 called translation, which just means

00:12:38 move everything three inches to the left.

00:12:40 And we want all of our theories

00:12:43 to be translation invariant.

00:12:45 What that means is that if you prove a theorem

00:12:46 about a thing that’s over here,

00:12:48 and then you move it three inches to the left,

00:12:51 it would be kind of weird if all of your theorems

00:12:53 like didn’t still work.

00:12:55 So this question of like, what are the symmetries

00:12:58 and which things that you want to study

00:12:59 are invariant under those symmetries

00:13:01 is absolutely fundamental.

00:13:02 Boy, this is getting a little abstract, right?

00:13:04 It’s not at all abstract.

00:13:05 I think this is completely central

00:13:08 to everything I think about

00:13:09 in terms of artificial intelligence.

00:13:11 I don’t know if you know about the MNIST dataset,

00:13:13 what’s handwritten digits.

00:13:15 And you know, I don’t smoke much weed or any really,

00:13:21 but it certainly feels like it when I look at MNIST

00:13:24 and think about this stuff, which is like,

00:13:26 what’s the difference between one and two?

00:13:28 And why are all the twos similar to each other?

00:13:32 What kind of transformations are within the category

00:13:37 of what makes a thing the same?

00:13:39 And what kind of transformations

00:13:40 are those that make it different?

00:13:42 And symmetries core to that.

00:13:44 In fact, whatever the hell our brain is doing,

00:13:46 it’s really good at constructing these arbitrary

00:13:50 and sometimes novel, which is really important

00:13:53 when you look at like the IQ test or they feel novel,

00:13:58 ideas of symmetry of like playing with objects,

00:14:02 we’re able to see things that are the same and not

00:14:07 and construct almost like little geometric theories

00:14:11 of what makes things the same and not

00:14:13 and how to make programs do that in AI

00:14:17 is a total open question.

00:14:19 And so I kind of stared and wonder

00:14:22 how, what kind of symmetries are enough to solve

00:14:27 the MNIST handwritten digit recognition problem

00:14:30 and write that down.

00:14:32 And exactly, and what’s so fascinating

00:14:33 about the work in that direction

00:14:35 from the point of view of a mathematician like me

00:14:38 and a geometer is that the kind of groups of symmetries,

00:14:42 the types of symmetries that we know of are not sufficient.

00:14:45 So in other words, like we’re just gonna keep on going

00:14:48 into the weeds on this.

00:14:51 The deeper, the better.

00:14:53 A kind of symmetry that we understand very well

00:14:55 is rotation.

00:14:56 So here’s what would be easy.

00:14:57 If humans, if we recognize the digit as a one,

00:15:01 if it was like literally a rotation

00:15:03 by some number of degrees or some fixed one

00:15:07 in some typeface like Palatino or something,

00:15:10 that would be very easy to understand.

00:15:12 It would be very easy to like write a program

00:15:13 that could detect whether something was a rotation

00:15:17 of a fixed digit one.

00:15:20 Whatever we’re doing when you recognize the digit one

00:15:22 and distinguish it from the digit two, it’s not that.

00:15:25 It’s not just incorporating one of the types of symmetries

00:15:30 that we understand.

00:15:32 Now, I would say that I would be shocked

00:15:36 if there was some kind of classical symmetry type formulation

00:15:40 that captured what we’re doing

00:15:43 when we tell the difference between a two and a three.

00:15:45 To be honest, I think what we’re doing

00:15:48 is actually more complicated than that.

00:15:50 I feel like it must be.

00:15:52 They’re so simple, these numbers.

00:15:53 I mean, they’re really geometric objects.

00:15:55 Like we can draw out one, two, three.

00:15:58 It does seem like it should be formalizable.

00:16:01 That’s why it’s so strange.

00:16:03 Do you think it’s formalizable

00:16:04 when something stops being a two and starts being a three?

00:16:06 Right, you can imagine something continuously deforming

00:16:09 from being a two to a three.

00:16:11 Yeah, but that’s, there is a moment.

00:16:15 Like I have myself written programs

00:16:17 that literally morph twos and threes and so on.

00:16:20 And you watch, and there is moments that you notice

00:16:23 depending on the trajectory of that transformation,

00:16:26 that morphing, that it is a three and a two.

00:16:32 There’s a hard line.

00:16:33 Wait, so if you ask people, if you showed them this morph,

00:16:36 if you ask a bunch of people,

00:16:37 do they all agree about where the transition happened?

00:16:39 Because I would be surprised.

00:16:40 I think so.

00:16:41 Oh my God, okay, we have an empirical dispute.

00:16:42 But here’s the problem.

00:16:44 Here’s the problem, that if I just showed that moment

00:16:48 that I agreed on.

00:16:50 Well, that’s not fair.

00:16:51 No, but say I said,

00:16:53 so I want to move away from the agreement

00:16:55 because that’s a fascinating actually question

00:16:57 that I want to backtrack from because I just dogmatically

00:17:02 said, because I could be very, very wrong.

00:17:04 But the morphing really helps that like the change,

00:17:09 because I mean, partially it’s because our perception

00:17:11 systems, see this, it’s all probably tied in there.

00:17:15 Somehow the change from one to the other,

00:17:18 like seeing the video of it allows you to pinpoint

00:17:21 the place where a two becomes a three much better.

00:17:23 If I just showed you one picture,

00:17:26 I think you might really, really struggle.

00:17:31 You might call a seven.

00:17:32 I think there’s something also that we don’t often

00:17:38 think about, which is it’s not just about the static image,

00:17:41 it’s the transformation of the image,

00:17:43 or it’s not a static shape,

00:17:45 it’s the transformation of the shape.

00:17:47 There’s something in the movement that seems to be

00:17:51 not just about our perception system,

00:17:53 but fundamental to our cognition,

00:17:55 like how we think about stuff.

00:17:57 Yeah, and that’s part of geometry too.

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

00:18:03 is this idea that maybe we would naively think

00:18:06 we’re gonna study, I don’t know,

00:18:08 like Poincare, we’re gonna study the three body problem.

00:18:10 We’re gonna study sort of like three objects in space

00:18:13 moving around subject only to the force

00:18:15 of each other’s gravity, which sounds very simple, right?

00:18:17 And if you don’t know about this problem,

00:18:18 you’re probably like, okay, so you just like put it

00:18:20 in your computer and see what they do.

00:18:21 Well, guess what?

00:18:22 That’s like a problem that Poincare won a huge prize for

00:18:25 like making the first real progress on in the 1880s.

00:18:27 And we still don’t know that much about it 150 years later.

00:18:32 I mean, it’s a humongous mystery.

00:18:34 You just opened the door and we’re gonna walk right in

00:18:38 before we return to symmetry.

00:18:40 What’s the, who’s Poincare and what’s this conjecture

00:18:44 that he came up with?

00:18:46 Why is it such a hard problem?

00:18:48 Okay, so Poincare, he ends up being a major figure

00:18:52 in the book and I didn’t even really intend for him

00:18:54 to be such a big figure, but he’s first and foremost

00:18:59 a geometer, right?

00:19:00 So he’s a mathematician who kind of comes up

00:19:02 in late 19th century France at a time when French math

00:19:07 is really starting to flower.

00:19:09 Actually, I learned a lot.

00:19:10 I mean, in math, we’re not really trained

00:19:11 on our own history.

00:19:12 We got a PhD in math, learned about math.

00:19:14 So I learned a lot.

00:19:15 There’s this whole kind of moment where France

00:19:18 has just been beaten in the Franco Prussian war.

00:19:22 And they’re like, oh my God, what did we do wrong?

00:19:23 And they were like, we gotta get strong in math

00:19:26 like the Germans.

00:19:27 We have to be like more like the Germans.

00:19:28 So this never happens to us again.

00:19:29 So it’s very much, it’s like the Sputnik moment,

00:19:31 like what happens in America in the 50s and 60s

00:19:34 with the Soviet Union.

00:19:35 This is happening to France and they’re trying

00:19:37 to kind of like instantly like modernize.

00:19:40 That’s fascinating that the humans and mathematics

00:19:43 are intricately connected to the history of humans.

00:19:46 The Cold War is I think fundamental to the way people

00:19:51 saw science and math in the Soviet Union.

00:19:55 I don’t know if that was true in the United States,

00:19:56 but certainly it was in the Soviet Union.

00:19:58 It definitely was, and I would love to hear more

00:20:00 about how it was in the Soviet Union.

00:20:01 I mean, there was, and we’ll talk about the Olympiad.

00:20:04 I just remember that there was this feeling

00:20:09 like the world hung in a balance

00:20:14 and you could save the world with the tools of science.

00:20:19 And mathematics was like the superpower that fuels science.

00:20:26 And so like people were seen as, you know,

00:20:30 people in America often idolize athletes,

00:20:32 but ultimately the best athletes in the world,

00:20:36 they just throw a ball into a basket.

00:20:40 So like there’s not, what people really enjoy about sports,

00:20:44 I love sports, is like excellence at the highest level.

00:20:48 But when you take that with mathematics and science,

00:20:51 people also enjoyed excellence in science and mathematics

00:20:54 in the Soviet Union, but there’s an extra sense

00:20:56 that that excellence would lead to a better world.

00:21:01 So that created all the usual things you think about

00:21:07 with the Olympics, which is like extreme competitiveness.

00:21:12 But it also created this sense that in the modern era

00:21:15 in America, somebody like Elon Musk, whatever you think

00:21:19 of him, like Jeff Bezos, those folks,

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

00:21:24 or a group of smart people can change the world.

00:21:27 Like not just be good at what they do,

00:21:29 but actually change the world.

00:21:30 Mathematics was at the core of that.

00:21:33 I don’t know, there’s a romanticism around it too.

00:21:36 Like when you read books about in America,

00:21:39 people romanticize certain things like baseball, for example.

00:21:42 There’s like these beautiful poetic writing

00:21:45 about the game of baseball.

00:21:47 The same was the feeling with mathematics and science

00:21:50 in the Soviet Union, and it was in the air.

00:21:53 Everybody was forced to take high level mathematics courses.

00:21:57 Like you took a lot of math, you took a lot of science

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

00:22:03 Like the level of education in Russia,

00:22:06 this could be true in China, I’m not sure,

00:22:09 in a lot of countries is in whatever that’s called,

00:22:14 it’s K to 12 in America, but like young people education.

00:22:18 The level they were challenged to learn at is incredible.

00:22:23 It’s like America falls far behind, I would say.

00:22:27 America then quickly catches up

00:22:29 and then exceeds everybody else as you start approaching

00:22:33 the end of high school to college.

00:22:35 Like the university system in the United States

00:22:37 arguably is the best in the world.

00:22:39 But like what we challenge everybody,

00:22:44 it’s not just like the good, the A students,

00:22:46 but everybody to learn in the Soviet Union was fascinating.

00:22:50 I think I’m gonna pick up on something you said.

00:22:52 I think you would love a book called

00:22:53 Dual at Dawn by Amir Alexander,

00:22:56 which I think some of the things you’re responding to

00:22:58 and what I wrote, I think I first got turned on to

00:23:01 by Amir’s work, he’s a historian of math.

00:23:02 And he writes about the story of Everest to Galois,

00:23:06 which is a story that’s well known to all mathematicians,

00:23:08 this kind of like very, very romantic figure

00:23:12 who he really sort of like begins the development of this

00:23:18 or this theory of groups that I mentioned earlier,

00:23:20 this general theory of symmetries

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

00:23:25 like all this stuff, mostly unpublished.

00:23:28 It’s a very, very romantic story that we all learn.

00:23:32 And much of it is true,

00:23:33 but Alexander really lays out just how much

00:23:37 the way people thought about math in those times

00:23:40 in the early 19th century was wound up with,

00:23:43 as you say, romanticism.

00:23:44 I mean, that’s when the romantic movement takes place

00:23:47 and he really outlines how people were predisposed

00:23:51 to think about mathematics in that way

00:23:52 because they thought about poetry that way

00:23:54 and they thought about music that way.

00:23:55 It was the mood of the era to think about

00:23:58 we’re reaching for the transcendent,

00:23:59 we’re sort of reaching for sort of direct contact

00:24:02 with the divine.

00:24:02 And part of the reason that we think of Gawa that way

00:24:06 was because Gawa himself was a creature of that era

00:24:08 and he romanticized himself.

00:24:10 I mean, now we know he wrote lots of letters

00:24:12 and he was kind of like, I mean, in modern terms,

00:24:14 we would say he was extremely emo.

00:24:16 Like we wrote all these letters

00:24:19 about his like florid feelings

00:24:21 and like the fire within him about the mathematics.

00:24:23 And so it’s just as you say

00:24:26 that the math history touches human history.

00:24:29 They’re never separate because math is made of people.

00:24:32 I mean, that’s what, it’s people who do it

00:24:35 and we’re human beings doing it

00:24:36 and we do it within whatever community we’re in

00:24:39 and we do it affected by the mores

00:24:42 of the society around us.

00:24:44 So the French, the Germans and Poincare.

00:24:47 Yes, okay, so back to Poincare.

00:24:48 So he’s, you know, it’s funny.

00:24:52 This book is filled with kind of mathematical characters

00:24:55 who often are kind of peevish or get into feuds

00:25:00 or sort of have like weird enthusiasms

00:25:03 because those people are fun to write about

00:25:05 and they sort of like say very salty things.

00:25:07 Poincare is actually none of this.

00:25:09 As far as I can tell, he was an extremely normal dude

00:25:12 who didn’t get into fights with people

00:25:15 and everybody liked him

00:25:16 and he was like pretty personally modest

00:25:18 and he had very regular habits.

00:25:20 You know what I mean?

00:25:21 He did math for like four hours in the morning

00:25:23 and four hours in the evening and that was it.

00:25:25 Like he had his schedule.

00:25:28 I actually, it was like, I still am feeling like

00:25:31 somebody’s gonna tell me now that the book is out,

00:25:33 like, oh, didn’t you know about this

00:25:34 like incredibly sordid episode?

00:25:37 As far as I could tell, a completely normal guy.

00:25:39 But he just kind of, in many ways,

00:25:44 creates the geometric world in which we live

00:25:47 and his first really big success is this prize paper

00:25:53 he writes for this prize offered by the King of Sweden

00:25:55 for the study of the three body problem.

00:26:01 The study of what we can say about, yeah,

00:26:04 three astronomical objects moving

00:26:07 in what you might think would be this very simple way.

00:26:09 Nothing’s going on except gravity.

00:26:12 So what’s the three body problem?

00:26:13 Why is it a problem?

00:26:15 So the problem is to understand

00:26:16 when this motion is stable and when it’s not.

00:26:20 So stable meaning they would sort of like end up

00:26:21 in some kind of periodic orbit.

00:26:23 Or I guess it would mean, sorry,

00:26:25 stable would mean they never sort of fly off

00:26:26 far apart from each other.

00:26:28 And unstable would mean like eventually they fly apart.

00:26:30 So understanding two bodies is much easier.

00:26:32 Yes, exactly.

00:26:33 When you have the third wheel is always a problem.

00:26:36 This is what Newton knew.

00:26:37 Two bodies, they sort of orbit each other

00:26:38 in some kind of either in an ellipse,

00:26:41 which is the stable case.

00:26:42 You know, that’s what the planets do that we know.

00:26:46 Or one travels on a hyperbola around the other.

00:26:49 That’s the unstable case.

00:26:50 It sort of like zooms in from far away,

00:26:51 sort of like whips around the heavier thing

00:26:54 and like zooms out.

00:26:56 Those are basically the two options.

00:26:58 So it’s a very simple and easy to classify story.

00:27:00 With three bodies, just the small switch from two to three,

00:27:04 it’s a complete zoo.

00:27:05 It’s the first, what we would say now

00:27:07 is it’s the first example of what’s called chaotic dynamics,

00:27:09 where the stable solutions and the unstable solutions,

00:27:13 they’re kind of like wound in among each other.

00:27:14 And a very, very, very tiny change in the initial conditions

00:27:17 can make the longterm behavior of the system

00:27:20 completely different.

00:27:21 So Poincare was the first to recognize

00:27:22 that that phenomenon even existed.

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

00:27:31 Right, so he also was one of the pioneers

00:27:36 of taking geometry, which until that point

00:27:41 had been largely the study of two

00:27:44 and three dimensional objects,

00:27:45 because that’s like what we see, right?

00:27:47 That’s those are the objects we interact with.

00:27:49 He developed the subject we now called topology.

00:27:53 He called it analysis situs.

00:27:55 He was a very well spoken guy with a lot of slogans,

00:27:57 but that name did not,

00:27:59 you can see why that name did not catch on.

00:28:01 So now it’s called topology now.

00:28:05 Sorry, what was it called before?

00:28:06 Analysis situs, which I guess sort of roughly means

00:28:09 like the analysis of location or something like that.

00:28:11 Like it’s a Latin phrase.

00:28:14 Partly because he understood that even to understand

00:28:19 stuff that’s going on in our physical world,

00:28:22 you have to study higher dimensional spaces.

00:28:24 How does this work?

00:28:25 And this is kind of like where my brain went to it

00:28:27 because you were talking about not just where things are,

00:28:29 but what their path is, how they’re moving

00:28:31 when we were talking about the path from two to three.

00:28:34 He understood that if you wanna study

00:28:36 three bodies moving in space,

00:28:39 well, each body, it has a location where it is.

00:28:44 So it has an X coordinate, a Y coordinate,

00:28:45 a Z coordinate, right?

00:28:46 I can specify a point in space by giving you three numbers,

00:28:49 but it also at each moment has a velocity.

00:28:53 So it turns out that really to understand what’s going on,

00:28:56 you can’t think of it as a point or you could,

00:28:58 but it’s better not to think of it as a point

00:29:01 in three dimensional space that’s moving.

00:29:03 It’s better to think of it as a point

00:29:04 in six dimensional space where the coordinates

00:29:06 are where is it and what’s its velocity right now.

00:29:09 That’s a higher dimensional space called phase space.

00:29:11 And if you haven’t thought about this before,

00:29:13 I admit that it’s a little bit mind bending,

00:29:15 but what he needed then was a geometry

00:29:20 that was flexible enough,

00:29:22 not just to talk about two dimensional spaces

00:29:24 or three dimensional spaces, but any dimensional space.

00:29:27 So the sort of famous first line of this paper

00:29:29 where he introduces analysis of Cetus

00:29:30 is no one doubts nowadays that the geometry

00:29:34 of n dimensional space is an actually existing thing, right?

00:29:37 I think that maybe that had been controversial.

00:29:39 And he’s saying like, look, let’s face it,

00:29:41 just because it’s not physical doesn’t mean it’s not there.

00:29:44 It doesn’t mean we shouldn’t study it.

00:29:46 Interesting.

00:29:46 He wasn’t jumping to the physical interpretation.

00:29:49 Like it can be real,

00:29:51 even if it’s not perceivable to the human cognition.

00:29:55 I think that’s right.

00:29:56 I think, don’t get me wrong,

00:29:58 Poincare never strays far from physics.

00:30:00 He’s always motivated by physics,

00:30:02 but the physics drove him to need to think about spaces

00:30:06 of higher dimension.

00:30:07 And so he needed a formalism that was rich enough

00:30:09 to enable him to do that.

00:30:10 And once you do that,

00:30:11 that formalism is also gonna include things

00:30:13 that are not physical.

00:30:14 And then you have two choices.

00:30:15 You can be like, oh, well, that stuff’s trash.

00:30:17 Or, and this is more of the mathematicians frame of mind,

00:30:21 if you have a formalistic framework

00:30:23 that like seems really good

00:30:24 and sort of seems to be like very elegant and work well,

00:30:27 and it includes all the physical stuff,

00:30:29 maybe we should think about all of it.

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

00:30:31 thinking maybe there’s some gold to be mined there.

00:30:34 And indeed, like, you know, guess what?

00:30:36 Like before long there’s relativity and there’s space time.

00:30:39 And like all of a sudden it’s like,

00:30:40 oh yeah, maybe it’s a good idea.

00:30:41 We already had this geometric apparatus like set up

00:30:43 for like how to think about four dimensional spaces,

00:30:47 like turns out they’re real after all.

00:30:48 As I said, you know, this is a story much told

00:30:51 right in mathematics, not just in this context,

00:30:53 but in many.

00:30:53 I’d love to dig in a little deeper on that actually,

00:30:55 cause I have some intuitions to work out.

00:31:00 Okay.

00:31:01 My brain.

00:31:02 Well, I’m not a mathematical physicist,

00:31:03 so we can work them out together.

00:31:05 Good.

00:31:06 We’ll together walk along the path of curiosity,

00:31:10 but Poincare conjecture.

00:31:13 What is it?

00:31:14 The Poincare conjecture is about curved

00:31:17 three dimensional spaces.

00:31:18 So I was on my way there.

00:31:21 I promise.

00:31:23 The idea is that we perceive ourselves as living in,

00:31:27 we don’t say a three dimensional space.

00:31:29 We just say three dimensional space.

00:31:30 You know, you can go up and down,

00:31:31 you can go left and right,

00:31:32 you can go forward and back.

00:31:33 There’s three dimensions in which we can move.

00:31:35 In Poincare’s theory,

00:31:36 there are many possible three dimensional spaces.

00:31:41 In the same way that going down one dimension

00:31:45 to sort of capture our intuition a little bit more,

00:31:48 we know there are lots of different

00:31:49 two dimensional surfaces, right?

00:31:51 There’s a balloon and that looks one way

00:31:54 and a donut looks another way

00:31:55 and a Mobius strip looks a third way.

00:31:57 Those are all like two dimensional surfaces

00:31:59 that we can kind of really get a global view of

00:32:02 because we live in three dimensional space.

00:32:03 So we can see a two dimensional surface

00:32:05 sort of sitting in our three dimensional space.

