How Did Geometry Create Modern Physics?

Geometry is one of the oldest disciplines in human history, yet the worlds it can describe extend far beyond its original use. What began thousands of years ago as a way to measure land and build pyramids was given rigor by Euclid in ancient Greece, became applied to curves and surfaces in the 19th century, and eventually helped Einstein understand the universe. Yang-Hui He sees geometry as a unifying language for modern physics, a mutual exchange in which each discipline can influence and shape the other. In the latest episode of The Joy of Why, He tells co-host Steven Strogatz how geometry evolved from its practical roots in ancient civilizations to its influence in the theory of general relativity and string theory — and speculates how AI could further revolutionize the field. They also discuss the tension between formal, rigorous mathematics and intuition-driven insight, and why there are two types of mathematicians — “birds” who have a broad overview of ideas from above, and “hedgehogs” who dig deep on one particular idea.

• 0:00Speaker 00

  Hi, I'm Terry Tao. My new book, Six Math Essentials, explores six concepts that have guided mathematicians like me throughout history. With a simple introduction to numbers, algebra, geometry, probability, analysis, and dynamics, I share how these ideas helped me think mathematically and can help everyone better understand our complex world. Mathematics isn't magic. It's a way of thinking anyone can learn. Please visit quantobooks.org to learn more.

• 0:31Speaker 01

  I'm Steve Strogatz.

• 0:32Speaker 02

  And I'm Janna Levin.

• 0:34Speaker 01

  And this is The Joy of Why, a podcast from Quantum Magazine, exploring some of the biggest unanswered questions in math and science today.

• 0:45Speaker 02

  Hey.

• 0:45Speaker 01

  Hey Jenna, good to see you.

• 0:47Speaker 02

  Good to see you.

• 0:48Speaker 01

  I wanted to talk to you about something I think you're gonna like.

• 0:51Speaker 02

  Okay, I'm intrigued already.

• 0:53Speaker 01

  Yeah, I think this is right up your alley, geometry.

• 0:56Speaker 02

  Oh, nice. Yeah, everywhere we look, there's geometry.

• 0:59Speaker 01

  Well, there is, right? My wife likes to do art, and she was asking me one time, if our room has a seven-foot ceiling and I have an eight-foot canvas, and I stick it in the corner, how far out is it gonna stick on the floor?

• 1:13Speaker 02

  Right, now, did you really impress her when you came up with a tangent?

• 1:17Speaker 01

  Pulling out Pythagoras?

• 1:18Speaker 02

  Yeah, that's good for a marriage.

• 1:20Speaker 01

  Pythagoras did help us out in our marriage a little bit, and, because she did wonder how cumbersome is it gonna be? Is this thing gonna take up a lot of floor space?

• 1:27Speaker 02

  Right, right, exactly. Have you ever seen those paintings that, when you look at them nearly edge on, the picture resolves?

• 1:34Speaker 01

  Yeah, yeah.

• 1:35Speaker 02

  Like, you see an image of a face?

• 1:36Speaker 01

  Right.

• 1:37Speaker 02

  And so these ideas of projecting geometries, I mean, I think that's probably in a lot of art practices, maybe it's not in the forefront of their mind that that's what they're doing, but that is ultimately geometry.

• 1:45Speaker 01

  Yeah, no, it's all part of our experience as visual beings.

• 1:49Speaker 02

  Oh, yeah. It's really funny, I think some friends of mine might imagine that we're just talking about triangles, but Einstein replaced the entire theory of gravity with a theory of geometry. It's very sophisticated. The universe is a geometry.

• 2:02Speaker 01

  Well, I'm not surprised that you're an easy sell on geometry.

• 2:05Speaker 02

  Right, exactly.

• 2:07Speaker 01

  But so I recently had a chance to speak with someone who feels the same way. He is a theoretical physicist, Yang Weihe, who loves geometry. And he sees geometry as a kind of unifying language for modern physics.

• 2:21Speaker 02

  Hmm, oh, neat.

• 2:22Speaker 01

  I think you're gonna enjoy hearing from him. This is Yang Weihe from London Institute of Mathematical Sciences and University of Oxford.

• 2:30Speaker 02

  Awesome. Here we go.

• 2:34Speaker 01

  Hi, Yang. Really happy to see you here.

• 2:36Speaker 03

  Pleasure. Great to see you.

• 2:37Speaker 01

  Tell me, where are you right now?

• 2:39Speaker 03

  I'm now at the London Institute for Mathematical Sciences, which is the rooms that Michael Faraday used to live and work in. So these are very special, very exciting places. I'm so privileged to be here.

• 2:50Speaker 01

  That's incredible. Michael Faraday, you wanna remind us, who was that?

• 2:55Speaker 03

  Michael Faraday is arguably the greatest scientist of the Victorian times. Faraday, Cage, the pioneer of the electric motor. He met Maxwell, who is my great hero, because Faraday was the experimentalist and Maxwell was the mathematician. And together, they sort of started this theory of electricity and magnetism. So very exciting place. They met just the room behind this wall. We have a beautiful set of exciting rooms to work in.

• 3:20Speaker 01

  Very exciting. I can already tell that you're interested in the history of science. I can only imagine how thrilling it must be for you to be in that kind of sacred territory.

