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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, he is 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 Terrence Tao. What was the first really difficult research-level math problem that you encountered? One that gave you pause maybe? Well, I mean in your undergraduate education, you learn about the really hard impossible problems like the Riemann hypothesis and the twin primes 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 on the boundary between what we can do relatively easily and what are hopeless. But what are problems where existing techniques can do like 90% of the job and then you just need that remaining 10%? I think as a PhD student, the CA problem certainly caught my eye, and it just actually got solved. It’s a problem I’ve worked on a lot in my early research. Historically, it came from a little puzzle by the Japanese mathematician Soji Kaya in like 1918 or so.

So the puzzle is that you have a needle on the plane. Think like driving on a road or something, and you want it to execute a U-turn. You want to turn the needle around, but you want to do it in as little space as possible. So you want to use as little area in order to turn it around. The needle is infinitely maneuverable, so you can imagine just spinning it around as a unit needle. You can spin it around its center.

I think that gives you a disc of area, I think pi over four. Or you can do a three-point U-turn, which is what they teach people in driving schools to do. That actually takes area pi over eight. 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 Mazikovich showed that in fact, you could actually turn the needle around using as little area as you wanted.

There was some really fancy multi-back-and-forth U-turn thing that you could do that would turn a needle around and in doing so would pass through every intermediate direction. Is this in the two-dimensional plane? This is in the two-dimensional plane, yeah. So we understand everything in two dimensions.

The next question is what happens in three dimensions. Suppose like the Hubble Space Telescope is a tube in space, and you want to observe every single star in the universe. So you want to rotate the telescope to reach every single direction. And 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 needle around to see every single star in the sky. How small a volume do you need to do that?

And so you can modify 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 two-dimensional construction. But the question is that if your telescope is not zero thickness but just very thin, some thickness delta, what is the minimum volume needed to be able to see every single direction as a function of delta.

So as delta gets smaller and you need gets thinner, the volume should go down. But how fast does it go down? The conjecture was that it goes down very very slowly, logarithmically, roughly speaking, and that was proved after a lot of work. So this seems like a puzzle. Why is it interesting?

So it turns out to be surprisingly connected to a lot of problems in partial differential equations, in number theory, in geometry, and comics. For example, in wave propagation, you splash some water around, you create water waves and they travel in various directions. But waves exhibit both particle and wave-type behavior.

So you can have what’s called a wave packet, which is like a very localized wave that is localized in space and moving in a certain direction in time. And so if you plot it in both space and time, it occupies a region that looks like a tube. And so what can happen is that you can have a wave which initially is very dispersed, but it all focuses at a single point later in time.

Like you can imagine dropping a pebble into a pond and ripples spread out. But then if you time-reverse that scenario, and the equations of wave motion are time-reversible, you can imagine ripples that are converging to a single point and then a big splash occurs, maybe even a singularity.

And so it’s possible to do that. Geometrically what’s going on is that there’s always a set of light rays. So like 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 on these light rays and they’re all focusing at this one point.

So 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 it separates. But potentially if the conjecture had a negative solution, what that meant is that there’s a very efficient way to pack tubes pointing in different directions into a very narrow region of very narrow volume.

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

In mathematical physics, we care a lot about whether certain equations in 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 the 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 around water 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 towards the belief that for certain very special initial configurations of, say, water, singularities can form. But people have not yet been able to actually establish this.

The Clay Foundation has these seven Millennium Prize Problems, which has a million-dollar prize for solving one of these problems. This is one of them. Of these seven, only one of them has been solved: the Poincaré Conjecture by Perelman. So the Navier-Stokes 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 neighbors? So the existence and smoothness, like you said, is a Millennium Prize Problem. You’ve made a lot of progress on this one. In 2016, you published a paper on finite-time blow-up for an averaged three-dimensional Navier-Stokes equation. So we’re trying to figure out if this thing usually doesn’t blow up, but can we say for sure it never blows up?

Yeah, so 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% of all situations are covered. So 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 that govern the field of fluid dynamics, trying to understand how fluid behaves. It actually turns out to be a really complicated thing to try to model. So it has practical importance. This Clay Prize problem concerns what’s called the incompressible Navier-Stokes, which governs things like water. There’s something called the compressible Navier-Stokes, which governs things like air. And that’s particularly important for weather prediction.

Weather prediction does a lot of computational fluid dynamics. A lot of it is actually just trying to solve the Navier-Stokes equations as best they can, also gathering a lot of data so that they can initialize the equation. 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 it not blowing up? The short answer is Maxwell’s demon. So Maxwell’s demon is a concept in thermodynamics. If you have a box of two gases, oxygen and hydrogen, and maybe you start with all the oxygen on one side and 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, like maybe there’s a microscopic demon called Maxwell’s demon that, every time an oxygen and nitrogen atom collide, will bounce off in such a way that the oxygen drifts onto one side and then goes to the other. improbable configuration emerge.

Which we never see, and 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. On the long term, you should see as many ones and twos and threes as fours and fives and sixes. 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, it biases one digit to 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.

Okay. So 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, and 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; like you splash around, there’s some turbulence and waves and so forth. But eventually, it settles down, and the lower the amplitude, the smaller the velocity, the more calm it gets.

But potentially there is some sort of demon that keeps pushing the energy of the fluid into a smaller and smaller scale, and it will move faster and faster. At faster speeds, the effective viscosity is relatively less. So it could happen that it creates some sort of self-similar blowup scenario where the energy of the fluid starts off at some large scale and then it all sort of transfers its energy into a smaller region of the fluid, which then 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, and then you could actually converge to all the energy concentrating in one point in a finite amount of time. That scenario is called finite blowup.

In practice, this doesn’t happen. So 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 into maybe three or four, and then those must split up into maybe three or four small eddies of their own. So the energy gets dispersed to the point where the viscosity can then keep that thing 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 blob can occur. There were papers claiming that you just need to take into account conservation of energy and just carefully use the viscosity, and you can keep everything under control for not just 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 blowup, that velocity say smooth, and it all failed. There was always some sign error or some subtle mistake and it couldn’t be salvaged. So what I was interested in doing was trying to explain why we were not able to disprove planet 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 naval, basically if I could turn off certain types of ways in which water interacts and only keep the ones that I want. So in particular, if there’s a fluid that 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. So 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? So, it provides what’s called an obstruction in mathematics.

So what I did was that basically if I turned off certain parts of the equation, usually when you turn off certain interactions, it makes it less nonlinear, more regular, and less likely to blow up. But I found that by turning off a very well-designed set of interactions, I could force all the energy to blow in finite time.

So 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 which my artificial equation does not satisfy. So it rules out certain approaches.

The thing about math 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 the 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 that these methods are going to work.

So having these counterexamples for nearby problems kind of 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 just some more general intuition you build up about mathematics?

So the key phenomenon that my technique exploits is what’s called supercriticality. In partial differential equations, often these equations are 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 that energy from one location of space can get transported because the fluid is in motion to other locations. And that’s a nonlinear effect, and that causes all the problems. There are these two competing terms in the Navier-Stokes 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. It’s 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. So at smaller and smaller scales, the transport terms are much stronger than the viscosity terms.

So the viscosity are the things that calm things down. This is why the problem is hard in two dimensions. The Soviet mathematician Ladishkaya, she in the 60s showed in two dimensions there is no blow-up, and in two dimensions the Navier equations are what’s called critical: the effect of transport and the effect of 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 proof of 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 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.

This is sort of 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 for being sort of nice and predictable, like planetary motion. I mean 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 two weeks into the future because it’s a supercritical equation.

Lots of really strange things are going on at very fine scales. So, whenever there is some huge source of nonlinearity, that can create a huge problem for predicting what’s going to happen. If the nonlinearity is somehow more and more featured and interesting at small scales, I mean there are many equations that are nonlinear, but in many equations you can approximate things by the bulk.

For example, planetary motion: if you want to understand the orbit of the moon or Mars or something, you don’t really need the microstructure of the seismology of the moon or exactly how the mass is distributed. You can almost approximate these planets by point masses, and just the aggregate behavior is important. But if you want to model a fluid, like the weather, you can’t just say in Los Angeles the temperature is this, the wind speed is this.

For supercritical equations, the fine confirmation is really important. If we can just linger on the Navier’s equations a little bit, you’ve suggested maybe you can describe that 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 computation 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 average equation that blew up. As part of how I had to do this, there is this 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. fast as possible. This is sort of the naive way to force blow up.

It turns out in five and high dimensions this works. But in three dimensions there was this funny phenomenon that I discovered that if you change the laws of physics you just always keep trying to push the energy into smaller scales. What happens is that the energy starts getting spread out into many scales at once. So that 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. And then it turns out that it makes it vulnerable for viscosity to come in and actually just damp out everything.

So it turns out this direct push doesn’t actually work. There was a separate paper by some other authors that actually showed this in three dimensions. So what I needed was to program a delay. So kind of like air locks. 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, and only after you pushed all the energy in then you sort of open the next gate and then you push that in as well.

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

So I actually thank my wife for this because she was trained as an electrical engineer. And she talked about how she had to design circuits and so forth. And you know 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 like capacitors and resistors and so forth, and you have to build a diagram.

These diagrams you can sort of follow 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. And I would stack them together in such a way that I would create something that would open one gate, and then there’ll be a clock that would, and then once the clock hits a certain threshold, it would close it kind of a Rube Goldberg type machine but described mathematically and this ended up working.

So what I realized is that if you could pull the same thing off for the actual equations. So if the equations of water support a computation, like if you can imagine kind of a steampunk but really water punk type of thing where modern computers are electronic, you know, 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 certain velocities and maybe it’s. They’re two different configurations corresponding to a bit being up or down. Probably if you had two 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 which are made completely out of water.

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, 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 one machine to Mars and this machine had the ability to mine the planet, create some more materials to 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, which is a robot, its purpose in life 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 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, and this is what I managed to accomplish for this average Navier-Stokes. 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, there are things called vortex rings that might possibly work, but analog computing is really nasty compared to digital computing. I mean, there’s 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 actually are pursuing much closer to the direct self-similar model, which 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 that you’re trying to get the smaller and smaller blob to now having a liquid Turing machine that gets 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. I mean, the thing about mathematics is that it’s really good at spotting connections between what you think of as completely different problems. But if the mathematical form is the same, you can draw a connection. So 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. 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 have these animations going, and they look very chaotic. In fact, they look 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 four or five cells which evolves and moves in a certain direction, similar to vortex rings. The Game of Life is kind of like a discrete equation, and the Navier-Stokes is a continuous equation, 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 three or four rules to operate, 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 one at a time. After a lot of effort, people managed to create AND gates and OR gates for gliders.

There’s this massive structure which, if you have a stream of gliders coming in one direction and a stream of gliders coming in another direction, you may produce a stream of gliders coming out. If both of the streams have gliders, there will be an output stream, but if only one of them does, then nothing comes out. 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 an enormous steampunk-type thing. They look ridiculous. But then people also generated self-replicating objects in the Game of Life—a massive machine, a “bomb machine,” which over a huge period of time always looks like glider guns inside doing these very steampunk calculations. It would create another version of itself which could replicate.

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. As I said, analog is much worse than digital. You can’t just directly take the constructions in the Game of Life and plunk them in, but 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 amendable to mathematical analysis? Do we have the tools to say something profound about that? The thing is you can get this emergent behavior in very complicated structures but only with very carefully prepared initial conditions.

Yeah. So these glider guns and gates and so forth, machines, if you just plunk down randomly some cells, 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 proving 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 like the digits of pi, which we believe is a good example.

However, there’s a very small number of things that have patterns. You can prove something has a pattern by 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 that, for example, most sequences of digits have no pattern.

If you just pick digits randomly, there’s something called law of large numbers. It tells you you’re going to get as many ones as twos 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, how can I show that this doesn’t have some weird pattern to it?

