In this Lex Fridman Podcast conversation, mathematician Terence Tao discusses some of the hardest open problems in mathematics and physics. He explains why the Navier-Stokes equations on fluid blow-up are so difficult to settle, using analogies from cellular automata and Maxwell's demon.
The conversation moves through infinity, the boundary between mathematics and physics, and the twin prime and Collatz conjectures. Tao also describes how the Lean proof assistant and AI tools may change mathematical collaboration, and offers his view on the limits of current AI.
Listeners will hear how mathematicians think about structure, randomness and proof.
In this episode Lex Fridman, Terence Tao
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The following is a conversation with Terence Tao, widely considered to be one of the greatest mathematicians in history, often referred to as the Mozart of math. He won the Fields Medal and the Breakthrough Prize in Mathematics, and has contributed groundbreaking work to a truly astonishing range of fields in mathematics and physics. This was a huge honor for me, for many reasons, including the humility and kindness that Terry showed to me throughout all our interactions.
It means the world. And now, a quick few-second mention of each sponsor. Check them out in the description or at lexfridman.com/sponsors. It's the best way to support this podcast. We've got Notion for teamwork, Shopify for selling stuff online, NetSuite for your business, Element for electrolytes, and AG1 for your health. Choose wisely, my friends. And now, onto the full ad reads. They're all here in one place. I do try to make them interesting by, uh, talking about some random things I'm reading or thinking about.
But if you skip, please still check out the sponsors. I enjoy their stuff. Maybe you will too. To get in touch with me for whatever reason, go to lexfridman.com/contact. All right, let's go. This episode is brought to you by Notion, a note-taking and team collaboration tool. I use Notion for everything: for personal notes, for planning this podcast, for collaborating with other folks, and for super boosting all of those things with AI, because Notion does a great job of integrating AI into the whole thing.
You know, what's fascinating is the mechanisms of human memory before we had widely adopted technologies and tools for writing and recording stuff, certainly before the computer. So you can look at medieval monks, for example, that would use the now-well-studied memory techniques, like the memory palace, the spatial memory techniques to memorize entire books. That is certainly the effect of technology started by Google Search and moving to all the other things like Notion, that we're offloading more and more and more of the task of memorization to the computers, which I think is probably a positive thing, because it frees more of our brain to do deep reasoning, whether that's deep dive, focused specialization, or the generalist type of thinking versus memorizing facts.
Although I do think that there's a kind of background model that's formed when you memorize a lot of things, and from there, from inspiration, arises discovery. So I don't know. It could be a great cost to offloading most of our memorization to the machines. But it is the way of the world. Try Notion AI for free when you go to notion.com/lex. That's all lowercase: notion.com/lex. To try the power of Notion AI today. This episode is also brought to you by Shopify, a platform designed for anyone to sell anywhere with a great-looking online store.
Our future friends has a lot of robots in it. Looking into that distant future, you have Amazon warehouses with millions of robots that move packages around. You have Tesla bots everywhere in the factories and in the home and on the streets and, uh, the baristas. All of that. That's our future. Right now, you have something like Shopify that connects a lot of humans in the digital space. But more and more, there will be an automated, digitized, AI-fueled connection between humans in the physical space.
Like a lot of futures, there's going to be negative things, and there's going to be positive things. And like a lot of possible futures, there's little we could do about stopping it. All we can do is steer it in the direction that, uh, enables human flourishing. Instead of hiding in fear or fear-mongering, be part of the group of people that are building the best possible trajectory of, uh, human civilization. Anyway, sign up for a $1 per month trial period at shopify.com/lex.
That's all lowercase. Go to shopify.com/lex to take your business to the next level today. This episode is also brought to you by NetSuite, an all-in-one cloud business management system. There's a lot of messy components to running a business, and I must ask, and I must wonder, at which point there's going to be an AI/AGI-like CFO of a company, an AI agent that handles most, if not all, of the financial responsibilities or all of the things that NetSuite is doing.
At which point will NetSuite increasingly leverage AI for those tasks? I think probably it will integrate AI into its tooling, but I think there's a lot of edge cases that we need the, the human wisdom, the human intuition grounded in, uh, years of experience in order to make the tricky decision around the edge cases. I suspect that running a company is a lot more difficult than people realize. But there's a lot of sort of paperwork-type stuff that could be automated, could be digitized, could be summarized, integrated, and used as a foundation for the said humans to make decisions.
Anyway, that's our future. Download the CFO's guide to AI and machine learning at net-suite.com/lex. That's net-suite.com/lex. This episode is also brought to you by Element, my daily zero-sugared and delicious electrolyte mix. You know, I run along the river, uh, often and, uh, get to meet some really interesting people. Uh, one of the, the people I met was preparing for his first ultra marathon. I believe he said it was 100 miles. And that, of course, sparked in me the thought that, uh, I need for sure to, uh, to do one myself.
Some time ago now, I was planning to do something with David Goggins, and I think that's still on the sort of to-do list between the two of us, to do some crazy, uh, physical feat. Of course, the thing that is, uh, crazy for me is, uh, daily activity for Goggins. But nevertheless, I think it's important in the physical domain, the mental domain, and, and all domains of life to challenge yourself. And athletic endeavors is one of the most sort of crisp, clear, well-structured way of challenging yourself.
But there's all kinds of things. Writing a book, to be honest, having kids and marriage and relationships and friendships, all of those, if you take it seriously, if you go all in and do it right, I think, uh, that's a serious challenge. Because most of us are not prepared for it. You can learn along the way. And if you have the rigorous feedback loop of improving constantly and growing as a person and really doing a great job of the thing, I think, uh, that might as well be an ultra marathon.
Anyway, get a sample pack for free with any purchase. Try it at drinkelement.com/lex. And finally, this episode is also brought to you by AG1, an all-in-one daily drink to support better health and peak performance. I drink it every day. I'm preparing for a conversation on, uh, drugs in the Third Reich. And funny enough, it's a kind of way to analyze Hitler's biography. It's to look at what he consumed throughout.
And, uh, Norman Oler does a great job of analyzing all of that and tells the story of Hitler and the Third Reich in a way that hasn't really been touched by historians before. It's always nice to look at, uh, key moments in history through a perspective that's not often taken. Anyway, I mention that because I think Hitler had a lot of stomach problems, and so that was the motivation for getting a doctor, the doctor that eventually, uh, would, uh, fill him up with all kinds of drugs.
But, uh, the doctor earned Hitler's trust by giving him probiotics, which is a kind of revolutionary thing at the time. And so that really helped deal with whatever stomach issues that, uh, Hitler was having. All of that is a reminder that wars waged by humans and humans are biological systems, and biological systems require fuel and supplements and all of that kind of stuff. And depending on what you put in your body will affect your performance in the short term and the long term. With meth, that's true with, uh, Hitler to his last days in the bunker in Berlin, all the cocktail of drugs that he was taking.
So I think I got myself somewhere deep, and I'm not sure how to get out of this. Uh, it deserves a multi-hour conversation versus a few seconds of mention. But yeah, all of that was sparked by my thinking of, uh, AG1 and how much I love it. I appreciate that you're listening to this and, uh, coming along for the wild journey that these, uh, ad reads are. Anyway, AG1 will give you a one-month supply of fish oil when you sign up at drinkag1.com/lex.
This is the Lex Fridman Podcast. To support it, please check out our sponsors in the description or at lexfridman.com/sponsors. And now, dear friends, here's Terence 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, um, education, you, you learn about the really hard and possible problems, like the Riemann hypothesis, the Trin-Primes conjecture. You can make problems arbitrarily difficult. That's not really a problem. In fact, there's even problems that we know to be unsolvable. What's really interesting are the problems just at the, on the boundary between what we can do relatively easily and what are hopeless. Um, but what are problems where, like, existing techniques can do like 90% of the job, and then you, you just need that remaining 10%. Um, I think as a PhD student, the Kocher problem certainly caught my eye, and it just got solved, actually.
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, Soichi Kocher, uh, in like 1918 or so. Um, so the puzzle is that you, you, you have, um, a needle, um, in, on the plane, um, or think like, like a, like driving on, on, on a road or something. And you, you wanted to execute a U-turn. You want to turn the needle around. Um, but you want to do it in as little space as possible. Uh, so you want to use this little area in order to turn it around.
So, um, but the needle's infinitely maneuverable. So you could imagine just spinning it around its, um, as a unit needle. You can spin it around its center. Um, and I think, um, that gives you a disc of, of area, I think, pi over four. Um, or you can do a three-point U-turn, which is what they re-reteach people in, in their driving schools to do. Uh, and that actually takes area pi over eight. Uh, so it's, it's a little bit more efficient than, um, a rotation. And so for a while, people thought that was the most efficient, uh, way to turn things around. But Vaskovich, uh, showed that, in fact, uh, you could actually, uh, turn the needle around using as little area as you wanted.
So point 001, there was some really fancy multi, um, uh, um, back-and-forth U-turn thing that you could, you could do that, that you could turn a needle around. And in so doing, it 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. So the next question is what happens in three dimensions. So 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. Uh, you want to occupy as little volume as possible in order to rotate your, your needle around in order to see every single star in the sky. Um, how small a volume do you need to do that?
And so you can modify Vaskovich's construction. And so if your telescope has zero thickness, then you, 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, but just very, 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, as the needle gets, gets thinner, the volume should go down. But, but how fast does it go down? Um, and the conjecture was that it goes down very, very slowly, um, like logarithmically, um, uh, 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, combinatorics. For example, in, in wave propagation, if you splash some, some water around, um, you create water waves, and they, they travel in various directions. Um, but waves exhibit both wa, both particle and wave-type behavior. So you can have what's called a wave packet, which is like a, a very localized wave that is localized in space and moving a certain direction in time.
And so if you plot it in both space and time, it occupies a region which looks like a tube. And so what can happen is that you can have a wave which initially is very dispersed, but it all comes, it all focuses at a single point later in time. Like, you can imagine dropping a, a pebble into a pond and the ripples spread out. But then if you time-reverse that, that, um, that scenario, and the equations of wave motion are time-reversible, you can imagine ripples that are converging, um, to a single point, and then a, a big splash occurs, um, maybe even a singularity.
Um, and so it's possible to do that. Uh, and geometrically, what's going on is that there's all this sort of light rays. Um, so like if, if, if this wave represents light, for example, um, you can imagine this wave as a superposition of photons, um, 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, it separates. But potentially, if the conjecture had a negative solution, so what that meant is that there's, there's a very efficient way to pack, um, tubes pointing in different directions into a very, very narrow region of, of, of, uh, very narrow volume, then you would also be able to create waves that start out, some there'll be some arrangement of waves that start out very, very dispersed, but they would concentrate not just at a single point, but, um, um, there'll be a, a large, um, uh, there'll be a lot of concentrations in space and time.
And, uh, um, and you could create what's called a blow-up, where these waves, their 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. Um, and so in mathematical physics, we care a lot about whether certain equations in, in, in wave equations are stable or not, uh, whether they can create, um, these singularities. There's a famous unsolved problem called the Navier-Stokes regularity problem. So the Navier-Stokes equations, equations that govern a fluid flow for incompressible fluids like water. The question asks, if you start with a smooth, velocity fluid of water, can it ever concentrate so much that, like, the velocity becomes infinite at some point?
That's called a singularity. We don't see that, um, in real life. You know, if you splash around water on the bathtub, it won't explode on you. Um, or, or have, have water leaving at the speed of light or anything. But potentially, it is possible. Um, and in fact, in recent years, the, the consensus has, has drifted towards the, uh, the, the belief that, uh, that in fact, for certain very special initial con, uh, configurations of, of, say, water, that singularities can form.
But people have not yet been able to, uh, to actually establish this. The Clay Foundation has these seven Millennium Prize problems. It 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. Uh, the Poincaré conjecture, like Perelman. So the Kocher conjecture is not directly, directly related to the Navier-Stokes problem, but understanding it would help us understand some aspects of things like wave concentration, which would indirectly probably help us understand the Navier-Stokes problem better.
Can you speak to the Navier-Stokes? So the existence and smoothness, like you said, Millennium Prize problem.
Right.
You've made a lot of progress on this one. In 2016, you published a paper, Finite Time Blow-Up, for an averaged three-dimensional Navier-Stokes equation.
Right.
So, so we're trying to figure out if this thing usually doesn't blow up.
Right.
But can we say for sure it never blows up?
Right. Yeah. So yeah, that is literally the, the million-dollar question. Yeah. So this is what distinguishes mathematicians from pretty much everybody else. Like, if, if, if something holds 99.99% of the time, um, that's good enough for most, you know, uh, for, for, for most things. But mathematicians are one of the few people who really care about whether every, like, 100%, really 100% of all, um, situations are covered by, by, um, yeah. So most fluids, most of the time, um, water does not blow up, but could you design a very special initial state that does this?
And maybe we should say that this is a, this is a set of equations that govern in the field of fluid dynamics.
Yes.
Trying to understand how fluid behaves, and it actually turns out to be a really comp, you know, fluid is.
Yeah.
Extremely complicated thing to try to model.
Yeah. So it has practical importance. So 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, it does a lot of computational fluid dynamics. A lot of it is actually just trying to solve the Navier-Stokes equations as, as best they can. Um, also gathering a lot of data so that they can get, they can initialize the equation. There's a lot of moving parts. So it's a very important problem practically.
Why is it difficult to prove general things about these set of equations, like it not, not blowing up?
Short answer is Maxwell's demon. Um, so Maxwell's demon is a concept in thermodynamics. Like, if you have a box with two gases in it, uh, oxygen and nitrogen, uh, and maybe you start with all the oxygen on one side and nitrogen on the other side, but there's no barrier between them, right? Then they will mix. Um, and they should stay mixed, right? There's, 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 that, that, um, like maybe there's a microscopic demon called Maxwell's demon that will, um, every time an oxygen and nitrogen atom collide, they'll bounce off in such a way that the oxygen sort of drifts onto one side and the nitrogen goes to the other.
And, uh, you could have an extremely improbable configuration emerge, uh, which we never see. Um, and, and we, we statistically, it's extremely unlikely, but mathematically, it's possible that this can happen, and we can't rule it out. Um, and this is a situation that shows up a lot in mathematics. Um, 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 as there should be no preference in the digits of pi to favor, let's say, seven over eight.
Um, but maybe there's some demon in the digits of pi that, that, like, every time you compute more and more digits, it sort of biases one digit to another. Um, and this is a conspiracy that should not happen. There's no reason it, it should happen, but, um, there's, there's, there's no way to prove it, uh, 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, uh, 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, it will, it will go to zero.
Um, and this is what happens, um, in, um, uh, when we actually experiment with water. Like, you splash around, there's, there's some turbulence and, and waves and so forth, but eventually, it, it settles down, and, and, and the, the lower the amplitude, the, the smaller the velocity, the, the more calm it gets. Um, but potentially, there is also a demon that keeps pushing the, uh, the energy of the fluid into a smaller and smaller scale, and it will move faster and faster. And at faster speeds, the effect of viscosity is relatively less. And so it could happen that, that it, it creates a, some sort of, um, um, what's called a self-similar blow-up scenario where, you know, um, the energy of the fluid starts off at some, um, large scale, and then it all sort of, um, transfers its energy into a smaller, um, region of, of, of the fluid, which then at a much faster rate, um, moves into, um, an even smaller, uh, region and so forth.
Um, and, and each time it does this, uh, it takes maybe half as, as long as, as the previous one. And then, uh, you, you could, you could actually, uh, converge to all the energy, um, concentrating in one point in a, a finite amount of time. Um, and that, that's, uh, that scenario is called Fahrenheit blow-up. Um, so in practice, this doesn't happen. Um, so water is what's called turbulent. Um, so it is true that, um, if you have a big eddy of water, it will tend to break up into smaller eddies, but it won't transfer all this energy from one big eddy into one smaller eddy.
It will transfer into maybe three or four, and then those ones split up into maybe three or four smaller eddies of their own. And so the energy gets dispersed to the point where the viscosity can, can then keep everything under control. Um, but if it can somehow, um, concentrate, um, all the energy, keep it all together, uh, and do it fast enough that the viscous effects don't have enough time to calm everything down, then this blow-up can occur. So there are papers who claim that, oh, 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, many types of equations like this.
And so 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 Fahrenheit blow-up, that a velocity stays smooth, and it all failed. There was always some sign error or some subtle mistake, and, and it, it couldn't be salvaged. Um, so what I was interested in doing was trying to explain why we were not able to disprove, um, Fahrenheit blow-up. I couldn't do it for the actual equations of fluids, which were too complicated. But if I could average the equations of motion of Navier-Stokes, basically, if, if, um, if I could turn off certain types of, of ways in which water interacts and only keep the ones that I want.
Um, so in particular, um, if, um, if there's a fluid and it could transfer its 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, this one, and, and direct it only into, um, this smaller eddy while still preserving the law of conservation of energy.
So you try and make a blow-up.
Yeah. Yeah. So I, I, I basically engineer, um, 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. Um, so, so what I did was that, uh, basically, if I turned off the, um, certain parts of the equation, so which usually when you turn off certain interactions, make it less nonlinear, it makes it more regular and less likely to blow up. But I found that by turning off a very well-designed set of, of, of, of, of, of, of interactions, I could force all the energy to blow up, uh, in finite time. So what that means is that if you wanted to prove, um, global regularity for Navier-Stokes, um, for the actual equation, you had, you must use some feature of the true equation, which, which my artificial equation, um, does not satisfy.
So it, it rules out certain, um, certain approaches. So, um, the thing about math is, is, is it's not just about finding, you know, taking a technique that is going to work and applying it, but you, you need to not take the techniques that don't work. Um, and for the problems that are really hard, often there are dozens of ways that you might think might apply to solve the problem, but, uh, 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, um, kind of rules out, um, uh, it, it saves you a lot of time because you, you, you're not wasting, um, energy on, on things that you now know cannot possibly ever work.
How deeply connected is it to that specific problem of fluid dynamics, or is it some more general intuition you build up about mathematics?
Right. Yeah. So the key phenomenon that, uh, my, my technique exploits is what's called supercriticality. So in partial differential equations, often these equations are like a tug-of-war between different forces. So in Navier-Stokes, there's the dissipation, um, force coming from viscosity, and it's very well understood. It's linear. It calms things down. If, if viscosity was all there was, then, then nothing bad would ever happen. Um, but there's also transport, um, that, that energy from in one location of space can get transported because the fluid is in motion to, to other locations.
Um, and that's a nonlinear effect, and that causes all the, all the problems. Um, so there are these two competing terms in the Navier-Stokes equation, uh, the dissipation term and the transport term. If the dissipation term dominates, if it's, if it's large, then basically you get regularity. And if, um, if the transport term dominates, then, uh, then we don't know what's going on. It's a very nonlinear situation. It's unpredictable. It's turbulent. So sometimes these forces are in balance at small scales, but not in balance at large scales or, or vice versa. Um, so Navier-Stokes is what's called supercritical.
So at, at smaller and smaller scales, the transport terms are much stronger than the viscosity terms. So the viscosity terms are the things that calm things down. Um, and so this is, um, um, this is why the problem is hard in two dimensions. So the Soviet mathematician, uh, Ladyshchinskaya, she in the '60s showed in two dimensions there was no blow-up, and in two dimensions, the Navier-Stokes equation is what's called critical. The effect of transport and the effect of viscosity are about the same strength, even at very, very small scales. And we have a lot of technology to handle critical and also subcritical equations and proof, um, regularity.
But for supercritical equations, it was not clear what was going on. And I did a lot of work, and then there's been a lot of follow-up showing that for many other types of supercritical equations, you can create all kinds of blow-up examples. Once the nonlinear effects dominate the linear effects at small scales, you can have all kinds of bad things happen. So this is sort of one of the main insights of this, this line of work is that supercriticality versus criticality and subcriticality, this, this makes a big difference. I mean, that's a key qualitative feature that distinguishes some equations from being sort of nice and predictable and, and, you know, like, like planetary motion.
And I mean, there's certain equations that, that you can predict for millions of years and, or, or thousands at least. Again, it's not really a problem, but, 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.
Yeah.
That can create a huge problem for predicting what's going to happen.
Yeah. And if the nonlinearity is somehow more and more featured and interesting at, at small scales. Um, I mean, there's, there's many equations that are nonlinear, but, um, in many, in many equations, you can approximate things by the bulk. Um, so for example, planetary motion, you know, if you want to understand the orbit of the Moon or Mars or something, you don't really need the microstructure of, uh, like the seismology of the Moon or, or like exactly how the mass is distributed. Um, you just basically, you can almost approximate these planets by point masses, uh, and just the aggregate behavior is important.
