#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI

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 a sponsor. Check them out in the description or at lexfreedman.com/sponsors. It's the best way to support this podcast. We got notion for teamwork, Shopify for selling stuff online, and that's weed for your business, element for electrolytes and the age you won for your health. Choose wise and my friends. And now onto the full ad reads, they're all here in one place. 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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 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. A more and more, there will be a 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 a little we could do about stopping it. All we can do is steer it in the direction that 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 human civilization. Anyway, sign up for a $1 per month trial period at Shopify.com/lex. That's all lower case. 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. And the 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 you will integrate AI into its tooling, but I think there's a lot of edge cases that we need the human wisdom, the human intuition grounded in 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. 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But nevertheless, I think it's important in the physical domain, the mental domain, 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 that's a serious challenge. 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 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 and all in one daily drink to support better health and peak performance. I drink it every day, I'm preparing for a conversation on drugs in the third rite. And funny enough, it's a kind of way to analyze Hitler's biography, is to look at what he consumed throughout in Norman Oler. There's a great job of analyzing all of that. And tells the story of Hitler and the third rite in a way that hasn't really been touched by historians before. It's always nice to look at key moments in history through a perspective that's not often taken. Anyway, I mentioned 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 would fill him up with all kinds of drugs, but 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 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 supplement and all of that kind of stuff. And depending on what you put in your body will affect your performance in the short term in the long term with meth, that's true with 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. It deserves a multi-hour conversation versus a few seconds of mention, but yeah, all of that was sparked by my thinking of AG1 and how much I love it. I appreciate that you're listening to this and coming along for the wild journey that these 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 Alex Friedman podcast to support it. Please check out our sponsors in the description or atlexf Friedman.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 term. education, you learn about the really hard and possible problems, like the women or fathers, the children, parents, 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 on the boundary between what we can do better easily and what are hopeless. But what are problems where like existing techniques can do like 90% of the job and then you just need that remaining 10%. I think as a PhD student, the Keir 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 Sujika Keir, in like 1918 or so. So the puzzle is that you have a needle on the plane or think like driving on a road and you want it to execute a uter, you want to turn the needle around. But you want to do it as little space as possible. So you want to use this little area in order to turn it around. But the needle is infinitely maneuverable. So you can imagine just spinning it around. It's as a unit needle. You can spin it around its center and I think that gives you a disc of area, I think, pi or four. Or you can do a three point uter, which is what we teach people in the driving schools to do. And that actually takes area pi over eight. So it's a little bit more efficient than a rotation. And so for a while, people thought that was the most efficient way to turn things around. But there's a coverage showed that in fact, you could actually turn the needle around using as little areas you wanted. So point zero zero one, there was some really fancy multi back and forth you turn thing that you could do that you could turn the needle around. And so doing it, it would pass through every intermediate direction. Is this in the international plane? This is an international 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 tube in space. And you want to observe every single star in the universe. So you want to rotate the telescope to every single direction. And he's unrealistic part. Suppose that space is at a premium, which totally is not. You want to occupy as little volume as possible in order to rotate your needle around in order to see every single star in the sky. How small of volume do you need to do that? And so you can modify it as the coverage is construction. And so if your telescope has zero thickness, then you can use as little volume as you need. That's a simple modification of the two-dimensional construction. But the question is that if your telescope is not zero thickness, but just very, 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 you need to get thinner, the volume should go down, but but how fast does it go down? And the conjecture was it goes down very, very slowly, like logarithically. Roughly speaking, and that was proved after a lot of work. So this seems like a puzzle-wise interesting. So it turns out to be surprisingly connected to a lot of problems in posh differential equations, in number theory, in geometry, commentarics. For example, in wave propagation, if you splash some water around, you create water waves, although they travel in various directions. But waves exhibit both both particle and wave type behavior. So you can have what's got a wave packet, which is like a very localized wave that is localized in space and moving at a certain direction in time. And so if you plot it into 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 pebble into a pond and the river spread out. But then if you time reverse that scenario and the equations of wave motion are time reversible, you can imagine ripples that are converging to a single point and then a big splash occurs, maybe even a singularity. And so it's possible to do that. And geometrical was going on is that there's also light rays. So like if this wave represents light, for example, you can imagine this wave as a superposition of photons, all traveling at the speed of light. They all travel on these light rays and they're all focusing at this one point. So you can have a very dispersed wave focus into a very constituted wave at one point in space and time. But then it defocuses again and it separates. But potentially if the conjecture had a negative solution. So what I mean is that there's a very efficient way to pack tubes pointing to different directions into a very, very narrow region of a very narrow volume. Then you would also be able to create waves that start out. There'll be some arrangement of waves that start out very, very dispersed. But they would concentrate, not just at a single point, but there'll be a large, there'll be a lot of concentrations in space and time. And you could create what's called a blow up where these waves, they have to do become so great that the laws of physics that they're governed by are no longer wave equations, but something more complicated and more linear. And so in mathematical physics, we care a lot about whether certain equations in wave equations are stable or not, whether they can create these singularities. There's a famous I saw a problem called the Navier Stokes regularity problem. So the Navier Stokes equations, equations that govern a fluid flow or incompressible fluid is 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 got a singularity. We don't see that in real life, you know, if you splash around water and the bathtub we want to explode on you or have water leaving a speed of light, I think, but potentially it is possible. And in fact, in recent years, the consensus has drifted towards the belief that in fact for certain very special initial configurations of say water, that singularities can form. But people have not yet been able to actually establish this. The Clay Foundation has these seven millennium prize problems as a million dollar prize for solving one of these problems. So this is one of them. Of these seven, only one of them has been solved. They are the point grade conjecture requirement. So the 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, millennial 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 average three-dimensional Navier Stokes equation. Right. 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 moving other question. Yeah. So this is what distinguishes mathematicians from pretty much everybody else. If something holds an act 9.99% of the time, that's good enough for most things. But mathematicians, I want a few people who really care about whether like 100%, really 100% of all situations are covered by. Yeah. So most of the time 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 set of equations that govern in the field of fluid dynamics. Yeah. Trying to understand how fluid behaves and it's actually turns out to be really complicated. You know, fluid is, yeah, extremely complicated thing to try and 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's just a lot of computational fluid dynamics. A lot of it is like you're just trying to solve the Navier Stokes equations as best they can. 