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

Every line below is reproduced as it was said and linked to the record it came from. Nothing here is summarised or generated. Directory · Search · Corrections

“Whether you live on a circle or a line segment, because if you live on a circle, if you walk a long way in one direction, you find yourself back where you started. And if you live in a line segment, you walk for a long enough one direction, you come to the end of the world. Or if you live on a line, like a whole line, an infinite line, then you walk in one direction for a long time. And, well, then there's not a sort of terminating algorithm to figure out whether you live on a line or a circle, but at least you sort of at least don't discover that you live on a circle. So all of those are intrinsic things, right? All of those are things that you can figure out about your world without leaving your world. On the other hand, ready, now we're going to go from intrinsic to extrinsic. Boy, did I not know we were going to talk about this, but why not?”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“Well, that's true. The visualization is harder, but in some sense, no, you're right, but the tools of mathematics are there. Sorry, I don't want to fight, but I was like, the tools of mathematics are exactly there to enable you to think about what you cannot visualize in this way. Always to make things easier, go downer dimension. Let's think about we live on a circle, okay? You can tell.”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“But that's exactly what topology does. Topology is what's called an intrinsic theory. That's what's so great about it. This question about the mug, you could answer it without ever leaving the mug, right? Because it's a question about... Loop drawn on the surface of the mug, and what happens if it never leaves that surface? So it's like always there.”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“I think there's somebody who thinks that there's like some kind of dodecahedral symmetry, or I mean, I remember reading something crazy about somebody saying that they saw the signature of that in the cosmic noise or what have you.”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“But what if your piece of string was the size of the universe? Like, what if your piece of string was like billions of light years long? How do you actually know?”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“Well, I can't speak to the universe, but what I can say is that Regular old space is not a mug. Regular old space, if you like, sort of actually physically have a loop of string.”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“If I draw a little circle on this mug, imagine this to be a loop of string. I could pull that loop of string closed on the surface of the mug. Right? That's definitely something I could do. I could shrink it, shrink it, shrink it until it's a point. On the other hand, if I draw a loop that goes around the handle, I can kind of zhit it up here and I can judge it down there and I can sort of slide it up and down the handle. But I can't pull it closed, can I? It's trapped. Not without breaking the surface of the mug, right? Not without going inside. The condition of being what's called simply connected. This is one of Bunker's inventions, says that any loop of string can be pulled shut. So it's a feature that the mug simply does not have. This is a non-simply connected mug and a simply connected mug would be a cup, right? You would burn your hand when you'd drank coffee out of it.”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“Right here in front of me a mug, yes, I might say it's a genus one surface, but we could also say it's a mug, same thing So if I were to draw a little circle on this mug, which way should I draw it so it's visible? Like here.”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“Hor We'd have to kind of have four-dimensional eyes, right? Which we don't. So we have to use our mathematical eyes. We have to envision. The Poincar ⁇ conjecture Says that there's a very simple way to determine whether a three dimensional space Is the standard one, the one that we're used to. And essentially, it's that it's what's called fundamental group has nothing interesting in it. And that I can actually say without saying what the fundamental group is, I can tell you what the criterion is. This would be good. Oh, look, I can even use a visual aid. So for the people watching this on YouTube, you will just see this. For the people on the podcast, you'll have to visualize it. So Lex has been nice enough to give me a surface with some interesting topology.”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“Pwangoi conjecture is about curved three dimensional spaces. So I wasn't my way there, I promise. The idea is that we perceive ourselves as living in, we don't say a three-dimensional space, we just say three-dimensional space. You know, you can go up and down, you can go left and right, you can go forward and back. There's three dimensions in which we can move. In Poincar ⁇'s theory, there are many possible three-dimensional spaces. In the same way that going down one dimension to sort of capture our intuition a little bit more, we know there are lots of different two-dimensional surfaces, right? There's a balloon and that looks one way and a donut looks another way and a mobia strip looks a third way. Those are all like two-dimensional surfaces that we can kind of really get a global view of because we live in three-dimensional space. So we can see a two-dimensional surface sort of sitting in our three-dimensional space. Well, to see a three-dimensional space.”