• 3:29Speaker 03

  I find myself extremely lucky because I'm either in this 18th century building here where Faraday is, or I'm in my college in Oxford, which is Merton College, where so many great people were there. And I'm into hero worship. That's why I love history so much.

• 3:46Speaker 01

  Well, I noticed you spent some time at Princeton, where there are also many great heroes of math and physics.

• 3:51Speaker 03

  Absolutely.

• 3:52Speaker 01

  But one in particular, I'm wondering if you may have made a pilgrimage to his house or anything like that.

• 3:57Speaker 03

  Oh, absolutely. In fact, the very first thing I ever did as a freshman in Princeton, like literally, I dropped off my bags and I ran to 112 Mercer Street to find Albert Einstein's old house and just went in and asked very boldly and say, is this a museum? The lady kindly opened the door and said, unfortunately, it's a private home. It was only years later that I found out that lady was actually the wife of Frank Wilczek. Whose condition of moving to Princeton was that he got Einstein's house. So years later, I actually made friends with the family and I told Frank that story and he thought it was very amusing.

• 4:41Speaker 01

  It is. Now, let me tell you. So when I was a freshman at Princeton.

• 4:45Speaker 03

  Oh, you were. I was already told, great, go Tigers.

• 4:48Speaker 01

  I also made a pilgrimage to Einstein's house and did the same thing as you. I knocked on the door and an elderly woman came to the door and said to me, this is a private residence.

• 5:00Speaker 03

  Yeah.

• 5:00Speaker 01

  But this was Einstein's secretary. This was Helen Dukas.

• 5:03Speaker 03

  Oh, no way. Oh, wow. You know, the other crazy thing I did, I spent many of my Sunday afternoons studying in the Princeton Cemetery. Because there's a plot of grass between Gödel and von Neumann, who were buried next to each other. So I would just go and did my quantum problem sheets or measure theory problem sheets in that plot of grass. Just trying to rub off some energy. I never got any.

• 5:30Speaker 01

  Are they literally side by side graves?

• 5:33Speaker 03

  They're literally side by side. Yeah.

• 5:35Speaker 01

  Oh, I didn't know this.

• 5:36Speaker 03

  It's an incredible little space there.

• 5:39Speaker 01

  Well, all right, I could talk to you all day long about all kinds of things.

• 5:42Speaker 03

  Yeah, I think we're gonna become very good friends.

• 5:44Speaker 01

  I would love to. So yes, this is our first time having a chance to meet and talk. But let's dive into the question of a very classical subject. Many of our listeners may be thinking of geometry from their high school geometry class. It's not something that is an obvious subject for today's research mathematician in the popular mind. Tell us a little about your own backstory. Did you know that you were interested in geometry from a young age?

• 6:10Speaker 03

  In the beginning, I was really drawn to mathematical physics because of Newton and Einstein. And also trying to understand and learn special relativity, general relativity. And then I found that we have to study advanced geometry. And the irony is that the more I got into it, the more and more I became interested in the actual geometry rather than in mathematical physics that I got started doing.

• 6:33Speaker 01

  So if I hear you, then, when you're speaking of geometry, you put that adjective in front of it, advanced geometry. So you're not talking about the triangles and parallel postulate geometry. Maybe you're talking about differential geometry or something?

• 6:47Speaker 03

  You're absolutely right. I wanted to study differential geometry when I was in my teens. This was a thing that I was told that I needed to study in order to understand things like general relativity and all these advanced theories of modern contemporary physics. And I didn't really have much of a taste for Euclidean geometry at the time. In this division between algebraist and geometry, I always found manipulating symbols in algebra like that more. But now, you know, I got into this path, studied differential geometry. And then, as I grew, and I realized the most beautiful thing is actually the crystal clear thing of Euclid's classical geometry,

• 7:29Speaker 03

  you know, absolute gold of mathematics in terms of axiomatic formulation. So it's a long journey.

• 7:35Speaker 01

  Oh, that is very interesting, that it was only with greater maturity that you began to appreciate the more elementary subject.

• 7:42Speaker 03

  Exactly. I think at the time, I was just too naive to see the absolute beauty of this.

• 7:47Speaker 01

  That's very interesting to me because I asked to be let out of my high school geometry course. I thought it was so tedious. I started to get bad grades because I was so bored. And the school was so nice that they said, you shouldn't be getting the grade you're getting in geometry. We think you're bored, so starting tomorrow, you're in pre-calculus. And so, to this day, there are certain basic things in geometry that I should know that I never actually studied.

• 8:15Speaker 03

  It's the same here. And I'm going back to pour over, you know, the classical axioms. And I find them just so clean and so beautiful.

• 8:23Speaker 01

  Well, so we've already used this bit of jargon, differential geometry. So help us understand what is that? And why do you regard it as an important subject?

• 8:32Speaker 03

  In terms of modern physics, by modern, I mean, post 20th century physics as opposed to Newtonian physics. One of the key ideas is to break away from Euclidean geometry, from planes and intersection of lines in planes according to Euclid maxims. And I think that was Einstein's great insight. You realize in that to formulate this consistent theory of general relativity, you needed this tool, which is non-Euclidean. You need curved spacetime, this continuum of spacetime. And then, of course, with Riemann and Lobachevsky and Bolyai, the three of them really formulated

• 9:11Speaker 03

  by the end of the 19th century, this idea of local geometry, curved geometry, which we now call differential geometry, which was completely crucial. And Einstein's insight was just to recognize that was exactly the tool he needed.

