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.