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 very important problem practically. Why is it difficult to prove general things about the set of equations like in that blowing up? Short answer is Maxwell's demon. So Maxwell's demon is a concept in thermodynamics. Like if you have a box of two gases in oxygen and hydrogen. And maybe you start with all the oxygen on one side and naturally on the other side. But there's no barrier between them. Then they will mix. And they should stay mixed. There's no reason why they should unmix. But in principle, because of all the collisions between them, there could be some sort of weird conspiracy like that. Like maybe there's a microscopic demon called Maxwell's demon that will every time a oxygen and hydrogen atom collide, they will bounce off in such a way that the oxygen sort of drifts onto one side and then naturally goes to the other. And you could have an extremely improbable configuration emerge, which we never see. And statistically, it's extremely unlikely. But mathematically, it's possible that this can happen and we can't wall it out. And this is a situation that shows up a lot in mathematics. A basic example is the digits of pi, three, four, one, four, one, five, nine, and so forth. The digits look like they have no pattern. And we believe they have no pattern. On the long term, you should see as many ones and twos and threes as fours and fives and sixes. There should be no preference in the digits of pi to favor, let's say seven over eight. But maybe there's some demon in the digits of pi that like every time you can be more more digits, it's a biases one digit to another. And this is a conspiracy that should not happen. There's no reason it should happen. But there's no way to prove it with our current technology. Okay, so getting back to Navier Stokes, fluid has a certain amount of energy. And because if fluid is in motion, the energy gets transported around. And what is also viscous? So if the energy is spread out over many different locations, the natural viscosity of fluid will just damp out the energy and it will go to zero. And this is what happens when we actually experiment with water. I get it, you splash around, there's some turbulence and waves and so forth, but eventually it settles down and the lower the amplitude, the smaller velocity the more calm it gets. But potentially there is some sort of demon that keeps pushing the energy of the fluid into a smaller and smaller scale and we move faster and faster and faster speeds the effective viscosity is relatively less. And so it could happen that it creates us some sort of, it's got a similar blob scenario where you know the end of the fluid starts off at some large scale and then it all sort of transfers energy into a smaller region of the fluid which then at a much faster rate moves into an even smaller region and so forth. And each time it does this it takes maybe half as long as as the previous one. And then you could actually converge to all the energy concentrating at one point in a finite amount of time. And that's an always good finite hand blow up. So in practice this doesn't happen. So water is what's called turbulent. So it is true that if you have a big Eddie of water it will tend to break up into smaller eddies but it won't transfer all this energy from one big Eddie into one smaller Eddie it will transfer into maybe three or four. And then those ones split up into maybe three or four small eddies of their own. And so the energy gets dispersed to the point where the viscosity can then keep a thing under control. But if it can somehow concentrate all the energy keep it all together and do it fast enough that the viscous effects don't have enough time to come everything down then this will up kind of go. So there are papers who had claimed that oh you just need to take into account conservation energy and just carefully use the viscosity and you can keep everything under control for not just the 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 finite hand blow up that have lost you say smooth and it all failed. There was always some sign error or some subtle mistake and it couldn't be salvaged. So what I was interested in doing was trying to explain why we were not able to disprove finite hand blow up. I couldn't do it for the actual equations of fluids which were too complicated but if I could average the equations of motion of Navier Stokes basically if I could turn off certain types of ways in which water interacts and only keep the ones that I want. So in particular 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 one and direct it only into this smaller eddy while still preserving the law of conservation energy. So you try and make a blow up yeah yeah so I basically engineer a blow up by changing laws of physics which is one thing that mathematicians are allowed to do we can change the equation. How does that help you get closer to the proof of something. Right so it provides what's called an obstruction in mathematics. So what I did was that basically if I turned off the certain parts of the equation 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 interactions I could force all the energy to blow in finite time. So what that means is that if you wanted to prove the regularity for Navier Stokes for the actual equation you had you must use some feature of the true equation which my artificial equation does not satisfy. So it rules out certain approaches. So the thing about math is it's not just about finding a technique that is going to work in and applying it but you need to not take the techniques that don't work and for the problem that are really hard often there are dozens of ways that you might think might apply to solve the problem but it's only after a lot of experience that you realize there's no way that these methods are going to work. So having these counter examples for nearby problems kind of rules out it saves you a lot of time because you're not wasting energy on things that you now know cannot possibly ever work. How deeply connected is it to that specific problem of fluid dynamics or is it some more general intuition you build up about mathematics? Right, so the key phenomenon that 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 force coming from viscosity and it's very well understood. It's linear, it calms things down. If viscosity was all there was then nothing bad would ever happen. But there's also transport that energy from in one location of space can get transported because of fluid in motion to other locations and that's a nonlinear effect and that causes all the problems. So there are these two competing terms in the Navier Stokes equation, the dissipation term and the transport term. If the dissipation term dominates, if it's large then basically you get regularity and if the transport term dominates then we don't know what's going on. It's a very nonlinear situation, it's unpredictable, it's tubular. So sometimes these forces are in balance at small scales but not in balance at large scales or vice versa. So Navier Stokes is what's called supercritical. So at smaller and smaller scales the transport terms are much stronger than the viscosity terms. So the viscosity terms are things that calm things down. And so this is why the problem is hard. In two dimensions, so the Soviet mathematician Skier, she in the sixties showed that in two dimensions there was no blow-up and in two dimensions the Navier Stokes equations is what's called critical. The effect of transport and the effect of viscosity are by the same strength even at very very small scales and we have a lot of technology to handle critical and also subcritical equations and prove regularity. But for supercritical equations it was not clear what was going on 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 line of work is that supercriticality versus criticality and subcriticality. This makes a big difference. I mean that that's a key qualitative feature that distinguishes some equations from being sort of nice and predictable and like like planetary motion. I mean there's certain equations that you can predict for millions of years or thousands at least. Again it's not really a problem but there's a reason why we can't predict the weather past two weeks into the future because it's a supercritical equation. Lots of really strange things are going on at very fine scales. So whenever there is some huge source of nonlinearity that can create a huge problem for predicting what's going to happen. Yeah and if nonlinearity is somehow more and more featured and interesting at small scales. I mean there's many equations that are nonlinear but in many equations you can approximate things by the bulk. So for example planetary motion you know if you wanted to understand the orbit of the moon or Mars or something you don't really need the microstructure of like the seismology of the moon or like exactly how the mass is distributed. You just basically you can also approximate these patterns by point masses and just the aggregate behavior is important. But if you want to model a fluid like the weather you can't just say in Los Angeles the temperature is this the wind speed is this for supercritical equations the science confirmation is really important. If we can just linger on the Navier Stokes equations a little bit. So you've suggested maybe you can describe it that one of the ways to 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 competition theory has consequences for fluid dynamics so show it in that way. Can you describe this? Yeah so this came out of all this work of constructing this average equation that blew up. So as part of how I had to do this so this naive way to do it you just keep pushing every time you get energy at one scale you push it immediately to the next scale as fast as possible. This is sort of the naive way to to force blow up. In terms of in five but high the machines this works. But in three dimensions there was this funny phenomenon that I discovered that if you if you if you change loads of physics you just always keep trying to push the energy into small, small scales. What happens is that the energy starts getting spread out into many scales at once. So you have energy at one scale you're pushing it into the next scale and then as soon as it enters that scale you also push it to the next scale but there's still some energy left over from the previous scale. You're trying to do everything at once and this spreads out the energy too much and then it turns out that that it makes it vulnerable for viscosity to come in and actually just damp out everything. So so it turns out this this directive or she doesn't it doesn't actually work. It was a separate paper by some other authors that I actually showed this in three dimensions. So what I needed was to program at the lathe so kind of like air locks. So I needed an equation which would start with a fluid doing something at one scale. It would push this energy into the next scale but it would stay there until all the energy from the from the larger scale got transferred and only after you pushed all the energy in then you sort of open the next gate. And then you push. in as well. So by doing that, the energy interest forward scaled by scale in such a way that it's always localized at one scale at a time. And then