• 9:26Speaker 01

  Hmm, there's so much good stuff here. Before we dismiss Euclidean geometry, I want to appreciate with you how much that did for the history of the world. You have an appreciation for geometry done in Mesopotamia, in Egypt, all around the world. So where did geometry come from originally? Why were people doing geometry in the really ancient times, maybe like even 3,000 B.C. or something?

• 9:51Speaker 03

  Yeah, it's an extremely good question. The word itself is Greek, literally from measuring the earth. But of course, the subject itself is predated, you know, the Babylonian tablets that you find now with Pythagoras's theorem and, you know, ancient Chinese carvings and Indian carvings as well. These very basic things that we just see the world around us. And I think it's very instinctively human. The first thing we see around us is inherently geometrical. We see shapes and sizes and distances. And gradually evolved this need to understand the relations. But I think really to truly formulate it from a rigorous point of view, that credit goes to the Greeks.

• 10:35Speaker 03

  Euclid, you know, he had his 13 books of the Elements, where he collected all of the then-known knowledge of ancient mathematics. And I think nine of the 13 books were devoted to geometry, because that's a very intuitive subject.

• 10:50Speaker 01

  Let me see if I get the distinction you're making there. There's geometry in the pre-Euclidean time of building sacred temples, whether in India or building the pyramids in Egypt. Geometry for architecture, for land measurement, surveying, maybe a little bit for astronomy. But then at some point, Euclid introduces this idea of rigor and axioms. High school teachers love to make a big deal about that, that geometry is where you learn about proof. You don't see so much about proof in more elementary courses, but in geometry, we all get exposed to it. Why is that such a big thing in the history of human thought?

• 11:29Speaker 03

  I mean, I think that's really what makes mathematics. Because, you know, mathematics is nothing unless you have rigorous and precise definitions and precise proofs and precise derivations. And prior to Euclid, of course, there's this entire school of thought. But Euclid somehow needed to collect everything and then formulate all of this in this precise language. And geometry was this very intuitive way toward that language. That's what makes the Elements so special. And that was really the foundations of what we now call mathematics. And we certainly, there are other parts of it, you know, elementary number theory, the famous proof of the infinite number of primes in the Elements.

• 12:12Speaker 03

  But somehow the geometric ones, you can see it, right? Literally, it's here's a triangle, and then I'm going to try to define this triangle that we've always known about in a rigorous way. And that's a truly amazing thing.

• 12:24Speaker 01

  That's interesting. So we were talking about Euclid and the legacy of Euclid on education and on human culture, certainly Western culture, but really world culture. You mentioned some names like Lobachevsky, Bolliay. For people who aren't so sure who those gentlemen were. Tell us about where geometry went after Euclid. So 300 B.C., we have Euclid. Then what?

• 12:48Speaker 03

  Yeah, and then, you know, every culture started catching up in some ways. And there must be some kind of communication between them. Interestingly, this classical Euclidean approach was central even to as far in time as Newton. When Newton was formulating his laws, when he was deriving the planetary motions, he certainly didn't have the notation of the calculus that we were using today. A lot of his derivations in Principia were purely geometrical. I think he was able to derive Gauss's law by summing up little pieces in a sort of brute force geometrical way, which is quite a feat. To me, that was a really rather definitive way for not only pure mathematics,

• 13:29Speaker 03

  but also for mathematical physics, this grounded way of thinking about a geometric approach. So Gauss was probably the greatest mathematician of all time. And Lobachevsky and Bélier were contemporaries to Gauss, and they had this idea even before Gauss. What if they were to relax one of the axioms of Euclid? You know, the so-called parallel postulate? That if I give you a fixed line and a point not on this line, in Euclidean geometry, there would exist one and only one line that is parallel to this that passes through that line. So they were thinking about trying to get rid of that postulate. They formulated possible alternative axioms to this,

• 14:12Speaker 03

  and they were able to formulate a new form of geometry. Gauss got very inspired by this, and he and then Riemann laid the foundations. In fact, I think Riemann's habilitación thesis defense was entitled "On the Foundations of Geometry", and Gauss was one of the examiners in that defense. Riemann was doing what we now call differential geometry, which is a generalization of this classical Euclidean approach to geometry.

• 14:38Speaker 01

  Uh huh. And so you mentioned, zooming a little bit closer to our own time, that Einstein

• 14:45Speaker 01

  Was somehow influenced by this geometry of Riemann. So what is it that Einstein is going to do with this math that comes from Riemann?

• 14:52Speaker 03

  This is truly Einstein's genius. He has this intuition that's almost superhuman, just how to find the right pieces of stuff. Special relativity came out of nowhere, which you needed in special relativity was essentially high school algebra. But he somehow realized, just by playing around with clocks and measuring rods in his Galavantian experiments, this was the right thing to do. I don't know the time sequence, did he have this intuition, then he approached a mathematician friend like Grossman, or maybe even his wife Mileva, who could teach him the right piece of mathematics so that he could deep dive. And that stuff happened to be formulated by Riemann.

