Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI | Lex Fridman Podcast #472

The following is a conversation with Terrence Tao. Widely considered to be one of the greatest mathematicians in history. Often referred to as the Mozart of math, he won the Fields Medal and the Breakthrough Prize in mathematics and has contributed groundbreaking work to a truly astonishing range of fields in mathematics and physics. This was a huge honor for me for many reasons, including the humility and kindness that Terry showed to me throughout all our interactions. It means the world. This is the Lex Freedman podcast. To support it, please check out our sponsors in the description or at lexfreedman.com/sponsors. And now, dear friends, here's Terren Tao.

What was the first really difficult, research-level math problem that you encountered—one that gave you pause, perhaps?

Terrence Tao

Well, in your undergraduate education, you learn about the really hard, seemingly impossible problems, like the Riemann hypothesis and the twin prime conjecture. You can make problems arbitrarily difficult; that's not really a problem. In fact, there are even problems that we know to be unsolvable.

What's really interesting are the problems just at the boundary between what we can do relatively easily and what is hopeless. What are the problems where existing techniques can do 90% of the job, and then you just need that remaining 10%?

I think, as a PhD student, the Kakeya problem certainly caught my eye, and it just got solved, actually. It's a problem I worked on a lot in my early research. Historically, it came from a little puzzle by the Japanese mathematician Sōichi Kakeya, in around 1918 or so.

The puzzle is that you have a needle on the plane. Think of it like driving on a road, and you want to execute a U-turn. You want to turn the needle around, but you want to do it in as little space as possible. You want to use as little area as possible in order to turn it around.

The needle is infinitely maneuverable, so you can imagine spinning it around its center as a unit. I think that gives you a disk of area π/4. Or you can do a three-point U-turn, which is what they teach people in driving school to do, and that actually takes area π/8. So it's a little bit more efficient than a rotation.

For a while, people thought that was the most efficient way to turn things around. But Besicovitch showed that, in fact, you could turn the needle around using as little area as you wanted. So, 0.0001. There was some really fancy, multi-U-turn, back-and-forth thing that you could do, where you could turn a needle around and, in so doing, have it pass through every intermediate direction.

Is this in the 2-dimensional plane?

Terrence Tao

This is in the 2-dimensional plane. We understand everything in 2 dimensions. So the next question is: What happens in 3 dimensions?

Suppose the Hubble Space Telescope is a tube in space, and you want to observe every single star in the universe. You want to rotate the telescope to reach every single direction. Here's the unrealistic part: Suppose that space is at a premium, which it totally is not. You want to occupy as little volume as possible in order to rotate your telescope around and see every single star in the sky. How small a volume do you need to do that?

You can modify the basic construction. If your telescope has zero thickness, then you can use as little volume as you need. That's a simple modification of the 2-dimensional construction. But the question is: If your telescope is not zero thickness, but just very, very thin—some thickness δ—what is the minimum volume needed to be able to see every single direction as a function of δ?

As δ gets smaller, as your needle gets thinner, the volume should go down. But how fast does it go down? The conjecture was that it goes down very, very slowly, roughly speaking, like a logarithm. That was proved after a lot of work.

This seems like a puzzle. Why is it interesting? It turns out to be surprisingly connected to a lot of problems in partial differential equations, number theory, geometry, and combinatorics.

For example, in wave propagation, you splash some water around, you create water waves, and they travel in various directions. Waves exhibit both particle-like and wave-like behavior. You can have what's called a wave packet, which is a very localized wave that is localized in space and moving in a certain direction in time. If you plot it in both space and time, it occupies a region that looks like a tube.

What can happen is that you can have a wave that is initially very dispersed, but it all focuses at a single point later in time. You can imagine dropping a pebble into a pond and ripples spreading out. But if you time-reverse that scenario—and the equations of wave motion are time-reversible—you can imagine ripples converging to a single point, and then a big splash occurs, maybe even a singularity.

Geometrically, what's going on is that there's always a set of light rays. If this wave represents light, for example, you can imagine this wave as a superposition of photons, all traveling at the speed of light. They all travel along these light rays, and they're all focusing at this one point.

You can have a very dispersed wave focus into a very concentrated wave at one point in space and time, but then it defocuses again and separates. Potentially, if the conjecture had a negative solution, what that would mean is that there's a very efficient way to pack tubes pointing in different directions into a very narrow region of very small volume.

Then you would also be able to create waves that start out very dispersed, but there would be some arrangement of waves that would concentrate not just at a single point, but there would be a lot of concentrations in space and time. You could create what's called a blowup, where the amplitude of these waves becomes so great that the laws of physics governing them are no longer described by wave equations, but by something more complicated and nonlinear.

In mathematical physics, we care a lot about whether certain equations, including wave equations, are stable or not—whether they can create these singularities. There's a famous unsolved problem called the Navier–Stokes regularity problem.

The Navier–Stokes equations govern fluid flow for incompressible fluids, like water. The question asks: If you start with a smooth velocity field of water, can it ever concentrate so much that the velocity becomes infinite at some point? That's called a singularity.

We don't see that in real life. If you splash water around in the bathtub, it won't explode on you or have water leaving at the speed of light, I think. But potentially, it is possible. In fact, in recent years, the consensus has drifted toward the belief that, for certain very special initial configurations of water, singularities can form. But people have not yet been able to establish this.

The Clay Mathematics Institute has 7 Millennium Prize Problems, with a million-dollar prize for solving each one. This is one of them. Of these 7, only 1 has been solved: the Poincaré conjecture, by Perelman.

The Kakeya conjecture is not directly related to the Navier–Stokes problem, but understanding it would help us understand some aspects of things like wave concentration, which would indirectly probably help us understand the Navier–Stokes problem better.

Can you speak to the Navier–Stokes existence and smoothness problem? As you said, it's a Millennium Prize Problem, and you've made a lot of progress on this one. In 2016, you published a paper, “Finite-Time Blowup for an Averaged Three-Dimensional Navier–Stokes Equation.”

We're trying to figure out whether this thing usually doesn't blow up, but can we say for sure that it never blows up?

Terrence Tao

Yes, that is literally the million-dollar question. This is what distinguishes mathematicians from pretty much everybody else. If something holds 99.99% of the time, that's good enough for most things, but mathematicians are one of the few people who really care about whether 100%—really 100%—of all situations are covered.

Most of the time, water does not blow up. But could you design a very special initial state that does this?

Maybe we should say that this is a set of equations in the field of fluid dynamics, trying to understand how fluid behaves. It turns out to be a really complicated thing to model. Fluid is extremely complicated to model.

Terrence Tao

Yes, so it has practical importance. This Clay Prize problem concerns what's called the incompressible Navier–Stokes equations, which govern things like water. There's something called the compressible Navier–Stokes equations, which govern things like air, and that's particularly important for weather prediction.

Weather prediction uses a lot of computational fluid dynamics. A lot of it is actually just trying to solve the Navier–Stokes equations as best they can, while also gathering a lot of data so that they can initialize the equations. There are a lot of moving parts, so it's very important practically.

Why is it difficult to prove general things about the set of equations, like that it doesn't blow up?

Terrence Tao

The short answer is Maxwell's demon. Maxwell's demon is a concept in thermodynamics. If you have a box of 2 gases—oxygen and nitrogen—and maybe you start with all the oxygen on one side and the nitrogen on the other side, but there's no barrier between them, then they will mix, and they should stay mixed. There's no reason why they should unmix.

But in principle, because of all the collisions between them, there could be some sort of weird conspiracy. Maybe there's a microscopic demon called Maxwell's demon that, every time an oxygen and nitrogen atom collide, makes them bounce off in such a way that the oxygen gradually drifts onto one side and then goes to the other. You could have an extremely improbable configuration emerge.

Which we never see. Statistically, it’s extremely unlikely, but mathematically it’s possible that this can happen, and we can’t rule it out. This is a situation that shows up a lot in mathematics. A basic example is the digits of pi: 3.14159 and so forth.

The digits look like they have no pattern, and we believe they have no pattern. In the long term, you should see as many 1s, 2s, and 3s as 4s, 5s, and 6s. There should be no preference in the digits of pi to favor, let’s say, 7 over 8. But maybe there’s some demon in the digits of pi that, every time you compute more digits, sort of biases one digit toward another.

This is a conspiracy that should not happen. There’s no reason it should happen, but there’s no way to prove it with our current technology.

Getting back to Navier–Stokes, a fluid has a certain amount of energy, and because a fluid is in motion, the energy gets transported around. Water is also viscous, so if the energy is spread out over many different locations, the natural viscosity of the fluid will just damp out the energy, and it will go to zero. This is what happens when we actually experiment with water: you splash it around, there’s some turbulence and waves, and so forth, but eventually it settles down.

The lower the amplitude and the smaller the velocity, the calmer it gets. But potentially, there is some sort of demon that keeps pushing the energy of the fluid into a smaller and smaller scale. It will move faster and faster, and at faster speeds, the effective viscosity is relatively less. So it could happen that it creates some sort of what’s called a self-similar blow-up scenario.

The energy of the fluid starts off at some large scale, and then it all transfers its energy into a smaller region of the fluid, which at a much faster rate moves into an even smaller region, and so forth. Each time it does this, it takes maybe half as long as the previous one. You could actually converge to all the energy concentrating in one point in a finite amount of time. That scenario is called finite-time blow-up.

In practice, this doesn’t happen. Water is what’s called turbulent. It is true that if you have a big eddy of water, it will tend to break up into smaller eddies, but it won’t transfer all the energy from one big eddy into one smaller eddy. It will transfer it into maybe 3 or 4, and then those must split up into maybe 3 or 4 small eddies of their own. The energy gets dispersed to the point where the viscosity can then keep everything under control.

But if it can somehow concentrate all the energy, keep it all together, and do it fast enough that the viscous effects don’t have enough time to calm everything down, then this blow-up can occur.

There were papers that had claimed that you just need to take into account conservation of energy and carefully use viscosity, and you can keep everything under control—not just for Navier–Stokes, but for many types of equations like this. In the past, there have been many attempts to try to obtain what’s called global regularity for Navier–Stokes, which is the opposite of finite-time blow-up: the velocity stays smooth.

They all failed. There was always some sign error or some subtle mistake, and it couldn’t be salvaged. What I was interested in doing was trying to explain why we were not able to disprove finite-time blow-up. I couldn’t do it for the actual equations of fluids, which were too complicated.

But if I could average the equations of motion of Navier–Stokes—basically, if I could turn off certain types of ways in which water interacts and only keep the ones that I want. In particular, if there’s a fluid and it could transfer energy from a large eddy into this small eddy or this other small eddy, I would turn off the energy channel that would transfer energy to this one and direct it only into this smaller eddy, while still preserving the law of conservation of energy.

So you’re trying to make it blow up.

Yeah. I basically engineer a blow-up by changing the laws of physics, which is one thing that mathematicians are allowed to do. We can change the equation.

How does that help you get closer to the proof of something?

Right. It provides what’s called an obstruction in mathematics. Basically, if I turned off certain parts of the equation—which usually, when you turn off certain interactions, makes it less nonlinear, more regular, and less likely to blow up—I found that by turning off a very well-designed set of interactions, I could force all the energy to blow up in finite time.

What that means is that if you wanted to prove global regularity for Navier–Stokes, for the actual equation, you must use some feature of the true equation that my artificial equation does not satisfy. So it rules out certain approaches.

The thing about mathematics is that it’s not just about finding a technique that is going to work and applying it. You need to not take the techniques that don’t work. For problems that are really hard, often there are dozens of ways that you might think might apply to solve the problem, but it’s only after a lot of experience that you realize there’s no way those methods are going to work.

Having these counterexamples for nearby problems kind of rules out certain approaches. It saves you a lot of time because you’re not wasting energy on things that you now know cannot possibly ever work.

How deeply connected is it to that specific problem of fluid dynamics, or is it just some more general intuition you build up about mathematics?

Right. Yeah. The key phenomenon that my technique exploits is what’s called supercriticality. In partial differential equations, these equations are often like a tug-of-war between different forces.

In Navier–Stokes, there’s the dissipation force coming from viscosity, and it’s very well understood. It’s linear; it calms things down. If viscosity was all there was, then nothing bad would ever happen. But there’s also transport: energy from one location in space can get transported to other locations because the fluid is in motion. That’s a nonlinear effect, and that causes all the problems.

There are 2 competing terms in this equation: the dissipation term and the transport term. If the dissipation term dominates, if it’s large, then basically you get regularity. If the transport term dominates, then we don’t know what’s going on. It’s a very nonlinear situation. It’s unpredictable and turbulent.

Sometimes these forces are in balance at small scales, but not in balance at large scales, or vice versa. Navier–Stokes is what’s called supercritical. At smaller and smaller scales, the transport terms are much stronger than the viscosity terms. Viscosity is the thing that calms things down.

This is why the problem is hard in 3 dimensions. The Soviet mathematician Olga Ladyzhenskaya showed in the 1960s that in 2 dimensions there is no blow-up, and in 2 dimensions the Navier–Stokes equations are what’s called critical. The effects of transport and viscosity are about the same strength, even at very, very small scales.

We have a lot of technology to handle critical and also subcritical equations and prove regularity, but for supercritical equations, it was not clear what was going on. I did a lot of work, and then there’s been a lot of follow-up showing that for many other types of supercritical equations, you can create all kinds of blow-up examples.

Once the nonlinear effects dominate the linear effects at small scales, you can have all kinds of bad things happen. So this is one of the main insights of this line of work: supercriticality versus criticality and subcriticality. This makes a big difference.

That’s a key qualitative feature that distinguishes some equations as being nice and predictable—like planetary motion. There are certain equations that you can predict for millions of years, or thousands at least. Again, it’s not really a problem, but there’s a reason why we can’t predict the weather past 2 weeks into the future: it’s a supercritical equation. Lots of really strange things are going on at very fine scales.

So whenever there’s some huge source of nonlinearity, that can create a huge problem for predicting what’s going to happen.

Yeah. And if the nonlinearity is somehow more and more pronounced and interesting at small scales, that can make a big difference. There are many equations that are nonlinear, but in many equations you can approximate things by the bulk.

For example, with planetary motion, if you want to understand the orbit of the moon or Mars or something, you don’t really need the microstructure, like the seismology of the moon or exactly how the mass is distributed. You can almost approximate these planets by point masses, and the aggregate behavior is what’s important.

But if you want to model a fluid, like the weather, you can’t just say, “In Los Angeles, the temperature is this and the wind speed is this.” For supercritical equations, the fine-scale information is really important.

If we can just linger on the Navier–Stokes equations a little bit, you’ve suggested maybe you can describe it: one of the ways to solve it, or to negatively resolve it, would be to construct a liquid—a kind of liquid computer, right? And then show that the halting problem from computability theory has consequences for fluid dynamics.

So show it in that way. Can you describe this?

Yeah. So this came out of this work of constructing this averaged equation that blew up. There’s a naive way to do it: you just keep pushing. Every time you get energy at one scale, you push it immediately to the next scale as fast as possible. This is sort of the naive way to force blow-up. It turns out in 5 and higher dimensions this works.

But in 3 dimensions, there was this funny phenomenon that I discovered: if you change the laws of physics and just always keep trying to push the energy into smaller and smaller scales, what happens is that the energy starts getting spread out into many scales at once. You have energy at one scale, you’re pushing it into the next scale, and then as soon as it enters that scale, you also push it to the next scale, but there’s still some energy left over from the previous scale.

You’re trying to do everything at once, and this spreads out the energy too much. Then it turns out that it makes it vulnerable for viscosity to come in and actually just damp out everything. So this direct push doesn’t actually work. There was a separate paper by some other authors that actually showed this in 3 dimensions.

What I needed was to program a delay, kind of like airlocks. I needed an equation which would start with a fluid doing something at one scale. It would push this energy into the next scale, but it would stay there until all the energy from the larger scale got transferred. Only after you pushed all the energy in would you sort of open the next gate, and then you would push that in as well.

By doing that, the energy inches forward scale by scale in such a way that it’s always localized at one scale at a time. Then it can resist the effects of viscosity because it’s not dispersed. In order to make that happen, I had to construct a rather complicated nonlinearity. It was basically like an electronic circuit.

I actually thank my wife for this because she was trained as an electrical engineer. She talked about how she had to design circuits and so forth. If you want a circuit that does a certain thing, like maybe have a light that flashes on and then turns off, and then on and then off, you can build it from more primitive components, capacitors and resistors and so forth.

You have to build a diagram, and you can sort of follow it with your eyeballs and say, “Oh, yeah, the current will build up here, and then it will stop, and then it will do that.” So I knew how to build the analog of basic electronic components, like resistors and capacitors and so forth.

I would stack them together in such a way that I would create something that would open one gate, and then there would be a clock. Once the clock hit a certain threshold, it would close it—a kind of Rube Goldberg-type machine, but described mathematically. This ended up working.

So what I realized is that if you could pull the same thing off for the actual equations—if the equations of water support a computation—then you could imagine kind of a steampunk, but really water-punk, type of thing. Modern computers are electronic: they’re powered by electrons passing through very tiny wires and interacting with other electrons and so forth.