it can resist the effects of viscosity, because this is not dispersed. So in order to make that happen, yeah, I had to construct a rather complicated non-linearity. And it was basically like, you know, like it was constructed like an other kind of circuit. So I actually thank my wife for this because she was trained as an electro engineer. And you know, she talked about, you know, she had to design circuits and so forth. And you know, if you want a circuit that does a certain thing, like maybe have a light that flashes on and then turns off and then on and 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 and these diagrams, you can sort of follow up your eyeballs and say, oh yeah, the current will build up here and then it will stop and then it will do that. So I knew how to build the analog of basic electronic components. You're like resistors and capacitors and so forth. And I would I would stack them together and start in such a way 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 us and threshold it would close it kind of a root go bird 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 like if you can imagine kind of a steam punk, but it's really water punk type of thing where, you know, so modern computers are electronic, you know, they're powered by electrons passing through very tiny wires and interacting with other electrons and so forth. But instead of electrons, you can imagine these pulses of water moving in certain velocity. And maybe it's there are two different configurations corresponding to a bit being up or down. Probably that if you had two of these moving bodies of water collide, they would come out with some new configuration which is which would be something like an AND gate or OR gate, you know, that the output would depend on a very critical way on the inputs. And like you could chain these together and maybe create a Turing machine and and then you could you have computers which are made completely out of water. And if you have computers, then maybe you can do robotics. So you know hydraulics and so forth. And so you could create some machine which is basically a fluid analog what's called a VONOMIN machine. So VONOMIN proposed, if you want to colonize Mars, the shear cost of transporting people machines, the Mars is just ridiculous. But if you could transport one machine to Mars and this machine had the ability to mine the planet, create some more materials, to smell them, and build more copies of the same machine, then you could colonize the whole planet over time. So if you could build a fluid machine which, yeah, so it's a fluid robot. And what they would do, 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 configured water would transfer all his energy into the smaller configuration and then power down. Okay, and then like clean yourself up. And then what's left is this newest set which would then turn on and do the same thing but smaller and faster. And then the equation has a certain skating symmetry. Once you do that, it can just keep iterating. So this in principle would create a blur for the actual Navier Stokes. And this is what I managed to accomplish for this average Navier Stokes. So it provided this sort of road map to solve the problem. Now this is a pipe dream because there are so many things that are missing for this to actually be a reality. So I can't create these basic logic gates. I don't have these special configurations of water. I mean, there's candidates that include vortex rings that might possibly work. But also, you know, analog computing is really nasty. Like a bit of digital computing. I mean, because there's always errors, 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 it doesn't interfere with the writing of the smaller machine. But everything in principle can happen. Like it doesn't contradict any of the laws of physics. So it's sort of evidence that this thing is possible. There are other groups who are now pursuing ways to make Navier Stokes blow up which are nowhere near as ridiculously complicated as this. They actually are pursuing much closer to the direct self-similar model, which can. It doesn't quite work as it 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, smaller blob to now having a liquid Turing machine gets smaller, smaller, 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, so the thing about mathematics is that it's really good at spotting connections between what you think of what you might think of as completely different problems. But if the mathematical form was the same, you can draw a connection. So there's a lot of previously, of course, cellular automata. The most famous of which is Conway's Game of Life. This is infinite to speak grid, and at any given time, the grid is occupied by a cell or it's empty. And there's a very simple rule that tells you how these cells evolve. Sometimes cells live, and sometimes they die. When I was a student, it was a very popular screen saber to actually just have these animations go on. And they look very chaotic. In fact, they look a little bit like turbulent flow, sometimes. But at some point, people discovered more and more interesting structures within this Game of Life. So for example, they discovered a single 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 these vortex rings. Yeah, so this is an analogy. The Game of Life is kind of a discrete equation, and the fluid Navi-SOX is a continuous equation, but mathematically, they have some similar features. And so all the 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 to do it, but you can design all kinds of interesting configurations inside it. There's some called a glider gun that does nothing of spit out gliders one at a time. And then after a lot of effort, people managed to create and gates and all gates for gliders. Like this is massive ridiculous structure, which if you have a stream of gliders coming in here and a stream of gliders coming in here, then you may produce a stream of gliders coming out. If maybe if both of the streams have gliders, then there will be an output stream. But if only one of them does, then nothing comes out. So they could build something like that. And once you could build and these basic gates, then just from software engineering, you can build almost anything, you can build a touring machine. I mean, it's like an enormous steam punk type of things. They look ridiculous. But then people also generated self-replicating objects in the Game of Life, a massive machine, a bono machine, which over a huge period of time, and they were always glider guns inside doing these steam punk calculations. They would create another version of itself, which could replicate. It's so incredible. A lot of this was like community crowdsourced by like amateur mathematicians, actually. So I knew about that, that that work. And so that is part of what inspired me to propose the same thing whenever you're stokes. Now if you're just a much, as I said, analog is much worse in digital. Like it's going to be, you can't just directly take the constructions in the Game of Life and plump them in. But again, it just, it shows as possible. You know, there's a kind of emergence that happens with these cellular automata. Local rules, maybe similar to fluids, I don't know, but local rules operating at scale can create these incredibly complex dynamic structures. Do you think any of that is amenable to mathematical analysis? Do we have the tools to say something profound about that? The thing is, you can get this emerging very complicated structures, but only with very carefully prepared initial conditions. So these glider guns and gates and sort of machines, if you just plunk down randomly, some cells, and that you will not see any of these. And that's the analogous situation with Navier Stokes again, that with typical initial conditions, you will not have any of this weird computation going on. But basically through engineering, you know, by, by, by, especially designing things in a very special way, you can pick 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 something interesting. This is, I've recurring challenge in mathematics that I call 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 there's a good example. But there's a very small number of things that have patterns. But 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 does something like repeat itself every so often. You can do that. And you can prove that, for example, you can prove that most sequences of digits have no pattern. So like if you just pick digits randomly, there's some little large numbers that tells you you're going to get as many ones as twos in the long run. But we have a lot fewer your tools 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 have spent a lot of time on is to prove or construct your theorems or inverse theorems that give tests for when something is very structured. So some functions are what's going to add to it. Like if I function the maximum numbers, the natural numbers, so maybe two maps to four, three maps to six and so forth. Some functions are what's going to add to it, which means that if you add two inputs together, the output gets added as well. For example, a multiply by constant. If you multiply a number by 10, if you multiply a plus b by 10, that's the same as one thing, a by 10, and b by 10 and then add them together. So some functions are additive. Some functions are kind of additive, but not completely additive. 