• 15:34Speaker 01

  Oh, you make me think of so many different directions for us to go now. Like, you talk about rigor in Euclid, and often people associate geometry with rigor, with proof, absolute certainty, theorems that stand for thousands of years and we know they'll never be overturned because they're proven. However, there's this other aspect of math and physics, intuition. Tell us about that.

• 15:58Speaker 03

  There is this wonderful essay, I think it is Milner, the great Phil's medalist.

• 16:04Speaker 01

  John Milner.

• 16:04Speaker 03

  Milner gives an excellent essay about how people thought mathematicians work by this very dry process of formulating and definition. And yet, he says, what mathematicians really do is have this intuition and then go sometimes backwards towards these definitions. So he had this very intuitive approach to mathematics, which I think Vladimir Arnold is a great fan of. In fact, I think Arnold even calls mathematics a branch of physics. He says mathematics is a branch of physics in which experiments are cheap.

• 16:39Speaker 03

  Which is a great saying. I think he said it partially as a rebellion against the rise of the Boubaki school. This purely hardcore formalization, dry definition way of mathematics that the French school came about. And I think Anand and Mionner as well were like, this is not where the fun in mathematics is. But this is not how most mathematicians work anyway.

• 17:05Speaker 01

  It's a very interesting sociological question because I feel like throughout the history of math, we see these two impulses. There are the people who say we should be intuitive and visual and there's always reaction. Like I think Lagrange makes a big point of saying that in his book on mechanics, you will not see a single diagram. He's proud of that. He doesn't want any diagrams because he thinks visualization is your enemy. You could convince yourself of something wrong from the picture. So there's always this drive and counter reaction, either towards more algebra or more geometry. So it's interesting to me that in recapitulating your own life, you say you started out and have always loved the algebraic manipulation side of things and then later have come to

• 17:50Speaker 01

  appreciate the more visual or geometric side.

• 17:54Speaker 03

  You know, you keep on oscillating in life. It's like what they say about music. When you're little, you listen to Mozart and then when you're older, when you're a teenager, you listen to all that romantic stuff. When you're mature, you listen to Bach. And then when you're about to die, you go back to listen to Mozart again. Which certainly happened to Mahler, apparently. You know, even on his deathbed, his last words were Mozart, Mozart.

• 18:18Speaker 01

  Really?

• 18:19Speaker 03

  So which has in some sense that purity of Mozart is like Euclidean geometry.

• 18:23Speaker 01

  It's so interesting because where does creativity come from? I feel like part of what we're talking about is the genesis of math in the human mind. Because as beings who get to move around in the world, who see things, we get to have intuition. We keep saying intuition. Let me ask you something hard. What do you mean by intuition?

• 18:42Speaker 03

  So you go back to the schools. I believe the intuitionist school of mathematics did exist. And then there is the formalist school of Hilbert. Hilbert and to the extreme, Russell and Whitehead, that kind of formalistic building up. But intuition is, yeah, I guess I'm gonna go to AI sooner or later. I was trying to think of this differences of what I call bottom up and top down mathematics.

• 19:09Speaker 01

  Yeah, what do you mean by those?

• 19:11Speaker 03

  So yeah, bottom up mathematics, this kind of Hilbert, Russell, Whitehead, building line by line, axiomatic, Euclidean, if you wish. This kind of way of building up mathematics from bottom up. And this top down is this intuition guided mathematical discovery, which is, well, I'm gonna look at all kinds of data, including formulae and sentences and papers. And I'm trying to get a rough idea of what's going on and then make some kind of a prejudice in my head.

• 19:45Speaker 03

  And then that'll guide me down some path. But then the great thing about mathematics is that I can check whether it's right or wrong. I can go back to this great example of a great intuition guided mathematical discovery. When Gauss plotted the prime counting function, the P of X, which is the number of prime numbers less than a given real positive X. And he just literally plotted it and he could look at this and it looks like X over log of X. It's very interesting that he got to this because there was no regression. I think historians have it that he actually invented the statistical method of regression in order to even formulate this. How did he do this?

• 20:30Speaker 01

  The way you're describing it, I mean, here's Gauss. He looks at examples. So like there are 25 prime numbers less than 100, turns out. Then we could say how many are less than 1,000? So we could keep track of how many prime numbers are less than a given number. And that's the thing you're saying Gauss was able to guess an approximate formula for that. And so you want to call that intuition because he looked at a lot of data. In fact, the legend or probably the true story is that he compiled enormous tables of prime numbers.

• 20:58Speaker 03

  By hand.

• 20:59Speaker 01

  By hand.

• 20:59Speaker 03

  I think he had to the tens of thousands in available data, but beyond 100,000, he didn't have anymore. And then he had to do that by hand. He's a poor guy.

• 21:10Speaker 01

  But you want to call that intuition rather than just like experimental evidence?

• 21:14Speaker 03

  But I think that is, the intuition is built upon that kind of experiment. But first of all, just to think of defining the prime counting function, why would you want to do something like this? It's like primes are not continuous things, but you just have this idea, I'm going to play around with this continuous thing. And then I'm going to plot it, calculate it, and then make a guess that should be roughly the shape of x over log x. That's the kind of almost divine approach to mathematics, but that happens to all the grades.