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“I think that's right. I think, don't get me wrong, Poincar ⁇ never strays far from physics. He's always motivated by physics. But the physics drove him to need to think about spaces of higher dimension, and so he needed a formalism that was rich enough to enable him to do that. And once you do that, that formalism is also going to include things that are not physical. And then you have two choices. You can be like, oh, well, that stuff's trash. Or, but, and this is more the mathematician's frame of mind. If you have a formalistic framework that like seems really good and sort of seems to be very elegant and work well and it includes all the physical stuff, maybe we should think about all of it. Maybe we should think about thinking, maybe there's some gold to be mined there. And indeed, guess what? Like before long, there's relativity and there's space time and like all of a sudden it's like, oh yeah, maybe it's a good idea. We already had this geometric apparatus set up for like how to think about four-dimensional spaces. Like turns out they're real after all. This is a”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“What he needed then was a geometry that was flexible enough, not just to talk about two-dimensional spaces or three-dimensional spaces, but any dimensional space, the sort of famous first line of this paper where he introduces analysis situs is no one doubts nowadays that the geometry of n-dimensional space is an actual existing thing, right? I think that maybe that had been controversial. And he's saying like, look, let's face it just because it's not physical doesn't mean it's not there. It doesn't mean we shouldn't study.”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“So it turns out that really to understand what's going on, you can't think of it as a point, or you could, but it's better not to think of it as a point in three-dimensional space that's moving. It's better to think of it as a point in six-dimensional space where the coordinates are where is it and what's its velocity right now. That's a higher dimensional space called phase space. And if you haven't thought about this before, I admit that it's a little bit. Mind bending”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“Analysis Which I guess sort of roughly means like the analysis of location or something like that. It's a Latin phrase. Partly because he understood that even to understand stuff that's going on in our physical world, you have to study higher dimensional spaces. How does this work? And this is kind of like where my brain went to it because you were talking about not just where things are, but what their path is, how they're moving when we were talking about the path from two to three. He understood that if you want to study three bodies moving in space, well, each body, it has a location where it is, so it has an x coordinate, a y coordinate, a z coordinate, right? I can specify a point in space by giving you three numbers. But it also at each moment has a velocity.”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“Right. So he also was one of the pioneers of taking geometry, which until that point had been largely the study of two and three-dimensional objects, because that's what we see, right? That's those are the objects we interact with. He developed the subject we now called topology. He called it analysis situs. He was a very well spoken guy with a lot of slogans, but that named, you can see why that name did not catch on. So now it's called topology now.”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“Two bodies. This is what Newton knew. Two bodies they sort of orbit each other in some kind of either in an ellipse, which is the stable case. You know, that's what the planets do that we know. Or one travels on a hyperbola around the other. That's the unstable case. It sort of like zooms in from far away, sort of like whips around the heavier thing and like zooms out. Those are basically the two options. So it's a very simple and easy to classify story. With three bodies, just the small switch from two to three, it's a complete zoo. It's the first example. What we would say now is it's the first example of what's called chaotic dynamics, where the stable solutions and the unstable solutions, they're kind of like wound in among each other in a very, very, very tiny change in the initial conditions can make the long-term behavior of the system completely different. So Poincare was the first to recognize that that phenomenon even existed.”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“So, the problem is to understand when this motion is stable and when it's not. So stable meaning they would sort of like end up in some kind of periodic orbital, or I guess it would mean, sorry, stable would mean they never sort of fly off far apart from each other and unstable would mean like eventually they fly apart.”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“In many ways, creates the geometric world in which we live. And his first really big success is this prize paper he writes for this prize offered by the King of Sweden for the study of the three-body problem, the study of what we can say about three astronomical objects moving and what you might think would be this very simple way. Nothing's going on except gravity relating to the three.”