Some other work that I spend a lot of time on is to prove 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 with natural numbers, for example, two maps to four, three maps to six, and so forth.

Some functions are additive, which means that if you add two inputs together, the output gets added as well. For example, multiplying by a constant: if you multiply a number by 10, multiplying a plus b by 10 is the same as multiplying a by 10 and b by 10 and then adding them together.

So some functions are additive, some are kind of additive but not completely additive. For example, if I take a number n, multiply it by the square root of two, and take the integer part of that, say 10 by the square root of two, it is like 14 point something. So 10 up to 14, 20 up to 28.

In that case, additively it is true then. So 10 + 10 is 20, and 14 + 14 is 28. But because of this rounding, sometimes there are roundoff errors. Sometimes when you add a plus b, this function doesn’t quite give you the sum of the two individual outputs, but the sum plus or minus one.

So 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, then there’s a reason for why it’s true. The reason is because there’s… there’s some other nearby function which is actually completely structured, which is explaining this sort of partial pattern that you have.

And so if you have these inverse theorems, it creates this sort of dichotomy that either the objects that you study 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 that there’s this old theorem in mathematics called Sim Theorem, proven in the 1970s. It concerns trying to find a certain type of pattern in a set of numbers.

The patterns that have progression are things like 3, 5, and 7 or 10, 15, and 20. Andreli proved that any set of numbers that are sufficiently big, what’s called positive density, has arithmetic progressions in it of any length you wish. For example, the odd numbers have a set of density 1/2, and they contain arithmetic progressions of any length. So 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.

But Zermelo’s theorem also applies to random sets. If I take the set of odd numbers and I flip a coin for each number, and I only keep the numbers for which I got heads, I just randomly take out half the numbers and keep one half. So 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’s 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, 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. But you need an extremely long random sequence for this to happen. I suppose that’s intuitive.

It’s just infinity. Yeah, infinity absorbs a lot of sins. How are we humans supposed to deal with infinity? Well, you can think of infinity as just 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?

And a way in which mathematicians formalize that is that mathematics has found a formalism to idealize something being extremely large or extremely small to actually being 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 got all kinds of real world effects, but you can idealize certain things to infinity and send certain things to zero. The mathematics becomes a lot simpler to work with there. I wonder how often using infinity forces us to deviate from the physics of reality.

Yeah. So, there’s 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 know, for example, a plus b is always b plus a. So, when you have a finite number of terms, 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. And so 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 what’s called finetizing them. So, 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 H to come out? That’s a more quantitative question, and 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. So, this is why you never see the monkeys create Hamilton. You can maybe see them create a four-letter word, but nothing that big. And so I personally find once you finitize an infinite statement, it does become much more intuitive, and it’s no longer so weird.

So, even if you’re working with infinity, it’s good to finitize so that you can have some intuition. Yeah. The downside is that the finite groups are just much messier. So, the infinite ones are found first, usually like decades earlier, and then later on people finetize 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, of seeing the world? Maybe we can throw in engineering; you mentioned your wife is an engineer, giving a new perspective on circuits, right?

So, this is a different way of looking at the world given that you’ve done mathematical physics. You’ve worn all the hats, right? I think science, in general, is an interaction between three things: there’s the real world, there’s what we observe of the world, our observations, and then our mental models as to how we think the world works.

We can’t directly access reality. All we have are the observations which are incomplete, and they have errors. There are many cases where we would want to know, for example, what is the weather like tomorrow, and we don’t yet have the observations we’d like to make a prediction. And then we have these simplified models, sometimes making unrealistic assumptions. Spherical cow type things. Those are the mathematical models. Mathematics is concerned with the models. Science collects the observations and it proposes the models that might explain these observations. What mathematics does is we stay within the model and we ask what are the consequences of that model? What predictions would the model make of future observations or past observations? Does it fit observed data?

So 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 up from that model. In almost any other discipline, you start with the conclusions. I want to do this; I want to build a bridge. I want to make money. I want to do this. Okay. And then you find the path to get there.

There’s a lot less speculation about, “Suppose I did this, what would happen?” You know, planning and modeling, speculative fiction maybe is one other place. But that’s about it, actually. Most of the things we do in life are conclusion-driven, including physics and science. They want to know where is this asteroid going to go? What is the weather going to be tomorrow?

But Bathe also has this other direction of going from the axioms. What do you think there is this tension in physics between theory and experiment? Mhm. What do you think is the more powerful way of discovering truly novel ideas about reality? Well, you need both top down and bottom up. Yeah, it’s a real interaction between all these things. So over time, the observations and the theory and the modeling should both get closer to reality.

Initially, this is always the case. You know they’re always far apart to begin with. But you need one to figure out where to push the other. So if your model is predicting anomalies that are not picked up by experiments, that tells experimenters where to look to find more data to refine the models.

Yeah, so 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. Like 99% of mathematics is theoretical mathematics and there’s a very tiny amount of experimental mathematics. I mean people do it. You know, like if they want to study prime numbers or whatever, they can just generate large data sets.

Once we had computers, we began to do it a little bit. Although even before, well like Gauss for example, he conjectured the most basic theorem in number theory called the prime number theorem, which predicts how many primes are up to a million, up to a trillion. It’s not an obvious question and basically what he did was that he computed, mostly by himself, but he also hired human computers—people whose professional job it was to do arithmetic—to compute the first 100,000 primes or something and made tables and made a prediction.

That was an early example of experimental mathematics. But until very recently, theoretical mathematics was just much more successful. I mean because doing complicated mathematical computations was just not feasible until very recently. Even nowadays, even though 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, Zodius, you want to study all possible subsets of the numbers 1 to 1000. There’s only 1000 numbers. How bad could it be? It turns out the number of different subsets of 1 to 1000 is 2 to the 1000, which is way bigger than any computer can currently enumerate. In fact, anybody ever will ever enumerate.

There are certain math problems that very quickly become just intractable to attack by direct brute force computation. Chess is another famous example. The number of chess positions we can’t get a computer to fully explore. But now we have AI; we have tools to explore this space not with 100% guarantees of success but with experiments. We can empirically solve chess now, for example. We have very good AIs that can, they don’t explore every single position in the game tree, but they have found some very good approximations, and people are using these chess engines to do experimental chess. They’re revisiting old chess theories about, when you have this type of opening, this is a good type of move, this is not, and they can use these chess engines to actually refine, in some cases overturn conventional wisdom about chess.

I do 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, there’s a similar thing in mathematics that I don’t believe is providing a kind of formal explanation of the different positions. It’s just saying which position is better or not that 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. So in case people don’t know, it’s where people are observing shadows of reality, not reality itself, and they 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? Well, there are these three ontological things. There’s actual reality, there’s 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; an initial model of the world is flat because it looks flat and that it’s big, and the rest of the universe, the skies are not. The sun, for example, looks really tiny. You start off with a model which is actually really far from reality, but it fits kind of the observations that you have. So 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, and so over time we had to realize that the Earth was round, that it spins, and it goes around the solar system. The solar system goes around the galaxy. So on and so forth. And the universe is expanding, the expansion itself is expanding, accelerating, and in fact very recently in this year.

So the acceleration of the universe itself is evidence that this is non-constant, and the explanation behind why that is, it’s catching up. It’s catching up. I mean, it’s still the dark matter or dark energy, this kind of thing. We have a model that sort of explains that fits the data really well. It just has a few parameters that you have to specify.

But so, people say that’s 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. So if you have a model with 10 parameters that explains 10 observations, that is a completely useless model. It’s what’s called overfitted. But if you have a model with two parameters and it explains a trillion observations, which is basically the dark matter model I think has like 14 parameters, and it explains petabytes of data that the astronomers have.

You can think of a theory, like one way to think about physical math theory is it’s a compression of the universe and data compression. So you know, you have these petabytes of observations. You’d like to compress it to a model which you can describe in five pages and specify a certain number of parameters, and if it can fit to reasonable accuracy almost all of your observations. I mean, the more compression that you make, the better your theory.

In fact, 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 do an equation like E=MC². There is actually some mathematical possible explanation for that.

There’s this phenomenon in mathematics called universality. So many complex systems at the macro scale are coming out of lots of tiny interactions at the micro scale, and normally because of the common form of explosion you would think that the macro scale equations must be like infinitely exponentially more complicated than the microscale ones. They are if you want to solve them completely exactly. If you want to model all the atoms in a box of air, that’s like Avogadro’s number is humongous. There’s a huge number of particles. If you actually have to track each one, it’ll be ridiculous.

But certain laws emerge at the microscopic 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 quintillion particles in a box, you just need to know its temperature, pressure, and volume, and a few parameters like five or six, and it models almost everything you need to know about these particles.

So we have, we don’t understand universality anywhere near 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 that explains why the bell curve shows up. Everywhere in nature, 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. Yes, you can go meta if you like. But there are many processes; for example, you can take lots and 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, and 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 this system fails. So universality is not a 100% reliable thing 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, that if you ask a population of 100,000 Americans with mortgages what proportion of them would default on their mortgages, if everything was decorated, it would be a 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.

Now I think there’s more awareness that systemic risk is actually a much bigger issue, and just because the model is pretty and nice, it may not match reality. Right. So 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. I mean that you need both.

Mathematics can help because it can, for example, with these central limit theorems, tell you that if you have certain assumptions like non-correlation, if all the inputs were 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. So if you have a mathematical understanding of the central limit theorem, and someone proposes to use this Gaussian copy or whatever to model default risk, if you’re mathematically trained, you would say okay, but what if there is systemic correlation between all your inputs?

Then you can ask the economists how much of a risk that is, and then you can go look for that. So 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.

So you’re the perfect person to ask: do you think there are threads that connect all the disparate areas of mathematics? Is there a kind of deep underlying structure to all of mathematics? There’s certainly a lot of connecting threads. and a lot of the progress of mathematics has can be represented by taking by stories of two fields of mathematics that were previously not connected and finding connections. An ancient example is geometry and number theory. So in the times of the ancient Greeks, these were considered different subjects. I mean, mathematicians worked on both. You could work both on geometry most famously but also on numbers. But they were not really considered related.

I mean, a little bit like you could say that this length was five times this length because you could take five copies of this length and so forth. But it wasn’t until Descartes who really realized that — who developed analytic geometry — that you can parameterize the plane, a geometric object, by two real numbers. Every point can be, and so geometric problems can be turned into problems about numbers. Today, this feels almost trivial, like there’s no content to this, like 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 two fields were unified.

This process has just gone on throughout mathematics over and over again. Algebra and geometry were separated and now we have a student algebraic geometry that connects them and over and over again. That’s certainly the type of mathematics that I enjoy the most.

So I think there’s sort of different styles to being a mathematician. I think hedgehogs and foxes— a fox knows many things a little bit, but a hedgehog knows one thing very, very well. In mathematics, there’s definitely both hedgehogs and foxes. Then there’s people who are kind of who can play both roles. I think like ideal collaboration between mathematicians involves diversity, like a fox working with many hedgehogs or vice versa.

So yeah, but I identify mostly as a fox. Certainly, I like arbitrage somehow, learning how one field works, learning the tricks of that field, and then going to another field which people don’t think is related, but I can adapt the tricks. I see the connections between the fields.

There are other mathematicians who are far deeper than I am, like who really— they’re really hedgehogs. They know everything about one field, and they’re much faster and more effective in that field. But I can give them these extra tools. I mean, you said that you can be both the hedgehog and the fox depending on the context, depending on the collaboration.

So what can you, if it’s at all possible, speak to the difference between those two ways of thinking about a problem? Say you’re encountering a new problem, searching for the connections versus like very singular focus. I’m much more comfortable with the fox paradigm. Yeah.

So, I like looking for analogies, narratives. I spend a lot of time — if there’s a result I see in one field and I like the result, it’s a cool result, but I don’t like the proof. 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 the exercise of doing so, I can say, oh 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, very doing crazy things in crazy fields and reinventing the wheel a lot. Yeah. Whereas the hedgehog style is I think 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. You have a very good understanding of exactly the strengths and weaknesses of each particular technique.