But instead of electrons, you can imagine these pulses of water moving at a certain velocity, and maybe there are 2 different configurations corresponding to a bit being up or down. Probably, if you had 2 of these moving bodies of water collide, it would come out with some new configuration, which would be something like an AND gate or OR gate.

The output would depend in a very predictable way on the inputs, and you could chain these together and maybe create a Turing machine. Then you could have computers made completely out of water, and if you have computers, then maybe you can do robotics—hydraulics and so forth.

You could create some machine which is basically a fluid analog of what’s called a von Neumann machine. So von Neumann proposed that if you want to colonize Mars, the sheer cost of transporting people and machines to Mars is just ridiculous.

But if you could transport 1 machine to Mars, and this machine had the ability to mine the planet, create some more materials, smelt them, and build more copies of the same machine, then you could colonize a whole planet over time.

If you could build a fluid machine—it’s a robot, okay?—what it would do, its purpose in life, is that it would be programmed so that it would create a smaller version of itself in some sort of cold state. It wouldn’t start just yet.

Once it’s ready, the big robot configuration of water would transfer all its energy into the smaller configuration and then power down. Then it would clean itself up. What’s left is this newest state, which would then turn on and do the same thing, but smaller and faster.

The equation has a certain scaling symmetry. Once you do that, it can just keep iterating. So this, in principle, would create a blow-up for the actual Navier–Stokes equations, and this is what I managed to accomplish for these averaged Navier–Stokes equations. So it provided the sort of roadmap to solve the problem.

Now, this is a pipe dream because there are so many things that are missing for this to actually be a reality. I can’t create these basic logic gates. I don’t have these special configurations of water. I mean, there are candidates—things called vortex rings that might possibly work—but analog computing is really nasty compared to digital computing.

There are always errors, and you have to do a lot of error correction along the way. I don’t know how to completely power down the big machine so that it doesn’t interfere with the running of the smaller machine. But everything, in principle, can happen; it doesn’t contradict any of the laws of physics. So it’s sort of evidence that this thing is possible.

There are other groups who are now pursuing ways to make Navier–Stokes blow up which are nowhere near as ridiculously complicated as this. They’re actually pursuing something much closer to the direct self-similar model. It doesn’t quite work as is, but there could be some simpler scheme than what I just described to make this work.

There is a real leap of genius here to go from Navier–Stokes to this Turing machine. It goes from the self-similar blob scenario, where you’re trying to get the smaller and smaller blob, to now having a liquid Turing machine get smaller and smaller and smaller, and somehow seeing how that could be used to say something about a blow-up. I mean, that’s a big leap.

So there’s precedent. The thing about mathematics is that it’s really good at spotting connections between what you might think of as completely different problems. But if the mathematical form is the same, you can draw a connection.

There’s a lot of work previously on what’s called cellular automata, the most famous of which is Conway’s Game of Life. There’s this infinite discrete grid, and at any given time, the grid is either occupied by a cell or it’s empty, and there’s a very simple rule that tells you how these cells evolve.

Sometimes cells live, and sometimes they die. When I was a student, it was a very popular screensaver to just have these animations going, and they looked very chaotic. In fact, they looked a little bit like turbulent flow sometimes.

But at some point, people discovered more and more interesting structures within this Game of Life. For example, they discovered this thing called a glider. A glider is a very tiny configuration of 4 or 5 cells which evolves, and it just moves in a certain direction.

That’s like these vortex rings. This is an analogy: the Game of Life is kind of like a discrete equation, and the fluid Navier–Stokes equations are continuous equations, but mathematically they have some similar features.

Over time, people discovered more and more interesting things you could build within the Game of Life. The Game of Life is a very simple system. It only has 3 or 4 rules, but you can design all kinds of interesting configurations inside it.

There’s something called a glider gun that does nothing but spit out gliders, 1 at a time. After a lot of effort, people managed to create AND gates and OR gates for gliders.

There’s this massive, ridiculous structure where, if you have a stream of gliders coming in here and a stream of gliders coming in here, then you may produce a stream of gliders coming out. If both of the streams have gliders, then there’ll be an output stream, but if only 1 of them does, then nothing comes out.

Mhm.

So they could build something like that. Once you could build these basic gates, then just from software engineering you can build almost anything. You can build a Turing machine. I mean, it’s like enormous steampunk-type things. They look ridiculous.

But then people also generated self-replicating objects in the Game of Life: a massive machine, a von Neumann machine, which, over a huge period of time, would create another version of itself that could replicate. It always looked like glider guns inside, doing these very steampunk calculations.

It's so incredible. A lot of this was community-crowdsourced by amateur mathematicians, actually. I knew about that work, and that is part of what inspired me to propose the same thing with Navier–Stokes, which, as I said, is much worse in analog than in digital. You can't just directly take the constructions in the Game of Life and plunk them in. But again, it shows it's possible.

There's a kind of emergence that happens with these cellular automata—local rules. Maybe it's similar to fluids; I don't know. But local rules operating at scale can create these incredibly complex, dynamic structures. Do you think any of that is amenable to mathematical analysis? Do we have the tools to say something profound about that?

The thing is, you can get this emergence in very complicated structures, but only with very carefully prepared initial conditions. These glider guns and gates and so forth—these machines—if you just plunk down some cells randomly, you will not see any of these. That's the analogous situation with Navier–Stokes: with typical initial conditions, you will not have any of this weird computation going on.

But basically, through engineering, by specially designing things in a very special way, you can make clever constructions. I wonder if it's possible to prove the negative—basically, prove that only through engineering can you ever create something interesting.

This is a recurring challenge in mathematics that I call the dichotomy between structure and randomness. Most objects that you can generate in mathematics are random. They look random, like the digits of pi, which we believe is a good example. But there's a very small number of things that have patterns.

You can prove something has a pattern by just constructing it. If something has a simple pattern and you have a proof that it does something like repeat itself every so often, you can do that. You can prove, for example, that most sequences of digits have no pattern. If you just pick digits randomly, the law of large numbers tells you that you're going to get as many 1s as 2s in the long run.

But we have a lot fewer tools to show that, if I give you a specific pattern like the digits of pi, this doesn't have some weird pattern to it. Some other work that I spend a lot of time on is proving what are called structure theorems or inverse theorems that give tests for when something is very structured.

Some functions are what's called additive. If you have a function that maps natural numbers to natural numbers, maybe 2 maps to 4, 3 maps to 6, and so forth. An additive function means that if you add 2 inputs together, the output gets added as well. For example, multiplying by a constant: if you multiply a number by 10, multiplying a + b by 10 is the same as multiplying a by 10 and b by 10 and then adding them together.

Some functions are kind of additive but not completely additive. For example, if I take a number n, multiply it by the square root of 2, and take the integer part of that, 10 times the square root of 2 is 14-point-something, so 10 maps to 14. Twenty maps to 28. In that case, additivity is sometimes true: 10 + 10 is 20, and 14 + 14 is 28. But because of this rounding, sometimes there are round-off errors, and sometimes when you add a + b, this function doesn't quite give you the sum of the 2 individual outputs, but the sum plus or minus 1. It's almost additive, but not quite additive.

There are a lot of useful results in mathematics, and I've worked a lot on developing things like this, to the effect that if a function exhibits some structure like this, there's basically a reason for why it's true. The reason is that there's some other nearby function which is actually completely structured, explaining this sort of partial pattern that you have.

These inverse theorems create this sort of dichotomy: the objects that you study either have no structure at all, or they are somehow related to something that is structured. In either case, you can make progress.

A good example of this is an old theorem in mathematics called Szemerédi's theorem, proven in the 1970s. It concerns trying to find a certain type of pattern in a set of numbers: patterns that make arithmetic progressions, things like 3, 5, and 7, or 10, 15, and 20. Szemerédi proved that any set of numbers that is sufficiently big—what's called positive density—has arithmetic progressions in it of any length you wish.

For example, the odd numbers have a density of 1/2, and they contain arithmetic progressions of any length. In that case, it's obvious because the odd numbers are really structured. I can just take 11, 13, 15, and 17. I can easily find arithmetic progressions in that set.

Szemerédi's theorem also applies to random sets. If I take the set of odd numbers and flip a coin for each number, keeping only the numbers for which I got heads, I just randomly take out half the numbers and keep one-half. That's a set that has no patterns at all. But just from random fluctuations, you will still get a lot of arithmetic progressions in that set.

Can you prove that there are arithmetic progressions of arbitrary length within a random set?

Yes. Have you heard of the infinite monkey theorem? Usually mathematicians give boring names to theorems, but occasionally they give colorful names.

The popular version of the infinite monkey theorem is that if you have an infinite number of monkeys in a room, each with a typewriter, and they type out text randomly, almost surely one of them is going to generate the entire script of Hamlet, or any other finite string of text. It will just take some time—quite a lot of time, actually—but if you have an infinite number, then it happens.

Basically, if you take an infinite string of digits or whatever, eventually any finite pattern you wish will emerge. It may take a long time, but it will eventually happen. In particular, arithmetic progressions of any length will eventually happen.

Okay. But you need an extremely long random sequence for this to happen. I suppose that's intuitive. It's just infinity.

Infinity absorbs a lot of sins. How are we humans supposed to deal with infinity?

You can think of infinity as an abstraction of a finite number for which you do not have a bound. Nothing in real life is truly infinite. But you can ask yourself questions like, "What if I had as much money as I wanted?" or "What if I could go as fast as I wanted?"

A way in which mathematicians formalize that is that mathematics has found a formalism to idealize—instead of something being extremely large or extremely small—to actually be exactly infinite or zero. Often, the mathematics becomes a lot cleaner when you do that.

In physics, we joke about assuming spherical cows. Real-world problems have all kinds of real-world effects, but you can idealize, send certain things to infinity, send certain things to zero, and the mathematics becomes a lot simpler to work with.

I wonder how often using infinity forces us to deviate from the physics of reality.

Yeah, there are a lot of pitfalls. We spend a lot of time in undergraduate math classes teaching analysis. Analysis is often about how to take limits and whether you can do certain things. For example, a + b is always b + a. When you have a finite number of terms and you add them, you can swap them and there's no problem.

But when you have an infinite number of terms, there are these sort of shell games you can play where you can have a series which converges to one value, but you rearrange it and it suddenly converges to another value. You can make mistakes. You have to know what you're doing when you allow infinity.

You have to introduce these epsilons and deltas, and there's a certain type of reasoning that helps you avoid mistakes. In more recent years, people have started taking results that are true in infinite limits and finitizing them. You know that something's true eventually, but you don't know when. Now give me a rate.

Okay. So if I don't have an infinite number of monkeys but a large finite number of monkeys, how long do I have to wait for Hamlet to come out?

That's a more quantitative question. This is something that you can attack by purely finite methods, and you can use your finite intuition. In this case, it turns out to be exponential in the length of the text that you're trying to generate.

This is why you never see the monkeys create Hamlet. You can maybe see them create a 4-letter word, but nothing that big. I personally find that once you finitized an infinite statement, it does become much more intuitive, and it's no longer so weird.

Even if you're working with infinity, it's good to finitize so that you can have some intuition.

Yeah. The downside is that finite groups are just much, much messier.

Yeah. The infinite ones are usually found first, like decades earlier, and then later on people finitize them.

Since we mentioned a lot of math and a lot of physics, what is the difference between mathematics and physics as disciplines, as ways of understanding or seeing the world? Maybe we can throw engineering in there. You mentioned your wife is an engineer, and she gives you a new perspective on circuits, right?

These are different ways of looking at the world. Given that you've done mathematical physics, you've worn all the hats.

I think science in general is an interaction between 3 things. There's the real world, there's what we observe of the real world—our observations—and then our mental models as to how we think the world works.

So we can't directly access reality. All we have are observations, which are incomplete and have errors. There are many cases where we want to know, for example, what the weather will be like tomorrow, and we don't yet have the observation. We'd like a prediction.

Then we have these simplified models, sometimes making unrealistic assumptions—spherical-cow-type things. Those are the mathematical models. Science collects the observations and proposes the models that might explain those observations.

What mathematics does is stay within the model and ask: What are the consequences of that model? What predictions would the model make of future observations or past observations? Does it fit the observed data?

There's definitely a symbiosis. I guess mathematics is unusual among other disciplines in that we start from hypotheses, like the axioms of a model, and ask what conclusions come from that model. In almost any other discipline, you start with the conclusions. You want to do something: “I want to build a bridge. I want to make money. I want to do this.” Then you find the path to get there.

There's a lot less speculation about, “Suppose I did this—what would happen?” Planning and modeling, and speculative fiction, maybe, are other places where that happens, but that's about it. Most of the things we do in life are conclusion-driven, including physics and science. You want to know where this asteroid is going to go or what the weather is going to be tomorrow, but there is also this other direction of going from the axioms.

Do you think there is this tension in physics between theory and experiment? What do you think is the more powerful way of discovering truly novel ideas about reality?

You need both top-down and bottom-up. It's a real interaction between all these things. Over time, the observations, the theory, and the modeling should all get closer to reality.

Initially, they are always far apart to begin with, but you need one to figure out where to push the other. If your model is predicting anomalies that are not picked up by experiment, that tells experimenters where to look to find more data and refine the models. It goes back and forth.

Within mathematics itself, there's also a theory and experimental component. It's just that until very recently, theory has dominated almost completely. About 99% of mathematics is theoretical mathematics, and there's a very tiny amount of experimental mathematics.

People do it. If they want to study prime numbers, they can just generate large data sets. Once we had computers, we were able to do it a little bit. Although even before that, Gauss, for example, discovered—or conjectured—the most basic theorem in number theory, called the prime number theorem, which predicts how many primes there are up to a million, up to a trillion.

It's not an obvious question. Basically, what he did was compute, mostly by himself, but also with the help of human computers—people whose professional job it was to do arithmetic—the first 100,000 primes or something. He made tables and made a prediction. That was an early example of experimental mathematics.

Until very recently, theoretical mathematics was just much more successful, because doing complicated mathematical computations was not feasible. Even nowadays, although we have powerful computers, only some mathematical things can be explored numerically.

There's something called the combinatorial explosion. If you want to study, for example, all possible subsets of the numbers 1 to 1,000, there's only 1,000 numbers. How bad could it be? It turns out the number of different subsets of the numbers 1 to 1,000 is 2^1,000, which is way bigger than any computer can currently—in fact, anybody ever will—enumerate.

There are certain math problems that very quickly become intractable to attack by direct, brute-force computation. Chess is another famous example. We can't get a computer to fully explore the number of chess positions.

Now we have AI, and we have tools to explore this space—not with 100% guarantees of success, but through experiment. We can empirically solve chess, for example. We have very good AIs that don't explore every single position in the game tree, but they have found some very good approximations.

People are actually using these chess engines to do experimental chess. They're revisiting old chess theories: “When you have this type of opening, this is a good type of move, and this is not.” They can use these chess engines to refine, and in some cases overturn, conventional wisdom about chess.

I hope that mathematics will have a larger experimental component in the future, perhaps powered by AI.

We'll of course talk about that. But in the case of chess—and there's a similar thing in mathematics—I don't believe it's providing a kind of formal explanation of the different positions. It's just saying which position is better or not, which you can intuit as a human being, and then from that we humans can construct a theory of the matter.

You've mentioned Plato's cave allegory. In case people don't know, it's where people observe shadows of reality, not reality itself, and believe what they're observing to be reality. Is that, in some sense, what mathematicians—and maybe all humans—are doing, looking at shadows of reality? Is it possible for us to truly access reality?

There are these 3 ontological things: actual reality, our observations, and our models. Technically, they are distinct, and I think they will always be distinct, but they can get closer over time.

The process of getting closer often means that you have to discard your initial intuitions. Astronomy provides great examples. Your initial model of the world is flat because it looks flat, and it's big. The rest of the universe—the sky—seems small. The Sun, for example, looks really tiny.

You start off with a model that's actually really far from reality, but it fits the observations that you have. Things look good, but over time, as you make more and more observations, you bring it closer to reality. The model gets dragged along with it.

Over time, we had to realize that the Earth was round, that it spins, that it goes around the Sun, that the solar system goes around the galaxy, and so on and so forth. The universe is expanding, and the expansion itself is accelerating.

In fact, very recently, this year, there has been evidence that even the acceleration of the universe itself is nonconstant, and the explanation behind why that is is catching up. It's catching up. We still have dark matter and dark energy and this kind of thing.

We have a model that sort of explains it and fits the data really well. It just has a few parameters that you have to specify. People say those are fudge factors: with enough fudge factors, you can explain anything.

The mathematical point of the model is that you want to have fewer parameters in your model than data points in your observational set. If you have a model with 10 parameters that explains 10 observations, that is a completely useless model. It's what's called overfitting.

But if you have a model with 2 parameters and it explains a trillion observations, which is basically what the dark matter model does—I think it has about 14 parameters—and it explains petabytes of data that the astronomers have, that's a very different thing.

You can think of a physical theory as a compression of the universe, like data compression. You have these petabytes of observations, and you'd like to compress them into a model that you can describe in 5 pages and specify with a certain number of parameters. If it can fit, to reasonable accuracy, almost all of your observations, then the more compression you make, the better your theory.