Um, but if you want to model a fluid, um, like, like the weather, you can't just say in Los Angeles, the temperature is this, the wind speed is this. For supercritical equations, the finest confirmation is, is really important.
If we can just linger on the Navier-Stokes, uh, equations a little bit. So you've suggested maybe you can describe it, that one of the ways to, uh, solve it or to negatively resolve it would be to sort of 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, uh.
Right.
Show it in that way. Can you describe this, this idea?
Right. Yeah. So this came out of, of this work of constructing this, this, this average equation that, that blew up. Um, so one, um, as, as part of how I had to, to do this, so there's still this naive way to do it. You just, you just keep pushing, um, um, every time you, you get energy at one scale, you, you push it immediately to the next scale as, as fast as possible. This is sort of the naive way to, to, to, to force blow-up. Um, it turns out in five and high dimensions, this works. Um, but in three dimensions, there was this funny phenomenon that I discovered that if you, if you keep, if, if you change the laws of physics, you just always keep trying to push, um, the energy into smaller, smaller scales.
Um, what happens is that the energy starts getting spread out into many, many scales at once. Um, so you have, you have energy at one scale, you're pushing it into the next scale, and then, um, 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. Um, you're trying to do everything at once. Um, and this spreads out the energy too much. Um, and then it turns out that, that, um, it makes it vulnerable for viscosity to come in and actually just damp out everything. So, um, so it turns out this, this direct abortion doesn't, doesn't actually work.
There was a separate paper by some other authors that actually showed this, um, in three dimensions. Um, so what I needed was to program a delay. Um, so kind of like airlocks. Uh, so, um, I, I needed an equation which would start with a fluid doing something at one scale. It would push its energy into the next scale, but it would stay there until all the energy from the, from the larger scale got transferred. And only after you pushed all the energy in, then you sort of open the next gate, uh, and, and then you, you push that in as well.
So, um, by doing that, it kind of, the energy inches forward scale by scale in such a way that it's always, um, localized at one scale at a time. Um, and then it can resist the effects of viscosity because it's, it's not dispersed. Um, so in order to make that happen, um, yeah, I had to construct a rather complicated nonlinearity. Um, and it was basically like, um, you know, like it was constructing like an electronic circuit. So I, I actually thanked my wife for this because she was trained as a, as a true engineer. Um, and, um, you know, she, she talked about, um, uh, you know, she had to design circuits and so forth.
And, you know, if you, if you want a circuit that does a certain thing, like maybe have a light that, that flashes on and then turns off and then on and then off, you can build it from, from more primitive components, you know, capacitors and resistors and so forth. And you have to build a diagram and you, um, and these diagrams, you can, you can sort of follow it with your eyeballs and say, oh, yeah, the, the, the current will, 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, you know, like resistors and capacitors and so forth. And, and I would, I would stack them together, um, in such, in such a way that, that I would create something that would open one gate and then there would be a clock that would, and then once the clock hits a certain threshold, it would close it.
Uh, 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, so, um, like if you can imagine kind of a steampunk, but it's really waterpunk, uh, type of thing where, um, you know, so modern computers are electronic, you know, they're, they're, they're, they're powered by, by electrons passing through very tiny wires and interacting with other, other electrons and, and so forth. But instead of electrons, you can imagine these pulses of, of water moving at a certain velocity.
And maybe it's, there are two different configurations corresponding to a bit being up or down. Probably if you had two of these moving bodies of water collide, uh, they would come out with some new configuration, which is, which would be something like an AND gate or OR gate, you know, that, uh, it would, the, the, the output would depend in a very predictable way on, on the inputs. And like you could chain these together and maybe create a Turing machine and, and then you could, you have computers, uh, which are made completely out of water. Um, and if you have computers, then maybe you can do robotics.
Uh, so, you know, hydraulics and so forth. Um, and so you could create some machine, which is basically a fluid analog of what's called a Von Neumann machine. Uh, so Von Neumann proposed, 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, smelt them, and build more copies of the same machine, um, then you could colonize the whole planet, um, over time.
Um, so, uh, if you could build a fluid machine, which, uh, yeah, so it's, it's, it's a, it's a robot, it's a fluid robot. Okay. And what it would do, it's, it's, it's purpose in life, it's programmed so that it would create a smaller version of itself in some sort of cold state. It wouldn't start just yet. Once it's ready, the big robot configuration of water would transfer all its energy into the smaller configuration and then power down. Okay. And then it clean itself up. And then what's left is this newer state, which would then turn on and do the same thing, but smaller and faster.
And then 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, uh, for the actual Navier-Stokes. And this is what I managed to accomplish for this average Navier-Stokes. So it provided this sort of roadmap to solve the problem. Now this is, uh, a pipe dream because, uh, there are so many things that are missing for this to actually be a reality. Um, so, um, I, I, I can't create these basic logic gates. Um, I, I don't, I don't have these, these special configurations of water. Um, I mean, there's candidates that include vortex rings that might possibly work, but, um, um, but also, you know, analog computing is really nasty, um, compared to digital computing.
I mean, 'cause there's always errors. Um, you, you have to, 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, the, the running of the smaller machine. But everything in principle can happen. Like it doesn't contradict any of the laws of physics. Um, so it's sort of evidence that this thing is possible. Um, there are other groups who are now pursuing ways to make Navier-Stokes blow up, which are nowhere near as ridiculously complicated as this. Um, um, they, they actually are pursuing much closer to the, the direct self-similar model, which can, it, it doesn't quite work as is, but there could be some simpler scheme than what I just described to make this work.
There is a real leap of genius here to go from Navier-Stokes to this Turing machine. So it goes from what the self-similar blob scenario that you're trying to get the smaller and smaller blob.
Mm-hmm.
To now having a liquid Turing machine that gets smaller and smaller and smaller and somehow seeing how that could be used to say something about a blow-up. I mean, that's a big leap.
So there's precedent. I mean, um, so the, the thing about mathematics is that it's, it's really good at, um, spotting connections between what you think of, what you might think of as completely different, um, problems. Um, but if, if the mathematical form is the same, you, you can, you, you can, you can draw a connection. Um, so, um, there's a lot of work previously on what's called cellular automator. Um, the most famous of which is Conway's game of life. This is infinite discrete grid and at any given time, the grid is either occupied by a cell or it's empty. And there's a very simple rule that, uh, tells you how these cells evolve.
So sometimes cells live and sometimes they, they die. Um, and there's, um, you know, um, when I was a, a student, uh, it was a very popular screensaver to actually just have these, these animations going on and, and they, 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. Um, so for example, they discovered this thing called a glider. So a glider is a very tiny configuration of like four or five cells, which evolves and it just moves at a certain direction. And that's like this, this vortex rings, this analog. Um, yeah, so this is an analogy.
The game of life is kind of like a discrete equation and, and, um, the fluid Navier-Stokes is a continuous equation, but mathematically they have some similar features. Um, and, um, so over time people discovered more and more interesting things that you could build within the game of life. The game of life is a very simple system. It only has like three or four rules, um, to, to do order, but, but you can design all kinds of interesting configurations inside it. Um, there's something called a glider gun that does nothing or spit out gliders one at a, one at a time. Um, and then after a lot of effort, people managed to create, um, AND gates and OR gates for gliders.
Like this is massive ridiculous structure, which if you, if a, if a, if you have a stream of gliders, um, coming in here and a stream of gliders coming in here, then you may produce a stream of gliders coming out if, maybe if both of, of the, um, streams, um, have gliders, then there'll be an output stream. But if only one of them does, then nothing comes out.
Mm-hmm.
So they could build something like that. And once you could build and, um, these basic gates, then just from software engineering, you can build almost anything. Um, you can build a Turing machine. I mean, it's, it's again, enormous steampunk type things. They look ridiculous, but then people also generated self-replicating objects in the game of life. A massive machine, a Von Neumann machine, which over a lot huge period of time and it always looked like glider guns inside doing these very steampunk calculations, it would create another version of itself, which could replicate.
That's so incredible.
A lot of this was like community crowdsourced by like amateur mathematicians actually. Um, so I knew about that, that, that work. And so that is part of what inspired me to propose the same thing for Navier-Stokes. Um, it was a much, as I say, analog is much worse than digital. Like it's gonna be, um, you can't just directly take the constructions in the game of life and plump them in. But again, it just, it shows it's possible.
You know, there's a kind of emergence that happens with these cellular automata, local rules.
Mm-hmm.
Maybe it's similar to fluids, I don't know, but local rules operating at scale can create these incredibly complex dynamic structures. Do you think any of that is amenable to mathematical analysis? Do we have the tools to say something profound about that?
The thing is, you can get these emergent very complicated structures, but only with very carefully prepared initial conditions. Yeah. So, so these, these, these glider guns and, and gates and, and software machines, if you just plunk on randomly some cells and you're looking at that, you will not see any of these. Um, and that's the analogous situation with Navier-Stokes again, you know, that, that with, with typical initial conditions, you would not, you would not have any of this weird computation going on. Um, but basically through engineering, you know, by, by, by specially designing things in a very special way, you can make clever constructions.
I wonder if it's possible to prove the sort of the negative of like basically prove that only through engineering can you ever create.
Yeah, yeah, yeah, yeah.
Something interesting.
This, this is a recurring challenge in mathematics that, um, I call it the dichotomy between structure and randomness, that most objects that you can generate in mathematics are random. They look like random, like the digits of pi. Well, we believe is a good example. Um, but there's a very small number of things that have patterns. Um, but, um, now you can prove something as a pattern by just constructing, you know, like if something has a simple pattern and you have a proof that it, it does something like repeat itself every so often, you can do that. But, um, and you, you can prove that, that for example, you can, you can prove that most sequences of, of digits have no pattern.
Um, so like if, if you just pick digits randomly, there's something called low large numbers. It tells you you're gonna get as many ones as, as twos in the long run. Um, but, um, we have a lot fewer tools to, to, to, 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 spent a lot of time on is to prove what called structure theorems or inverse theorems that give tests for when something is, is very structured. So some functions are what's called additive. Like if you have a function that maps the natural numbers to the natural numbers, so maybe, um, you know, two maps to four, three maps to six, and so forth.
Um, some functions are what's called additive, which means that if you add, if you add two inputs together, the output gets, gets added as well. Uh, for example, a multiplying by a constant. If you multiply a number by 10, um, if you, if you multiply A plus B by 10, that's the same as multiplying A by 10 and B by 10 and then adding them together. So some, um, functions are additive. Some functions are kind of additive, but not completely additive. Um, so for example, if I take a number N, I multiply by the square root of two, and I take the integer part of that. So 10 by square root of two is like 14 point something.
So 10 will map to 14. Um, 20 will map to 28. Um, so in that case, additivity is true then. So 10 plus 10 is 20 and 14 plus 14 is 28. But because of this rounding, uh, sometimes there's round off errors and, and sometimes when you, um, add A plus B, this function doesn't quite give you the sum of, of the two individual outputs, but the sum plus or minus one. Um, so it's almost additive, but not quite additive. Um, so there's a lot of useful results in mathematics, and I've worked a lot in developing things like this to the effect that if, if a function exhibits some structure like this, then, um, it's basically, there's a reason for why it's true.
And the reason is because there's, there's some other nearby function, which is actually, um, completely structured, which is explaining this sort of partial pattern that you have. Um, and so if you have these sort of inverse theorems, it, um, it creates this sort of dichotomy that, that either the objects that you study are either have no structure at all, or they're somehow related to something that is structured. Um, and in either way, in either, um, uh, in either case, you can make progress. Um, a good example of this is that there's this old theorem in mathematics called Simmeredis theorem, uh, proven in the 1970s.
It concerns trying to find a certain type of pattern in a set of numbers. The pattern is arithmetic progression, things like three, five, and seven, or, or, or 10, 15, and 20. And Simmeredis, Andrea Simmeredis proved that, um, any set of numbers that are sufficiently big, um, what's called positive density, has, um, arithmetic progressions in it of, of any length you wish. Um, so for example, um, the odd numbers have a set of density one half, um, and they contain arithmetic progressions of any length. Um, so in that case, it's obvious because the, the, the odd numbers are really, really structured.
I can just take, uh, 11, 13, 15, 17, I can just, I can, I can easily find arithmetic progressions in, in, in that set. Um, but, um, Simmeredis theorem also applies to random sets. If I take the set of odd numbers and I flip a coin, um, and I, uh, for each number, and I only keep the numbers which, for which I got aheads. Okay. So I just flip coins, I just randomly take out half the numbers, I keep one half. So that's a set that has no, no patterns at all. But just from random fluctuations, you will still get a lot of, um, um, of arithmetic progressions in that set.
Can you prove that there's arithmetic progressions of arbitrary length within a random?
Yes. Um, have you heard of the infinite monkey theorem? Usually mathematicians give boring names to theorems, but occasionally they, they give colorful names.
Yes.
The popular version of the infinite monkey theorem is that if you have an infinite number of monkeys in a room with each of a typewriter, they type out, uh, text randomly, almost surely one of them is going to generate the entire script of Hamlet or any other finite string of text. Uh, it will just take some time, quite a lot of time actually, but if you have an infinite number, then it happens. Um, so, um, basically the theorem says that if you take an infinite string of, of digits or whatever, um, eventually any finite pattern you wish will emerge. Uh, it may take a long time, but it will eventually happen.
Um, in particular, arithmetic progressions of any length will eventually happen. Okay. But you need that, 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.
Yeah. How are we humans supposed to deal with infinity?
Well, you can think of infinity as, as just an abstraction of, um, a finite number for which you, you do not have a bound for. Um, that, uh, you know, I mean, so nothing in real life is truly infinite. Um, but, you know, you can, um, you know, you can ask yourself questions like, you know, what if I had as much money as I wanted, you know, or what if I could go as fast as I wanted? And a way in which mathematicians formalize that is mathematics has found a formalism to idealize instead of something being extremely large or extremely small to actually be exactly infinite or zero.
Um, and often the, the mathematics becomes a lot cleaner when you do that. I mean, in physics, we, we joke about, uh, assuming spherical cows. Um, you know, like real world problems have got all kinds of real world effects, but you can idealize, send some things to infinity, send some things to zero. Um, and, um, and the mathematics becomes a lot simpler to work with then.
I wonder how often using infinity, uh, forces us to deviate from, um, the physics of reality.
Yeah. So there's a lot of pitfalls. Um, so, you know, we, we spend a lot of time, you know, undergraduate math classes teaching analysis. Um, and analysis is often about how to take limits and, and, and, and whether, you know, so for example, A plus B is always B plus A. Um, so when, when you have a finite number of terms and you add them, you can swap them and there's, there's no problem. But when you have an infinite number of terms, there, these sort of show 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.
Um, you have to introduce these epsilons and deltas and, and, and there's, there's a certain type of way of reasoning that helps you avoid mistakes. Um, in more recent years, um, people have started taking results that are true in, in infinite limits and what's called, what's called finalizing them. Um, so you know that something's true eventually, but, um, you don't know when. Now give me a rate. Okay. So such that if I don't have an infinite number of monkeys, but, but a large finite number of monkeys, how long do I have to wait for Hamlet to come out?
Um, and, um, that's a more quantitative question. Um, and this is something that you can, you can, um, attack by purely finite methods and you can use your finite intuition. Um, and in this case, it turns out to be exponential in the length of the text that you're, you're trying to generate. Um, so if, um, and so this is why you'd never see the monkeys create Hamlet. You can maybe see them create a four letter word, but nothing that big. And so I personally find once you finalize an infinite statement, it's, it does become much more intuitive. Uh, and it's no longer so, so weird.
Um.
So even if you're working with infinity, it's good to finalize so that you can have some intuition.
Yeah. The downside is that the finalized proofs are just much, much messier.
Yeah.
And, and, uh, yeah. So, so the infinite ones are found first, usually like decades earlier. Uh, and then later on people finalize them.
So since we mentioned a lot of math and a lot of physics.
Mm-hmm.
Uh, what do you use the difference between mathematics and physics as disciplines, as ways of understanding, of seeing the world? Maybe we can throw in engineering in there. You mentioned your wife is an engineer. Give it new perspective on circuits.
Right.
So this different way of looking at the world, given that you've done mathematical physics, so you, you've, you've worn all the hats.
Right. So I think science in general is interaction between three things. Um, there's the real world. Um, there's what we observe of the real world, our observations, and then our mental models as to how we think the world works. Um, so, um, we can't directly access reality. Okay. Uh, all we have are the observations, which are incomplete and they, they have errors. Um, and, um, there are many, many cases where we would, um, uh, we want to know, for example, what is the weather like tomorrow?
And we don't yet have the observation that we'd like to a prediction. Um, and then we have these simplified models, sometimes making unrealistic assumptions, you know, spherical cow type things. Those are the mathematical models.
Mm-hmm.
Mathematics is concerned with the models. Science collects the observations and it proposes the models that might explain these observations. What mathematics does is, uh, we, we stay within the model and we ask what are the consequences of that model? What observations would, what predictions would the model make of the, of future observations? Um, or past observations, does it fit observed data? Um, so there's definitely a symbiosis. Um, it's math, I guess mathematics is, is unusual among other disciplines is that we start from hypotheses like the axioms of a model and ask what conclusions come out from that, that model.
Um, in almost any other discipline, uh, you start with the conclusions, you know, I want to do this. I want to build a bridge, you know, I, I want to, to make money. I want to do this. Okay. And then you, you, you find the path to get there. Um, a lot there's, there's a lot less sort of speculation about, suppose I did this, what would happen? Um, you know, planning and, and, and modeling. Um, uh, speculative fiction maybe is, is one other place. Uh, but, uh, that's about it actually. Most of the things we do in life is conclusions driven, including physics and science.
You know, I mean, they want to know, you know, where is this asteroid gonna go? You know, what, what's, what, what is the weather gonna be tomorrow? Um, but, um, mathematics also has this other direction of, of, of going from the, uh, the axioms.
What do you think? There is this tension in physics between theory and experiment.
Mm-hmm.
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. Um, yeah, it's just a, it's, it's a really interaction between all these things. So over time, the observations and the theory and the modeling should both get closer to reality. But initially, and this is, I mean, uh, this is, um, this is always the case, you know, they're, they're always far apart to begin with. Um, but you need one to figure out where, where to push the other, you know? So, um, if your model is predicting anomalies, um, that are not picked up by experiment, that tells experimenters where to look, you know, um, to, to, to, to, to find more data to refine the models.
Um, you know, so it, it, it goes, it goes back and forth. Um, within mathematics itself, there's, 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. Um, I mean, people do, do it, you know, like if they wanna study prime numbers or whatever, they can just generate large data sets and with a computer. So once we had computers, um, we began to do it a little bit. Um, although even before, well, like Gauss, for example, he discovered, he conjectured the most basic theorem in, in number theory, which is called the prime number theorem, which predicts how many primes there are up to a million, up to a trillion.
It's not an obvious question. And basically what he did was that he computed, uh, I mean, mostly, um, by himself, but also hired human computers. Um, people who, whose professional job it was to do arithmetic, um, to compute the first hundred thousand primes or something and made tables and made a prediction. Um, and that was an early example of experimental mathematics. Um, but until very recently, it was not, um, yeah, I mean, theoretical mathematics was just much more successful. I mean, because doing complicated mathematical computations is, uh, was just not, not feasible, uh, until very recently.
Uh, and even nowadays, you know, even though we have powerful computers, only some mathematical things can be, um, explored numerically. There's something called the combinatorial explosion. If you want to study, for example, Simmeredis theorem, you wanna study all possible subsets of the numbers one to a thousand, there's only 1,000 numbers. How bad could it be? It turns out the number of different subsets of, of one to a thousand is two to the power of 1,000, which is way bigger than, than, than any computer can currently can, can enumerate, any computer ever, whoever can, um, enumerate. Um, so you have to, you have to be, um, there are certain math problems that very quickly become just intractable to attack by direct brute force computation.
Uh, chess is another, um, famous example. The number of chess positions, uh, we can't get a computer to fully explore. But now we have AI, um, um, we have tools to explore this space, not with 100% guarantees of success, but with experiment, you know? So like, um, we can empirically solve chess now. Uh, for example, uh, we have, we have a, a very, very good AIs that, that can, you know, they don't explore every single position in, in the game tree, but they have found some very good approximation.