Yeah, I think you rely a lot more on sort of calculation than trying to find narratives. So yeah, I mean I can do that too, but there are other people who are extremely good at that. Let’s step back and maybe look at a bit of a romanticized version of mathematics. Mhm. So, 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 just this amazing way of thinking about all kinds of things in a way that you would normally think of. So he thought of proofs themselves as occupying some sort of space, you know.

So if you want to prove something, let’s say that there’s infinitely many primes, you avoid different proofs, but you could rank them in different axes like some proofs are elegant, some are long, some proofs are elementary and so forth. And so there’s this cloud. The space of all proofs itself has some sort of shape. And so he was interested in extreme points of this shape; like out of all these proofs, what is one that is the shortest at the extent of everything else or the most elementary 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 just found that really eye-opening that it’s not just getting a proof for a result was interesting, but once you have that proof, you know, trying to optimize it in various ways. That proof itself had some craftsmanship to it. It certainly informed my writing style.

But you know, like when you do your math assignments as an undergraduate, your homework and so forth, you’re sort of encouraged to just write down any proof that works, okay, and hand it in and get 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 to generalize to other things.

It’s the same in many other disciplines, like coding. There’s a lot of analogies between math and coding. I like analogies if you haven’t noticed. But you know, like you can code something, spaghetti code, that works for a certain task and 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. on and has fewer bugs and whatever.

And there’s similar things with mathematics. So yeah, first of all, there’s so many beautiful things there. It is one of the great minds in mathematics ever and computer science.

Just even considering the space of proofs. Yeah. 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. 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. Then I believe there’s even competitions on this. Yeah. And it’s also a nice way to stress test not just the programs or in this case the proofs but also the different languages. Maybe that’s the different notation or whatever to use to accomplish a different task.

Yeah, you learn a lot. I mean 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 the simplicity. So if you look at E=MC², when a few concepts come together, that’s why the Euler identity is often considered the most beautiful equation in mathematics. Do you find beauty in that one and the Euler identity?

Yeah. Well, as I said, what I find most appealing is the connections between different things. If ei = -1, yeah, people use all the fundamental constants. Okay, that’s cute, but to me, the exponential function was introduced by Euler to measure exponential growth. You know, compound interest or decay—anything which is continuously growing, continuously decreasing, growth and decay or dilation or contraction is modeled by the exponential function.

Whereas pi comes around from circles and rotation. If you want to rotate a needle, for example, 180°, you need to rotate by pi radians. Complex numbers represent the swing between imagine axes of a 90° rotation, so 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, now instead of motion in the same direction as your current position, the motion is at right angles to composition.

So rotation and then e^(iπ) = -1 tells you that if you rotate for time pi, 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 in mathematics.

Yeah. Yeah. Dynamic structure and complex. The complex numbers, they all considered almost 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 it just a frivolous side effect, or do you think there is legitimate value in when the notation, all our old friends, come together?

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 your model is again 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; I mean, but it turns out okay.

Actually, with physics, E=mc² is one of the big things. When Aristotle first came up with his laws of motion, then Galileo or Newton and so forth, they saw the things they could measure. They could measure mass, acceleration, and force, and so forth. Newtonian mechanics, for example, F=ma, was the famous Newton’s second law of motion. Those were the primary objects, so they gave them the central building in the theory.

It was only later, after people started analyzing these equations, that there always seemed to be these quantities that were conserved, such as momentum and energy. It’s not obvious that energy happens; it’s not something you can directly measure 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 describe completely the dynamics, like what happens to all the states. It really was a central actor, which was not obvious initially.

This change of perspective really helped when quantum mechanics came along. Because the early physicists who studied quantum mechanics had a lot of trouble trying to adapt their Newtonian thinking, because everything was a particle and so forth to quantum mechanics. I think it was because it was a wave; it just looked really weird. You ask what is the quantum version of F equals ma? It’s 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 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 Schrodinger’s equation that tells you exactly how quantum systems evolve once you have a Hamiltonian. Side by side, they look completely different, like so one involves particles and one involves waves and so forth. But with this centrality, you could 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 was here. That corresponds to conservation of momentum. If I turn around by some angle again, I experience the same laws of physics. This corresponds to the conservation of angular momentum. If I wait for 10 minutes, I still have the same laws of physics. Translation variance. This corresponds to the low conservation of energy.

So, there’s this fundamental connection between symmetry and conservation. And that’s also true in quantum mechanics. Even though the equations are completely different, 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 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. There has to be, you know, we kind of know that at very tiny scales, there’s going to be quite fluctuations of space-time foam. Trying to use Cartesian coordinates is just a non-starter, but we don’t know how to replace it with.

We don’t actually have the mathematical concepts, the analog Hamiltonian that sort of organized everything. Does your gut say that there is a theory of everything? So this is even possible to unify to find this language that unifies general relativity and quantum mechanics. I believe so. I mean, the history of physics has been about 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 the earth and so forth. So, it should happen. It’s just that, again, to go back to this model of the observations and theory, part of our problem is that physics is a victim of its own success. Our two big theories of physics, general relativity and quantum mechanics, are so good now that together they cover 99.9% of all the observations we can make.

You have to either go to extremely insane particle accelerations or the early universe or things that are really hard to measure in order to get any deviation from either of these two 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 that develops the theory of everything? What often happens is that when the physicists need some of mathematics, there’s often some precursor that the mathematicians worked out earlier. So when Einstein started realizing that space was curved, he went to some mathematician and asked if there was some theory of curved space that the mathematicians already came up with that could be useful.

He said, oh yeah, I think Riemann came up with something. Riemann had developed Riemannian geometry, which is precisely a theory of spaces that occurred in various general ways which turned out to be almost exactly what was needed for Einstein’s theory. This goes back to Wigner’s unreasonable effectiveness of mathematics. 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 sort of both ways of organizing data in useful ways. It just feels like you might need to go some weird land that’s very hard to intuit, like 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 cognition to intuit what that reality really is like.

This is why analogies are so important. I mean, the round earth is not intuitive because we’re stuck on it, but round objects in general, we have pretty good intuition over, and 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 round earth and round moon models. You can just take a basketball, a golf ball, and a light source and actually do these things yourself.

The intuition is there, but yeah, you have to transfer it. That is a big leap intellectually for us to go from flat to round earth because our life is mostly lived in flat land. To load that information, we all take it for granted. We take so many things for granted because science has established a lot of evidence for this kind of thing. But you know, we’re on a round rock, flying through space.

It’s a big leap, and you have to take a chain of those leaps the more we progress. Right? Yeah. So modern science is maybe, again, a victim of its own success, in that 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 suspicious because of that.

We need more grounding. I think there are scientists who do excellent outreach, but there’s lots of science things that you can do at home. There’s lots of YouTube videos. I did a YouTube video recently with Grant Sanderson. We talked about how the ancient Greeks were able to measure things like the distance to the moon and the distance to the earth, using techniques that you could also replicate yourself.

It doesn’t all have to be like fancy space telescopes and very intimidating mathematics. 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, shrouded in mystery. You know, 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 environment. world. Yeah. Yeah. And try to make some kind of general statements about distance to places. Change your perspective; it 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 from some other time period. Make hypotheses, spherical cows, speculate. This is what mathematicians do and somewhat what 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 right when you don’t have much to figure out. Stuff like if you propose axioms, then mathematics lets you follow those 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 from a sort of mathematical standpoint? What aspects of general relativity are intriguing to you, challenging to you? I have worked on some equations. There’s something called the wave maps equation or the sigma field model, which is not quite the equation of space-time gravity itself but of certain fields that might exist on top of space-time. 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, there’s control fields, and there’s this whole hierarchy of different equations of which Einstein is considered one of the most nonlinear and difficult.

But relatively low in the hierarchy was this thing called the wave maps equation. It’s a wave which at any given point is fixed to be like on a sphere. I can think of a bunch of arrows in space and time, and the arrows point 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. And 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?

So the equation I considered was actually what’s called a critical equation where the behavior at all scales is roughly the same. I was able barely 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 did disperse a little bit, it would stay regular. This was back in 2000. That was part of why I got interested in narrows afterwards, actually.

So I developed some techniques to solve that problem. Part of it was that this problem is really nonlinear because of the curvature of the sphere. There was a certain nonlinear effect, which was a non-perturbative effect. When you sort of 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. So the equation is kind of like an evolution of heaves of wheat, and they’re all bending back and forth, and so there’s a lot of motion. 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, and I was trying to understand the dynamics of all these fields, and I couldn’t do it with pen and paper. I had not enough facility with computers to do any computer simulations. So I ended up closing my eyes, being on the floor, and just imagining myself to actually be 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 she was asking what I was doing. It’s complicated, is the answer. And yeah, my aunt said, “Okay, fine. You’re a young man. I don’t ask questions, but I have to ask about how you approach solving difficult problems.”

She asked, “What if it’s possible to go inside your mind when you’re thinking? Are you visualizing in your mind the mathematical objects, symbols maybe? 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 sort of cheating strategically. The beauty of mathematics is that you get to change the rules, change the problem, change the rules as you wish.

You don’t get to do this for any other field. If you’re an engineer and someone says build a bridge over this, you can’t say you want to build this bridge over here instead, or you want to build it out of paper instead of steel. But a mathematician? You can do whatever you want.

It’s like trying to solve a computer game where there are unlimited cheat codes available. You can set this. So, there’s a dimension that’s too large? I’ll set it to one. I’ll solve the one-dimensional problem first. There’s a main term and an error term. I’m going to make a spherical car assumption. I’ll assume the error term is zero.

The way you should solve these problems is not in sort of this iron man mode where you make things maximally difficult. Actually, the way you should approach any reasonable math problem is that if there are ten things making your life difficult, find a version of the problem that turns off nine of the difficulties but only keeps one of them.

So, then you install nine cheats. Okay. You install ten cheats, and then the game is trivial. You solve nine cheats; you solve one problem that teaches you how to deal with that particular difficulty, and then you turn that one off. You turn someone else on, and then you solve that one.

After you know how to solve the ten problems 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 from a culture. One thing is that every time there was a fight scene, maybe the hero would get swarmed by a hundred bad guy goons or whatever. But it would always be choreographed so that he’d… Always be only fighting one person at a time, and then he would defeat that person and move on. Because of that, he could defeat all of them, right? But 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 pen and paper? Are you working with computer and latte? I’m mostly pen and paper actually. So in my office, I have four giant blackboards. Sometimes I just have to write everything I know about the problem on the four blackboards and then sit on my couch and just sort of 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. I mean, the beauty of a blackboard is 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 that, you know, if I wanted to plot a function before, which is moderately complicated as some iteration or something, I’d have to remember how to set up a Python program and how does a for loop work and debug it. It would take two hours and so forth. Now I can do it in 10 to 15 minutes. Yeah, 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 and 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 like Python and C and so forth, except that in most languages, the focus is on producing executable code. Lines of code do things; they flip bits or make a robot move or deliver text on the internet or something. Lean is a language that can also do that. It can be run as a standard traditional language, but it can also produce certificates. For example, a software like Python might do a computation and give you that the answer is seven. That is, 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 seven as 3 + 4 and 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 for which you can create quite complicated, intricate mathematical proofs. They produce these certificates that give a 100% guarantee that your arguments are correct, if you trust the compiler. They made the compiler really small, and you can find several different compilers available for the same purpose.

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 mathematical argument, like you might want to. Introduce a variable. You want to prove a contradiction. There are various standard things that you can do and it’s written so ideally it should be like a one correspondence. In practice, it isn’t because Lean is like explaining a proof to an extremely pedantic colleague who will point out, “Okay, did you really mean this? What happens if this is zero?”