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 points, something. So 10 will be up to 14, 20 will be up to 28. So in that case, add it to these two then. So 10 plus 10 is 20, and 14 plus 20 is 28. But because of this rounding, sometimes there's round of errors and sometimes when you add a plus b, this function doesn't quite give you the sum of the two individual outputs, but the sum plus minus one. So it's almost additive, but not quite additive. So there's a lot of useful results in mathematics, and I've worked a lot on developing things like this, to the effect that if I function and exhibit some structure like this, then it's basically, there's a reason for why it's true, and the reason is because there's some other nearby function, which is actually completely structured, which is explaining this sort of partial pattern that you have. And so if you have these two inverse theorems, it creates this sort of dichotomy that either the objects that you study, either have no structure at all, or they are somehow related to some other structure. And in either way, in either case, you can make progress. A good example of this is that this is old theorem in mathematics called semi-radi theorem, proven in the 1970s. It concerns trying to find a certain type of pattern in a set of numbers, but the pattern has make progression, things like 35 and more, or 10, 15 and 20. And the already-onjury has already proved that any set of numbers that are sufficiently big, let's go to positive density, has ethnic progressions in it of any length you wish. So for example, the odd numbers have a set of density one-half, and they contain ethnic progressions of any length. So in that case, it's obvious because the odd numbers are really, really structured. I can just take 11, 13, 15, 17, I can easily find ethnic progressions in that set. But generally, some also applies to random sets. If I take this set of odd numbers and I flip a coin for each number, and I only keep the numbers for which I got a heads. So I just flip coins. I just randomly take out half the numbers, I keep one half. So that's the set that has no patterns at all. But just from random fluctuations, you will still get a lot of ethnic progressions in that set. Can you prove that there's arithmetic progressions of arbitrary length within a random- Yes, I mean, that of the infinite monkey theorem. Usually, mathematicians give boring names to theorists, but occasionally, they give colorful names. The popular version of the infinite monkey theorem is that if you have an infinite number of monkeys in a room with each of a typewriter, they type out text randomly, almost surely one of them is going to generate the entire school of hamlet or any other finite string of text. It will just take some time, quite a lot of time actually. But if you have an infinite number, then it happens. So basically, the theme says that if you take an infinite string of digits or whatever, eventually any finite pattern you wish or you merge, it may take a long time, but it will eventually happen. In particular, ethnic progressions of any length will eventually happen, but you put it in an extremely long random sequence for this to happen. I suppose that's intuitive. It's just infinity. Yeah. Inventic absorbs a lot of sins. Yeah. How are we humans supposed to deal with infinity? Well, you can think of infinity as an abstraction of a finite number of which you do not have a bound for. So nothing in real life is truly infinite. But you know, you can ask these old questions, like, what have I had as much money as I wanted? 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. And often, the mathematics becomes a lot cleaner when you do that. I mean, in physics, we joke about assuming spherical chaos. You know, like reward problems have got all kinds of reward effects, but you can idealize, send some things to infinity, send some things to zero. And the mathematics becomes a lot simpler to work with it. I wonder how often using infinity forces us to deviate from the physics of reality? Yeah. So there's a lot of pitfalls. So, you know, we spend a lot of time, you know, undergraduate math classes, teaching analysis, and analysis. It's often about how to take limits and whether, you know, so for example, 8 plus B is always B plus A. So when you have a finite number of terms, you add them, you can swap them, and there's no problem. But when you have an infinite number of terms, they're these sort of show games you can play where you can have a series which converges to one value, but you rearrange it and suddenly converges to another value. And so you can make mistakes. You have to know what you're doing when you allow infinity. You have to introduce these epsilons and deltas and there's a certain type of wave of reasoning that helps you avoid mistakes. In more recent years, people have started taking results that are true in infinite limits and what's going on and what's going to finalizing them. So you know that's something true eventually, but 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 a large finite number of monkeys, how long do I have to wait for Hamlet to come out? And that's a more quantitative question. And this is something that you can you can attack by purely finite methods and you can use your finite intuition. And in this case, it turns out to be exponential in the length of the text that you're trying to generate. And so this is why you never see the monkeys create a fuller word, but nothing that big. And so I personally find once you find it high, say infinite statement, it's just a much more intuitive and it's no longer so weird. So even if you're working with infinity, it's good to find it out so that you can have some intuition. Yeah. The downside is that the finite ice groups are just much, much messier. And so the infinite ones are found first, usually like decades earlier and then later on, people find it high. So since we mentioned a lot of math and a lot of physics, what is the difference between mathematics and physics as disciplines, as ways of understanding of seeing the world? Maybe we can throw an engineering in there. You mentioned your wife is an engineer, give it new perspective on circuits. So this is a different way of looking at the world given that you've done mathematical physics. So you've worn all the hats. Right. So I think science in general is interaction between three things. There's the real world. There's what we observe are observations. And then our mental models as to how we think the world works. So we can't directly access reality. Okay. All we have are observations, which are incomplete and they have errors. And there are many, many cases where we would we want to know, for example, what is the weather like tomorrow? We don't have the observation and we'd like to predict. And then we have these simplified models, sometimes making unrealistic assumptions, you know, spherical cow type things. Those are the mathematical models. Mathematics is concerned with the models. Science collects the observations and it proposes the models that might explain these observations. What mathematics does it, we stay within the model and ask what are the consequences of that model? What observations, what predictions will the model make of the future observations or past observations to test it fit observed data. So there's definitely a symbiosis. I guess mathematics 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 model. And almost any other discipline, you start with conclusions. I want to do this. I want to board a bridge. I want to make money. I want to do this. Okay. Then you find the paths to get there. There's a lot less sort of speculation about it. Suppose I did this. What would happen? Planning and modeling. Specular defection maybe is one other place. But that's what I did. Actually, most of the things we do in life is conclusions driven, including physics. I mean, they want to know where is this asteroid going to go? What is the weather going to be tomorrow? But mathematics also has this other direction of going from the axioms. What do you think there is this tension in physics between theory and experiment? What do you think is the more powerful way of discovering truly novel ideas about reality? Well, you need both. Top down and bottom up. It's a really an interaction with you in all these things. So over time, the observations and the theory and the modeling should go both get closer to reality. But initially, this is the case out there. They're always far apart to begin with. but you need one to figure out where to push the other. So if your model is predicting anomalies that are not picked up by experiment, that those experimenters were to look to find more data, to refine the models. So it goes back and forth. Within mathematical itself, there's also a theory and experimental component. It's just that until very recently, theory has dominated almost completely like 99% of mathematics is theoretical mathematics. And there's a very tiny amount of experimental mathematics. I mean, people do do it. If they want to study prime numbers or whatever, they can just generate large data sets and with a computer. So once we had a computer, we'd be able to do it a little bit. Although even before, like Gals, for example, he discovered he conjectured the most basic theorem in number theory to call the prime number theorem, which predicts how many primes that are up to a million, up to a trillion. It's not obvious question. And basically what he did was that he computed, I mean, mostly by himself, but also hired human computers, people whose professional job it was to do arithmetic. To compute the first 100,000 primes or something and made tables and made a prediction. And that was an early example of experimental mathematics. But until very recently, it was not, I mean, theoretical mathematics was just much more successful. Because doing complicated mathematical computations was just not feasible until very recently. And even nowadays, you know, even though we have powerful computers, only some mathematical things can be explored numerically, there's some called the combinatorial explosion. If you want to study, for example, Xiamodesh's theory, we want to study all possible subsets of numbers 1 to 1,000, there's only 1,000 numbers, how bad could it be? It turns out the number of different subsets of 1 to 1,000 is 2 to the power 1,000, which is way bigger than the any computer can currently, in fact, any computer ever will have an enumerate. So you have to be, there are certain math problems that very quickly become just intractable to attack by direct brute force computation. Chess is another famous example and the number of chess positions, we can't get a computer to fully explore. But now we have AI, we have tools to explore this space, not with 100% guarantees of success, but with experiment, you know, so we can empirically solve chess now, for example, we have very, very good AI's that they don't explore every single position in the game tree, but they have found some very good approximation. And people are using actually these chess engines to do experimental chess, that they're revisiting old chess theories about, oh, you know, this type of opening, this is a good type of move this is not. And they can use these chess engines to actually refine, in some case, overturn Commission Wisdom about chess. And I do hope that mathematics will have a larger experimental component in the future, perhaps powered by AI. Well, of course, talk about that, in the case of chess, and there's a similar thing in mathematics, I don't believe it's providing a kind of formal explanation of the different positions. It's just saying which position is better or not, that you can intuit as a