Um, and people are using actually these chess engines, uh, to make, uh, to do experimental chess. Um, that, uh, they, they're revisiting old chess theories about, oh, you know, when you this type of opening, you know, this is a good, this is a good type of move, this is not. And they can use these chess engines to actually, uh, refine, uh, in some cases overturn, um, um, conventional wisdom about chess. And I, I do hope that, uh, that mathematics will, will have a larger experimental component in the future, perhaps powered by AI.
We'll of course talk about that, but in the case of chess, and there's a similar thing in mathematics, that I don't believe it's providing a kind of formal explanation of the different positions.
No.
It's just saying which position is better or not, and you can intuit as a human being. And then from that, we humans can construct.
Yes.
A theory of the matter. You've mentioned the Plato's cave allegory.
Mm-hmm.
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 is, um, 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, our models. Um, and technically they are distinct, and I think they will always be distinct. Um.
Right.
But they can get closer, um, over time. Um, you know, so, um, and the process of getting closer often means that you, you have to discard your initial intuitions. Um, so, um, like astronomy provides great examples, you know, like, you know, like your, an initial model of the world is, is flat because it, it looks flat, you know, uh, and, um, and that it's, and it's big, you know, and the rest of the universe, the skies is not, you know, like the sun, for example, looks really tiny. Um, and so you start off with a model which is actually really far from reality.
Um, but it fits kind of the observations that you have. Um, you know, so, you know, so things look good, you know, but, but over time, as you make more and more observations, bringing it closer to, to, to reality, um, the model gets dragged along with it, you know, and so over time we had to realize that the earth is round, that it spins, it goes around this whole system, this whole system goes around the galaxy, and so on and so forth. And the guys about the universe, the universe expanding, um, the expansion is self-expanding, accelerating. And in fact, very recently in this year or so, there's, uh, even the acceleration of the universe itself is, uh, there's evidence now that it's non-constant.
And, uh, the explanation behind why that is.
It's catching up. Um.
It's catching up. I mean, it's still, you know, the dark matter, dark energy, this, this kind of thing.
Yes. We have, we have a model that sort of explains, that fits the data really well. It just has a few parameters that, um, uh, you have to specify. Um, but so, you know, people say, oh, that's fudge factors, you know, with, with enough fudge factors, you can, you can explain anything. Um, yeah, but, but, uh, the mathematical point of the model is that, um, 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, 10 observations, that is a completely useless model. It's what's called overfitted. But like if you have a model with, with, you know, two parameters and it explains a trillion observations, which is basically, uh, so, uh, yeah, the, the, the dark matter model, I think it has like 14 parameters and it explains petabytes of data, um, that, that, that, that the astronomers have.
Um, you can think of, of a theory, uh, like one way to think about, um, uh, a physical mathematical theory, a theory is, it's, it's a compression of, of the universe, um, and a, a 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, you know, 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.
Yeah.
It's the unreasonable effectiveness of mathematics.
Yeah. Einstein had a quote like that. The, the most incomprehensible thing about the universe is that it is comprehensible.
Right. And not just comprehensible, you can do an equation like E equals mc2.
There is actually some mathematical possible explanation for that. Um, so 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 macro scale. And normally, because of the combinatory explosion, you would think that, uh, the macro scale equations must be like infinitely exponentially more complicated than, than the, uh, the micro scale ones. And they are, if you want to solve them completely exactly, like if you want to model, um, all the atoms in a box of, of air, uh, that's like Avogadro's number is humongous, right?
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 macro scale or only depend on a very small number of parameters. So if you wanna model a gas, um, of, you know, quintillion particles in a box, you just need to know its temperature and pressure and volume and a few parameters like five or six, and it models almost everything you need to know about these 10 to 23 or whatever particles. Um, so we, we have, um, we, we don't understand universality any way as we would like mathematically, but there are much simpler toy models where we do, um, have a good understanding of why universality occurs.
Um, uh, most basic one is, is the central limit theorem that explains why the bell curve shows up everywhere in nature, that so many things are distributed by what's called a Gaussian distribution, a famous bell curve. There's now even a meme with this curve.
And even the meme applies broadly.
Yeah.
The universality to the meme.
Yeah. Yes, you can call meta, uh, if you like. But there are many, many processes. For example, you, you can take lots of independent, um, random variables and average them together, um, uh, in, in various ways. You can take a simple average or more complicated average. And we can prove in various cases that, that these, these bell curves, these Gaussians emerge, and it is a satisfying, satisfying explanation. Um, sometimes they don't. Um, so, so 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 show up. Uh, and this is also important to know when the central fails.
So universality is not a 100% reliable thing to rely on. That, um, um, that the global financial crisis was a, a famous example of this. Uh, people thought that, uh, um, mortgage defaults, um, uh, had this sort of, um, Gaussian type behavior that, that if you, if you ask if a population of, of, of, uh, you know, a hundred thousand Americans with mortgages, ask what, what proportion of them will default on their mortgages, um, if everything was decorrelated, it'll be a nice bell curve and, and like you, you can, you can manage risk with options and derivatives and so forth.
And, um, and it is a very beautiful theory. Um, but if there are systemic shocks in the economy, uh, that can push everybody to default at the same time, uh, that's very non-Gaussian behavior. Um, and, uh, this wasn't fully accounted for in, uh, 2008. Um, now I think there's some more awareness that this is, uh, systemic risk is actually a much bigger issue. And, uh, just because the model is pretty, uh, and nice, uh, it may not match reality. Uh, so, so the mathematics of working out what models do is really important.
Um, but, um, also the, the science of validating when the models, uh, fit reality and when they don't. Um, I mean, that you need both. Um, and, but mathematics can help because it, it can, for example, the central limit theorems, it told you that, that if you have certain axioms like, like, like, uh, non-correlation, that if all the inputs were not correlated to each other, um, then you have this Gaussian behavior that things are fine. It, 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 these Gaussian copulas, or whatever, to, to model, um, default risk, um, if you're mathematically, um, trained, you would say, okay, but what are the systemic correlation between all your inputs?
And so then that, then you can ask the economists, you know, how, how, how much of a risk is that? Um, and then you can, you can, you can go look for that. So there's always this, this, this synergy between science and, and mathematics.
A little bit on the topic of universality.
Mm-hmm.
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.
Ha.
Uh, he's a colleague of yours.
Oh yeah. Good friend.
But anyway, so you, 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, areas of mathematics? Is there a kind of deep underlying structure, uh, to all of mathematics?
There's certainly a lot of connecting threads. Um, 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. Um, an ancient example is, um, geometry and number theory, you know? So, so in the times of the ancient Greeks, these were considered different subjects. Um, I mean, mathematicians worked on both, you know, Euclid, uh, worked both on, on geometry most famously, but also on numbers. Um, but they were not really considered related.
Um, I mean, a little bit like, you know, you, you could say that, that, 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, uh, who developed what we now call analytic geometry, that you can, you can parameterize the plane, a geometric object by, um, by two real numbers, right? Every point can be, and so geometric problems can be turned into, into problems about numbers. Um, and the, the today this feels almost trivial, like there's, there's, there's no content to this.
Like, of course, uh, you, you know, um, a plane is X, X and Y, and because that's what we teach and it's internalized. Um, but it was an important development that these, these two fields are, are, were unified. Um, and this process has, has just gone on throughout mathematics over and over again. Algebra and geometry were separated, and now we have this field algebraic geometry that connects them and, and over and over again. And that's certainly the type of mathematics that, that I enjoy the most. So I think there's sort of different styles to being a mathematician. I think hedgehogs and fox, a fox knows many things a little bit, but a hedgehog knows one thing very, very well.
Um, and in mathematics, there's definitely both hedgehogs and foxes. Um, and then there's people who are kind of, uh, who can play both roles. Um, and I think ideal collaboration between mathematicians involves very, you need some diversity, like, um, a fox working with many hedgehogs or, or vice versa. So yeah, but, but I identify mostly as a fox. Uh, certainly I, I, I like, uh, arbitrage somehow, you know, like, like, um, 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, I can adapt the tricks.
So see the connections between the fields.
Yeah. Yeah. So there are other mathematicians who are far deeper than I am. Like they're really, they're really hedgehogs. They, they, they know everything about one field and they're much faster and, and, and more effective in that field. But I can, I can give them these extra tools.
I mean, you said that you can be both the hedgehog and, and the fox depending on the context.
Yeah.
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, you know, searching for the connections versus like very singular focus.
I'm much more comfortable with, with the, uh, the, uh, the fox paradigm. Yeah. So, um, yeah, I, I like looking for analogies, narratives. Um, I, I spend a lot of time, if there's a result, I see it in one field and I like the result. It's a cool result, but I don't like the proof. Like it uses types of mathematics that I'm not super familiar with. Um, I often try to reprove it myself using the tools that I favor. Um, often my proof is worse. Um, but, um, by the exercise of doing so, um, I can say, oh, now I can see what the other proof was trying to do.
Um, and from that, I can get some understanding of, of the tools that are used in, in, in that field. So it's very exploratory, very doing crazy things in crazy fields and, and like reinventing the wheel a lot. Yeah. Whereas sort of the hedgehog style is, uh, I think much more scholarly. You know, you, you, you're very knowledge based. You, you, you, you stay up to speed on like all the developments in this field. You, you know all the history. Um, you have a very good understanding of, of exactly the strengths and weaknesses of, of each particular, uh, technique. Um, yeah, uh, I think you, you'd rely a lot more on sort of calculation than sort of trying to find narratives.
Um, so yeah, I mean, I can do that too, but, uh, there are other people who are extremely good at that.
Let's step back and, uh, uh, maybe look at the, the, a bit of a romanticized version of mathematics.
Mm-hmm.
So, uh, I think you've said that early on in your life, uh, math was more like a puzzle solving activity when you were.
Mm-hmm.
Uh, 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. Um, when I came to graduate school, uh, in Princeton, um, so John Conway was there at the time. He passed away a few years ago, but, uh, I remember one of the very first research talks I, I went to was a talk by Conway on what he called extreme proof. So Conway had just had this, this amazing way of, of thinking about all kinds of things in, in a way that you wouldn't normally think of. So, um, he thought of proofs themselves as occupying some sort of space, you know? So, so, um, if you want to prove something, let's say that there's infinitely many primes, okay? You have all your different proofs, but you could, you could rank them in different axes.
Like some proofs are elegant, some proofs are long, some proofs are, are, um, are elementary and so forth. Um, and so there's this cloud, so the space of all proofs itself has some sort of shape. Um, and so he was interested in, in extreme points of this shape. Like out of all, all these proofs, what is the one that is the shortest at the ex, at the expense of everything, everything else or, or the most elementary or, or whatever. Um, and so he gave some examples of well-known theorems and then he would give what he thought was, was the extreme proof, um, in these different aspects.
Um, and I, I just found that really eye-opening, um, that, that, um, you know, it's, it's, it's not just getting a proof for was, was interesting, but, but once you have that proof, you know, trying to, to, uh, to optimize it in various ways, um, that, that proof, um, uh, proofing itself had some craftsmanship to it. Um, it, it certainly informed my writing style, um, that, you know, like when you do your, your math assignments and as an undergraduate, your homework and so forth, you, you're sort of encouraged to just write down any proof that works, okay?
And hand it in and get a, get as long as it gets a tick mark, you, you move on. Um, but if you want your, your, your results to actually be influential and be read by people, um, it can't just be correct. It should also, um, be a pleasure to read, you know, um, motivated, um, be adaptable to, to generalize to other, um, things. Um, it's, it's the same in, in many other disciplines like, like coding. There's a, there's a, there's a lot of analogies between math and coding. I like analogies if you haven't noticed. Um, but, um, you know, like you can code something spaghetti code that works for a certain task and it's quick and, and dirty and it works.
But, uh, there's lots of good principles for, for, um, writing, uh, code well so that other people can use it, build upon it and so on and has fewer bugs and whatever. Um, and there's similar things with mathematics. So.
Yeah. The, first of all, there's so many beautiful things there and, and Conway is one of the great minds, uh, in mathematics ever and computer science, uh, just even considering the space of proofs.
Yeah.
And saying, okay, what does this space look like and what are the extremes? Uh, like you mentioned, coding as an analogy is interesting because there's also this activity called the code golf.
Oh yeah. Yeah. Yeah.
Which I also find beautiful and fun, uh, where people use different programming languages to try to write the shortest possible program that accomplishes a particular task.
Yeah.
And I believe there's even competitions on this.
Yeah. Yeah. Yeah. Yeah.
And, uh, it's also a nice way to stress test not just the sort of the programs or in this case the proofs, but also the different languages. Maybe that's a different notation or whatever to use to, to accomplish a different task.
Yeah. Yeah. You learn a lot. I mean, it, it may seem like a frivolous exercise, but it, it can generate all these insights, which if you didn't have this artificial, um, objective to, to, to pursue, you, you might not see.
What do you use the most beautiful or elegant equation in mathematics? I mean, one of the things that people often look to in, in beauty is the simplicity. So if you look at E equals MC squared.
Mm-hmm.
So when, when a few concepts come together, that's why the Euler identity is often considered, uh, the most beautiful equation in mathematics.
Mm-hmm. Mm-hmm.
Do you, do you find beauty in that one in the Euler identity?
Yeah. Well, as I said, I mean, what I find most appealing is, is connections between different things that you, um, so the, if you, uh, E to the pi I equals minus one. Um, so yeah, people, oh, he uses all the formula constants. Okay. That, that, that's, I mean, that's cute. Um, but, but to me, so the exponential function. Was interested by Euler to measure exponential growth, you know, something compound interest or decay or anything which is continuously growing, continuously decreasing growth and decay or dilation or contraction is modeled by the exponential function. Um, whereas pi, uh, comes around from circles and rotation, right?
If you want to rotate a needle, for example, 180 degrees, uh, you need to rotate by pi radians. And I, complex numbers, represents this swapping between one imaginary axis of a 90-degree rotation. So a change in direction. So the exponential function represents growth and decay in the direction that you already are. Um, when you stick an eye in the exponential, it, it, it now, it's, it's instead of motion in the same direction as your current position, it's motion as a right angle as to your current position. So rotation. Um, and then so E to the pi I equals minus one tells you that if you rotate for a time pi, you end up at the other direction.
So it unifies geometry through dilation and exponential growth or dynamics through this act of, of complexification, rotation by, by, by I. So it, it connects together all these fields of mathematics. Yeah. Yeah. Dynamics, geometry and complex and complex and, um, the complex numbers, they're all considered almost, yeah, they're all next door neighbors in mathematics because of this identity.
Do, do you think the thing you mentioned is cute, the, the, the, the collision of notations from these disparate fields, um, is just a frivolous side effect? Or do you think there is legitimate like value in when the notation, all the our old friends come together in the night?
Right. Well, it's, it's, it's confirmation that you have the right concepts. Um, so when you first study anything, um, you, you have to measure things and give them names. Um, and initially sometimes you're, because your, 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, what's really important.
Physicists can do this sometimes. I mean, but it turns out, okay.
So actually with physics, so E equals MC squared. Okay. So, uh, one of the, the big things was the E, right? So when, when Aristotle first came up with his laws of, of motion and then, and then, um, Galileo and Newton and so forth, you know, they saw the things they could, they could measure. They could measure mass and acceleration and force and so forth. And so Newtonian mechanics, for example, F equals ma was the famous Newton's second law of motion. So those were the, 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.
Um, so in particular momentum and energy. Um, uh, and it's not obvious that things happen energy. Like it's not something you can directly measure the same way you can measure mass and, and, 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, the dynamics, like what, what happens to, to, to all the states.
Like it's, um, it, it really was a central actor, which was not obvious initially. Um, and this, uh, helped actually, uh, this change of perspective really helped when quantum mechanics came along. Uh, because, um, the early physicists who studied quantum mechanics, they had a lot of trouble trying to adapt their Newtonian thinking because, you know, everything was a particle and so forth to, to, to quantum mechanics, you know, because now everything because it was a wave, it just looked really, really weird. Um, like you ask what is the quantum equals ma?
And it's really, really hard to, to give an answer to that. Um, but it turns out that the Hamiltonian, which was so, um, secretly behind the scenes in classical mechanics, also is the key, uh, object in, um, um, in quantum mechanics that there's, there's also an object called Hamiltonian. It's a different type of object. It's what's called an operator rather than than a function. But, um, and, um, but again, once you specify it, you specify the entire dynamics. So there's something called Schrödinger's equation that tells you exactly how quantum systems evolve once you have the Hamiltonian. So side by side, they look completely different objects, you know, like one involves particles, one involves waves and so forth.
But with this centrality, you could start actually transferring a lot of intuition and facts from classical mechanics to quantum mechanics. So for example, in, in classical mechanics, there's this thing called Newton's theorem. Every time there's a symmetry in a physical system, there was a conservation law. So the laws of physics are translation invariant. Like if I move 10 steps to the left, I experience the same laws of physics as if I was here. And that corresponds to conservation momentum. Um, if I turn around by, by some angle, again, I experience the same laws of physics. Uh, this corresponds to conservation of angular momentum.
If I wait for 10 minutes, um, I still have the same laws, laws of physics. Um, so there's time translation invariance. This corresponds to the law of conservation of energy. Um, so there's this fundamental connection between symmetry and conservation. Um, and that's also true in quantum mechanics, even though the equations are completely different, but because they're both coming from the Hamiltonian, the Hamiltonian controls everything. Um, every time the Hamiltonian has a symmetry, the equations will, will have a conservation law. Um, so it's, it's, it's, it's, it's once you have the right language, it actually makes things, um, uh, a lot, a lot cleaner.
One of the problems why we can't unify quantum mechanics and general relativity yet, we haven't figured out what the fundamental objects are. Like, for example, we have to give up the notion of space and time being these almost Euclidean type spaces. And it has to be, um, you know, and, you know, we kind of know that at very tiny scales, uh, um, there's going to be quantum fluctuations. There's a space, space-time foam. Um, and trying to, to use Cartesian coordinates X, Y, Z is going to be, it's, it's just, it's, it's a nonstarter. But we don't know how to, what to replace it with. Um, we don't actually have the mathematical, um, um, concepts, the analog of the 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 that of unification, much like mathematics, um, over the years, you know, electricity and magnetism were separate theories and then Maxwell unified them. You know, Newton unified the, the motions of the heavens with the motions of, of objects on the earth and so forth. So it should happen. Uh, it's just that the, um, uh, again, to go back to this model of the, of the observations and, and, and theory, part of our problem is that physics is a victim of its own success, that our two big theories of, of, of physics, general relativity and quantum mechanics are so, are so good now is that together they cover 99.9% of sort of all the observations we can make.
Um, and you have to like either go to extremely insane particle accelerations or, or the early universe or, or, or things that are really hard to measure, um, in order to get any deviation from either of these two theories to the point where you can actually figure out how to, how to combine them together. Um, but I have faith that we, you know, we've, 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, uh, theory of everything?
What often happens is that when the physicists need, uh, um, some theory of mathematics, there's often some precursor that the mathematicians, um, worked out earlier. So when Einstein started realizing that space was curved, he went to some mathematician and asked, you know, is there, is there some theory of curved space that the mathematicians already came up with that could be useful? And he said, oh, yeah, there's, uh, I think, uh, Riemann came up with something. Um, and so, yeah, Riemann had developed Riemannian geometry, um, which is precisely, um, you know, a, a, a theory of spaces that are curved in, in various general ways, which turned out to be almost exactly what was needed, um, for Einstein's theory.
This is going back to doing this 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, um, in, in, in, in useful ways.
It just feels like you might need to go some weird land that's very hard to, to intuit. Like, you know, you have like string theory.
Yeah. That, that's, that was, that was a leading candidate for many decades. It's, I think it's slowly falling out of fashion 'cause it's, it's not matching experiment. Uh.
So one of the big challenges, of course, like you said, is experiment is very tough.
Yes.
Because of the how effective.
Yeah.
Both theories are. But the other is like just, you know, you're talking about, you're not just deviating from space-time. You're going into like some crazy number of dimensions.
Yeah.
You're doing all kinds of weird stuff that to us, we've gone so far from this flat earth that we started at, like you mentioned.
Yeah. Yeah, yeah, yeah.
Now we're just, it's, it's very hard to use our limited a descendants of a, a cognition to intuit what that reality really is like.