00:32:07 Well, to see a three dimensional space whole,

00:32:11 we’d have to kind of have four dimensional eyes, right?

00:32:13 Which we don’t.

00:32:14 So we have to use our mathematical eyes.

00:32:15 We have to envision.

00:32:17 The Poincare conjecture says that there’s a very simple way

00:32:22 to determine whether a three dimensional space

00:32:26 is the standard one, the one that we’re used to.

00:32:29 And essentially it’s that it’s what’s called

00:32:31 fundamental group has nothing interesting in it.

00:32:34 And that I can actually say without saying

00:32:36 what the fundamental group is,

00:32:36 I can tell you what the criterion is.

00:32:39 This would be good.

00:32:39 Oh, look, I can even use a visual aid.

00:32:40 So for the people watching this on YouTube,

00:32:42 you will just see this for the people on the podcast,

00:32:45 you’ll have to visualize it.

00:32:46 So Lex has been nice enough to like give me a surface

00:32:49 with an interesting topology.

00:32:50 It’s a mug right here in front of me.

00:32:52 A mug, yes.

00:32:53 I might say it’s a genus one surface,

00:32:55 but we could also say it’s a mug, same thing.

00:32:58 So if I were to draw a little circle on this mug,

00:33:03 which way should I draw it so it’s visible?

00:33:04 Like here, okay.

00:33:06 If I draw a little circle on this mug,

00:33:07 imagine this to be a loop of string.

00:33:09 I could pull that loop of string closed

00:33:12 on the surface of the mug, right?

00:33:14 That’s definitely something I could do.

00:33:15 I could shrink it, shrink it, shrink it until it’s a point.

00:33:18 On the other hand,

00:33:19 if I draw a loop that goes around the handle,

00:33:21 I can kind of zhuzh it up here

00:33:23 and I can zhuzh it down there

00:33:24 and I can sort of slide it up and down the handle,

00:33:25 but I can’t pull it closed, can I?

00:33:27 It’s trapped.

00:33:28 Not without breaking the surface of the mug, right?

00:33:30 Not without like going inside.

00:33:32 So the condition of being what’s called simply connected,

00:33:37 this is one of Poincare’s inventions,

00:33:39 says that any loop of string can be pulled shut.

00:33:42 So it’s a feature that the mug simply does not have.

00:33:45 This is a non simply connected mug

00:33:48 and a simply connected mug would be a cup, right?

00:33:51 You would burn your hand when you drank coffee out of it.

00:33:53 So you’re saying the universe is not a mug.

00:33:56 Well, I can’t speak to the universe,

00:33:59 but what I can say is that regular old space is not a mug.

00:34:05 Regular old space,

00:34:06 if you like sort of actually physically have

00:34:07 like a loop of string,

00:34:09 you can pull it shut.

00:34:11 You can always pull it shut.

00:34:12 But what if your piece of string

00:34:14 was the size of the universe?

00:34:14 Like what if your piece of string

00:34:16 was like billions of light years long?

00:34:18 Like how do you actually know?

00:34:20 I mean, that’s still an open question

00:34:21 of the shape of the universe.

00:34:22 Exactly.

00:34:25 I think there’s a lot,

00:34:26 there is ideas of it being a torus.

00:34:28 I mean, there’s some trippy ideas

00:34:30 and they’re not like weird out there controversial.

00:34:33 There’s legitimate at the center of a cosmology debate.

00:34:38 I mean, I think most people think it’s flat.

00:34:40 I think there’s some kind of dodecahedral symmetry

00:34:42 or I mean, I remember reading something crazy

00:34:43 about somebody saying that they saw the signature of that

00:34:45 in the cosmic noise or what have you.

00:34:48 I mean.

00:34:49 To make the flat earthers happy,

00:34:51 I do believe that the current main belief is it’s flat.

00:34:56 It’s flat ish or something like that.

00:34:59 The shape of the universe is flat ish.

00:35:01 I don’t know what the heck that means.

00:35:03 I think that has like a very,

00:35:06 how are you even supposed to think about the shape

00:35:09 of a thing that doesn’t have any thing outside of it?

00:35:14 I mean.

00:35:14 Ah, but that’s exactly what topology does.

00:35:16 Topology is what’s called an intrinsic theory.

00:35:19 That’s what’s so great about it.

00:35:20 This question about the mug,

00:35:22 you could answer it without ever leaving the mug, right?

00:35:26 Because it’s a question about a loop drawn

00:35:28 on the surface of the mug

00:35:29 and what happens if it never leaves that surface.

00:35:31 So it’s like always there.

00:35:33 See, but that’s the difference between the topology

00:35:37 and say, if you’re like trying to visualize a mug,

00:35:42 that you can’t visualize a mug while living inside the mug.

00:35:46 Well, that’s true.

00:35:47 The visualization is harder, but in some sense,

00:35:49 no, you’re right.

00:35:50 But if the tools of mathematics are there,

00:35:51 I, sorry, I don’t want to fight,

00:35:53 but I think the tools of mathematics are exactly there

00:35:55 to enable you to think about

00:35:56 what you cannot visualize in this way.

00:35:58 Let me give, let’s go, always to make things easier,

00:36:00 go down to dimension.

00:36:03 Let’s think about we live in a circle, okay?

00:36:05 You can tell whether you live on a circle or a line segment,

00:36:11 because if you live in a circle,

00:36:12 if you walk a long way in one direction,

00:36:13 you find yourself back where you started.

00:36:15 And if you live in a line segment,

00:36:17 you walk for a long enough one direction,

00:36:18 you come to the end of the world.

00:36:20 Or if you live on a line, like a whole line,

00:36:22 infinite line, then you walk in one direction

00:36:25 for a long time and like,

00:36:27 well, then there’s not a sort of terminating algorithm

00:36:28 to figure out whether you live on a line or a circle,

00:36:30 but at least you sort of,

00:36:33 at least you don’t discover that you live on a circle.

00:36:35 So all of those are intrinsic things, right?

00:36:37 All of those are things that you can figure out

00:36:39 about your world without leaving your world.

00:36:42 On the other hand, ready?

00:36:43 Now we’re going to go from intrinsic to extrinsic.

00:36:45 Boy, did I not know we were going to talk about this,

00:36:46 but why not?

00:36:48 Why not?

00:36:48 If you can’t tell whether you live in a circle

00:36:52 or a knot, like imagine like a knot

00:36:55 floating in three dimensional space.

00:36:56 The person who lives on that knot, to them it’s a circle.

00:36:59 They walk a long way, they come back to where they started.

00:37:01 Now we, with our three dimensional eyes can be like,

00:37:04 oh, this one’s just a plain circle

00:37:05 and this one’s knotted up,

00:37:06 but that has to do with how they sit

00:37:09 in three dimensional space.

00:37:10 It doesn’t have to do with intrinsic features

00:37:12 of those people’s world.

00:37:13 We can ask you one ape to another.

00:37:14 Does it make you, how does it make you feel

00:37:17 that you don’t know if you live in a circle

00:37:19 or on a knot, in a knot,

00:37:24 inside the string that forms the knot?

00:37:28 I don’t even know how to say that.

00:37:29 I’m going to be honest with you.

00:37:30 I don’t know if, I fear you won’t like this answer,

00:37:34 but it does not bother me at all.

00:37:37 I don’t lose one minute of sleep over it.

00:37:39 So like, does it bother you that if we look

00:37:41 at like a Mobius strip, that you don’t have an obvious way

00:37:46 of knowing whether you are inside of a cylinder,

00:37:49 if you live on a surface of a cylinder

00:37:51 or you live on the surface of a Mobius strip?

00:37:55 No, I think you can tell if you live.

00:37:58 Which one?

00:37:59 Because what you do is you like tell your friend,

00:38:02 hey, stay right here, I’m just going to go for a walk.

00:38:04 And then you like walk for a long time in one direction

00:38:06 and then you come back and you see your friend again.

00:38:08 And if your friend is reversed,

00:38:09 then you know you live on a Mobius strip.

00:38:10 Well, no, because you won’t see your friend, right?

00:38:13 Okay, fair point, fair point on that.

00:38:17 But you have to believe the stories about,

00:38:19 no, I don’t even know, would you even know?

00:38:24 Would you really?

00:38:25 Oh, no, your point is right.

00:38:26 Let me try to think of a better,

00:38:28 let’s see if I can do this on the fly.

00:38:29 It may not be correct to talk about cognitive beings

00:38:33 living on a Mobius strip

00:38:35 because there’s a lot of things taken for granted there.

00:38:37 And we’re constantly imagining actual

00:38:39 like three dimensional creatures,

00:38:42 like how it actually feels like to live in a Mobius strip

00:38:47 is tricky to internalize.

00:38:50 I think that on what’s called the real protective plane,

00:38:52 which is kind of even more sort of like messed up version

00:38:54 of the Mobius strip, but with very similar features,

00:38:57 this feature of kind of like only having one side,

00:39:01 that has the feature that there’s a loop of string

00:39:04 which can’t be pulled closed.

00:39:06 But if you loop it around twice along the same path,

00:39:09 that you can pull closed.

00:39:11 That’s extremely weird.

00:39:12 Yeah.

00:39:14 But that would be a way you could know

00:39:16 without leaving your world

00:39:17 that something very funny is going on.

00:39:20 You know what’s extremely weird?

00:39:21 Maybe we can comment on,

00:39:23 hopefully it’s not too much of a tangent is,

00:39:26 I remember thinking about this,

00:39:29 this might be right, this might be wrong.

00:39:31 But if we now talk about a sphere

00:39:35 and you’re living inside a sphere,

00:39:37 that you’re going to see everywhere around you,

00:39:41 the back of your own head.

00:39:44 That I was,

00:39:46 cause like I was,

00:39:47 this is very counterintuitive to me to think about,

00:39:50 maybe it’s wrong.

00:39:51 But cause I was thinking of like earth,

00:39:54 your 3D thing sitting on a sphere.

00:39:57 But if you’re living inside the sphere,

00:40:00 like you’re going to see, if you look straight,

00:40:02 you’re always going to see yourself all the way around.

00:40:05 So everywhere you look, there’s going to be

00:40:07 the back of your own head.

00:40:09 I think somehow this depends on something

00:40:10 of like how the physics of light works in this scenario,

00:40:13 which I’m sort of finding it hard to bend my.

00:40:14 That’s true.

00:40:15 The sea is doing a lot of work.

00:40:16 Like saying you see something is doing a lot of work.

00:40:19 People have thought about this a lot.

00:40:20 I mean, this metaphor of like,

00:40:22 what if we’re like little creatures

00:40:24 in some sort of smaller world?

00:40:26 Like how could we apprehend what’s outside?

00:40:27 That metaphor just comes back and back.

00:40:29 And actually I didn’t even realize like how frequent it is.

00:40:32 It comes up in the book a lot.

00:40:33 I know it from a book called Flatland.

00:40:35 I don’t know if you ever read this when you were a kid.

00:40:37 A while ago, yeah.

00:40:38 An adult.

00:40:39 You know, this sort of comic novel from the 19th century

00:40:42 about an entire two dimensional world.

00:40:46 It’s narrated by a square.

00:40:48 That’s the main character.

00:40:49 And the kind of strangeness that befalls him

00:40:53 when one day he’s in his house

00:40:55 and suddenly there’s like a little circle there

00:40:57 and they’re with him.

00:40:59 But then the circle like starts getting bigger

00:41:02 and bigger and bigger.

00:41:04 And he’s like, what the hell is going on?

00:41:06 It’s like a horror movie, like for two dimensional people.

00:41:08 And of course what’s happening

00:41:09 is that a sphere is entering his world.

00:41:12 And as the sphere kind of like moves farther and farther

00:41:15 into the plane, it’s cross section.

00:41:16 The part of it that he can see.

00:41:18 To him, it looks like there’s like this kind

00:41:20 of bizarre being that’s like getting larger

00:41:22 and larger and larger

00:41:24 until it’s exactly sort of halfway through.

00:41:27 And then they have this kind of like philosophical argument

00:41:29 where the sphere is like, I’m a sphere.

00:41:30 I’m from the third dimension.

00:41:31 The square is like, what are you talking about?

00:41:32 There’s no such thing.

00:41:33 And they have this kind of like sterile argument

00:41:36 where the square is not able to kind of like

00:41:39 follow the mathematical reasoning of the sphere

00:41:40 until the sphere just kind of grabs him

00:41:42 and like jerks him out of the plane and pulls him up.

00:41:45 And it’s like now, like now do you see,

00:41:47 like now do you see your whole world

00:41:50 that you didn’t understand before?

00:41:52 So do you think that kind of process is possible

00:41:55 for us humans?

00:41:56 So we live in the three dimensional world,

00:41:58 maybe with the time component four dimensional

00:42:01 and then math allows us to go high,

00:42:06 into high dimensions comfortably

00:42:08 and explore the world from those perspectives.

00:42:13 Like, is it possible that the universe

00:42:19 is many more dimensions than the ones

00:42:23 we experience as human beings?

00:42:25 So if you look at the, you know,

00:42:28 especially in physics theories of everything,

00:42:32 physics theories that try to unify general relativity

00:42:35 and quantum field theory,

00:42:37 they seem to go to high dimensions to work stuff out

00:42:42 through the tools of mathematics.

00:42:44 Is it possible?

00:42:46 So like the two options are,

00:42:47 one is just a nice way to analyze a universe,

00:42:51 but the reality is, is as exactly we perceive it,

00:42:54 it is three dimensional, or are we just seeing,

00:42:58 are we those flatland creatures

00:43:00 that are just seeing a tiny slice of reality

00:43:03 and the actual reality is many, many, many more dimensions

00:43:08 than the three dimensions we perceive?

00:43:10 Oh, I certainly think that’s possible.

00:43:14 Now, how would you figure out whether it was true or not

00:43:17 is another question.

00:43:20 And I suppose what you would do

00:43:22 as with anything else that you can’t directly perceive

00:43:25 is you would try to understand

00:43:29 what effect the presence of those extra dimensions

00:43:33 out there would have on the things we can perceive.

00:43:36 Like what else can you do, right?

00:43:39 And in some sense, if the answer is

00:43:42 they would have no effect,

00:43:44 then maybe it becomes like a little bit

00:43:46 of a sterile question,

00:43:47 because what question are you even asking, right?

00:43:49 You can kind of posit however many entities that you want.

00:43:53 Is it possible to intuit how to mess

00:43:56 with the other dimensions

00:43:58 while living in a three dimensional world?

00:44:00 I mean, that seems like a very challenging thing to do.

00:44:03 The reason flatland could be written

00:44:06 is because it’s coming from a three dimensional writer.

00:44:11 Yes, but what happens in the book,

00:44:13 I didn’t even tell you the whole plot.

00:44:15 What happens is the square is so excited

00:44:17 and so filled with intellectual joy.

00:44:19 By the way, maybe to give the story some context,

00:44:22 you asked like, is it possible for us humans

00:44:25 to have this experience of being transcendentally jerked

00:44:28 out of our world so we can sort of truly see it from above?

00:44:30 Well, Edwin Abbott who wrote the book

00:44:32 certainly thought so because Edwin Abbott was a minister.

00:44:35 So the whole Christian subtext of this book,

00:44:37 I had completely not grasped reading this as a kid,

00:44:41 that it means a very different thing, right?

00:44:43 If sort of a theologian is saying like,

00:44:45 oh, what if a higher being could like pull you out

00:44:48 of this earthly world you live in

00:44:50 so that you can sort of see the truth

00:44:51 and like really see it from above as it were.

00:44:54 So that’s one of the things that’s going on for him.

00:44:56 And it’s a testament to his skill as a writer

00:44:58 that his story just works whether that’s the framework

00:45:01 you’re coming to it from or not.

00:45:05 But what happens in this book and this part,

00:45:07 now looking at it through a Christian lens,

00:45:08 it becomes a bit subversive is the square is so excited

00:45:13 about what he’s learned from the sphere

00:45:16 and the sphere explains to him like what a cube would be.

00:45:18 Oh, it’s like you but three dimensional

00:45:20 and the square is very excited

00:45:21 and the square is like, okay, I get it now.

00:45:23 So like now that you explained to me how just by reason

00:45:26 I can figure out what a cube would be like,

00:45:27 like a three dimensional version of me,

00:45:29 like let’s figure out what a four dimensional version

00:45:31 of me would be like.

00:45:32 And then the sphere is like,

00:45:33 what the hell are you talking about?

00:45:34 There’s no fourth dimension, that’s ridiculous.

00:45:36 Like there’s three dimensions,

00:45:37 like that’s how many there are, I can see.

00:45:39 Like, I mean, it’s this sort of comic moment

00:45:40 where the sphere is completely unable to conceptualize

00:45:44 that there could actually be yet another dimension.

00:45:47 So yeah, that takes the religious allegory

00:45:49 like a very weird place that I don’t really

00:45:51 like understand theologically, but.

00:45:53 That’s a nice way to talk about religion and myth in general

00:45:57 as perhaps us trying to struggle,

00:46:00 us meaning human civilization, trying to struggle

00:46:03 with ideas that are beyond our cognitive capabilities.

00:46:08 But it’s in fact not beyond our capability.

00:46:10 It may be beyond our cognitive capabilities

00:46:13 to visualize a four dimensional cube,

00:46:16 a tesseract as some like to call it,

00:46:18 or a five dimensional cube, or a six dimensional cube,

00:46:20 but it is not beyond our cognitive capabilities

00:46:23 to figure out how many corners

00:46:26 a six dimensional cube would have.

00:46:28 That’s what’s so cool about us.

00:46:29 Whether we can visualize it or not,

00:46:31 we can still talk about it, we can still reason about it,

00:46:33 we can still figure things out about it.

00:46:36 That’s amazing.

00:46:37 Yeah, if we go back to this, first of all, to the mug,

00:46:41 but to the example you give in the book of the straw,

00:46:44 how many holes does a straw have?

00:46:49 And you, listener, may try to answer that in your own head.

00:46:54 Yeah, I’m gonna take a drink while everybody thinks about it

00:46:56 so we can give you a moment.

00:46:57 A slow sip.

00:46:59 Is it zero, one, or two, or more than that maybe?

00:47:04 Maybe you can get very creative.

00:47:06 But it’s kind of interesting to each,

00:47:10 dissecting each answer as you do in the book

00:47:13 is quite brilliant.

00:47:14 People should definitely check it out.

00:47:15 But if you could try to answer it now,

00:47:18 think about all the options

00:47:21 and why they may or may not be right.

00:47:23 Yeah, and it’s one of these questions

00:47:25 where people on first hearing it think it’s a triviality

00:47:28 and they’re like, well, the answer is obvious.

00:47:29 And then what happens if you ever ask a group of people

00:47:31 that something wonderfully comic happens,

00:47:33 which is that everyone’s like,

00:47:34 well, it’s completely obvious.

00:47:36 And then each person realizes that half the person,

00:47:38 the other people in the room

00:47:39 have a different obvious answer for the way they have.

00:47:42 And then people get really heated.

00:47:44 People are like, I can’t believe

00:47:46 that you think it has two holes

00:47:47 or like, I can’t believe that you think it has one.

00:47:49 And then, you know, you really,

00:47:50 like people really learn something about each other

00:47:52 and people get heated.

00:47:54 I mean, can we go through the possible options here?

00:47:57 Is it zero, one, two, three, 10?

00:48:01 Sure, so I think, you know, most people,

00:48:04 the zero holders are rare.

00:48:06 They would say like, well, look,

00:48:07 you can make a straw by taking a rectangular piece of plastic

00:48:10 and closing it up.

00:48:11 A rectangular piece of plastic doesn’t have a hole in it.

00:48:14 I didn’t poke a hole in it when I,

00:48:16 so how can I have a hole?

00:48:18 They’d be like, it’s just one thing.

00:48:19 Okay, most people don’t see it that way.

00:48:21 That’s like a…

00:48:23 Is there any truth to that kind of conception?

00:48:25 Yeah, I think that would be somebody who’s account, I mean,

00:48:33 what I would say is you could say the same thing

00:48:39 about a bagel.

00:48:40 You could say, I can make a bagel by taking like a long

00:48:43 cylinder of dough, which doesn’t have a hole

00:48:45 and then schmushing the ends together.

00:48:47 Now it’s a bagel.

00:48:49 So if you’re really committed, you can be like, okay,

00:48:50 a bagel doesn’t have a hole either.

00:48:51 But like, who are you if you say a bagel doesn’t have a hole?

00:48:54 I mean, I don’t know.

00:48:54 Yeah, so that’s almost like an engineering definition of it.

00:48:57 Okay, fair enough.

00:48:59 So what about the other options?

00:49:02 So, you know, one whole people would say…

00:49:07 I like how these are like groups of people.

00:49:09 Like we’ve planted our foot, this is what we stand for.

00:49:12 There’s books written about each belief.

00:49:16 You know, I would say, look, there’s like a hole

00:49:17 and it goes all the way through the straw, right?

00:49:19 It’s one region of space, that’s the hole.

00:49:21 And there’s one.

00:49:22 And two whole people would say like, well, look,

00:49:24 there’s a hole in the top and a hole at the bottom.

00:49:28 I think a common thing you see when people

00:49:34 argue about this, they would take something like this

00:49:35 bottle of water I’m holding and go open it and they say,

00:49:40 well, how many holes are there in this?

00:49:41 And you say like, well, there’s one hole at the top.

00:49:44 Okay, what if I like poke a hole here

00:49:46 so that all the water spills out?

00:49:48 Well, now it’s a straw.

00:49:50 Yeah.

00:49:51 So if you’re a one holder, I say to you like,

00:49:53 well, how many holes are in it now?

00:49:56 There was one hole in it before

00:49:57 and I poked a new hole in it.