human being. And from that, we humans can construct a theory of the matter. You've mentioned the Plato's cave algorithm, 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 looking at shadows of reality? Is it possible for us to truly access reality? Well, there are these three ontological things. This actual reality is observations and models. And technically they are distinct, and I think they will always be distinct, but they can get closer over time. So, and the process of getting closer often means that you have to discard your initial intuitions. So astronomy provides great examples, you know, like an initial model of the world is flat because it looks flat and is big. And the rest of the universe, the sky is not like the sun, for example, looks really tiny. And so you start off with a model which is actually really far from reality, but it fits kind of the observations that you have. So things look good, but over time as you make more and more observations, bringing it closer to reality, the model gets dragged along with it. And so over time we had to realize that the Earth was round, that it spins, it goes around the solar system, the solar system goes on the galaxy, and so on and so forth. And the guys part of the universe is expanding, expansions is self-expanding, accelerating. And in fact, very recently in this year or so, this even the acceleration of the universe itself is this evidence that it's non-constant. And the explanation behind why that is, it's catching up, it's catching up. I mean, it's still, you know, the dark matter, dark energy is this kind of thing. Yes. We have a model that sort of explains that fits the data really well, it just has a few parameters that you have to specify. But so people say all that's fudge factors, you know, with enough fudge factors, you can explain anything. But the mathematical point over the model is that you want to have fewer parameters in your model than data points in your observational set. So if you have a model with 10 parameters that explains 10 up to observations, that is completely useless model. It's what's called overfitted. But like if you have a model with two parameters and it explains a trillion observations, which is basically, so the dark matter model, I think it has like 14 parameters and it explains petabytes of data that the electron must have. You can think of all the theory, like one way to think about physical mathematical theory, theory is a compression of the universe and a data compression. So 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 it can fit to reasonable accuracy, you know, almost all of the 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. That's the unreasonable effect and it's the mathematics. Yeah. I'm excited to quote like that. The most incompressible thing about the universe is that it is comprehensible, right. And not just comprehensible. You can do an equation like E equals empty squared. There is actually some mathematical possible explanation for that. So there's this phenomenon in mathematical universality. So many complex systems at the macro scale are coming out of lots of tiny new interactions at the macro scale. And normally because of the common form of explosion, you would think that the macro scale equations must be like infinitely exponentially more complicated than the macro scale ones. And they are, if you want to solve them completely exactly, like if you want to model all the atoms in a box of air, like I've got other numbers, humongous, like 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 want to model a gas of, you know, quintillion particles in a box, you just need to know temperature and pressure and volume in a few parameters, like five or six. And it models almost everything you need to know about these 10th or 23 or whatever particles. And so we have, we don't understand universality anywhere new as we would like mathematically. But there are much simpler toy models where we do have a good understanding of why universality occurs. Most basic one is the central limit theorem that explains why the bell curve shows up everywhere in nature. That so many things are distributed by, I was going to gas distribution, famous bell curve. There's now even a meme with this curve. And even the meme applies broadly, the universality to the meme. Yes, you can go matter if you like. But there are many, many processes, for example, you can take a lot of independent random variables and average them together in various ways, you can take a simple average or more complicated average. And we can prove in various cases that these bell curves, these calcium emerge. And it is a satisfying explanation. Sometimes they don't. So if you have many different inputs and they were correlated in some systemic way, then you can get something very far from a bell curve, show up. And this is also important to know when the situation is really fails. So universality is not a 100% reliable thing to rely on, that global financial crisis was a famous example of this. People thought that mortgage defaults had this sort of calcium type behavior that if you ask if a population of 100,000 Americans with mortgages asked what proportion would default in the mortgages, if everything was decarolated, it would be a nice bell curve and you can manage risk of options and derivatives and so forth. And it is a very beautiful theory. But if there are systemic shocks in the economy, that can push everybody at default at the same time, that's very non-gasting behavior. And this wasn't fully accounted for in 2008. Now I think there's some more awareness that this is a systemic risk as a key up a much bigger issue. And just because the model is pretty and nice, it may not match reality. So the mathematics of working at what models do is really important. Also, the size of validating when the models fit reality and when they don't, I mean, that you need both. But mathematics can help because it can, for example, these central limit themes, it tells you that if you have certain axioms like non-correlation, that if all the inputs were not correlated to each other, then you have these class behaviors that things are fine, it tells you where to look for weaknesses in the model. So if you have a mathematical understanding of central limit theorem and someone proposes to use these Gaussian copulators or whatever to model, default risk, if you're mathematically trained, you would say, okay, but what are the systemic correlation between all your inputs? And so then you can ask the economists, you know, how much risk is that? And then you can co-look for that. So there's always this synergy between science and mathematics. A little bit on the topic of universality. You're known and celebrated for working across an incredible breadth of mathematics, reminiscent of Hilbert, a century ago. In fact, the great field metal-winning mathematician Tim Gowers has said that you are the closest thing we get to Hilbert. He's a colleague of yours, but anyway, so you are known for this ability to go both deep and broad in mathematics. So you're the perfect person to ask, do you think there are threads that connect all the disparate areas of mathematics? Is there a kind of deep underlying structure to all of mathematics? There's certainly a lot of connecting threads and a lot of the progress of mathematics can be represented by taking by stories of two few mathematics that were previously not connected and finding connections. An ancient example is geometry and number theory. So in the times of ancient Greeks, these were considered different subjects. I mean, mathematicians worked on both. You could work both on geometry, most famously, but also on numbers. But they were not really considered related. I mean, a little bit like, you could say that this length was five times this length because you could take five copies of this length and so forth. But it wasn't until Descartes, you really realized that you could develop the geometry. You can parameterize the plane, a geometric object by two real numbers at every point can be. And so geometric problems can be turned into problems about numbers. And today, this feels almost trivial, like there's no content to list. Of course, the plane is XX and Y, and of course, that's what we teach and it's internalized. But it was an important development that these two fields were unified. And this process has just gone on throughout mathematics over and over again. Algebra and geometry were separated, and now we have a suitable algebraic geometry that connects them and over and over again. And that's certainly the type of mathematics that I enjoy the most. So I think there's sort of different styles to being a mathematician. I think Hedgehogs and Fox. Fox knows many things a little bit, but a Hedgehog knows one thing very, very well. And in mathematics, there's definitely both Hedgehogs and Foxes. And then there's people who can play both roles. And I think, I do a collaboration between mathematicians involves a very, you need some diversity. You don't like a Fox working with many Hedgehogs or vice versa. So yeah, but I identify mostly as a Fox, certainly. I like arbitrage somehow, you know, like learning how one field works, learning the tricks of that wheel and going to another field, which people don't think it is related, but I can adapt the tricks. So see the connections between the fields. Yeah. So there are other mathematicians who are far deeper, but I am like, they're really, they're really Hedgehogs. They know everything about one field and they're much faster and more effective in that field. But I can, I can give them these extra tools. I mean, you said that you can be both a Hedgehog and the Fox, depending on the context, depending on the collaboration. So what can you, if it's at all possible, speak to the difference between those two ways of thinking about a problem, say you're encountering a new problem, you know, searching for the connections versus like very singular focus. I'm much more comfortable with the, the, the Fox paradigm. Yeah. So yeah, I like looking for analogies, narratives. 