This is why analogies are so important, you know? I mean, so, yeah, the round earth is not intuitive 'cause we're, we're stuck on it. Um, but, you know, but you, you, you know, but round objects in general, we have pretty good intuition over, uh, and we have intuition about lightworks and so forth. And like it's, it's actually a good exercise to actually work out how eclipses and phases of, of the sun and the moon and so forth can be really easily explained by, by, by, by round earth and round moon, you know, um, and models. Um, and, and you can just take, you know, a basketball and a golf ball and, and a, and a light source and actually do these things yourself.
Um, so the intuition is there. Um, but you have to transfer it.
That is a big leap intellectually for us to go from flat to round earth.
Mm-hmm.
Because, you know, our life is mostly lived in flat land.
Yeah.
To load that information. And we're all like 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 r-round rock.
Yeah.
Flying through space.
Yeah. Yeah.
That's a big leap. And you have to take a chain of those leaps the more and more and more we progress.
Right. Yeah. So modern science is maybe again a victim of its own success is that, you know, in order to be more accurate, it has to, to move further and further away from your initial intuition. And so, um, for someone who hasn't gone through the whole process of science education, it looks more and more suspicious.
Yeah.
Because of that. So, you know, we, we, we need, we need more grounding. I mean, I, I think, um, I mean, you know, there are, there are scientists who do excellent outreach. Um, but there's, there's, there's, there's, there's lots of science things that you can do at home. I, there's lots of YouTube videos. I did a YouTube video recently of Guns Anderson. We talked about this earlier that, uh, you know, how the ancient Greeks were able to measure things like the distance to the moon, distance to the earth, and, you know, using techniques that you, you could also replicate yourself. Um, it doesn't all have to be like fancy space telescopes and, and very intimidating mathematics.
Yeah, that's, uh, I highly recommend that. I believe you gave a lecture and you also did an incredible video with Grant. It's a beautiful experience to try to put yourself in the mind of a person from that time.
Mm-hmm.
Shrouded in mystery.
Right.
You know, you're like on this planet, you don't know the shape of it, the size of it. You see some stars, you see some, you see some things and you try to like localize yourself in this world.
Yeah, yeah.
And try to make some kind of general statements about distance to places.
Change of perspective is really important. You say travel borders the mind. This is intellectual travel, you know, put yourself in the mind of the ancient Greeks or, or some other person, some other time period, make hypotheses, spherical cows, whatever, you know, speculate. Um, and, you know, this is, this is what mathematicians do and some sort of 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 with.
If you propose axioms, then the mathematics lets you follow those axioms to, to their conclusions. And sometimes you can get quite a, quite a long way from, you know, initial hypotheses.
If we can stay in the land of the weird, you mentioned general relativity. You've, uh, you've contributed, uh, to the mathematical understanding of Einstein's field equations. Can you explain this work and, uh, from a sort of mathematical standpoint, uh, what aspects of general relativity are intriguing to you, challenging to you?
I have worked on some equations. There's something called the, 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. Um, so Einstein's equations of relativity just describes space and time itself. Um, but then there's other fields that live on top of that. Uh, there's the electromagnetic field. Um, there's things called Young-Mills fields. And there's this whole hierarchy of different equations, of which Einstein's considered one of the most nonlinear and difficult, but relatively low on the hierarchy was this thing called the wave maps equation.
So it's a wave which at any given point, uh, is fixed to be like on a sphere. Um, so, uh, I can think of a bunch of arrows in space and time and, and, and, and, yeah, it's pointing in, in different directions. Um, but they propagate like waves. If, if, if you wiggle an arrow, it'll propagate and create and make all the arrows move kind of like sheaves of wheat in the wheat field. And I was interested in the global regularity problem again for this question. Like, is it possible for, for all the energy here to, to, to collect at a point? So the equation I considered was actually what's called a critical equation, where it's actually the behavior at all scales is roughly the same.
Um, and I was able barely to show that, um, that you couldn't actually force a scenario where all the energy concentrated at one point, that the energy had to disperse a little bit, and the moment it dispersed a little bit, it would, it would, it would stay regular. Yeah, this was back in 2000. That was part of why I got interested in Navier-Stokes afterwards, actually. Yeah. So I developed some techniques to, um, to solve that problem. So part of it is it was, um, this problem is really nonlinear, uh, because of the curvature of the sphere. Um, there's, there was a certain nonlinear effect, which was a non-perturbative effect.
It was when you sort of looked at it normally, it, it looked larger than the linear effects of the wave equation. Um, and so it was hard to, to keep things under control, even when your energy was small. But I developed what's called a Gauss transformation. So the equation is kind of like an evolution of, of, of sheaves of wheat, and, and they're all bending back and forth. And so there's a lot of motion. Um, but like if you imagine like 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, and under this sort of stabilized flow, the flow becomes a lot more linear.
I discovered a way to transform the, the equation to reduce the amount of, of nonlinear effects. Um, and then I was able to, to, to, 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, I couldn't do it with pen and paper. Um, and I had not enough facility of 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, to see how to change coordinates in such a way that somehow things in all directions would behave in a reasonably linear fashion.
And, uh, yeah, my aunt walked in on me while I was doing that, and she was asking, what am I, what am I doing, doing this?
It's complicated is the answer.
Yeah. Yeah. And, you know, she said, okay, fine. You know, you're a young man. I don't ask questions.
I, I, I have to ask about the, you know, um, how do you approach solving difficult problems? 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?
Um, a lot of pen and paper. One thing you pick up as a mathematician is sort of, uh, a collective cheating strategically. Um, so, uh, the, the beauty of mathematics is that, is that you get to change the rule, change the problem, change the rules as you wish. Uh, like this, you don't get to do this for any other field. Like, you know, if, if you're an engineer and someone says build a bridge over this, this river, you kind of say, I wanna build this bridge over here instead, or I wanna build it out of paper instead of steel. Um, but a mathematician, you can, you can do whatever you want. Um, it's, it's like trying to solve a computer game where you can, there's unlimited cheat codes available.
Uh, and so, you know, you, you can, 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. Or there's a main term, an error term. I'm gonna make a spherical cow assumption. I'll assume the error term is zero. And so the way you should solve these problems is, is not in sort of this Iron Man mode where you make things maximally difficult. Um, but actually the way you should, you should approach any reasonable math problem is that you, if, if there are 10 things that are making your life difficult, find a version of the problem that turns off 9 of the difficulties, but only keeps one of them.
Um, and solve that. Um, and then that just, so you, you, you, you install nine cheats. Okay. If you install 10 cheats, then, then the game is trivial. But you install nine cheats, you solve one problem. That, that, that, that teaches you how, how to deal with that particular difficulty. And then you turn that one off and you turn someone else, something else on, and then you solve that one. And after you, you know how to solve the 10 problems, 10 difficulties separately, then you have to start merging them a few at a time. Um, I, I, as a kid, I watched a lot of these Hong Kong action movies. Um, from a culture.
Um, and, uh, one thing is that every time there's a fight scene, you know, so maybe the, the hero gets swarmed by a hundred bad guy goons or whatever, but it'll always be choreographed so that you'd always be only fighting one person at a time, and then you would defeat that person and move on. And, and because of that, he could, he could defeat all of them, right? But whereas if they had fought a bit more intelligently, yeah, and just swarmed the guy at once, uh, it would make for much, uh, much worse choreo, um, cinema, but, uh, but they would win.
Are you usually, uh, pen and paper? Are you working, uh, with computer and LaTeX?
I'm mostly pen and paper, actually. So in my office, I have four giant blackboards. Um, and 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 is there some drawings?
Oh, there's a lot of drawing and a lot of bespoke doodles that, that only make sense to me. Um, I mean, and, and this beautiful blackboard is erased. And it's, it's a very organic thing. Um, I'm beginning to use more and more computers, um, 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, has some iteration or something, you know, I'd had to, to remember how to set up a Python program and, and, and, and, and how does a for loop work and, and, and debug it. And it would take two hours and so forth.
And, and now I can do it in 10, 15 minutes as much. Um, yeah, I'm, I'm using more and more, uh, computers to do simple explorations.
Let's talk about AI a little bit if we could. So, um, 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, uh, it has helped you?
So, um, lean is a computer language, um, much like sort of 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, you know, they, they flip bits or, or they make a robot move or, or they, they deliver you text on the internet or something. Um, so lean is a language that can also do that. Uh, it can also be run as a standard, uh, traditional language, but it can also produce certificates. So a software language like, like Python might do a computation and give you that the answer is seven.
Okay. That does the sum of three plus four is equal to seven. But, uh, lean can produce not just the answer, but, but a proof that, uh, how it got the, the answer of seven as three plus four and all the steps involved in, in, in, in, um, so it's a, it creates these more complicated objects, not just statements, but statements with proofs attached to them. Um, and, um, every line of code is just a way of piecing together previous statements to, to create new ones. So the idea is not new. These things are called proof assistants. And so they provide languages for which you, you can create quite complicated, um, intricate mathematical proofs.
And, um, they produce these certificates that give a 100%, um, guarantee that your arguments are correct if you trust the compiler of lean. But they made the compiler really small and you can, there are several different compilers available for the same language.
Can you give people some intuition about the, the difference between writing on pen and paper versus using lean programming language? How hard is it to formalize a statement?
Right. So lean, a lot of mathematicians were involved in the design of lean. So it's, it's designed so that, um, individual lines of code resemble individual lines in a mathematical argument. Like you might want to introduce a variable, you want to, want to prove that contradiction, you, you know, um, um, there are various standard things that you can do and, and it's, it's written so that ideally it's like a one-to-one correspondence. In practice, it isn't because lean is like explaining a proof to an extremely pedantic colleague who will, will point out, okay, did you really mean this? Like what, what happens if this is zero?
Okay. Um, did you, how do you justify this? Um, so lean has a lot of automation in it, um, to try to, to, uh, to be less annoying. Um, so for example, um, every mathematical object has to come with a type. Like if I, if I talk about X, is X a real number or, um, a, a natural number or, or a function or something? Um, if you write things informally, um, it's often in a certain context. You say, you know, um, clearly X is equal to, uh, 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.
Um, so lean can do a lot of that. Um, but every so often it, it says, wait a minute, uh, can you tell me more about what this object is? Uh, what, what type of object it is? You have to think more, um, at a philosophical level about not just sort of the computations that you're doing, but sort of what each object actually, um, is in some sense.
Is it using something like LLMs to do, uh, the type inference or like you mentioned with the real number?
It's, 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 algorithm to match one tree to another tree.
So it's actually doable to figure out if something is a, a real number or a natural number.
Yeah. Every object sort of comes with a history of where it came from and you can, you can kind of trace.
Oh, I see.
Um, yeah. So it's, it's, it's designed for reliability. So, uh, modern AIs are not used in, it's a disjoint technology. People are beginning to use AIs on top of lean. So when a mathematician tries to program, um, a proof in lean, um, often there's a step. Okay. Now I want to use, um, the fundamental thing of calculus, say, okay, to do the next step. So the lean developers have built this, this massive project called MethLib, a collection of tens of thousands of useful facts about mathematical objects. And somewhere in there is the fundamental thing of calculus, but you need to find it.
So a lot of the bottleneck now is actually Llama search. You know, there's a tool that, that you know is in there somewhere and you need to find it. Um, and so you can, there are various search engines specialized for MethLib that you can do. Um, but there's now these large language models that you can say, um, I need the fundamental thing of calculus at this point. And it was like, okay, uh, um, uh, for example, um, when I code, I have GitHub Copilot installed as a plugin to my IDE and it scans my text and it sees what I need. It says, you know, I might even type, okay, now I need to use the fundamental thing of calculus.
Okay. And then it might suggest, okay, try this. And like maybe 25% of the time it works exactly. And then another 10, 15% of the time it doesn't quite work, but it, it's close enough that I can say, oh, yeah, if I just change it here and here, it'll work. And then like half the time it gives me complete rubbish. Um, so, but people are beginning to use AIs a little bit on top, um, mostly on the level of basically fancy autocomplete. Um, that, uh, you can type half of one line of a proof and it will find, it will tell you.
Yeah. But, but a fancy, especially fancy with the sort of capital letter F is, uh, uh, remove some of the friction.
Yeah.
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, to write it out. Yeah. So it's doable, but, uh, you don't, it's, 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, if that was the only aspect of it. Okay. But, um, okay. There, there's some, there's some cases where it's actually more pleasant to do things formally. So there was a, there was a theorem I formalized and there was a certain constant 12, um, that, that came out at, um, in, in the final statement. And so this 12 had to be carried all through the proof. Um, and like everything had to be checked that it goes, all the, all these other numbers that had to be consistent with this final number 12. And then 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.
And when this happens with pen and paper, um, 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 on number 12 that you didn't even realize that you were taking advantage of. So a proof can break down at a subtle place. Um, so we had formalized the proof with this constant 12. And then when this, this new paper came out, uh, we said, oh, let's, so that took like three weeks to formalize and, and like 20 people to formalize this, this, this original proof. I said, oh, but now, now let's, let's, um, uh, uh, let's update the 12 to 11.
And what you can do with lean is that you just, in your headline theorem, you, you change the 12 to an 11, you run the compiler and like of the thousands of lines of code you have, 90% of them still work. And there's a couple that are lined in red. Now I can't justify these, these steps, but it, it immediately isolates which steps you need to change, but you can skip over everything which, which works just fine. Um, and if you program things correctly, um, with sort of good programming practices, most of your lines will not be red. Um, and there'll just be a few places where you, I mean, if, if you don't hard code your constants, but you sort of, uh, um, um, you use smart tactics and so forth, yeah, you can, you can localize, um, the things you need to change to, to a very small, um, period of time.
So it's like within a day or two, we had updated our proof because this is a very quick process here. Um, you make a change, there are 10 things now that don't work. For each one, you make a change and now there's five more things that don't work, but, 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, how, what's, what's the, uh, versus paper and.
Yeah. So the proofs are longer, but each individual piece is easier to read. So, um, if you take a math paper and you jump to page 27 and you look at paragraph six and you have a line of, of, of text of math, I often can't read it immediately because it assumes various definitions, which I have to, to go back and, and, and maybe on 10 pages earlier this was defined and this, um, the proof is scattered all over the place and you basically are forced to read fairly sequentially. Um, it's, it's not like, say, a novel where like, you know, in theory you could, you know, open up a novel halfway through and start reading.
There's a lot of context. But when a proof in lean, if you put your cursor on a line of code, every single object there, you can hover over it and it would, it would say what it is, where it came from, where the step 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, um, when you want to collaborate with another mathematician, um, either you do it as a blackboard where you, um, you can really interact, but if you're doing it sort of by email or something, um, basically, yeah, you have to segment it.
Say I'm going to, I'm going to finish section three, you do section four, but, uh, you can't really sort of work on the same thing, uh, collaboratively at the same time. But with lean, you can be trying to formalize some portion of the proof and say, uh, I got stuck at line 67 here. I need to prove this thing, but it doesn't quite work. Here's the, like the three lines of code I'm having trouble with. Um, but because all the context is there, someone else can say, oh, okay, I recognize what you need to do. You need to, to, to apply this trick or this tool. And you can do extremely atomic level conversations. So because of lean, I can collaborate, you know, with dozens of people across the world, most of whom I don't have never met in person.
Um, and I may not know actually even whether they're, um, how reliable they are in, in, in their, um, um, in, in the proof they give me, but lean gives me a certificate of, of, of trust. Um, so I can do, I can do trustless mathematics.
So there's so many interesting questions there. So one, you're, 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 brain, are you focused on a particular part and you're brainstorming?
There's always a brainstorming process first. Yeah. So math research projects sort of by their nature, when you start, you don't really know how to do the problem. Um, it's not like an engineering project where somehow the theory has been established for decades and it's, it's implementation is the main difficulty. You have to figure out even what is the right path. Uh, so, so this is what I said about, about cheating first, you know, um, it's like, um, to go back to the bridge building analogy, you know, so first assume you have infinite budget and, and like unlimited amounts of, of, of workforce and so forth. Now can you, can you build this bridge?
Okay. Okay. Now have infinite budget, but only finite workforce. Right. Now can you do that and so forth? Um, so, uh, I mean, of course, you know, no, no engineer can actually do this. Like I say, they have fixed requirements. Yeah. So there's this sort of jam sessions always at the beginning where you try all kinds of crazy things and you, you, you make all these assumptions that are unrealistic, but you plan to fix later. Um, and you try to see if there's even some skeleton of an approach that might work. Um, and then hopefully that breaks up the problem into smaller sub-problems, which you don't know how to do, but then you, uh, you focus on, on, on the sub ones.
Uh, and sometimes different collaborators are better at, at working on, on certain things. Um, so one of my theorems I'm known for is a theorem of Ben Green, which is now called the Green Towel Theorem. Um, it's a statement that the primes contain arithmetic progressions of any length. So it was a modification of this theorem of Semeretti. And the way we collaborated was that Ben had already proven a similar result for progressions of length three. Um, he showed that just like the primes contain loss-minus progressions of length three, um, even, and even, um, subsets of the primes, certain subsets do.
Um, but his techniques only worked for, um, for length three progressions. They didn't work for longer progressions. Um, but I had these techniques coming from ergodic theory, which is something that I had been playing with and, and, uh, I knew better than Ben at the time. Um, and so, um, if I could justify certain randomness properties of some set relating to the primes, like there's, there's a certain technical condition which if I could have it, if, if Ben could supply me this fact, I could give, I could conclude the theorem. But I, what I asked was a really difficult question in number theory, which, um, he said, no, there's no way we can prove this.
Can you, so he said, can you prove your part of the theorem using a weaker hypothesis that I have a chance to prove it? And he proposed something which he could prove, but it was too weak for me. Uh, I can't use this. Um, so there's this, there was this conversation going back and forth. Um.
Different cheats too.
Yeah, yeah. I want to cheat more and he wants to cheat less. But eventually we found a, a, a, a, a, a property which A, he could prove and B, I could use. Um, and then we, we could prove our theorem. And, um, yeah, so there's, there's a, there are all kinds of dynamics, you know, I mean, it's, it's, it's every, every, um, collaboration has a, has a, has some story. No two are the same.
And then on, on the flip side of that, like you mentioned with lean programming, now that's almost like a different story because you can do, you can create, I think you've mentioned a kind of a blueprint for a problem and 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.
Yeah. So it makes everything compatible and, uh, yeah, and, and trustable. Um, yeah. So 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 blue, a blueprint in a sense. It is taking a difficult statement, like a big theorem and breaking it up into a hundred little lemma's. Um, 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 as, as much detail as, as, as, as possible and to try to make each step kind of self-contained, um, and, or depending on only a very specific number of, of previous statements that have been proven.
So that each node of this blueprint graph that gets generated can be tackled independently of, of the others. And you don't even need to know how the whole thing works. Um, so it's like a modern supply chain, you know, like if you want to create an iPhone or, or some other complicated object, um, no one person can, can build up, um, a single object, but you can have specialists who, who just, if they're given some widgets from some other company, they can combine them together to form a slightly bigger widget.
I think that's a really exciting possibility 'cause you can have, if you can find problems that could be broken down in this way, then you can have, you know, thousands of contributors, right?
Yes.
That become 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, and other software tools, uh, so, um, GitHub and things like that, um, allow, they will allow experimental mathematics to be, to scale up, um, to a much greater degree than we can do now. So right now, if you want to, um, um, do any mathematical exploration, uh, of some mathematical pattern or something, you need some code to write out the pattern. And I mean, sometimes there are some computer algebra packages that can help, but often it's just one mathematician coding lots and lots of Python or whatever.
And 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. Um, so it's, these are, uh, so you get these bespoke, uh, spaghetti code that written by non, not professional programmers, but mathematicians, you know, and they're clunky and, and, and slow. And, um, and so because of that, it's, it's, it's hard to, to really mass produce experimental results. Um, but, um, yeah, but 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.
So I have this project called the Equational Theories Project. Basically, we generated about 22 million little problems in abstract algebra. Maybe I should back up and, and tell you what, what the project is. Okay. So abstract algebra studies operations like multiplication and addition and their abstract properties. Okay. So multiplication, for example, is commutative. X times Y is always Y times X, at least for numbers. Um, and it's also associative. X times Y times Z is the same as X times Y times Z. Um, so, um, these operations obey some laws that don't obey others. For example, X times X is not always equal to X.
So that law is not always true. So given any, any operation, it obeys some laws and not others. Um, and so we generated about 4,000 of these possible laws of algebra that certain operations could satisfy. And our question is, which laws imply which other ones? Um, so for example, does commutativity imply associativity? And the answer is no, because it turns out you can describe an operation which obeys the commutative law, but doesn't obey the associative law. So by producing an example, you can, you can show that commutativity does not imply associativity, but some other laws do imply other laws by substitution and so forth.