00:49:59 And then you think there’s still one hole

00:50:01 even though there was one hole and I made one more?

00:50:04 Clearly not, this is two holes.

00:50:06 Yeah.

00:50:08 And yet if you’re a two holder, the one holder will say like,

00:50:10 okay, where does one hole begin and the other hole end?

00:50:13 Yeah.

00:50:16 And in the book, I sort of, you know, in math,

00:50:18 there’s two things we do when we’re faced with a problem

00:50:20 that’s confusing us.

00:50:22 We can make the problem simpler.

00:50:24 That’s what we were doing a minute ago

00:50:25 when we were talking about high dimensional space.

00:50:27 And I was like, let’s talk about like circles

00:50:28 and line segments.

00:50:29 Let’s like go down a dimension to make it easier.

00:50:31 The other big move we have is to make the problem harder

00:50:35 and try to sort of really like face up

00:50:36 to what are the complications.

00:50:37 So, you know, what I do in the book is say like,

00:50:39 let’s stop talking about straws for a minute

00:50:41 and talk about pants.

00:50:42 How many holes are there in a pair of pants?

00:50:46 So I think most people who say there’s two holes in a straw

00:50:48 would say there’s three holes in a pair of pants.

00:50:51 I guess, I mean, I guess we’re filming only from here.

00:50:54 I could take up, no, I’m not gonna do it.

00:50:56 You’ll just have to imagine the pants, sorry.

00:50:58 Yeah.

00:50:59 Lex, if you want to, no, okay, no.

00:51:01 That’s gonna be in the director’s cut.

00:51:04 That’s that Patreon only footage.

00:51:06 There you go.

00:51:07 So many people would say there’s three holes

00:51:09 in a pair of pants.

00:51:10 But you know, for instance, my daughter, when I asked,

00:51:11 by the way, talking to kids about this is super fun.

00:51:14 I highly recommend it.

00:51:16 What did she say?

00:51:17 She said, well, yeah, I feel a pair of pants

00:51:21 like just has two holes because yes, there’s the waist,

00:51:23 but that’s just the two leg holes stuck together.

00:51:26 Whoa, okay.

00:51:28 Two leg holes, yeah, okay.

00:51:29 I mean, that really is a good combination.

00:51:31 So she’s a one holder for the straw.

00:51:32 So she’s a one holder for the straw too.

00:51:34 And that really does capture something.

00:51:39 It captures this fact, which is central

00:51:42 to the theory of what’s called homology,

00:51:44 which is like a central part of modern topology

00:51:46 that holes, whatever we may mean by them,

00:51:49 they’re somehow things which have an arithmetic to them.

00:51:51 They’re things which can be added.

00:51:53 Like the waist, like waist equals leg plus leg

00:51:57 is kind of an equation,

00:51:58 but it’s not an equation about numbers.

00:52:00 It’s an equation about some kind of geometric,

00:52:02 some kind of topological thing, which is very strange.

00:52:05 And so, you know, when I come down, you know,

00:52:09 like a rabbi, I like to kind of like come up

00:52:11 with these answers and somehow like dodge

00:52:13 the original question and say like,

00:52:14 you’re both right, my children.

00:52:15 Okay, so.

00:52:17 Yeah.

00:52:19 So for the straw, I think what a modern mathematician

00:52:23 would say is like, the first version would be to say like,

00:52:27 well, there are two holes,

00:52:29 but they’re really both the same hole.

00:52:31 Well, that’s not quite right.

00:52:32 A better way to say it is there’s two holes,

00:52:34 but one is the negative of the other.

00:52:37 Now, what can that mean?

00:52:39 One way of thinking about what it means is that

00:52:41 if you sip something like a milkshake through the straw,

00:52:44 no matter what, the amount of milkshake

00:52:48 that’s flowing in one end,

00:52:49 that same amount is flowing out the other end.

00:52:53 So they’re not independent from each other.

00:52:55 There’s some relationship between them.

00:52:57 In the same way that if you somehow

00:53:00 could like suck a milkshake through a pair of pants,

00:53:05 the amount of milkshake,

00:53:06 just go with me on this thought experiment.

00:53:08 I’m right there with you.

00:53:09 The amount of milkshake that’s coming in

00:53:11 the left leg of the pants,

00:53:13 plus the amount of milkshake that’s coming in

00:53:15 the right leg of the pants,

00:53:16 is the same that’s coming out the waist of the pants.

00:53:20 So just so you know, I fasted for 72 hours

00:53:24 the last three days.

00:53:25 So I just broke the fast with a little bit of food yesterday.

00:53:27 So this sounds, food analogies or metaphors

00:53:32 for this podcast work wonderfully

00:53:33 because I can intensely picture it.

00:53:35 Is that your weekly routine or just in preparation

00:53:37 for talking about geometry for three hours?

00:53:39 Exactly, this is just for this.

00:53:41 It’s hardship to purify the mind.

00:53:44 No, it’s for the first time,

00:53:45 I just wanted to try the experience.

00:53:46 Oh, wow.

00:53:47 And just to pause,

00:53:50 to do things that are out of the ordinary,

00:53:52 to pause and to reflect on how grateful I am

00:53:55 to be just alive and be able to do all the cool shit

00:53:59 that I get to do, so.

00:54:00 Did you drink water?

00:54:01 Yes, yes, yes, yes, yes.

00:54:03 Water and salt, so like electrolytes

00:54:05 and all those kinds of things.

00:54:07 But anyway, so the inflow on the top of the pants

00:54:10 equals to the outflow on the bottom of the pants.

00:54:14 Exactly, so this idea that,

00:54:18 I mean, I think, you know, Poincare really had this idea,

00:54:21 this sort of modern idea.

00:54:22 I mean, building on stuff other people did,

00:54:25 Betty is an important one,

00:54:26 of this kind of modern notion of relations between holes.

00:54:29 But the idea that holes really had an arithmetic,

00:54:32 the really modern view was really Emmy Noether’s idea.

00:54:35 So she kind of comes in and sort of truly puts the subject

00:54:40 on its modern footing that we have now.

00:54:43 So, you know, it’s always a challenge, you know,

00:54:45 in the book, I’m not gonna say I give like a course

00:54:48 so that you read this chapter and then you’re like,

00:54:50 oh, it’s just like I took like a semester

00:54:51 of algebraic anthropology.

00:54:53 It’s not like this and it’s always a challenge

00:54:55 writing about math because there are some things

00:55:00 that you can really do on the page and the math is there.

00:55:03 And there’s other things which it’s too much

00:55:05 in a book like this to like do them all the page.

00:55:07 You can only say something about them, if that makes sense.

00:55:12 So, you know, in the book, I try to do some of both.

00:55:14 I try to do, I try to, topics that are,

00:55:18 you can’t really compress and really truly say

00:55:22 exactly what they are in this amount of space.

00:55:27 I try to say something interesting about them,

00:55:28 something meaningful about them

00:55:30 so that readers can get the flavor.

00:55:31 And then in other places,

00:55:34 I really try to get up close and personal

00:55:36 and really do the math and have it take place on the page.

00:55:40 To some degree be able to give inklings

00:55:44 of the beauty of the subject.

00:55:45 Yeah, I mean, there’s a lot of books that are like,

00:55:48 I don’t quite know how to express this well.

00:55:49 I’m still laboring to do it,

00:55:51 but there’s a lot of books that are about stuff,

00:55:57 but I want my books to not only be about stuff,

00:56:01 but to actually have some stuff there on the page

00:56:03 in the book for people to interact with directly

00:56:05 and not just sort of hear me talk about

00:56:07 distant features of it.

00:56:10 Right, so not be talking just about ideas,

00:56:13 but the actually be expressing the idea.

00:56:16 Is there, you know, somebody in the,

00:56:18 maybe you can comment, there’s a guy,

00:56:21 his YouTube channel is 3Blue1Brown, Grant Sanderson.

00:56:25 He does that masterfully well.

00:56:27 Absolutely.

00:56:28 Of visualizing, of expressing a particular idea

00:56:31 and then talking about it as well back and forth.

00:56:34 What do you think about Grant?

00:56:37 It’s fantastic.

00:56:37 I mean, the flowering of math YouTube

00:56:40 is like such a wonderful thing

00:56:41 because math teaching, there’s so many different venues

00:56:47 through which we can teach people math.

00:56:48 There’s the traditional one, right?

00:56:51 Where I’m in a classroom with, depending on the class,

00:56:55 it could be 30 people, it could be a hundred people,

00:56:57 it could, God help me, be a 500 people

00:56:59 if it’s like the big calculus lecture or whatever it may be.

00:57:01 And there’s sort of some,

00:57:02 but there’s some set of people of that order of magnitude

00:57:05 and I’m with them, we have a long time.

00:57:06 I’m with them for a whole semester

00:57:08 and I can ask them to do homework and we talk together.

00:57:10 We have office hours, if they have one on one questions,

00:57:12 a lot of, it’s like a very high level of engagement,

00:57:14 but how many people am I actually hitting at a time?

00:57:17 Like not that many, right?

00:57:20 And you can, and there’s kind of an inverse relationship

00:57:22 where the more, the fewer people you’re talking to,

00:57:27 the more engagement you can ask for.

00:57:29 The ultimate of course is like the mentorship relation

00:57:32 of like a PhD advisor and a graduate student

00:57:35 where you spend a lot of one on one time together

00:57:38 for like three to five years.

00:57:41 And the ultimate high level of engagement to one person.

00:57:46 Books, this can get to a lot more people

00:57:50 that are ever gonna sit in my classroom

00:57:52 and you spend like however many hours it takes

00:57:57 to read a book.

00:57:58 Somebody like Three Blue One Brown or Numberphile

00:58:01 or people like Vi Hart.

00:58:03 I mean, YouTube, let’s face it, has bigger reach than a book.

00:58:07 Like there’s YouTube videos that have many, many,

00:58:09 many more views than like any hardback book

00:58:13 like not written by a Kardashian or an Obama

00:58:15 is gonna sell, right?

00:58:16 So that’s, I mean,

00:58:20 and then those are, some of them are like longer,

00:58:24 20 minutes long, some of them are five minutes long,

00:58:26 but they’re shorter.

00:58:27 And then even some of you look like Eugenia Chang

00:58:29 who’s a wonderful category theorist in Chicago.

00:58:31 I mean, she was on, I think the Daily Show or is it,

00:58:33 I mean, she was on, she has 30 seconds,

00:58:35 but then there’s like 30 seconds

00:58:37 to sort of say something about mathematics

00:58:38 to like untold millions of people.

00:58:41 So everywhere along this curve is important.

00:58:43 And one thing I feel like is great right now

00:58:46 is that people are just broadcasting on all the channels

00:58:49 because we each have our skills, right?

00:58:51 Somehow along the way, like I learned how to write books.

00:58:53 I had this kind of weird life as a writer

00:58:55 where I sort of spent a lot of time

00:58:57 like thinking about how to put English words together

00:58:59 into sentences and sentences together into paragraphs,

00:59:01 like at length,

00:59:03 which is this kind of like weird specialized skill.

00:59:06 And that’s one thing, but like sort of being able to make

00:59:09 like winning, good looking, eye catching videos

00:59:13 is like a totally different skill.

00:59:15 And probably somewhere out there,

00:59:16 there’s probably sort of some like heavy metal band

00:59:19 that’s like teaching math through heavy metal

00:59:21 and like using their skills to do that.

00:59:23 I hope there is at any rate.

00:59:25 Their music and so on, yeah.

00:59:26 But there is something to the process.

00:59:28 I mean, Grant does this especially well,

00:59:31 which is in order to be able to visualize something,

00:59:36 now he writes programs, so it’s programmatic visualization.

00:59:39 So like the things he is basically mostly

00:59:42 through his Manum library and Python,

00:59:46 everything is drawn through Python.

00:59:49 You have to truly understand the topic

00:59:54 to be able to visualize it in that way

00:59:58 and not just understand it,

00:59:59 but really kind of think in a very novel way.

01:00:04 It’s funny because I’ve spoken with him a couple of times,

01:00:07 spoken to him a lot offline as well.

01:00:09 He really doesn’t think he’s doing anything new,

01:00:14 meaning like he sees himself as very different

01:00:17 from maybe like a researcher,

01:00:20 but it feels to me like he’s creating something totally new.

01:00:26 Like that act of understanding and visualizing

01:00:29 is as powerful or has the same kind of inkling of power

01:00:33 as does the process of proving something.

01:00:36 It doesn’t have that clear destination,

01:00:39 but it’s pulling out an insight

01:00:42 and creating multiple sets of perspective

01:00:44 that arrive at that insight.

01:00:46 And to be honest, it’s something that I think

01:00:49 we haven’t quite figured out how to value

01:00:53 inside academic mathematics in the same way,

01:00:55 and this is a bit older,

01:00:56 that I think we haven’t quite figured out

01:00:57 how to value the development

01:00:59 of computational infrastructure.

01:01:01 We all have computers as our partners now

01:01:02 and people build computers that sort of assist

01:01:07 and participate in our mathematics.

01:01:09 They build those systems

01:01:10 and that’s a kind of mathematics too,

01:01:12 but not in the traditional form

01:01:14 of proving theorems and writing papers.

01:01:16 But I think it’s coming.

01:01:17 Look, I mean, I think, for example,

01:01:20 the Institute for Computational Experimental Mathematics

01:01:23 at Brown, which is like, it’s a NSF funded math institute,

01:01:27 very much part of sort of traditional math academia.

01:01:29 They did an entire theme semester

01:01:31 about visualizing mathematics,

01:01:33 looking at the same kind of thing that they would do

01:01:34 for like an up and coming research topic.

01:01:37 Like that’s pretty cool.

01:01:38 So I think there really is buy in

01:01:40 from the mathematics community

01:01:43 to recognize that this kind of stuff is important

01:01:45 and counts as part of mathematics,

01:01:47 like part of what we’re actually here to do.

01:01:50 Yeah, I’m hoping to see more and more of that

01:01:52 from like MIT faculty, from faculty,

01:01:54 from all the top universities in the world.

01:01:57 Let me ask you this weird question about the Fields Medal,

01:02:00 which is the Nobel Prize in Mathematics.

01:02:02 Do you think, since we’re talking about computers,

01:02:05 there will one day come a time when a computer,

01:02:11 an AI system will win a Fields Medal?

01:02:16 No.

01:02:16 Of course, that’s what a human would say.

01:02:19 Why not?

01:02:20 Is that like, that’s like my captcha?

01:02:23 That’s like the proof that I’m a human?

01:02:24 Is that like the lie that I know?

01:02:25 Yeah.

01:02:26 What is, how does he want me to answer?

01:02:28 Is there something interesting to be said about that?

01:02:31 Yeah, I mean, I am tremendously interested

01:02:34 in what AI can do in pure mathematics.

01:02:37 I mean, it’s, of course, it’s a parochial interest, right?

01:02:40 You’re like, why am I interested in like,

01:02:41 how it can like help feed the world

01:02:43 or help solve like real social problems?

01:02:44 I’m like, can it do more math?

01:02:46 Like, what can I do?

01:02:47 We all have our interests, right?

01:02:49 But I think it is a really interesting conceptual question.

01:02:53 And here too, I think it’s important to be kind of historical

01:02:59 because it’s certainly true that there’s lots of things

01:03:02 that we used to call research in mathematics

01:03:04 that we would now call computation.

01:03:07 Tasks that we’ve now offloaded to machines.

01:03:09 Like, you know, in 1890, somebody could be like,

01:03:12 here’s my PhD thesis.

01:03:13 I computed all the invariants of this polynomial ring

01:03:18 under the action of some finite group.

01:03:19 Doesn’t matter what those words mean,

01:03:21 just it’s like some thing that in 1890

01:03:24 would take a person a year to do

01:03:26 and would be a valuable thing that you might wanna know.

01:03:28 And it’s still a valuable thing that you might wanna know,

01:03:29 but now you type a few lines of code

01:03:32 in Macaulay or Sage or Magma and you just have it.

01:03:37 So we don’t think of that as math anymore,

01:03:40 even though it’s the same thing.

01:03:41 What’s Macaulay, Sage and Magma?

01:03:43 Oh, those are computer algebra programs.

01:03:45 So those are like sort of bespoke systems

01:03:46 that lots of mathematicians use.

01:03:48 That’s similar to Maple and…

01:03:49 Yeah, oh yeah, so it’s similar to Maple and Mathematica,

01:03:51 yeah, but a little more specialized, but yeah.

01:03:54 It’s programs that work with symbols

01:03:56 and allow you to do, can you do proofs?

01:03:58 Can you do kind of little leaps and proofs?

01:04:01 They’re not really built for that.

01:04:02 And that’s a whole other story.

01:04:04 But these tools are part of the process of mathematics now.

01:04:07 Right, they are now for most mathematicians, I would say,

01:04:09 part of the process of mathematics.

01:04:11 And so, you know, there’s a story I tell in the book,

01:04:14 which I’m fascinated by, which is, you know,

01:04:17 so far, attempts to get AIs

01:04:22 to prove interesting theorems have not done so well.

01:04:27 It doesn’t mean they can.

01:04:28 There’s actually a paper I just saw,

01:04:29 which has a very nice use of a neural net

01:04:32 to find counter examples to conjecture.

01:04:34 Somebody said like, well, maybe this is always that.

01:04:37 And you can be like, well, let me sort of train an AI

01:04:39 to sort of try to find things where that’s not true.

01:04:43 And it actually succeeded.

01:04:44 Now, in this case, if you look at the things that it found,

01:04:48 you say like, okay, I mean, these are not famous conjectures.

01:04:53 Okay, so like somebody wrote this down, maybe this is so.

01:04:58 Looking at what the AI came up with, you’re like,

01:05:00 you know, I bet if like five grad students

01:05:03 had thought about that problem,

01:05:04 they wouldn’t have come up with that.

01:05:05 I mean, when you see it, you’re like,

01:05:06 okay, that is one of the things you might try

01:05:08 if you sort of like put some work into it.

01:05:10 Still, it’s pretty awesome.

01:05:12 But the story I tell in the book, which I’m fascinated by

01:05:15 is there is, okay, we’re gonna go back to knots.

01:05:21 There’s a knot called the Conway knot.

01:05:23 After John Conway, maybe we’ll talk about

01:05:25 a very interesting character also.

01:05:26 Yeah, it’s a small tangent.

01:05:28 Somebody I was supposed to talk to

01:05:29 and unfortunately he passed away

01:05:30 and he’s somebody I find as an incredible mathematician,

01:05:35 incredible human being.

01:05:36 Oh, and I am sorry that you didn’t get a chance

01:05:38 because having had the chance to talk to him a lot

01:05:40 when I was a postdoc, yeah, you missed out.

01:05:44 There’s no way to sugarcoat it.

01:05:45 I’m sorry that you didn’t get that chance.

01:05:46 Yeah, it is what it is.

01:05:47 So knots.

01:05:50 Yeah, so there was a question and again,

01:05:52 it doesn’t matter the technicalities of the question,

01:05:54 but it’s a question of whether the knot is slice.

01:05:56 It has to do with something about what kinds

01:05:59 of three dimensional surfaces and four dimensions

01:06:02 can be bounded by this knot.

01:06:03 But nevermind what it means, it’s some question.

01:06:06 And it’s actually very hard to compute

01:06:08 whether a knot is slice or not.

01:06:12 And in particular, the question of the Conway knot,

01:06:16 whether it was slice or not, was particularly vexed

01:06:23 until it was solved just a few years ago

01:06:24 by Lisa Piccarillo, who actually,

01:06:26 now that I think of it, was here in Austin.

01:06:27 I believe she was a grad student at UT Austin at the time.

01:06:29 I didn’t even realize there was an Austin connection

01:06:31 to this story until I started telling it.

01:06:34 In fact, I think she’s now at MIT,

01:06:35 so she’s basically following you around.

01:06:38 If I remember correctly.

01:06:38 The reverse.

01:06:39 There’s a lot of really interesting richness to this story.

01:06:42 One thing about it is her paper was rather,

01:06:45 was very short, it was very short and simple.

01:06:48 Nine pages of which two were pictures.

01:06:51 Very short for like a paper solving a major conjecture.

01:06:54 And it really makes you think about what we mean

01:06:55 by difficulty in mathematics.

01:06:57 Like, do you say, oh, actually the problem wasn’t difficult

01:06:59 because you could solve it so simply?

01:07:00 Or do you say like, well, no, evidently it was difficult

01:07:03 because like the world’s top topologists,

01:07:05 many, you know, worked on it for 20 years

01:07:06 and nobody could solve it, so therefore it is difficult.

01:07:08 Or is it that we need sort of some new category

01:07:10 of things that about which it’s difficult

01:07:12 to figure out that they’re not difficult?

01:07:15 I mean, this is the computer science formulation,

01:07:18 but the sort of the journey to arrive

01:07:22 at the simple answer may be difficult,

01:07:24 but once you have the answer, it will then appear simple.

01:07:28 And I mean, there might be a large category.

01:07:30 I hope there’s a large set of such solutions,

01:07:37 because, you know, once we stand

01:07:41 at the end of the scientific process

01:07:43 that we’re at the very beginning of,

01:07:46 or at least it feels like,

01:07:47 I hope there’s just simple answers to everything

01:07:50 that we’ll look and it’ll be simple laws

01:07:53 that govern the universe,

01:07:55 simple explanation of what is consciousness,

01:07:58 what is love, is mortality fundamental to life,

01:08:02 what’s the meaning of life, are humans special

01:08:07 or we’re just another sort of reflection

01:08:09 of all that is beautiful in the universe

01:08:13 in terms of like life forms, all of it is life

01:08:16 and just has different,

01:08:18 when taken from a different perspective

01:08:19 is all life can seem more valuable or not,

01:08:22 but really it’s all part of the same thing.