One of the great surprises of our universe, and of everything in it, is that it's compressible at all. It's the unreasonable effectiveness of mathematics.

Einstein had a quote like that: “The most incomprehensible thing about the universe is that it is comprehensible,” right? And not just comprehensible—you can write an equation like E = mc². There is actually a mathematical explanation for that.

There's a phenomenon in mathematics called universality. Many complex systems at the macroscale come out of lots of tiny interactions at the microscale.

Normally, because of the combinatorial explosion, you would think that the macroscale equations must be exponentially more complicated than the microscale ones. They are, if you want to solve them completely and exactly. If you want to model all the atoms in a box of air, that's Avogadro's number, which is humongous. There's a huge number of particles, and if you actually have to track each one, it would be ridiculous.

But certain laws emerge at the macroscopic scale that almost don't depend on what's going on at the microscale, or only depend on a very small number of parameters. So if you want to model a gas of, you know, quintillion particles in a box, you just need to know its temperature, pressure, volume, and a few other parameters—five or six—and it models almost everything you need to know about these 10^23 or whatever particles.

We don't understand universality anywhere near as well as we would like mathematically, but there are much simpler toy models where we do have a good understanding of why universality occurs. The most basic one is the central limit theorem, which explains why the bell curve shows up everywhere in nature, why so many things are distributed by what's called a Gaussian distribution—the famous bell curve.

There's now even a meme with this curve, and even the meme applies broadly: universality to the meme.

Yeah.

Yes, you can go meta if you like. But there are many processes—for example, you can take lots of independent random variables and average them together in various ways. You can take a simple average or a more complicated average, and we can prove in various cases that these bell curves, these Gaussians, emerge. It is a satisfying explanation.

Sometimes they don't. If you have many different inputs and they're all correlated in some systemic way, then you can get something very far from a bell curve showing up. This is also important to know when the system fails. Universality is not 100% reliable as something to rely on.

The global financial crisis was a famous example of this. People thought that mortgage defaults had this sort of Gaussian-type behavior: if you ask a population of 100,000 Americans with mortgages what proportion of them would default on their mortgages, if everything was uncorrelated, it would be an exact bell curve. You can manage risk with options and derivatives and so forth, and there's a very beautiful theory.

But if there are systemic shocks in the economy, that can push everybody to default at the same time. That's very non-Gaussian behavior, and this wasn't fully accounted for in 2008. I think there's now more awareness that systemic risk is actually a much bigger issue. Just because the model is pretty and nice, it may not match reality.

Right. The mathematics of working out what models do is really important, but also the science of validating when the models fit reality and when they don't. You need both. Mathematics can help because, for example, these central limit theorems tell you that if you have certain assumptions, like noncorrelation—if all the inputs are not correlated to each other—then you have this kind of behavior and things are fine.

It tells you where to look for weaknesses in the model. If you have a mathematical understanding of the central limit theorem and someone proposes using this Gaussian copula or whatever to model default risk, if you're mathematically trained, you would say, “Okay, but what if there's this systemic correlation between all your inputs?” Then you can ask the economists, “How much of a risk is that?” and go look for it. There's always this synergy between science and mathematics.

A little bit on the topic of universality. You're known and celebrated for working across an incredible breadth of mathematics, reminiscent of Hilbert a century ago. In fact, the great Fields Medal–winning mathematician Tim Gowers has said that you are the closest thing we get to Hilbert. He's a colleague of yours.

Oh, yeah. Good friend.

But anyway, you are known for this ability to go both deep and broad in mathematics. You're the perfect person to ask: Do you think there are threads that connect all the disparate areas of mathematics? Is there a deep underlying structure to all of mathematics?

There's certainly a lot of connecting threads. A lot of the progress of mathematics can be represented by stories of 2 fields of mathematics that were previously not connected and finding connections.

An ancient example is geometry and number theory. In the times of the ancient Greeks, these were considered different subjects. Mathematicians worked on both—you could work on geometry, most famously, but also on numbers—but they were not really considered related. You could say that this length was 5 times this length because you could take 5 copies of this length and so forth, but it wasn't until Descartes, who developed analytic geometry, that people really realized that you can parameterize the plane, a geometric object, by 2 real numbers. Every point can be represented that way, and so geometric problems can be turned into problems about numbers.

Today, this feels almost trivial. There's no content to this—of course a plane is x and y—because that's what we teach and it's internalized. But it was an important development that these 2 fields were unified. Algebra and geometry were separated, and now we have a subject called algebraic geometry that connects them. This has happened over and over again, and that's certainly the type of mathematics that I enjoy the most.

I think there are different styles to being a mathematician. A fox knows many things a little bit, but a hedgehog knows 1 thing very, very well. In mathematics, there are definitely both hedgehogs and foxes, and then there are people who can play both roles. An ideal collaboration between mathematicians involves diversity: a fox working with many hedgehogs, or vice versa.

I identify mostly as a fox. I like arbitrage somehow: learning how 1 field works, learning the tricks of that field, and then going to another field that people don't think is related, adapting the tricks, and seeing the connections between the fields.

There are other mathematicians who are far deeper than I am, who are really hedgehogs. They know everything about 1 field and are much faster and more effective in that field, but I can give them these extra tools.

You said that you can be both the hedgehog and the fox depending on the context, depending on the collaboration. If it's at all possible, can you speak to the difference between those 2 ways of thinking about a problem? Say you're encountering a new problem: searching for the connections versus having a very singular focus.

I'm much more comfortable with the fox paradigm. I like looking for analogies and narratives. I spend a lot of time thinking about this: if there's a result I see in 1 field and I like the result—it's a cool result—but I don't like the proof because it uses types of mathematics that I'm not super familiar with, I often try to reprove it myself using the tools that I favor.

Often my proof is worse, but by doing so, I can say, “Now I can see what the other proof was trying to do.” From that, I can get some understanding of the tools that are used in that field. So it's very exploratory—doing crazy things in crazy fields and reinventing the wheel a lot.

The hedgehog style is much more scholarly. You're very knowledge-based. You stay up to speed on all the developments in this field, you know all the history, and you have a very good understanding of exactly the strengths and weaknesses of each particular technique. You rely a lot more on calculation than on trying to find narratives.

I can do that too, but there are other people who are extremely good at it.

Let's step back and maybe look at a bit of a romanticized version of mathematics. I think you've said that, early on in your life, math was more like a puzzle-solving activity when you were young. When did you first encounter a problem or proof where you realized math can have a kind of elegance and beauty to it?

That's a good question. When I came to graduate school in Princeton, John Conway was there at the time. He passed away a few years ago, but I remember one of the very first research talks I went to was a talk by Conway on what he called “extreme proof.”

Conway had this amazing way of thinking about all kinds of things in a way that you wouldn't normally think of. He thought of proofs themselves as occupying some sort of space. If you want to prove something—let's say that there are infinitely many primes—you have different proofs, but you can rank them along different axes. Some proofs are elegant, some are long, some are elementary, and so forth.

There is this cloud, so the space of all proofs itself has some sort of shape. He was interested in the extreme points of this shape. Out of all these proofs, what is the shortest one, at the extreme of everything else? What is the most elementary one, or whatever? He gave some examples of well-known theorems, and then he would give what he thought was the extreme proof in these different aspects.

I found that really eye-opening. It's not just that getting a proof for a result is interesting; once you have that proof, you can try to optimize it in various ways.

That proof itself had some craftsmanship to it. It certainly informed my writing style. When you do your math assignments as an undergraduate, your homework and so forth, you're encouraged to just write down any proof that works and hand it in. As long as it gets a tick mark, you move on.

But if you want your results to actually be influential and be read by people, it can't just be correct. It should also be a pleasure to read, motivated, and adaptable enough to generalize to other things.

It's the same in many other disciplines, like coding. There are a lot of analogies between math and coding. I like analogies, if you haven't noticed. You can code something—spaghetti code—that works for a certain task. It's quick and dirty, and it works, but there are lots of good principles for writing code well, so that other people can use it, build upon it, and so on, and so that it has fewer bugs.

There are similar things with mathematics.

So, yeah, first of all, there are so many beautiful things there, and he is one of the great minds ever in mathematics and computer science. Just considering the space of proofs and saying, “Okay, what does this space look like, and what are the extremes?”

Coding as an analogy is interesting because there's also this activity called code golf.

Manjul Bhargava

Oh, yeah. Yeah. Yeah.

Which I also find beautiful and fun, where people use different programming languages to try to write the shortest possible program that accomplishes a particular task. I believe there are even competitions on this.

Manjul Bhargava

Yeah, you learn a lot. It may seem like a frivolous exercise, but it can generate all these insights which, if you didn't have this artificial objective to pursue, you might not see.

What, to you, is the most beautiful or elegant equation in mathematics? I mean, one of the things that people often look to in beauty is simplicity. If you look at E = mc², when a few concepts come together, that's why Euler's identity is often considered the most beautiful equation in mathematics. Do you find beauty in that one, the Euler identity?

Yeah. Well, as I said, what I find most appealing is connections between different things. So, e^(iπ) = −1—people often say it uses all the fundamental constants. That's cute.

To me, the exponential function was introduced by Euler to measure exponential growth. Compound interest or decay—anything which is continuously growing or continuously decreasing, growth and decay, or dilation and contraction—is modeled by the exponential function. Whereas π comes around from circles and rotation, right? If you want to rotate a needle, for example, 180°, you need to rotate by π radians.

And i, in complex numbers, represents the swing between the real and imaginary axes of a 90° rotation—a change in direction. The exponential function represents growth and decay in the direction where you really are. When you stick an i in the exponential, instead of motion in the same direction as your current position, the motion is at right angles to your position. So, rotation.

Then e^(iπ) = −1 tells you that if you rotate for a time of π, you end up in the other direction. So it unifies geometry through dilation and exponential growth, or dynamics through this act of complexification—rotation by i. It connects together all these tools of mathematics.

Yeah. Dynamical structures and complex numbers—they were all next-door neighbors in mathematics because of this identity. Do you think the thing you mentioned is cute—the collision of notations from these disparate fields—is just a frivolous side effect, or do you think there is legitimate value in having all our old friends come together?

Manjul Bhargava

Well, it's confirmation that you have the right concepts. When you first study anything, you have to measure things and give them names. Initially, sometimes, because your model is too far off from reality, you give the wrong things the best names, and you only find out later what's really important.

Physicists can do this sometimes. With physics, E = mc²—one of the big things was the E. When Aristotle first came up with his laws of motion, and then Galileo or Newton and so forth, they saw the things they could measure: mass, acceleration, force, and so forth. In Newtonian mechanics, for example, F = ma was Newton's famous second law of motion. Those were the primary objects, so they gave them the central building blocks in the theory.

It was only later, after people started analyzing these equations, that there always seemed to be quantities that were conserved—momentum and energy. Energy isn't something you can directly measure in the same way you can measure mass and velocity and so forth, but over time people realized that this was actually a really fundamental concept.

Hamilton eventually, in the 19th century, reformulated Newton's laws of physics into what's called Hamiltonian mechanics, where the energy, which is now called the Hamiltonian, was the dominant object. Once you know how to measure the Hamiltonian of any system, you can completely describe the dynamics—what happens to all the states. It really was a central actor that was not obvious initially.

This change of perspective really helped when quantum mechanics came along. The early physicists who studied quantum mechanics had a lot of trouble trying to adapt their Newtonian thinking—everything was a particle and so forth—to quantum mechanics. I think because it was a wave, it just looked really weird. You ask, “What is the quantum version of F = ma?” and it's really, really hard to give an answer to that.

But it turns out that the Hamiltonian, which was secretly behind the scenes in classical mechanics, is also the key object in quantum mechanics. There's also an object called a Hamiltonian. It's a different type of object; it's what's called an operator rather than a function. But again, once you specify it, you specify the entire dynamics.

There's something called Schrödinger's equation that tells you exactly how quantum systems evolve once you have a Hamiltonian. Side by side, they look like completely different objects: one involves particles, one involves waves, and so forth. But with this centrality, you can start transferring a lot of intuition and facts from classical mechanics to quantum mechanics.

For example, in classical mechanics, there's this thing called Noether's theorem. Every time there's a symmetry in a physical system, there is a conservation law. The laws of physics are translation-invariant. If I move 10 steps to the left, I experience the same laws of physics as if I were here, and that corresponds to conservation of momentum.

If I turn around by some angle, again, I experience the same laws of physics. This corresponds to conservation of angular momentum. If I wait for 10 minutes, I still have the same laws of physics. This time-translation invariance corresponds to the conservation of energy.

There's this fundamental connection between symmetry and conservation. That's also true in quantum mechanics. Even though the equations are completely different, because they're both coming from the Hamiltonian—the Hamiltonian controls everything—every time the Hamiltonian has a symmetry, the equations will have a conservation law. Once you have the right language, it actually makes things a lot cleaner.

One of the problems with why we can't unify quantum mechanics and general relativity yet is that we haven't figured out what the fundamental objects are. For example, we have to give up the notion of space and time being these almost Euclidean-type spaces. At very tiny scales, we kind of know that there's going to be quantum fluctuations of spacetime—spacetime foam.

Trying to use Cartesian coordinates x, y, z is just a nonstarter, but we don't know what to replace them with. We don't actually have the mathematical concepts—the analogue of the Hamiltonian—that organize everything.

Does your gut say that there is a theory of everything? Is it even possible to unify—to find this language that unifies general relativity and quantum mechanics?

I believe so. The history of physics has been one of unification, much like mathematics, over the years. Electricity and magnetism were separate theories, and then Maxwell unified them. Newton unified the motions of the heavens with the motions of objects on Earth, and so forth.

So it should happen. It's just that, again, to go back to this model of observations and theory, part of our problem is that physics is a victim of its own success. Our 2 big theories of physics, general relativity and quantum mechanics, are so good that together they cover 99.9% of all the observations we can make.

You have to either go to extremely high particle accelerations, or the early universe, or things that are really hard to measure, in order to get any deviation from either of these 2 theories to the point where you can actually figure out how to combine them together.

But I have faith that we've been doing this for centuries, and we've made progress before. There's no reason why we should stop.

Do you think it will be a mathematician who develops a theory of everything?

What often happens is that when physicists need some mathematics, there's often some precursor that mathematicians worked out earlier. So when Einstein started realizing that space was curved, he went to some mathematician and asked, “Is there some theory of curved space that the mathematicians already came up with that could be useful?” And he said, “Oh, yeah, I think Riemann came up with something.”

Riemann had developed Riemannian geometry, which is precisely a theory of spaces that are curved in various general ways, and which turned out to be almost exactly what was needed for Einstein's theory. This is going back to Eugene Wigner's “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” I think the theories that work well to explain the universe tend to also involve the same mathematical objects that work well to solve mathematical problems. Ultimately, they're just both ways of organizing data in useful ways.

It just feels like you might need to go to some weird land that's very hard to intuit. You have string theory.

Yeah, that was a leading candidate for many decades. I think it's slowly falling out of fashion because it's not matching experiment. So one of the big challenges, of course, like you said, is that experiment is very tough.

Yes, because of how effective both theories are. But the other is that you're not just deviating from spacetime. You're going into some crazy number of dimensions. You're doing all kinds of weird stuff that, to us, we've gone so far from this flat Earth that we started at. Now we're just—it's very hard to use our limited ape-descended cognition to intuit what that reality really is like.

This is why analogies are so important. The round Earth is not intuitive because we're stuck on it, but round objects in general—we have pretty good intuition for them. We have intuition about how light works and so forth. It's actually a good exercise to work out how eclipses and phases of the Sun and the Moon can be really easily explained by a round Earth and a round Moon.

You can just take a basketball, a golf ball, and a light source and actually do these things yourself. So the intuition is there, but you have to transfer it.

That is a big leap intellectually for us, to go from a flat Earth to a round Earth, because our life is mostly lived in flat land. We all take that information for granted. We take so many things for granted because science has established a lot of evidence for this kind of thing. But we're on a round rock, flying through space.

Yeah, it's a big leap, and you have to take a chain of those leaps the more and more we progress.

Right. Modern science is maybe, again, a victim of its own success. In order to be more accurate, it has to move further and further away from your initial intuition. For someone who hasn't gone through the whole process of science education, it looks more and more suspicious because of that.

So we need more grounding. I think there are scientists who do excellent outreach, but there are lots of science things that you can do at home. There are lots of YouTube videos. I did a YouTube video recently with Grant Sanderson—we talked about this earlier—about how the ancient Greeks were able to measure things like the distance to the Moon, the distance to the Sun, and so on, using techniques that you could also replicate yourself. It doesn't all have to be fancy space telescopes and very intimidating mathematics.

Yeah, I highly recommend that. I believe you give a lecture, and you also did an incredible video with Grant. It's a beautiful experience to try to put yourself in the mind of a person from that time.

Mm-hm. Shrouded in mystery, right? You're on this planet, you don't know the shape of it, the size of it. You see some stars, you see some things, and you try to localize yourself in this world.

Yeah. And try to make some kind of general statements about distance to places.