Uh, and you can write down some, some algebraic proof. So we looked at all the pairs between these 4,000 laws and there's about 22 million of these pairs. And for each pair, we asked, does this law imply this law, um, law? If so, give a, give, uh, give a proof. If not, give a counterexample. Um, so 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, out of these 22 million, there are like a hundred or so that are really quite hard. Okay. But a lot are easy. And the project was just to, to work out, to determine the entire graph, like, like 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. You know, I mean, the state of the art in the literature was like, you know, 15 equations and sort of how they imply. That's sort of at the limit of what a human with pen and paper can do. So, so you need to scale it up. So you need to crowdsource, but you also need to trust all the, um, I mean, no one person can check 22 million of these proofs. You need to be computerized. And so it only became possible with, with lean. Um, we were hoping to use a lot of AI as well. Um, so the project is almost complete. Um, so of these 22 million, orbit two have been settled.
Um, and, uh, well, actually, and of those two, we have a pen and paper proof of the two, uh, and we, we're formalizing it. In fact, I was, this morning I was working on finishing it. Um, so we're almost done on this. Um, it's, it's.
Incredible.
It's, yeah, fantastic.
How many people were able to get?
About 50. Um, which in mathematics is, is considered a huge number.
It's a huge number. That's crazy.
Yeah. So we're gonna have a paper with 50 authors, uh, and a big appendix of who contributed what.
Here's an interesting question, not to maybe speak even more generally about it. When you have this pool of people, is there a way to, uh, organize the contributions by level of expertise of the people, of the contributors? Now, okay, uh, I'm asking you a lot of pothead 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 like a gamification of this?
The beauty of, of these lean projects is that automatically you get all this data, you know, so like, like everything's uploaded to this GitHub and GitHub tracks who contributed what. Um, so you could generate statistics from at any, at any later point in time, you can say, oh, this person contributed this many, this many lines of code or whatever. I mean, these are very crude metrics. Um, I would, I would definitely not want this to become like, you know, part of your tenure review or something. Uh, um, but, um, I mean, I think already in, in, in enterprise computing, right? People do use some of these metrics as part of, of the assessment of, of, uh, performance of a, of an employee.
Um, again, this is the direction which is a bit scary for academics to go down. We, we, we don't like metrics so much.
And yet academics use metrics. They just use old ones. Number of papers.
Yeah. Yeah. It's true. It's true that, yeah, I mean, um.
It feels like this is a metric while flawed is, 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, you can do studies of, of, of whether these are better predictors. Um, there's this problem called Goodhart's Law. If a statistic is actually used to incentivize performance, it becomes gamed. Um, and then it is no longer a useful measure.
Oh, humans always game.
Yeah. Yeah. No, I mean, it's, it's rational. So what we've done for this project is, is self-report. So, um, there are actually these standard categories, um, from the sciences of what types of contributions people give. So there's, there's a concept and validation and resources and, and, and, and, and coding and so forth. Um, so we, we, we, there's a standard list of 12 or so categories. Um, and we just ask each contributor to, there's a big matrix of all the, of all the authors and all the categories just to tick the boxes where they think that they contributed. Um, and just to give a rough idea, you know, like, oh, so you did some coding and, and, uh, and you provided some compute, but you didn't do any of the pen and paper verification or whatever.
And 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, you know, lead author and second author and so forth, like, which we're proud of. You know, we make all the authors equal status, but it doesn't quite scale to this size. So a decade ago, I was involved in these things called Polymath Projects. It was a crowdsourcing mathematics, but without the lean component. So it was limited by, you needed a human moderator to actually check that all the contributions coming in were actually valid. Um, and this was a huge bottleneck actually.
Um, but still we had projects that were, you know, 10 authors or so, but we had decided at the time, um, not to try to decide who did what, um, but to have a single pseudonym. Uh, so we created this fictional character called DHJ Polymath in the spirit of Bob Ikey. Bob Ikey is, is the, the pseudonym for a famous group of mathematicians in the 20th century. But, um, and so the paper was authored on the pseudonym. So none of us got the author credit. Um, this actually turned out to be not so great for a couple of reasons.
So, so one is that if you actually wanted to be considered for tenure or whatever, you could not use this paper in your, uh, uh, as you submitted as one of your publications because it was, you didn't have the formal author credit. Um, um, but the other thing that we've recognized much later is that when people referred to these projects, they naturally referred to the most famous person who was involved in the project. Oh, so this was Tim Gao's Polymath Project. This was Terence Tao's Polymath Project, and not mentioned the, the other 19 or whatever people that were involved.
Ah, yeah.
So we're trying something different this time around where we have everyone's an author, um, but we will have an, an appendix with this matrix and we'll see how that works.
I mean, uh, so both projects are incredible, just the fact that you're involved in such huge collaborations. But I think I saw a talk from Kevin Buzzard about, uh, the lean programming languages a few years ago, and he was saying that, uh, this might be the future of mathematics. And so it's also exciting that you're embracing, uh, one of the greatest mathematicians in, in the world embracing this, what seems like the paving of the future of mathematics. Um, so I have to ask you here about the integration of AI into this whole process.
So DeepMind's Alpha Proof was trained using reinforcement learning on both failed and successful formal lean proofs of IML problems.
Yes.
So 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 gradual level, uh, problems?
Yeah. The difficulty increases exponentially with the, the number of steps involved in the proof. It's a combinatorial explosion, right? So the thing of large language models is, is that they make mistakes. And so if a proof has got 20 steps and your large language model has a 10% failure rate, um, at each step, um, of, of going in the wrong direction, like, uh, it's, it's just extremely unlikely to actually, uh, reach the end.
Actually, uh, just to take a small tangent here, is how hard is the problem of mapping from natural language to the formal program?
Oh, yeah. It's extremely hard actually. Um, natural language, you know, it's very fault tolerant. Um, like you can make a few minor grammatical errors and a speaker in the second language can get some idea of what you're saying. Um, yeah, but, but formal language, yeah, you know, if, if you get one little thing wrong, um, like the whole thing is, is, is, is nonsense. Um, even formal to formal is, is, is very hard. There are, there are different incompatible, um, uh, prefaces and languages. Uh, there's lean, but also Koch and Isabel and, and so forth. And actually even converting from a formal language to formal language, um, is, uh, is an unsolved, basically unsolved problem.
That is fascinating. Okay. So, uh, but once you have an informal language, they're using, uh, their RL trained model. So something akin to AlphaZero that they used to go to then 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 this system and, um, what do you think is the gap?
Yeah. We talked earlier about things that are amazing over time become kind of normalized. Um, so yeah, now somehow it's, oh, of course geometry is a solvable problem.
Right. That's true. That's true. I mean, it's still beautiful.
Yeah, yeah. No, it's a great work. It shows what's possible. I mean, um, it's, it, um, the approach doesn't scale currently. Yeah. Three days of Google's server is server time due to one, uh, high school math formula. This, this is not a scalable, uh, prospect, um, especially with the exponential increase in, um, as, as the complexity, um, increases.
We should mention that they got a silver medal performance.
The equivalent of, I mean.
Equivalent of a silver medal.
So first of all, they took way more time than was, uh, a lot of audit. Um, and they had this assistance with where the humans started helped by, by formalizing. Um, but, uh, yeah, also they, they're giving themselves full marks for the solution, which I guess is formally verified. So I guess that that's, that's fair. Um, yeah. Um, there, there are efforts, there was, there will be a proposal at some point to actually have an, an AI math olympiad where at the same time as the human contestants get the, the actual Olympiad, um, problems, AIs will also be given the same problems with the same time period.
Um, and the outputs will have to be created by the same judges. Um, um, and which means that it will have to be written in natural language rather than formal language.
Oh, I hope that happens. I hope that this IML happens. I hope, I hope next one.
It's, it won't happen this IML. The performance is not good enough in, in, in the time period. And, and, uh, um, but there are smaller competitions. Uh, there are competitions where the, the answer is a, is a number rather than a long form proof. Um, and that's, that's, um, AIs are actually a lot better at, um, problems where there's a specific numerical answer. Um, 'cause it's, it's, it's easy to, to, to, uh, to reinforce, to reinforce learning on it. Yeah. You got the right answer, you got the wrong answer. Uh, it's, it's, it's a very clear signal, but a long form proof either has to be formal and then the lean can give it a thumbs up, thumbs down, or it's informal.
Um, but then you need a human to grade it to tell. Uh, and if you're trying to do billions of, of reinforcement learning, um, you know, um, um, runs, you're, you're not, you can't hire enough humans to, uh, to grade those. Um, I mean, it's already hard enough for, for the large language models to do reinforcement learning on, on just the regular text that, that people get. But now if you actually hire people not just give thumbs up, thumbs down, but actually check the, the output mathematically. Yeah. Uh, that's too expensive.
So if we, uh, just explore this possible future, what, what, what is the thing that humans do that's most special in, uh, in mathematics? So that you could see AI, uh, not cracking for a while. So inventing new theories, so coming up with new conjectures versus, uh, proving the conjectures, building new abstractions, new representations, maybe, uh, an AI Terence Tao with, uh, seeing new connections between disparate fields.
That's a good question. Um, I think the nature of what mathematicians do over time has changed a lot. Um, you know, um, so a thousand years ago, mathematicians had to compute the date of Easter. Uh, and there's really complicated, uh, calculations, you know, but it's all automated, been automated for centuries. Uh, we don't need that anymore. You know, they used to navigate to do spherical navigation, spherical trigonometry to navigate how to get from, from, um, the old world to the new. So very complicated calculations again, been, been automated. Um, you know, even a lot of undergraduate mathematics, even before AI, um, like Wolfram Alpha, for example, uh, it's, 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, a problem and, um, and say, here's a problem in partial factorial equations. Could you solve it using any of the 20 standard techniques? Um, and the AI will say, yes, I've tried all 20 and here are the 100 different permutations and, and here's my results. Um, and that type of thing, I think it will work very well. Um, type of scaling to once you solve one problem to, to make the AI attack a hundred adjacent problems. Um, the things that humans do still, yeah.
So, so where the AI really struggles right now, um, is knowing when it's made a wrong turn. Um, that it can say, oh, I'm gonna solve this problem. I'm gonna split up this problem into, um, into these two cases. I'm gonna try this technique. And, um, 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, it will get, it will have a problem, it will, it would propose an approach which is just complete nonsense. Um, and, but like it looks like a proof. Um, so this is one annoying thing about LLM generated mathematics.
So, um, we, we, we've had human generated mathematics that's very low quality. Um, uh, like, you know, submissions 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. Uh, and it's partly because that's what the reinforcement learning has actually trained them to do, uh, to, to make things, to, to produce text that looks like, um, uh, what is correct, which for many applications is good enough. Um, uh, so the errors are often really subtle.
And then when you spot them, they're really stupid. Um, like, you know, like, like no human would have actually made that mistake.
Yeah. It's actually really frustrating in the programming context 'cause I, I program a lot and yeah, when a human makes when a low quality code, there's something called code smell, right? You can tell, you can tell immediately like, okay, there's signs. But with, with AI generate code.
Yeah. All of us.
And then you're right. Eventually you find an obvious dumb thing that just looks like good code.
Yeah. So, um.
It's very tricky to, and frustrating for some reason to have to work.
Yeah. So the sense of smell.
Okay.
This, this is, this is one thing that humans have. Um, and there's a metaphorical mathematical smell that, uh, there's, it's not clear how to get the AI to duplicate that eventually. Um, I mean, so the, the way, um, AlphaZero and so forth, they make progress on Go and, and chess and so forth is, is in some sense they have developed a sense of smell for Go and chess positions. You know, that, that this position is good for white, it's good for black. Um, they can't enunciate why. Um, but just having that, that sense of smell lets them strategize.
So if AIs gain that ability to sort of a sense of viability of certain proof strategies, so you can say, I'm gonna try to, to break up this problem into two smaller subtasks and it can say, oh, this looks good. Two tasks look like they're simpler tasks than, than your main task, and they still got a good chance of being true. Um, so this is good to try. Or no, you've, you've, you 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, uh, thing to try. Normally you actually, it's very easy to transform a problem into an even harder problem.
Very rarely do you transform to a simpler problem. Um, yeah. So if they can pick up a sense of smell, then they could maybe start competing with, uh, uh, human level mathematicians.
So, so this is a hard question, but not competing, but collaborating.
Yeah.
If, okay, hypothetical, if I gave you an oracle that was able to do some aspect of what you do and you could just collaborate with it.
Yeah, yeah, yeah.
What would that oracle, what would you like that oracle to be able to do? Would you like it to, uh, maybe be a verifier, like check through the code smell like you're, yes, uh, a professor Tao, this is the correct, this is a good, this is a promising fruitful direction.
Yeah, yeah, yeah.
Or, or would you like it to, uh, generate possible proofs and then you see which one is the right one? Um, or would you like it to maybe generate different representation, different, totally different ways of seeing this problem?
Yeah. I think all of the above. Um, a lot of it is, we don't know how to use these tools because it's a paradigm that it's not, um, yeah, we have not had in the past assistants that are competent enough to understand complex instructions, um, that can work at massive scale, but are also unreliable. Uh, like it's, it's an interesting, uh, but unreliable in subtle ways whilst we, whilst providing superficially good output. Um, it's a interesting combination. Um, you know, I mean, you have, you have like graduate students that you work with who like kind of like this, but not at scale.
Um, you know, and, and, and we had previous software tools that, um, can work at scale, but, but very narrow. Um, so we have to figure out how to, how to use, um, I mean, um, so Tim Gao is actually, you imagine he actually foresaw like in, in 2000, he was envisioning what mathematics would look like in, in actually two and a half decades.
That's funny.
Yeah. He, he wrote in his, in his article like a, a hypothetical conversation between a mathematical assistant of the future, um, and himself, you know, he's trying to solve a problem and they would have a conversation that sometimes the human would propose an idea and the AI would, would evaluate it. Uh, and sometimes the AI would propose an idea. Um, and, uh, and sometimes a competition was required and AI would just go and say, okay, I've, I've checked the, the 100 cases needed here. Or, um, uh, the first, uh, you, you set the situation for all n up to n up to 100. Um, and it looks good so far.
Or hang on, there's a problem at n equals 46. Yeah. So just a free form conversation where you don't know in advance where things are gonna go, but just based on, on, I think ideas get proposed on both sides. Calculations get proposed on both sides. I've had conversations with AI where I say, okay, let's, we're gonna collaborate to solve this math problem. And it's a problem that I already know the solution to. So I, I try to prompt it. Okay. So here's the problem. I, I suggest using this tool and it, it'll find this lovely argument using a completely different tool, which eventually goes, you know, into the weeds and say, no, no, no, try using this. Okay.
And, and I start using this and then it'll go back to the tool that it wanted to do before. Um, and like you have to keep railroading it, um, onto the path you want. And I, I could eventually force it to give the proof I wanted. Um, but it was like herding cats. Um, like, and the amount of personal effort I had to take to not just sort of prompt it, but also check its output. 'Cause it, a lot of what it looks like is gonna work. I know there's a problem on line 17 and basically arguing with it. Um, like it was more exhausting than doing it, uh, unassisted.
So like it, but that's the current state of the art.
I wonder if there's, there's a phase shift that happens to where it's no longer feels like herding cats and maybe you'll surprise us how quickly that comes.
I, I, I believe so. Um, 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 is, and also just better tooling, um, the lean, um, um, developers are doing a great job adding more and more features and making it user-friendlier. It's going from nine to eight to seven. Okay. No big deal. But one day it will drop below one. Um, and that's a phase shift because suddenly, um, it makes sense when you write a paper to, to write it in lean first, uh, or through a conversation with AI who's generating lean, um, on the fly with you.
And it becomes natural for journals to accept, uh, you know, maybe they'll offer expedite refereeing, you know, if, if a paper has already been formalized in, in lean, um, they'll just ask the referee to comment on, on the significance of the results and how it connects to literature and not worry so much about the correctness, um, 'cause that's been certified. Um, papers are getting longer and longer in mathematics and actually it's harder and harder to get good refereeing for, um, the, the really long ones, unless they're really important. Uh, it, it is actually an issue which the formalization is coming in at just the right time for this to be.
And the easier and easier to guess because of the tooling and all the other factors, then you're gonna see much more like math lib will grow potentially exponentially. It's a, it's a, it's a virtuous, uh, cycle. Okay.
I mean, one phase shift of this type that happened in the past was, uh, the adoption of LaTeX. So LaTeX is this type setting language that all mathematicians use now. So in the past, people used all kinds of word processors and typewriters and, and whatever, but at some point LaTeX became easier to use than all other competitors and that people would switch, you know, within a few years. Like, like it was just a dramatic, um, phase shift.
It's a wild out there question, but what, what year, how far away are we from a, uh, AI system being a collaborator on a proof that wins the Fields Medal? So that level.
Okay. Um, well, it depends on the level of collaboration. I mean.
No, like it deserves to be, to get the Fields Medal. Like, so half and half.
Already, like I, I can imagine if it was a medal winning paper having some AI systems in writing it, you know, uh, just, you know, like the autocomplete alone is already, I, I, I use it. Like it speeds up my, my own writing. Um, um, like, you know, you, you, you can have a theorem and you have a proof and the proof has three cases and I, I, I write down the proof of the first case and the autocomplete just suggests, okay, now here's how the proof of the second case could work. And like it was exactly correct. That was great. Saved me like five, 10 minutes of, uh, of, of typing.
But in that case, the AI system doesn't get the Fields Medal.
No.
Uh, are we talking 20 years, 50 years, a hundred years? What do you think?
Okay. So I, I gave a prediction in print by 2026, which is now next year. Um, there will be math collaborations with AI. So not Fields Medal winning, but, but like actual research level math papers.
Like, like published ideas that are in part generated by AI.
Um, maybe not the ideas, but at least, uh, some of the computations, um, um, verifications. Yeah. I mean.
Has that already happened?
Has it already happened? Yeah. There, there are, there are problems that were solved, uh, by a complicated process conversing with, with AI to propose things and the human goes and tries it and it, and the contract doesn't work. Uh, but it might propose a different idea. Um, it, it's, it's hard to disentangle exactly. Um, there are certainly math results which could only have been accomplished because there was a math, human mathematician and an AI involved. Um, but it's hard to sort of disentangle credit.
Um, I mean, these tools, they, they do not, uh, replicate all the skills needed to do mathematics, but they can replicate sort of some non-trivial percentage of them, you know, 30, 40%. So they can fill in gaps. Um, you know, so, uh, coding is, is, is a, is a good example, you know? So I, I, um, um, it's annoying for me to, to code in Python. I'm not, I'm not a native, um, I'm not a professional, um, programmer. Um, but, um, the, with AI, the, the, the friction cost of, of doing it is, is, is much reduced.
Uh, so it, it fills in that gap for me. Um, AI is getting quite good at literature review. Um, I mean, there's still a problem with, um, hallucinating, you know, references that don't exist. Um, but this I think is a solvable problem. Uh, if you train in the right way and so forth, you can, you can, and, um, and verify, um, you know, using the internet, um, you, you know, you, um, you should in a few years get to the point where you, you have a, a lemma that you need and, uh, say has anyone proven this lemma before? And it will do basically a fancy web search AI assistant and say, yeah, yeah, there are these six papers where something similar has happened.
Uh, and I mean, you can ask it right now and it'll give you six papers of which maybe one is, is legitimate and relevant, one exists, but is not relevant and four are hallucinated. Um, it has a non-zero success rate right now, but, uh, there's so much garbage, uh, so much, the signal to noise ratio is so poor that it's, it's, um, it's most helpful when you already somewhat know the literature. Um, and you just need to be prompted to be reminded of a paper that was already subconsciously in your memory.
Versus helping you discover new you were not even aware of, but is the correct citation.
Yeah, that's, yeah, that it can sometimes do, but, but when it does, it's, it's buried in, in a list of options to which the other.
That are bad.
Yeah.
I mean, being able to automatically generate a related work section that is correct.
Yeah.
That's actually a beautiful thing that might be another phase shift because it assigns credit correctly.
Yeah.
It does, it breaks you out of the silos of.
Yeah, yeah, yeah.
Thought, you know.
Yeah. No, there's a big hump to overcome right now. I mean, it's, it's, it's like self-driving cars.
Right.
You know, the safety margin has to be really high.