01:08:24 All those will have a nice, like two equations,

01:08:26 maybe one equation, but.

01:08:28 Why do you think you want those questions

01:08:30 to have simple answers?

01:08:32 I think just like symmetry

01:08:35 and the breaking of symmetry is beautiful somehow.

01:08:39 There’s something beautiful about simplicity.

01:08:41 I think it, what is that?

01:08:42 So it’s aesthetic.

01:08:43 It’s aesthetic, yeah.

01:08:45 Or, but it’s aesthetic in the way

01:08:47 that happiness is an aesthetic.

01:08:49 Like, why is that so joyful

01:08:53 that a simple explanation that governs

01:08:57 a large number of cases is really appealing?

01:09:01 Even when it’s not, like obviously we get

01:09:05 a huge amount of trouble with that

01:09:07 because oftentimes it doesn’t have to be connected

01:09:11 with reality or even that explanation

01:09:13 could be exceptionally harmful.

01:09:15 Most of like the world’s history that has,

01:09:18 that was governed by hate and violence

01:09:21 had a very simple explanation at the core

01:09:23 that was used to cause the violence and the hatred.

01:09:26 So like we get into trouble with that,

01:09:28 but why is that so appealing?

01:09:30 And in this nice forms in mathematics,

01:09:33 like you look at the Einstein papers,

01:09:36 why are those so beautiful?

01:09:38 And why is the Andrew Wiles proof

01:09:40 of the Fermat’s last theorem not quite so beautiful?

01:09:43 Like what’s beautiful about that story

01:09:45 is the human struggle of like the human story

01:09:48 of perseverance, of the drama,

01:09:51 of not knowing if the proof is correct

01:09:53 and ups and downs and all of those kinds of things.

01:09:56 That’s the interesting part.

01:09:57 But the fact that the proof is huge

01:09:58 and nobody understands, well,

01:10:00 from my outsider’s perspective,

01:10:01 nobody understands what the heck it is,

01:10:04 is not as beautiful as it could have been.

01:10:06 I wish it was what Fermat originally said,

01:10:09 which is, you know, it’s not,

01:10:13 it’s not small enough to fit in the margins of this page,

01:10:17 but maybe if he had like a full page

01:10:19 or maybe a couple of post it notes,

01:10:20 he would have enough to do the proof.

01:10:22 What do you make of,

01:10:23 if we could take another of a multitude of tangents,

01:10:27 what do you make of Fermat’s last theorem?

01:10:29 Because the statement, there’s a few theorems,

01:10:31 there’s a few problems that are deemed by the world

01:10:35 throughout its history to be exceptionally difficult.

01:10:37 And that one in particular is really simple to formulate

01:10:42 and really hard to come up with a proof for.

01:10:46 And it was like taunted as simple by Fermat himself.

01:10:51 Is there something interesting to be said about

01:10:53 that X to the N plus Y to the N equals Z to the N

01:10:57 for N of three or greater, is there a solution to this?

01:11:02 And then how do you go about proving that?

01:11:04 Like, how would you try to prove that?

01:11:08 And what do you learn from the proof

01:11:09 that eventually emerged by Andrew Wiles?

01:11:12 Yeah, so right, so to give,

01:11:13 let me just say the background,

01:11:14 because I don’t know if everybody listening knows the story.

01:11:17 So, you know, Fermat was an early number theorist,

01:11:21 at least sort of an early mathematician,

01:11:23 those special adjacent didn’t really exist back then.

01:11:27 He comes up in the book actually,

01:11:28 in the context of a different theorem of his

01:11:31 that has to do with testing,

01:11:32 whether a number is prime or not.

01:11:34 So I write about, he was one of the ones who was salty

01:11:37 and like, he would exchange these letters

01:11:39 where he and his correspondents would like

01:11:41 try to top each other and vex each other with questions

01:11:44 and stuff like this.

01:11:44 But this particular thing,

01:11:47 it’s called Fermat’s Last Theorem because it’s a note

01:11:50 he wrote in his copy of the Disquisitiones Arithmetic I.

01:11:57 Like he wrote, here’s an equation, it has no solutions.

01:12:00 I can prove it, but the proof’s like a little too long

01:12:03 to fit in the margin of this book.

01:12:05 He was just like writing a note to himself.

01:12:07 Now, let me just say historically,

01:12:08 we know that Fermat did not have a proof of this theorem.

01:12:11 For a long time, people were like this mysterious proof

01:12:15 that was lost, a very romantic story, right?

01:12:17 But a fair amount later,

01:12:21 he did prove special cases of this theorem

01:12:24 and wrote about it, talked to people about the problem.

01:12:27 It’s very clear from the way that he wrote

01:12:29 where he can solve certain examples

01:12:30 of this type of equation

01:12:32 that he did not know how to do the whole thing.

01:12:35 He may have had a deep, simple intuition

01:12:39 about how to solve the whole thing

01:12:41 that he had at that moment

01:12:43 without ever being able to come up with a complete proof.

01:12:47 And that intuition maybe lost the time.

01:12:50 Maybe, but you’re right, that is unknowable.

01:12:54 But I think what we can know is that later,

01:12:56 he certainly did not think that he had a proof

01:12:59 that he was concealing from people.

01:13:00 He thought he didn’t know how to prove it,

01:13:04 and I also think he didn’t know how to prove it.

01:13:06 Now, I understand the appeal of saying like,

01:13:10 wouldn’t it be cool if this very simple equation

01:13:12 there was like a very simple, clever, wonderful proof

01:13:16 that you could do in a page or two.

01:13:17 And that would be great, but you know what?

01:13:18 There’s lots of equations like that

01:13:20 that are solved by very clever methods like that,

01:13:22 including the special cases that Fermat wrote about,

01:13:24 the method of descent,

01:13:25 which is like very wonderful and important.

01:13:26 But in the end, those are nice things

01:13:31 that like you teach in an undergraduate class,

01:13:34 and it is what it is,

01:13:35 but they’re not big.

01:13:38 On the other hand, work on the Fermat problem,

01:13:41 that’s what we like to call it

01:13:42 because it’s not really his theorem

01:13:44 because we don’t think he proved it.

01:13:45 So, I mean, work on the Fermat problem

01:13:49 developed this like incredible richness of number theory

01:13:52 that we now live in today.

01:13:54 Like, and not, by the way,

01:13:56 just Wiles, Andrew Wiles being the person

01:13:58 who, together with Richard Taylor,

01:13:59 finally proved this theorem.

01:14:01 But you know how you have this whole moment

01:14:03 that people try to prove this theorem

01:14:05 and they fail,

01:14:06 and there’s a famous false proof by LeMay

01:14:08 from the 19th century,

01:14:10 where Kummer, in understanding what mistake LeMay had made

01:14:14 in this incorrect proof,

01:14:16 basically understands something incredible,

01:14:18 which is that a thing we know about numbers

01:14:20 is that you can factor them

01:14:24 and you can factor them uniquely.

01:14:26 There’s only one way to break a number up into primes.

01:14:30 Like if we think of a number like 12,

01:14:32 12 is two times three times two.

01:14:35 I had to think about it.

01:14:38 Or it’s two times two times three,

01:14:39 of course you can reorder them.

01:14:41 But there’s no other way to do it.

01:14:43 There’s no universe in which 12 is something times five,

01:14:46 or in which there’s like four threes in it.

01:14:47 Nope, 12 is like two twos and a three.

01:14:49 Like that is what it is.

01:14:50 And that’s such a fundamental feature of arithmetic

01:14:54 that we almost think of it like God’s law.

01:14:56 You know what I mean?

01:14:57 It has to be that way.

01:14:58 That’s a really powerful idea.

01:15:00 It’s so cool that every number

01:15:02 is uniquely made up of other numbers.

01:15:05 And like made up meaning like there’s these like basic atoms

01:15:10 that form molecules that get built on top of each other.

01:15:15 I love it.

01:15:16 I mean, when I teach undergraduate number theory,

01:15:18 it’s like, it’s the first really deep theorem

01:15:22 that you prove.

01:15:23 What’s amazing is the fact

01:15:25 that you can factor a number into primes is much easier.

01:15:28 Essentially Euclid knew it,

01:15:30 although he didn’t quite put it in that way.

01:15:33 The fact that you can do it at all.

01:15:34 What’s deep is the fact that there’s only one way to do it

01:15:38 or however you sort of chop the number up,

01:15:40 you end up with the same set of prime factors.

01:15:44 And indeed what people finally understood

01:15:49 at the end of the 19th century is that

01:15:51 if you work in number systems slightly more general

01:15:54 than the ones we’re used to,

01:15:56 which it turns out are relevant to Fermat,

01:16:01 all of a sudden this stops being true.

01:16:04 Things get, I mean, things get more complicated

01:16:07 and now because you were praising simplicity before

01:16:10 you were like, it’s so beautiful, unique factorization.

01:16:12 It’s so great.

01:16:13 Like, so when I tell you

01:16:14 that in more general number systems,

01:16:16 there is no unique factorization.

01:16:18 Maybe you’re like, that’s bad.

01:16:19 I’m like, no, that’s good

01:16:20 because there’s like a whole new world of phenomena

01:16:22 to study that you just can’t see

01:16:24 through the lens of the numbers that we’re used to.

01:16:26 So I’m for complication.

01:16:29 I’m highly in favor of complication

01:16:32 because every complication is like an opportunity

01:16:34 for new things to study.

01:16:35 And is that the big kind of one of the big insights

01:16:40 for you from Andrew Wiles’s proof?

01:16:42 Is there interesting insights about the process

01:16:46 that you used to prove that sort of resonates

01:16:49 with you as a mathematician?

01:16:51 Is there an interesting concept that emerged from it?

01:16:54 Is there interesting human aspects to the proof?

01:16:57 Whether there’s interesting human aspects

01:16:59 to the proof itself is an interesting question.

01:17:02 Certainly it has a huge amount of richness.

01:17:05 Sort of at its heart is an argument

01:17:07 of what’s called deformation theory,

01:17:12 which was in part created by my PhD advisor, Barry Mazer.

01:17:18 Can you speak to what deformation theory is?

01:17:20 I can speak to what it’s like.

01:17:21 How about that?

01:17:22 What does it rhyme with?

01:17:24 Right, well, the reason that Barry called it

01:17:27 deformation theory, I think he’s the one

01:17:29 who gave it the name.

01:17:30 I hope I’m not wrong in saying it’s a name.

01:17:32 In your book, you have calling different things

01:17:35 by the same name as one of the things

01:17:37 in the beautiful map that opens the book.

01:17:40 Yes, and this is a perfect example.

01:17:42 So this is another phrase of Poincare,

01:17:44 this like incredible generator of slogans and aphorisms.

01:17:46 He said, mathematics is the art

01:17:47 of calling different things by the same name.

01:17:49 That very thing we do, right?

01:17:52 When we’re like this triangle and this triangle,

01:17:53 come on, they’re the same triangle,

01:17:55 they’re just in a different place, right?

01:17:56 So in the same way, it came to be understood

01:18:00 that the kinds of objects that you study

01:18:06 when you study Fermat’s Last Theorem,

01:18:10 and let’s not even be too careful

01:18:12 about what these objects are.

01:18:13 I can tell you there are gaol representations

01:18:15 in modular forms, but saying those words

01:18:18 is not gonna mean so much.

01:18:19 But whatever they are, they’re things that can be deformed,

01:18:23 moved around a little bit.

01:18:25 And I think the insight of what Andrew

01:18:28 and then Andrew and Richard were able to do

01:18:31 was to say something like this.

01:18:33 A deformation means moving something just a tiny bit,

01:18:36 like an infinitesimal amount.

01:18:39 If you really are good at understanding

01:18:41 which ways a thing can move in a tiny, tiny, tiny,

01:18:44 infinitesimal amount in certain directions,

01:18:46 maybe you can piece that information together

01:18:49 to understand the whole global space in which it can move.

01:18:52 And essentially, their argument comes down

01:18:54 to showing that two of those big global spaces

01:18:57 are actually the same, the fabled R equals T,

01:19:00 part of their proof, which is at the heart of it.

01:19:05 And it involves this very careful principle like that.

01:19:09 But that being said, what I just said,

01:19:12 it’s probably not what you’re thinking,

01:19:14 because what you’re thinking when you think,

01:19:16 oh, I have a point in space and I move it around

01:19:18 like a little tiny bit,

01:19:22 you’re using your notion of distance

01:19:26 that’s from calculus.

01:19:28 We know what it means for like two points

01:19:29 on the real line to be close together.

01:19:32 So yet another thing that comes up in the book a lot

01:19:37 is this fact that the notion of distance

01:19:41 is not given to us by God.

01:19:42 We could mean a lot of different things by distance.

01:19:44 And just in the English language, we do that all the time.

01:19:46 We talk about somebody being a close relative.

01:19:49 It doesn’t mean they live next door to you, right?

01:19:51 It means something else.

01:19:52 There’s a different notion of distance we have in mind.

01:19:54 And there are lots of notions of distances

01:19:57 that you could use.

01:19:58 In the natural language processing community and AI,

01:20:01 there might be some notion of semantic distance

01:20:04 or lexical distance between two words.

01:20:06 How much do they tend to arise in the same context?

01:20:08 That’s incredibly important for doing autocomplete

01:20:13 and like machine translation and stuff like that.

01:20:15 And it doesn’t have anything to do with

01:20:16 are they next to each other in the dictionary, right?

01:20:17 It’s a different kind of distance.

01:20:19 Okay, ready?

01:20:20 In this kind of number theory,

01:20:21 there was a crazy distance called the peatic distance.

01:20:25 I didn’t write about this that much in the book

01:20:26 because even though I love it

01:20:27 and it’s a big part of my research life,

01:20:28 it gets a little bit into the weeds,

01:20:29 but your listeners are gonna hear about it now.

01:20:32 Please.

01:20:34 What a normal person says

01:20:35 when they say two numbers are close,

01:20:37 they say like their difference is like a small number,

01:20:40 like seven and eight are close

01:20:41 because their difference is one and one’s pretty small.

01:20:44 If we were to be what’s called a two attic number theorist,

01:20:48 we’d say, oh, two numbers are close

01:20:50 if their difference is a multiple of a large power of two.

01:20:55 So like one and 49 are close

01:21:00 because their difference is 48

01:21:02 and 48 is a multiple of 16,

01:21:04 which is a pretty large power of two.

01:21:06 Whereas one and two are pretty far away

01:21:09 because the difference between them is one,

01:21:12 which is not even a multiple of a power of two at all.

01:21:14 That’s odd.

01:21:15 You wanna know what’s really far from one?

01:21:17 Like one and 1 64th

01:21:21 because their difference is a negative power of two,

01:21:24 two to the minus six.

01:21:25 So those points are quite, quite far away.

01:21:28 Two to the power of a large N would be two,

01:21:33 if that’s the difference between two numbers

01:21:35 then they’re close.

01:21:37 Yeah, so two to a large power is in this metric

01:21:40 a very small number

01:21:41 and two to a negative power is a very big number.

01:21:44 That’s two attic.

01:21:45 Okay, I can’t even visualize that.

01:21:48 It takes practice.

01:21:49 It takes practice.

01:21:50 If you’ve ever heard of the Cantor set,

01:21:51 it looks kind of like that.

01:21:54 So it is crazy that this is good for anything, right?

01:21:57 I mean, this just sounds like a definition

01:21:58 that someone would make up to torment you.

01:22:00 But what’s amazing is there’s a general theory of distance

01:22:05 where you say any definition you make

01:22:08 to satisfy certain axioms deserves to be called a distance

01:22:11 and this.

01:22:12 See, I’m sorry to interrupt.

01:22:13 My brain, you broke my brain.

01:22:15 Awesome.

01:22:16 10 seconds ago.

01:22:18 Cause I’m also starting to map for the two attic case

01:22:21 to binary numbers.

01:22:23 And you know, cause we romanticize those.

01:22:25 So I was trying to.

01:22:26 Oh, that’s exactly the right way to think of it.

01:22:27 I was trying to mess with number,

01:22:29 I was trying to see, okay, which ones are close.

01:22:31 And then I’m starting to visualize

01:22:33 different binary numbers and how they,

01:22:35 which ones are close to each other.

01:22:37 And I’m not sure.

01:22:38 Well, I think there’s a.

01:22:39 No, no, it’s very similar.

01:22:40 That’s exactly the right way to think of it.

01:22:41 It’s almost like binary numbers written in reverse.

01:22:44 Because in a binary expansion, two numbers are close.

01:22:47 A number that’s small is like 0.0000 something.

01:22:50 Something that’s the decimal

01:22:51 and it starts with a lot of zeros.

01:22:53 In the two attic metric, a binary number is very small

01:22:56 if it ends with a lot of zeros and then the decimal point.

01:23:01 Gotcha.

01:23:02 So it is kind of like binary numbers written backwards

01:23:04 is actually, I should have said,

01:23:05 that’s what I should have said, Lex.

01:23:07 That’s a very good metaphor.

01:23:08 Okay, but so why is that interesting

01:23:12 except for the fact that it’s a beautiful kind of framework,

01:23:18 different kind of framework

01:23:19 of which to think about distances.

01:23:20 And you’re talking about not just the two attic,

01:23:23 but the generalization of that.

01:23:24 Why is that interesting?

01:23:25 Yeah, the NEP.

01:23:25 And so that, because that’s the kind of deformation

01:23:27 that comes up in Wiles’s proof,

01:23:31 that deformation where moving something a little bit

01:23:34 means a little bit in this two attic sense.

01:23:36 Trippy, okay.

01:23:38 No, I mean, it’s such a,

01:23:38 I mean, I just get excited talking about it

01:23:40 and I just taught this like in the fall semester that.

01:23:43 But it like reformulating, why is,

01:23:49 so you pick a different measure of distance

01:23:53 over which you can talk about very tiny changes

01:23:56 and then use that to then prove things

01:23:59 about the entire thing.

01:24:02 Yes, although, honestly, what I would say,

01:24:05 I mean, it’s true that we use it to prove things,

01:24:07 but I would say we use it to understand things.

01:24:09 And then because we understand things better,

01:24:11 then we can prove things.

01:24:12 But the goal is always the understanding.

01:24:14 The goal is not so much to prove things.

01:24:16 The goal is not to know what’s true or false.

01:24:18 I mean, this is something I write about

01:24:19 in the book, Near the End.

01:24:20 And it’s something that,

01:24:21 it’s a wonderful, wonderful essay by Bill Thurston,

01:24:25 kind of one of the great geometers of our time,

01:24:27 who unfortunately passed away a few years ago,

01:24:29 called on proof and progress in mathematics.

01:24:32 And he writes very wonderfully about how,

01:24:35 we’re not, it’s not a theorem factory

01:24:38 where you have a production quota.

01:24:39 I mean, the point of mathematics

01:24:40 is to help humans understand things.

01:24:43 And the way we test that

01:24:45 is that we’re proving new theorems along the way.

01:24:46 That’s the benchmark, but that’s not the goal.

01:24:49 Yeah, but just as a kind of, absolutely,

01:24:51 but as a tool, it’s kind of interesting

01:24:54 to approach a problem by saying,

01:24:56 how can I change the distance function?

01:24:59 Like what, the nature of distance,

01:25:03 because that might start to lead to insights

01:25:07 for deeper understanding.

01:25:08 Like if I were to try to describe human society

01:25:12 by a distance, two people are close

01:25:15 if they love each other.

01:25:17 Right.

01:25:17 And then start to do a full analysis

01:25:21 on the everybody that lives on earth currently,

01:25:23 the 7 billion people.

01:25:25 And from that perspective,

01:25:27 as opposed to the geographic perspective of distance.

01:25:30 And then maybe there could be a bunch of insights

01:25:32 about the source of violence,

01:25:35 the source of maybe entrepreneurial success

01:25:39 or invention or economic success or different systems,

01:25:42 communism, capitalism start to,

01:25:44 I mean, that’s, I guess what economics tries to do,

01:25:47 but really saying, okay, let’s think outside the box

01:25:50 about totally new distance functions

01:25:52 that could unlock something profound about the space.

01:25:57 Yeah, because think about it.

01:25:58 Okay, here’s, I mean, now we’re gonna talk about AI,

01:26:01 which you know a lot more about than I do.

01:26:02 So just start laughing uproariously

01:26:05 if I say something that’s completely wrong.

01:26:07 We both know very little relative

01:26:09 to what we will know centuries from now.

01:26:12 That is a really good humble way to think about it.

01:26:15 I like it.

01:26:16 Okay, so let’s just go for it.

01:26:18 Okay, so I think you’ll agree with this,

01:26:20 that in some sense, what’s good about AI

01:26:23 is that we can’t test any case in advance,

01:26:26 the whole point of AI is to make,

01:26:27 or one point of it, I guess, is to make good predictions

01:26:30 about cases we haven’t yet seen.

01:26:32 And in some sense, that’s always gonna involve

01:26:34 some notion of distance,

01:26:35 because it’s always gonna involve

01:26:37 somehow taking a case we haven’t seen

01:26:40 and saying what cases that we have seen is it close to,

01:26:43 is it like, is it somehow an interpolation between.

01:26:47 Now, when we do that,

01:26:49 in order to talk about things being like other things,

01:26:52 implicitly or explicitly,

01:26:53 we’re invoking some notion of distance,

01:26:55 and boy, we better get it right.

01:26:57 If you try to do natural language processing

01:26:59 and your idea of distance between words

01:27:01 is how close they are in the dictionary,

01:27:03 when you write them in alphabetical order,

01:27:04 you are gonna get pretty bad translations, right?

01:27:08 No, the notion of distance has to come from somewhere else.

01:27:11 Yeah, that’s essentially what neural networks are doing,

01:27:14 that’s what word embeddings are doing is coming up with.