Did you how do you justify this? So 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 or a natural number or a function or something? If you write things informally, it’s up in terms of context. You say, “Clearly, x is equal to let x be 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 it is?”

You see, you have to think more at a philosophical level. Well, not just sort of the computations you’re doing, but sort of what each object actually is in some sense. Is Lean using something like LLMs to do the type inference, or like 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 sort of comes with a history of where it came from, and you can kind of trace. Oh, I see. So it’s designed for reliability. Modern AIs are not used in 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, “Okay, now I want to use the fundamental thing of calculus,” say, 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, and somewhere in there is the fundamental theme of calculus, but you need to find it. A lot of the bottleneck now is actually lemma search. You know there’s a tool that is in there somewhere, and you need to find it. There are various search engines specialized for math that you can do, but there’s now these large language models that you can say, “I need the fundamental calculus at this point,” and it says, “Okay, for example, when I code, I have GitHub Copilot installed as a plugin to my IDE, and it scans my text and sees what I need.”

It says, “I might even type here, okay, now I need to use the final thing with calculus.” Then it might suggest, “Okay, try this,” and like maybe 25% of the time it works exactly. Then another 10-15% of the time it doesn’t quite work, but it’s close enough that I can say, “Oh, if I just change it here and here it will work.”

Half the time it gives me complete rubbish, but people are beginning to use AI a little bit on top, mostly on the level of basically fancy autocomplete. You can type half of one line of a proof, and it will find it. It will tell you, “Yeah, but a fancy especially fancy with the sort of capital letter F removes some of the friction.” a mathematician might feel when they move from pen and paper to formalizing. Yes.

Yeah. So, right now I estimate that the effort time and effort taken to formalize a proof is about 10 times the amount taken to write it out. Yeah. So, it’s doable, but it’s annoying. But doesn’t it like kill the whole vibe of being a mathematician?

Yeah. So, I mean having a pedantic coworker, right? Yeah. If that was the only aspect of it. Okay. But there are some cases where it’s actually more pleasant to do things formally. So there was a theorem I formalized, and there was a certain constant 12 that came out in the final statement, and so this 12 had to be carried all through the proof and everything had to be checked that all these other numbers had to be consistent with this final number 12.

So we wrote a paper through this theorem with this number 12, and then a few weeks later someone said, “Oh, we can actually improve this 12 to an 11 by reworking some of these steps.” When this happens with pen and paper, like every time you change a parameter, you have to check line by line that every single line of your proof still works, and 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.” And what you can do with Lean is that you just in your headline theorem change a 12 to 11. You run the compiler and, like, 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, and you can skip over everything which works just fine.

If you program things correctly, with sort of good programming practices, most of your lines will not be red. There’ll just be a few places where you, I mean, if you don’t hard code your constants, but you use smart tactics and so forth. Yeah, you can localize the things you need to change to a very small period of time. So, like within a day or two, we had updated our proof. This is a very quick process.

You make a change, there are 10 things now that don’t work. For each one, you make a change, and now there are five more things that don’t work, but the process converges much more smoothly than with pen and paper.

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

Yeah, so the proofs are longer, but each individual piece is easier to read. So, if you take a math paper and you jump to page 27 and look at paragraph 6 and you have a line of math, I often can’t read it immediately because it assumes various definitions which I have to go back and maybe 10 pages earlier this was defined.

The proof is scattered all over the place, and you basically are forced to read fairly sequentially. It’s not like, say, a novel where, in theory, you can jump around. you could open up a novel halfway through and start reading. There’s a lot of context. But when a proven lean, if you put your cursor on a line of code, every single object there, you can hover over it and it would say what it is, where it came from, where stuff is justified. You can trace things back much easier than sort of flipping through a math paper.

So, one thing that lean really enables is actually collaborating on proofs at a really atomic scale that you really couldn’t do in the past. So traditionally, with pen and paper, when you want to collaborate with another mathematician, either you do it as a blackboard where you can really interact, but if you’re doing it sort of by email or something, basically, yeah, you have to segment it, say I’m going to finish section three, you do section four, but you can’t really sort of 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 like the three 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,” and you can do extremely atomic level conversations.

So, because of lean, I can collaborate with dozens of people across the world, most of whom I’ve never met in person, and I may not know actually even whether they’re reliable in the process, but lean gives me a certificate of trust. So I can do trustless mathematics.

There are so many interesting questions. So one, you’re known for being a great collaborator. So 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 you’re brainstorming?

There’s always a brainstorming process first. So math research projects, sort of by their nature, 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 its implementation is the main difficulty. You have to figure out even what is the right path.

So 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 finite workforce. Right now, can you do that and so forth?

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

Then hopefully, that breaks up the problem into smaller subproblems which you don’t know how to do, but then you focus on sub ones, and sometimes different collaborators are better at working on certain things. One of my themes I’m known for is a theorem. of Ben Green, which is called the Green Tower Theorem. It’s a statement that the primes contain arithmetic progressions of any length. So 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 three. He showed that sets like the primes contain lots and lots of progressions of length three. Even certain subsets of the primes do, but his techniques only worked for length three progressions. They didn’t work for longer progressions.

But I had these techniques coming from ergodic theory, which is something that I had been playing with, and I knew better than Ben at the time. If I could justify certain randomness properties of some set relating to primes—there’s a certain technical condition—which if I could have it, if Ben could supply me this fact, I could conclude the theorem. But what I asked was a really difficult question in number theory, which he said there’s no way we can prove. So he said, can you prove your part of the theorem using a weaker hypothesis that I have a chance to prove it? He proposed something which he could prove, but it was too weak for me; I can’t use this.

So there was this conversation going back and forth. Different approaches. Yeah. I want to cheat more; he wants to cheat less. But eventually, we found a property which he could prove and I could use, and then we could prove our view. So, there are all kinds of dynamics. I mean, every collaboration has a story; no two are the same.

On the flip side of that, like you mentioned with Lean programming, that’s almost like a different story because you can create, I think you’ve mentioned, a kind of a blueprint for a problem. Then you can really do a divide and conquer with Lean where you’re working on separate parts, and they’re using the computer system proof checker essentially to make sure that everything is correct along the way. So, 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 basically is a blueprint in a sense. It is taking a difficult statement like Fermat’s Last Theorem and breaking it up into 100 little lemmas, but often not all written with enough detail that each one can be sort of 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. It’s trying to make each step kind of self-contained, depending on only a very specific number of previous statements that have been proven so that each node of this 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. So it’s like a modern supply chain, you know. If you want to create an iPhone or some other complicated object, no one person can build up a single object, but you can have a specialist who, if they’re given some widgets from some other company, can combine them together to form a slightly bigger widget. I think that’s a really exciting possibility because you can have thousands of contributors distributed.

So I told you before about the split between theoretical and experimental mathematics, and right now most mathematics is theoretical, and when you type it, it’s experimental. I think the platform that Lean and other software tools, like GitHub and things, allow will scale up experimental mathematics to a much greater degree than we can do now.

So 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 some 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 because if one of the modules has a bug in it, the whole thing is unreliable.

So you get these bespoke spaghetti code that is written by not professional programmers, but by mathematicians, and they’re clunky and slow. Because of that, it’s hard to really mass-produce experimental results. But yeah, I think with Lean, I mean, so I’m already starting some projects where we are not just experimenting with data but experimenting with proofs.

I have this project called the Equation 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. Okay. So abstract algebra studies operations like multiplication and addition and the 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 that don’t obey others. For example, x * x is not always equal to x. That law is not always true.

So 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 some algebraic proof. So we look at all the pairs between these 4,000 laws and the 22 million pairs. For each pair, we ask, does this law imply this law? If so, give a proof. If not, give a counterexample.

So we have 22 million problems, each one of which you could give to like an undergraduate algebra student, and they had a decent chance of solving the problem, although there are a few of these 22 million. There are like 100 or so that are really quite hard. Okay. But 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. So it would not have been feasible. I mean, the state-of-the-art in the literature was 15 equations and sort of how they apply. That’s sort of at the limit of what a human-represented paper can do. So you need to scale it up. You need to crowdsource, but you also need to trust all the—no one person can check 22 million of these proofs. You needed to be computerized, and so it only became possible with Lean. We were hoping to use a lot of AI as well.

So the project is almost complete. Of these 22 million, all but two had been settled. Wow, and well actually, of those two, we have a pen and paper proof of the two, and we’re formalizing it. In fact, this morning I was working on finishing it. So we’re almost done on this, incredible, yeah, fantastic.

How many people were able to get about 50, which in mathematics is considered a huge number? It’s a huge number. That’s crazy.

So we kind of have a paper with 50 authors and a big appendix of who contributed to what. Here’s an interesting question. Now to maybe speak even more generally about it, when you have this pool of people, is there a way to organize the contributions by level of expertise of the contributors?

Now, okay, I’m asking you a lot of questions here, but I’m imagining a bunch of humans and maybe in the future some AIs. Can there be like an ELO rating type of situation, where there’s a gamification of this? The beauty of these Lean projects is that automatically you get all this data, you know, so everything has to be uploaded for this GitHub, and GitHub tracks who contributed what.

You could generate statistics from any later point in time. You can say, “Oh, this person contributed this many lines of code,” or whatever. I mean, these are very crude metrics. I would definitely not want this to become part of your tenure review or something.

But I mean, I think already in enterprise computing, right, people do use some of these metrics as part of the assessment of performance of an employee. Again, this is a direction which is 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. Number of papers. Yeah.

It’s true. It’s true that, yeah. I mean, it feels like this is a metric, while flawed, is going in the more in the right direction. Right. Yeah. It’s an interesting—at least it’s a very interesting metric.

Yeah. I think it’s interesting to study. I mean, 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 is no longer a useful measure.

Oh, humans always. Yeah. I know. It’s rational. So what we’ve done for this project is self-report. So there are actually standard categories from the sciences of what types of contributions people give. So there’s concept and… validation and resources and and and and coding and so forth. There’s a standard list of troll categories, and we just ask each contributor to there’s a big matrix of all the authors in all the categories just to tick the boxes where they think that they contributed.

They just give a rough idea, oh so you did some coding and you provided some compute but you didn’t do any of the pen and paper verification or whatever. I think that that works out. Traditionally, mathematicians just order alphabetically by surname, so we don’t have this tradition as in the sciences of lead author, second author, and so forth, which we’re proud of. We make all the authors equal status, but it doesn’t quite scale to this size.

A decade ago, I was involved in these things called polymath projects. It was crowd-sourcing mathematics but without the lean component, so it was limited by needing a human moderator to actually check that all the contributions coming in were valid, and this was a huge bottleneck. Still, we had projects that were you know, 10 authors or so. But we had decided at the time not to try to decide who did what but to have a single pseudonym.

So we created this fictional character called DHJ Polymath in the spirit of Bwaki. Baki 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 to be not so great for a couple of reasons.

One is that if you actually 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 refer to the most famous person who was involved in the project.

Oh, so this was Tim Gow’s P project. This was Ter’s project and not mention the other 19 or whatever people that were involved. So we’re trying something different this time around where everyone’s an author, but we will 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’re embracing one of the greatest mathematicians in the world 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, and maybe what is the gap between this system that is able to prove the high school level problems versus graduate level problems? Yeah, the difficulty increases exponentially with the number of steps involved in the proof. It’s a combinatorial explosion, right?

The thing with large language models is… that they make mistakes. And so if a proof has got 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 actually 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? Oh yeah, it’s extremely hard actually. Natural language is very fault tolerant. You can make a few minor grammatical errors and a speaker in the second language can get some idea of what you’re saying.

But formal language, yeah, if you get one little thing wrong, like the whole thing is nonsense. Even formal to formal is very hard. There are different incompatible proofist languages. There’s Lean, but also Coq and Isabelle, and so forth. Actually, even converting from a formal language to formal language is basically an unsolved problem. That is fascinating.