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“Yes, okay, so back to Buenco Rey. So, You know, it's funny, this book is filled with kind of mathematical characters who often are kind of peevish or get into feuds or sort of have like weird enthusiasms because those people are fun to write about and they sort of like say very salty things. Point ar is actually none of this as far as I can tell. He was an extremely normal dude who didn't get into fights with people and everybody liked him and he was like pretty personally modest and he had very regular habits. You know what I mean? He did math for like four hours in the morning and four hours in the evening and that was it. Like he had his schedule. I actually was like, I still am feeling like somebody's going to tell me now the book is out. Like, oh, didn't you know about this incredibly sordid episode of this? As far as I can tell, a completely normal guy. But he just kind of...”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“The math history touches human history. They're never separate because math is made of people. I mean, that's what it's people who do it and we're human beings doing it and we do it within whatever community we're in and we do it affected by the mores of the society around us.”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“And much of it is true, but Alexander really lays out just how much the way people thought about math in those times, in the early 19th century, was wound up with, as you say, romanticism. I mean, that's when the romantic movement takes place. And he really outlines how people were predisposed to think about mathematics in that way because they thought about poetry that way and they thought about music that way. It was the mood of the era to think about we're reaching for the transcendent, we're sort of reaching for sort of direct contact with the divine. And so part of the reason that we think of Gawa that way was because Gawa himself was a creature of that era and he romanticized himself. I mean now we know he like wrote lots of letters and like he was kind of like, I mean in modern terms we would say he was extremely emo. We wrote all these letters about his florid feelings and like the fire within him about the mathematics and you know so he so it's just as you say that”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“Think I'm going to pick up on something you said. I think you would love a book called Duel at Dawn by Amir Alexander, which I think some of the things you're responding to, what I wrote, I think I first got turned onto by Amir's work. He's a historian of math. And he writes about the story of Everest Galois, which is a story that's well known to all mathematicians, this kind of very, very romantic figure who he really sort of like begins the development of this theory of groups that I mentioned earlier, this general theory of symmetries, and then dies in a duel in his early 20s, like all this stuff mostly unpublished. It's a very, very romantic story that we all learn.”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“Moment where France has just been beaten in the Franco Prussian War. And they're like, oh my God, what did we do wrong? And they were like, we got to get strong in math like the Germans. We have to be more like the Germans. So this never happens to us again. So it's very much, it's like the Sputnik moment, you know, like what happens in America in the 50s and 60s with the Soviet Union. This is happening to France and they're trying to kind of like instantly like modernize.”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“Okay, so Poincar ⁇, he ends up being a major figure in the book. And I didn't even really intend for him to be such a big figure, but he's. First and foremost, a geometer, right? So he's a mathematician who kind of comes up in late 19th century France at a time when French math is really starting to flower. Actually, I learned a lot. I mean, you know, in math, we're not really trained on our own history when we got a PhD in math. What about math? So I learned a lot. There's this whole kind of...”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“Yeah. And, you know, that's part of geometry too. And in fact, again, another insight of modern geometry is this idea that maybe we would naively think we're going to study, I don't know, like Poincar ⁇, we're going to study the three-body problem. We're going to study sort of like three objects in space moving around subject only to the force of each other's gravity, which sounds very simple, right? And if you don't know about this problem, you're probably like, okay, so you just like put it in your computer and see what they do. Well, guess what? That's like a problem that Poincar ⁇ won a huge prize for like making the first real progress on in the 1880s. And we still don't know that much about it 150 years later. I mean, it's a... Humongous myst”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“So, if you ask people, if you show them this morph, if you ask a bunch of people, do they all agree about where the transition happened? Because I would be surprised”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“Do you think it's formalizable when something stops being a two and starts being a three, right? You can imagine something continuously deforming from being a 2 to a three.”