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.

Listen on Apple Podcasts, Spotify, TuneIn or your favorite podcasting app, or you can stream it from Quanta.

Transcript

[Music plays]

STEVE STROGATZ: I’m Steve Strogatz

JANNA LEVIN: And I am Janna Levin

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

LEVIN: Hey!

STROGATZ: Hey, Janna. Good to see you.

LEVIN: Good to see you.

STROGATZ: I wanted to talk to you about something I think you’re going to like.

LEVIN: Okay, I’m intrigued already.

STROGATZ: Yeah, I think this is right up your alley: Geometry.

LEVIN: Oh, nice. Yeah, everywhere we look there’s geometry.

STROGATZ: 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 going to stick on the floor?

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

STROGATZ: Pulling out Pythagoras?

LEVIN: Yeah. That’s good for a marriage.

STROGATZ: Pythagoras did help us out in our marriage a little bit, cause she did wonder, how, you know, cumbersome is it going to be? Is this thing going to take up a lot of floor space?

LEVIN: Right. Exactly. Have you ever seen those paintings that when you look at them nearly edge on the picture resolves, like you see an image of a face?

STROGATZ: Yeah, yeah, right.

LEVIN: And so these ideas of like projecting geometries, I mean, I think that it’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.

STROGATZ: Yeah, no, it’s all part of our experience as visual beings.

LEVIN: 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.

STROGATZ: Well, I’m not surprised that you are an easy sell on geometry.

LEVIN: Right, exactly.

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

LEVIN: Hmm. Oh, neat.

STROGATZ: I think you’re going to enjoy hearing from him. This is Yang Hui He from London Institute of Mathematical Sciences and University of Oxford.

LEVIN: Awesome. Here we go.

[Music plays]

STROGATZ: Hi, Yang. Really happy to see you here.

YANG-HUI HE: Pleasure. Great to see you.

STROGATZ: Tell me, where are you right now?

HE: 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.

STROGATZ: That’s incredible. Michael Faraday, you want to remind us who was that?

HE: Michael Faraday is arguably the greatest scientist of the Victorian times. The 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 kind 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.

STROGATZ: 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.

HE: I find myself extremely lucky because I’m either in this 18th-century building here where Faraday is, or I am 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.

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

HE: Absolutely.

STROGATZ: But one in particular; I’m wondering if you may have made a pilgrimage to his house or anything like that.

HE: Oh, well absolutely. In fact, the very first thing that I ever did as a freshman in Princeton, like literally I dropped off my bags and 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’s 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.

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

HE: Oh, you were? Oh, right, oh great. Go Tigers.

STROGATZ: 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, but this was Einstein’s secretary. This was Helen Dukas.

HE: 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.

STROGATZ: Oh really?

HE: 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 try to rub off some energy. I never got any.

STROGATZ: Are they literally side-by-side graves?

HE: They’re literally side-by-side, yeah.

STROGATZ: Oh, I didn’t know this.

HE: It’s an incredible little space there.

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

HE: I think we’re going to become very good friends.

STROGATZ: 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?

HE: In the beginning, I was really drawn to mathematical physics because of Newton and Einstein. And I was 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 mathematical physics that I got started doing.

STROGATZ: 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?

HE: You’re absolutely right. I wanted to study differential geometry when I was in my teens. This was the thing that I was told that I needed to study, you know, 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 geometer, I always found manipulating symbols in algebra I liked that more. But now, you know, I got into this path, studied differential geometry. And then, as I grew, I never realized the most beautiful thing is actually the crystal-clear thing of Euclid’s classical geometry. And you know, absolute gold of mathematics in terms of axiomatic formulation. So it’s a long journey.

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

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

STROGATZ: 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’s certain basic things in geometry that I should know that I never actually studied.

HE: I mean, 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.

STROGATZ: Uh-huh. 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?

HE: 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 axioms.

And I think that was Einstein’s great insight. In realizing that to formulate this consistent theory of general relativity, you needed this tool, which is non-Euclidean. You need curved space time, this continuum of space-time.

And then of course, with Riemann and Lobachevsky and Bolyai, the three of them really formulated 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.

STROGATZ: 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?

HE: Yeah, it’s an extremely good question. The word itself is Greek, literally from measuring the earth. But of course, the subject itself predated it. 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 in 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.

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.

STROGATZ: 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?

HE: 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 foundation of what we now call mathematics.

I mean, certainly there are other parts of it, you know, elementary number theory, the famous proof of the infinite number of primes in the elements. But somehow the geometric ones, you can see it, right? Literally, 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.