Yeah.
For it to be, um, uh, to be feasible. So yeah, so there's a last mile problem, um, with a lot of AI applications, um, that, uh, you know, they can develop tools that work 20%, 80% of the time, but it's still not good enough. Um, and in fact, even worse than good in some ways.
I mean, another way of asking the Fields Medal question is, what year do you think you'll wake up and be like real surprised? You read the headline, the news of something happened that AI did like, you know, real breakthrough something. It doesn't, you know, like Fields Medal or even hypothesis. It could be like really just this alpha zero moment would go that kind of thing.
Right. Right. Um, yeah, this, this decade I can, I can see it like making a conjecture between two unrelated, two, two things that people would thought was unrelated.
Oh, interesting. Generating a conjecture that's a beautiful conjecture.
Yeah. And, and actually has a real chance of being correct and, and, and meaningful. And, um.
Because that's actually kind of doable, I suppose. But the, where the data is, it's, yeah.
Yeah.
No, that would be truly amazing.
Um, the current model struggle a lot. I mean, so, um, a version of this is, um, I mean, the, the physicists have a dream of getting the AIs to discover new, new laws of physics. Um, uh, you know, the, the dream is you just feed it all this data. Okay. Uh, uh, and, and this is a, here's a new patent that we didn't see before, but it actually even struggle, the current state of the art even struggles to discover old laws of physics, um, from the data. I mean, uh, or if it does, uh, there's a big concern of contamination that it did it only because it's like somewhere in its training data it already somehow knew, um, you know, Boyle's law or whatever ball you're trying to, to, to reconstruct.
Um, part of it is we don't have the right type of training data for this. Um, you know, so for laws of physics, like we, we don't have like a million different universes with a million different laws of nature. Um, and, um, like a lot of what we are missing in math is actually the negative space of, so we have published things of things that people have been able to prove, um, and conjecture that ended up being verified, um, or where the counterexamples produced, but, um, we don't have data on, on things that were proposed and they're kind of a good thing to try, but then people quickly realized that it was the wrong conjecture and then they, they, they said, oh, but we, we should actually change, um, our claim to modify it in this way to actually make it more plausible.
Um, there's this, there's a trial and error process, which is a real integral part of human mathematical discovery, which we don't record 'cause it's, it's embarrassing. Uh, we make mistakes and, and we only like to publish our, our wins. Um, and, uh, the AI has no access to this data to train on. Um, I sometimes joke that basically we get an AI has to go through, um, a grad school and actually, you know, go to grad courses, do the assignments, go to office hours, make mistakes, um, get advice on how to correct the mistakes and learn from that.
Let me, uh, ask you if I may about, uh, Gregory Perlman.
Mm-hmm.
You mentioned that you try to be careful in your work and not let a problem completely consume you. Just you've really fallen in love with the problem and really cannot rest until you solve it. But you also hasted to add that sometimes this approach actually can be very successful.
Mm-hmm.
An example you gave is Gregory Perlman, who proved the Poincaré conjecture and did so by working alone for seven years with basically little contact with the outside world. Can you explain this one Millennium Prize problem that's been solved, Poincaré conjecture, and maybe speak to the journey that Gregory Perlman's been on?
All right. So it's, it's a question about curved spaces. Earth is a good example. So if you think of it as a 2D surface, interesting round view could maybe be a torus with a hole in it or it can have many holes. And there, there are many different topologies a priori that, that a surface could have. Um, even if you assume that it's, it's bounded and, and, uh, and smooth and so forth. So we, we have figured out how to classify surfaces. As a first approximation, uh, everything's determined. It's not called the genus how many holes it has. So a sphere has genus zero, a donut has genus one and so forth. And one way you can tell these surfaces apart, probably the sphere has, which is called simply connected.
If you take any closed loop on the sphere, like a big closed loop rope, you can contract it to a point and while staying on the surface. And the sphere has this property, but a torus doesn't. And if you're on a torus and you take a rope that goes around, say the, the outer diameter torus, there's no way it, it can't get through the hole. There's no way to contract it to a point. So it turns out that the, the, the sphere is the only surface with this property of contractability up to like continuous deformations of the sphere. So, um, so things that I want to go topologically, um, equivalent to the sphere.
So Poincaré asked the same question in higher dimensions. Um, so this, it becomes hard to visualize, uh, because, um, 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, to, to live in. And then there are also 3D spaces that can't even fit into four dimensions. You need five or six or, or higher. But anyway, uh, mathematically you can still pose this question that if you have a bounded three-dimensional space now, which is also has this simply connected property that every loop can be contracted, can you turn it into a three-dimensional version of the sphere?
And so this is the Poincaré conjecture. Weirdly, in higher dimensions, four and five it was actually easier. So, uh, it was solved first in higher dimensions. There's somehow more room to do the deformation. It's easier to, to, to move things around to, to a sphere, but three was really hard. So people tried many approaches. There's sort of commentary approaches where you chop up the, the surface into little triangles or, or tetrahedron and you, you, you just try to argue based on how the faces interact with each other. Um, there were, um, algebraic approaches. Uh, there's, there's various algebraic objects, uh, like things called the fundamental group that you can attach to these homology and chromology and, and, and, and all these very fancy tools.
Um, they also didn't quite work. Um, but Richard Hamilton's proposed a, um, partial differential equations approach. So you take, um, you take, so the problem is that you, so you have this object which is sort of secretly a sphere, but it's given to you in a, in a really, um, in a, in a weird way. So it's like, I think of a ball that's been kind of crumpled up and twisted and it's not obvious that it's a ball. Um, but, um, like if you, if you have some sort of surface which is, which is a deformed sphere, you could, um, uh, you could, for example, think of it as the surface of a balloon.
You could try to inflate it. You blow it up. Um, and naturally, as you fill it with air, um, the, the wrinkles will sort of smooth out and it will turn into, um, um, a nice round sphere. Um, uh, unless of course it was a torus or something in which case it would get stuck at some point. Like if you inflate a torus, it would, there'll be a point in the middle. When the inner ring shrinks to zero, you get, 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 now called Richie flow, which is a way of taking an arbitrary surface or, or space and smoothing it out to make it rounder and rounder to make it look like a sphere.
And he wanted to show that, that either, uh, this process would give you a sphere or it would create a singularity. Um, like you very much like how PDEs, either they have global regularity or finite and blow up. Like it, it basically it's almost exactly, exactly the same thing. It's all connected. Um, and so, and, and he showed that for two dimensions, two dimensional surfaces, um, uh, if you started something connected, no singularities ever formed. Um, you, you never ran into trouble and you could flow and, and it would give you a sphere. And it, so he, he got a new proof of the two-dimensional result.
But by the way, that's a beautiful explanation of Richie flow and its application in this context. How difficult is the mathematics here? Like for the 2D case? Is it?
Yeah, these are quite sophisticated equations on par with the Einstein equations. Slightly simpler, but, um, um, yeah, but, but they were considered hard nonlinear equations to solve. Um, and there's lots of special tricks in 2D that, that, that helped. But in 3D, the problem was that, uh, this equation was actually super critical. It has the same problems as Nabier-Stokes. As you blow up, um, maybe the curvature could get concentrated in finer and smaller, smaller regions, and it, um, it looked more and more nonlinear and things just looked worse and worse.
And there could be all kinds of singularities that showed up. Um, some singularities, um, like if, if, uh, there's these things called neck pinches where, where the, uh, the surface sort of behaves like, like a, like a, a barbell and it, it, it pinches at a point. Some, 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, there was the, the con, the prospect that there's some really nasty, like knotted singularities showed up that you, you couldn't see how to, um, resolve in any way, uh, that you couldn't do any surgery to.
Um, so you need to classify all the singularities, like what are all the possible ways that things can go wrong. Um, so what Perlman did was first of all, he, he made the problem, he turned the problem from a super critical problem to a critical problem. Um, I said before about how, um, the invention of the, of, of energy, the Hamiltonian, like really clarified, um, Newtonian mechanics. Um, uh, so he introduced, uh, something which is now called Perlman's reduced volume and Perlman's entropy. Um, he introduced new quantities, kind of like energy that looked 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.
Um, and then he had to solve, he still had to analyze the singularities of this critical problem. Uh, and that itself was a problem similar to this wake-up thing I worked on actually. Um, so on the, on the level of difficulty of that, so he managed to classify all the singularities of this problem and show how to apply surgery to each of these and through that was able to, to resolve the Poincaré conjecture. So, um, quite, uh, like a lot of really ambitious steps. Um, and like, like nothing that a large language model today, for example, could, I mean, um, at best, uh, I could imagine a model proposing this idea as one of hundreds of different things to try.
Um, but the other 99 would be complete dead ends, but you'd only find out after months of, of work. He must have had some sense that this was the right track to pursue because like, you know, it takes years to get from, from A to B.
So you've done, like you said, actually, you see even strictly mathematically, but more broadly in terms of the process, you've done similarly difficult things. What, what can you infer from the process he was going through? 'Cause he was doing it alone. What are some low points in a process like that when you start to like, you've mentioned hardship, like, uh, AI doesn't know when it's failing. What, what happens to you? You're sitting in your office when you realize the thing you did for the last few days, maybe weeks.
Yeah.
Is a failure.
Well, for me, I switched to a different problem. Uh, so, uh, I'm, I'm, I'm, I'm a fox. I'm not a hedgehog.
But you'll generally, that is a break that you can take is, is to step away and look at a different problem.
Yeah. Yeah. You can modify the problem too. Um, I mean, um, yeah, you can ask them cheat if, if there's a specific thing that's blocking you that, that this, um, some bad case keeps showing up that, that, that for which your tool doesn't work, you can just assume by fiat this, this bad case doesn't occur. So you, you do some magical thinking, um, for the, you know, but, but, but strategically, okay, for the point to see if the rest of the argument goes through. Um, if there's multiple problems, uh, with, with, with your approach, then maybe you just give up. Okay. But if this is the only problem that, you know, and everything else checks out, then it's still worth fighting.
Um, so yeah, you have to do some, some, some forward reconnaissance sometimes to, uh, you know.
And that is sometimes productive to assume like, okay, we'll figure it out.
Oh yeah, yeah, yeah. Um, sometimes actually it's, it's even productive to make mistakes. So, um, one of the, I mean, um, there was a project which actually, uh, we won some prizes for actually with four other people. Um, we worked on this PDE problem again, actually this blow up regularity type problem. Um, and it was considered very hard. Um, Jean Bourgain, um, uh, was another field methodist who worked on a special case of this, but he could not solve the general case. Um, and we worked on this problem for two months and we found, we thought we solved it.
We, we had this, this cute argument that if everything fit and we were excited, uh, we were planning celebrationally, um, to all get together and have champagne or something. Um, and we started writing it up. Um, and one of, one of us, not me actually, but another co-author said, oh, um, in this, in this lemma here, we, um, we have to estimate these 13 terms that, that show up in this expansion. And we estimate 12 of them, but in our notes, I can't find the, the estimation of the 13th. Can you, can someone supply that? And I said, sure, I'll look at this. And, and 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. Um, in fact, we could not estimate this term. Um, and we tried for a few more months and, and all different permutations. And there was always this one thing, one term that we could not control. Um, and so like, um, this was very frustrating. Um, but because we had already invested months and months of effort in this already, um, we stuck at this, which we tried increasingly desperate things and, and crazy things. Um, and after two years, we found an approach which was somewhat different, but quite a bit from our initial, um, strategy, which did actually, didn't generate these problematic terms and, and, and actually solve the problem.
So we, we solved the problem after two years, but if we hadn't had that initial false dawn of nearly solving the problem, we would've given up by month two or something and, and worked on an easier problem. Um, yeah, if we had known it would take two years, not sure we would've started the project. You know, sometimes actually having the incorrect, you know, it's, it's like Columbus traveling in the New World. They had an incorrect version of a measurement of the size of the earth. Um, he thought he was going to find a new trade route to India, uh, or at least that was how he sold it in his prospectus.
I mean, it, it could be that he actually secretly knew, but.
There's some of the psychological element. Do you have like emotional or like self-doubt that just overwhelms you moments like that? You know, 'cause it, this stuff, it feels like math, it's, it's so engrossing that like it can break you when you like invest so much yourself in the problem and then it turns out wrong, you could start to, similar way chess has broken some people.
Yeah. Um, I, I think different mathematicians have different levels of emotional investment in what they do. I mean, I think for some people it's just a job. You know, you, you have a problem and if it doesn't work out, you, you work on, you work on the next one. Um, yeah. So the fact that you can always move on to another problem, um, it reduces the emotional connection. I mean, there are cases, you know, so there are certain problems that are what I call mathematical diseases where, where, where just latch onto that one problem and they spend years and years thinking about nothing but that one problem.
And, um, you know, maybe their career suffers and so forth. They say, oh, but I've got this big win. This will, you know, once I, once I finish this, this problem, I will, I will make up for all the years of, of, of, of lost opportunity. And that's, that's, I mean, occasionally, occasionally it works, but I, um, I really don't recommend it for people who have the right fortitude. Yeah. So I, I've never been super invested in any one problem. Um, one thing that helps is that we don't need to call our problems in advance.
Uh, um, well, uh, when we do grant proposals, uh, we sort of say we will, we will study this set of problems, but even there, we don't promise definitely by five years I will supply a proof of all these things. You know, we, um, you promise to make some progress or discover some interesting phenomena. Uh, and maybe you don't solve the problem, but you find some related problem that you, you can say something new about. Uh, and that's, that's a much more feasible task.
But I'm sure for you, there's problems like this. You have, you have, uh, made so much progress towards the hardest problems in the history of mathematics. So is there, is there a problem that just haunts you? It sits there in the dark corners, you know, twin prime conjecture, Riemann hypothesis, Goldbach conjecture.
Twin prime. That sounds again. So, I mean, the problems like the Riemann hypothesis, those are so far out of reach.
Oh, you think so?
Yeah. There, there's no even viable strategy. Like even if I activate all my, all the cheats that I know of in this problem, like it, there's just still no way to get me to be. Um, like it's, it's, um, I think it needs a breakthrough in another area of mathematics to happen first. And for someone to recognize that it, that would be a useful thing to transport into this problem.
So we, 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, uh, atoms provide?
So the natural numbers have two basic operations attached to them, addition and multiplication. Um, so if you wanna 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 number. If you wanna generate multiplicatively, you can take all the prime numbers 2, 3, 5, 7 and multiply them all together. Um, and together they, they gives you all the, the, 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 a multiplicative point of view.
Um, and separately they're not so bad. Um, so like any question about that natural number that only involves addition is relatively easy to solve. And any question that only involves multiplication is relatively easy to solve. Um, but what has been frustrating is that you combine the two together, um, and suddenly you get this extremely rich, I mean, we know that there are statements in number theory that are actually as, uh, undesirable. There are certain polynomials and some number of variables. Is there a solution in the natural numbers? And the answer depends on, on an undesirable statement. Um, like, like whether, um, the axioms of, of mathematics are consistent or not.
Um, but, um, yeah, but even the, the simplest problems that combine something multiplicative such as the primes with something additive such as shifting by two, uh, separately we understand both of them well. But if you ask, when you shift a prime by two, do you, can you get a, how often can you get another prime? We, it's been amazingly hard to relate the two.
And we should say that the twin prime conjecture is just that it posits that there are infinitely many pairs of prime numbers that differ by two.
Yes.
Now, the interesting thing is that you have been very successful at pushing forward the field in answering these complicated questions, uh, of this variety. Like you mentioned the green tile theorem, it proves that prime numbers contain arithmetic progressions of any length.
Right.
Which is mind blowing that you could prove something like that.
Right. Yeah. So what we've realized because of this, this, this type of, of research is that there's different patterns have different levels of, uh, intersectability. Um, so, so what makes the twin prime problem hard is that if you take all the primes in the world, you know, 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, um, redact the primes to get rid of, to get rid of the, um, these twins, like the twins, they show up and they're infinitely many of them, but they're actually reasonably sparse.
Um, not there's, there's not, I mean, initially there's quite a few, but once you go to the millions, the trillions, they become rarer and rarer. And you could actually just, you know, if, if, if someone was given access to the database of primes, just edit out a few, a few primes here and there, they could make the twin prime conjecture false by just removing like 0.01% of the primes or something. Um, just world, world chosen to, to, um, 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, it, it obeys things like the prime number theorem and other things about the primes, but doesn't contain any twin primes anymore.
Um, and this is a real obstacle to the twin prime conjecture. It means that any proof strategy to actually find twin primes in the actual primes must fail when applied to these slightly edited primes. And so it, it must use some very, um, subtle, delicate feature of the primes that you can't just get from like, like aggregate statistical analysis.
Okay. So that's out.
Yeah. On the other hand, arithmetic progression has turned out to be much more robust. Um, like you can take the primes and you can eliminate 99% of the primes actually, you know, and you can take, take any 99% you want. And, uh, it turns out, and another thing we prove is that you still get arithmetic progressions. Um, arithmetic progressions are much, you know, they're like cockroaches.
Of arbitrary length though.
Yes. Yes.
That's crazy. I mean, so, so this, uh, for, for people who don't know, arithmetic progressions is a sequence of numbers that differ by some fixed amount.
Yeah. But it's again, like it's, it's an infinite monkey type phenomenon. For any fixed length of your set, you don't get arbitrary length progressions. You only get quite short progressions.
But you're saying twin primes are not an infinite monkey phenomenon. I mean, it's a very subtle monk. It's still an infinite monkey phenomenon.
Right. Yeah. If the primes were really genuinely random, if the primes were generated by monkeys, um, then yes, in fact, the infinite monkey theorem would.
Oh, but you're saying that twin prime is, it doesn't, you can't use the same tools like the, it doesn't appear random almost.
Well, we don't know. Yeah. We, we, we, we believe the primes behave like a random set. And so the reason why we care about the twin prime conjecture is, is it's a test case for whether we can genuinely confidently say with, 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. Um, at least with, with, with 100% probability, uh, or probably tending to 100% as you go out further and further. Um, yeah. So the primes we believe that they're random. Um, the reason why arithmetic progressions are indestructible is that regardless of whether you look random or looks, um, structured like periodic, in both cases, um, arithmetic progressions appear, but for different reasons.
Um, and this is basically all the ways in which the things, there are many proofs of, of these sort of arithmetic progression type theorems, 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. Um, but in twin primes, if, if the primes are random, then you are happy, you win. But if your primes are structured, they could be structured in, in a specific way that eliminates the twins, uh, and we can't rule out that one conspiracy.
And yet you were able to make a, as I understand, progress on the K-tuple version.
Right. Yeah. So, um, the, the one funny thing about conspiracies is that any one conspiracy theory is really hard to disprove.
Mm-hmm.
That, you know, if, if you believe the world is won by, by lizards, you say, here's some evidence that, that it, it's not won by lizards world, but that, that evidence was planted by the lizards.
Yeah.
Right. You might have encountered, uh, uh, this kind of phenomenon. Yeah. So, so like, like, um, a pure, like there's, there's almost no way to, to, um, definitively rule out a conspiracy. And the same is true in mathematics that a conspiracy that is solely devoted to eliminating twin primes, you know, like you would, you would have to also infiltrate other areas of mathematics to sort of, but, but like it could be made consistent at least as far as we know. But there's a weird phenomenon that you can make one, um, uh, one conspiracy rule out other conspiracies. So, you know, if the, if the world is, is won by lizards, they can't also be won by aliens.
Right. Right.
So one unreasonable thing is, is, is, is hard to disprove, but, but more than one, there are, there are tools. Um, so yeah. So for example, we, we know there's infinitely many primes that are, um, uh, no two of which had, um, so there are infinitely many pairs of primes which differ by at most, uh, um, 246 actually is, is, is the, is the current record.
Oh, so there's like a bound.
Yes.
On that.
Right. So like there's twin primes, there's things called cousin primes that differ by, by four. Um, there's things called sexy primes that differ by six.
Uh, what are sexy primes?
Primes that differ by six. The name, the name is much less, the concept is much less exciting than the name suggests.
Got it.
Um, so you can make a conspiracy rule out one of these, but like once you have like 50 of them, it turns out that you can't rule out all of them at once. It just, it requires too much energy somehow in this conspiracy space. Um.
How do you do the bound part? How do you, how do you develop a bound for the difference between the primes that.
Okay. So, um.
That there's an infinite number of.