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 cause out, but I don't like the proof. I get users types of mathematics that I'm not super familiar with. I often try to re-prove it myself using the tools that I favor of my proof is worse. But by the exercise you're doing so, I can say, oh, now I can see what the other proof was trying to do. And from that, I can get some understanding of, of the tools that are used in, in that field. So it's very exploratory, very, doing crazy things, and crazy fields, and like reinventing the wheel a lot. Yeah. Whereas the Hedgehog style is, I think what's more scholarly, you know, you, you, you 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, you have a very good understanding of, of exactly the strength and weaknesses of each particular technique. Yeah. I think you, you rely a lot more on sort of calculation than sort of trying to find narratives. So yeah, I mean, I could do that too, but there are other works extremely good at that. Let's start back and maybe look at the, the, the, a bit of a romanticized version of mathematics. So I think you've said that early on in your life, math was more like a puzzle solving activity when you were young. When did you first encounter a problem or proof where you realize math can have a kind of elegance and beauty to it? That's a good question. When I came to graduate school in Princeton, so John Conway was there at the time, he passed away a few years ago, but I remember one of the very first research talks I went to was a talk by Conway on what he called extreme proof. So Conway had just said this is an amazing way of thinking about all kinds of things in a way that you would normally think of. So he thought of proofs themselves as occupying some sort of space, you know, so, so if you want to prove something, let's say that there's infinitely many primes, okay, you're always different proofs, but you could, you could rank them in different axes, like some proofs are elegance, some proofs are long, some proofs are elementary and so forth. And so this is cloud, so this space of all proofs itself has some sort of shape. And so he was interested in extreme points of this shape, like all these proofs, what's one of those? These shortest at the expense of everything else or the most elementary or whatever. And so he gave some examples of well-known theorems and then he would give what he thought was the extreme proof in these different aspects. I just found out really eye-opening that, you know, it's not just getting a proof for what was interesting, but, but once you have that proof, you know, trying to, to, to optimize it in various ways, that proof, proofing itself had some craftsmanship to it. It's something for my writing style, that, you know, like when you do your math assignments and as you undergraduate, your homework and so forth, you're sort of encouraged to just write down any proof that works, okay, and the hand is in, as long as it gets a tick mark, you move on. But if you want your, your, your results to actually be influential and be read by people, it can't just be correct. It should also be a pleasure to read, you know, motivated, be adaptable to, to generalize other things. It's the same in many other disciplines, like coding, there's a, there's a lot of analogies between math and coding, I like analogies, if you haven't noticed, but, you know, like, you can code something spaghetti code that works for a certain task and it's quick and dirty and it works, but there's lots of good principles for, for writing code well, so that other people can use it, board upon it, and so on, that has fewer bugs and whatever. And there's some of the things with mathematics, so yeah, the, first of all, there's so many beautiful things there. And comma is one of the great minds in mathematics ever and computer science. Just even considering the space of proofs, yeah, and saying, okay, what is the space to look like and what are the extremes? Like you mentioned, coding is an analogies, interesting, because there's also this activity called code golf, which I also find beautiful and fun, where people use different programming languages to try to write the shortest possible program that accomplishes a particular task. Then I believe there's even competitions on this. It's also a nice way to stress tests, not just the, sort of for the programs or in this case, the proofs, but also the different languages, maybe that's a different notation or whatever, to use the catalog opposite of the task. Yeah, you learn a lot. I mean, it may seem like a frivolous exercise, but it can generate all these insights, which, if you didn't have this artificial objective to pursue, you might not see. What to use the most beautiful or elegant equation in mathematics? I mean, one of the things that people often look to in beauty is the simplicity. So if you look at E equals obviously squared, so when a few concepts come together, that's why the Euler idea. identity is often considered the most beautiful equation mathematics. Do you find beauty in that one in the oil identity? - Yeah, well, as I said, I mean, what I find most appealing is connections between different things that you like. So if you eat the pie I equals minus one. So yeah, people, oh, this is all the fundamental constants, okay, that's cute. But to me, so the exponential function was, to measure exponential growth. I think compound interest or decay or anything which is continuously growing, continuously decreasing growth in decay or dilation or contraction is modeled by the exponential function. Whereas pie comes around from circles and rotation. If you want to rotate the needle, for example, 100 degrees, you need to rotate by pie radians. And I, complex numbers, represents the swapping which we would imagine axes of a 90 degree rotation. So a change in direction. So the exponential function represents growth in decay in the direction where you really are. When you stick an eye in the exponential, now it's instead of motion in the same direction as your composition, it's a motion as a right angle to your composition, so rotation. And then so if the pie 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 complexification, but by the way, so it connects together all these two as mathematics. Yeah, the time was different complex and complex and the complex numbers, they were considered almost, yeah, they were all next to all neighbors in mathematics because of this identity. - Did you think the thing you mentioned is cute, the collision of notations from these disparate fields is just a frivolous side effect. Or do you think there is legitimate like value in when the notation, all the old friends come together at night, but it's confirmation that you have the right concepts. So when you first study anything, you have to measure things and give them names. And initially sometimes you're, because your model is again too far off from reality, you give the wrong things the best names and you only find out later what's really important. - Physicists can do this sometimes. I mean, but it turns out okay. - So actually the physics of it, so it equals empty squared. Okay, so one of the big things was the E, right? So when Aristotle first came up with his laws of motion and then Galileo and Newton and so forth, you know, they saw the things they could measure. They could measure mass and acceleration and force and so forth. And so Newtonian mechanics, for example, Ethicals MA was the famous Newton signal of motion. So those were the primary objects. So as they gave them the central building in the theory, it was only later after people started analyzing these equations that they always seem to be these quantities that were conserved. So a particular momentum in energy. And it's not obvious that things happen energy. Like it's not something you can directly measure at the same way you can measure mass and velocity. So both, but all the time people realized that this was actually a really fundamental concept. Hamilton, eventually in 19th century, reformulated Newton's laws of physics into what it'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 just completely detect the dynamics, like what happens to it or to all the states, like it's, it really was a central actor, which was not obvious initially. And this helped actually, this change of perspective really helped when quantum mechanics came along. Because the early physicists who studied quantum mechanics, they had a lot of trouble trying to adapt in Newtonian thinking because the other thing was particle and so forth to quantum mechanics. Because I think because it was a wave, it just looked really, really weird. Like what is the quantum version of F equals M A? And it's really, really hard to give an answer to that. But it turns out that the Hamiltonian, which was so secretly behind the scenes in classical mechanics, also is the key object in quantum mechanics, that there's also an object called Hamiltonian. It's a different type of object, it's also called an operator rather than a function, but again, once you specify it, you specify the tie dynamics. So the sun goes through in this equation. That tells you exactly how quantum systems evolve once you have a Hamiltonian. So side by side, they look completely different objects. 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 classical mechanics, there's this thing called nervous 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 ten steps to the left, I experience the same laws of physics as I was here. And that corresponds to conservation momentum. If I turn around by some angle, again, I experience the same laws of physics, this corresponds to the conservation angle momentum. If I wait for 10 minutes, I still have the same laws of physics. So there's time transition to variance, this corresponds to the law of conservation energy. So there's this fundamental connection between symmetry and conservation. And that's also true in quantum mechanics. Even though the equations are completely different, but because they're both coming from the Hamiltonian, the Hamiltonian controls everything. Every time the Hamiltonian is a symmetry, the equations will have a conservation law. So it's, it's, it's, it's, it's, once you have the right language, it actually makes things a lot, a lot cleaner. One of the problems is why we can't unify quantum mechanics and general relativity yet. We haven't figured out what the fundamental object is like. For example, we have to give up the notion of space and time being these almost clean enough spaces. And it has to be, you know, and, you know, we kind of know that at very tiny scales, there's going to be quantum fluctuations. There's space, space time foam. And trying to use Cartesian coordinates x, y, z is going to be, it's just, it's a non-starter. But we don't know how to, what to replace it with. We don't actually have the mathematical with concepts. 