YouSaid · the spoken record

Jordan Ellenberg

lines on the record: 193
first: 2021-06-13
most recent: 2021-06-13
sittings or episodes: 1
sources: podcast

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“Yeah, I mean, I love that subject. I haven't really done research in it myself. I've played around with it. I'll send you a fun blog post I wrote where I made some cool texture patterns from Cellar Automata that I, um,”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“These ideas that maybe we should be doing math in this more restrictive way, where even a thing that, you know, because look, the origin of all this is like. Number represents a magnitude, like the length of a line. So, I mean, the idea that there's a continuum, Pretty old, but that you know, just because something is old doesn't mean we can't reject it if we want to.”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“That's a great point, actually. I think in some sense this flavor of doing math saying we shouldn't talk about things that we cannot specify in a finite amount of time. There's something very computational in flavor about that. And it's probably not a coincidence that it becomes popular. In the 30s and 40s, which is also like kind of like the dawn of ideas about formal computation, right? You probably know the timeline better than I do.”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“People were queasy about it, and they weren't wrong to be queasy about it, right? From a modern perspective, it was not really well formed. There's this very famous critique of Newton by Bishop Berkeley, where he says, like, what these things you define, like, you know, they're not zero, but they're smaller than any number. Are they the ghosts of departed quantities? That was this ultra burn And on the one hand He was right. It wasn't really rigorously modern standards. On the other hand, Newton was out there doing calculus and other people were not, right? It works. I think a sort of intuitionist view, for instance, I would say would express serious doubt. And it's not, by the way, it's not just infinity. It's like saying, I think we would express serious doubt that like the real numbers exist. Most people are comfortable”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“Talk to us and remember this mysterious thing. And what is it? What is it? Well, he'd say like, well, it's like you sort of, um... Divide the length of this line segment by the length of this other line segment, and then you make them a little shorter and you divide again, and then you make them a little shorter and you divide again. And then you just keep on doing that until they're infinitely short, and then you divide them again. These quantities that are like, they're not zero, but they're also smaller than any... Actual number, these infinitesimals.”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“Well, so first of all, okay, if you say, is there a serious way of doing mathematics that doesn't really treat infinity as a real thing or maybe as kind of agnostic and is like, I'm not really going to make a firm statement about whether it's a real thing or not? Yeah, that's called most of the history of mathematics. So it's only after Cantor, right, that we really are sort of, okay, we're going to have a notion of the cardinality of an infinite set and do something that you might call like the modern theory of infinity. That said, obviously everybody was drawn to this notion and no, not everybody was comfortable with it. Look, I mean, this is what happens with Newton, right? I mean, so Newton understands that to talk about tangents and to talk about instantaneous philosophy, he has to do something that we would now call taking a limit, right? The fable DY over dx, if you sort of go back to your calculus class for those who have thinking.”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“I bet Ultra Fantasy came second. I'll bet it's like when there's like a hardcore scene, and then one guy is like, Oh, now there's a lot of people in this scene. I have to find a way to be more hardcore than the hardcore people.”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“Is that like the ultimate sentence of the modern age? Can I just comment? Because I read the Wikipedia page. That sums up our moment.”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“I mean, I think, okay, so first of all, I'm not an expert in. I couldn't even tell you what the difference is between those three terms finitism, ultrafinitism, and intuitionism, although I know they're related and I tend to associate them with the Netherlands in the 1930s.”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“Yeah, because I feel look, I'm sorry, but I feel like you have more insights about people if you think of them as like beings that have wants and needs and desires and do stuff on purpose. Even if that's not true, you still understand better what's going on by treating them in that way. Don't you look, what you work on machine learning, don't you find yourself sort of talking about what the machine is trying to do in a certain instance? Do you not find yourself drawn to that language? It knows this. It's trying to do that. It's learning that”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“Like, you are talking about prime numbers, and you're like, but prime numbers are completely deterministic. And I'm saying like, well, but let's treat them like a random process. And then you say, but you're just saying something that's not true. They're not a random process. They're deterministic. And I'm like, okay, great. You hold to your insistence that it's not a random process. Meanwhile, I'm generating insight about the primes that you're not because I'm willing to sort of pretend that there's something that they're not in order to understand what's going on.”