• 21:45Speaker 01

  Then the distinction is that bottom up, in your way of using the term, means building from the foundations. The solid definitions, the axioms, you're like a machine grinding out theorems, always on a secure footing because everything comes from the foundations. Whereas top down is more like you're a person in the world or a being in the world who sees things happening. Maybe you're a little kid putting stuff in your mouth when you're a baby. And you start to learn about different shapes because of the way it feels when they touch your lips. Like you get experience by being a person or by looking at tables of numbers like Gauss. And gradually that gives you a feeling about the way the world works. And that's what you're going to call top down intuitive style of math, not building from definitions.

• 22:30Speaker 03

  Exactly. That's the way I would put it in it. I guess I'm influenced again by the Russians. I think they describe it as birds and hedgehogs.

• 22:37Speaker 01

  Ah, tell us what you're talking about. There's Freeman Dyson's essay about birds and frogs.

• 22:42Speaker 03

  Exactly. I think it all originated with a Greek parable of foxes and hedgehogs. The fox knows a bit of this, bit of that, and hedgehog really digs deep. But the version that I first heard was actually from my Russian professors in Princeton. I went to Princeton in the time when it was just after Parastroyka. America was completely flooded by really top notch former Soviet mathematicians and physicists. So in my generation, my professors, I was so lucky to have people like Yakhov Sennai, Sasha Paljakov, Sally Sasha McDowell, these people. So not only they teach us, you know, a lot of stuff, which to be fair, was completely over our head at the time.

• 23:26Speaker 03

  But they also taught a lot about the philosophy because this Soviet school is very lively. I think it was very much influenced by Long Tao and Arnold and people like this, these great thinkers. So the version I heard from people like Sennai and McDowell were they're the birds or eagles who fly very quickly and they have this glimpse of an overview of mathematics from the top. And then they're the hedgehogs that dig deep. And that kind of influenced my thought with, I mean, they're both absolutely necessary. In all of science, we have these two and they're absolutely necessary. It's not just, you know, there's a school of this and there's a school of that. All of us as researchers, we must play both roles in different times, right?

• 24:07Speaker 03

  I guess I'm trying to say is sometimes the intuition comes by not digging deep, but just stepping back a bit. But of course, you can only do that after you've dug deep to be even to understand what the depth of the subject is about.

• 24:22Speaker 02

  I think it's really interesting to hear mathematicians talk about themselves. I don't think necessarily mathematicians do that that often, self reflect, right? There's the work and the work is all consuming and there's not always a lot of self analysis. You know, it doesn't really lend itself to that kind of a disposition. It's also kind of reassuring that it's not this immediacy. There's some wandering in the dark, there's some intuition and then there's some burrowing. I think it's really an interesting analogy. You had heard of Freeman Dyson's discussing this. I was not aware of this.

• 24:54Speaker 01

  Oh yeah, Freeman Dyson has a nice essay about this, Birds and Frogs. It's close in spirit to foxes and hedgehogs. They're a little bit different animals being talked about, but the birds fly overhead. The birds are often what people refer to as theory builders. They see a whole subject, but they're not interested in minutia. They're not interested in solving specific, concrete problems. They're interested in global structures that unify whole disciplines. Whereas the frogs are in the mud. Down low, solving very concrete problems, one by one. And out of that, a theory may emerge. But they're not interested in building theories. So you sometimes hear the distinction of theory builders versus problem solvers. Birds and frogs.

• 25:36Speaker 02

  I hadn't really heard that breakdown before, but it makes sense. I can start to scan my neighborhood and say, oh, yeah, it's a theory builder now.

• 25:45Speaker 01

  Yeah, it's a charming essay. One of the funniest things that Dyson says is he has his own take on various famous mathematicians. And if I remember right, he says von Neumann was a frog who thought he was a bird.

• 26:00Speaker 01

  I mean, isn't that an interesting thing to say about someone?

• 26:03Speaker 02

  Yeah, that's hilarious. I also just one other thing that appeals to me is that there's different types and that the mathematics community needs all types.

• 26:12Speaker 01

  Right, it's a big tent, room for lots of different styles. We're going to be talking about another style of doing math after the break that is the way our artificially intelligent friends might do it.

• 26:22Speaker 02

  Hmm, oh, I'm intrigued.

• 26:24Speaker 01

  Yeah, we'll be right back.

• 26:42Speaker 01

  Welcome back to the joy of why we're here with mathematical physicist Yang Wei Ha.

• 26:50Speaker 01

  Let us switch gears here to start talking about our silicon friends, the AIs. As everyone knows, like 2022 ChatGPT took the world by storm. We started to see artificial intelligence, not just as a thing to read about in magazines, but it was right there on your laptop and you could play with it. It's astonishing what it could do. But what about its relationship to physics or math? Do you see some role for AI in those domains?

• 27:20Speaker 03

  Oh, absolutely. I mean, I think that really the future of mathematical discovery or theoretical discovery is a combination of human intelligence, human expertise with AI assisted discovery. It's already happening. I got into this whole business back in 2017. Before this, I knew nothing about AI or machine learning and all that stuff. But I happened to be working on geometrical problems related to string theory.