Changing your perspective is really important. You say travel broadens the mind. This is intellectual travel. Put yourself in the mind of the ancient Greeks or some other person in some other time period. Make hypotheses, spherical cows, whatever. Speculate. This is what mathematicians do, and some artists do, actually.

It's just incredible that, given the extreme constraints, you could still say very powerful things. That's why it's inspiring looking back in history. How much can be figured out when you don't have much to figure things out with?

Stuff like, if you propose axioms, then mathematics lets you follow those axioms to their conclusions, and sometimes you can get quite a long way from your initial hypothesis.

If we can stay in the land of the weird, you mentioned general relativity. You've contributed to the mathematical understanding of Einstein's field equations. Can you explain this work and, from a mathematical standpoint, what aspects of general relativity are intriguing to you and challenging to you?

I have worked on some equations. There's something called the wave maps equation, or the sigma model, which is not quite the equation of spacetime gravity itself, but of certain fields that might exist on top of spacetime.

Einstein's equations of relativity just describe space and time itself, but then there are other fields that live on top of that. There's the electromagnetic field, scalar fields, and this whole hierarchy of different equations. Einstein's is considered one of the most nonlinear and difficult. Relatively low in the hierarchy was this thing called the wave maps equation.

So it's a wave which, at any given point, is fixed to be on a sphere. I can think of a bunch of arrows in space and time, with the arrows pointing in different directions, but they propagate like waves. If you wiggle an arrow, it will propagate and make all the arrows move, kind of like sheets of wheat in a wheat field.

I was interested in the global regularity problem, again, for this question: Is it possible for all the energy here to collect at a point? The equation I considered was actually what's called a critical equation, where the behavior at all scales is roughly the same. I was barely able to show that you couldn't actually force a scenario where all the energy concentrated at one point, that the energy had to disperse a little bit, and the moment it dispersed a little bit, it would stay regular. This was back in 2000. That was part of why I got interested in Navier–Stokes afterward, actually.

So I developed some techniques to solve that problem. Part of it is that this problem is really nonlinear because of the curvature of the sphere. There was a certain nonlinear effect, which was a nonperturbative effect. When you looked at it normally, it looked larger than the linear effects of the wave equation, and so it was hard to keep things under control even when the energy was small.

But I developed what's called a gauge transformation. The equation is kind of like an evolution of a field of wheat, and it's all bending back and forth, so there's a lot of motion. But if you imagine stabilizing the flow by attaching little cameras at different points in space, which are trying to move in a way that captures most of the motion, under this stabilized flow, the flow becomes a lot more linear. I discovered a way to transform the equation to reduce the amount of nonlinear effects, and then I was able to solve the equation.

I found this transformation while visiting my aunt in Australia. I was trying to understand the dynamics of all these fields, and I couldn't do it with pen and paper. I didn't have enough facility with computers to do any computer simulations, so I ended up closing my eyes, lying on the floor, and just imagining myself actually being this vector field and rolling around to try to see how to change coordinates in such a way that somehow things in all directions would behave in a reasonably linear fashion.

My aunt walked in on me while I was doing that and asked, “What am I doing?” “It's complicated,” was the answer. She said, “Okay, fine. You're a young man. I don't ask questions.”

I have to ask about how you approach solving difficult problems. If it's possible to go inside your mind when you're thinking, are you visualizing the mathematical objects, symbols? What are you visualizing in your mind usually when you're thinking?

A lot of pen and paper. One thing you pick up as a mathematician is what I call cheating strategically. The beauty of mathematics is that you get to change the rules and change the problem as you wish. You don't get to do this in any other field. If you're an engineer and someone says, “Build a bridge over this,” you can't say, “I want to build this bridge over here instead,” or, “I want to build it out of paper instead of steel.”

But as a mathematician, you can do whatever you want. It's like trying to solve a computer game where there are unlimited cheat codes available.

And so you can set this. If there's a dimension that's too large, I'll set it to 1 and solve the 1-dimensional problem first. There's a main term and an error term. I'll make a spherical cow assumption and assume the error term is 0.

The way you should solve these problems is not in this Iron Man mode where you make things maximally difficult. The way you should approach any reasonable math problem is: if there are 10 things that are making your life difficult, find a version of the problem that turns off 9 of the difficulties but keeps 1 of them.

You install 9 cheats. If you install 10 cheats, then the game is trivial. You solve 1 problem that teaches you how to deal with that particular difficulty, then you turn that one off and turn something else on, and solve that one. After you know how to solve the 10 problems—the 10 difficulties—separately, then you have to start merging them a few at a time.

As a kid, I watched a lot of these Hong Kong action movies. It's from a culture. One thing is that every time there was a fight scene, maybe the hero would get swarmed by 100 bad-guy goons or whatever, but it would always be choreographed so that he'd only be fighting 1 person at a time. Then he would defeat that person and move on, and because of that he could defeat all of them. Whereas if they had fought a bit more intelligently and just swarmed the guy at once, it would make for much worse cinema, but they would win.

Are you usually working with pen and paper? Are you working with a computer and LaTeX?

I'm mostly pen and paper, actually. In my office, I have 4 giant blackboards, and sometimes I just have to write everything I know about the problem on the 4 blackboards and then sit on my couch and see the whole thing.

Is it all symbols, like notation, or are there some drawings?

Oh, there's a lot of drawing and a lot of bespoke doodles that only make sense to me. The beauty of a blackboard is that you erase, and it's a very organic thing.

I'm beginning to use more and more computers, partly because AI makes it much easier to do simple coding things. If I wanted to plot a function before that was moderately complicated and had some iteration or something, I'd have to remember how to set up a Python program, how a for loop works, debug it, and so forth. It would take 2 hours. Now I can do it in 10 or 15 minutes, so I'm using more and more computers to do simple explorations.

Let's talk about AI a little bit, if we could. Maybe a good entry point is just talking about computer-assisted proofs in general. Can you describe the Lean formal proof programming language, how it can help as a proof assistant, and maybe how you started using it and how it has helped you?

Lean is a computer language, much like standard languages such as Python and C, except that in most languages the focus is on producing executable code. Lines of code do things: they flip bits, make a robot move, deliver text on the internet, or something. Lean is a language that can also do that. It can also be run as a standard, traditional language, but it can also produce certificates.

Software like Python might do a computation and tell you that the answer is 7. It can tell you that the sum of 3 + 4 is equal to 7. But Lean can produce not just the answer, but a proof of how it got the answer of 7 from 3 + 4, including all the steps involved. It creates these more complicated objects—not just statements, but statements with proofs attached to them. Every line of code is just a way of piecing together previous statements to create new ones.

The idea is not new. These things are called proof assistants, and they provide languages with which you can create quite complicated, intricate mathematical proofs. They produce certificates that give you a 100% guarantee that your arguments are correct if you trust the compiler. They made the compiler really small, and there are several different compilers available for Lean.

Can you give people some intuition about the difference between writing on pen and paper versus using the Lean programming language? How hard is it to formalize a statement?

A lot of mathematicians were involved in the design of Lean, so it's designed so that individual lines of code resemble individual lines of a mathematical argument. You might want to introduce a variable, prove a contradiction, or do various standard things. Ideally, it's written so that there's a one-to-one correspondence.

In practice, it isn't, because Lean is like explaining a proof to an extremely pedantic colleague who will point out, "Did you really mean this? What happens if this is 0? How do you justify this?"

Lean has a lot of automation in it to try to be less annoying. For example, every mathematical object has to come with a type. If I talk about x, is x a real number, a natural number, a function, or something else?

If you write things informally, it's understood in terms of context. You say, "Clearly, x is equal to the sum of y and z," and y and z were already real numbers, so x should also be a real number. Lean can do a lot of that, but every so often it says, "Wait a minute. Can you tell me more about what this object is? What type of object is it?"

You have to think more at a philosophical level—not just about the computations you're doing, but about what each object actually is in some sense.

Is it using something like LLMs to do the type inference? You mentioned the real number.

It's using much more traditional, what's called good old-fashioned AI. You can represent all these things as trees, and there's always an algorithm to match one tree to another tree. So it's actually doable to figure out if something is a real number or a natural number. Every object comes with a history of where it came from, and you can trace it.

Oh, I see.

Yeah, so it's designed for reliability. Modern AIs are not used in it; it's a disjoint technology. People are beginning to use AIs on top of Lean.

When a mathematician tries to program a proof in Lean, often there's a step where they want to use the fundamental theorem of calculus, for example, to do the next step. The Lean developers have built this massive project called Mathlib, a collection of tens of thousands of useful facts about mathematical objects. Somewhere in there is the fundamental theorem of calculus, but you need to find it.

The bottleneck now is actually lemma search. You know there's a tool in there somewhere, and you need to find it. There are various search engines specialized for Mathlib that you can use.

There are now large language models that you can say, "I need the fundamental theorem of calculus at this point," and they say, "Okay, try this." For example, when I code, I have GitHub Copilot installed as a plug-in to my IDE. It scans my text and sees what I need. I might even type, "Okay, now I need to use the fundamental theorem of calculus," and it might suggest something.

Maybe 25% of the time it works exactly. Another 10% or 15% of the time it doesn't quite work, but it's close enough that I can say, "If I just change it here and here, it will work." Half the time it gives me complete rubbish.

People are beginning to use AI a little bit on top, mostly at the level of fancy autocomplete. You can type half of 1 line of a proof, and it will find the rest.

But "fancy," especially fancy with a capital F, removes some of the friction a mathematician might feel when moving from pen and paper to formalizing.

Yes. Right now, I estimate that the time and effort taken to formalize a proof is about 10 times the amount taken to write it out.

So it's doable, but you don't like it. Doesn't it kill the whole vibe of being a mathematician?

Yeah, having a pedantic coworker, right? If that were the only aspect of it, okay. But there are some cases where it was actually more pleasant to do things formally.

There was a theorem I formalized, and there was a certain constant, 12, that came out in the final statement. This 12 had to be carried all through the proof, and everything had to be checked so that all these other numbers were consistent with this final number, 12.

We wrote a paper proving this theorem with the number 12, and then a few weeks later someone said, "We can actually improve this 12 to an 11 by reworking some of these steps." When this happens with pen and paper, every time you change a parameter, you have to check line by line that every single line of your proof still works. There can be subtle things that you didn't quite realize—some properties of the number 12 that you didn't even realize you were taking advantage of—so a proof can break down at a subtle place.

We had formalized the proof with this constant 12, and then when this new paper came out, we said, "Okay, let's update the 12 to 11."

What you can do with Lean is that, in your headline theorem, you just change a 12 to 11. You run the compiler, and of the thousands of lines of code you have, 90% of them still work, and there are a couple that are lined in red. Now, I can’t justify these steps, but it immediately isolates which steps you need to change. You can skip over everything that works just fine.

If you program things correctly, with good programming practices, most of your lines will not be red. There’ll just be a few places where you—if you don’t hard-code your constants, but instead use smart tactics and so forth—you can localize the things you need to change to a very small period of time. Within a day or two, we had updated our proof. This is a very quick process: you make a change, there are 10 things that don’t work; for each one, you make a change, and now there are 5 more things that don’t work. The process converges much more smoothly than with pen and paper.

So that’s for writing. Are you able to read it? If somebody else sends a proof, are you able to read it? What’s the difference versus paper?

The proofs are longer, but each individual piece is easier to read. If you take a math paper and jump to page 27, look at paragraph 6, and see a line of mathematical text, I often can’t read it immediately because it assumes various definitions that I have to go back to. Maybe 10 pages earlier, something was defined. The proof is scattered all over the place, and you’re basically forced to read fairly sequentially.

It’s not like a novel, where, in theory, you could open it halfway through and start reading. There’s a lot of context. But in a proof in Lean, if you put your cursor on a line of code, every single object there can be hovered over, and it will say what it is, where it came from, and where it’s justified. You can trace things back much more easily than by flipping through a math paper.

One thing that Lean really enables is collaborating on proofs at a really atomic scale, which you couldn’t really do in the past. Traditionally, with pen and paper, when you want to collaborate with another mathematician, either you do it at a blackboard, where you can really interact, or, if you’re doing it by email or something, you have to segment it: “I’m going to finish section 3; you do section 4.” You can’t really work on the same thing collaboratively at the same time.

But with Lean, you can be trying to formalize some portion of the proof and say, “I got stuck at line 67 here. I need to prove this thing, but it doesn’t quite work. Here are the 3 lines of code I’m having trouble with.” Because all the context is there, someone else can say, “Oh, okay. I recognize what you need to do. You need to apply this trick or this tool.” You can have extremely atomic-level conversations.

Because of Lean, I can collaborate with dozens of people across the world, most of whom I’ve never met in person. I may not even know how reliable they are in the process, but Lean gives me a certificate of trust. I can do trustless mathematics.

There are so many interesting questions. You’re known for being a great collaborator. What is the right way to approach solving a difficult problem in mathematics when you’re collaborating? Are you doing a divide-and-conquer type of thing, or are you focusing on a particular part and brainstorming?

There’s always a brainstorming process first. Math research projects, by their nature, are such that when you start, you don’t really know how to do the problem. It’s not like an engineering project, where somehow the theory has been established for decades and implementation is the main difficulty. You have to figure out even what the right path is.

This is what I said about cheating first. To go back to the bridge-building analogy, first assume you have an infinite budget and unlimited amounts of workforce and so forth. Now, can you build this bridge? Okay. Now have an infinite budget but only a finite workforce. Can you do that? And so forth.

Of course, no engineer can actually do this. They have fixed requirements. But there are always these jam sessions at the beginning where you try all kinds of crazy things and make all these assumptions that are unrealistic, but you plan to fix them later. You try to see if there’s even some skeleton of an approach that might work.

Hopefully, that breaks up the problem into smaller subproblems, which you don’t know how to do. Then you focus on the subproblems, and sometimes different collaborators are better at working on certain things.

One of the themes I’m known for is a theorem of Ben Green called the Green–Tao theorem. It’s a statement that the primes contain arithmetic progressions of any length. It was a modification of this theorem, and the way we collaborated was that Ben had already proven a similar result for progressions of length 3. He showed that sets like the primes contain lots and lots of progressions of length 3.

Even certain subsets of the primes do, but his techniques only worked for length-3 progressions. They didn’t work for longer progressions. I had these techniques coming from ergodic theory, which is something that I had been playing with and knew better than Ben at the time.

If I could justify certain randomness properties of some set relating to the primes—there’s a certain technical condition—if I could have that, and if Ben could supply me with this fact, I could conclude the theorem. But what I asked was a really difficult question in number theory. He said, “There’s no way we can prove this. Can you prove your part of the theorem using a weaker hypothesis that I have a chance to prove?”

He proposed something that he could prove, but it was too weak for me. I couldn’t use it. So there was this conversation going back and forth: different cheats. I wanted to cheat more; he wanted to cheat less. Eventually, we found a property that he could prove and that I could use, and then we could prove our theorem.

There are all kinds of dynamics. Every collaboration has a story. No two are the same.

On the flip side of that, with Lean programming, that’s almost a different story because you can create, I think you’ve mentioned, a kind of blueprint for a problem. You can really do a divide-and-conquer with Lean, where you’re working on separate parts, and they’re using the computer system, the proof checker, essentially to make sure that everything is correct along the way.

Yes. It makes everything compatible and trustable. Currently, only a few mathematical projects can be cut up in this way. At the current state of the art, most of the Lean activity is on formalizing proofs that have already been proven by humans.

A math paper is basically a blueprint in a sense. It takes a difficult statement, like a big theorem, and breaks it up into 100 little lemmas, but often not all of them are written with enough detail that each one can be directly formalized. A blueprint is like a really pedantically written version of a paper, where every step is explained in as much detail as possible, and you try to make each step self-contained or dependent only on a very specific number of previous statements that have been proven.

That way, each node of the blueprint graph that gets generated can be tackled independently of the others, and you don’t even need to know how the whole thing works. It’s like a modern supply chain. If you want to create an iPhone or some other complicated object, no one person can build a single object. But you can have a specialist who, if they’re given some widgets from another company, can combine them together to form a slightly bigger widget.

I think that’s a really exciting possibility, because if you can find problems that can be broken down this way, then you can have thousands of contributors distributed across the world. I told you before about the split between theoretical and experimental mathematics. Right now, most mathematics is theoretical, and when you type it, it’s experimental.

I think the platform that Lean and other software tools, such as GitHub and things like that, provide will allow experimental mathematics to scale up to a much greater degree than we can do now. Right now, if you want to do any mathematical exploration of some mathematical pattern or something, you need some code to write out the pattern.

Sometimes there are computer algebra packages that help, but often it’s just one mathematician coding lots and lots of Python or whatever. Because coding is such an error-prone activity, it’s not practical to allow other people to collaborate with you on writing modules for your code. If one of the modules has a bug in it, the whole thing is unreliable.

So you get this bespoke spaghetti code written not by professional programmers but by mathematicians. It’s clunky and slow, and because of that, it’s hard to really mass-produce experimental results. I’m already starting some projects where we are not just experimenting with data but experimenting with proofs.

I have this project called the Equational Theories Project. Basically, we generated about 22 million little problems in abstract algebra. Maybe I should back up and tell you what the project is.