01:27:17 In the case of word embeddings, literally,

01:27:18 literally what they are doing is learning a distance.

01:27:21 But those are super complicated distance functions,

01:27:23 and it’s almost nice to think

01:27:26 maybe there’s a nice transformation that’s simple.

01:27:31 Sorry, there’s a nice formulation of the distance.

01:27:34 Again with the simple.

01:27:36 So you don’t, let me ask you about this.

01:27:41 From an understanding perspective,

01:27:43 there’s the Richard Feynman, maybe attributed to him,

01:27:45 but maybe many others,

01:27:48 is this idea that if you can’t explain something simply

01:27:52 that you don’t understand it.

01:27:56 In how many cases, how often is that true?

01:28:00 Do you find there’s some profound truth in that?

01:28:05 Oh, okay, so you were about to ask, is it true?

01:28:07 To which I would say flatly, no.

01:28:09 But then you said, you followed that up with,

01:28:11 is there some profound truth in it?

01:28:13 And I’m like, okay, sure.

01:28:14 So there’s some truth in it.

01:28:15 It’s not true. But it’s not true.

01:28:16 It’s just not.

01:28:17 That’s such a mathematician answer.

01:28:22 The truth that is in it is that learning

01:28:25 to explain something helps you understand it.

01:28:29 But real things are not simple.

01:28:33 A few things are, most are not.

01:28:36 And to be honest, we don’t really know

01:28:40 whether Feynman really said that right

01:28:41 or something like that is sort of disputed.

01:28:43 But I don’t think Feynman could have literally believed that

01:28:46 whether or not he said it.

01:28:47 And he was the kind of guy, I didn’t know him,

01:28:49 but I’ve been reading his writing,

01:28:51 he liked to sort of say stuff, like stuff that sounded good.

01:28:55 You know what I mean?

01:28:55 So it’s totally strikes me as the kind of thing

01:28:57 he could have said because he liked the way saying it

01:29:00 made him feel, but also knowing

01:29:02 that he didn’t like literally mean it.

01:29:04 Well, I definitely have a lot of friends

01:29:07 and I’ve talked to a lot of physicists

01:29:09 and they do derive joy from believing

01:29:12 that they can explain stuff simply

01:29:14 or believing it’s possible to explain stuff simply,

01:29:17 even when the explanation is not actually that simple.

01:29:20 Like I’ve heard people think that the explanation is simple

01:29:23 and they do the explanation.

01:29:25 And I think it is simple,

01:29:27 but it’s not capturing the phenomena that we’re discussing.

01:29:30 It’s capturing, it’s somehow maps in their mind,

01:29:33 but it’s taking as a starting point,

01:29:35 as an assumption that there’s a deep knowledge

01:29:38 and a deep understanding that’s actually very complicated.

01:29:41 And the simplicity is almost like a poem

01:29:45 about the more complicated thing

01:29:46 as opposed to a distillation.

01:29:48 And I love poems, but a poem is not an explanation.

01:29:51 Well, some people might disagree with that,

01:29:55 but certainly from a mathematical perspective.

01:29:57 No poet would disagree with it.

01:29:59 No poet would disagree.

01:30:01 You don’t think there’s some things

01:30:02 that can only be described imprecisely?

01:30:06 As an explanation.

01:30:07 I don’t think any poet would say their poem

01:30:09 is an explanation.

01:30:10 They might say it’s a description.

01:30:11 They might say it’s sort of capturing sort of.

01:30:14 Well, some people might say the only truth is like music.

01:30:20 Not the only truth,

01:30:20 but some truths can only be expressed through art.

01:30:24 And I mean, that’s the whole thing

01:30:26 we’re talking about religion and myth.

01:30:27 And there’s some things

01:30:28 that are limited cognitive capabilities

01:30:32 and the tools of mathematics or the tools of physics

01:30:35 are just not going to allow us to capture.

01:30:37 Like it’s possible consciousness is one of those things.

01:30:39 And.

01:30:42 Yes, that is definitely possible.

01:30:44 But I would even say,

01:30:46 look, I mean, consciousness is a thing about

01:30:47 which we’re still in the dark

01:30:48 as to whether there’s an explanation

01:30:50 we would understand it as an explanation at all.

01:30:53 By the way, okay.

01:30:54 I got to give yet one more amazing Poincare quote

01:30:56 because this guy just never stopped coming up

01:30:57 with great quotes that,

01:31:00 Paul Erdős, another fellow who appears in the book.

01:31:02 And by the way,

01:31:03 he thinks about this notion of distance

01:31:05 of like personal affinity,

01:31:07 kind of like what you’re talking about,

01:31:08 the kind of social network and that notion of distance

01:31:11 that comes from that.

01:31:12 So that’s something that Paul Erdős.

01:31:13 Erdős did?

01:31:14 Well, he thought about distances and networks.

01:31:16 I guess he didn’t probably,

01:31:16 he didn’t think about the social network.

01:31:17 Oh, that’s fascinating.

01:31:18 And that’s how it started that story of Erdős number.

01:31:20 Yeah, okay.

01:31:20 It’s hard to distract.

01:31:22 But you know, Erdős was sort of famous for saying,

01:31:25 and this is sort of long lines we’re saying,

01:31:26 he talked about the book,

01:31:28 capital T, capital B, the book.

01:31:31 And that’s the book where God keeps the right proof

01:31:33 of every theorem.

01:31:34 So when he saw a proof he really liked,

01:31:36 it was like really elegant, really simple.

01:31:38 Like that’s from the book.

01:31:39 That’s like you found one of the ones that’s in the book.

01:31:43 He wasn’t a religious guy, by the way.

01:31:44 He referred to God as the supreme fascist.

01:31:46 He was like, but somehow he was like,

01:31:48 I don’t really believe in God,

01:31:49 but I believe in God’s book.

01:31:50 I mean, it was,

01:31:53 but Poincare on the other hand,

01:31:55 and by the way, there were other managers.

01:31:57 Hilda Hudson is one who comes up in this book.

01:31:58 She also kind of saw math.

01:32:01 She’s one of the people who sort of develops

01:32:05 the disease model that we now use,

01:32:06 that we use to sort of track pandemics,

01:32:08 this SIR model that sort of originally comes

01:32:10 from her work with Ronald Ross.

01:32:11 But she was also super, super, super devout.

01:32:14 And she also sort of on the other side

01:32:17 of the religious coin was like,

01:32:18 yeah, math is how we communicate with God.

01:32:20 She has a great,

01:32:21 all these people are incredibly quotable.

01:32:22 She says, you know, math is,

01:32:24 the truth, the things about mathematics,

01:32:26 she’s like, they’re not the most important of God thoughts,

01:32:29 but they’re the only ones that we can know precisely.

01:32:32 So she’s like, this is the one place

01:32:34 where we get to sort of see what God’s thinking

01:32:35 when we do mathematics.

01:32:37 Again, not a fan of poetry or music.

01:32:39 Some people will say Hendrix is like,

01:32:41 some people say chapter one of that book is mathematics,

01:32:44 and then chapter two is like classic rock.

01:32:46 Right?

01:32:48 So like, it’s not clear that the…

01:32:51 I’m sorry, you just sent me off on a tangent,

01:32:52 just imagining like Erdos at a Hendrix concert,

01:32:54 like trying to figure out if it was from the book or not.

01:32:59 What I was coming to was just to say,

01:33:00 but what Poincaré said about this is he’s like,

01:33:03 you know, if like, this is all worked out

01:33:07 in the language of the divine,

01:33:08 and if a divine being like came down and told it to us,

01:33:12 we wouldn’t be able to understand it, so it doesn’t matter.

01:33:15 So Poincaré was of the view that there were things

01:33:17 that were sort of like inhumanly complex,

01:33:19 and that was how they really were.

01:33:21 Our job is to figure out the things that are not like that.

01:33:23 That are not like that.

01:33:25 All this talk of primes got me hungry for primes.

01:33:29 You wrote a blog post, The Beauty of Bounding Gaps,

01:33:32 a huge discovery about prime numbers

01:33:35 and what it means for the future of math.

01:33:39 Can you tell me about prime numbers?

01:33:40 What the heck are those?

01:33:41 What are twin primes?

01:33:42 What are prime gaps?

01:33:43 What are bounding gaps and primes?

01:33:46 What are all these things?

01:33:47 And what, if anything,

01:33:49 or what exactly is beautiful about them?

01:33:52 Yeah, so, you know, prime numbers are one of the things

01:33:57 that number theorists study the most and have for millennia.

01:34:02 They are numbers which can’t be factored.

01:34:06 And then you say, like, five.

01:34:08 And then you’re like, wait, I can factor five.

01:34:09 Five is five times one.

01:34:11 Okay, not like that.

01:34:13 That is a factorization.

01:34:14 It absolutely is a way of expressing five

01:34:16 as a product of two things.

01:34:18 But don’t you agree there’s like something trivial about it?

01:34:20 It’s something you could do to any number.

01:34:22 It doesn’t have content the way that if I say

01:34:24 that 12 is six times two or 35 is seven times five,

01:34:27 I’ve really done something to it.

01:34:28 I’ve broken up.

01:34:29 So those are the kind of factorizations that count.

01:34:31 And a number that doesn’t have a factorization like that

01:34:34 is called prime, except, historical side note,

01:34:38 one, which at some times in mathematical history

01:34:42 has been deemed to be a prime, but currently is not.

01:34:46 And I think that’s for the best.

01:34:47 But I bring it up only because sometimes people think that,

01:34:49 you know, these definitions are kind of,

01:34:52 if we think about them hard enough,

01:34:53 we can figure out which definition is true.

01:34:56 No.

01:34:57 There’s just an artifact of mathematics.

01:34:58 So it’s a question of which definition is best for us,

01:35:03 for our purposes.

01:35:04 Well, those edge cases are weird, right?

01:35:06 So it can’t be, it doesn’t count when you use yourself

01:35:11 as a number or one as part of the factorization

01:35:15 or as the entirety of the factorization.

01:35:19 So you somehow get to the meat of the number

01:35:22 by factorizing it.

01:35:24 And that seems to get to the core of all of mathematics.

01:35:27 Yeah, you take any number and you factorize it

01:35:29 until you can factorize no more.

01:35:31 And what you have left is some big pile of primes.

01:35:33 I mean, by definition, when you can’t factor anymore,

01:35:36 when you’re done, when you can’t break the numbers up

01:35:39 anymore, what’s left must be prime.

01:35:40 You know, 12 breaks into two and two and three.

01:35:45 So these numbers are the atoms, the building blocks

01:35:48 of all numbers.

01:35:50 And there’s a lot we know about them,

01:35:52 or there’s much more that we don’t know about them.

01:35:53 I’ll tell you the first few.

01:35:54 There’s two, three, five, seven, 11.

01:35:59 By the way, they’re all gonna be odd from then on

01:36:00 because if they were even, I could factor out

01:36:02 two out of them.

01:36:03 But it’s not all the odd numbers.

01:36:04 Nine isn’t prime because it’s three times three.

01:36:06 15 isn’t prime because it’s three times five,

01:36:08 but 13 is.

01:36:09 Where were we?

01:36:09 Two, three, five, seven, 11, 13, 17, 19.

01:36:13 Not 21, but 23 is, et cetera, et cetera.

01:36:15 Okay, so you could go on.

01:36:17 How high could you go if we were just sitting here?

01:36:19 By the way, your own brain.

01:36:20 If continuous, without interruption,

01:36:23 would you be able to go over 100?

01:36:25 I think so.

01:36:27 There’s always those ones that trip people up.

01:36:29 There’s a famous one, the Grotendeek prime 57,

01:36:31 like sort of Alexander Grotendeek,

01:36:33 the great algebraic geometer was sort of giving

01:36:35 some lecture involving a choice of a prime in general.

01:36:38 And somebody said, can’t you just choose a prime?

01:36:41 And he said, okay, 57, which is in fact not prime.

01:36:43 It’s three times 19.

01:36:45 Oh, damn.

01:36:46 But it was like, I promise you in some circles

01:36:49 it’s a funny story.

01:36:50 But there’s a humor in it.

01:36:55 Yes, I would say over 100, I definitely don’t remember.

01:36:59 Like 107, I think, I’m not sure.

01:37:02 Okay, like, I mean.

01:37:03 So is there a category of like fake primes

01:37:08 that are easily mistaken to be prime?

01:37:12 Like 57, I wonder.

01:37:14 Yeah, so I would say 57 and 51 are definitely

01:37:20 like prime offenders.

01:37:21 Oh, I didn’t do that on purpose.

01:37:23 Oh, well done.

01:37:24 Didn’t do it on purpose.

01:37:25 Anyway, they’re definitely ones that people,

01:37:28 or 91 is another classic, seven times 13.

01:37:30 It really feels kind of prime, doesn’t it?

01:37:32 But it is not.

01:37:34 Yeah.

01:37:35 But there’s also, by the way,

01:37:36 but there’s also an actual notion of pseudo prime,

01:37:39 which is a thing with a formal definition,

01:37:41 which is not a psychological thing.

01:37:43 It is a prime which passes a primality test

01:37:47 devised by Fermat, which is a very good test,

01:37:50 which if a number fails this test,

01:37:52 it’s definitely not prime.

01:37:54 And so there was some hope that,

01:37:55 oh, maybe if a number passes the test,

01:37:57 then it definitely is prime.

01:37:58 That would give a very simple criterion for primality.

01:38:00 Unfortunately, it’s only perfect in one direction.

01:38:03 So there are numbers, I want to say 341 is the smallest,

01:38:09 which pass the test but are not prime, 341.

01:38:12 Is this test easily explainable or no?

01:38:14 Yes, actually.

01:38:16 Ready, let me give you the simplest version of it.

01:38:18 You can dress it up a little bit, but here’s the basic idea.

01:38:22 I take the number, the mystery number,

01:38:25 I raise two to that power.

01:38:29 So let’s say your mystery number is six.

01:38:32 Are you sorry you asked me?

01:38:33 Are you ready?

01:38:34 No, you’re breaking my brain again, but yes.

01:38:37 Let’s do it.

01:38:38 We’re going to do a live demonstration.

01:38:40 Let’s say your number is six.

01:38:43 So I’m going to raise two to the sixth power.

01:38:45 Okay, so if I were working on it,

01:38:46 I’d be like that’s two cubes squared,

01:38:48 so that’s eight times eight, so that’s 64.

01:38:51 Now we’re going to divide by six,

01:38:53 but I don’t actually care what the quotient is,

01:38:54 only the remainder.

01:38:57 So let’s see, 64 divided by six is,

01:39:01 well, there’s a quotient of 10, but the remainder is four.

01:39:05 So you failed because the answer has to be two.

01:39:08 For any prime, let’s do it with five, which is prime.

01:39:13 Two to the fifth is 32.

01:39:15 Divide 32 by five, and you get six with a remainder of two.

01:39:23 With a remainder of two, yeah.

01:39:24 For seven, two to the seventh is 128.

01:39:26 Divide that by seven, and let’s see,

01:39:29 I think that’s seven times 14, is that right?

01:39:32 No.

01:39:33 Seven times 18 is 126 with a remainder of two, right?

01:39:40 128 is a multiple of seven plus two.

01:39:43 So if that remainder is not two,

01:39:46 then it’s definitely not prime.

01:39:49 And then if it is, it’s likely a prime, but not for sure.

01:39:53 It’s likely a prime, but not for sure.

01:39:54 And there’s actually a beautiful geometric proof

01:39:56 which is in the book, actually.

01:39:57 That’s like one of the most granular parts of the book

01:39:58 because it’s such a beautiful proof, I couldn’t not give it.

01:40:00 So you draw a lot of like opal and pearl necklaces

01:40:05 and spin them.

01:40:06 That’s kind of the geometric nature

01:40:07 of this proof of Fermat’s Little Theorem.

01:40:11 So yeah, so with pseudo primes,

01:40:13 there are primes that are kind of faking it.

01:40:14 They pass that test, but there are numbers

01:40:16 that are faking it that pass that test,

01:40:17 but are not actually prime.

01:40:20 But the point is, there are many, many,

01:40:26 many theorems about prime numbers.

01:40:28 There’s a bunch of questions to ask.

01:40:32 Is there an infinite number of primes?

01:40:34 Can we say something about the gap between primes

01:40:37 as the numbers grow larger and larger and larger and so on?

01:40:40 Yeah, it’s a perfect example of your desire

01:40:43 for simplicity in all things.

01:40:44 You know what would be really simple?

01:40:46 If there was only finitely many primes

01:40:48 and then there would be this finite set of atoms

01:40:51 that all numbers would be built up.

01:40:53 That would be very simple and good in certain ways,

01:40:56 but it’s completely false.

01:40:58 And number theory would be totally different

01:41:00 if that were the case.

01:41:01 It’s just not true.

01:41:03 In fact, this is something else that Euclid knew.

01:41:04 So this is a very, very old fact,

01:41:07 like much before, long before we’ve had anything

01:41:10 like modern number theory.

01:41:11 The primes are infinite.

01:41:12 The primes that there are, right.

01:41:14 There’s an infinite number of primes.

01:41:15 So what about the gaps between the primes?

01:41:17 Right, so one thing that people recognized

01:41:20 and really thought about a lot is that the primes,

01:41:22 on average, seem to get farther and farther apart

01:41:25 as they get bigger and bigger.

01:41:27 In other words, it’s less and less common.

01:41:29 Like I already told you of the first 10 numbers,

01:41:31 two, three, five, seven, four of them are prime.

01:41:32 That’s a lot, 40%.

01:41:34 If I looked at 10 digit numbers,

01:41:38 no way would 40% of those be prime.

01:41:40 Being prime would be a lot rarer.

01:41:42 In some sense, because there’s a lot more things

01:41:43 for them to be divisible by.

01:41:45 That’s one way of thinking of it.

01:41:47 It’s a lot more possible for there to be a factorization

01:41:49 because there’s a lot of things

01:41:50 you can try to factor out of it.

01:41:52 As the numbers get bigger and bigger,

01:41:53 primality gets rarer and rarer, and the extent

01:41:58 to which that’s the case, that’s pretty well understood.

01:42:01 But then you can ask more fine grained questions,

01:42:03 and here is one.

01:42:07 A twin prime is a pair of primes that are two apart,

01:42:11 like three and five, or like 11 and 13, or like 17 and 19.

01:42:17 And one thing we still don’t know

01:42:18 is are there infinitely many of those?

01:42:21 We know on average, they get farther and farther apart,

01:42:24 but that doesn’t mean there couldn’t be occasional folks

01:42:28 that come close together.

01:42:30 And indeed, we think that there are.

01:42:33 And one interesting question, I mean, this is,

01:42:37 because I think you might say,

01:42:38 well, how could one possibly have a right

01:42:41 to have an opinion about something like that?

01:42:44 We don’t have any way of describing a process

01:42:46 that makes primes.

01:42:49 Sure, you can look at your computer

01:42:51 and see a lot of them, but the fact that there’s a lot,

01:42:53 why is that evidence that there’s infinitely many, right?

01:42:55 Maybe I can go on the computer and find 10 million.

01:42:57 Well, 10 million is pretty far from infinity, right?

01:42:59 So how is that evidence?

01:43:01 There’s a lot of things.

01:43:02 There’s like a lot more than 10 million atoms.

01:43:04 That doesn’t mean there’s infinitely many atoms

01:43:05 in the universe, right?

01:43:06 I mean, on most people’s physical theories,

01:43:07 there’s probably not, as I understand it.

01:43:10 Okay, so why would we think this?

01:43:13 The answer is that it turns out to be like incredibly

01:43:17 productive and enlightening to think about primes

01:43:21 as if they were random numbers,

01:43:23 as if they were randomly distributed

01:43:24 according to a certain law.

01:43:26 Now they’re not, they’re not random.

01:43:27 There’s no chance involved.

01:43:28 There it’s completely deterministic

01:43:30 whether a number is prime or not.

01:43:31 And yet it just turns out to be phenomenally useful

01:43:35 in mathematics to say,

01:43:38 even if something is governed by a deterministic law,

01:43:41 let’s just pretend it wasn’t.

01:43:43 Let’s just pretend that they were produced

01:43:44 by some random process and see if the behavior

01:43:46 is roughly the same.

01:43:47 And if it’s not, maybe change the random process,

01:43:49 maybe make the randomness a little bit different

01:43:51 and tweak it and see if you can find a random process

01:43:53 that matches the behavior we see.

01:43:55 And then maybe you predict that other behaviors

01:44:00 of the system are like that of the random process.

01:44:02 And so that’s kind of like, it’s funny

01:44:04 because I think when you talk to people

01:44:05 at the twin prime conjecture,

01:44:07 people think you’re saying,

01:44:09 wow, there’s like some deep structure there

01:44:12 that like makes those primes be like close together

01:44:15 again and again.

01:44:16 And no, it’s the opposite of deep structure.

01:44:18 What we say when we say we believe the twin prime conjecture

01:44:20 is that we believe the primes are like sort of

01:44:22 strewn around pretty randomly.

01:44:24 And if they were, then by chance,

01:44:26 you would expect there to be infinitely many twin primes.

01:44:28 And we’re saying, yeah, we expect them to behave

01:44:29 just like they would if they were random dirt.

01:44:33 The fascinating parallel here is,

01:44:36 I just got a chance to talk to Sam Harris

01:44:38 and he uses the prime numbers as an example.

01:44:41 Often, I don’t know if you’re familiar with who Sam is.

01:44:44 He uses that as an example of there being no free will.

01:44:50 Wait, where does he get this?

01:44:52 Well, he just uses as an example of,

01:44:54 it might seem like this is a random number generator,

01:44:58 but it’s all like formally defined.

01:45:01 So if we keep getting more and more primes,

01:45:05 then like that might feel like a new discovery

01:45:09 and that might feel like a new experience, but it’s not.