So it's ultimately based on, uh, what's called the pigeonhole principle. Um, so the pigeonhole principle, uh, it's a statement that if you have a number of pigeons and they all have to go into, 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. So for instance, if you have a hundred numbers and they all range from one to a thousand, um, two of them have to be at most 10 apart.
Mm-hmm.
Because you, you can divide up the numbers from one to a hundred into 100 pigeonholes. Let's, let's say if you have a hundred, if you have a hundred and one numbers, if you have a hundred and one numbers, then two of them have to be, uh, distance less than 10 apart because two of them have to belong to the same pigeonhole. So it, it's a basic, um, basic feature of, uh, a basic principle in mathematics. Um, so it doesn't quite work with the primes directly because the primes get sparser and sparser as you go out. That, that fewer and fewer numbers are prime. But it turns out that there's a way to assign weights to the, to, to numbers like, um, so there are numbers that are kind of almost prime, but they're not, they, they don't have no factors at all other than themselves and one, but they have very few factors.
Um, and it turns out that we understand almost primes a lot better than we understand primes. Um, and so, for example, it was known for a long time that there were twin almost primes. This has been worked out. So almost primes are something we can't understand. So you can actually restrict attention to a, a suitable set of almost primes. And, uh, whereas the primes are very sparse overall, uh, relative to the almost primes, they're actually are, are much less sparse. They may, um, you can set up a, a set of almost primes where the primes have density like, say, 1%.
Um, and that gives you a shot at proving by applying some sort of pigeonhole principle that, that there's pairs of primes that are distantly 100, 100 apart. But in order to prove the twin prime conjecture, you need to get the density of primes inside the almost primes up to, up to a threshold of 50%. Um, once you get up to 50%, you will get twin primes. But, uh, unfortunately, there are barriers. Um, we know that, 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. Um, and I would love to find, yeah, so one of my long-term dreams is to find a way to breach that barrier because it would open up not only the twin prime conjecture, the go back conjecture, and many other problems in number theory are currently blocked because our current techniques would require improv going beyond this theoretical, um, parity barrier.
It's like, it's like going faster than the speed of light.
Yeah. So we should say twin prime conjecture, one of the biggest problems in the history of mathematics, go back conjecture also. Um, they feel like next door neighbors. Uh, has there been days when you felt, you saw the path?
Oh, yeah. Um, um, yeah. Uh, sometimes you try something and it works super well. Um, you, you again, again, the sense of mathematical smell, uh, we talked about earlier, uh, you learn from experience when things are going too well because there are certain difficulties that you sort of have to encounter. Um, um, I think the way of a colleague might put it is that, um, you know, like if, if you are on the streets of New York and you put in a blindfold and you put in a car and, and, um, after some hours, um, you, the blindfold is off and, and you're in Beijing.
Um, you know, I mean, that was too easy somehow. Like, like, there was no ocean being crossed. Even if you don't know exactly what, how, what, what was done, uh, you're suspecting that there's something that wasn't right.
But is that still in the back of your head to, do you return to these, to the prime, 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 tired, which is I get busy with so many things these days. But yeah, when I have free time and I'm not, and I'm too frustrated to, to work on my sort of real research projects and I also don't want to do my administrative stuff or I don't want to do some errands for my family. Um, I can play with these, these things, um, for fun. Uh, and usually you get nowhere. Yeah. You have to, you have to learn to just say, okay, fine. Once again, nothing happened. I will, I will move on. Um, yeah. Very occasionally one of these problems I actually solved, uh, or sometimes as you say, you think you solved it and then you're euphoric for, uh, maybe 15 minutes and then you think, I should check this 'cause this is too easy, too good to be true.
And it usually is.
What's your gut say about when these problems would be, uh, solved? Twin prime and go back.
Prime, I think we'll keep getting, keep getting more partial results. Um, it does need at least one. This parity barrier is, is the biggest remaining obstacle. Um, there are simpler versions of the conjecture where we are getting really close. Um, so I think we will, in 10 years, we will have many more, much closer results. We may not have the whole thing. Um, yeah. So twin primes is somewhat close. Yeah. Riemann hypothesis, I have no, I mean, it has happened by accident, I think.
So the Riemann hypothesis is a kind of more general conjecture about the distribution of prime numbers, right?
Right. Yeah. It's, it's, it's safe to sort of view it multiplicatively, like for questions only involving multiplication, no addition. The primes really do behave as randomly as, as you could hope. So there's a phenomenon in probability called square root cancellation that, um, you know, like if you wanna poll, say, America upon, on some issue, um, and you, you ask one or two voters and you may have sampled a bad sample and then you get, you get a really imprecise, um, measurement of, of the full average. But if you sample more and more people, the accuracy gets better and better and the accuracy improves like the square root of the number of, of people you, you sample.
So yeah, if you sample, um, a thousand people, you can get like a two, 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. It's called the Riemann zeta function and it fluctuates up and down. But in some sense, um, as you keep averaging more and more, if you sample more and more, the fluctuation should go down as if they were random. And there's a very precise way to quantify that. And the Riemann hypothesis is a very elegant way that captures this. But, um, as with many other areas in mathematics, we have very few tools to show that something really genuinely behaves like really random.
And this is actually not just a little bit random, but it's, it's asking that it behaves as random as an actually random set. It's, this, this square root cancellation. And we know actually because of things related to the parity problem actually that, that most of us usual techniques cannot hope to settle this question. Um, the proof has to come out of left field. Um, yeah. But, uh, what that is, yeah, no one has any serious proposal. Um, yeah. And, and there's, there's various ways of, as I said, you can modify the primes a little bit and you can destroy the Riemann hypothesis.
Um, so like it has to be very delicate. Um, you can't apply something that has huge margins of error. It has to just barely work. Um, and like, um, there's like all these pits, pitfalls that you have to like dodge very adeptly to.
The prime numbers are just fascinating.
Yeah.
Yeah. Yeah. What, what to you is, um, most mysterious about the prime numbers?
That's a good question. So like conjecturally, we have a good model of them. I mean, like, as I said, I mean, they have certain patterns, like the primes are usually odd, for instance, but apart from these sort of obvious patterns, they behave very randomly. And just assuming that they behave. So there's something called the Cramer random model of the primes that, that, that after a certain point, primes just behave like a random set. Um, and there's various slight modifications to this model, but this has been a very good model. It matches the numerics. It tells us what to predict. Like I can tell you with complete certainty the twin prime conjecture is true. Uh, the random model gives overwhelming odds that it's true.
I just can't prove it. Most of our mathematics is optimized for solving things with patterns in them. Um, and the primes have this anti-pattern, um, as do almost everything really, but we can't prove that. Yeah. I guess it's not mysterious that the primes behave random. It's kind of random 'cause there's sort of no reason for them to be, um, uh, to have any kind of secret pattern. But what is mysterious is what is the mechanism that really forces the randomness to happen. Uh, and this is just absent.
Another incredibly surprisingly difficult problem is the Kollatz conjecture.
Oh, yes.
Simple to state.
Mm-hmm.
Beautiful to visualize.
Yes.
In its simplicity, and yet extremely, uh, difficult to solve. And yet you have been able to make progress. Uh, uh, Paul Erdar said about the Kollatz 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.
Yeah.
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, it's, it's a problem that you can explain. Um, yeah. It, um, it helps with some, um, visual aids, but yeah. So you take any natural number, like say 13, and you apply the following procedure to it. So if it's even, you divide it by two, and if it's odd, you multiply it by three and add one. So even numbers get smaller, odd numbers get bigger. So 13, uh, will become 40 'cause 13 times three 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. And then you put it together, it's still reasonably simple.
Um, but then you ask what happens when you iterate it. You, you take the output that you just got and feed it back in. So 13 becomes 40. 40 is now even. Divide by two is 20. 20 is still even. Divide by two, 10, five, and then five times three plus one is 16. And then 8, 4, 2, 1. So, uh, and then from one it goes 1, 4, 2, 1, 4, 2, 1. It cycles forever. So this sequence I just described, um, you know, 13, 40, 20, 10, and so forth. Uh, these are also called hailstone sequences because there's an oversimplified model of, of hailstone formation, you know, which is not actually quite correct, but it's still somehow taught to high school students as a first approximation is that, um, like a, a little nugget of ice gets, gets a nice crystal that forms in a cloud and it goes up and down because of the wind.
And, and sometimes when it's cold, it acquires a bit more mass and maybe it melts a little bit. And this process of going up and down creates this sort of partially melted ice, which eventually becomes this hailstone, and eventually it falls down to the earth. So the conjecture is that no matter how high you start up, like you take a number which is in the millions or billions, you go, there's this process that, that goes up if you're odd and down if you're even, it eventually, um, goes down to, to, to earth all the time.
No matter where you start with this very simple algorithm, you end up at one.
Right.
And you might climb for a while.
Right. Yeah. So it's.
Come down.
Yeah. If you plot it, um, these sequences, they look like Brownian motion. Um, they look like the stock market. You know, they just go up and down in a, in a seemingly random pattern. And in fact, usually that's what happens that, that if you plug in a random number, you're supposed to prove at least initially that it would look like, um, uh, random walk. Um, and that's actually a random walk with, uh, a downward drift. Um, it's like if you're always gambling on, on a roulette at, at the casino with odds slightly weighted against you. So sometimes you, you win, sometimes you lose, but over in the long run, you lose a bit more than you win.
Um, and so normally your wallet will hit, will go to zero, um, if you just keep playing over and over again.
So statistically it makes sense.
Yes.
It would go.
So, so the result that I, I proved roughly speaking is such that, that statistically like 99% of all inputs would, would drift down to maybe not all the way to one, but to be much, much smaller than what you started. So it's, it's like if I told you that if you go to a casino, most of the time you end up, if you keep playing for long enough, you end up with a smaller amount in your wallet than when you started. Um, that's kind of like the result that I proved.
So why is that result? Like, can you continue down that thread to prove the full conjecture?
Well, the, the problem is that, um, my, I, I used arguments from probability theory. Um, and there's always this exceptional event. So, you know, so in probability we have these, these low large numbers, um, which tells you things like if you play a casino over, um, a game at a casino over losing, um, expectation over time, you are guaranteed or almost surely with probability, probability as close to 100% as you wish, you're guaranteed to lose money. But there's always this exceptional outlier. Like it is mathematically possible that even in, in the game is, is the 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, you know, um, most of the time, um, 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, um, um, a special number that you stick in that shoots off to infinity while all other numbers crash to earth, uh, crash to one.
Yeah.
Um, in fact, um, there's some mathematicians that, um, who've, Alex Kantorovich for instance, who've proposed that, um, that actually, um, these Kollatz, uh, iterations are like these cellular order meter. Um, actually if you look at what they happen on, on in binary, they do actually look a little bit like, like these game of life type patterns.
Mm-hmm.
Um, and in an analogy to how the game of life can create these, these massive like 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, which is just whose job it is, is to encode, is to create a, a version of itself which is larger.
Heavier than air machine encoded in a number.
Yeah.
That flies forever.
Yeah. So Conway in fact worked, worked on this problem as well.
Oh wow.
So Conway, um, so similar, in fact, that was what were my inspirations for the Navier-Stokes project, that Conway studied generalizations of the Kollatz problem where instead of multiplying by three and adding one or dividing by two, you, you have more complicated branching rules. But, but instead of having two cases, maybe you have 17 cases and then you go up and down. And he showed that once your iteration gets complicated enough, you can actually encode Turing machines and you can actually make these problems undecidable and, and do things like this. In fact, he invented a programming language for, uh, these kind of fractional linear transformations.
He called it a FactTrad as a play on, on Fortrad. Uh, and he showed that, that you, you could, um, you, you can program, it was Turing complete. You could, you could, you could, uh, um, you could make a program that if, if your number you insert in was encoded as a prime, it would sink to zero, it would go down, otherwise it would go up, uh, and things like that. Um, so the general class of problems is, is really, uh, as complicated as all the mathematics.
Some of the mystery of the cellular automaton that we talked about, uh, having a mathematical framework to say anything about cellular automaton, maybe the same kind of framework is required.
Yeah. Yeah.
For Kollatz conjecture.
Yeah. If you want to do it not statistically, but you really want 100% of all inputs to, to fall to earth. Yeah. So what might be feasible is, is, yeah, statistically 99%, you know, going to go to one, but like everything, yeah, that looks hard.
What would you say is out of these within reach famous problems is the hardest problem we have today? Is the Riemann hypothesis?
Riemann is up there. Um, P cos np is, is a good one because like, uh, that's a, that's a meta problem. Like if you solve that in the, um, in the positive sense that you can find a P cos np algorithm, then potentially this solves a lot of other problems as well.
And we should mention some of the conjectures we've been talking about. You know, a lot of stuff is built on top of them now.
Yeah.
There's ripple effects. P cos np has more ripple effects than basically any other.
Right. If the Riemann hypothesis is disproven, um, that'd be a big mental shock to the number theorists. Uh, but it would have follow-on effects for, um, cryptography. Um.
Uh-huh.
Because a lot of cryptography uses number theory, um, uses number theory constructions involving primes and so forth. And, um, it relies very much on the intuition that number theories are built over many, many years of what operations involving primes behave randomly and what ones don't. Um, and in particular, uh, encryption, um, methods are designed to turn text with information on it into text which is indistinguishable from, um, from random noise. So, um, and hence we believe to be almost impossible to crack, um, at least mathematically.
Um, but, uh, if something as core to our beliefs as the Riemann hypothesis is, is wrong, it means that there are, there are actual patterns of the primes that we're not aware of. And if there's one, there's probably gonna be more. Um, and suddenly a lot of our crypto systems are in doubt. Uh.
Yeah. But then how do you then say stuff about the primes?
Yeah.
Like you're going towards the Kollatz conjecture again. Um, because I, I, I, you, you want it to be random, right?
Yes.
You want it to be random.
Yeah. So more broadly, I'm just looking for more tools, more ways to show that, that.
Yeah.
That things are random. How do you prove a conspiracy doesn't happen?
Right. Is there any chance to you that P equals np? Is there some, can you imagine a possible universe?
It is possible. I mean, there's, there's various, uh, scenarios. I mean, there, there's one where it is technically possible, but in fact it's never actually implementable. The evidence is sort of slightly pushing in favor of no, that we probably P is not equal to np.
I mean, it seems like it's one of those cases similar, similar to Riemann hypothesis. I think the evidence is leaning pretty heavily on the no.
Certainly more on the no than on the, on the yes. The funny thing about P cos np is that we have also a lot more obstructions than we do for almost any other problem. Um, so while there's evidence, uh, we also have a lot of results ruling out many, many types of approaches to the problem. Uh, this is the one thing that the computer scientists have actually been very good at. It's actually saying that, that certain approaches cannot work. No go theorems. It could be undecidable. We don't, yeah, 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, uh, you know, what are you gonna do now that you've won this prestigious award? And then you just quickly, very humbly said that, you know, this, a shiny medal is not gonna solve any of the problems I'm currently working on. So I'm just, I'm gonna keep, I'm gonna 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, um, it just shows your humility. But anyway, uh, maybe you could speak to the Fields Medal, but it's another way for me to ask, uh, about, uh, Grigori Perelman.
What do you think about him famously declining the Fields Medal and the Millennium Prize, which came with a $1 million of 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, he's somewhat of an outlier, um, even among mathematicians who tend to, uh, to have, uh, somewhat idealistic views. Um, I've never met him. I, I think I'd be interested in meeting one day, but I, I never had the chance. I, I know people who met him. He's always had strong views about certain things. Um, you know, I mean, it's, it's not like he was completely isolated from the math community. I mean, he would, he would give talks and, and write papers and so forth. Um, but at some point he just decided not to engage with the rest of the community. He was, he was disillusioned or something. Um, I, I don't know. Um, and he decided to, to, uh, uh, to peace out, uh, and, you know, collect mushrooms in St.
Petersburg or something. And, and that's, that's fine. You know, and you can, you can do that. Um, I mean, that's another sort of flip side. I mean, we are not, a lot of problems that we solve, you know, they, some of them do have practical application and that's, that's great. But, uh, like if you stop thinking about a problem, you, you know, so he's, he hasn't published since in, in this field, but that's fine. There's many, many other people who've done so as well. Um, yeah. So I guess one thing I didn't realize initially with the Fields Medal is that it, it sort of makes you part of the establishment.
Um, you know, so, you know, most mathematicians, you know, there's, uh, just career mathematicians, you know, you just focus on publishing the next paper, maybe getting one to promote to one, one rank, you know, and, and starting a few projects, maybe taking some students or something. Yeah. But then suddenly people want your opinion on things and, uh, you have to think a little bit about the, you know, things that you might just so foolishly say because you know no one's gonna listen to you. Uh, it's, 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.
Well, uh.
Play with ideas?
I have a lot less free time than I had previously. Um, I mean, mostly by choice. I mean, I, I, I, I can always see I have the option to sort of, uh, decline. So I decline a lot of things. I, I could decline even more. Um, or I could acquire a reputation being so unreliable that people don't even ask anymore. Uh, this is this.
I love, I love the different algorithms here. This is great.
This is, it's always an option. Um, but you know, um, there are things that are, you know, like, I mean, so, I mean, I, I, I don't spend as much time as I do as a postdoc, you know, just, just working on one problem at a time or, um, 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, um, yeah, so it's, uh, as a postdoc, as publisher or perish, you're, you're incentivized to basically focus on, on proving very technical theorems, so prove yourself, um, as well as prove the theorems.
Um, but then as, as you get more senior, you have to start, you know, mentoring and, and, and, and giving interviews, uh, and, uh, and trying to shape, um, direction of the field, both research-wise and, and, you know, uh, sometimes you have to, uh, uh, you know, do various administrative things. And it's kind of the right social contract because you, you need to, to work in the trenches to see what can help mathematicians.
The other side of the establishment, sort of the, the, the really positive thing is that, um, 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. And it's like.
Yeah, yeah.
It's just how the human mind works. This is where I would probably, uh, say that I like the Fields Medal, that it does inspire a lot of young people somehow. I don't, this is just how human brains work.
Yeah.
At the same time, I also wanna give sort of respect to somebody like Grigori Perelman, who is critical of awards in his mind. Those are his principles. And any human that's able for their principles to like do the thing that most humans would not be able to do. It's beautiful to see.
Some recognition is, is necessary and important. Uh, but yeah, it's, it's also important to not let these things take over your life.
Yeah.
Um, and like only be concerned about, uh, getting the next big award or whatever. Um, I mean, yeah, yeah. So again, you see these people try to only solve like a really big math problems and, and not work on, on, on things that are less, uh, sexy if you wish, but, but, but actually, uh, still interesting and, and instructive. As you say, like the way the human mind works, it's, um, we understand things better when they're attached to humans. Um, and also, uh, if they're attached to a small number of humans, like there's, there's the way our humans, um, mind is, is wired, we can comprehend the relationships between 10 or 20 people.
Uh, but once you get beyond like a hundred people, like there's, there's a, there's a limit. I forget there's a name for it. Um, beyond which, uh, it just becomes the other.
Yeah.
Um, and, uh, so we have, you have to simplify the whole mass of, you know, 99.9% of humanity becomes the other. Um, and, uh, and often these models are, are, are incorrect and this causes all kinds of problems. But, um, so yeah, so to humanize a subject, you know, if you identify a small number of people and say, you know, these are representative people of a subject, you know, role models, for example, um, that has some role. Um, but it can also be, um, uh, yeah, too much of it can be harmful because it's, I'll be the first to say that my own career track path is not that of a typical mathematician.
Um, I had a very accelerated education. I skipped a lot of classes. Um, I think I was had very fortunate mentoring opportunities. Um, and I think I was at the right place at the right time. Just because someone does, doesn't have my, um, trajectory, you know, doesn't mean that they can't be good mathematicians. I mean, they might be mathematicians in a very different style. Uh, and we need people with a different style. Um, and, you know, even if, and sometimes too much focus is given on the, on the person who does the last step to complete, um, a project in mathematics or elsewhere that's, that's really taken, you know, centuries or decades with lots and lots of, building lots of previous work.
Um, but that's a story that's difficult to tell, um, if you're not an expert, 'cause you know, it's easier to just say one person did this one thing, you know, it makes for a much simpler history.
I think on the whole, it, um, is a hugely positive thing to, to talk about Steve Jobs.
Hmm.
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, like pie.
Yeah.
It's Steve Jobs.
Yeah, yeah.
Pie.
As, as a starting point, you know, as a first approximation.
Yeah.
That's how you put it.
And then read some biographies and then look into much deeper first approximation.
Yeah.