• 27:47Speaker 03

  I had been primarily working on spaces called Calabi-Yau manifolds. Because string theory is in 10 dimensions and we live in four dimensions. We need to reduce the six extra dimensions. One of the solutions for the six dimensions that will ultimately produce Einstein's equation was this paper by Witton, Horowitz, Kandelis and Strominger. When they realized that these are particular shapes. That was in the mid eighties. And then for 20 years, mathematicians and physicists have been compiling databases of this geometry. The current most complete database of Calabi-Yau manifolds is about half billion. I happened to be working on these databases. My motivation at the time was to try to sift through them and find which one will give

• 28:32Speaker 03

  the particle content of the standard model of four dimensional physics. So that was a fairly specialized task. But 2017, my son was born. That meant for at least three months I didn't sleep. Like literally, maybe an hour a day or something. I was completely out of it. One night, I remember very well, the kids asleep, my wife asleep. And I can't sleep, but have this data of tens of thousands, even millions of such manifolds. People were talking a lot about data science and big data. The word AI hasn't really seeped into. But all I knew is at the time, my former PhD students and postdocs were entering this

• 29:15Speaker 03

  field to learn about machine learning. I said, well, let me try to play around with some of the algorithms to see how some generic neural network would start computing invariants, which are quite complicated stuff. The topological invariance of these manifolds are not easy to compute. Would it just like kind of curve fit and give an answer?

• 29:37Speaker 01

  So if I'm following the story, you're saying there's a lot of different ways of dealing with those six extra dimensions and they're all somehow encoded in various Calabi-Yau manifolds?

• 29:48Speaker 01

  And you mentioned that there were invariants, numbers, maybe even whole numbers or other nice structures associated to these different manifolds. Were those already computed, or was part of your job to compute them?

• 30:02Speaker 03

  So part of my job was to compute. When I was a postdoc, when I was a PhD student, I was doing a lot of this computation. Because as a young scientist, these were things you can translate to precise things, like the number of generations of particles, the number of BPS particles in a gauge there. So they're very precise physical things like Betty numbers or Hodge numbers. They literally count the number of holes in these high dimensional things. And so I was involved in computing some of them, getting these quantities. But what I took was there are these manifolds whose representation is some way. And then attach this topological invariant, which is, in this case, hole numbers. And that computation is hard, because you have to really go through all the sequence

• 30:45Speaker 03

  chase, basically all that stuff in Hartshorn and all that stuff that Bobaki has told us that he cried about to try to compute. I mean, luckily, a lot of this has been at least automated by computers. So but then, because I was half hallucinating.

• 30:58Speaker 01

  Right, it's the middle of the night.

• 31:00Speaker 03

  So I was literally taking a manifold and representing it as a picture. Because this is algebraic variety, so you can record the multi-degrees. You can get them into some kind of tensor representation. But once you have this tensor representation, it's essentially a picture. So now I have a labeled picture problem. And so, oh, that looks cool. So I just fed it into some neural network, MNIST neural network, the standard ones. And I was expecting to get complete crap. Because in order to predict these things, you would really have to know some quite subtle algebraic geometry.

• 31:33Speaker 01

  Okay, I don't know where this story is going, but I like it.

• 31:36Speaker 03

  Right, but then you train 50% and you validate. I mean, I was doing the first part, and it was getting to very quickly 95% accuracy. And I was like, you've got to be kidding me. This is not possible. It can't be happening. The the neural network clearly doesn't know any geometry, doesn't even know any math. It's just basically doing handwriting recognition. And it was getting the right answers. And now people have improved this. They're using sophisticated architectures to get into ninety nine point nine nine percent accuracy in kind of prediction. That's really bizarre because you can't know what it's doing because I never told it what to do.

• 32:13Speaker 01

  It's a very good guesser. It's very good at spotting patterns and and extrapolating.

• 32:18Speaker 03

  Exactly. So that's what got me thinking about this whole idea about what is intuition, what is mathematics, or what is this primitive version of AI doing?

• 32:27Speaker 01

  Whoa. So this is circa twenty seventeen AI doing this, and you say today's AI is much more accurate on this kind of problem. It's really helping people in string theory or other parts of algebraic geometry or whatever.

• 32:43Speaker 03

  So at the time, I became a fanatic. I was like, this is awesome. Let's take, you know, every data set in mathematics you can get your hand on. So I made like a hundred friends. Just like, can I have some of your data? And can you explain to me from what branch of mathematics this is coming from? But I am very grateful to being trained as a string theorist. Because I know string theory is now a quite controversial thing. But whatever it is, being trained as a string theorist, what it gave him was the opportunities to talk to specialists in different branches of mathematics with somewhat of an ease. I had to learn a little bit of group theory, a little bit of representation theory, a little bit of number theory, a little bit of this, a little bit of that. And of course, not all string theorists are trained this way.

• 33:24Speaker 03

  But somehow, my upbringing, and especially my PhD supervisor, Ami Ha Hanani, he always told me to just look at different problems, you know, just be curious all the time. So I basically spent like the last seven years just making friends and just asking them, can you give me a data? Can you explain to me why is it significant to your field of mathematics? And let's see whether a neural network can do better than your computation. And why would it be doing better than you? And the general mathematics, you know, very open, very receptive to this idea.