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“Well, not in order to prove stuff about them so much as to figure out what we expect to be true and then try to prove that. Because here's what you don't want to do. Try really hard to prove something that's false. That makes it really hard to prove the thing if it's false. You certainly want to have some heuristic ways of guessing, making good guesses about what's true. So, yeah, here's what I would say you're going to be imaginary, Sam Harris now.”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“Of the system are like that of the random process. And so that's kind of like, it's funny because I think when you talk to people about the twin prime conjecture, People think you're saying, wow, there's like some deep structure there that makes those primes be like close together again and again. And no, it's the opposite of deep structure. What we say when we say we believe the twin prime conjecture is that we believe the primes are like sort of strewn around pretty randomly. And if they were, then by chance you would expect there to be infinitely many twin primes. And we're saying, yeah, we expect them to behave just like they would if they were random dirt.”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“Why would we think this? The answer is that it turns out to be incredibly productive and enlightening to think about primes as if they were random numbers, as if they were randomly distributed according to a certain law. Now they're not. They're not random. There's no chance involved. It's completely deterministic whether a number is prime or not. And yet it just turns out to be phenomenally useful in mathematics to say even if something is governed by a deterministic law, let's just pretend it wasn't. Let's just pretend that they were produced by some random process and see if the behavior is roughly the same. And if it's not, maybe change the random process. Maybe make the randomness a little bit different and tweak it and see if you can find a random process that matches the behavior we see. And then maybe you predict that other behaviors”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“We don't have any way of describing a process that makes primes. Like, sure, you can look at your computer and see a lot of them, but the fact that there's a lot, why is that evidence that there's infinitely many, right? Maybe I can go on the computer and find 10 million. Well, 10 million is pretty far from infinity, right? So how is that evidence? There's a lot of things. There's like a lot more than 10 million atoms. That doesn't mean there's infinitely many atoms in the universe, right? I mean, on most people's physical theories, there's probably not, as I understand it. Okay, so”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“And the extent to which that's the case, that's pretty well understood. But then you can ask more fine grained questions. And here is one. A twin prime is a pair of primes that are two apart. Three and five, or like 11 and 13, or like 17 and 19. And one thing we still don't know is are there infinitely many of those? We know on average they get farther and farther apart, but that doesn't mean there couldn't be like occasional That come close together. And indeed, we think that there are. And one interesting question. Think you might say, well, why, how could one possibly have a right to have an opinion about something like that?”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“Right, so one thing that people recognized and really thought about a lot is that the primes on average seem to get farther and farther apart as they get bigger and bigger. In other words, it's less and less common. I already told you of the first 10 numbers, two, three, five, seven, four of them are prime. That's a lot, 40%. If I looked at 10 digit numbers, no way would 40% of those be prime. Being prime would be a lot rarer. In some sense, because there's a lot more things for them to be divisible by. That's one way of thinking of it. It's a lot more possible for there to be a factorization because there's a lot of things you can try to factor out of it. As the numbers get bigger and bigger, primality gets rarer and rarer.”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“Yeah, it's a perfect example of your desire for simplicity in all things. You know what would be really simple if there was only finitely many primes? And then there would be this finite set of atoms that all numbers would be built up. That would be very simple and good in certain ways, but it's completely false. And number three would be totally different if that were the case. It's just not true. In fact, this is something else that Euclid knew. So this is a very, very old fact, like much before, long before we had anything like modern number theory.”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“It's likely a problem for not for sure. And there's actually a beautiful geometric proof which is in the book actually. That's like one of the most granular parts of the book because it's such a beautiful proof I couldn't not give it. So you draw a lot of opal and pearl necklaces and spin them. That's kind of the geometric nature of this proof of Fairmaux's little theorem. So, yeah, so with pseudo primes, there are primes that are kind of faking that they pass that test, but they're, or numbers that are faking it that pass that test but are not actually prime But the point is There are many, many, many theorems about prime numbers.”