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

Yeah, we talked earlier about things that are amazing, over time become kind of normalized. So yeah, now somehow it’s, oh of course geometry is a solved problem. Right, that’s true. I mean, it’s still beautiful. These are great works, it shows what’s possible.

I mean, the approach doesn’t scale currently. It’s three days of Google’s server time to solve one high school math problem. This is not a scalable prospect, especially with the exponential increase in complexity.

We should mention that they got a silver medal performance, the equivalent of, I mean, yeah, equivalent of a silver. So first of all, they took way more time than was allotted, and they had this assistance where the humans helped by formalizing. But also they’re giving us those full marks for the solution, which I guess is formally verified. So I guess that’s fair.

Yeah, 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 will have to be written in natural language rather than formal language.

Oh, I hope that happens. I hope that this IMO happens. I hope the next one it won’t happen. This IMO, the performance is not good enough in the time period. But there are smaller competitions. There are competitions where the answer is a number rather than a long form proof, and AI are actually a lot better at problems where there’s a specific numerical answer because it’s easy to reinforce learning on it.

Yeah, 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 to tell. If you’re trying to do… billions of reinforcement learning runs. You’re not you can’t hire enough humans to grade those. It’s already hard enough for the last language to do reinforcement learning on just the regular text that people get. But now if you actually hire people not just give thumbs up, thumbs down, but actually check the output mathematically. Yeah, that’s too expensive.

So 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 for a while. So inventing new theories. So coming up with new conjectures versus proving the conjectures, right? Building new abstractions, new representations, maybe an AI turn style with 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.

So a thousand years ago mathematicians had to compute the date of Easter and there was 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 how to get from the old world to the new or very complicated calculations. Again, we’ve been automated. Even a lot of undergraduate mathematics, even before AI, like Wolfram Alpha for example. It’s not a language model, but it can solve a lot of undergraduate level math tasks.

So on the computational side, verifying routine things like having a problem and say here’s a problem in partial equations. Could you solve it using any of the 20 standard techniques? And they say yes, I’ve tried all 20 and here are the 100 different permutations and here’s my results. And that type of thing I think will work very well, type of scaling to once you solve one problem to make the AI tackle 100 adjacent problems.

The things that humans do still. So where the AI really struggles right now is knowing when it’s made a wrong turn. That it can say, “Oh, I’m going to solve this problem. I’m going to split up this problem into these two cases. I’m going to try this technique.” And sometimes if you’re lucky and it’s a simple problem, it’s the right technique and you solve the problem and sometimes it will have a problem it would propose an approach which is just complete nonsense.

And it looks like a proof. So this is one annoying thing about LM generated mathematics. So yeah, we’ve had human generated mathematics as 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. But the AI generated proofs, they can look superficially flawless and that’s partly because that’s what the reinforcement learning has trained them to do, right? To produce text that looks like what is correct, which for many applications is good enough.

So 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 and yeah, when a human… makes when low-quality code, there’s something called code smell, right? You can tell immediately, like, okay, there’s signs. But with generated code, eventually you find an obvious dumb thing that just looks like good code. Yeah. So, it’s very tricky and frustrating for some reason to work. Yeah. So the sense of smell. Okay, 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 that eventually. I mean, so the way Alpha Zero and so forth make progress on Go and chess is, in some sense, they have developed a sense of smell for Go and chess positions. You know that this position is good for white, is good for black. They can’t initiate why. But just having that sense of smell lets them strategize.

So if AIs gain that ability to sort of have a sense of viability of certain proof strategies, you can say I’m going to try to break up this problem into two small subtasks. They can say, well, this looks good; two tasks look like they’re simpler than your main task, and they still got a good chance of being true. So this is good to try or no, you’ve made the problem worse because each of the two subproblems is actually harder than your original problem, which is actually what normally happens if you try a random thing to try. Normally, it’s very easy to transform a problem into an even harder problem. Very rarely do you transform a simpler problem.

So if they can pick up a sense of smell, then they could maybe start competing with human-level mathematicians. So, this is a hard question, but not competing, but collaborating.

If 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 that oracle — what would you like that oracle to be able to do? Would you like it to maybe be a verifier? Like check. Do the codes like you’re a professor; this is the correct, this is a good, this is a 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?

Yeah, I think all of the above. A lot of it is 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.

It’s an interesting bit — unreliable in subtle ways while providing sufficiently good output. It’s an interesting combination. You know, I mean, you have graduate students that you work with who kind of like this but not at scale, you know? And we have previous software tools that can work at scale but very narrow. So we have to figure out how to use — I mean, so Tim C actually imagined, he foresaw like in 2000 he was envisioning what mathematics would look like in actually two and a half decades.

That’s funny. He wrote in his article like a hypothetical conversation between a… mathematical assistant of the future and himself trying to solve a problem. They would have a conversation that sometimes the human would propose an idea and the AI would evaluate it. Sometimes the AI would propose an idea, and sometimes that computation was required. The AI would just go and say, “Okay, I’ve checked the 100 cases needed here,” or “The first 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 equals 46.”

Just a free form conversation where you don’t know in advance where things are going to go. But just based on ideas proposed on both sides, calculations get proposed on both sides. I’ve had conversations with AI where I say, “Okay, we’re going to collaborate to solve this math problem,” and it’s a problem that I already know the solution to. So I try to prompt it, “Okay, so here’s the problem. I suggest using this tool,” and then it finds this lovely argument using a totally different tool, which eventually goes into the weeds and says, “No, no, no. If I use this, and it might start using this, and then it’ll go back to the tool that I wanted to before.”

You have to keep railroading it onto the path you want. I could eventually force it to give the proof I wanted, but it was like herding cats. The amount of personal effort I had to take, not just sort of prompting it, but also checking its output because a lot of what it looked like was going to work, I know there’s a problem online with 17 and basically arguing with 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 that happens to 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 also just better tooling. The Lean developers are doing a great job adding more and more features and making it user-friendly.

It’s going up from 9 to 8 to 7. Okay, no big deal. But one day it will drop below one. 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 who is generally on the fly with you. It becomes natural for journals to accept. 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 it connects to literature, not worry so much about the correctness because that’s been certified.

Papers are getting longer and longer in mathematics, and actually, 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 the formalization is coming in at just the right time for this to be. The easier it is to guess because of the tooling and all the other factors, then you’re going to see much more like Mathlib will grow potentially exponentially. It’s a virtuous cycle. One facet of this type that happened in the past was the adoption of LaTeX. LaTeX is this typesetting language that… All means use now. So in the past, people used all kinds of word processors and typewriters and whatever, but at some point, Latte became easier to use than all other competitors and that people just switched within a few years. It was just a dramatic shift.

It’s a wild out there question, but what year, how far away are we from an AI system being a collaborator on a proof that wins the Fields medal?

So that level. Okay. Well, it depends on the level of collaboration. I mean, it deserves to get the Fields Medal. Half and half already, I can imagine if it was a winning paper having some AI systems in writing it. The order complete alone is already something I use. It speeds up my writing. You can have a theorem, you have a proof, and the proof has three 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,” and it was exactly correct. That was great; saved me like 5 to 10 minutes of typing. But in that case, the AI system doesn’t get the Fields Medal.

Are we talking 20 years, 50 years, 100 years? What do you think? Okay, so I gave a prediction in print. By 2026, which is now next year, there will be math collaborations, you know, where the AI—so not Fields Medal winning—but actual research-level math, like published ideas that are in part generated by AI. Maybe not the ideas, but at least some of the computations and the verifications.

Yeah. I mean, has that already happened? There are problems that were solved by a complicated process conversing with AI to propose things, and the human goes and tries it, and the contract doesn’t work, but it might propose a different idea.

It’s hard to disentangle exactly. There are certainly math results that could only have been accomplished because there was a method, a human mathematician, and an AI involved. But it’s hard to sort of disentangle credit.

These tools do not replicate all the skills needed to do mathematics, but they can replicate some non-trivial percentage of them, you know, 30 to 40%. They can fill in gaps.

Coding is a good example. It’s annoying for me to code in Python. I’m not a native, I’m not a professional programmer. But with AI, the friction cost of doing it is much reduced. It fills in that gap for me.

AI is getting quite good at literature review. I mean, there’s still a problem with hallucinating references that don’t exist. But this, I think, is a solvable problem. If you train in the right way and so forth, you can verify using the internet.

In a few years, you should get to the point where you have a lemma that you need, and we can ask, “Has anyone proven this lemma before?” It will do basically a fancy web search and say, “Yeah, there are these six papers where something similar has happened.”

You can ask it right now, and it’ll give you six papers, of which maybe one is legitimate and relevant, one exists but is not relevant, and four are hallucinated. It has a non-zero success rate right now, but it’s… 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. You just need to be prompted to be reminded of a paper that was already subconsciously in your memory versus helping you discover something new that you were not even aware of but is the correct citation.

Yeah, that can sometimes happen. But when it does, it’s buried in a list of options, with many other options that are not helpful. I mean, 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.

It does break you out of the silos of thought. There’s a big hump to overcome right now. I mean, it’s like self-driving cars, you know. The safety margin has to be really high for it to be feasible. So yeah, there’s a last mile problem with a lot of AI applications. They can develop tools that work 20% or 80% of the time, but it’s still not good enough, and in fact, it can be even worse in some ways.

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, the news of something that happened that AI did, like a real breakthrough. It could feel like a Fields medal winning hypothesis. It could be like this alpha zero moment with Go, that kind of thing.

This decade, I can see it making a conjecture between two unrelated things that people thought were unrelated. Oh, interesting, generating a conjecture that’s a beautiful conjecture. And actually, it has a real chance of being correct and meaningful because that’s actually kind of doable, I suppose. But the worth of the data is a concern.

No, that would be truly amazing. The current models struggle a lot. A version of this is the physicists have a dream of getting AI to discover new laws of physics. You know, the dream is you just feed it all this data. Okay, here’s a new pattern that we didn’t see before. But it struggles; the current state of the art even struggles to discover old laws of physics from the data. If it does, there’s a big concern about contamination, that it did so only because somewhere in its training data it encountered some new law, like Boyle’s Law or whatever.

Part of the issue 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 math is actually the negative space. We have published things that people have been able to prove, conjectures that ended up being verified, or maybe counterexamples produced.

However, we don’t have data on things that were proposed as a good thing to try but were quickly realized to be the wrong conjecture. Those people then said, “Oh, but we should change our claim to modify it in this way to actually make it more plausible.” There’s this trial-and-error process which is a real integral part of human mathematical discovery. don’t record cuz it’s embarrassing. We make mistakes and we only like to publish our wins. The AI has no access to this data to train on. I sometimes joke that basically AI has to go through grad school and actually go to grad 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 Gregori Pearlman. You mentioned that you try to be careful in your work and not let a problem completely consume you. Just you really fall in love with the problem and really cannot rest until you solve it. But you also hastened to add that sometimes this approach actually can be very successful. An example you gave is Gregori Pearlman who proved the point conjecture and did so by working alone for 7 years with basically little contact with the outside world. Can you explain this one millennial prize problem that’s been solved, point conjecture, and maybe speak to the journey that Gregori Pearlman’s been on?

All right. So it’s a question about curb spaces. Earth is a good example. So you can think of a 2D surface in being round, could maybe be a torus with a hole in it, or it can have many holes. There are many different topologies a surface could have. Even if you assume that it’s bounded and smooth and so forth. So 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.

So a sphere has genus 0, a donut has genus one, and so forth. One way you can tell these surfaces apart is by the property called simply connected. If you take any closed loop on the sphere, like a big closed little rope, you can contract it to a point while staying on the surface. The sphere has this property, but a torus doesn’t. If on a torus you take a rope that goes around, say, the outer diameter of the torus, there’s no way it can get through the hole—there’s no way to contract it to a point.