STROGATZ: Yeah, it’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, Bolyai. 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?

HE: 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’re 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. I mean, that was a really rather way for, not only pure mathematics, but also for mathematical physics, this grounded way of thinking about a geometric approach.

So Gauss, probably the greatest mathematician of all time, and Lobachevsky and Bolyai, they 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 the 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. 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 habilitation 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.

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

HE: 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. What 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 Gedanken 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.

STROGATZ: 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?

HE: There is this wonderful essay. I think it is Milnor, the great Fields Medalist.

STROGATZ: John Milnor.

HE: Milnor 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 you have 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. Which is a great saying.

I think he said it partially as a rebellion against the rise of the Bourbaki school, this purely hardcore formalization, dry definition way of mathematics that the French school came about. I think Arnold and Milner as well, were like this is not where the fun in mathematics is. But this is not how most mathematicians work anyway.

STROGATZ: 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 appreciate the more visual or geometric side.

HE: You know, I keep on oscillating in life. It’s like what they say about music. When you’re, you listen to Mozart. And then when you’re older, 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 listening to Mozart again.

Which certainly happened to Mahler apparently. You know, even on his deathbed, his last words were, “Mozart, Mozart.”

STROGATZ: Really?

HE: So, it’s just in some sense that purity of Mozart is like Euclidean geometry.

STROGATZ: 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?

HE: 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 going to end up going to AI sooner or later. I was trying to think of this difference of what I call bottom-up and top-down mathematics.

STROGATZ: Okay, yeah. What do you mean by those?

HE: So yeah, bottom-up mathematics is 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 going to 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, 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?

STROGATZ: 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, by hand.

HE: By hand. I think he had to the tens of thousands in available data, but beyond a hundred thousand, he didn’t have any more. And then he had to do that by hand. This poor guy.

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

HE: 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 greats.

STROGATZ: 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.

HE: Exactly. That’s a great way of putting it. I guess I’m influenced again by the Russians. I think they describe it as birds and hedgehogs.

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

HE: Exactly. I think it all originated with a Greek parable of foxes and hedgehogs. The fox knows that 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 perestroika. 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 Yakov Sinai, Sasha Polyakov, Sasha Migdal, 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. But they also taught a lot about the philosophy, because this Soviet school was very lively. I think it was very much influenced by Landau and Arnold and people like this, these great thinkers.

So, the version I heard from people like Sinai 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 there are the hedgehogs that dig deep. And that kind of influenced my thought.

Well, I mean, they’re both absolutely necessary. In all of science, we have these two and they’re absolutely necessary. It’s not just a you know, there’s a school of this and there’s a school of them.

All of us as researchers, we must play both roles in different times, right? 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 understand what the depth of the subject is about.

[Music plays]

LEVIN: I think it’s really interesting to hear mathematicians talk about themselves. Like, 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 discussing this? I was not aware of this.

STROGATZ: 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 minutiae. 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 theory builders versus problem solvers. Birds and frogs, yeah.

LEVIN: Yeah, I hadn’t really heard that breakdown before, but it makes sense. I can start to scan my neighborhood and say, “Oh, yeah, there’s a theory builder down there.”

STROGATZ: 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. I mean, isn’t that an interesting thing to say about someone?

LEVIN: That’s hilarious, yeah, that’s hilarious. I also just, one other thing is that what appeals to me is that there’s different types and that the mathematics community needs all types.

STROGATZ: 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.

LEVIN: Hmm. Oh, I’m intrigued.

STROGATZ: Yeah, we’ll be right back.

[Music plays] 

STROGATZ: Welcome back to The Joy of Why. We’re here with mathematical physicist Yang-Hui He.

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?

HE: 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 happen to be working on geometrical problems related to string theory.

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 Witten, Horowitz, Candelas and Strominger when they realized that these are particular shapes. That was in the mid-’80s. 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 happen to be working on these databases. My motivation at the time was to try to sift through them and find which one will give the particle content of the standard model of four-dimensional physics. And 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 kid’s asleep, my wife’s asleep, and I can’t sleep. I 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 hadn’t really seeped into it, but all I knew at the time, my former PhD students and postdocs were entering this 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. You know, topological invariants of these manifolds are not easy to compute. Would it just like, kind of, curve-fit and give an answer?

STROGATZ: 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. And you mentioned that there were invariance 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?

HE: 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, you know, the number of generations of particles, the number of BPS particles in a gauge theory. So they’re very precise physical things like, Betti numbers or Hodge numbers. You know, they literally count the number of holes in these high-dimensional things. And so I was involved in computing some of them, you know, getting these quantities.