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“2 to the seventh is 128. Divide that by seven. And let's see. I think that's seven times 14. Is that right? No. Is 126 with a remainder of two, right? 128 is a multiple of seven plus two. So if that remainder is not 2,”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“Let's do it. We're going to do a live demonstration. Let's say your number is six. So I'm going to raise two to the sixth power. Okay, so if I were working out, I'd be like, that's two cubed squared, so that's eight times eight. So that's 64. Now we're going to divide by six, but I don't actually care what the quotient is, only the remainder. Let's see, 64 divided by 6 is, well, there's a quotient of 10, but the remainder is 4. So you failed because the answer has to be For any prime Let's do it with five, which is To the fifth is thirty two divide thirty two by five and you get six with a remainder of two.”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“Yes, actually. Ready, let me give you the simplest version of it. You can dress it up a little bit, but here's the basic idea. I take the number, the mystery number. I raise two to that power. So let's say your mystery number is six. Are you sorry you asked me? Are you ready to thumb”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“Didn't do it on purpose. Anyway, they're definitely ones that people, or 91 is another classic seven times 13. It really feels kind of prime, doesn't it? But it is not. But there's also, by the way, but there's also an actual notion of pseudo prime, which is a thing with a formal definition, which is not a psychological thing. It is a prime which passes a primality test devised by Fermat, which is a very good test, which if a number fails this test, it's definitely not prime. And so there was some hope that, oh, maybe if a number passes the test, then it definitely is prime. That would give a very simple criterion for primality. Unfortunately, it's only perfect in one direction. So there are numbers, I want to say three hundred forty one is the smallest, which pass the test but are not prime 341.”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“I think so. There's always those ones that trip people up. There's a famous one, the Grotendieck Prime 57, like sort of Alexander Grotendieck, the great algebraic geometer, was sort of giving some lecture involving a choice of a prime in general and somebody said, like, can't you just choose a prime? And he said, okay, 57, which is in fact not prime. It's three times 19.”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“By the way, they're all going to be odd from the non because they were even, I could factor a two out of them, but it's not all the odd numbers. Nine isn't prime because it's 3 times 3. 15 isn't prime because it's 3 times 5, but 13 is. Where were we? 2, 3, 5, 7, 11, 13, 17, 19, not 21, but 23 is, et cetera, et cetera. Okay, so you could go on”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“You take any number and you factorize it until you can factorize no more and what you have left is some big pile of primes. I mean, by definition, when you can't factor anymore, when you're done or you can't break the numbers up anymore, what's left must be prime. You know, 12 breaks into two and two and three. So these numbers are the atoms, the building blocks of all numbers. There's a lot we know about them, but there's much more that we don't know them. I'll tell you the first few there's two, three, five, seven, eleven.”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“Which at some times in mathematical history has been deemed to be a prime, but currently is not. And I think that's for the best. But I bring it up only because sometimes people think that these definitions are kind of, if we think about them hard enough, we can figure out which definition is true.”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“Yeah, so prime numbers are one of the things that number theorists study the most and have for millennia. They are numbers which can't be factored. And then you say like five. And then you're like, wait, I can factor five. Five is five times one. Okay, not like that. That is a factorization. It absolutely is a way of expressing five as a product of two things. But don't you agree there's like something trivial about it? It's something you can do to any number. It doesn't have content the way that if I say that 12 is six times two or thirty-five is seven times five. I've really done something to it. I've broken up. So those are the kind of factorizations that count. And a number that doesn't have a factorization like that is called prime, except historical side note one.”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“I'm sorry, you just sent me off on a tangent, just imagining like Erdish Adah Hendricks concert trying to sort of figure out if it was from The book or not. What I was coming to was just to say, but one Poincar ⁇ said about this is he's like, you know. This is all worked out in the language of the divine and of a divine being came down and told it to us, we wouldn't be able to understand it. So it doesn't matter. So Punk Array was of the view that there were things that were sort of like inhumanly complex. And that was how they really were. Our job is to figure out the things that are not like that.”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“But Poincar ⁇, on the other hand, and by the way, there were other maps, Hilda Hudson is one who comes up in this book. She also kind of saw math. She's one of the people who sort of develops the disease model that we now use, that we use to sort of track pandemics, this SIR model that sort of originally comes from her work with Ronald Ross. But she was also super, super, super devout. And she also sort of from the other side of the religious coin was like, yeah, math is how we communicate with God. She has a great, all these people are incredibly quotable. She says, you know, math is the truth. The things about mathematics, she's like, they're not the most important of God thoughts, but they're the only ones that we can know precisely.”