Abstract algebra studies operations like multiplication and addition and their abstract properties. Multiplication, for example, is commutative: x * y is always y * x, at least for numbers. It’s also associative: (x * y) * z is the same as x * (y * z). These operations obey some laws and not others. For example, x * x is not always equal to x, so that law is not always true.

Given any operation, it obeys some laws and not others. We generated about 4,000 of these possible laws of algebra that certain operations can satisfy. Our question is: Which laws imply which other ones? For example, does commutativity imply associativity? The answer is no, because it turns out you can describe an operation that obeys the commutative law but doesn’t obey the associative law.

By producing an example, you can show that commutativity does not imply associativity. But some other laws do imply other laws by substitution and so forth, and you can write down an algebraic proof. So we look at all the pairs between these 4,000 laws—22 million of these pairs—and for each pair we ask, “Does this law imply this law?” If so, give a proof. If not, give a counterexample.

Mhm.

So, 22 million problems, each one of which you could give to an undergraduate algebra student, and they’d have a decent chance of solving the problem. Although there are a few of these 22 million—around 100—that are really quite hard. A lot are easy, and the project was just to work out, to determine, the entire graph: which ones imply which other ones.

That’s an incredible project, by the way. Such a good idea. Such a good test of the very thing we’ve been talking about, at a scale that’s remarkable.

Yeah. It would not have been feasible. The state of the art in the literature was about 15 equations and how they apply. That’s sort of at the limit of what a human-written paper can do.

You need to scale it up. You need to crowdsource, but you also need to trust all the proofs. No one person can check 22 million of them. They needed to be computerized, and so it only became possible with Lean. We were hoping to use a lot of AI as well.

The project is almost complete. Of these 22 million, all but 2 had been settled.

Wow.

Actually, of those 2, we have a pen-and-paper proof, and we’re formalizing it. In fact, I was working on finishing it this morning. So we’re almost done with this.

Incredible. Yeah, fantastic. How many people were able to get involved?

About 50, which in mathematics is considered a huge number.

It’s a huge number. That’s crazy.

Yeah. We kind of have a paper with 50 authors and a big appendix detailing who contributed to what.

Here’s an interesting question, to maybe speak even more generally about it. When you have this pool of people, is there a way to organize the contributions by the level of expertise of the contributors? I’m asking you a lot of naive questions here, but I’m imagining a bunch of humans and maybe, in the future, some AIs. Can there be an Elo-rating-type situation, like a gamification of this?

The beauty of these Lean projects is that automatically you get all this data. Everything has to be uploaded to GitHub, and GitHub tracks who contributed what. You could generate statistics from that at any later point in time. You could say, “This person contributed this many lines of code,” or whatever.

These are very crude metrics. I would definitely not want this to become part of your tenure review or something. But I think already in enterprise computing, people do use some of these metrics as part of the assessment of an employee’s performance.

Again, this is a direction that’s a bit scary for academics to go down. We don’t like metrics so much, and yet academics use metrics; they just use old ones. The number of papers.

Yeah. It’s true that this is a metric that, while flawed, is going in more of the right direction, right?

Yeah. It’s an interesting—at least, it’s a very interesting metric. I think it’s interesting to study. I think you can do studies of whether these are better predictors.

There’s this problem called Goodhart’s law: If a statistic is actually used to incentivize performance, it becomes gamed, and then it’s no longer a useful measure.

Oh, humans always.

Yeah. I know. It’s rational.

What we’ve done for this project is self-reporting. There are actually standard categories from the sciences describing the types of contributions people give. There’s conceptualization, validation, resources, coding, and so forth. There’s a standard list of 12 or so categories.

We have a big matrix of all the authors and all the categories, and we just ask each contributor to tick the boxes where they think they contributed. It gives a rough idea: You did some coding and provided some compute, but you didn’t do any of the pen-and-paper verification, or whatever.

Traditionally, mathematicians just order authors alphabetically by surname. We don’t have the tradition in the sciences of having a lead author, a second author, and so forth. We’re proud of that; we make all the authors equal in status. But it doesn’t quite scale to this size.

A decade ago, I was involved in these things called Polymath projects. It was crowdsourced mathematics, but without the Lean component, so it was limited because you needed a human moderator to check that all the contributions coming in were valid. This was a huge bottleneck, actually. But still, we had projects that were 10 authors or so.

We had decided at the time not to try to decide who did what, but to have a single pseudonym. We created this fictional character called D. H. J. Polymath, in the spirit of Bourbaki. Bourbaki is the pseudonym for a famous group of mathematicians in the 20th century.

The paper was authored under the pseudonym, so none of us got the author credit. This actually turned out not to be so great for a couple of reasons. If you wanted to be considered for tenure or whatever, you could not use this paper as one of your publications because it didn’t have the formal author credit.

The other thing that we recognized much later is that when people referred to these projects, they naturally referred to the most famous person who was involved. “Oh, this was Tim Gowers’s project. This was Terry Tao’s project,” without mentioning the other 19 or whatever people who were involved.

Yeah.

So we’re trying something different this time around, where everyone is an author. We’ll have an appendix with this matrix, and we’ll see how that works.

Both projects are incredible, just the fact that you’re involved in such huge collaborations. I think I saw a talk from Kevin Buzzard about the Lean programming language just a few years ago, and he was saying that this might be the future of mathematics.

It’s also exciting that you, one of the greatest mathematicians in the world, are embracing what seems like the paving of the future of mathematics. I have to ask you here about the integration of AI into this whole process.

DeepMind’s AlphaProof was trained using reinforcement learning on both failed and successful formal Lean proofs of IMO problems. This is sort of high-level, high-school—

Oh, very high level.

Yes, very high-level, high-school-level mathematics problems. What do you think about the system? What is the gap between this system, which is able to prove high-school-level problems, and graduate-level problems?

The difficulty increases exponentially with the number of steps involved in the proof. It’s a combinatorial explosion. The thing with large language models is that they make mistakes. If a proof has 20 steps and your model has a 10% failure rate at each step—of going in the wrong direction—it’s just extremely unlikely to reach the end.

Actually, just to take a small tangent here: How hard is the problem of mapping from natural language to the formal program?

It’s extremely hard, actually. Natural language is very fault-tolerant. You can make a few minor grammatical errors, and a speaker of a second language can still get some idea of what you’re saying.

With formal language, if you get one little thing wrong, the whole thing is nonsense. Even formal language to formal language is very hard. There are different, incompatible proof-assistant languages. There’s Lean, but also Coq and Isabelle, and so forth. Even converting from one formal language to another is basically an unsolved problem.

But once you have an informal language, they’re using their RL-trained model—something akin to AlphaZero—that they use to then try to come up with proofs. They also have a model—I believe it’s a separate model—for geometric problems. What impresses you about the system, and what do you think is the gap?

We talked earlier about how things that are amazing over time become normalized. So now, somehow, it’s, “Of course geometry is a solved problem,” right?

That’s true. That’s true. I mean, it’s still beautiful.

These are great works. It shows what’s possible. The approach doesn’t scale currently. It’s 3 days of Google’s server time to solve 1 high-school math problem.

This is not a scalable prospect, especially with the exponential increase as the complexity increases. We should mention that they got a silver medal performance—the equivalent of a silver. First of all, they took way more time than was allotted, and they had this assistance where the humans started by helping to formalize. But they’re also giving us full marks for the solution, which I guess is formally verified. So I guess that’s fair.

There are efforts—there will be a proposal at some point—to actually have an AI Math Olympiad, where at the same time as the human contestants get the actual Olympiad problems, AIs will also be given the same problems with the same time period, and the outputs will have to be graded by the same judges. Which means that they will have to be written in natural language rather than formal language.

I hope that happens. I hope that this IMO happens.

I hope the next one—it won’t happen at this IMO. The performance is not good enough in the time period. But there are smaller competitions, competitions where the answer is a number rather than a long-form proof, and AIs are actually a lot better at problems where there’s a specific numerical answer because it’s easy to do reinforcement learning on them. You got the right answer; you got the wrong answer. It’s a very clear signal.

But a long-form proof either has to be formal, and then Lean can give it a thumbs-up or thumbs-down, or it’s informal, but then you need a human to grade it and tell you. If you’re trying to do billions of reinforcement-learning runs, you can’t hire enough humans to grade those. It’s already hard enough for large language models to do reinforcement learning on just the regular text that people get. But if you actually hire people not just to give a thumbs-up or thumbs-down, but to check the output mathematically, that’s too expensive.

If we just explore this possible future, what is the thing that humans do that’s most special in mathematics, so that you could see AI not cracking it for a while?

Inventing new theories, coming up with new conjectures versus proving the conjectures, right? Building new abstractions, new representations, maybe an AI will turn out to be useful in seeing new connections between disparate fields.

It’s a good question. I think the nature of what mathematicians do over time has changed a lot. A thousand years ago, mathematicians had to compute the date of Easter, and there were really complicated calculations. But it’s all been automated for centuries; we don’t need that anymore. They used to navigate—to do spherical navigation, spherical trigonometry—to navigate from the Old World to the New. Very complicated calculations, again, that have been automated.

Even a lot of undergraduate mathematics, even before AI—for example, Wolfram Alpha, which is not a language model—can solve a lot of undergraduate-level math tasks. On the computational side, verifying routine things, like having a problem in partial differential equations and saying, “Here’s a problem in partial differential equations. Could you solve it using any of the 20 standard techniques?” And it says, “Yes, I’ve tried all 20, and here are the 100 different permutations, and here are my results.” That type of thing, I think, will work very well.

The type of scaling where, once you solve one problem, you make the AI attack 100 adjacent problems—that will work very well. The things that humans do still—

Yeah. So where the AI really struggles right now is knowing when it’s made a wrong turn. It can say, “I’m going to solve this problem. I’m going to split up this problem into these 2 cases. I’m going to try this technique.” Sometimes, if you’re lucky and it’s a simple problem, it’s the right technique and you solve the problem. Sometimes it will have a problem; it will propose an approach which is just complete nonsense, but it looks like a proof.

This is one annoying thing about language-model-generated mathematics. We’ve had human-generated mathematics that’s very low quality—submissions from people who don’t have the formal training and so forth. But if a human proof is bad, you can tell it’s bad pretty quickly. It makes really basic mistakes. AI-generated proofs can look superficially flawless, and that’s partly because that’s what the reinforcement learning has trained them to do: to produce text that looks like what is correct, which for many applications is good enough.

The errors are often really subtle, and then when you spot them, they’re really stupid. No human would have actually made that mistake.

Yeah, it’s actually really frustrating in the programming context because I program a lot. When a human makes low-quality code, there’s something called code smell, right? You can tell immediately. There are signs. But with AI-generated code, eventually you find an obvious, dumb thing that just looks like good code.

Yeah, it’s very tricky and frustrating to work with.

The sense of smell. There you go. This is one thing that humans have. There’s a metaphorical mathematical smell that it’s not clear how to get the AI to duplicate eventually.

The way AlphaZero and so forth make progress on Go and chess is, in some sense, that they have developed a sense of smell for Go and chess positions. They know that this position is good for White or good for Black. They can’t articulate why, but just having that sense of smell lets them strategize.

If AIs gain that ability—a sense of the viability of certain proof strategies, say—you can say, “I’m going to try to break up this problem into 2 smaller subtasks,” and they can say, “Well, this looks good. The 2 tasks look like simpler tasks than your main task, and they still have a good chance of being true.” So this is good to try. Or: “No, you’ve made the problem worse because each of the 2 subproblems is actually harder than your original problem,” which is actually what normally happens if you try a random thing. It’s normally very easy to transform a problem into an even harder problem. Very rarely do you transform a problem into a simpler problem.

If they can pick up a sense of smell, then they could maybe start competing with human-level mathematicians.

This is a hard question, but not competing—collaborating. If, hypothetically, I gave you an oracle that was able to do some aspect of what you do, and you could just collaborate with it, what would you like that oracle to be able to do? Would you like it to maybe be a verifier, to check the code—to be your professor, to say, “This is correct. This is a good, promising, fruitful direction”? Or would you like it to generate possible proofs, and then you see which one is the right one? Or would you like it to maybe generate different representations, totally different ways of seeing this problem?

I think all of the above. A lot of it is that we don’t know how to use these tools because it’s a paradigm that we have not had in the past: systems that are competent enough to understand complex instructions, that can work at massive scale, but are also unreliable—unreliable in subtle ways—while providing sufficiently good output. It’s an interesting combination.

You have graduate students that you work with who are kind of like this, but not at scale. We have previous software tools that can work at scale, but they’re very narrow. So we have to figure out how to use these tools.

Tim Gowers actually imagined—he foresaw—what mathematics would look like in two and a half decades. That’s funny. He wrote in his article a hypothetical conversation between a mathematical assistant of the future and himself, trying to solve a problem. They would have a conversation where sometimes the human would propose an idea and the AI would evaluate it, and sometimes the AI would propose an idea.

Sometimes computation was required, and the AI would just go and say, “Okay, I’ve checked the 100 cases needed here.” Or, “You said this is true for all n. I’ve checked for n up to 100, and it looks good so far.” Or, “Hang on, there’s a problem at n = 46.” It’s just a free-form conversation where you don’t know in advance where things are going to go, but ideas get proposed on both sides and calculations get proposed on both sides.

I’ve had conversations with AI where I say, “Okay, let’s collaborate to solve this math problem.” It’s a problem that I already know the solution to, so I try to prompt it: “Here’s the problem. I suggest using this tool, and then you’ll find this lovely argument.” But it goes into the weeds. I say, “No, no, no. If I use this—” and it might start using that, then go back to the tool that I wanted to use before.

You have to keep railroading it onto the path you want. I could eventually force it to give me the proof I wanted, but it was like herding cats. The amount of personal effort I had to take—not just to prompt it, but also to check its output—was enormous, because a lot of what it produced looked like it was going to work.

I know there’s a problem on line 17, and I’m basically arguing with it. It was more exhausting than doing it unassisted. But that’s the current state of the art.

I wonder if there’s a phase shift where it no longer feels like herding cats, and maybe it’ll surprise us how quickly that comes.

I believe so. In formalization, I mentioned before that it takes 10 times longer to formalize a proof than to write it by hand. With these modern AI tools and better tooling—the Lean developers are doing a great job adding more and more features and making it user-friendly—it’s going from 9 to 8 to 7. No big deal. But one day it will drop below 1.

That’s a phase shift, because suddenly it makes sense, when you write a paper, to write it in Lean first, or through a conversation with an AI that’s working with you on the fly. It becomes natural for journals to accept it. Maybe they’ll offer expedited refereeing if a paper has already been formalized in Lean. They’ll just ask the referee to comment on the significance of the results and how they connect to the literature, and not worry so much about correctness, because that’s been certified.

Papers are getting longer and longer in mathematics, and it’s harder and harder to get good refereeing for the really long ones unless they’re really important. It is actually an issue, and formalization is coming in at just the right time for this. As it gets easier and easier because of the tooling and all the other factors, you’re going to see much more of this. Mathlib will potentially grow exponentially. It’s a virtuous cycle.

One facet of this that happened in the past was the adoption of LaTeX. LaTeX is this typesetting language that all mathematicians use now. In the past, people used all kinds of word processors, typewriters, and whatever, but at some point LaTeX became easier to use than all the other competitors, and people just switched within a few years. It was a dramatic phase shift.

It’s a wild, out-there question, but what year—or how far away—are we from an AI system being a collaborator on a proof that wins the Fields Medal? At that level.

It depends on the level of collaboration. I mean, if it deserves to get the Fields Medal, half-and-half is already something I can imagine. If it was a winning paper with some AI systems involved in writing it, the autocomplete alone is already something I use. It speeds up my own writing.

You can have a theorem, you have a proof, and the proof has 3 cases. I write down the proof of the first case, and the autocomplete just suggests, “All right, now here’s how the proof of the second case could work.” It was exactly correct. That was great; it saved me 5 or 10 minutes of typing. But in that case, the AI system doesn’t get the Fields Medal.

No. Are we talking 20 years, 50 years, 100 years? What do you think?

I gave a prediction in print: by 2026, which is now next year, there will be math collaborations where the AI is involved. Not Fields Medal-winning, but actual research-level mathematics—published ideas that are in part generated by AI. Maybe not the ideas, but at least some of the computations and the verifications.

Has that already happened?

Yeah. There are problems that were solved by a complicated process of conversing with AI to propose things. The human goes and tries them, and the conjecture doesn’t work, but it might propose a different idea. It’s hard to disentangle exactly. There are certainly mathematical results that could only have been accomplished because there was a human mathematician and an AI involved, but it’s hard to disentangle credit.

These tools do not replicate all the skills needed to do mathematics, but they can replicate some nontrivial percentage of them—30% or 40%. They can fill in gaps. Coding is a good example. It’s annoying for me to code in Python. I’m not a professional programmer, but with AI, the friction cost of doing it is much reduced, so it fills in that gap for me.

AI is getting quite good at literature review. There’s still a problem with hallucinating references that don’t exist, but I think this is a solvable problem. If you train it in the right way and verify using the internet, in a few years you should get to the point where you have a lemma that you need and say, “Has anyone proven this lemma before?” It will do a fancy web search as an AI assistant and say, “Yeah, there are these 6 papers where something similar has happened.”