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 out of unification, much like mathematics over the years. You know, electricity and magnetism was separate theories. And then backs will unify them. You know, and you can unify the motions of heavens for the motions of objects on the earth and so forth. So it should happen. It's just that the, again, to go back to this model of the observations and theory. Part of our problem is that physics is a victim of its own success. That of two big theories of physics, general relativity, and quantum mechanics are so good now. So together, they cover 99.9% of sort of all the observations we can make. And you have to, like, either go to extremely insane particle celebrations or the early universe or things that are really hard to measure in order to get any deviation from either of these two theories to the point where you're going to keep figure out how to come together. But I have faith that we've been doing this for centuries. We've made progress before. And there's no reason why we should stop. Do you think you will be a mathematician that develops a theory of everything? What often happens is that when the physicists need something of mathematics, there's often some precursor that the mathematicians worked out earlier. So when Einstein started realizing that space was curbed, he went to some mathematician and asked, is there some theory of curve space that the mathematicians already came up with that could be useful? And he said, oh, yeah, I think we mind came up with something. And so we might have developed a remaining geometry, which is precisely a theory of spaces that are curbed in various general ways, which turn out to be almost exactly what was needed by Einstein's theory. This has been effective to witness unreasonable effectiveness on mathematics. I think the theories that work well fix them in universe tend to also involve the same mathematical objects that work well to solve mathematical problems. Ultimately, there's just both ways of organizing data in useful ways. It just feels like you might need to go to some weird land that's very hard to into it. You have string theory. Yeah, that was a leading candidate for many decades. It's something is slowly pulling out of fashion, because it's not matching experiment. So one of the big challenges, of course, like you said, is experiment is very tough, because of the how effective both theories are. But the other is just you're talking about you're not just deviating from space time. You're going into some crazy number of dimensions. You're doing all kinds of weird stuff that to us, we've gone so far from this flat earth that we started. Yeah, yeah, yeah, yeah, we're just, it's very hard to use. Our limited a descendants of a cognition to into it. What that reality really is like, this is why analogies are so important. I mean, so yeah, the round earth is not intuitive, because we're stuck on it. But round objects in general, we have pretty good intuition. And we've introduced about light works and so forth. And it's actually a good exercise to work out how eclipses and phases of the sun and the moon and so forth. It can be really easy to explain by round earth and round moon and models. And you can just take a basketball and a golf ball and a light source and actually do these things yourself. So the intuition is there, but you have to transfer it. There is a big leap into lecture for us to go to from flat to round earth, because 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 emptiness for this kind of thing, but you know, we're in a rough 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 leap progress. Right. Yeah. So modern science is maybe, again, a victim of his own success is that, yeah, in order to be more accurate, it has to move further further away from your initial intuition. And so for someone who hasn't gone through the whole process of science education, it looks more more suspicious because of that. So, you know, we need we need more grounding. I mean, I think, I mean, you know, there are scientists who do excellent outreach. But there's this, there's lots of science things that you can do at home at this, lots of YouTube videos. I did a YouTube video recently with Grant Sanderson, we talked about this earlier, that, you know, how the ancient Greeks were able to measure things like the distance of the moon, distance of the earth. And, you know, using techniques that you could also replicate yourself, it doesn't all have to be like fancy space telescopes and very intimidating mathematics. Yeah, that's, I highly recommend that. I believe you have a lecture and you also did an incredible video with Grant. It's a beautiful experience to try to put yourself in the mind of a person from that time shrouded in a mystery. 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 your 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 some other persons, some other time period, make hypotheses, spherical cows, whatever, you know, speculate. And, you know, this is, this is what mathematicians do and some other 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. We don't have much to figure out stuff with. If you propose axioms, then mathematics lets you follow those axioms to it to their conclusions. And sometimes you can get quite a lot, quite a long way from, you know, initial hypotheses. If you're going to stay in the land of the weird, you mentioned general relativity. You've contributed to the mathematical understanding, lifestyles, field equations. Can you explain this work and from a sort of mathematical standpoint, what aspects of general relativity are intriguing to you, challenging to you? I have worked on some equations. There's something called the wave maps equation, all of 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. So, all right, times equations of relativity just describe space-time itself. But then there's other fields that live on top of that. There's the electromagnetic field, there's things like Yang-Mills fields. And there's this whole hierarchy of different equations, of which aren't signs considered one of the most nonlinear and difficult. But relatively low on a hierarchy was this thing called the wave maps equation. So it's a wave, which at any given point is fixed to be like on a sphere. So I can think of a bunch of arrows in space and time and yeah, so it's pointing in different directions. But they propagate like waves. If you wiggle an arrow, it will propagate and make all the arrows move kind of like a sheep's or wheat in a wheat field. And I was interested in the global or clouded problem again for this question. Is it possible for all the energy here to collect at a point? So equation, I considered what exactly was called a critical equation, where it's actually the behavior at all scales is roughly the same. And I was able barely to show that you couldn't actually force a scenario where all the energy concentrated at one point. But at the end, you had to dismiss a little bit and moment it was a little bit, it would stay regular. Yeah, this was back in 2000. That was part of why I got into the Narysox afterwards, actually. Yeah, so I developed some techniques to solve that problem. So part of it was, this problem is really nonlinear because of the curvature of the sphere. There was a certain nonlinear effect, which was a non perturbative. It was when you sort of looked at it normally, it looked larger than the linear effects of the wave equation. And so it was hard to keep things under control, even when the energy was small. But I developed what's called a gauge transformation. So the equation is kind of like an evolution of of sheep's or wheat and they're all bending back and forth. And so there's a lot of motion. 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 equation to reduce the amount of nonlinear effects and then I was able to solve the equation. I found this transformation while visiting my art in Australia. And I was trying to understand the dynamics of all these fields, and I couldn't do a pen and paper. And I had not necessarily computers to do any computer simulations. So I ended up closing my eyes on the floor and just imagining myself to actually be the spectrophil and rolling around to try to see how to change coordinates in such a way that somehow things in order directions would behave in a reasonably linear fashion. And my art walked in while I was doing that. And he was asking, what am I doing doing this? It's complicated. Okay, fine. You're a young man. I don't ask questions. I have to ask about the, you know, 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? A lot of pen and paper. One thing you pick up as a mathematician is sort of a colored cheating strategically. So the beauty of mathematics is that you get to change the vote, change the problem, change the rules as you wish. You don't get to do this for any other field. Like, you know, if you're an engineer and someone says, it brought a bridge over this river, you can say, I want to build this bridge over here instead or I want to put out a paper instead of steel. But a mathematician, you can do whatever you want. It's like trying to solve a computer game where you can, there's unlimited cheat codes available. And so, you know, you can, you can set this, so there's a dimension that's large. I've set it to one. I'd solve the one-dimensional problem first, so there's a main term and an error term. I'm going to make a spherical call assumption as I'll assume the error term is zero. And so the way you solve these problems is not in sort of this iron man mode where you make things maximally difficult. But actually, the way you should approach any reasonable math problem is that you, if there are 10 things that are making it like difficult, find a version of the problem that turns off 9 of the difficulties, but only keeps one of them. And solve that. And then that just, so you, you, you, you, you install nine sheets. Okay, you saw 10 sheets then then the game is trivial, but you saw nine sheets. You saw one problem that, that, that, that, that, that, that teaches you how to do all that particular difficulty. And then you turn that one off and you see someone else, someone else else on and then you saw that one. And after you, you know, how to solve the 10 problems, 10 difficulty separately, then you have to start merging them a few at a time. I, I was a kid, I watched a lot of these Hong Kong action movies, this is from a culture. And one thing is that every time it's the fight scene, you know, so maybe the hero gets swarmed by 100 bad guy goons or whatever. But it'll always be choreographs, so that you'd always be only fighting one person at a time, and then it would defeat that person and move on. And because of that, they