• 33:55Speaker 01

  But I know that there are some mathematicians who feel that this could be a real distortion of our shared culture. Right. The great legacy of Euclid is that we can understand what we're doing. When we build up from the axioms, we have clear insight into why theorems are true. We're not in Babylon, just empirically noticing the Pythagorean theorem. We're coming up with very clear proofs. And a lot of people champion this as the best thing about math is insight. And so I have friends, and I'm sure you do too, who feel that insight is the ultimate goal in math. For instance, there's an old quote that the goal of numerical analysis is not numbers, it's insight.

• 34:37Speaker 01

  And like you mentioned, being in an office close to Michael Farady, I'm in the office where Bill Thurston was, the great topologist.

• 34:44Speaker 03

  Oh, really? Oh, wow.

• 34:46Speaker 01

  Yeah. So he wrote this nice article, "Proof and Progress in Mathematics", arguing that it's not about churning out one theorem after another like a machine, like Hilbert would have us do. It's about one human being conveying insight to another human being. That's what we're trying to do. These AIs are not giving us insight. They're giving us patterns, but they can't explain themselves yet. They don't tell us how to think. So I'm not sure every mathematician likes this development. What's your take on all that?

• 35:15Speaker 03

  You're absolutely right. So the initial experiments and the biggest criticism that I got from various people is that, well, fine, you've given me some neural network that can compute 99.99%. Not 100, it still gets it wrong sometimes, but 99.99% of a, say, a topology computation. So what? I learned absolutely nothing from this. It's not giving me any new tool that I can use. So because of this, over the years, gosh, eight years, oh, geez.

• 35:47Speaker 01

  Yeah, how old is your kid now?

• 35:49Speaker 03

  My son is now eight.

• 35:50Speaker 01

  Yeah, he's eight.

• 35:51Speaker 03

  Exactly. I don't want to think about this. But that is why with my colleague here, Misha Burzov, we tried to formulate a more precise and useful thing. So this came from a conference that I was running in 2023 in Isaac Newton Institute in Cambridge. We basically got some quantum field theorists, we got some number theorists, and we tried to run a three-month workshop to get everybody together, to have a conversation about what's happening here. And during this conference, we were listening to a lecture given by Brian Birch.

• 36:31Speaker 03

  And we wrote this quick correspondence for Nature, which was entitled The Birch Test. With all this AI guided discovery, we need something that's a little bit better than what ChatGPT can do. You know, ChatGPT passes the Turing test. So we felt we needed something stronger for math. We called it the AIN for AI discovery. So AIN are the three components of the Birch test. It has to be A for automaticity, so that the conjecture raised by your neural network can't be influenced by humans during its process. It has to be I interpretable. So how do you interpret an actual human piece of mathematics out of it? And then N has to be non trivial in the sense that it has to galvanize an actual research topic precise enough for humans to work on.

• 37:21Speaker 03

  So so far nothing has passed the Birch test in in its entirety. But of course, there there are things that come close to it. There was this beautiful paper by the DeepMind collaboration in twenty twenty two, where they had this knot in variant. Right.

• 37:39Speaker 03

  They were using saliency test and they got a new formula for the Jones polynomial. So that one is automatic because they got different things that did the saliency test. It was certainly interpretable because they got a formula out of it. It wasn't non trivial enough in the sense that a conjecture that they raised, they were able to prove themselves within a month. So you test different things like this murmuration thing that I was working with Oliver and Lee and Posniakov, where we raised the conjecture in the distribution of L functions. That one passed the interpretability test because it was a precise formula. It was non trivial in the sense that it's still open. The parts of this conjecture have been proven, but not all of it. But it fails the A test because we were kind of mucking around.

• 38:24Speaker 03

  We didn't say, well, here's the data for three million elliptic curves and hit the return button, do whatever we were guiding it. So we were we were choosing different algorithms, but ultimately it led to a conjecture that was very counterintuitive. It was beyond what we could intuit because it was such high dimensions. Even specialists like Peter Sonak, the god of analytic number theory. When we wrote a letter to him, Sonak said, I've never seen a conjecture like this. If Sonak doesn't know the proof of this, that's probably a hard one. So he's still working on it with his team of students and we're still working on it together. So it's kind of interesting. So to your question about what merits a truly good AI assisted discovery? I think, you know, at least it must pass the Birch test.

• 39:06Speaker 01

  I had not heard about the Birch test before. Well, as you may know, if you follow any of the discussions about math education, there's a movement these days to introduce more data science into education. People make the argument that what you need to be an informed citizen in the 21st century is an ability to work with numbers, read charts, have a feeling about probability and risk and uncertainty. And by this line of reasoning, since there's only so many hours in the day, something in the classical math education may need to go. A lot of people are pointing their finger at things like trigonometry

• 39:47Speaker 01

  and even maybe geometry in favor of a more data oriented curriculum. What do you think about any of that?

• 39:56Speaker 03

  I think it is certainly important to learn data science. To the modern human, it's inevitable. An education won't be complete without a basic understanding of machine learning algorithms and little bit of statistical inference in from data. But there's something about teaching mathematical rigor, not necessarily at a university level, but just at a generalist level. There is value in teaching this mathematical rigor because it sharpens the the the human mind and teaches us about where we came from, about how we are human. You know, I had this wonderful conversation with the great Andrew Wiles about the advance of AI, is when AI going to replace mathematicians?