So it turns out that the sphere is the only surface with this property of contractability up to continuous deformations of the sphere. So things that I want to call topologically equivalent to the sphere. So point asked the same question in higher dimensions. This becomes hard to visualize because a surface you can think of as embedded in three dimensions, but a curved free space, we don’t have good intuition of 4D space to live in.

There are also 3D spaces that can’t even fit into four dimensions; you need five, six, or higher. But anyway, mathematically you can still pose this question that if you have a bounded three-dimensional space now which also has this simply connected property that every loop can be contracted, can you turn it into a three-dimensional version of a sphere? And so this is the point conjecture.

Weirdly in higher dimensions, four and five, 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 to a sphere. But three was really hard. So people tried many approaches. There were sort of commentary approaches where you chop up the surface into little triangles or tetrahedra, and you just try to argue based on how the faces interact with each other.

There were algebraic approaches. There’s various… Algebraic objects like things called the fundamental group that you can attach to these homology and coology and all these very fancy tools. They also didn’t quite work. But Richard Hamilton proposed a partial differential equations approach.

So you take the problem. The problem is that you have this object which secretly is a sphere, but it’s given to you in a really weird way. So think of a ball that’s been kind of crumpled up and twisted, and it’s not obvious that it’s a ball. But if you have some sort of surface which is a deformed sphere, you could think of it as a surface of a balloon. You could try to inflate it.

You blow it up, and naturally, as you fill it with air, the wrinkles will sort of smooth out and it will turn into a nice round sphere. Unless, of course, it was a torus or something, in which case it would get stuck at some point. If you instead had a torus, there would be a point in the middle when the inner ring shrinks to zero. You get a singularity and you can’t blow up any further. You can’t flow any further.

So he created this flow which is called Ricci flow, which is a way of taking an arbitrary surface or space and smoothing it out to make it rounder and rounder, to make it look like a sphere. He wanted to show that this process would either give you a sphere or it would create a singularity.

Actually, very much like how PDEs either have global regularity or finite blow-up, basically it’s almost exactly the same thing. It’s all connected. And he showed that for two-dimensional surfaces, if you started simply connected, no singularities ever formed. You never ran into trouble and you could flow, and it would give you a sphere.

So he got a new proof of the two-dimensional result, but 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—slightly simpler—but yeah, they were considered hard nonlinear equations to solve.

There’s lots of special tricks in 2D that helped, but in 3D, the problem was that this equation was actually supercritical. It had the same problems as Navier-Stokes. As you blow up, maybe the curvature could get constrained in finer, 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, like if there’s this thing called neck pinches where the surface sort of behaves like a barbell, pinch at a point. Some singularities are simple enough that you can sort of see what to do next. You just make a snip and then you can turn one surface into two and evolve them separately.

But there was the prospect that there were some really nasty, knotted singularities showing up that you couldn’t see how to resolve in any way that you couldn’t do any surgery to. You need to classify all the singularities—what are all the possible ways that things can go wrong.

What Perelman did was, first of all, he turned the problem into a supercritical… the energy the Hamiltonian clarified Newtonian mechanics. He introduced something which is now called permanence reduced volume and permanence 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 actually suddenly looked a lot less scary than they did before. He still had to analyze the singularities of this critical problem, and that itself was a problem similar to this wake-up thing I worked on.

So he managed to classify all the singularities of this problem and show how to apply surgery to each of these. Through that, he was able to resolve the point Cray conjecture. Quite a lot of really ambitious steps, and nothing that a large language model today could propose. At best, I could imagine a model proposing this idea as one of hundreds of different things to try, but the other 99 would be complete dead ends. 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.

So, you’ve done as you said. Even strictly mathematically, but more broadly in terms of the process he’s done similarly difficult things. What can you infer from the process he was going through? He was doing it alone. What are some low points in a process like that when 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 the thing you did for the last few days, maybe weeks, is a failure?

Well, for me, I switch to a different problem. As I said, I’m a fox. I’m not a hedgehog. But legitimately, that is a break that you can take; it’s to step away and look at a different problem. You can modify the problem too. You can ask some cheat if there’s a specific thing that’s blocking you, that this bad case keeps showing up for which your tool doesn’t work. You can just assume by fiat this bad case doesn’t occur.

You do some magical thinking 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, but 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 to assume like, okay, we’ll figure it out eventually. Sometimes, it’s even productive to make mistakes.

One project, which we won some prizes for, involved a group of four other people. We worked on this PD problem again, this blowup regularity type problem, and it was considered very hard. Sean Bain, who was another field methodist, worked on a special case of this but could not solve the general case. We worked on this problem for two months and thought we solved it. We had this cute argument that if… everything fit and we were excited. We were planning celebrationally to all get together and have champagne or something. We started writing it up. One of us, not me actually, but another co-author said, “Oh, in this lema here, we have to estimate these 13 terms that show up in this expansion.” We estimated 12 of them, but in our notes, I can’t find the estimation of the 13th. Can someone supply that? And I said, “Sure, I’ll look at this.”

Actually, yeah, we didn’t cover; we completely omitted this term and this term turned out to be worse than the other 12 terms put together. In fact, we could not estimate this term, and we tried for a few more months and all different permutations, and there was always this one term that we could not control. This was very frustrating, but because we had already invested months and months of effort into this already, we stuck at this. We tried increasingly desperate and crazy things. After two years, we found an approach which was actually somewhat different by quite a bit from our initial strategy.

It actually didn’t generate these problematic terms and actually solved the problem. So we solved a problem after two years, but if we hadn’t had that initial false dawn of nearly solving a problem, we would have given up by month two or something and worked on an easier problem. If we had known it would take two years, I’m not sure we would have started the project. Sometimes, having the incorrect measurement is like Columbus’s incorrect version 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.

I mean, it could be that he actually secretly knew, but just for the psychological element. Do you have emotional or self-doubt that just overwhelms you in moments like that? Because 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 similar to how chess has broken some people.

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 go on to the next one. The fact that you can always move on to another problem reduces the emotional connection. There are cases where there are certain problems that I call “back diseases,” where people latch on to that one problem and spend years thinking about nothing but that one problem.

Maybe their career suffers and so forth, but they think, “Okay, this big win 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. 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 by five years I will supply a solution. 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’s problems like this. You have made so much progress towards the hardest problems in the history of mathematics. So is there a problem that just haunts you? It sits there in the dark corners, twin prime conjecture, Riemann hypothesis, global conjecture. Twin prime, that sounds again. So, I mean, the problem is like a Riemann hypothesis; those are so far out of reach. Why do you think so?

Yeah, there’s not even a viable strategy. Even if I activate all the cheats that I know of in this problem, there’s just still no way to get me to be… I think it needs a breakthrough in another area of mathematics to happen 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.

Okay. So they’re often referred to as the atoms of mathematics. Can you just speak to the structure that these atoms, the natural numbers, have two basic operations attached to them? Addition and multiplication. So if you want to generate the natural numbers, you can do one of two things. You can just start with one and add one to itself over and over again, and that generates you the natural numbers.

So additively, they’re very easy to generate: 1, 2, 3, 4, 5. Or you can take the prime; if you want to generate multiplicatively, you can take all the prime numbers, 2, 3, 5, and multiply them all together, and together that gives you all the natural numbers except maybe for one. So there are these two separate ways of thinking about the natural numbers: from an additive point of view and from a multiplicative point of view.

And separately, they’re not so bad. Any question about addition is relatively easy to solve, and any question that involves multiplication is easy to solve. But what has been frustrating is that you combine the two together, and suddenly you get this extremely rich… I mean we know that there are statements in number theory that are actually undecidable. There are certain polynomials in some number of variables. You know, is there a solution in the natural numbers? The answer depends on an undecidable statement, like whether the aims of mathematics are consistent or not.

But yeah, even the simplest problems that combine something multiplicative, such as the primes, with something additive, such as shifting by two, we understand both of them well, but if you ask, when you shift the prime by two, can you get another prime? It’s been amazingly hard to relate the two. We should say that the twin prime conjecture posits that there are infinitely many pairs of prime numbers that differ by two.

Now, the interesting thing is that you have been very successful at pushing forward the field in answering these complicated questions of this variety. Like 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, right? Yeah. So, what we’ve realized… because of this type of research is that there’s different patterns have different levels of indestructibility. So what makes the twin prime problem hard is that if you take all the primes in the world, 3, 5, 7, 11, so forth, there are some twins in there. 11 and 13 is a twin prime pair of twin primes and so forth.

But you could easily, if you wanted to redact the primes to get rid of these twins, like the twins they show up and there are infinitely many of them, but they’re actually reasonably sparse. Initially, there’s quite a few, but once you got to the millions the trillions they become rarer and rarer, and you could actually, if someone was given access to the database of primes, just edit out a few primes here and there. They could make the twin prime conjecture false by just removing like 0.1% of the primes or something well chosen to do this.

And so you could present a censored database of the primes which passes all of the statistical tests of the primes. You know that it obeys things like the parallel theorem and other tests about the primes but doesn’t contain any true 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 edited primes must fail when applied to these slightly edited primes.

And so it must be some very subtle delicate feature of the primes that you can’t just get from aggregate statistical analysis. Okay. So that’s all. On the other hand, progressions have turned out to be much more robust. You can take the primes, and you can eliminate 99% of the primes actually, and you can take any 99% you want, and it turns out another thing we proved is that you still get arithmetic progressions.

Arithmetic progressions are much, you know, they’re like cockroaches of arbitrary length. Yes, that’s crazy. I mean, for people who don’t know, arithmetic progressions are a sequence of numbers that differ by some fixed amount. Yeah, but it’s again like it’s infinite monkey type phenomenon for any fixed length of your set. You don’t get arbitrary progressions. You only get quite short progressions.

But you’re saying twin prime is not an infinite monkey phenomenon. I mean, it’s a very subtle monkey. It’s still an infinite monkey phenomenon. Yeah. If the primes were really genuinely random, if the primes were generated by monkeys, then yes. In fact, the infinite monkey theorem would apply, but you’re saying that twin primes, you can’t use the same tools. It doesn’t appear random almost.

Well, we don’t know. Yeah, we believe the primes behave like a random set. And so the reason why we care about the twin prime conjecture is a test case for whether we can genuinely confidently say with 0% chance of error that the primes behave like a random set. Okay. Random. Yeah. 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.

So the primes we believe that they’re random. The reason why arithmetic progressions are indestructible is that regardless of whether it looks random or looks structured, like periodic in both cases, arithmetic regressions appear but for different reasons. And this is basically all the ways in which the primes and progressions interact with respect to randomness. there are many proofs of these sort of arithmetic region epithems and 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 two 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. And we can’t rule out that one conspiracy and yet you were able to make, as I understand, progress on the Kupal version. Right. Yeah.

So the one funny thing about conspiracies is that any one conspiracy theory is really hard to disprove. If you believe the water is run by lizards, you say here’s some evidence that it’s not run by lizards. Well, that evidence was planted by the lizards. You may have encountered this kind of phenomenon. So there’s almost no way to definitively rule out a conspiracy and the same is true in mathematics. A conspiracy solely devoted to learning twin primes means you would also have to infiltrate other areas of mathematics to sort of. But it could be made consistent at least as far as we know.

There’s a weird phenomenon that you can make one conspiracy rule out other conspiracies. So, you know, if the world is run by lizardists, it can’t also be run by others. Right. So one unreasonable thing is hard to dispute, but more than one can have tools. So, for example, we know there’s infinitely many primes that are no two, which means there are infinitely many pairs of primes which differ by at most 246. Actually, that is the current bound.

So like there’s twin primes, this thing called cousin primes that differ by four. There’s also sexy primes that differ by six. What are sexy primes? Primes that differ by six. The name is much less exciting than the concept suggests. You can make a conspiracy rule out one 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? Okay, so there’s an infinite number of primes, so it’s ultimately based on what’s called the pigeonhole principle. The pigeonhole principle states that if you have a number of pigeons and they all have to go into pigeonholes, and you have more pigeons than pigeonholes, then one of the pigeonholes has to have at least two pigeons in it. So there has to be two pigeons that are close together.