01:45:12 It was always written in the cards.

01:45:14 But it’s funny that you say that

01:45:15 because a lot of people think of like randomness,

01:45:19 the fundamental randomness within the nature of reality

01:45:23 might be the source of something

01:45:25 that we experience as free will.

01:45:27 And you’re saying it’s like useful to look at prime numbers

01:45:30 as a random process in order to prove stuff about them.

01:45:35 But fundamentally, of course, it’s not a random process.

01:45:38 Well, not in order to prove some stuff about them

01:45:40 so much as to figure out what we expect to be true

01:45:43 and then try to prove that.

01:45:44 Because here’s what you don’t want to do.

01:45:45 Try really hard to prove something that’s false.

01:45:48 That makes it really hard to prove the thing if it’s false.

01:45:51 So you certainly want to have some heuristic ways

01:45:53 of guessing, making good guesses about what’s true.

01:45:55 So yeah, here’s what I would say.

01:45:56 You’re going to be imaginary Sam Harris now.

01:45:58 Like you are talking about prime numbers

01:46:00 and you are like,

01:46:01 but prime numbers are completely deterministic.

01:46:04 And I’m saying like,

01:46:04 well, but let’s treat them like a random process.

01:46:06 And then you say,

01:46:08 but you’re just saying something that’s not true.

01:46:09 They’re not a random process, they’re deterministic.

01:46:10 And I’m like, okay, great.

01:46:11 You hold to your insistence that it’s not a random process.

01:46:13 Meanwhile, I’m generating insight about the primes

01:46:15 that you’re not because I’m willing to sort of pretend

01:46:17 that there’s something that they’re not

01:46:18 in order to understand what’s going on.

01:46:20 Yeah, so it doesn’t matter what the reality is.

01:46:22 What matters is what framework of thought

01:46:28 results in the maximum number of insights.

01:46:30 Yeah, because I feel, look, I’m sorry,

01:46:32 but I feel like you have more insights about people.

01:46:34 If you think of them as like beings that have wants

01:46:37 and needs and desires and do stuff on purpose,

01:46:40 even if that’s not true,

01:46:41 you still understand better what’s going on

01:46:43 by treating them in that way.

01:46:44 Don’t you find, look, when you work on machine learning,

01:46:46 don’t you find yourself sort of talking

01:46:48 about what the machine is trying to do

01:46:51 in a certain instance?

01:46:52 Do you not find yourself drawn to that language?

01:46:54 Well, it knows this, it’s trying to do that,

01:46:57 it’s learning that.

01:46:58 I’m certainly drawn to that language

01:47:00 to the point where I receive quite a bit of criticisms

01:47:03 for it because I, you know, like.

01:47:05 Oh, I’m on your side, man.

01:47:07 So especially in robotics, I don’t know why,

01:47:09 but robotics people don’t like to name their robots.

01:47:14 They certainly don’t like to gender their robots

01:47:17 because the moment you gender a robot,

01:47:18 you start to anthropomorphize.

01:47:20 If you say he or she, you start to,

01:47:22 in your mind, construct like a life story.

01:47:27 In your mind, you can’t help it.

01:47:29 There’s like, you create like a humorous story

01:47:31 to this person.

01:47:32 You start to, this person, this robot,

01:47:35 you start to project your own.

01:47:37 But I think that’s what we do to each other.

01:47:38 And I think that’s actually really useful

01:47:40 for the engineering process,

01:47:42 especially for human robot interaction.

01:47:44 And yes, for machine learning systems,

01:47:46 for helping you build an intuition

01:47:48 about a particular problem.

01:47:49 It’s almost like asking this question,

01:47:53 you know, when a machine learning system fails

01:47:55 in a particular edge case, asking like,

01:47:57 what were you thinking about?

01:47:59 Like, like asking, like almost like

01:48:02 when you’re talking about to a child

01:48:04 who just did something bad, you want to understand

01:48:08 like what was, how did they see the world?

01:48:12 Maybe there’s a totally new, maybe you’re the one

01:48:13 that’s thinking about the world incorrectly.

01:48:16 And yeah, that anthropomorphization process,

01:48:19 I think is ultimately good for insight.

01:48:21 And the same is, I agree with you.

01:48:23 I tend to believe about free will as well.

01:48:26 Let me ask you a ridiculous question, if it’s okay.

01:48:28 Of course.

01:48:30 I’ve just recently, most people go on like rabbit hole,

01:48:34 like YouTube things.

01:48:35 And I went on a rabbit hole often do of Wikipedia.

01:48:39 And I found a page on

01:48:43 finiteism, ultra finiteism and intuitionism

01:48:49 or into, I forget what it’s called.

01:48:51 Yeah, intuitionism.

01:48:52 Intuitionism.

01:48:53 That seemed pretty, pretty interesting.

01:48:55 I have it on my to do list actually like look into

01:48:58 like, is there people who like formally attract,

01:49:00 like real mathematicians are trying to argue for this.

01:49:03 But the belief there, I think, let’s say finiteism

01:49:07 that infinity is fake.

01:49:11 Meaning, infinity might be like a useful hack

01:49:16 for certain, like a useful tool in mathematics,

01:49:18 but it really gets us into trouble

01:49:22 because there’s no infinity in the real world.

01:49:26 Maybe I’m sort of not expressing that fully correctly,

01:49:30 but basically saying like there’s things

01:49:32 that once you add into mathematics,

01:49:37 things that are not provably within the physical world,

01:49:41 you’re starting to inject to corrupt your framework

01:49:45 of reason.

01:49:47 What do you think about that?

01:49:49 I mean, I think, okay, so first of all, I’m not an expert

01:49:51 and I couldn’t even tell you what the difference is

01:49:54 between those three terms, finiteism, ultra finiteism

01:49:58 and intuitionism, although I know they’re related

01:49:59 and I tend to associate them with the Netherlands

01:50:01 in the 1930s.

01:50:02 Okay, I’ll tell you, can I just quickly comment

01:50:04 because I read the Wikipedia page.

01:50:06 The difference in ultra.

01:50:07 That’s like the ultimate sentence of the modern age.

01:50:10 Can I just comment because I read the Wikipedia page.

01:50:12 That sums up our moment.

01:50:14 Bro, I’m basically an expert.

01:50:17 Ultra finiteism.

01:50:19 So, finiteism says that the only infinity

01:50:22 you’re allowed to have is that the natural numbers

01:50:25 are infinite.

01:50:27 So, like those numbers are infinite.

01:50:29 So, like one, two, three, four, five,

01:50:32 the integers are infinite.

01:50:35 The ultra finiteism says, nope, even that infinity is fake.

01:50:41 I’ll bet ultra finiteism came second.

01:50:43 I’ll bet it’s like when there’s like a hardcore scene

01:50:44 and then one guy’s like, oh, now there’s a lot of people

01:50:47 in the scene.

01:50:48 I have to find a way to be more hardcore

01:50:49 than the hardcore people.

01:50:50 It’s all back to the emo, Doc.

01:50:52 Okay, so is there any, are you ever,

01:50:54 because I’m often uncomfortable with infinity,

01:50:58 like psychologically.

01:50:59 I have trouble when that sneaks in there.

01:51:04 It’s because it works so damn well,

01:51:06 I get a little suspicious,

01:51:09 because it could be almost like a crutch

01:51:12 or an oversimplification that’s missing something profound

01:51:15 about reality.

01:51:17 Well, so first of all, okay, if you say like,

01:51:20 is there like a serious way of doing mathematics

01:51:24 that doesn’t really treat infinity as a real thing

01:51:29 or maybe it’s kind of agnostic

01:51:30 and it’s like, I’m not really gonna make a firm statement

01:51:32 about whether it’s a real thing or not.

01:51:33 Yeah, that’s called most of the history of mathematics.

01:51:36 So it’s only after Cantor that we really are sort of,

01:51:41 okay, we’re gonna like have a notion

01:51:43 of like the cardinality of an infinite set

01:51:45 and like do something that you might call

01:51:49 like the modern theory of infinity.

01:51:51 That said, obviously everybody was drawn to this notion

01:51:54 and no, not everybody was comfortable with it.

01:51:55 Look, I mean, this is what happens with Newton.

01:51:57 I mean, so Newton understands that to talk about tangents

01:52:01 and to talk about instantaneous velocity,

01:52:04 he has to do something that we would now call

01:52:06 taking a limit, right?

01:52:08 The fabled dy over dx, if you sort of go back

01:52:11 to your calculus class, for those who have taken calculus

01:52:13 and remember this mysterious thing.

01:52:14 And you know, what is it?

01:52:17 What is it?

01:52:18 Well, he’d say like, well, it’s like,

01:52:19 you sort of divide the length of this line segment

01:52:24 by the length of this other line segment.

01:52:25 And then you make them a little shorter

01:52:26 and you divide again.

01:52:27 And then you make them a little shorter

01:52:28 and you divide again.

01:52:28 And then you just keep on doing that

01:52:29 until they’re like infinitely short

01:52:30 and then you divide them again.

01:52:32 These quantities that are like, they’re not zero,

01:52:36 but they’re also smaller than any actual number,

01:52:42 these infinitesimals.

01:52:43 Well, people were queasy about it

01:52:46 and they weren’t wrong to be queasy about it, right?

01:52:48 From a modern perspective, it was not really well formed.

01:52:50 There’s this very famous critique of Newton

01:52:52 by Bishop Berkeley, where he says like,

01:52:54 what these things you define, like, you know,

01:52:57 they’re not zero, but they’re smaller than any number.

01:53:00 Are they the ghosts of departed quantities?

01:53:02 That was this like ultra burn of Newton.

01:53:06 And on the one hand, he was right.

01:53:10 It wasn’t really rigorous by modern standards.

01:53:11 On the other hand, like Newton was out there doing calculus

01:53:14 and other people were not, right?

01:53:15 It works, it works.

01:53:17 I think a sort of intuitionist view, for instance,

01:53:20 I would say would express serious doubt.

01:53:23 And by the way, it’s not just infinity.

01:53:25 It’s like saying, I think we would express serious doubt

01:53:28 that like the real numbers exist.

01:53:31 Now, most people are comfortable with the real numbers.

01:53:36 Well, computer scientists with floating point number,

01:53:39 I mean, floating point arithmetic.

01:53:42 That’s a great point, actually.

01:53:44 I think in some sense, this flavor of doing math,

01:53:48 saying we shouldn’t talk about things

01:53:51 that we cannot specify in a finite amount of time,

01:53:53 there’s something very computational in flavor about that.

01:53:55 And it’s probably not a coincidence

01:53:57 that it becomes popular in the 30s and 40s,

01:54:01 which is also like kind of like the dawn of ideas

01:54:04 about formal computation, right?

01:54:06 You probably know the timeline better than I do.

01:54:07 Sorry, what becomes popular?

01:54:09 These ideas that maybe we should be doing math

01:54:12 in this more restrictive way where even a thing that,

01:54:16 because look, the origin of all this is like,

01:54:18 number represents a magnitude, like the length of a line.

01:54:22 So I mean, the idea that there’s a continuum,

01:54:26 there’s sort of like, it’s pretty old,

01:54:30 but just because something is old

01:54:31 doesn’t mean we can’t reject it if we want to.

01:54:34 Well, a lot of the fundamental ideas in computer science,

01:54:36 when you talk about the complexity of problems,

01:54:41 to Turing himself, they rely on an infinity as well.

01:54:45 The ideas that kind of challenge that,

01:54:47 the whole space of machine learning,

01:54:48 I would say, challenges that.

01:54:51 It’s almost like the engineering approach to things,

01:54:53 like the floating point arithmetic.

01:54:54 The other one that, back to John Conway,

01:54:57 that challenges this idea,

01:55:00 I mean, maybe to tie in the ideas of deformation theory

01:55:06 and limits to infinity is this idea of cellular automata

01:55:13 with John Conway looking at the game of life,

01:55:17 Stephen Wolfram’s work,

01:55:19 that I’ve been a big fan of for a while, cellular automata.

01:55:22 I was wondering if you have,

01:55:23 if you have ever encountered these kinds of objects,

01:55:26 you ever looked at them as a mathematician,

01:55:29 where you have very simple rules of tiny little objects

01:55:34 that when taken as a whole create incredible complexities,

01:55:37 but are very difficult to analyze,

01:55:39 very difficult to make sense of,

01:55:41 even though the one individual object, one part,

01:55:45 it’s like what we were saying about Andrew Wiles,

01:55:47 you can look at the deformation of a small piece

01:55:49 to tell you about the whole.

01:55:51 It feels like with cellular automata

01:55:54 or any kind of complex systems,

01:55:57 it’s often very difficult to say something

01:55:59 about the whole thing,

01:56:01 even when you can precisely describe the operation

01:56:05 of the local neighborhoods.

01:56:09 Yeah, I mean, I love that subject.

01:56:10 I haven’t really done research on it myself.

01:56:12 I’ve played around with it.

01:56:13 I’ll send you a fun blog post I wrote

01:56:15 where I made some cool texture patterns

01:56:17 from cellular automata that I, but.

01:56:20 And those are really always compelling

01:56:22 is like you create simple rules

01:56:24 and they create some beautiful textures.

01:56:25 It doesn’t make any sense.

01:56:26 Actually, did you see, there was a great paper.

01:56:28 I don’t know if you saw this,

01:56:28 like a machine learning paper.

01:56:30 Yes.

01:56:31 I don’t know if you saw the one I’m talking about

01:56:32 where they were like learning the texture

01:56:33 as like let’s try to like reverse engineer

01:56:35 and like learn a cellular automaton

01:56:37 that can reduce texture that looks like this

01:56:39 from the images.

01:56:40 Very cool.

01:56:41 And as you say, the thing you said is I feel the same way

01:56:44 when I read machine learning paper

01:56:45 is that what’s especially interesting

01:56:47 is the cases where it doesn’t work.

01:56:49 Like what does it do when it doesn’t do the thing

01:56:51 that you tried to train it to do?

01:56:53 That’s extremely interesting.

01:56:54 Yeah, yeah, that was a cool paper.

01:56:56 So yeah, so let’s start with the game of life.

01:56:58 Let’s start with, or let’s start with John Conway.

01:57:02 So Conway.

01:57:03 So yeah, so let’s start with John Conway again.

01:57:06 Just, I don’t know, from my outsider’s perspective,

01:57:08 there’s not many mathematicians that stand out

01:57:11 throughout the history of the 20th century.

01:57:13 And he’s one of them.

01:57:15 I feel like he’s not sufficiently recognized.

01:57:18 I think he’s pretty recognized.

01:57:20 Okay, well.

01:57:21 I mean, he was a full professor at Princeton

01:57:24 for most of his life.

01:57:25 He was sort of certainly at the pinnacle of.

01:57:27 Yeah, but I found myself every time I talk about Conway

01:57:30 and how excited I am about him,

01:57:33 I have to constantly explain to people who he is.

01:57:36 And that’s always a sad sign to me.

01:57:39 But that’s probably true for a lot of mathematicians.

01:57:41 I was about to say,

01:57:42 I feel like you have a very elevated idea of how famous.

01:57:44 This is what happens when you grow up in the Soviet Union

01:57:46 or you think the mathematicians are like very, very famous.

01:57:49 Yeah, but I’m not actually so convinced at a tiny tangent

01:57:53 that that shouldn’t be so.

01:57:54 I mean, there’s, it’s not obvious to me

01:57:57 that that’s one of the,

01:57:59 like if I were to analyze American society,

01:58:01 that perhaps elevating mathematical and scientific thinking

01:58:05 to a little bit higher level would benefit the society.

01:58:08 Well, both in discovering the beauty of what it is

01:58:11 to be human and for actually creating cool technology,

01:58:15 better iPhones.

01:58:16 But anyway, John Conway.

01:58:18 Yeah, and Conway is such a perfect example

01:58:20 of somebody whose humanity was,

01:58:22 and his personality was like wound up

01:58:24 with his mathematics, right?

01:58:25 And so it’s not, sometimes I think people

01:58:26 who are outside the field think of mathematics

01:58:28 as this kind of like cold thing that you do

01:58:31 separate from your existence as a human being.

01:58:33 No way, your personality is in there,

01:58:34 just as it would be in like a novel you wrote

01:58:37 or a painting you painted

01:58:38 or just like the way you walk down the street.

01:58:40 Like it’s in there, it’s you doing it.

01:58:41 And Conway was certainly a singular personality.

01:58:46 I think anybody would say that he was playful,

01:58:50 like everything was a game to him.

01:58:54 Now, what you might think I’m gonna say,

01:58:56 and it’s true is that he sort of was very playful

01:58:59 in his way of doing mathematics,

01:59:01 but it’s also true, it went both ways.

01:59:03 He also sort of made mathematics out of games.

01:59:06 He like looked at, he was a constant inventor of games

01:59:08 or like crazy names.

01:59:10 And then he would sort of analyze those games mathematically

01:59:15 to the point that he,

01:59:16 and then later collaborating with Knuth like,

01:59:19 created this number system, the serial numbers

01:59:22 in which actually each number is a game.

01:59:25 There’s a wonderful book about this called,

01:59:26 I mean, there are his own books.

01:59:27 And then there’s like a book that he wrote

01:59:28 with Berlekamp and Guy called Winning Ways,

01:59:31 which is such a rich source of ideas.

01:59:35 And he too kind of has his own crazy number system

01:59:41 in which by the way, there are these infinitesimals,

01:59:44 the ghosts of departed quantities.

01:59:45 They’re in there now, not as ghosts,

01:59:47 but as like certain kind of two player games.

01:59:53 So, he was a guy, so I knew him when I was a postdoc

02:00:00 and I knew him at Princeton

02:00:01 and our research overlapped in some ways.

02:00:03 Now it was on stuff that he had worked on many years before.

02:00:05 The stuff I was working on kind of connected

02:00:07 with stuff in group theory,

02:00:08 which somehow seems to keep coming up.

02:00:13 And so I often would like sort of ask him a question.

02:00:16 I would sort of come upon him in the common room

02:00:17 and I would ask him a question about something.

02:00:19 And just anytime you turned him on, you know what I mean?

02:00:23 You sort of asked the question,

02:00:25 it was just like turning a knob and winding him up

02:00:28 and he would just go and you would get a response

02:00:31 that was like so rich and went so many places

02:00:35 and taught you so much.

02:00:37 And usually had nothing to do with your question.

02:00:40 Usually your question was just a prompt to him.

02:00:43 You couldn’t count on actually getting the question answered.

02:00:44 Yeah, those brilliant, curious minds even at that age.

02:00:47 Yeah, it was definitely a huge loss.

02:00:51 But on his game of life,

02:00:54 which was I think he developed in the 70s

02:00:56 as almost like a side thing, a fun little experiment.

02:00:59 His game of life is this, it’s a very simple algorithm.

02:01:05 It’s not really a game per se

02:01:07 in the sense of the kinds of games that he liked

02:01:09 where people played against each other.

02:01:12 But essentially it’s a game that you play

02:01:16 with marking little squares on the sheet of graph paper.

02:01:20 And in the 70s, I think he was like literally doing it

02:01:22 with like a pen on graph paper.

02:01:24 You have some configuration of squares.

02:01:26 Some of the squares in the graph paper are filled in,

02:01:28 some are not.

02:01:29 And there’s a rule, a single rule that tells you

02:01:33 at the next stage, which squares are filled in

02:01:36 and which squares are not.

02:01:38 Sometimes an empty square gets filled in,

02:01:39 that’s called birth.

02:01:40 Sometimes a square that’s filled in gets erased,

02:01:43 that’s called death.

02:01:43 And there’s rules for which squares are born

02:01:45 and which squares die.

02:01:50 The rule is very simple.

02:01:51 You can write it on one line.

02:01:53 And then the great miracle is that you can start

02:01:56 from some very innocent looking little small set of boxes

02:02:00 and get these results of incredible richness.

02:02:04 And of course, nowadays you don’t do it on paper.

02:02:05 Nowadays you do it in a computer.

02:02:07 There’s actually a great iPad app called Golly,

02:02:09 which I really like that has like Conway’s original rule

02:02:12 and like, gosh, like hundreds of other variants

02:02:15 and it’s a lightning fast.

02:02:16 So you can just be like,

02:02:17 I wanna see 10,000 generations of this rule play out

02:02:21 like faster than your eye can even follow.

02:02:23 And it’s like amazing.

02:02:24 So I highly recommend it if this is at all intriguing to you

02:02:26 getting Golly on your iOS device.

02:02:29 And you can do this kind of process,

02:02:30 which I really enjoy doing,

02:02:32 which is almost from like putting a Darwin hat on

02:02:35 or a biologist hat on and doing analysis

02:02:38 of a higher level of abstraction,

02:02:41 like the organisms that spring up.

02:02:43 Cause there’s different kinds of organisms.

02:02:45 Like you can think of them as species

02:02:46 and they interact with each other.

02:02:48 They can, there’s gliders, they shoot different,

02:02:51 there’s like things that can travel around.

02:02:54 There’s things that can,

02:02:55 glider guns that can generate those gliders.

02:02:59 You can use the same kind of language

02:03:01 as you would about describing a biological system.

02:03:04 So it’s a wonderful laboratory

02:03:06 and it’s kind of a rebuke to someone

02:03:07 who doesn’t think that like very, very rich,

02:03:10 complex structure can come from very simple underlying laws.

02:03:16 Like it definitely can.

02:03:18 Now, here’s what’s interesting.

02:03:20 If you just pick like some random rule,

02:03:24 you wouldn’t get interesting complexity.

02:03:26 I think that’s one of the most interesting things

02:03:28 of these, one of these most interesting features

02:03:31 of this whole subject,

02:03:32 that the rules have to be tuned just right.