• 40:38Speaker 03

  And he gave a very good answer, which is, he says, well, you know, the calculator. Has made completely effectively obsolete any need for arithmetic. Why would you ever want to do that? And yet, we still teach four and five year olds arithmetic because that's what makes us human. And of course, there's some other stuff that can do arithmetic better than us. And yet, just in terms of shaping the neural pathways of our minds, teaching kids that is important.

• 41:11Speaker 01

  It's a deep answer. Well, our show is The Joy of Why. What is it about your research or about being a mathematician and a physicist that brings you joy?

• 41:20Speaker 03

  Well, I mean, I'm going to give, of course, the star caster, the beauty of the subject, the joy of understanding. It's interesting now that I got into this whole field as a teenager, trying to understand the workings of the universe. But now I think the universe is so vast. I've given up trying to understand it. But I've gotten more and more interested in just the pure beauty of the mathematics itself. What I do understand is, well, here's a theorem that I can try to prove. Here's a calculation that is very inherently beautiful. And I think the only thing more beautiful than doing mathematics is making a lot of friends to do mathematics together. Like you're on a blackboard. You make a new friend or you catch up with an old friend just on this board.

• 42:03Speaker 03

  And you just, you talk about this beautiful thing that Euclid would have done with chalk from centuries ago. And that sense of human community is also an amazing feeling.

• 42:12Speaker 01

  It is community, right? And as you say, community across the centuries that you can feel like, in a way, you're communing with Euclid and Pythagoras and the Babylonians.

• 42:21Speaker 03

  Yeah, like when you write on your blackboard, you know, that was the blackboard that Thurston wrote on. That's a beautiful thing to do.

• 42:28Speaker 01

  Well, we've been speaking here with Yang Weiha. I'm really glad to get to know you and wish we could have talked for hours more. Thanks again for being with us on The Joy of Why.

• 42:37Speaker 03

  Thank you very much. It's been great fun.

• 42:42Speaker 02

  That's a very sweet image, working at the chalkboard. It's still the best technology. It really is. Chalk on a nice, smooth board is magical. And the old boards, I don't know if you know this, but I actually salvaged one

• 42:56Speaker 01

  Yeah.

• 42:57Speaker 02

  when Columbia was tearing apart some building, are pressed glass. It's like a bottled glass, like crushed.

• 43:03Speaker 01

  Really?

• 43:04Speaker 02

  And they're so smooth to write on, and it's not like chalkboard paint or anything like that. They're just a pleasure, and that is still the best technology.

• 43:11Speaker 01

  I'd be curious to hear your reaction to this question of artificial intelligence and what it will do for math and, let's say, very arcane parts of physics. Suppose the scenario does come to pass that the best theorems some decades from now, or maybe even sooner, are theorems proved and even created by machines. The machines have left us so far in the dust. Will it still be fun to do math? Will it be worth doing?

• 43:39Speaker 02

  Aye, aye. I think that I do math that other people understand until I understand it. Right? So it's not trying to learn just something as basic as geometry, as we were discussing. I'm not satisfied with the fact that the world understands geometry, therefore I don't have to. That doesn't really work like that. I still try to learn it. I still try to understand it. I still try to acquire it. It's not as though, oh, well, somebody else has it in their brain, therefore it doesn't need to be in my brain. It's almost the opposite. It should kind of be touching all of our minds, right? You know, it's this one thing we all can share. And just by doing these thought experiments of trying to understand a circle, or a triangle, or a square.

• 44:24Speaker 02

  So I guess if a superintelligence solves some quantum gravity problem, or understands the spacetime geometry of the whole universe, I'm not done. I've got to understand it, too.

• 44:36Speaker 01

  All right. Well, good luck. That's optimistic. And I'll see you back here next time.

• 44:43Speaker 02

  All right. See you next time.

• 44:49Speaker 02

  Thanks for listening. If you're enjoying The Joy of Y and you're not already subscribed, hit the subscribe or follow button where you're listening. You can also leave a review for the show. It helps people find this podcast. Find articles, newsletters, videos, and more at quantamagazine.org.

• 45:07Speaker 01

  The Joy of Y is a podcast from Quanta Magazine, an editorially independent publication supported by the Simons Foundation. Funding decisions by the Simons Foundation have no influence on the selection of topics, guests, or other editorial decisions in this podcast or in Quanta Magazine. The Joy of Y is produced by PRX Productions. The production team is Caitlin Folds, Livia Brock, Genevieve Sponsler, and Merritt Jacob. The executive producer of PRX Productions is Jocelyn Gonzalez. Edwin Ochoa is our project manager. From Quanta Magazine, Simon France and Samir Patel provided editorial guidance with support from Matt

• 45:48Speaker 01

  Carlstrom, Samuel Velasco, Simone Barr, and Michael Kanjungalo. Samir Patel is Quanta's editor-in-chief. Our theme music is from APM Music. The episode art is by Peter Greenwood and our logo is by Jackie King and Christina Armitage. Special thanks to the Columbia Journalism School and the Cornell Broadcast Studios. I'm your host, Steve Strogatz. If you have any questions or comments for us, please email us at [email protected].