But what I took was there are these manifolds, whose representation is some way, and then you attach these topological invariants, which is, in this case, whole numbers.

And that computation is hard because you have to really go through all the sequence chase, basically all that stuff in Hartshorne and all that stuff that Bourbaki has told us that you cry 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.

STROGATZ: Right? It’s the middle of the night.

HE: So, I was literally taking a manifold and representing it as a picture because these are algebraic varieties, 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.

STROGATZ: Okay, I don’t know where this story is going, but I like it.

HE: Right? But then you train 50 percent and you validate. I mean, I was doing a first pass and it was getting to, very quickly, 95 percent accuracy. And I was like, you gotta be kidding me. This is not possible. It can’t be happening. 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 99.99 percent accuracy kind of prediction. That’s really bizarre because it can’t know what it’s doing because I never told it what to do.

STROGATZ: It’s a very good guesser. It’s very good at spotting patterns and extrapolating.

HE: Exactly. So that’s what got me thinking about this whole idea about what is intuition? What is mathematics? What is this primitive version of AI doing?

STROGATZ: Whoa, so this is circa-2017 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.

HE: 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 hands on. So I made like a hundred friends. Just like, can I have some of your data? And then can you explain to me from what branch of mathematics this is coming from?

And I am very grateful to be 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 me 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. But somehow, my upbringing and especially my PhD supervisor, Amihay Hanany, 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 data? Can you explain to me why is it significant to your field of mathematics? And let’s see whether a newer 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.

STROGATZ: Hmm. 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. And, like you mentioned, being in an office close to Michael Faraday, I’m in the office where Bill Thurston was, the great topologist.

HE: Oh, really? Oh, wow.

STROGATZ: 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?

HE: 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 percent — not a hundred, it still gets it wrong sometimes. But 99.99 percent of a say, a topology computation. So what, I learn absolutely nothing from this. It’s not giving me any new tool that I can use. So because of this, over the years, you gosh, eight years, oh gosh.

STROGATZ: Yeah, how old is your kid now?

HE: My son is now eight.

STROGATZ: Yeah, he’s now eight.

HE: Exactly. Oh, I don’t want to think about this. Well, but that is why with my colleague here, Misha Burtsev, we try to formulate a more precise and useful thing. So this came from a conference that I was running in 2023 in the Isaac Newton Institute in Cambridge. We basically got some quantum field theorists, we got some number theorists, and we try 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 Bryan Birch.

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 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.

So, so far, nothing has passed the Birch Test in its entirety. Of course, there are things that come close to it. There was this beautiful paper by the DeepMind Collaboration in 2022, where they had this knot invariance.

STROGATZ: Right.

HE: They were using saliency tests, and they got a new formula for the Jones polynomial.

So that one is automatic because they got different things that did a 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 Pozdnyakov, 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, 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. We didn’t say, well, here’s the data for 3 million elliptic curves, and hit the return button, do whatever, we were guiding it.

So 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 Sarnak, the god of analytic number theory, when we wrote a letter to him, Sarnak said, I’ve never seen a conjecture like this. If Sarnak 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.

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

HE: 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 inferencing 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 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 AI going to replace mathematicians? 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.

STROGATZ: 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?

HE: Well, I mean I’m going to give, of course, the stock answer: 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 that, 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, here on a blackboard. You make a new friend, or you catch up with an old friend, just on this board 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.

STROGATZ: 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.

HE: 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.

STROGATZ: Well, we’ve been speaking here with Yang-Hui He. 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.

HE: Thank you very much. This has been great fun.

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LEVIN: 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 when Columbia was tearing apart some building — are pressed glass. It’s like a bottled glass, like crushed.

STROGATZ: Really?

LEVIN: 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.

STROGATZ: 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?

LEVIN: Uh, ay. I, 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, right? 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, or a triangle or a square. So, I guess if a super-intelligence solves some quantum gravity problem or understands the space-time geometry of the whole universe I’m, I’m not done. I’ve gotta understand it too.

STROGATZ: All right, well, good luck. It’s optimistic, and I’ll see you back here next time.

LEVIN: All right. See you next time.

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LEVIN: Thanks for listening. If you’re enjoying The Joy of Why 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

STROGATZ: The Joy of Why is a podcast from Quanta Magazine, a 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 Why is produced by PRX productions.

The production team is Caitlin Faulds, 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 Frantz and Samir Patel provided editorial guidance with support from Matt Carlstrom, Samuel Velasco, Simone Barr, and Michael Kanyongolo 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 Jaki King and Kristina 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].