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“I would be shocked if there was some kind of classical symmetry type formulation that captured what we're doing when we tell the difference between the two and a three, to be honest. I think what we're doing is actually more complicated than that. I feel like it must be.”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“You know, a kind of symmetry that we understand very well is rotation. Yeah. Right. So here's what would be easy if humans, if we recognized a digit as a one, if it was like literally a rotation by some number of degrees of some fixed One in some typeface like Palatino or something. That would be very easy to understand, right? It would be very easy to write a program that could detect whether something was a rotation of a fixed digit one. Whatever we're doing when you recognize the digit one and distinguish it from the digit two, it's not that. It's not just incorporating one of the types of symmetries that we understand. Now I would say that”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“Exactly, and what's so fascinating about the work in that direction from the point of view of a mathematician like me and a geometer is that the kind of groups of symmetries, the types of symmetries that we know of are not sufficient, right? So in other words, like we're just going to keep on going into the weeds on this.”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“And I have a triangle of the exact same dimensions, but it's over here. Are those the same or different? Well, you might say, like, well, look, there's two different things. This one's over here, this one's over there. On the other hand, if you prove a theorem about this one, it's probably still true about this one if it has like all the same side lanes and angles and looks exactly the same. The term of art, if you want it, you would say they're congruent. But one way of saying it is there's asymmetry called translation, which just means move everything three inches to the left. And we want all of our theories to be translation invariant. What that means is that if you prove a theorem about a thing that's over here, and then you move it three inches to the left, it would be kind of weird if all of your theorems didn't still work. So this question of what are the symmetries and which things that you want to study are invariant under those symmetries is absolutely fundamental. Boy, this is getting a little abstract, right?”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“A name. And just as you can sort of study which kinds of objects are symmetrical under the operations of switching left and right or switching top and bottom or rotating 40 degrees or what have you, you could study what kinds of things are preserved by this kind of scrunch symmetry. And this kind of More general idea of what a symmetry can be. Let me put it this way a fundamental mathematical idea. In some sense, I might even say the idea that dominates contemporary mathematics, or by contemporary, by the way, I mean like the last like 150 years. We're on a very long time scale in math. I don't mean like yesterday. I mean like a century or so. Up till now is this idea that it's a fundamental question of when do we consider two things to be the same. That might seem like a complete triviality. It's not. For instance, if I have a triangle,”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“We could use a symmetry to refer to any kind of transformation of an image or a space or an object. So what I talk about in the book is take a figure and stretch it vertically. Make it twice as big vertically and make it half as wide. That I would call asymmetry. It's not a symmetry in the classical sense, but it's a well-defined transformation that has an input and an output. I give you some shape, and it gets kind of, I call this in the book a scrunch. I just made to make up some sort of funny sounding name for it because it doesn't really have.”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“Well, it's an absolutely fundamental concept, and it starts with the word symmetry in the way that we usually use it when we're just talking English and not talking mathematics, right? Sort of something is when we say something is symmetrical, we usually Or more, and that can take you in a lot of different directions, the abstract study of what the possible combinations of symmetries there are, a subject which is called group theory, was actually one of my first loves in mathematics when I thought about a lot when I was in college. But the notion of symmetry is actually much more general than the things that we would call symmetry if we were looking at like a classical building or a painting or something like that. You know, nowadays in math”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“The intertwining of algebra and geometry, this algebraic fact that, well, in the instance 8 times 6 is equal to 6 times 8, but in general, that whatever two numbers you have, you multiply them one way and it's the same as if you multiply them in the other order. It attaches it to this geometric fact about a rectangle, which in some sense makes it true. So, you know, who knows? Maybe I was always faded to be an algebraic geometer, which is what I am as a researcher.”