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“You know, Erdish was sort of famous for saying this is sort of one line you were saying, he talked about the book, capital T, capital B, the book. And that's the book where God keeps the right proof of every theorem. So when he saw a proof he really liked, it was really elegant, really simple. That's from the book. That's like you found one of the ones that's in the book. He wasn't a religious guy, by the way. He referred to God as the supreme fascist. He was like, but somehow he was like, I don't really believe in God, but I believe in God's book. I mean, there was a, um,”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“Yes, that is definitely possible, but I would even say, look, consciousness is a thing about which we're still in the dark as to whether there's an explanation we would. We would understand it as an explanation at all. By the way, okay, I got to give yet one more amazing Poincar ⁇ quote because this guy just never stopped coming out with great quotes that Paul Erdish, another fellow who appears in the book. And by the way, he thinks about this notion of distance, of like personal affinity, kind of like what you're talking about, the kind of social network and that notion of distance that comes from that. So that's something that politicians.”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“That explanation. I don't think any poem would, I don't think any poet would say their poem is an explanation. They might say it's a description. They might say it's sort of capturing sort of.”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“Such a mathematician answered. The truth that is in it Is that learning to explain something helps you understand it? But real things are not simple. Few things are, most are not, and I don't to be honest, I don't, I mean, we don't really know whether Feynman really said that right or something like that is sort of disputed, but I don't think Feynman could have literally believed that whether or not he said it. And, you know, he was the kind of guy, I didn't know him, but I'm rereading his writing. He liked to sort of say stuff like stuff that sounded good. You know what I mean? So it totally strikes me as the kind of thing he could have said because he liked the way saying it made him feel. Also, knowing that he didn't like literally mean it”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“Oh, okay. So you were about to ask is it true to which I would say flatly no? But then you said you followed that up with is there some profound truth in it? And I'm like, okay, sure. So there's some truth in it. But it's not true.”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“If you try to do natural language processing and your idea of distance between words is how close they are in the dictionary when you write them in alphabetical order, you are going to get pretty bad translations, right? No, the notion of distance has to come from somewhere else.”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“That is a really good, humble way to think about it. I like it. Okay, so let's just go for it. Okay, so I think you'll agree with this, that in some sense, what's good about AI is that We can't test any case in advance. The whole point of AI is to make our one point of it, I guess, is to make good predictions about cases we haven't yet seen. And in some sense, that's always going to involve some notion of distance because it's always going to involve somehow taking a case we haven't seen and saying what cases that we have seen, is it close to? Is it like? Is it somehow an interpolation between? Now, when we do that in order to talk about things being like other things implicitly or explicitly, we're invoking some notion of distance. And boy, we better get it right.”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“Yeah, because think about it. Okay, here's, I mean, now we're going to talk about AI, which you know a lot more about than I do, so just, you know, start laughing uproariously if I say something that's completely wrong.”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“Yes, although, you know, honestly, what I would say I mean, it's true that we use it to prove things, but I would say we use it to understand things. And then because we understand things better, then we can prove things. But, you know, the goal is always the understanding. The goal is not so much to prove things. The goal is not to know what's true or false. I mean, this is the thing I read about in the book near the end. It's something that's a wonderful, wonderful essay by Bill Thurston, kind of one of the great geometers of our time, who unfortunately passed away a few years ago, called on proof and progress in mathematics. And he writes very wonderfully about how it's not a theorem factory where we have a production quota. I mean, the point of mathematics is to help humans understand things.”