02:03:34 Like a sort of typical rule set

02:03:36 doesn’t generate any kind of interesting behavior.

02:03:38 But some do.

02:03:40 And I don’t think we have a clear way of understanding

02:03:44 which do and which don’t.

02:03:45 Maybe Steven thinks he does, I don’t know.

02:03:47 No, no, it’s a giant mystery where Steven Wolfram did is,

02:03:53 now there’s a whole interesting aspect to the fact

02:03:56 that he’s a little bit of an outcast

02:03:57 in the mathematics and physics community

02:03:59 because he’s so focused on a particular,

02:04:02 his particular work.

02:04:03 I think if you put ego aside,

02:04:05 which I think unfairly some people

02:04:08 are not able to look beyond,

02:04:09 I think his work is actually quite brilliant.

02:04:11 But what he did is exactly this process

02:04:13 of Darwin like exploration.

02:04:15 He’s taking these very simple ideas

02:04:17 and writing a thousand page book on them,

02:04:19 meaning like, let’s play around with this thing.

02:04:22 Let’s see.

02:04:23 And can we figure anything out?

02:04:25 Spoiler alert, no, we can’t.

02:04:28 In fact, he does a challenge.

02:04:31 I think it’s like rule 30 challenge,

02:04:33 which is quite interesting,

02:04:34 just simply for machine learning people,

02:04:36 for mathematics people,

02:04:39 is can you predict the middle column?

02:04:41 For his, it’s a 1D cellular automata.

02:04:45 Can you, generally speaking,

02:04:48 can you predict anything about

02:04:50 how a particular rule will evolve just in the future?

02:04:55 Very simple.

02:04:56 Just looking at one particular part of the world,

02:04:59 just zooming in on that part,

02:05:02 100 steps ahead, can you predict something?

02:05:04 And the challenge is to do that kind of prediction

02:05:08 so far as nobody’s come up with an answer.

02:05:10 But the point is like, we can’t.

02:05:13 We don’t have tools or maybe it’s impossible or,

02:05:16 I mean, he has these kind of laws of irreducibility

02:05:19 that he refers to, but it’s poetry.

02:05:21 It’s like, we can’t prove these things.

02:05:22 It seems like we can’t.

02:05:24 That’s the basic.

02:05:26 It almost sounds like ancient mathematics

02:05:28 or something like that, where you’re like,

02:05:30 the gods will not allow us to predict the cellular automata.

02:05:34 But that’s fascinating that we can’t.

02:05:37 I’m not sure what to make of it.

02:05:39 And there’s power to calling this particular set of rules

02:05:43 game of life as Conway did, because not exactly sure,

02:05:47 but I think he had a sense that there’s some core ideas here

02:05:51 that are fundamental to life, to complex systems,

02:05:55 to the way life emerge on earth.

02:05:59 I’m not sure I think Conway thought that.

02:06:01 It’s something that, I mean, Conway always had

02:06:03 a rather ambivalent relationship with the game of life

02:06:05 because I think he saw it as,

02:06:11 it was certainly the thing he was most famous for

02:06:12 in the outside world.

02:06:14 And I think that he, his view, which is correct,

02:06:18 is that he had done things

02:06:19 that were much deeper mathematically than that.

02:06:22 And I think it always aggrieved him a bit

02:06:24 that he was the game of life guy

02:06:26 when he proved all these wonderful theorems

02:06:28 and created all these wonderful games,

02:06:32 created the serial numbers.

02:06:33 I mean, he was a very tireless guy

02:06:36 who just did an incredibly variegated array of stuff.

02:06:40 So he was exactly the kind of person

02:06:42 who you would never want to reduce to one achievement.

02:06:45 You know what I mean?

02:06:46 Let me ask you about group theory.

02:06:50 You mentioned it a few times.

02:06:51 What is group theory?

02:06:53 What is an idea from group theory that you find beautiful?

02:06:58 Well, so I would say group theory sort of starts

02:07:01 as the general theory of symmetries,

02:07:04 that people looked at different kinds of things

02:07:08 and said, as we said, oh, it could have,

02:07:12 maybe all there is is symmetry from left to right,

02:07:16 like a human being, right?

02:07:17 That’s roughly bilaterally symmetric, as we say.

02:07:21 So there’s two symmetries.

02:07:23 And then you’re like, well, wait, didn’t I say

02:07:24 there’s just one, there’s just left to right?

02:07:26 Well, we always count the symmetry of doing nothing.

02:07:30 We always count the symmetry

02:07:31 that’s like there’s flip and don’t flip.

02:07:33 Those are the two configurations that you can be in.

02:07:35 So there’s two.

02:07:37 You know, something like a rectangle

02:07:40 is bilaterally symmetric.

02:07:41 You can flip it left to right,

02:07:42 but you can also flip it top to bottom.

02:07:45 So there’s actually four symmetries.

02:07:47 There’s do nothing, flip it left to right

02:07:50 and flip it top to bottom or do both of those things.

02:07:52 And then a square, there’s even more,

02:07:59 because now you can rotate it.

02:08:01 You can rotate it by 90 degrees.

02:08:03 So you can’t do that.

02:08:03 That’s not a symmetry of the rectangle.

02:08:04 If you try to rotate it 90 degrees,

02:08:06 you get a rectangle oriented in a different way.

02:08:08 So a person has two symmetries,

02:08:11 a rectangle four, a square eight,

02:08:14 different kinds of shapes

02:08:15 have different numbers of symmetries.

02:08:18 And the real observation is that

02:08:19 that’s just not like a set of things, they can be combined.

02:08:25 You do one symmetry, then you do another.

02:08:27 The result of that is some third symmetry.

02:08:31 So a group really abstracts away this notion of saying,

02:08:38 it’s just some collection of transformations

02:08:41 you can do to a thing

02:08:42 where you combine any two of them to get a third.

02:08:44 So, you know, a place where this comes up

02:08:45 in computer science is in sorting,

02:08:48 because the ways of permuting a set,

02:08:50 the ways of taking sort of some set of things

02:08:52 you have on the table

02:08:53 and putting them in a different order,

02:08:54 shuffling a deck of cards, for instance,

02:08:56 those are the symmetries of the deck.

02:08:57 And there’s a lot of them.

02:08:58 There’s not two, there’s not four, there’s not eight.

02:09:00 Think about how many different orders

02:09:01 the deck of card can be in.

02:09:02 Each one of those is the result of applying a symmetry

02:09:06 to the original deck.

02:09:07 So a shuffle is a symmetry, right?

02:09:09 You’re reordering the cards.

02:09:10 If I shuffle and then you shuffle,

02:09:12 the result is some other kind of thing.

02:09:16 You might call it a double shuffle,

02:09:17 which is a more complicated symmetry.

02:09:19 So group theory is kind of the study

02:09:22 of the general abstract world

02:09:24 that encompasses all these kinds of things.

02:09:27 But then of course, like lots of things

02:09:29 that are way more complicated than that.

02:09:31 Like infinite groups of symmetries, for instance.

02:09:33 So they can be infinite, huh?

02:09:35 Oh yeah.

02:09:35 Okay.

02:09:36 Well, okay, ready?

02:09:37 Think about the symmetries of the line.

02:09:41 You’re like, okay, I can reflect it left to right,

02:09:45 you know, around the origin.

02:09:46 Okay, but I could also reflect it left to right,

02:09:49 grabbing somewhere else, like at one or two

02:09:52 or pi or anywhere.

02:09:54 Or I could just slide it some distance.

02:09:56 That’s a symmetry.

02:09:57 Slide it five units over.

02:09:58 So there’s clearly infinitely many symmetries of the line.

02:10:01 That’s an example of an infinite group of symmetries.

02:10:03 Is it possible to say something that kind of captivates,

02:10:06 keeps being brought up by physicists,

02:10:09 which is gauge theory, gauge symmetry,

02:10:12 as one of the more complicated type of symmetries?

02:10:14 Is there an easy explanation of what the heck it is?

02:10:18 Is that something that comes up on your mind at all?

02:10:21 Well, I’m not a mathematical physicist,

02:10:23 but I can say this.

02:10:24 It is certainly true that it has been a very useful notion

02:10:29 in physics to try to say like,

02:10:31 what are the symmetry groups of the world?

02:10:34 Like what are the symmetries

02:10:35 under which things don’t change, right?

02:10:36 So we just, I think we talked a little bit earlier

02:10:39 about it should be a basic principle

02:10:40 that a theorem that’s true here is also true over there.

02:10:44 And same for a physical law, right?

02:10:45 I mean, if gravity is like this over here,

02:10:47 it should also be like this over there.

02:10:49 Okay, what that’s saying is we think translation in space

02:10:52 should be a symmetry.

02:10:54 All the laws of physics should be unchanged

02:10:56 if the symmetry we have in mind

02:10:57 is a very simple one like translation.

02:10:59 And so then there becomes a question,

02:11:03 like what are the symmetries of the actual world

02:11:07 with its physical laws?

02:11:09 And one way of thinking, this isn’t oversimplification,

02:11:12 but like one way of thinking of this big shift

02:11:18 from before Einstein to after

02:11:22 is that we just changed our idea

02:11:25 about what the fundamental group of symmetries were.

02:11:29 So that things like the Lorenz contraction,

02:11:31 things like these bizarre relativistic phenomenon

02:11:34 or Lorenz would have said, oh, to make this work,

02:11:37 we need a thing to change its shape

02:11:44 if it’s moving nearly the speed of light.

02:11:47 Well, under the new framework, it’s much better.

02:11:50 You say, oh, no, it wasn’t changing its shape.

02:11:51 You were just wrong about what counted as a symmetry.

02:11:54 Now that we have this new group,

02:11:55 the so called Lorenz group,

02:11:57 now that we understand what the symmetries really are,

02:11:59 we see it was just an illusion

02:12:00 that the thing was changing its shape.

02:12:02 Yeah, so you can then describe the sameness of things

02:12:05 under this weirdness that is general relativity,

02:12:08 for example.

02:12:10 Yeah, yeah, still, I wish there was a simpler explanation

02:12:16 of like exact, I mean, gauge symmetries,

02:12:19 pretty simple general concept about rulers being deformed.

02:12:26 I’ve actually just personally been on a search,

02:12:31 not a very rigorous or aggressive search,

02:12:34 but for something I personally enjoy,

02:12:37 which is taking complicated concepts

02:12:40 and finding the sort of minimal example

02:12:44 that I can play around with, especially programmatically.

02:12:47 That’s great, I mean,

02:12:48 this is what we try to train our students to do, right?

02:12:50 I mean, in class, this is exactly what,

02:12:52 this is like best pedagogical practice.

02:12:54 I do hope there’s simple explanation,

02:12:57 especially like I’ve in my sort of drunk random walk,

02:13:02 drunk walk, whatever that’s called,

02:13:04 sometimes stumble into the world of topology

02:13:08 and like quickly, like, you know when you go into a party

02:13:11 and you realize this is not the right party for me?

02:13:14 It’s, so whenever I go into topology,

02:13:16 it’s like so much math everywhere.

02:13:20 I don’t even know what, it feels like this is me

02:13:23 like being a hater, I think there’s way too much math.

02:13:25 Like there are two, the cool kids who just want to have,

02:13:29 like everything is expressed through math.

02:13:31 Because they’re actually afraid to express stuff

02:13:33 simply through language.

02:13:34 That’s my hater formulation of topology.

02:13:37 But at the same time, I’m sure that’s very necessary

02:13:39 to do sort of rigorous discussion.

02:13:41 But I feel like.

02:13:42 But don’t you think that’s what gauge symmetry is like?

02:13:44 I mean, it’s not a field I know well,

02:13:45 but it certainly seems like.

02:13:46 Yes, it is like that.

02:13:47 But my problem with topology, okay,

02:13:50 and even like differential geometry is like,

02:13:55 you’re talking about beautiful things.

02:13:59 Like if they could be visualized, it’s open question

02:14:02 if everything could be visualized,

02:14:03 but you’re talking about things

02:14:05 that can be visually stunning, I think.

02:14:09 But they are hidden underneath all of that math.

02:14:13 Like if you look at the papers that are written

02:14:16 in topology, if you look at all the discussions

02:14:18 on Stack Exchange, they’re all math dense, math heavy.

02:14:22 And the only kind of visual things

02:14:25 that emerge every once in a while,

02:14:27 is like something like a Mobius strip.

02:14:30 Every once in a while, some kind of simple visualizations.

02:14:33 Every once in a while, some kind of simple visualizations.

02:14:36 Every once in a while, some kind of simple visualizations.

02:14:37 Well, there’s the vibration, there’s the hop vibration

02:14:40 or all those kinds of things that somebody,

02:14:42 some grad student from like 20 years ago

02:14:45 wrote a program in Fortran to visualize it, and that’s it.

02:14:48 And it’s just, you know, it’s makes me sad

02:14:51 because those are visual disciplines.

02:14:53 Just like computer vision is a visual discipline.

02:14:56 So you can provide a lot of visual examples.

02:14:59 I wish topology was more excited

02:15:03 and in love with visualizing some of the ideas.

02:15:07 I mean, you could say that, but I would say for me,

02:15:09 a picture of the hop vibration does nothing for me.

02:15:11 Whereas like when you’re like, oh,

02:15:13 it’s like about the quaternions.

02:15:14 It’s like a subgroup of the quaternions.

02:15:16 And I’m like, oh, so now I see what’s going on.

02:15:17 Like, why didn’t you just say that?

02:15:18 Why were you like showing me this stupid picture

02:15:20 instead of telling me what you were talking about?

02:15:22 Oh, yeah, yeah.

02:15:25 I’m just saying, no, but it goes back

02:15:26 to what you were saying about teaching

02:15:27 that like people are different in what they’ll respond to.

02:15:29 So I think there’s no, I mean, I’m very opposed

02:15:32 to the idea that there’s a one right way to explain things.

02:15:34 I think there’s like a huge variation in like, you know,

02:15:37 our brains like have all these like weird like hooks

02:15:40 and loops and it’s like very hard to know

02:15:42 like what’s gonna latch on

02:15:43 and it’s not gonna be the same thing for everybody.

02:15:46 So I think monoculture is bad, right?

02:15:49 I think that’s, and I think we’re agreeing on that point

02:15:51 that like, it’s good that there’s like a lot

02:15:53 of different ways in and a lot of different ways

02:15:55 to describe these ideas because different people

02:15:57 are gonna find different things illuminating.

02:15:59 But that said, I think there’s a lot to be discovered

02:16:04 when you force little like silos of brilliant people

02:16:11 to kind of find a middle ground

02:16:15 or like aggregate or come together in a way.

02:16:20 So there’s like people that do love visual things.

02:16:23 I mean, there’s a lot of disciplines,

02:16:25 especially in computer science

02:16:27 that they’re obsessed with visualizing,

02:16:28 visualizing data, visualizing neural networks.

02:16:31 I mean, neural networks themselves are fundamentally visual.

02:16:34 There’s a lot of work in computer vision that’s very visual.

02:16:36 And then coming together with some folks

02:16:39 that were like deeply rigorous

02:16:41 and are like totally lost in multi dimensional space

02:16:43 where it’s hard to even bring them back down to 3D.

02:16:48 They’re very comfortable in this multi dimensional space.

02:16:50 So forcing them to kind of work together to communicate

02:16:53 because it’s not just about public communication of ideas.

02:16:57 It’s also, I feel like when you’re forced

02:16:59 to do that public communication like you did with your book,

02:17:02 I think deep profound ideas can be discovered

02:17:05 that’s like applicable for research and for science.

02:17:08 Like there’s something about that simplification

02:17:10 or not simplification, but distillation or condensation

02:17:15 or whatever the hell you call it,

02:17:17 compression of ideas that somehow

02:17:19 actually stimulates creativity.

02:17:22 And I’d be excited to see more of that

02:17:25 in the mathematics community.

02:17:27 Can you?

02:17:28 Let me make a crazy metaphor.

02:17:29 Maybe it’s a little bit like the relation

02:17:31 between prose and poetry, right?

02:17:32 I mean, if you, you might say like,

02:17:33 why do we need anything more than prose?

02:17:35 You’re trying to convey some information.

02:17:36 So you just like say it.

02:17:38 Well, poetry does something, right?

02:17:40 It’s sort of, you might think of it as a kind of compression.

02:17:43 Of course, not all poetry is compressed.

02:17:44 Like not all, some of it is quite baggy,

02:17:47 but like you are kind of, often it’s compressed, right?

02:17:53 A lyric poem is often sort of like a compression

02:17:55 of what would take a long time

02:17:57 and be complicated to explain in prose

02:18:00 into sort of a different mode

02:18:03 that is gonna hit in a different way.

02:18:05 We talked about Poincare conjecture.

02:18:10 There’s a guy, he’s Russian, Grigori Perlman.

02:18:14 He proved Poincare’s conjecture.

02:18:16 If you can comment on the proof itself,

02:18:19 if that stands out to you as something interesting

02:18:21 or the human story of it,

02:18:23 which is he turned down the field’s metal for the proof.

02:18:28 Is there something you find inspiring or insightful

02:18:32 about the proof itself or about the man?

02:18:36 Yeah, I mean, one thing I really like about the proof

02:18:40 and partly that’s because it’s sort of a thing

02:18:42 that happens again and again in this book.

02:18:45 I mean, I’m writing about geometry and the way

02:18:46 it sort of appears in all these kind of real world problems.

02:18:50 But it happens so often that the geometry

02:18:52 you think you’re studying is somehow not enough.

02:18:56 You have to go one level higher in abstraction

02:18:59 and study a higher level of geometry.

02:19:01 And the way that plays out is that Poincare asks a question

02:19:05 about a certain kind of three dimensional object.

02:19:07 Is it the usual three dimensional space that we know

02:19:10 or is it some kind of exotic thing?

02:19:13 And so, of course, this sounds like it’s a question

02:19:15 about the geometry of the three dimensional space,

02:19:17 but no, Perelman understands.

02:19:20 And by the way, in a tradition that involves

02:19:21 Richard Hamilton and many other people,

02:19:23 like most really important mathematical advances,

02:19:26 this doesn’t happen alone.

02:19:27 It doesn’t happen in a vacuum.

02:19:28 It happens as the culmination of a program

02:19:30 that involves many people.

02:19:31 Same with Wiles, by the way.

02:19:32 I mean, we talked about Wiles and I wanna emphasize

02:19:34 that starting all the way back with Kummer,

02:19:36 who I mentioned in the 19th century,

02:19:38 but Gerhard Frey and Mazer and Ken Ribbit

02:19:42 and like many other people are involved

02:19:45 in building the other pieces of the arch

02:19:47 before you put the keystone in.

02:19:48 We stand on the shoulders of giants.

02:19:50 Yes.

02:19:53 So, what is this idea?

02:19:56 The idea is that, well, of course,

02:19:57 the geometry of the three dimensional object itself

02:19:59 is relevant, but the real geometry you have to understand

02:20:02 is the geometry of the space

02:20:04 of all three dimensional geometries.

02:20:07 Whoa, you’re going up a higher level.

02:20:10 Because when you do that, you can say,

02:20:12 now let’s trace out a path in that space.

02:20:18 There’s a mechanism called Ricci flow.

02:20:19 And again, we’re outside my research area.

02:20:21 So for all the geometric analysts

02:20:23 and differential geometers out there listening to this,

02:20:25 if I, please, I’m doing my best and I’m roughly saying it.

02:20:29 So the Ricci flow allows you to say like,

02:20:32 okay, let’s start from some mystery three dimensional space,

02:20:35 which Poincare would conjecture is essentially

02:20:37 the same thing as our familiar three dimensional space,

02:20:39 but we don’t know that.

02:20:41 And now you let it flow.

02:20:44 You sort of like let it move in its natural path

02:20:47 according to some almost physical process

02:20:50 and ask where it winds up.

02:20:51 And what you find is that it always winds up.

02:20:54 You’ve continuously deformed it.

02:20:55 There’s that word deformation again.

02:20:58 And what you can prove is that the process doesn’t stop

02:21:00 until you get to the usual three dimensional space.

02:21:02 And since you can get from the mystery thing

02:21:04 to the standard space by this process

02:21:06 of continually changing and never kind of

02:21:09 having any sharp transitions,

02:21:12 then the original shape must’ve been the same

02:21:16 as the standard shape.

02:21:17 That’s the nature of the proof.

02:21:18 Now, of course, it’s incredibly technical.

02:21:20 I think as I understand it,

02:21:21 I think the hard part is proving

02:21:23 that the favorite word of AI people,

02:21:25 you don’t get any singularities along the way.

02:21:29 But of course, in this context,

02:21:30 singularity just means acquiring a sharp kink.

02:21:34 It just means becoming non smooth at some point.

02:21:37 So just saying something interesting about formal,

02:21:41 about the smooth trajectory

02:21:42 through this weird space of geometries.

02:21:45 But yeah, so what I like about it

02:21:46 is that it’s just one of many examples of where

02:21:49 it’s not about the geometry you think it’s about.

02:21:51 It’s about the geometry of all geometries, so to speak.

02:21:55 And it’s only by kind of like being jerked out of flatland.

02:21:59 Same idea.

02:22:00 It’s only by sort of seeing the whole thing globally at once

02:22:04 that you can really make progress on understanding

02:22:05 the one thing you thought you were looking at.

02:22:08 It’s a romantic question,

02:22:09 but what do you think about him

02:22:11 turning down the Fields Medal?

02:22:12 Is that just, are Nobel Prizes and Fields Medals

02:22:17 just the cherry on top of the cake

02:22:19 and really math itself, the process of curiosity,

02:22:25 of pulling at the string of the mystery before us?

02:22:28 That’s the cake?

02:22:29 And then the awards are just icing

02:22:33 and clearly I’ve been fasting and I’m hungry,

02:22:37 but do you think it’s tragic or just a little curiosity

02:22:44 that he turned down the medal?