For instance, if you have 100 numbers and they all range from one to a thousand, two of them have to be at most 10 apart because you can divide up the numbers from one to 100 into 100 pigeonholes. Let’s say if you have 101 numbers, then two of them have to be a distance less than 10 apart because two of them have to belong to the same pigeonhole. So 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 the… Numbers are like that. There are numbers that are kind of almost prime, but they’re not. They don’t have no factors at all other than themselves and one, but they 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. Almost primes are something we can understand. You can actually restrict attention to a suitable set of almost primes, whereas the primes are very sparse overall relative to the almost primes, which actually are much less sparse. You can set up a set of almost primes where the primes have density, say, 1%. That gives you a shot at proving, by applying some original principle, that those pairs of primes are just only 100 apart.

But in order to work with the twin prime conjecture, you need to get the density of primes inside the almost size up to a first of 50%. Once you get up to 50%, you will get twin primes. But 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 because it would open up not only the trip conjecture, the Goldbach conjecture, and many other problems in number theory that are currently blocked because our current techniques would require improvement going beyond this theoretical parity barrier. It’s like pulling past the speed of light.

The twin prime conjecture is one of the biggest problems in the history of mathematics, and the Goldbach conjecture also feels like next-door neighbors. Has there been days when you felt you saw the path? Oh yeah. Sometimes you try something and it works super well. You again get the sense of déjà vu we talked about earlier; you learn from experience when things are going too well because there are certain difficulties that you have to encounter.

A colleague might put it this way: if you are on the streets in New York and you put on a blindfold and get in a car, after some hours, the blindfold’s off and you’re in Beijing. You know that was too easy; somehow, there was no ocean being crossed. Even if you don’t know exactly what was done, you’re suspecting that something wasn’t right.

But is that still in the back of your head? Do you return to the prime numbers every once in a while to see? Yeah, when I have nothing better to do, which is less and less often as I get busy with so many things these days. But yeah, when I have free time and I’m frustrated to work on my sort of real research projects and I also don’t want to do my administrative stuff or 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, one of these problems I actually solve, or sometimes, as you say, you think you 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’s your gut say about when these problems would be solved when prime and go back? Prime I think we’ll keep getting more partial results. It does need at least one this parody barrier, which is the biggest remaining obstacle. There are simpler versions of the conjecture where we are getting really close. So I think in 10 years we will have many much closer results. May not have the whole thing. Yeah, so the Riemann hypothesis is somewhat close. I have no, 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 states are sort of viewed multiplicatively, like for questions only involving multiplication, no addition. The primes really do behave as randomly as you could hope. There’s a phenomenon in probability called square root cancellation that, if you want to poll say America on some issue, and you ask one or two voters, you may have sampled a bad sample and then you 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 actually improves like the square root of the number of people you sample.

So, yeah, if you sample a thousand people, you can get like a 2-3% margin of error. So 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’s data function, and it fluctuates up and down. But in some sense, as you keep averaging more and more, if you sample more and more, the fluctuation should go down as if they were random. There’s a very precise way to quantify that, and the Riemann hypothesis is a very elegant way that captures this.

But, as with many others in mathematics, we have very few tools to show that something really genuinely behaves like truly random. And this is actually not just a little bit random, but it’s asking that it behaves as randomly as it actually is. This square root cancellation, and we know actually because of things related to the parity problem that most of us usual techniques cannot hope to settle this question. The proof has to come out of left field.

Yeah, but what that is, no one has any serious proposal. There are various ways to sort of, as I said, you can modify the primes a little bit and you can 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’s all these pitfalls that you have to dodge very adeptly. The prime numbers are just fascinating.

Yeah, what to you is most mysterious about the prime numbers? So that’s a good question. Conjecturally we have a good model of them. I mean, as I said, they have certain patterns like the primes are usually odd, for instance. But apart from these obvious patterns, they behave very randomly. Assuming that they behave… there’s something called the random model of the primes that after a certain point, primes just behave like a random set.

And there are various slight modifications to this model, but this has been a very good model. It matches the numeric. It tells us what to predict. Like I can tell you with complete certainty the truth is… True. The random model gives overwhelming odds 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 be kind of random because there’s no reason for them to have any kind of secret pattern. But what is mysterious is what is the mechanism that really forces the randomness to happen. This is just absent.

Another incredibly surprisingly difficult problem is the Collatz conjecture. Oh yes. 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. This is in 2010, out of reach of present-day mathematics, and yet you have made some progress. Why is it so difficult to make? 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 yeah, so you take any natural number like say 13. You apply the following procedure to it. If it’s even, you divide it by two, and if it’s odd, you multiply by three and add one. So, even numbers get smaller, odd numbers get bigger. So, 13 will become 40 because 13 * 3 is 39. Add one, you get 40. So, it’s a simple process for odd numbers and even numbers. They’re both very easy operations.

Then you put it together. 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. 40 is now even, divide by 2 is 20. 20 is still even, divide by 2 is 10. 10 is even, divide by 2 is 5, and then 5 * 3 + 1 is 16. And then 8, 4, 2, 1. From 1, it goes 1, 4, 2, 1. It cycles forever.

This sequence I just described, 13, 40, 20, 10, these are also called hailstone sequences because there’s an oversimplified model of hailstone formation, which is not actually correct. However, it’s somehow taught to high school students as a first approximation. A little nugget of ice gets an ice crystal forming 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 kind of partially melted ice, which eventually hails and eventually it falls to the earth. So, the conjecture is that no matter how high you start, like you take a number which is in the millions or billions, you go through this process that goes up if you’re odd and down if you’re even. Eventually, it goes down to earth all the time. No matter where you start with this very simple algorithm, you end up at one. You might climb for a while, right?

If you plot it, these sequences 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. That’s actually a random walk with a downward drift. It’s like if you’re always gambling on roulette at the casino with odds slightly weighted against you. Sometimes you win, sometimes you lose, but over in the long run you lose a bit more than you win. And so normally your wallet will go to zero if you just keep playing over and over again. Statistically, it makes sense. Yes. So the result that I proved, roughly speaking, is such that statistically, like 99% of all inputs would drift down to maybe not all the way to one, but to be much smaller than what you started.

So, 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 the result that I proved. So why is that result? Can you continue down that thread to prove the full conjecture? Well, the problem is that I used arguments from probability theory, and there’s always this exceptional event.

So in probability, we have this law of large numbers which tells you things like if you play a game at a casino with a losing expectation over time, you are guaranteed, or almost surely with probability as close to 100% as you wish, you’re guaranteed to lose money. But there’s always this exceptional outlier. It is mathematically possible that even when the game’s odds are not in your favor, you could just keep winning slightly more often than you lose.

Very much like how in Navier-Stokes, there could be, most of the time, your waves can disperse. There could be just one outlier choice of initial conditions that would lead you to blow up. And there could be one outlier choice of a special number that they stick in that shoots off to infinity while all other numbers crash to one.

In fact, there’s some mathematicians, like Alex Kovvich for instance, who’ve proposed that actually these collatz iterations are like the similar automata. If you look at what they happen on in binary, they do actually look a little bit like these Game of Life type patterns. And 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 which is actually encoding this machine whose job it is to create a version of itself which is a larger, heavier-than-air machine encoded in a number that flies forever.

Yeah. So Conway in fact worked on this problem as well. Oh, wow. Conway, so similar, in fact, that was one of the inspirations for the Navier-Stokes project that Conway studied—generalizations of the collapse problem where instead of multiplying by three and adding one or dividing by two, you have a more complicated branch.

But instead of having two 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 and you can actually 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 Factrat, as a play on forrat.

He showed that you could program it, it was too incomplete. You could make a program that if the number you insert in was encoded as a prime, it would sync to zero. It would go down. Otherwise it would go up and things like that. So 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.

Yeah, if you want to do it not statistically but you really want 100% of all inputs to fall to earth, so what might be feasible is statistically 99% go to one. But like everything, that looks hard.

What would you say is out of these within reach famous problems is the hardest problem we have today? Is there a Riemann hypothesis? We want is up there. POS MP is a good one because that’s a meta problem. If you solve that in the positive sense that you can find a PMP 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. Now there’s ripple effects. P equ= 1 P has more ripple effects than basically any other. If the Riemann hypothesis is disproven, that would be a big mental shock to the number theorist, but it would have follow-on effects for cryptography because a lot of cryptography uses number theory.

It uses constructions involving primes and so forth, and it relies very much on the intuition that number theories are built over many years of what operations involving primes behave randomly and what 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.

We believe it to be almost impossible to crack, at least mathematically. But if something has core to our belief as a human hypothesis is wrong, it means that there are actual patterns of the primes that we’re not aware of. If there’s one, there’s probably going to be more. Suddenly a lot of our crypto systems are in doubt.

But then how do you say stuff about the primes? You’re going towards the conjecture again. Because if you want it to be random, you want it to be random. More broadly, I’m just looking for more tools, more ways to show that things are random.

How do you prove a conspiracy doesn’t happen, right? Is there any chance to you that P equals NP? Can you imagine a possible universe? It is possible. I mean, there’s various scenarios. There’s one where it is technically possible but in practice is never actually implementable.

The evidence is slightly pushing in favor of “no,” that we probably is not equal to NP. It seems like it’s one of those cases similar to the Riemann hypothesis that I think the evidence is leaning pretty heavily on the no. Certainly more on the no than on yes.

The funny thing about P versus NP is that we have also a lot more obstructions than we do for almost any other problem. So while there’s evidence, we also have a lot of results ruling out many types of approaches to the problem. This is the one thing that the computer scientists have actually been very good at. It’s actually saying that certain approaches cannot work—no-go theorems. It could be undecidable. We don’t. 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? Then you just quickly, very humbly said that this shiny metal is not going to solve any of the problems I’m currently working on. So, I’m just going to keep working on them.

It’s just, first of all, it’s funny to me that you would answer an email in that context, and second of all, it just shows your humility. But anyway, maybe you could speak to the Fields Medal, but 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 a $1 million 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.

Yeah. No, 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. I mean, 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.

He decided to peace out and, you know, collect mushrooms in St. Petersburg or something. And then that’s fine. You know, you can do that. I mean, that’s another sort of flip side. A lot of our problems that we solve do have practical applications, and that’s great, but if you stop thinking about a problem, he hasn’t published since in this field, but that’s fine. There are many other people who’ve done so as well.

So I guess one thing I didn’t realize initially with the Fields Medal is that it sort of makes you part of the establishment. You know, most mathematicians, they’re just career mathematicians. You just focus on publishing the next paper, maybe getting one rank up, starting a few projects, maybe taking some students or something. Yeah. But then suddenly people want your opinion on things, and you have to think a little bit about things that you might just so foolishly say because you know no one’s going to listen to you.

It’s more important now. Is it constraining to you? Are you able to still have fun and be a rebel and try crazy stuff and play with ideas? I have a lot less free time than I had previously. I mean, 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. This is always an option. But you know, there are things that are like, I mean, 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 yeah as you advance in your career somehow the more soft skills so math somehow frontloads all the technical skills to the early stages of your career.

So yeah, as a post office publisher or parish, you’re incentivized to basically focus on proving very technical themselves as well as proof the theorems. But then as you get more senior you have to start mentoring and giving interviews and trying to shape the direction of the field both research-wise and you know sometimes you have to do various administrative things and 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, sort of the really positive thing is that you get to be a light that’s an inspiration to a lot of young mathematicians or young people that are just interested in mathematics. It’s just how the human mind works. This is where I would probably say that I like the field’s metal, that it does inspire a lot of young people somehow. I don’t this just how human brains work.

Yeah. At the same time, I also want to give sort of respect to somebody like Gregoria Pearlman who is critical of awards in his mind. Those are his principles and any human that’s able for their principles to do the thing that most humans would not be able to do, it’s beautiful to see. Some recognition is necessarily important. But yeah, it’s also important to not let these things take over your life and only be concerned about getting the next big award or whatever.