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“Okay. No, I mean, it's such a, I mean, I just get excited talking about it, and I just taught this like in the fall semester that”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“Yeah, the MEP. And so, because that's the kind of deformation that comes up. Wiles' proof that Darformation were moving something a little bit means a little bit in this to add extension”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“No, no, it's very similar. That's exactly the right way to think of it. It's almost like binary numbers written in reverse. Because in a binary expansion, two numbers are close, a number that's small is like 0.000 something. Thing that's the decimal one, it starts with a lot of zeros. In the two attic metric, a binary number is very small. If it ends with a lot of zeros and then the decimal point. So it is kind of like binary numbers written backwards is actually, I should have said that's what I should have said, Lex. That's a very good metaphor. Okay”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“It takes practice. It takes practice. If you've ever heard of the cantor set, it looks kind of like that. So it is crazy that this is good for anything, right? I mean, this just sounds like a definition that someone would make up to torment you. But what's amazing is there's a general theory of distance where you say any definition you make that satisfies certain axioms deserves to be called a distance”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“If we were to be what's called a two addict number theorist, we'd say, oh, two numbers are close if their difference is a multiple of a large power of two. So one and 49 are close because their difference is 48 and 48 is a multiple of sixteen, which is a pretty large power of two. Whereas one and two are pretty far away because the difference between them is one, which is not even a multiple of a power of two at all. It's odd. You want to know what's really far from one? Like one and 164th. Because their difference is a negative power of two, two to the minus six. So those points are quite, quite far from the two.”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“There's a different notion of distance we have in mind, and there are lots of notions of distances that you could use in the natural language processing community in AI. There might be some notion of semantic distance or lexical distance between two words. How much do they tend to arise in the same context? That's incredibly important for doing autocomplete and like machine translation and stuff like that. And it doesn't have anything to do with are they next to each other in the dictionary, right? It's a different kind of distance. Okay, ready? In this kind of number theory, there was a crazy distance called the Piatic distance. I didn't write about this that much in the book because even though I love it and it's a big part of my research. It gets a little bit into the weeds, but your listeners are going to hear about it now. Please. What a normal person says when they say two numbers are close. They say their difference is like a small number, like seven and eight are close because their difference is one and one's pretty small.”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“Which is at the heart of it. And it involves this very careful principle like that. But that being said, what I just said. It's probably not what you're thinking because what you're thinking when you think, oh, I have a point in space and I move it around like a little tiny bit. You're using Your notion of distance that's from calculus. We know what it means for like two points on the real line to be close together. Yet another thing that comes up in the book a lot is this fact that the notion of distance is not given to us by God. We could mean a lot of different things by distance. And just in the English language, we do that all the time. We talk about somebody being a close relative. It doesn't mean they live next door to you, right? It means something else.”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“When you study Ferm's theorem, and let's not even be too careful about what these objects are. I can tell you they're Gao representations in modular forms, but saying those words is not going to mean so much. But whatever they are, they're things that can be deformed, moved around a little bit. And I think the insight of what Andrew and then Andrew and Richard were able to do was to say something like this. Deformation means moving something just a tiny bit like an infinitesimal amount. If you really are good at understanding which ways a thing can move in a tiny, tiny, tiny infinitesimal amount in certain directions, maybe you can piece that information together to understand the whole global space in which it can move. And essentially their argument comes down to showing that two of those big global spaces are actually the same, the fabled r equals t part of their proof.”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“Yes, and this is a perfect example. So, this is another phrase of Poincar ⁇, this incredible generator of slogans and aphorisms. He said, mathematics is the art of calling different things by the same name. That very thing we do, right, when we're like this triangle and this triangle, come on, they're the same triangle. They're just in a different place, right? So in the same way, it came to be understood that the kinds of objects that you study”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“Right. Well, the reason that Barry called it deformation theory, I think he's the one who gave it the name. I hope I'm not wrong in saying this on Dave.”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“Whether there's interesting human aspects to the proof itself is an interesting question. Certainly, it has a huge amount of richness sort of at its heart is an argument of what's called deformation theory. Which was in part created by my PhD advisor, Barry Mazar.”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source