You can ask it right now, and it’ll give you 6 papers, of which maybe 1 is legitimate and relevant, 1 exists but is not relevant, and 4 are hallucinated. It has a nonzero success rate right now, but there’s so much garbage. The signal-to-noise ratio is so poor that it’s most helpful when you already somewhat know the literature and just need to be prompted to remember a paper that was already subconsciously in your memory, versus helping you discover something you weren’t even aware of but that is the correct citation.

Yeah, it can sometimes do that.

But when it does, it’s buried in a list of options, most of which are bad.

Being able to automatically generate a related-work section that is correct—that’s actually a beautiful thing. That might be another phase shift, because it assigns credit correctly.

Yeah, it does. It breaks you out of the silos of thought.

There’s a big hump to overcome right now. It’s like self-driving cars: the safety margin has to be really high for it to be feasible. There’s a last-mile problem with a lot of AI applications. They can develop tools that work 20% to 80% of the time, but it’s still not good enough—in fact, in some ways, it’s even worse than good.

Another way of asking the Fields Medal question is: what year do you think you’ll wake up and be really surprised? You read the headline in the news that something happened that AI did—a real breakthrough. It doesn’t have to be the Fields Medal, even a hypothesis. It could be an AlphaZero moment with Go, that kind of thing.

This decade. I can see it making a conjecture between 2 unrelated things that people thought were unrelated.

Oh, interesting.

Generating a conjecture—that’s a beautiful conjecture, and it actually has a real chance of being correct and meaningful. That’s actually kind of doable, I suppose.

No, that would be truly amazing.

The current models struggle a lot. Physicists have a dream of getting AI to discover new laws of physics. The dream is that you just feed it all this data, and it says, “Here’s a new pattern that we didn’t see before.” But the current state of the art even struggles to discover old laws of physics from the data.

Or, if it does, there’s a big concern about contamination: it did it only because somewhere in the training data it saw Boyle’s law, or whatever law you’re trying to reconstruct. Part of it is that we don’t have the right type of training data for this. For laws of physics, we don’t have a million different universes with a million different laws of nature.

A lot of what we’re missing in mathematics is actually the negative space. We have published things that people have been able to prove, and conjectures that ended up being verified, or maybe counterexamples were produced. But we don’t have data on things that were proposed and seemed like a good thing to try, but then people quickly realized that it was the wrong conjecture. Then they said, “Oh, but we should actually change our claim and modify it in this way to make it more plausible.”

There’s a trial-and-error process that’s an integral part of human mathematical discovery, which we don’t record because it’s embarrassing. We make mistakes, and we only like to publish our wins. AI has no access to this data to train on. I sometimes joke that AI has to go through graduate school: take graduate courses, do the assignments, go to office hours, make mistakes, get advice on how to correct the mistakes, and learn from that.

Let me ask you, if I may, about Grigori Perelman. You mentioned that you try to be careful in your work and not let a problem completely consume you. You really fall in love with the problem and can’t rest until you solve it. But you also hastened to add that sometimes this approach can actually be very successful.

An example you gave is Grigori Perelman, who proved the Poincaré conjecture by working alone for 7 years, with basically little contact with the outside world. Can you explain this Millennium Prize problem that was solved—the Poincaré conjecture—and maybe speak to the journey that Grigori Perelman has been on?

It’s a question about curved spaces. Earth is a good example. You can think of a 2D surface as being round. It could be a torus with a hole in it, or it can have many holes, and there are many different topologies a priori that a surface could have.

Even if you assume that it’s bounded and smooth and so forth, we have figured out how to classify surfaces. As a first approximation, everything is determined by something called the genus—how many holes it has. A sphere has genus 0, a donut has genus 1, and so forth.

One way you can tell these surfaces apart is that the sphere is simply connected. If you take any closed loop on the sphere, like a big closed loop of rope, you can contract it to a point while staying on the surface. The sphere has this property, but a torus doesn’t. If you take a rope that goes around the outer diameter of a torus, there’s no way to get it through the hole, so there’s no way to contract it to a point.

It turns out that the sphere is the only surface with this property of contractibility, up to continuous deformations. These are the things that we would call topologically equivalent to the sphere. Poincaré asked the same question in higher dimensions.

This becomes hard to visualize because you can think of a surface as embedded in 3 dimensions, but for a curved 3-dimensional space, we don’t have good intuition of 4-dimensional space to live in. There are also 3-dimensional spaces that can’t even fit into 4 dimensions; you need 5 or 6 or higher. Mathematically, you can still pose this question: If you have a bounded 3-dimensional space that also has this simply connected property, where every loop can be contracted, can you turn it into a 3-dimensional version of a sphere? This is the Poincaré conjecture.

Weirdly, in higher dimensions—4 and 5—it was actually easier. It was solved first in higher dimensions. There’s somehow more room to do the deformation; it’s easier to move things around into a sphere. But 3 was really hard.

People tried many approaches. There were combinatorial approaches, where you chop up the surface into little triangles or tetrahedra and try to argue based on how the faces interact with each other. There were algebraic approaches as well. There are various algebraic objects, like the fundamental group, homology, cohomology, and all these very fancy tools. They also didn’t quite work.

But Richard Hamilton proposed a partial differential equations approach. You have this object that secretly is a sphere, but it’s given to you in a really weird way. Think of a ball that’s been crumpled up and twisted, so it’s not obvious that it’s a ball.

If you have some sort of surface that’s a deformed sphere, you could think of it as the surface of a balloon and try to inflate it. As you fill it with air, the wrinkles will naturally smooth out and it will turn into a nice, round sphere. Unless, of course, it was a torus or something. In that case, there would be a point in the middle where the inner ring shrinks to 0; you get a singularity and can’t blow it up any further. You can’t flow it any further.

He created this flow, which is called Ricci flow. It’s a way of taking an arbitrary surface or space and smoothing it out, making it rounder and rounder, so that it looks like a sphere. He wanted to show that either this process would give you a sphere or it would create a singularity. It’s very much like how partial differential equations either have global regularity or finite-time blowup. It’s almost exactly the same thing; it’s all connected.

He showed that for 2-dimensional surfaces, if you started with something simply connected, no singularities ever formed. You never ran into trouble, and you could flow it until it became a sphere. So he got a new proof of the 2-dimensional result.

By the way, that’s a beautiful explanation of Ricci flow and its application in this context. How difficult is the mathematics here? For the 2D case, is it—

Yeah. These are quite sophisticated equations, on par with the Einstein equations, though slightly simpler. They were considered hard nonlinear equations to solve. There are lots of special tricks in 2D that helped, but in 3D the problem was that this equation was actually supercritical—the same problem as with Navier–Stokes.

As you blow up, maybe the curvature could get concentrated in finer and smaller regions, and it looked more and more nonlinear. Things just looked worse and worse, and there could be all kinds of singularities that showed up.

Some singularities are simple enough that you can see what to do next. There are things called neck pinches, where the surface behaves like a barbell and pinches at a point. You just make a snip, and then you can turn 1 surface into 2 and evolve them separately.

But there was the prospect that some really nasty, knotted singularities could show up, and you couldn’t see how to resolve them in any way. You couldn’t do any surgery on them. So you need to classify all the singularities—all the possible ways that things can go wrong.

What Perelman did, first of all, was turn the problem from a supercritical problem into a critical problem. I said before how the invention of energy—the Hamiltonian—really clarified Newtonian mechanics. He introduced something that is now called Perelman’s reduced volume and Perelman’s entropy. He introduced new quantities, kind of like energy, that look the same at every single scale, and turned the problem into a critical one where the nonlinearities suddenly looked a lot less scary than they did before.

He still had to analyze the singularities of this critical problem. That itself was a problem similar to this blowup problem I worked on, actually, at about the same level of difficulty. He managed to classify all the singularities of this problem and show how to apply surgery to each of them. Through that, he was able to resolve the Poincaré conjecture.

That is quite a lot of really ambitious steps. It’s nothing that a large language model today, for example, could do. At best, I could imagine a model proposing this idea as 1 of hundreds of different things to try, but the other 99 would be complete dead ends, and you’d only find out after months of work.

He must have had some sense that this was the right track to pursue, because it takes years to get from A to B. You’ve done, as you said, similarly difficult things, strictly mathematically, but also more broadly in terms of the process. What can you infer from the process he was going through, given that he was doing it alone?

What are some low points in a process like that? You’ve mentioned hardship. AI doesn’t know when it’s failing. What happens to you when you’re sitting in your office and realize that the thing you worked on for the last few days, maybe weeks, is a failure?

For me, I switch to a different problem. As I said, I’m a fox, not a hedgehog. Legitimately, that is a break that you can take: step away and look at a different problem.

You can modify the problem, too. You can sometimes cheat. If there’s a specific thing that’s blocking you, some bad case that keeps showing up for which your tool doesn’t work, you can just assume by fiat that this bad case doesn’t occur. So you do some magical thinking, but strategically, to see if the rest of the argument goes through.

If there are multiple problems with your approach, then maybe you just give up. But if this is the only problem that you know of and everything else checks out, then it’s still worth fighting. You have to do some sort of forward reconnaissance sometimes, and that can be productive: assuming, “Okay, we’ll figure it out eventually.”

Sometimes it’s even productive to make mistakes. There was a project for which we actually won some prizes, with 4 other people. We worked on this PDE problem again, this blowup-regularity-type problem, and it was considered very hard. Jean Bourgain, who was another Fields Medalist, had worked on a special case of this, but he could not solve the general case.

We worked on this problem for 2 months, and we thought we solved it. We had this cute argument where everything fit, and we were excited. We were planning a celebration, all getting together and having champagne or something, and we started writing it up.

One of us—not me, but another co-author—said, “In this lemma here, we have to estimate these 13 terms that show up in this expansion. We estimate 12 of them, but in our notes I can’t find the estimation of the 13th. Can someone supply that?” I said, “Sure, I’ll look at this.”

Actually, we didn’t cover it. We completely omitted this term, and it turned out to be worse than the other 12 terms put together. In fact, we could not estimate this term. We tried for a few more months, with all different permutations, and there was always this 1 term that we could not control.

This was very frustrating, but because we had already invested months and months of effort into it, we stuck with it. We tried increasingly desperate and crazy things. After 2 years, we found an approach that was quite different from our initial strategy. It didn’t generate these problematic terms, and it actually solved the problem.

So we solved a problem after 2 years, but if we hadn't had that initial false dawn of nearly solving the problem, we would have given up by month two or something and worked on an easier problem. If we had known it would take 2 years, I'm not sure we would have started the project.

Sometimes, actually having the incorrect version—it's like Columbus's incorrect measurement of the size of the Earth. He thought he was going to find a new trade route to India, or at least that was how he sold it in his prospectus. It could be that he actually secretly knew, but just on the psychological element, do you have emotional or self-doubt that just overwhelms you in moments like that?

This stuff feels like math is so engrossing that it can break you when you invest so much of yourself in the problem and then it turns out wrong. You could start to feel that way, similar to the way chess has broken some people.

Yeah. I think different mathematicians have different levels of emotional investment in what they do. For some people, it's just a job. You have a problem, and if it doesn't work out, you move on to the next one. The fact that you can always move on to another problem reduces the emotional connection.

There are certain problems that are what I call black holes, where people just latch on to that one problem and spend years and years thinking about nothing but that one problem. Maybe their career suffers and so forth, but they think, “This big win—once I finish this problem, I will make up for all the years of lost opportunity.” Occasionally, it works, but I really don't recommend it for people without the right fortitude.

Yeah.

I've never been super invested in any one problem. One thing that helps is that we don't need to call our problems in advance. When we do grant proposals, we say we will study this set of problems, but even then we don't promise, “Definitely, in 5 years, I will supply a proof of all these things.”

You promise to make some progress or discover some interesting phenomena. Maybe you don't solve the problem, but you find some related problem that you can say something new about, and that's a much more feasible task.

But I'm sure for you, there are problems like this. You have made so much progress toward the hardest problems in the history of mathematics. Is there a problem that just haunts you? It sits there in the dark corners—the twin prime conjecture, the Riemann hypothesis, the Goldbach conjecture.

Twin prime—that sounds interesting. I mean, problems like the Riemann hypothesis are so far out of reach.

Why do you think so?

There's not even a viable strategy. Even if I activate all the tricks that I know of in this problem, there's still no way forward. I think it needs a breakthrough in another area of mathematics first, and for someone to recognize that it would be a useful thing to transport into this problem.

So we should maybe step back for a little bit and just talk about prime numbers. They're often referred to as the atoms of mathematics. Can you just speak to the structure that these atoms—the natural numbers—have, with 2 basic operations attached to them: addition and multiplication?

If you want to generate the natural numbers, you can do 1 of 2 things. You can start with 1 and add 1 to itself over and over again, and that generates the natural numbers. Additively, they're very easy to generate: 1, 2, 3, 4, 5.

Or, if you want to generate them multiplicatively, you can take all the prime numbers—2, 3, 5, 7—and multiply them all together. Together, that gives you all the natural numbers except maybe for 1. So there are these 2 separate ways of thinking about the natural numbers: from an additive point of view and from a multiplicative point of view.

Separately, they're not so bad. Any question that only involves addition is relatively easy to solve, and any question that only involves multiplication is easy to solve. But what has been frustrating is that when you combine the 2 together, suddenly you get this extremely rich field.

We know that there are statements in number theory that are actually undecidable. There are certain polynomials in some number of variables, and whether there is a solution in the natural numbers depends on an undecidable statement, like whether the axioms of mathematics are consistent or not.

But even the simplest problems that combine something multiplicative, such as the primes, with something additive, such as shifting by 2, are difficult. Separately, we understand both of them well, but if you ask, when you shift a prime by 2, can you get another prime, or how often can you get another prime, it's been amazingly hard to relate the 2.

The twin prime conjecture is just that it posits there are infinitely many pairs of prime numbers that differ by 2.

Yes.

The interesting thing is that you've been very successful at pushing the field forward in answering these complicated questions of this variety. You mentioned the Green–Tao theorem. It proves that prime numbers contain arithmetic progressions of any length, right? Which is mind-blowing—that you can prove something like that.

What we've realized because of this type of research is that different patterns have different levels of indestructibility. What makes the twin prime problem hard is that if you take all the primes in the world—3, 5, 7, 11, and so forth—there are some twins in there. 11 and 13 are a twin-prime pair, and so forth.

But you could easily, if you wanted to, redact the primes to get rid of these twins. The twins show up, and even if there are infinitely many of them, they're actually reasonably sparse. Initially there are quite a few, but once you get to the millions and trillions, they become rarer and rarer. You could just edit out a few primes here and there; you could make the twin prime conjecture false by removing 0.1% of the primes, or something, choosing them well to do this.

You could present a censored database of the primes that passes all of the statistical tests of the primes. It obeys things like the prime number theorem and other tests about the primes, but it doesn't contain any twin primes anymore. This is a real obstacle for the twin prime conjecture. It means that any proof strategy to actually find twin primes in the actual primes must fail when applied to these slightly edited primes.

On the other hand, progressions have turned out to be much more robust. You can take the primes and eliminate 99% of the primes, and you can take any 99% you want, and it turns out—another thing we prove is that—you still get arithmetic progressions. Arithmetic progressions are like cockroaches of arbitrary length.

Yes, that's crazy. For people who don't know, an arithmetic progression is a sequence of numbers that differ by some fixed amount.

Yeah, but again, it's like an infinite-monkey-type phenomenon. For any fixed length of your set, you don't get arbitrary-length progressions. You only get quite short progressions.

But you're saying twin primes are not an infinite-monkey phenomenon?

I mean, it's a very subtle monkey. It's still an infinite-monkey phenomenon. If the primes were really genuinely random—if the primes were generated by monkeys—then yes, in fact, the infinite monkey theorem would—

Oh, but you're saying that with twin primes, you can't use the same tools. It doesn't appear random, almost.

Well, we don't know. We believe the primes behave like a random set. The reason we care about the twin prime conjecture is as a test case for whether we can genuinely and confidently say, with 0% chance of error, that the primes behave like a random set.

Random versions of the primes, we know, contain twins, at least with 100% probability—or probably tending to 100% as you go out further and further. We believe that the primes are random. The reason arithmetic progressions are indestructible is that regardless of whether your set looks random or looks structured, like periodic, in both cases arithmetic progressions appear, but for different reasons.

This is basically how the many proofs of these sorts of arithmetic progression theorems work. They're all proven by some sort of dichotomy where your set is either structured or random, and in both cases you can say something, and then you put the 2 together.

But in twin primes, if the primes are random, then you're happy—you win. But if your primes are structured, they could be structured in a specific way that eliminates the twins. We can't rule out that 1 conspiracy.

And yet, you were able to make progress on the k-tuple version, right?

Yeah. The one funny thing about conspiracies is that any one conspiracy theory is really hard to disprove. If you believe the world is run by lizards, and someone says, “Here's some evidence that it's not run by lizards,” you can say, “That evidence was planted by the lizards.”