I mean yeah, so again you see these people try to only solve a really big math problem and not work on things that are less sexy if you wish but actually still interesting and instructive. As you say, like the way the human mind works, we understand things better when they’re attached to humans and also if they’re attached to a small number of humans.

This way our human mind is wired; we can comprehend the relationships between 10 or 20 people, you know, but once you get beyond like 100 people there’s a limit, I think there’s a name for it, beyond which it just becomes the other. We have to simplify the pole master; you know 99.9% of humanity becomes the other and often these models are incorrect and this causes all kinds of problems.

But so yeah, 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 it can also be too much of it can be harmful because it’s…

I’ll be the first to say that my own career path is not that of a typical mathematician. The very accelerated education, I skipped a lot of classes. I think I was very fortunate with 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.

I mean they can be good mathematicians in a very different style. And we need people of a different style. Even if sometimes too much focus is given on 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, lots of 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 is 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, the incredible engineering teams, just the individual humans on those teams. They’re not a team. They’re individual humans on a team. And there’s a lot of brilliance there. But it’s just a nice shorthand, like a very simple way to refer to it.

Steve Jobs. As a starting point, as a first approximation, that’s how you, and then read some biographies and then look into much deeper. First approximation. That’s right. So you mentioned you were at Princeton with Andrew Wiles at that time. He’s a professor there. It’s a funny moment how history is just all interconnected. And at that time he announced that he proved the Fermat’s Last Theorem. What did you think maybe looking back now with more context about that moment in math history? Yes. So I was a graduate student at the time. I mean, I vaguely remember there was press attention and we all had the same pigeon holes in the same mail room. So we all picked our mail and suddenly Andrew Wiles’s mailbox exploded to be overflowing.

That’s a good metric. Yeah, we all talked about it at tea and so forth. I mean, we didn’t understand. Most of us didn’t understand the proof. We understood sort of high-level details. In fact, there’s an ongoing project to formalize it in Lean.

Right, Kevin Buzzard. Can we take that small tangent? How difficult does that seem? As I understand, the proof for Fermat’s Last Theorem has super complicated objects. Yeah, really difficult to formalize. You’re right that the objects that they use can be defined. So they’ve been defined in Lean. Just defining what they are can be done. That’s really not trivial, but it’s been done. But there’s a lot of really basic facts about these objects that have taken decades to prove, and that they’re in all these different math papers. So lots of these have to be formalized as well. Kevin Buzzard’s goal actually, he has a five-year grant to formalize Fermat’s Last Theorem, and his aim is that he doesn’t think he will be able to get all the way down to the basic axioms. But he wants to formalize it to the point where the only things that he needs to rely on as black boxes are things that were known by 1980 to number theorists at the time.

Then some other person, some other work would have to be done to get from there. It’s a different area of mathematics than 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, and things that are within the scope of a high school math education to at least define. But then there’s this very advanced algebraic. the side of number theory where people have been building structures upon structures for quite a while. It’s a very sturdy structure. It’s been very 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 about what is going on at level six of this tower, you have to spend quite a bit of time before they can get to the point where you can see something you recognize.

What inspires you about his journey that we talked about seven years, mostly working in secret? That is a romantic journey. So it fits with the romantic image I think people have of mathematicians to the extent they think of them at all as these kind of eccentric wizards or something. So that certainly kind of accentuated that perspective. I mean, it’s a great achievement. His style of solving problems is so different from my own, but which is great. I mean, we need people to speak to it.

In terms of the collaborative aspect, I like moving on from a problem if it’s giving too much trouble. 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 either, but they’re convinced they have the third, fourth, and fifth approaches that will work. 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 the incredible productivity, just the number of papers, which are all 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. I mean, there are also people who are very productive and they focus very deeply. I think everyone has to find their own workflow.

One thing which is a shame in mathematics is that we have a sort of 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, they don’t find their native math language until very late, or usually too late, so they stop doing mathematics and have a bad experience with a teacher who’s trying to teach them one way to do mathematics. They don’t like it.

My theory is that humans, evolution has not given us a math center of the brain directly. We have a vision center and a language center and some other centers, which evolution has honed, but we don’t have an innate sense of mathematics. But our other centers are sophisticated enough that different people can repurpose other areas of our brain to do mathematics. Some people have figured out how to use the visual center to do mathematics, and they think very visually when they do mathematics. Some people have repurposed their language center and they think very symbolically. competitive and they like gaming. There’s a type, there’s this part of your brain that’s very good at solving puzzles and games, and that can be repurposed. But when I talked about the mathematicians, they don’t quite think they can tell that they’re using some different styles of thinking than I am. I mean, not disjoint, but they may prefer visual. I don’t actually prefer visual 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 are thinking in different ways.

But you can tell there’s a different set of subsystems being used in the thinking process. 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 I mean, the way we educate, unless you have a personalized tutor or something, education sort of just by natural scale has to be mass-produced. You have to teach to 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 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? What would you say? Yeah, 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 already were math competitions, and you know there are also popular math books in the library.

But now you have, you know, YouTube; there are forums just devoted to solving math puzzles, and math shows up in other places. For example, there are hobbyists who play poker for fun, and they are interested in very specific probability questions. They actually know there’s a community of amateur theorists in poker, in chess, in baseball. I mean, there’s math all over the place.

I’m hoping actually with these new sorts of tools for learning that we can incorporate the broader public into math research projects. This doesn’t happen at all currently. So in the sciences, there’s some scope for citizen science. There are amateur astronomers who discover comets, and there are biologists, people 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 due to error checking and everything. One thing about these formalization projects is that they are bringing together more people.

I’m sure there are high school students who’ve already contributed to some of these formalizing projects, who have contributed to math libraries. You know you don’t need to be a PhD holder to just work on one atomic thing. There’s something about the formalization here that also acts 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 just the feeling, but it feels more accessible to folks than math. Math is seen as this extreme, especially modern mathematics, 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 you can get results. You can print a hello world pretty quickly.

If programming was taught as almost entirely a theoretical subject where you just taught the computer science, the theory of functions and routines and so forth, and outside of some very specialized homework assignments, you’re not actually programming like on the weekend for fun. They would be considered as hard as math. So, as I said, there are communities of non-mathematicians where they’re deploying math for some very specific purpose, like optimizing their poker game. 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. There’s a lot less certainty now in the world. I mean, 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 go to college, you get an education, you pick one profession and you stick to it. That is becoming much more a thing of the past, so I think you just have to be adaptable and flexible. I think people have to get skills that are transferable, like learning one specific programming language or one specific subject of mathematics or something.

It’s that itself is not a super transferable skill but sort of knowing how to reason with abstract concepts or how to problem-solve when things go wrong. So these are things which I think we will still need even as our tools get better. You would be working with AI and so forth. But actually, you’re an interesting case study. I mean, you’re one of the great living mathematicians, right? You had a way of doing things, and then all of a sudden you start learning— first of all, you kept learning new fields, but you learned lean. That’s not a non-trivial thing to learn; for a lot of people, that’s an extremely uncomfortable leap to take.

Mathematicians have 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 four or five 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 Schulzer had just formalized one of his new theorems, and there were some other developments in computer-assisted proof that look quite. interesting. And I said, “Oh, we should run a workshop on this. This be a good idea.”

And then I was a bit too enthusiastic about this idea. So I got volunteered. So I did with a bunch of other people, Kevin Bisard and Jordan Ellenburg and a bunch of other people. It was a nice success. We brought together a bunch of mathematicians and computer scientists and other people and we got up to speed and state and it was really interesting developments that most mathematicians didn’t know were going on. Lots of nice proofs of concept, just sort of hints of what was going to happen.

This was just before chat GBD, but there was even then one talk about language models and the potential capability of those in the future. So that got me excited about the subject. So I started giving talks about this is something we should more of us should start looking at now that I arranged to run this conference, and then chat GPT came out and suddenly AI was everywhere. I got interviewed a lot about this topic, in particular the interaction between AI and formal proof assistance. I said yeah they should be combined; this is perfect synergy to happen here.

At some point I realized that I have 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. There’s a limit to how much I can just rely on authority and saying, “I’m a mathematician, just trust me,” when I say that this is going to change athletics and I’m not doing it any when I don’t do any of it myself. So I felt like I had to actually justify it.

A lot of what I get into actually I don’t quite see in advance as how much time I’m going to spend on it, and it’s only after I’m sort of waste 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, right? Or have some of the challenges that a beginner would, right? New concepts, new ways of thinking also, you know, sucking at a thing that others I think in that talk you could be a Fields Medal winning mathematician and an undergrad knows something better than you.

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 just sort of 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.

So it does keep us honest. I mean not, I mean you can still, it’s not a perfect panacea, but I think we do have more of a culture of admitting error than 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, Raman, Hilbert. So, first of all, as mentioned before, there’s some time dependent on the day. Like if you plot cumulatively over time, for example, Euclid is one of the leading. Contenders. And 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? Just oh yeah, directly of everything that’s happened in the 20th century.

Hilbert spaces, we have lots of things that are named after him of course. Just the arrangement of mathematics and just the introduction of certain concepts. I mean, 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, I mean this is bystander effect in everywhere. If no one says you should do X, everyone just moves around waiting for somebody else to do something. And like nothing gets done.

It’s one thing that actually you have to teach undergraduates in mathematics, which 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 they see none of their standard techniques obviously applies, and the common reaction is then just paralysis. “I don’t know what to do,” or “I think there’s a quote from the Simpsons: I’ve tried nothing and I’m all out of ideas.”

So, the next step then is to try anything, like no matter how stupid. In fact, almost as stupid is better, which you know. One technique which 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. Do you imagine a thing you don’t want to do more? Yes. That’s worse than that. And then in that way, you procrastinate by not doing the thing that’s worse.

Yeah. It’s a nice hack. It actually works. This, I mean with anything, psychology is really important. You talk to athletes like marathon runners and so forth, and they talk about what’s the most important thing: is it their training regimen or the diet and so forth? Actually, so much of it is really psychology—tricking yourself to think 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, I sort of as a mathematician, I mean 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, even with the help of mathematics?

Well, okay. I mean, 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 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, right? Yeah. We’re already a collective intelligence. Yes. So, 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 yeah, so, mathematical community plural is an incredibly super intelligent entity that no single human mathematician can come closer to replicating. You see it a little bit on these question analysis sites. So this math overflow, which is the math version of stack overflow, sometimes you get very quick responses to very difficult questions from the community. It’s a pleasure to watch actually, as an expert. I’m a fan 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 fun.

What gives you hope about this whole thing we have going on, human civilization? I think the younger generation is always really creative and enthusiastic and inventive. It’s a pleasure working with young students.

You know, the progress of science tells us that the problems that used to be really difficult can become extremely trivial to solve. I mean, it was like navigation. 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. I guess it’s a completely solved problem. So, things that seem unfeasible for us now could be just sort of homework exercises.

But 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. 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 know what I mean? Just the internet and our AI. Again, they’ve been beginning to be internalized, and yes, of course, an AI can understand our voice and give reasonable, slightly incorrect answers to any question. But yeah, this was mind-blowing even two years ago, and at the moment, it’s hilarious to watch 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 argue with each other about it. But when you look at the arc of things, just even in the progress of robotics, take a step back and be like, “Wow, this is beautiful that we humans are able to create this.” Infrastructure and the culture is healthy. The community of humans can be so much more intelligent and 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’s a lot of really smart people there. And 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.

Oh, thank you. It was a pleasure.

Thanks for listening to this conversation with Terrence Tao. To support this podcast, please check out our sponsors in the description or at lexfreedman.com/sponsors.

And now, let me leave you with some words from Galileo Galilei. Mathematics is a language with which God has written the universe.

Thank you for listening and hope to see you next time.