That's right. Uh, so you mentioned you were a princeton to, um, Andrew Wiles at that time.
Oh yeah.
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.
Oh yeah.
What did you think, maybe looking back now with more context about that moment in, uh, math history?
Yeah. So I was a graduate student at the time. I mean, I, I vaguely remember, you know, there was press attention and, uh, um, we all had the same, um, uh, we all had pigeonholes in the same mailroom, you know, so we all picked up mail and, and like suddenly Andrew Wiles's mailbox exploded to be overflowing.
That's a good, that's a good metric.
Yeah. Um, you know, so yeah, we, we all talked about it at, at tea and so forth. I mean, we, we didn't understand, most of us still didn't understand the proof. Um, we understand sort of high-level details. Um, in fact, there's an ongoing project to formalize it in Lean, right? Kevin Buzzard is actually.
Yeah. Can, can we take that small tangent? Is it, is it, how difficult is that? 'Cause as, as I understand the Fermat's last, the, the proof for, uh, Fermat's last theorem has like super complicated objects.
Yeah.
Really difficult to formalize now.
Yeah. I guess, yeah, you're right. The, the objects that they use, um, you can define them. Uh, so they've been defined in Lean. Okay. So, so just defining what they are can be done. Uh, that's already not trivial, but it's been done there. But there's a lot of really basic facts about, um, these objects that have taken decades to prove and that, that, that in all these different math papers. And so lots of these have to be formalized as well. Um, Kevin's, uh, 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, um, to number theorists at the time.
Um, and then some other person or some other work would have he done to, to, to get from there. Um, so it's, it's a different area of mathematics than, um, the type of mathematics I'm used to. Um, um, in analysis, which is kind of my area, um, the objects we study are kind of much closer to the ground. We study, I study things like prime numbers and, and, and functions and, and things that are within scope of a high school, um, uh, math education to at least, uh, define. Um, yeah, but then this is very advanced algebraic side of number theory where people have been building structures upon structures for quite a while.
Um, and it's, it's a very sturdy structure. There's, it's, it's, it's been, it's been very, um, at the base at least, it's extremely well developed in the textbooks and so forth. But, um, um, it does get to the point where, um, if you're, if you haven't taken these years of study and you wanna ask about what, what is going on at, um, like level six of, of this tower, you have to spend quite a bit of time before they can even get to the point where you can see, you see something you recognize.
What inspires you about his journey that was similar, as we talked about, seven years, mostly working in secret?
Yeah. Uh, yes, that is a romantic, uh, yeah. So it kind of fits with sort of the, the romantic image I think people have of mathematicians to the extent that they think of them at all as these kind of eccentric, uh, you know, wizards or something. Um, so that certainly kind of, uh, uh, accentuated that perspective. You know, I mean, it's, it is a great achievement. Yeah. His style of solving problems is so different from my own, um, but which is great. I mean, we, we need people like that.
Can you speak to it? Like what, uh, in, in terms of like, uh, you like the collaborative.
I like moving on from a problem if it's giving too much difficulty.
Got it.
Um.
Got it.
But you need the people who have the tenacity and the, the fearlessness. Um, you know, I, I've, I've collaborated with, with people like that where, where I wanna give up, uh, 'cause the, the first approach that we tried didn't work and the second one didn't approach, but they're convinced and they have third, fourth, and the fifth approach works. Um, and I have to eat my words. Okay. I didn't think this was gonna 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. Yeah. I mean, but there are also people who are very productive and they, they focus very deeply on, yeah. I think everyone has to find their own workflow. Um, like one thing which is a shame in mathematics is that we have mathematics, there's sort of a one size fits all approach to teaching, teaching mathematics. Um, and, you know, so we have a certain curriculum and so forth. I mean, you know, maybe like if you do math competitions or something, you get a slightly different experience. But, um, I think many people, um, they don't find their, their native math language, uh, until very late, or usually too late.
So they, they, they stop doing mathematics and they have a bad experience with a teacher who's trying to teach them one way to do mathematics that they, they don't like it. Um, my theory is that, um, humans don't come, evolution has not given us a math center of a brain directly. We have a vision center and a language center and some other centers, um, which have evolution has honed, but we, it doesn't, we don't have an innate sense of mathematics. Um, but our other centers are sophisticated enough that different people, we, we, we can repurpose other areas of our brain to do mathematics.
So some people have figured out how to use the visual center to do mathematics. And so they think, think very visually when they do mathematics. Some people have repurposed their, their language center and they think very symbolically. Um, you know, um, some people like if, if they are very competitive and they're like gaming, there's a type, there's a part of your brain that's very good at, at, at, uh, at solving puzzles and games and, and, and, and that can be repurposed. But like when I talk to other mathematicians, you know, they don't quite think that I can tell that they're using somehow different styles of, of thinking than I am.
I mean, not, not disjoint, but they, they may prefer visual. Like I'm, I, I, I don't actually prefer visual so much. I, I need lots of visual aids myself. Um, you know, mathematics provides a common language, so we can still talk to each other even if we are thinking in, in different ways.
But you can tell there's a different set of subsystems being used in the thinking process.
Yeah. Like, like, they, they take different paths. They're very quick at things that I struggle with and vice versa. Um, and yet they still get to the same goal.
Yeah.
Um.
That's beautiful.
And yeah, but I mean, the way we educate, unless you have like a personalized tutor or something, I mean, education sort of just by nature of scale has to be mass produced. You know, you have to teach the 30 kids and, you know, they have 30 different styles. You can't, you can't teach 30 different ways.
On that topic, what advice would you give to students, uh, young students who are struggling with math and, but are interested in it and would like to get better? Is there something in this?
Yeah. Um.
In this complicated educational context, what, what would you?
Yeah, it's a tricky problem. One nice thing is that there are now lots of sources for math enrichment outside the classroom.
Mm-hmm.
Um, so in, in, in my day, there already there were math competitions. Um, and you know, there are also like popular math books in the library. Um, you know, but, but now you have, you know, YouTube, uh, there, there are forums just devoted to solving, you know, math puzzles and, um, and math shows up in, in other places, you know, like, um, for example, there, there are hobbyists who play poker, uh, for fun. Uh, and, um, they, they, you know, they're for very specific reasons. I'm interested in very specific probability questions. Um, and, and they actually, you know, there's a community of amateur probabilists in, in, in, in poker.
Mm-hmm.
Um, in chess, in baseball. I mean, there's, there's, there's, uh, yeah, um, there's math all over the place. Um, and I'm, I'm, I'm hoping actually with the, with these new sort of tools for, for Lean and so forth, that actually we can incorporate the broader public into math research projects. Um, like this is almost, is, uh, is doesn't happen at all currently. So in the sciences, there is some scope for citizen science, like astronomers, uh, looking at the amateurs who would discover comets and there's biologists, there are people who could identify butterflies and so forth.
Um, and in math, in, um, there are a small number of activities where, um, amateur mathematicians can like discover new primes and so forth. But, but previously, because we have to verify every single contribution, um, like most mathematical research projects, it would not help to have input from the general public. In fact, it would, it would just be, be time consuming 'cause just error checking and everything. Um, but, you know, one thing about these formalization projects is that they are bringing together more, bringing in more people. So I'm sure there are high school students who've already contributed to some of these, these formalizing projects who contributed to MathLib.
Um, 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 at, at, as a very first step opens it up to the programming community too.
Yes.
The people who are already comfortable.
Yes.
With programming. It seems like programming is somehow maybe just a feeling, but it feels more accessible to folks than math.
Yeah.
Math is seen as this like extreme, especially modern mathematics is seen as this extremely difficult to enter area and programming is not. So that could be just an entry point.
You can execute code and you can get results. You know, you can print out the world pretty quickly.
Yeah.
Um, you know, like if, uh, if programming was taught as an almost entirely theoretical subject where you're just taught the, the computer science, the, the theory of functions and, and, and, uh, and, and routines and so forth and, and outside of some, some very specialized homework assignments, you're not actually programmed like on the weekend for fun.
Yeah.
Or, yeah, they would be as considered as hard as math.
Mm-hmm.
Um, yeah. So as I said, you know, there are communities of non-mathematicians where they're deploying math for some very specific purpose, you know, like, like optimizing their poker game and, and for them, then math becomes fun for them.
Uh, what advice would you give in general to young people, how to pick a career, how to find themselves? Like what they could be good at?
Yeah. That's a, that's a tough, tough, tough question. Yeah. So, um, there's a lot less certainty now in the world, you know, I mean, I, there was this period after the war where, uh, at least in the West, you know, if you came from a good demographic, you, uh, you know, like you, there was a very stable path through it to a good career. You go to college, you get an education, you pick one profession and, and you, you stick to it. This is becoming much more I think of the past. So I think you just have to be adaptable and flexible. I think people have to get skills that are transferable, you know, like, like learning one specific programming language or one specific subject of mathematics or something.
It's, it's, it's, that itself is not a super transferable skill, but sort of knowing how to, um, reason with, with abstract concepts or how to problem solve and things go wrong and so forth. Anyway, these are things which I think we will still need even as our tools get, get better and, you know, you'll, you'll be working with AI, sports and so forth.
But actually, you're an interesting case study. I mean, you're like a one of the great living mathematicians, right? And then you had a way of doing things and then all of a sudden you start learning. I mean, first of all, you kept learning new fields.
Yeah.
But you learn Lean. That's not, that's a non-trivial thing to learn. Like that's a.
Yeah.
That's a, for, for a lot of people, that's an extremely uncomfortable leap to take, right?
Yeah. So.
A lot of mathematicians.
Um, first of all, I've, I've always been interested in new ways to do mathematics. I, I, I feel like a lot of the ways we do things right now are inefficient. Um, like, I, I, I spend, like many of my colleagues, we 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. Uh, and why can't we search and get a quick response and so forth? So 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.
And at the time, Peter Schultz had just, uh, formalized one of his, his, um, new theorems. And, um, there were some other developments in computer-assisted proof that look quite interesting. And I said, oh, we should, we should, uh, um, we should run a workshop on this. This would be a good idea. Um, and then I, I was a bit too enthusiastic about this idea. And so I, I, I got volun- voluntold to actually run it. Um, so I did with a bunch of other people, Kevin Buzzard and Jordan Ellenberg and, and, uh, a bunch of other people. Um, and it was, it was a, a, a nice success.
We brought together a bunch of mathematicians and computer scientists and other people and, and we got up speed on state of the art. Um, and it was really interesting, um, developments that most mathematicians didn't know what was going on. Um, that lots of nice proofs of concept, you know, just sort of hints of, of what was going to happen. This was just before ChatGPT, but there was even then there was one talk about language models and the potential, um, capability of, of those in the future. So that got me excited about the subject. So I started giving talks, um, about this is something we should, more of us should start looking at.
Um, now that I, I arranged to run this conference and then ChatGPT came out and like suddenly AI was everywhere. And so, uh, I got interviewed a lot, um, about, about this topic. Um, and in particular, um, the interaction between AI and formal proof assistants. And I said, yeah, they should be combined. This, this is, this is, um, this perfect synergy to happen here. And at some point I realized that I have to actually do not just talk the talk, but walk the walk.
Mm-hmm.
You know, like, you know, I don't work in machine learning and I don't work in proof formalization. And there's a limit to how much I can just rely on authority and saying, you know, I, I, I'm a, I'm a well-known mathematician. Just trust me, you know, when I say that this is gonna change mathematics and I'm not doing it any when I don't do any of it myself. So I felt like I, I had to actually, uh, uh, justify it. You know, a lot of what I get into actually, um, I don't quite see in advance as how much time I'm gonna spend on it. And it's only after I'm sort of waist deep in, in, in, in a project that I, I, 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 sort of challenges that a beginner would, right?
Yeah.
New concepts, new ways of thinking, also, you know, sucking at a thing that others, I think, I think in that talk, you know, you could be a Fields Medal winning mathematician and undergrad knows something better than you.
Yeah. Um, I think mathematics inherently, I mean, mathematics is so huge these days that nobody knows all of modern mathematics. Um, and inevitably we make mistakes and, um, you know, uh, you can't cover up your mistakes with just sort of bravado and, and, uh, I mean, 'cause people will ask for your proofs and if you don't have the proofs, you don't have the proofs. Um.
I don't love math.
Yeah. So it, it does keep us honest. I mean, not, not, I mean, you can still, it's not a perfect, uh, panacea, but I think, uh, we do have more of a culture of admitting error than 'cause 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. Uh, who are the candidates? Euler, Gauss, Newton, Ramanujan, Hilbert.
So first of all, as, as, as mentioned before, like there's a, there's some time dependence, but.
On the day.
Yeah. Like, like if, if you, if you, if you park cumulatively over time, for example, Euclid like, like sort of like.
Yeah.
Is, is, is one of the leading contenders. Um, and then maybe some unnamed, anonymous mathematicians before that. Um, you know, whoever came up with the concept of odd numbers, you know, you know, um.
Do mathematicians today still feel the impact of Hilbert? Just.
Oh yeah.
Directly of what everything that's happened in the 20th century?
Yeah. 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 the declaring which problems.
Yeah.
Are hard to solve.
Yeah.
The statement of the open problems.
Yeah. I mean, this is bystander effect in everywhere, right? Like if, if no one says you should do X, everyone just sort of mills around waiting for somebody else to, to, uh, to do something and, and like nothing gets done. Um, so, and, and like, like, it's the one thing that actually, uh, you have to teach undergraduates in mathematics is that you should always try something. So, um, you see a lot of paralysis, um, in an undergraduate trying a math problem. If they recognize that there's a certain technique that, that can be applied, they will try it.
But there are problems 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. I, or, um, or I think there's a quote from The Simpsons, "I've tried nothing and I'm all out of ideas." Um, so, you know, like the next step then is to try anything, like no matter how stupid, um, and in fact, how almost the stupider the better, um, which, you know, I want a technique which is almost guaranteed to fail, but the way it fails is gonna be instructive. Um, like it, it fails because you, you, you're not at all taking into account this hypothesis.
Oh, this hypothesis must be useful. That's a clue.
I, I think you also suggested somewhere this, this fascinating approach which really stuck with me as they're using and it really works. I think you said it's called structured procrastination.
No, yes.
It's when you really don't wanna do a thing, then you imagine a thing you don't wanna do more.
Yes. Yes.
And that's worse than that. And then in that way you procrastinate by not doing the thing that's worse.
Yeah. Yeah.
That's a nice, it's a nice hack. It actually works.
Yeah. Yeah. It is. Um, I mean, with anything that I, you know, I mean, like you, um, psychology is really important. Like you, you, you, you, you talk to athletes like marathon runners and so forth and, you know, and they talk about what's the most important thing is their training regimen or their diet and so forth. Actually, so much of it is actually psychology. Um, you know, just tricking yourself to, to think that the problem is feasible, um, so that you're motivated to do it.
Is there something our human mind will never be able to comprehend?
Well, I sort of, I guess a mathematician, I mean, you know, like it's reduction. I, I, that's a very large, there must be some such large number that you can't understand. That was the first thing that came to mind.
So that, but even broadly, is there, are we, is there something about our mind that's we're going to be limited even with the help of mathematics?
Well, okay. I mean, it's like how much augmentation are you willing? Like, like for example, if, if I didn't even have pen and paper, um, like 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. Uh, it's, uh, it's one that's been very internalized.
So you're right. They're really, the, the, the formulation of the problem is incorrect 'cause there really is no longer a, just a solo human. We're already augmented in extremely complicated, intricate ways, right?
Yeah. Yeah, yeah.
So we're already like a collective intelligence.
Yes. Yeah. Yes. It's, so humanity plural has much more intelligence in principle on its good days than, than, than the individual humans put together. It can all have less. Okay. But, uh, um, yeah. So yeah, mathemat, math, the mathematical community plural is, is, is incredibly super intelligent, uh, entity. Um, that, uh, no single human mathematician can, can come close to, to, to replicating. You see it a little bit on these like question analysis sites. Um, uh, so this math overflow, which is the math version of stack overflow.
Mm-hmm.
And like sometimes you get like this very quick responses to very difficult questions from the community. Um, and it is, it's, it's a pleasure to watch actually as an, as an expert.
I'm a fan spectator of that, uh, of that site, just seeing the brilliance of the different people there. Um, the depth of knowledge that some people have and the, the willingness to engage in the, in the rigor and the nuance of the particular question. It's pretty cool to watch. It's fun, it's almost like just fun to watch.
Yeah.
Um, what gives you hope about this whole thing we have going on, human civilization?
I think, uh, yeah, um, the younger generation is always like, like really creative and enthusiastic and, and inventive. Um, it's a pleasure working with, with, uh, with, uh, with young students. Um, you know, the, uh, the progress of science tells us that the problems that used to be really difficult can become extremely, you know, can become like trivial to solve, you know? And I mean, like it was like navigation, you know, just, just knowing where you were on the planet was this horrendous problem.
People, people died, um, you know, or, or lost fortunes because they couldn't navigate, you know? And we have devices in our pockets that do this automatically for us, like as a completely solved problem, you know? So things that seem unfeasible for us now could be maybe just sort of homework exercises for me.
Yeah. 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, you know? That, 'cause in the next 100 years, 200 years, just imagine showing, showing up in 200 years.
Yeah. Well, already plenty has happened, you know, like if, if you could go back in time and, and talk to your, your teenage self or something, you know what I mean?
Yeah.
Just the internet and, and, and our AI, I mean, again, they've, they've been, they're getting internalized and say, yeah, of course, an AI can understand our voice and, and give reasonable, you know, slightly incorrect answers to, to any question. But, you know, this was mind blowing even two years ago.
And in the moment, it's hilarious to watch on the internet and so on. The, the drama, uh, people take everything for granted very quickly and then they, we humans seem to entertain ourselves with drama. Well, out of anything that's created, somebody needs to take one opinion, another person needs to take an opposite opinion, argue with each other about it. But when you look at the arc of things, I mean, just even in progress of robotics.
Yeah.
Just to take a step back and be like, wow, this is beautiful that we humans are able to create this.
Yeah. When the infrastructure and the culture is, is healthy, you know, the community of humans can be so much more intelligent and mature and, and, 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, uh, of course, for, uh, for putting those ideas out on the blog. And it's, I can't tell you how, uh, honored I am that you would spend your time with me today. I was, uh, looking forward to this for a long time, Terry. I'm a huge fan. Um, you inspire me. You inspire millions of people. Thank you so much for talking to me.
Thank you. It was a pleasure.
Thanks for listening to this conversation with Terence Tao. To support this podcast, please check out our sponsors in the description or at lexfridman.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.
Terence Tao is widely considered to be one of the greatest mathematicians in history. He won the Fields Medal and the Breakthrough Prize in Mathematics, and has contributed to a wide range of fields from fluid dynamics with Navier-Stokes equations to mathematical physics & quantum mechanics, prime numbers & analytics number theory, harmonic analysis, compressed sensing, random matrix theory, combinatorics, and progress on many of the hardest problems in the history of mathematics.
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OUTLINE:
(00:00) – Introduction
(00:36) – Sponsors, Comments, and Reflections
(09:49) – First hard problem
(15:16) – Navier–Stokes singularity
(35:25) – Game of life
(42:00) – Infinity
(47:07) – Math vs Physics
(53:26) – Nature of reality
(1:16:08) – Theory of everything
(1:22:09) – General relativity
(1:25:37) – Solving difficult problems
(1:29:00) – AI-assisted theorem proving
(1:41:50) – Lean programming language
(1:51:50) – DeepMind’s AlphaProof
(1:56:45) – Human mathematicians vs AI
(2:06:37) – AI winning the Fields Medal
(2:13:47) – Grigori Perelman
(2:26:29) – Twin Prime Conjecture
(2:43:04) – Collatz conjecture
(2:49:50) – P = NP
(2:52:43) – Fields Medal
(3:00:18) – Andrew Wiles and Fermat’s Last Theorem
(3:04:15) – Productivity
(3:06:54) – Advice for young people
(3:15:17) – The greatest mathematician of all time
PODCAST LINKS:
– Podcast Website: https://lexfridman.com/podcast
– Apple Podcasts: https://apple.co/2lwqZIr
– Spotify: https://open.spotify.com/episode/2iH3xRUnROJ1kLD2HA8tRl
– RSS: https://lexfridman.com/feed/podcast/
– Podcast Playlist: https://www.youtube.com/playlist?list=PLrAXtmErZgOdP_8GztsuKi9nrraNbKKp4
– Clips Channel: https://www.youtube.com/lexclips