Right. You may have encountered this kind of phenomenon.

A pure conspiracy has almost no way to definitively rule out a conspiracy, and the same is true in mathematics. A conspiracy solely devoted to eliminating twin primes would have to also infiltrate other areas of mathematics. It could be made consistent, at least as far as we know, but there's a weird phenomenon that you can make 1 conspiracy rule out other conspiracies. If the world is run by lizard people, it can't also be run by—

Right.

Right. So 1 unreasonable thing is hard to dispute, but more than 1, there are tools. For example, we know there are infinitely many pairs of primes that differ by at most 246, actually. That is the current bound.

So there’s a bound on the difference, right?

There are twin primes, and there’s this thing called cousin primes that differ by 4. There are called sexy primes that differ by 6.

What are sexy primes?

Primes that differ by 6. The name is much less exciting than the concept is.

Got it.

You can make 1 conspiracy rule out 1 of these, but once you have, like, 50 of them, it turns out that you can’t rule out all of them at once. It just requires too much energy somehow in this conspiracy space.

How do you do the bound part? How do you develop a bound for the difference between the primes?

It’s ultimately based on what’s called the pigeonhole principle. The pigeonhole principle is a statement that if you have a number of pigeons and they all have to go into pigeonholes, and you have more pigeons than pigeonholes, then 1 of the pigeonholes has to have at least 2 pigeons in it. There have to be 2 pigeons that are close together.

For instance, if you have 101 numbers and they all range from 1 to 1,000, 2 of them have to be at most 10 apart, because you can divide up the numbers from 1 to 1,000 into 100 pigeonholes. Two of the numbers have to belong to the same pigeonhole. It’s a basic principle in mathematics.

It doesn’t quite work with the primes directly because the primes get sparser and sparser as you go out. Fewer and fewer numbers are prime. But it turns out that there’s a way to assign weights to numbers. There are numbers that are kind of almost prime, but they don’t have many factors other than themselves and 1; they just have very few factors.

It turns out that we understand almost primes a lot better than primes. For example, it was known for a long time that there were twin almost primes. This has been worked out. You can actually restrict attention to a suitable set of almost primes, and whereas the primes are very sparse overall, the almost primes are much less sparse.

You can set up a set of almost primes where the primes have density of, say, 1%. That gives you a shot at proving, by applying some sort of pigeonhole principle, that those pairs of primes are only 100 apart. But in order to prove the twin prime conjecture, you need to get the density of primes inside the almost primes up to at least 50%. Once you get up to 50%, you will get twin primes.

Unfortunately, there are barriers. We know that no matter what kind of good set of almost primes you pick, the density of primes can never get above 50%. It’s called the parity barrier. I would love to find a way to breach that barrier.

One of my long-term dreams is to find a way to breach that barrier, because it would open up not only the twin prime conjecture and the Goldbach conjecture, but many other problems in number theory that are currently blocked because our current techniques would require going beyond this theoretical parity barrier.

It’s like pulling past the speed of light. So we just say the twin prime conjecture is 1 of the biggest problems in the history of mathematics. The Goldbach conjecture also—they feel like next-door neighbors. Have there been days when you felt you saw the path?

Oh, yeah. Sometimes you try something and it works super well. You gain, again, the sense of mathematical smell we talked about earlier. You learn from experience when things are going too well, because there are certain difficulties that you sort of have to encounter.

I think the way a colleague might put it is that if you are on the streets in New York and you put on a blindfold, and you get in a car, and after some hours the blindfold comes off and you’re in Beijing, that was too easy somehow. There was no ocean being crossed. Even if you don’t know exactly what was done, you suspect that something wasn’t right.

Is that still in the back of your head? Do you return to the prime numbers every once in a while to see—

When I have nothing better to do, which is less and less often. I get busy with so many things these days, but when I have free time and I’m too frustrated to work on my real research projects, and I also don’t want to do my administrative stuff or run errands for my family, I can play with these things for fun.

Usually you get nowhere. You have to learn to just say, “Okay, fine. Once again, nothing happened. I will move on.” Very occasionally, I actually solve 1 of these problems. Sometimes, as you say, you think you’ve solved it, and then you’re euphoric for maybe 15 minutes. Then you think, “I should check this, because this is too easy, too good to be true,” and it usually is.

What does your gut say about when these problems would be solved—when the twin prime and Goldbach conjectures would be solved?

I think we’ll keep getting more partial results. We need at least 1 breakthrough on this parity barrier; it’s the biggest remaining obstacle. There are simpler versions of the conjecture where we’re getting really close.

I think in 10 years we will have many more, much closer results. We may not have the whole thing.

So the twin prime conjecture is somewhat close. The Riemann hypothesis—I have no idea. I mean, it has to happen by accident, I think. The Riemann hypothesis is a kind of more general conjecture about the distribution of prime numbers, right?

Yeah. It’s sort of viewed multiplicatively. For questions only involving multiplication, with no addition, the primes really do behave as randomly as you could hope.

There’s a phenomenon in probability called square-root cancellation. If you want to poll, say, America on some issue, and you ask 1 or 2 voters, you may have sampled a bad sample and get a really imprecise measurement of the full average. But if you sample more and more people, the accuracy gets better and better, and it improves like the square root of the number of people you sample.

If you sample 1,000 people, you can get a 2–3% margin of error. In the same sense, if you measure the primes in a certain multiplicative sense, there’s a certain type of statistic you can measure, and it’s called the Riemann zeta function. It fluctuates up and down, but as you keep averaging more and more, the fluctuation should go down as if the primes were random.

There’s a very precise way to quantify that, and the Riemann hypothesis is a very elegant way to capture this. But, as with many other things in mathematics, we have very few tools to show that something genuinely behaves like it’s random. This is not just a little bit random; it’s asking that it behaves as randomly as an actually random set, with this square-root cancellation.

We know, because of things related to the parity problem, that most of our usual techniques cannot hope to settle this question. The proof has to come out of left field. No one has any serious proposal for what that is.

There are various ways to modify the primes a little bit and destroy the Riemann hypothesis, so it has to be very delicate. You can’t apply something that has huge margins of error. It has to just barely work, and there are all these pitfalls that you have to dodge very adeptly.

The prime numbers are just fascinating. What, to you, is most mysterious about the prime numbers?

That’s a good question. Conjecturally, we have a good model of them. They have certain patterns—the primes are usually odd, for instance—but apart from these obvious patterns, they behave very randomly.

If you just assume that they behave randomly, there’s something called the Cramér random model of the primes: after a certain point, primes just behave like a random set. There are various slight modifications of this model, but it has been a very good model. It matches the numerics, and it tells us what to predict.

I can tell you with complete certainty that the twin prime conjecture is true. The random model gives overwhelming odds that it is true. I just can’t prove it.

Most of our mathematics is optimized for solving things with patterns in them. The primes have this anti-pattern, as do almost everything, really. But we can’t prove that.

I guess it’s not mysterious that the primes would be kind of random, because there’s no reason for them to have any kind of secret pattern. What is mysterious is the mechanism that really forces the randomness to happen. This mechanism is just absent.

Another incredibly surprisingly difficult problem is the Collatz conjecture. It’s simple to state, beautiful to visualize in its simplicity, and yet extremely difficult to solve. You have been able to make progress. Paul Erdős said about the Collatz conjecture that mathematics may not be ready for such problems. Others have stated that it is an extraordinarily difficult problem, completely out of reach of present-day mathematics. And yet you have made some progress.

Why is it so difficult to make progress? Can you actually even explain what it is?

Oh, yeah.

So, it’s a problem that you can explain. It helps with some visual aids, but you take any natural number, like 13, and apply the following procedure to it. If it’s even, you divide it by 2, and if it’s odd, you multiply it by 3 and add 1.

Even numbers get smaller; odd numbers get bigger. So 13 becomes 40, because 13 × 3 is 39. Add 1, and you get 40. It’s a simple process for odd numbers and even numbers. They’re both very easy operations. Then you put them together, and it’s still reasonably simple.

But then you ask what happens when you iterate it. You take the output that you just got and feed it back in. So 13 becomes 40. Forty is now even, so divide by 2 and you get 20. Twenty is still even, so divide by 2 and you get 10. Then 5, and then 5 × 3 + 1 is 16. Then 8, 4, 2, 1.

From 1, it goes 1, 4, 2, 1, 4, 2, 1. It cycles forever. So the sequence I just described is 13, 40, 20, 10, 5, 16, 8, 4, 2, 1.

These are also called hailstone sequences because there’s an oversimplified model of hailstone formation, which is not actually quite correct, but is somehow taught to high school students as a first approximation. A little nugget of ice forms in a cloud, and it goes up and down because of the wind. Sometimes, when it’s cold, it acquires a bit more mass, and maybe it melts a little bit. This process of going up and down creates this ball of partially melted ice, which eventually becomes a hailstone and falls out of the sky.

The conjecture is that no matter how high you start—you take a number that’s in the millions or billions—you go through this process that goes up if you’re odd and down if you’re even, and eventually it goes down to Earth. All the time, no matter where you start with this very simple algorithm, you end up at 1.

You might climb for a while. If you plot these sequences, they look like Brownian motion. They look like the stock market. They just go up and down in a seemingly random pattern. Usually, that’s what happens: if you plug in a random number, you can actually prove, at least initially, that it would look like a random walk.

It’s a random walk with a downward drift. It’s like if you’re always gambling on roulette at a casino with odds slightly weighted against you. Sometimes you win, sometimes you lose, but over the long run, you lose a bit more than you win. Normally, your wallet will go to zero if you just keep playing over and over again.

So statistically, it makes sense.

Yes. The result that I proved, roughly speaking, says that statistically, around 99% of all inputs would drift down to maybe not all the way to 1, but to be much, much smaller than what you started with.

It’s like if I told you that if you go to a casino, most of the time, if you keep playing for long enough, you end up with a smaller amount in your wallet than when you started. That’s kind of like the result that I proved.

So why is that result—can you continue down that thread to prove the full conjecture?

The problem is that I used arguments from probability theory, and there’s always this exceptional event. In probability, we have the law of large numbers, which tells you things like, if you play a casino game with a losing expectation over time, you are guaranteed—or almost surely, with probability as close to 100% as you wish—to lose money.

But there’s always this exceptional outlier. It is mathematically possible that even when the odds are not in your favor, you could just keep winning slightly more often than you lose. It’s very much like how in Navier–Stokes, most of the time your waves can disperse, but there could be just 1 outlier choice of initial conditions that would lead you to blow up.

There could be 1 outlier choice of a special number that shoots off to infinity while all other numbers crash to Earth, crash to 1.

In fact, there are some mathematicians, Alex Kontorovich, for instance, who have proposed that Collatz iterations are like cellular automata. If you look at what happens in binary, they do look a little bit like these Game of Life-type patterns.

In analogy to how the Game of Life can create these massive self-replicating objects and so forth, possibly you could create some sort of heavier-than-air flying machine—a number that is actually encoding this machine, whose job it is to create a version of itself that is larger, a heavier-than-air machine encoded in a number that flies forever.

Yeah. So Conway, in fact, worked on this problem as well.

Conway worked on generalizations of the Collatz problem. Instead of multiplying by 3 and adding 1 or dividing by 2, you have a more complicated branch. Instead of having 2 cases, maybe you have 17 cases, and then you go up and down.

He showed that once your iteration gets complicated enough, you can actually encode Turing machines. You can make these problems undecidable and do things like this.

In fact, he invented a programming language for these kinds of fractional linear transformations. He called it FRACTRAN, as a play on Fortran. He showed that you could program it—it was Turing-complete. You could make a program such that, if the number you inserted was encoded as a prime, it would sink to zero. It would go down; otherwise, it would go up, and things like that.

The general class of problems is really as complicated as all of mathematics.

Some of the mystery of the cellular automata that we talked about—having a mathematical framework to say anything about cellular automata—maybe the same kind of framework is required here?

Yeah, in the conjecture, if you want to do it not statistically, but really want 100% of all inputs to fall to 1, that looks hard.

What might be feasible is that statistically, 99% go to 1. But everything? That looks hard.

What would you say, out of these famous problems within reach, is the hardest problem we have today? Is the Riemann hypothesis up there?

P versus NP is a good one because that’s a meta-problem. If you solve that in the positive sense—that you can find a P versus NP algorithm—that potentially solves a lot of other problems as well.

We should mention some of the conjectures we’ve been talking about. A lot of stuff is built on top of them, and now there are ripple effects. P equals NP has more ripple effects than basically any other, right?

If the Riemann hypothesis is disproven, that would be a big mental shock to number theorists, but it would have follow-on effects for cryptography, because a lot of cryptography uses number theory. It uses number-theoretic constructions involving primes, and so forth, and relies very much on the intuition that number theorists have built over many years about which operations involving primes behave randomly and which ones don’t.

In particular, our encryption methods are designed to turn text with information on it into text that is indistinguishable from random noise, and hence we believe it to be almost impossible to crack, at least mathematically.

But if something as core to our belief as the Riemann hypothesis is wrong, it means that there are actual patterns of the primes that we’re not aware of. If there’s 1, there’s probably going to be more. Suddenly, a lot of our cryptosystems are in doubt.

Yeah. But then how do you say stuff about the primes? You’re going towards the Collatz conjecture again, because you want it to be random, right? You want it to be random.

Yeah. So more broadly, I’m just looking for more tools, more ways to show that things are random.

How do you prove a conjecture doesn’t hold? Is there any chance to you that P equals NP? Can you imagine a possible universe?

It is possible. I mean, there are various scenarios. There’s 1 where it is technically possible, but in practice is never actually implementable.

The evidence is slightly pushing in favor of no, that P is probably not equal to NP. It seems like one of those cases, similar to the Riemann hypothesis, where I think the evidence is leaning pretty heavily toward no—certainly more toward no than toward yes.

The funny thing about P versus NP is that we also have a lot more barriers than we do for almost any other problem. So while there’s evidence, we also have a lot of results ruling out many, many types of approaches to the problem.

This is the 1 thing that computer scientists have actually been very good at: saying that certain approaches cannot work. No-go theorems.

It could be undecidable.

We don’t know.

There’s a funny story I read that when you won the Fields Medal, somebody from the internet wrote you and asked, “What are you going to do now that you’ve won this prestigious award?” You quickly and very humbly said, “This shiny medal is not going to solve any of the problems I’m currently working on, so I’m just going to keep working on them.”

First of all, it’s funny to me that you would answer an email in that context. Second of all, it just shows your humility. But anyway, maybe you could speak to the Fields Medal. It’s another way for me to ask about Grigori Perelman. What do you think about him famously declining the Fields Medal and the Millennium Prize, which came with $1 million in prize money? He stated that, “I’m not interested in money or fame.”

The prize is completely irrelevant for me. If the proof is correct, then no other recognition is needed.

He's somewhat of an outlier, even among mathematicians, who tend to have somewhat idealistic views. I've never met him. I think I'd be interested to meet him one day, but I never had the chance. I know people who met him, but he's always had strong views about certain things.

It's not like he was completely isolated from the math community. He would give talks and write papers and so forth, but at some point he just decided not to engage with the rest of the community. He was disillusioned or something—I don't know—and he decided to peace out and collect mushrooms in St. Petersburg or something.

That's fine. You can do that. A lot of the problems that we solve do have practical applications, and that's great, but if you stop thinking about a problem, that's fine. He hasn't published since then in this field, but that's fine; there are many other people who've done so as well.

One thing I didn't realize initially with the Fields Medal is that it sort of makes you part of the establishment. Most mathematicians—you have career mathematicians who just focus on publishing the next paper, maybe getting promoted one rank, starting a few projects, taking some students, or something.

But then suddenly people want your opinion on things, and you have to think a little bit about things that you might have just foolishly said because no one was going to listen to you. It's more important now.

Is it constraining to you? Are you able to still have fun, be a rebel, try crazy stuff, and play with ideas?

I have a lot less free time than I had previously, mostly by choice. I obviously have the option to decline, so I decline a lot of things. I could decline even more, or I could acquire a reputation for being so unreliable that people don't even ask anymore.

This is—I love the different algorithms here. This is great. It's always an option.

There are things that—I don't spend as much time as I did as a postdoc just working on one problem at a time or fooling around. I still do that a little bit, but as you advance in your career, somehow the more soft skills—math somehow front-loads all the technical skills to the early stages of your career.

As a postdoc, you're incentivized to basically focus on proving very technical theorems. But then as you get more senior, you have to start mentoring, giving interviews, and trying to shape the direction of the field, both research-wise and sometimes administratively.

It's kind of the right social contract because you need to work in the trenches to see what can help mathematicians. The other side of the establishment—the really positive thing—is that you get to be a light and an inspiration to a lot of young mathematicians or young people who are just interested in mathematics.

It's just how the human mind works. This is where I would probably say that I like the Fields Medal: it does inspire a lot of young people somehow. I don't know; it's just how human brains work.

At the same time, I want to give some respect to somebody like Grigori Perelman, who is critical of awards. In his mind, those are his principles, and any human who's able to hold to their principles and do the thing that most humans would not be able to do—it's beautiful to see.