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“I knew that the timestable was symmetric, but I didn't know why that was the case until that moment. And in that moment, I could see like, oh, I didn't have to have somebody tell me that. That's information that you can just directly access. That's a really amazing moment. And as math teachers, that's something that we're really trying to bring to our students. And I was one of those who did not love the kind of Euclidean geometry, ninth grade class of like prove that an isosceles triangle has equal angles at the base, like this kind of thing. It didn't vibe with me the way that algebra and numbers did. But if you go back to that moment, from my adult perspective, looking back at what happened with that rectangle, I think that is a very geometric moment. In fact, that moment exactly encapsulates.”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“Exactly. And it's just as you say, you know, I knew that 6 times 8 was the same as 8 times 6, right? I knew my times table. Like I knew that that was a fact. But did I really know it until that moment? That's the question.”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“I was like, well, there's six rows of eight holes each. But there's also eight columns of six holes each. So eight sixes and six eights. It's just like the dissection boost you were just talking about. But it's the same holes. It's the same 48 holes. That's how many there are, no matter whether you count them as rows or count them as columns. And this was unbelievable to me. A lot to cost on your podcast. I don't know if that's Okay. It was fucking unbelievable. Okay, that's the last time. Get it in there. This story merits it.”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“There's something special about it We're going to walk around in circles a little bit, but we'll get there. You asked me. How I fell in love with math. I have a story about this. When I was a small child, I don't know, maybe like I was six or seven, I don't know, I'm from the 70s. I think you're from a different decade than that. But in the 70s, we had a cool wooden box around your stereo. That was the look. Everything was dark wood. And the box had a bunch of holes in it to let the sound out. The holes were in this rectangular array, a six by eight array of holes. And I was just kind of like, you know, zoning out in the living room as kids do, looking at this six by eight rectangular array of holes. And if you like just by kind of like focusing in and out, just by kind of looking at this box, looking at this rectangle.”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“Wow, you've given me a lot to say. And certainly the experience that you describe is so typical, but there's two versions of it. You know, one thing I say in the book is that geometry is the cilantro of math. People are not neutral about it. There's people who are like, who like you are like, the rest of it I could take or leave. But then at this one moment, it made sense. This class made sense. Why wasn't it all like that? There's other people, I can tell you, because they come and talk to me all the time who are like, I understood all the stuff we were trying to figure out what X was or some mystery you're trying to solve it, X is a number I figured it out. But then there was this geometry, like what was that? What happened that year? Like, I didn't get it. I was like, lost the whole year and I didn't understand why we even spent the time doing that. But what everybody agrees on is that it's somehow different.”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“There's some dispute about exactly how accurate that is. So then that's an interesting question. If your proof is a diagram, if your proof is a picture, or even if your proof is like a movie of the same pieces, like coming together in two different formations to make two different things, is that language? I'm not sure I have a good answer. What do you think?”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“That's a really interesting question. And one thing reminds me of is one thing I talk about in the book is dissection proofs, these very beautiful proofs of geometric propositions. There's a very famous one by Baskara of the Pythagorean theorem. Proofs which are purely visual. Proofs where you show that two quantities are the same by taking the same pieces and putting them together one way and making one shape and putting them together another way and making a different shape. And then observing that those two shapes must have the same area because they were built out of the same pieces. You know, there's a famous story and it's a little bit disputed about how accurate this is, but then in Basco's manuscript, he sort of gives this proof, just gives the diagram, and then the entire verbal content of the proof is he just writes under it, behold, that's it.”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source
“That's a really interesting question. You're getting me to reflect on this question of whether the feeling of producing mathematical output, if you want, is like the process of uttering language or producing linguistic output. Think it feels something like that, and it's certainly the case. Let me put it this way it's hard to imagine doing mathematics in a completely non-linguistic way. It's hard to imagine doing mathematics without talking about mathematics and sort of thinking and propositions. But, you know, maybe it's just because that's the way I do mathematics that maybe I can't imagine it any other way, right?”
2021-06-13 · Lex Fridman Podcast · #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries · IDENTIFIED FROM THE TRANSCRIPT · source