02:22:46 Well, it’s interesting because on the one hand,

02:22:48 I think it’s absolutely true that right,

02:22:50 in some kind of like vast spiritual sense,

02:22:55 like awards are not important,

02:22:57 like not important the way that sort of like

02:22:59 understanding the universe is important.

02:23:01 On the other hand, most people who are offered that prize

02:23:04 accept it, so there’s something unusual

02:23:07 about his choice there.

02:23:11 I wouldn’t say I see it as tragic.

02:23:14 I mean, maybe if I don’t really feel like

02:23:16 I have a clear picture of why he chose not to take it.

02:23:19 I mean, he’s not alone in doing things like this.

02:23:22 People sometimes turn down prizes for ideological reasons,

02:23:26 but probably more often in mathematics.

02:23:28 I mean, I think I’m right in saying that

02:23:30 Peter Schultz turned down sort of some big monetary prize

02:23:33 because he just, you know, I mean, I think he,

02:23:36 at some point you have plenty of money

02:23:39 and maybe you think it sends the wrong message

02:23:41 about what the point of doing mathematics is.

02:23:45 I do find that there’s most people accept.

02:23:47 You know, most people give it a prize.

02:23:48 Most people take it.

02:23:49 I mean, people like to be appreciated,

02:23:50 but like I said, we’re people.

02:23:53 Not that different from most other people.

02:23:54 But the important reminder that that turning down

02:23:57 a prize serves for me is not that there’s anything wrong

02:24:01 with the prize and there’s something wonderful

02:24:03 about the prize, I think.

02:24:04 The Nobel prize is trickier

02:24:07 because so many Nobel prizes are given.

02:24:10 First of all, the Nobel prize often forgets

02:24:12 many, many of the important people throughout history.

02:24:15 Second of all, there’s like these weird rules to it

02:24:18 that it’s only three people

02:24:20 and some projects have a huge number of people.

02:24:22 And it’s like this, it, I don’t know.

02:24:26 It doesn’t kind of highlight the way science is done

02:24:31 on some of these projects in the best possible way.

02:24:33 But in general, the prizes are great.

02:24:34 But what this kind of teaches me and reminds me

02:24:37 is sometimes in your life, there’ll be moments

02:24:41 when the thing that you would really like to do,

02:24:47 society would really like you to do,

02:24:50 is the thing that goes against something you believe in,

02:24:53 whatever that is, some kind of principle.

02:24:56 And standing your ground in the face of that

02:24:59 is something I believe most people will have

02:25:03 a few moments like that in their life,

02:25:05 maybe one moment like that, and you have to do it.

02:25:07 That’s what integrity is.

02:25:09 So like, it doesn’t have to make sense

02:25:10 to the rest of the world, but to stand on that,

02:25:12 like to say no, it’s interesting, because I think.

02:25:16 But do you know that he turned down the prize

02:25:17 in service of some principle?

02:25:20 Because I don’t know that.

02:25:20 Well, yes, that seems to be the inkling,

02:25:22 but he has never made it super clear.

02:25:24 But the inkling is that he had some problems

02:25:26 with the whole process of mathematics that includes awards,

02:25:30 like this hierarchies and the reputations

02:25:33 and all those kinds of things,

02:25:34 and individualism that’s fundamental to American culture.

02:25:37 He probably, because he visited the United States quite a bit

02:25:41 that he probably, it’s all about experiences.

02:25:47 And he may have had some parts of academia,

02:25:51 some pockets of academia can be less than inspiring,

02:25:54 perhaps sometimes, because of the individual egos involved,

02:25:57 not academia, people in general, smart people with egos.

02:26:01 And if you interact with a certain kinds of people,

02:26:05 you can become cynical too easily.

02:26:07 I’m one of those people that I’ve been really fortunate

02:26:10 to interact with incredible people at MIT

02:26:12 and academia in general, but I’ve met some assholes.

02:26:15 And I tend to just kind of,

02:26:17 when I run into difficult folks,

02:26:19 I just kind of smile and send them all my love

02:26:21 and just kind of go around.

02:26:23 But for others, those experiences can be sticky.

02:26:26 Like they can become cynical about the world

02:26:29 when folks like that exist.

02:26:31 So he may have become a little bit cynical

02:26:35 about the process of science.

02:26:37 Well, you know, it’s a good opportunity.

02:26:38 Let’s posit that that’s his reasoning

02:26:40 because I truly don’t know.

02:26:42 It’s an interesting opportunity to go back

02:26:43 to almost the very first thing we talked about,

02:26:46 the idea of the Mathematical Olympiad,

02:26:48 because of course that is,

02:26:50 so the International Mathematical Olympiad

02:26:52 is like a competition for high school students

02:26:54 solving math problems.

02:26:55 And in some sense, it’s absolutely false

02:26:59 to the reality of mathematics,

02:27:00 because just as you say,

02:27:02 it is a contest where you win prizes.

02:27:07 The aim is to sort of be faster than other people.

02:27:11 And you’re working on sort of canned problems

02:27:13 that someone already knows the answer to,

02:27:15 like not problems that are unknown.

02:27:18 So, you know, in my own life,

02:27:20 I think when I was in high school,

02:27:21 I was like very motivated by those competitions.

02:27:24 And like, I went to the Math Olympiad and…

02:27:26 You won it twice and got, I mean…

02:27:28 Well, there’s something I have to explain to people

02:27:30 because it says, I think it says on Wikipedia

02:27:32 that I won a gold medal.

02:27:33 And in the real Olympics,

02:27:35 they only give one gold medal in each event.

02:27:37 I just have to emphasize

02:27:38 that the International Math Olympiad is not like that.

02:27:40 The gold medals are awarded

02:27:42 to the top 112th of all participants.

02:27:44 So sorry to bust the legend or anything like that.

02:27:47 Well, you’re an exceptional performer

02:27:48 in terms of achieving high scores on the problems

02:27:51 and they’re very difficult.

02:27:53 So you’ve achieved a high level of performance on the…

02:27:56 In this very specialized skill.

02:27:57 And by the way, it was a very Cold War activity.

02:28:00 You know, in 1987, the first year I went,

02:28:02 it was in Havana.

02:28:04 Americans couldn’t go to Havana back then.

02:28:06 It was a very complicated process to get there.

02:28:08 And they took the whole American team on a field trip

02:28:10 to the Museum of American Imperialism in Havana

02:28:14 so we could see what America was all about.

02:28:17 How would you recommend a person learn math?

02:28:22 So somebody who’s young or somebody my age

02:28:26 or somebody older who’ve taken a bunch of math

02:28:29 but wants to rediscover the beauty of math

02:28:32 and maybe integrate it into their work

02:28:34 more solid in the research space and so on.

02:28:38 Is there something you could say about the process of…

02:28:44 Incorporating mathematical thinking into your life?

02:28:47 I mean, the thing is,

02:28:48 it’s in part a journey of self knowledge.

02:28:50 You have to know what’s gonna work for you

02:28:53 and that’s gonna be different for different people.

02:28:55 So there are totally people who at any stage of life

02:28:59 just start reading math textbooks.

02:29:01 That is a thing that you can do

02:29:03 and it works for some people and not for others.

02:29:06 For others, a gateway is, I always recommend

02:29:09 the books of Martin Gardner,

02:29:10 another sort of person we haven’t talked about

02:29:12 but who also, like Conway, embodies that spirit of play.

02:29:16 He wrote a column in Scientific American for decades

02:29:19 called Mathematical Recreations

02:29:20 and there’s such joy in it and such fun.

02:29:23 And these books, the columns are collected into books

02:29:26 and the books are old now

02:29:27 but for each generation of people who discover them,

02:29:29 they’re completely fresh.

02:29:31 And they give a totally different way into the subject

02:29:33 than reading a formal textbook,

02:29:36 which for some people would be the right thing to do.

02:29:40 And working contest style problems too,

02:29:42 those are bound to books,

02:29:43 especially like Russian and Bulgarian problems.

02:29:45 There’s book after book problems from those contexts.

02:29:47 That’s gonna motivate some people.

02:29:50 For some people, it’s gonna be like watching

02:29:51 well produced videos, like a totally different format.

02:29:54 Like I feel like I’m not answering your question.

02:29:56 I’m sort of saying there’s no one answer

02:29:57 and it’s a journey where you figure out

02:30:00 what resonates with you.

02:30:01 For some people, it’s the self discovery

02:30:04 is trying to figure out why is it that I wanna know?

02:30:06 Okay, I’ll tell you a story.

02:30:07 Once when I was in grad school,

02:30:09 I was very frustrated with my lack of knowledge

02:30:11 of a lot of things as we all are

02:30:13 because no matter how much we know,

02:30:14 we don’t know much more and going to grad school

02:30:15 means just coming face to face

02:30:17 with the incredible overflowing vault of your ignorance.

02:30:20 So I told Joe Harris, who was an algebraic geometer,

02:30:23 a professor in my department,

02:30:26 I was like, I really feel like I don’t know enough

02:30:27 and I should just take a year of leave

02:30:29 and just read EGA, the holy textbook,

02:30:32 Elements de Géométrie Algebraique,

02:30:34 the Elements of Algebraic Geometry.

02:30:36 I’m just gonna, I feel like I don’t know enough

02:30:38 so I’m just gonna sit and read this like 1500 page

02:30:42 many volume book.

02:30:46 And he was like, and Professor Harris was like,

02:30:48 that’s a really stupid idea.

02:30:49 And I was like, why is that a stupid idea?

02:30:50 Then I would know more algebraic geometry.

02:30:52 He’s like, because you’re not actually gonna do it.

02:30:53 Like you learn.

02:30:55 I mean, he knew me well enough to say like,

02:30:57 you’re gonna learn because you’re gonna be working

02:30:58 on a problem and then there’s gonna be a fact from EGA

02:31:01 that you need in order to solve your problem

02:31:03 that you wanna solve and that’s how you’re gonna learn it.

02:31:05 You’re not gonna learn it without a problem

02:31:06 to bring you into it.

02:31:08 And so for a lot of people, I think if you’re like,

02:31:10 I’m trying to understand machine learning

02:31:12 and I’m like, I can see that there’s sort of

02:31:14 some mathematical technology that I don’t have,

02:31:19 I think you like let that problem

02:31:22 that you actually care about drive your learning.

02:31:26 I mean, one thing I’ve learned from advising students,

02:31:28 math is really hard.

02:31:32 In fact, anything that you do right is hard.

02:31:38 And because it’s hard, like you might sort of have some idea

02:31:41 that somebody else gives you, oh, I should learn X, Y and Z.

02:31:44 Well, if you don’t actually care, you’re not gonna do it.

02:31:46 You might feel like you should,

02:31:47 maybe somebody told you you should,

02:31:48 but I think you have to hook it to something

02:31:51 that you actually care about.

02:31:52 So for a lot of people, that’s the way in.

02:31:54 You have an engineering problem you’re trying to handle,

02:31:57 you have a physics problem you’re trying to handle,

02:31:59 you have a machine learning problem you’re trying to handle.

02:32:02 Let that not a kind of abstract idea

02:32:05 of what the curriculum is, drive your mathematical learning.

02:32:08 And also just as a brief comment that math is hard,

02:32:12 there’s a sense to which hard is a feature, not a bug,

02:32:15 in the sense that, again,

02:32:17 maybe this is my own learning preference,

02:32:19 but I think it’s a value to fall in love with the process

02:32:24 of doing something hard, overcoming it,

02:32:27 and becoming a better person because of it.

02:32:29 Like I hate running, I hate exercise,

02:32:32 to bring it down to like the simplest hard.

02:32:35 And I enjoy the part once it’s done,

02:32:39 the person I feel like in the rest of the day

02:32:41 once I’ve accomplished it, the actual process,

02:32:44 especially the process of getting started in the initial,

02:32:47 like it really, I don’t feel like doing it.

02:32:49 And I really have, the way I feel about running

02:32:51 is the way I feel about really anything difficult

02:32:55 in the intellectual space, especially in mathematics,

02:32:58 but also just something that requires

02:33:01 like holding a bunch of concepts in your mind

02:33:04 with some uncertainty, like where the terminology

02:33:08 or the notation is not very clear.

02:33:10 And so you have to kind of hold all those things together

02:33:13 and like keep pushing forward through the frustration

02:33:16 of really like obviously not understanding certain like

02:33:20 parts of the picture, like your giant missing parts

02:33:23 of the picture and still not giving up.

02:33:26 It’s the same way I feel about running.

02:33:28 And there’s something about falling in love

02:33:32 with the feeling of after you went through the journey

02:33:36 of not having a complete picture,

02:33:38 at the end having a complete picture,

02:33:40 and then you get to appreciate the beauty

02:33:42 and just remembering that it sucked for a long time

02:33:46 and how great it felt when you figured it out,

02:33:48 at least at the basic.

02:33:49 That’s not sort of research thinking,

02:33:52 because with research, you probably also have to

02:33:55 enjoy the dead ends with learning math

02:34:00 from a textbook or from video.

02:34:02 There’s a nice.

02:34:03 I don’t think you have to enjoy the dead ends,

02:34:04 but I think you have to accept the dead ends.

02:34:06 Let’s put it that way.

02:34:08 Well, yeah, enjoy the suffering of it.

02:34:11 So the way I think about it, I do, there’s an.

02:34:17 I don’t enjoy the suffering.

02:34:18 It pisses me off.

02:34:19 You have to accept that it’s part of the process.

02:34:21 It’s interesting.

02:34:22 There’s a lot of ways to kind of deal with that dead end.

02:34:24 There’s a guy who’s the ultra marathon runner,

02:34:26 Navy SEAL, David Goggins, who kind of,

02:34:30 I mean, there’s a certain philosophy of like,

02:34:34 most people would quit here.

02:34:37 And so if most people would quit here and I don’t,

02:34:42 I’ll have an opportunity to discover something beautiful

02:34:45 that others haven’t yet.

02:34:46 And so like any feeling that really sucks,

02:34:52 it’s like, okay, most people would just like,

02:34:56 go do something smarter.

02:34:58 And if I stick with this,

02:35:01 I will discover a new garden of fruit trees that I can pick.

02:35:06 Okay, you say that, but like,

02:35:07 what about the guy who like wins

02:35:09 the Nathan’s hot dog eating contest every year?

02:35:11 Like when he eats his 35th hot dog,

02:35:13 he like correctly says like,

02:35:14 okay, most people would stop here.

02:35:17 Are you like lauding that he’s like,

02:35:18 no, I’m gonna eat the 35th hot dog.

02:35:20 I am, I am.

02:35:21 In the long arc of history, that man is onto something.

02:35:26 Which brings up this question.

02:35:28 What advice would you give to young people today,

02:35:30 thinking about their career, about their life,

02:35:34 whether it’s in mathematics, poetry,

02:35:37 or hot dog eating contest?

02:35:40 And you know, I have kids,

02:35:41 so this is actually a live issue for me, right?

02:35:43 I actually, it’s not a thought experiment.

02:35:45 I actually do have to give advice

02:35:47 to two young people all the time.

02:35:48 They don’t listen, but I still give it.

02:35:53 You know, one thing I often say to students,

02:35:55 I don’t think I’ve actually said this to my kids yet,

02:35:56 but I say it to students a lot is,

02:35:59 you know, you come to these decision points

02:36:03 and everybody is beset by self doubt, right?

02:36:06 It’s like, not sure like what they’re capable of,

02:36:09 like not sure what they really wanna do.

02:36:14 I always, I sort of tell people like,

02:36:16 often when you have a decision to make,

02:36:20 one of the choices is the high self esteem choice.

02:36:22 And I always tell them, make the high self esteem choice.

02:36:24 Make the choice, sort of take yourself out of it

02:36:26 and like, if you didn’t have those,

02:36:29 you can probably figure out what the version of you

02:36:31 that feels completely confident would do.

02:36:35 And do that and see what happens.

02:36:36 And I think that’s often like pretty good advice.

02:36:40 That’s interesting.

02:36:40 Sort of like, you know, like with Sims,

02:36:44 you can create characters.

02:36:45 Create a character of yourself

02:36:47 that lacks all the self doubt.

02:36:50 Right, but it doesn’t mean,

02:36:51 I would never say to somebody,

02:36:52 you should just go have high self esteem.

02:36:56 You shouldn’t have doubts.

02:36:57 No, you probably should have doubts.

02:36:58 It’s okay to have them.

02:36:59 But sometimes it’s good to act in the way

02:37:01 that the person who didn’t have them would act.

02:37:04 That’s a really nice way to put it.

02:37:08 Yeah, that’s like from a third person perspective,

02:37:13 take the part of your brain that wants to do big things.

02:37:16 What would they do?

02:37:18 That’s not afraid to do those things.

02:37:20 What would they do?

02:37:21 Yeah, that’s really nice.

02:37:24 That’s actually a really nice way to formulate it.

02:37:26 That’s very practical advice.

02:37:27 You should give it to your kids.

02:37:31 Do you think there’s meaning to any of it

02:37:32 from a mathematical perspective, this life?

02:37:36 If I were to ask you,

02:37:39 we talked about primes, talked about proving stuff.

02:37:43 Can we say, and then the book that God has,

02:37:47 that mathematics allows us to arrive

02:37:49 at something about in that book.

02:37:51 There’s certainly a chapter

02:37:52 on the meaning of life in that book.

02:37:54 Do you think we humans can get to it?

02:37:57 And maybe if you were to write cliff notes,

02:37:59 what do you suspect those cliff notes would say?

02:38:01 I mean, look, the way I feel is that mathematics,

02:38:04 as we’ve discussed, it underlies the way we think

02:38:07 about constructing learning machines.

02:38:09 It underlies physics.

02:38:11 It can be used.

02:38:12 I mean, it does all this stuff.

02:38:15 And also you want the meaning of life?

02:38:17 I mean, it’s like, we already did a lot for you.

02:38:18 Like, ask a rabbi.

02:38:22 No, I mean, I wrote a lot in the last book,

02:38:25 How Not to Be Wrong.

02:38:27 I wrote a lot about Pascal, a fascinating guy who is

02:38:32 a sort of very serious religious mystic,

02:38:35 as well as being an amazing mathematician.

02:38:37 And he’s well known for Pascal’s wager.

02:38:38 I mean, he’s probably among all mathematicians.

02:38:40 He’s the one who’s best known for this.

02:38:42 Can you actually like apply mathematics

02:38:44 to kind of these transcendent questions?

02:38:49 But what’s interesting when I really read Pascal

02:38:53 about what he wrote about this,

02:38:54 I started to see that people often think,

02:38:56 oh, this is him saying, I’m gonna use mathematics

02:39:00 to sort of show you why you should believe in God.

02:39:03 You know, mathematics has the answer to this question.

02:39:07 But he really doesn’t say that.

02:39:08 He almost kind of says the opposite.

02:39:11 If you ask Blaise Pascal, like, why do you believe in God?

02:39:15 He’d be like, oh, cause I met God.

02:39:16 You know, he had this kind of like psychedelic experience.

02:39:20 It’s like a mystical experience where as he tells it,

02:39:23 he just like directly encountered God.

02:39:24 It’s like, okay, I guess there’s a God, I met him last night.

02:39:26 So that’s it.

02:39:27 That’s why he believed.

02:39:29 It didn’t have to do with any kind.

02:39:30 You know, the mathematical argument was like

02:39:33 about certain reasons for behaving in a certain way.

02:39:36 But he basically said, like, look,

02:39:38 like math doesn’t tell you that God’s there or not.

02:39:41 Like, if God’s there, he’ll tell you.

02:39:43 You know, you don’t even.

02:39:45 I love this.

02:39:46 So you have mathematics, you have, what do you have?

02:39:50 Like a way to explore the mind, let’s say psychedelics.

02:39:53 You have like incredible technology.

02:39:56 You also have love and friendship.

02:39:59 And like, what the hell do you want to know

02:40:01 what the meaning of it all is?

02:40:02 Just enjoy it.

02:40:03 I don’t think there’s a better way to end it, Jordan.

02:40:07 This was a fascinating conversation.

02:40:08 I really love the way you explore math in your writing.

02:40:14 The willingness to be specific and clear

02:40:18 and actually explore difficult ideas,

02:40:21 but at the same time stepping outside

02:40:23 and figuring out beautiful stuff.

02:40:25 And I love the chart at the opening of your new book

02:40:30 that shows the chaos, the mess that is your mind.

02:40:33 Yes, this is what I was trying to keep in my head

02:40:35 all at once while I was writing.

02:40:38 And I probably should have drawn this picture

02:40:40 earlier in the process.

02:40:41 Maybe it would have made my organization easier.

02:40:43 I actually drew it only at the end.

02:40:45 And many of the things we talked about are on this map.

02:40:48 The connections are yet to be fully dissected, investigated.

02:40:52 And yes, God is in the picture.

02:40:56 Right on the edge, right on the edge, not in the center.

02:41:00 Thank you so much for talking to me.

02:41:01 It is a huge honor that you would waste

02:41:03 your valuable time with me.

02:41:05 Thank you, Lex.

02:41:06 We went to some amazing places today.

02:41:07 This was really fun.

02:41:09 Thanks for listening to this conversation

02:41:11 with Jordan Ellenberg.

02:41:12 And thank you to Secret Sauce, ExpressVPN, Blinkist,

02:41:16 and Indeed.

02:41:17 Check them out in the description to support this podcast.

02:41:21 And now let me leave you with some words from Jordan

02:41:24 in his book, How Not To Be Wrong.

02:41:26 Knowing mathematics is like wearing a pair of X ray specs

02:41:30 that reveal hidden structures underneath the messy

02:41:33 and chaotic surface of the world.

02:41:35 Thank you for listening and hope to see you next time.