Some recognition is necessarily important, but it's also important not to let these things take over your life and only be concerned about getting the next big award or whatever.

Again, you see these people trying to solve only really big math problems and not work on things that are less sexy, if you wish, but are actually still interesting and instructive.

As you say, the way the human mind works is that we understand things better when they're attached to humans, and also when they're attached to a small number of humans. This is the way our human mind is wired. We can comprehend the relationships between 10 or 20 people, but once you get beyond 100 people, there's a limit. I think there's a name for it.

Beyond that, it just becomes the other. You have to simplify the whole mass: 99.9% of humanity becomes the other. Often these models are incorrect, and this causes all kinds of problems.

To humanize a subject, if you identify a small number of people and say, "These are representative people of the subject," role models, for example, that has some role. But too much of it can be harmful, because I'll be the first to say that my own career path is not that of a typical mathematician.

I had a very accelerated education and skipped a lot of classes. I think I had very fortunate mentoring opportunities, and I think I was at the right place at the right time. Just because someone doesn't have my trajectory, it doesn't mean that they can't be good mathematicians. They may be good mathematicians in a very different style, and we need people of different styles.

Sometimes too much focus is given to the person who does the last step to complete a project in mathematics or elsewhere that's really taken centuries or decades, with lots and lots of building on previous work. But that's a story that's difficult to tell if you're not an expert, because it's easier to just say one person did this one thing. It makes for a much simpler history.

I think, on the whole, it's a hugely positive thing to talk about Steve Jobs as a representative of Apple, when I personally know—and of course everybody knows—the incredible design and engineering teams, just the individual humans on those teams. They're not a team; they're individual humans on a team. There's a lot of brilliance there, but it's just a nice, pithy shorthand: Steve Jobs.

Yeah. As a starting point, as a first approximation, that's how you approach it. Then read some biographies and look into it much more deeply.

First approximation.

Yeah.

Yeah. That's right. So you mentioned you were a Princeton postdoc. Andrew Wiles was a professor there. It's a funny moment, how history is all interconnected. At that time, he announced that he had proved Fermat's Last Theorem. What did you think, looking back now with more context, about that moment in math history?

Yes. I was a graduate student at the time. I vaguely remember there was press attention, and we all had pigeonholes in the same mailroom, so we all picked up our mail. Suddenly, Andrew Wiles's mailbox exploded and overflowed.

That's a good metric.

Yeah. We all talked about it at tea and so forth. Most of us didn't understand the proof; we understood the high-level details. In fact, there's an ongoing project to formalize it in Lean.

Right, Kevin Buzzard? Can we take that small tangent? Is it difficult? Because, as I understand it, the proof of Fermat's Last Theorem has super-complicated objects. It's really difficult to formalize, right?

I guess you're right. The objects that they use are really complicated. You can define them; they've been defined in Lean. Just defining what they are can be done. That's really not trivial, but it's been done.

There are a lot of really basic facts about these objects that have taken decades to prove, and they're in all these different math papers, so lots of these have to be formalized as well.

Kevin's goal—actually, he has a 5-year grant to formalize Fermat's Last Theorem—is that he doesn't think he'll be able to get all the way down to the basic axioms. He wants to formalize it to the point where the only things he needs to rely on as black boxes are things that were known by 1980 to number theorists at the time.

Then some other work would have to be done to get from there. It's a different area of mathematics from the type of mathematics I'm used to.

In analysis, which is kind of my area, the objects we study are much closer to the ground. I study things like prime numbers and functions, things that are within the scope of a high school math education to at least define.

But then there's this very advanced algebraic side of number theory where people have been building structures upon structures for quite a while. It's a very sturdy structure. At the base, at least, it's extremely well developed in the textbooks and so forth.

But it does get to the point where, if you haven't taken these years of study and you want to ask what's going on at level 6 of this tower, you have to spend quite a bit of time before you can even get to the point where you see something you recognize.

What inspires you about his journey, which is similar to what we talked about—7 years mostly working in secret?

That is romantic. Yeah, so it kind of fits with the romantic image I think people have of mathematicians, to the extent they think of them at all, as these eccentric wizards or something. So that certainly accentuated that perspective. It is a great achievement. His style of solving problems is so different from my own, which is great. I mean, we need people like that.

In terms of collaboration, you like the collaborative approach; I like moving on from a problem if it's taking too much effort.

But you need the people who have the tenacity and the fearlessness. I've collaborated with people like that, where I want to give up because the first approach that we tried didn't work and the second one didn't work, but they're convinced, and then the third, fourth, or fifth approach works. And I have to eat my words: “Okay, I didn't think this was going to work, but yes, you were right all along.”

And we should say, for people who don't know, not only are you known for the brilliance of your work, but also for your incredible productivity—the sheer number of papers, all of which are of very high quality. So there's something to be said about being able to jump from topic to topic.

Yeah, it works for me. There are also people who are very productive and focus very deeply on one topic. I think everyone has to find their own workflow.

One thing that is a shame in mathematics is that we have a one-size-fits-all approach to teaching mathematics. We have a certain curriculum and so forth. Maybe if you do math competitions or something, you get a slightly different experience, but I think many people don't find their native mathematical language until very late, or usually too late. They stop doing mathematics and have a bad experience with a teacher who's trying to teach them one way to do mathematics, and they don't like it.

My theory is that evolution has not given us a mathematical center in the brain directly. We have a vision center, a language center, and some other centers that evolution has honed, but we don't have an innate sense of mathematics. Our other centers are sophisticated enough that different people can repurpose different areas of the brain to do mathematics.

Some people have figured out how to use the visual center to do mathematics, so they think very visually when they do mathematics. Some people have repurposed their language center, and they think very symbolically. If they're very competitive and they like gaming, there's a part of the brain that's very good at solving puzzles and games, and that can be repurposed.

When I talk about mathematicians, I can tell that they're using some different styles of thinking than I am. They're not disjoint, but they may prefer visual thinking. I don't actually prefer visual thinking so much; I need lots of visual aids myself. Mathematics provides a common language, so we can still talk to each other even if we're thinking in different ways.

You can tell that there's a different set of subsystems being used in the thinking process, as if they take different paths. They're very quick at things that I struggle with, and vice versa, and yet they still get to the same goal.

That's beautiful. But the way we educate—unless you have a personalized tutor or something—education, by natural necessity, has to be mass-produced. You have to teach 30 kids, and if they have 30 different styles, you can't teach 30 different ways.

On that topic, what advice would you give to students, young students who are struggling with math but are interested in it and would like to get better? Is there something in this complicated educational context that you would recommend?

It's a tricky problem. One nice thing is that there are now lots of sources for mathematical enrichment outside the classroom. In my day, there were already math competitions and popular math books in the library. But now you have YouTube, and there are forums devoted to solving math puzzles.

Math shows up in other places, too. For example, there are hobbyists who play poker for fun and, for very specific reasons, are interested in very specific probability questions. There is a community of amateur probabilists in poker, in chess, and in baseball. There's math all over the place.

I'm hoping that with these new tools for Lean and so forth, we can incorporate the broader public into math research projects. This almost doesn't happen at all currently. In the sciences, there's some scope for citizen science: amateur astronomers discover comets, and there are biologists who can identify butterflies and so forth.

In mathematics, amateur mathematicians can discover new primes and so forth, but previously, because we have to verify every single contribution, most mathematical research projects would not benefit from input from the general public. In fact, it would just be time-consuming because of all the error-checking and everything.

One thing about these formalization projects is that they are bringing in more people. I'm sure there are high school students who've already contributed to some of these formalization projects and to mathlib. You don't need to be a PhD holder to work on one atomic thing.

There's something about the formalization here that, as a very first step, opens it up to the programming community too—the people who are already comfortable with programming. It seems like programming is somehow—maybe it's just the feeling—but it feels more accessible to folks than math.

Math, especially modern mathematics, is seen as this extremely difficult-to-enter area, and programming is not. So that could be just an entry point. You can execute code and get results. You can print “Hello, world!” pretty quickly.

If programming were taught as an almost entirely theoretical subject, where you just taught computer science—the theory of functions and routines and so forth—and outside of some very specialized homework assignments, you weren't actually programming on the weekend for fun, it would be considered as hard as math.

Yeah, it would be considered as hard as math. As I said, there are communities of non-mathematicians who deploy math for some very specific purpose, like optimizing their poker game, and for them math becomes fun.

What advice would you give in general to young people about how to pick a career and how to find themselves? That's a tough question.

It's a tough problem. There's a lot less certainty now in the world. There was this period after the war where, at least in the West, if you came from a good demographic, there was a very stable path to a good career. You went to college, got an education, picked one profession, and stuck to it. That's becoming more and more a thing of the past.

I think you just have to be adaptable and flexible. People have to get skills that are transferable. Learning one specific programming language or one specific subject of mathematics is not, in itself, a super-transferable skill. But knowing how to reason with abstract concepts or how to problem-solve when things go wrong—these are things that I think we'll still need, even as our tools get better and we're working with AI support and so forth.

Actually, you're an interesting case study. You're one of the great living mathematicians, right? You had a way of doing things, and then all of a sudden you started learning new fields. First of all, you kept learning new fields, but you learned Lean. That's not a trivial thing to learn. For a lot of people, that's an extremely uncomfortable leap to take.

Yeah. Mathematicians—first of all, I've always been interested in new ways to do mathematics. I feel like a lot of the ways we do things right now are inefficient. My colleagues and I spend a lot of time doing very routine computations or doing things that other mathematicians would instantly know how to do, and we don't know how to do them. Why can't we search and get a quick response? That's why I've always been interested in exploring new workflows.

About 4 or 5 years ago, I was on a committee where we had to ask for ideas for interesting workshops to run at a math institute. At the time, Peter Scholze had just formalized one of his new theorems, and there were some other developments in computer-assisted proof that looked quite interesting. I said, “We should run a workshop on this. This would be a good idea.”

Then I was a bit too enthusiastic about this idea, so I got volunteered. I did it with a bunch of other people—Kevin Buzzard, Jordan Ellenberg, and a bunch of other people. It was a nice success. We brought together a bunch of mathematicians, computer scientists, and other people, and we got up to speed on the state of the art.

It was really interesting. There were developments that most mathematicians didn't know were going on.

There were lots of nice proofs of concept, just hints of what was going to happen. This was just before ChatGPT, but even then there was 1 talk about language models and their potential capabilities in the future. So that got me excited about the subject. I started giving talks about how this is something more of us should start looking at.

Then ChatGPT came out, and suddenly AI was everywhere. I got interviewed a lot about this topic, and in particular, the interaction between AI and formal proof assistants. I said, “Yeah, they should be combined. This is a perfect synergy waiting to happen here.”

At some point, I realized that I had to actually do—not just talk the talk, but walk the walk. I don’t work in machine learning, and I don’t work in proof formalization, and there’s a limit to how much I can rely on authority and say, “I’m a working mathematician; just trust me when I say that this is going to change mathematics.” If I’m not doing any of it myself, I felt like I had to actually justify it.

A lot of what I get into, actually, I don’t quite see in advance how much time I’m going to spend on it. It’s only after I’m waist-deep in a project that I realize by that point I’m committed.

Well, that’s deeply admirable—that you’re willing to go into the fray, be in some small way a beginner, and have some of the challenges that a beginner would: new concepts, new ways of thinking. I think in that context, you could be a Fields Medal–winning mathematician, and an undergrad knows something better than you.

Yeah. I think mathematics inherently—I mean, mathematics is so huge these days that nobody knows all of modern mathematics. Inevitably, we make mistakes, and you can’t cover up your mistakes with bravado because people will ask for your proofs, and if you don’t have the proofs, you don’t have the proofs.

I don’t love math.

Yeah, so it does keep us honest. You can still—it’s not a perfect panacea—but I think we do have more of a culture of admitting error because we’re forced to all the time.

Big, ridiculous question. I’m sorry for it once again. Who is the greatest mathematician of all time? Maybe one who’s no longer with us.

Who are the candidates? Euler, Gauss, Newton, Ramanujan, Hilbert.

First of all, as mentioned before, there’s some time dependence on the day. If you plot cumulatively over time, for example, Euclid is one of the leading contenders. Then maybe some unnamed, anonymous mathematicians before that—whoever came up with the concept of numbers.

Do mathematicians today still feel the impact of Hilbert directly, of everything that’s happened in the 20th century?

Yeah, directly. Hilbert spaces, for example. We have lots of things named after him, of course—the arrangement of mathematics and the introduction of certain concepts. The 23 problems have been extremely influential. There’s some strange power to declaring which problems are hard to solve.

The statement of the open problems.

Yeah. This is the bystander effect everywhere. If no one says you should do X, everyone just moves around waiting for somebody else to do something, and nothing gets done.

One thing that you actually have to teach undergraduates in mathematics is that you should always try something. You see a lot of paralysis in an undergraduate trying a math problem. If they recognize that there’s a certain technique that can be applied, they will try it, but there are problems for which none of their standard techniques obviously apply. The common reaction is then just paralysis: “I don’t know what to do.”

There’s a quote from The Simpsons: “I’ve tried nothing and I’m all out of ideas.” The next step is to try anything, no matter how stupid—in fact, the stupider, the better. One technique is almost guaranteed to fail, but the way it fails is going to be instructive. It fails because you’re not at all taking into account this hypothesis. “Oh, this hypothesis must be useful.” That’s a clue.

I think you also suggested somewhere this fascinating approach, which really stuck with me. I started using it, and it really works. I think you said it’s called structured procrastination.

Yes. It’s when you really don’t want to do a thing, so you imagine a thing you don’t want to do more—something that’s worse than that. Then, in that way, you procrastinate by not doing the thing that’s worse.

Yeah.

It’s a nice hack. It actually works. Psychology is really important. You talk to athletes, like marathon runners and so forth, and they talk about what’s most important: their training regimen, their diet, and so forth. Actually, so much of it is psychology—tricking yourself into thinking that the problem is feasible so that you’re motivated to do it.

Is there something our human mind will never be able to comprehend?

Well, as a mathematician, there must be some stuff that you can’t understand. That was the first thing that came to mind.

But even broadly, is there something about our mind that we’re going to be limited in, even with the help of mathematics?

Well, okay, how much augmentation are you willing to consider? For example, if I didn’t even have pen and paper, if I had no technology whatsoever—

Okay, so I’m not allowed a blackboard, pen, and paper, right?

You’re already much more limited than you would be.

Incredibly limited. Even language—the English language—is a technology.

It’s one that’s been very internalized.

So you’re right. The formulation of the problem is incorrect, because there really is no longer just a solo human. We’re already augmented in extremely complicated, intricate ways. We’re already like a collective intelligence.

Humanity, plural, has much more intelligence in principle, on its good days, than the individual humans put together. It can also have less.

Okay, but the mathematical community, plural, is an incredibly superintelligent entity that no single human mathematician can come close to replicating.

You see it a little bit on these question-and-answer sites. MathOverflow is the math version of Stack Overflow, and sometimes you get these very quick responses to very difficult questions from the community. It’s a pleasure to watch, actually. As an expert, I’m a fascinated spectator of that site, just seeing the brilliance of the different people, the depth of knowledge that people have, and the willingness to engage in the rigor and nuance of the particular question. It’s pretty cool to watch. It’s almost just fun to watch.

What gives you hope about this whole thing we have going on—human civilization?

I think the younger generation is always really creative, enthusiastic, and inventive. It’s a pleasure working with young students.

The progress of science tells us that problems that used to be really difficult can become extremely trivial to solve. Navigation, for example—just knowing where you were on the planet—was this horrendous problem. People died or lost fortunes because they couldn’t navigate, and we have devices in our pockets that do this automatically for us. It’s a completely solved problem.

Things that seem unfeasible for us now could perhaps just be homework exercises for future generations.

One of the things I find really sad about the finiteness of life is that I won’t get to see all the cool things we create as a civilization. In the next 100 or 200 years, just imagine showing up in 200 years.

Well, already plenty has happened. If you could go back in time and talk to your teenage self, you’d see how much has happened.

Yeah, and just the internet and our AI. Again, they’re beginning to be internalized, so we say, “Yeah, of course an AI can understand our voice and give reasonable, slightly incorrect answers to any question.” But this was mind-blowing even 2 years ago.

At the moment, it’s hilarious to watch the drama on the internet. People take everything for granted very quickly, and then we humans seem to entertain ourselves with drama out of anything that’s created. Somebody needs to take one opinion, another person needs to take an opposite opinion, and they argue with each other about it.

But when you look at the arc of things, even the progress of robotics, you can take a step back and be like, “Wow, this is beautiful, that we humans are able to create this.”

Terrence Tao

When the infrastructure and the culture are healthy, the community of humans can be so much more intelligent, mature, and rational than the individuals within it.

Well, one place I can always count on rationality is the comment section of your blog, which I’m a big fan of. There are a lot of really smart people there.

Thank you, of course, for putting those ideas out on the blog. I can’t tell you how honored I am that you would spend your time with me today.

I was looking forward to this for a long time, Terry. I'm a huge fan. You inspire me. You inspire millions of people. Thank you so much for talking.

Terrence Tao

Oh, thank you. It was a pleasure.