could, they could defeat all of them. But whereas if they had fought a bit more intelligently, and just swarmed the guy once, it would make for much, much, much worse coil and cinema, but, but they would win. Are you usually pen and paper? Are you working with computer and late tech? I'm mostly pen and paper actually. So in my office, I have four giant blackboards. And sometimes I just have to write everything I know about the problem on the full blackboards and then sit 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 only makes sense to me. I mean, and that's a bit of a blackboard you raise. It's a very organic thing. I'm beginning to use more more computers, partly because AI makes it much easier to do simple coding things. That, you know, if I wanted to plot a function before which is moderately complicated as some iteration or something, I'd have to remember how to set up a Python program and how does a full loop work and debug it and it would take two hours and so forth. And now I can do it in 10, 15 minutes as much. Yeah, I'm using more and more computers to do simple explorations. Let's talk about AI a little bit if we could. So maybe a good entry point is just talking about computer-assisted proofs in general. Can you describe the lean formal proof programming language in how it can help as a proof assistant and maybe how you started using it and how it has helped you? So lean is a computer language, much like sort of standard languages like Python and C and so forth. Except in most languages, the focus is on using executable code. Lines of code do things. You know, they flip bits or they make a real one move or they deliver you text or anything or something. So lean is a language that can also do that. It can also be run as a standard traditional language but it can also produce certificates. So a software like Python might do a computation and give you the answer is seven. Okay, that it does the sum of people was equal to seven. But lean can produce not just the answer, but a proof that how it got the answer of 7 as 3 plus 4 and all the steps involved in. So it creates these more complicated objects, not just statements, but statements with proofs attached to them. And every line of code is just a way of piecing together previous statements to create new ones. So the idea is not new. These things are called proof assistance. And so they provide languages for which you can create quite complicated intricate mathematical proofs. And they produce these certificates that give it 100% guarantee that your arguments are correct. If you trust the compiler, obviously, but they make the compiler really small. And you can, there are several different compilers available for the same level. Can you give people some intuition about the difference between writing on pen and paper versus using lean programming language? How hard is it to formalize statement? So lean, a lot of mathematicians were involved in the design of lean. So it's it's designed for individual lines of code, resemble individual lines of mathematical argument. You might want to introduce a variable. You want to want to improve our contradiction. There are various standard things that you can do. And it's written so ideally, it should be like a one-to-one correspondence. In fact, it isn't because lean is like explaining a proof to an extremely pedantic colleague who will point out, okay, did you really mean this? Like, what happens if this is zero? Okay, how do you justify this? So lean has a lot of automation in it to try to to to be less annoying. So for example, every mathematical object has to come with a type. Like if I talk about X, is X a rule number or a natural number or a function or something. If you write things informally, it's something in firm context. You say, clearly X is equal to, let X be the sum of Y and Z and Y and Z were already rule numbers. So X should also be a rule number. So lean can do a lot of that. But every so often, it says, wait a minute, can you tell me more about what this object is? What type of object it is? You have to think more at a philosophical level, not just sort of computations you're doing, but sort of what each object actually is in some sense. Is it using something like LLMs to do the type inference or like you match with the real number? It's it's using much more traditional or good or fashion to 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 real number or a natural number. Yeah, every object sort of comes with a history of what it came from and you can kind of trace all I see. Yeah, so it's it's it's designed for reliability. So modern AI's are not used in it's a disjointed technology. People are beginning to use AI's on top of lean. So when a mathematician tries to program a proof in lean, often there's a step. Okay, now I want to use the fundamental thing we'll call this say to do the next step. So the lean developers have built this massive project called methylib collection of tens of thousands of useful facts about methodical objects. And somewhere in there is the fundamental thing with calculus. But you need to find it. So a lot of the bottleneck now is actually LLMs search. You know, there's a tool that that you know is in there somewhere and you need to find it. And so you can there are various search engines specialized for methylib that you can do. But there's now these large language models that you can say, I need the fundamental thing with calculus at this point. And I said, okay, for example, when I code I have GitHub code pilot installed as a plugin to my IDE and it scans my text and it sees what I need. It says, you know, I'm not even typing. Okay, now I need to use the fundamental thing with calculus. Okay, and then it might just 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's close enough that I can say, oh, if I just change it here and here it will work. And then like half the time it gives me complete rubbish. So but people are beginning to use AI's a little bit on top. Most of the level of basically fancy autocomplete that you can type half of one line of a proof and it will find you'll tell you what a fancy especially fans with the sort of capital that are f is removed some of the friction. Yeah. Mathematician might feel when they move from pattern paper to formalizing. Yes, yeah. So right now I estimate that the effort time and effort taken to formalize the proof is about 10 times the amount taken to write it out. Yeah, so it's doable. But 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 worker. Right. Yeah, if that was the only aspect of it. Okay, but okay, there's something there's something because it was actually more pleasant to do this formally. So there's a theory of my formalized and there was a certain constant 12 that came out of it in the final statement. And so this 12 had be carried all through the proof. And like everything had to be checked that it goes all the all these other numbers that had be consistent with this final number 12. And then so we want 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 we working some of these steps. And when this happens with pen and paper, every time you change your 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 problems with number 12 that you didn't even realize that you were taking advantage of. So a proof can break down at a subtle place. So we had formalized the proof with this constant 12. And then when this new paper came out, we said, oh, okay, so that took like three weeks to formalize and like 20 people to formalize this original proof. I said, oh, but now let's update the total 11. And what you can do with lean, so you just in your headline theorem, you change your 12 to 11, you run the compiler. And like of the thousands of lines that code you have, 90% of them still work. And there's a couple that are line and red. Now I can't justify these steps, but it means it isolates which steps you need to change. But you can skip over everything, which works just fine. And if you program things correctly, with good programming practices, most of your lines will not be red. And there'll just be a few places where you, I mean, if you don't hard code your constants, but you sort of, you smart tactics and so forth, you can you can localize the things you need to change to a very small period of time. So it's like within a day or two, we had updated our proof, because this is a very quick process, you make a change. There are 10 things now that don't work for each one. You make a change. And now there's five more things that don't work, but the process converges much more smoothly than with pen and paper. So that's for writing. Are you able to read it? Like if somebody else doesn't approve, they're able to like, how, what's the versus paper? And yeah, so the proofs are longer. But each individual piece is easier to read. So if you take a math paper and you jump to page 27 and you look at paragraph six and you have a line of text or math, I often can't read it immediately, because it assumes various definitions, which I have to go back and maybe on 10 pages earlier, this was defined. And the proof is scattered all over the place and you basically are forced to read fairly sequentially. It's not like say a novel where like, you know, in a theory, you could open up a novel halfway through it and start reading. There's a lot of context. But when I've proven lean, if you put your cursor on a line code, every single object there, you can hover over it. And it would say what it is, what it came from, where it's 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, pen and paper, when you want to collaborate with another mathematician, either you do it at a blackboard where you can really interact. But if you're doing it sort of by email or something, basically, yeah, you have to segment it. I'm going to finish section three, you do section four, but you can't really sort of work on the same thing, collaborate at the same time. But with lean, you can be trying to formalize some portion of the proof and say, I got stuck at line 67 here. I need to prove this thing, but it doesn't quite work. Here's like the three lines of code I've been trouble with. But because all the context is there, someone else can say, oh, okay, I recognize what you need to do. You need to apply this trick or this tool. And you can do extremely atomic level conversations. So because of lean, I can collaborate with dozens of people across the world. Most of them I don't have never met in person. And I may not know, actually, even whether they're reliable there are in the proofs they can make. But lean gives me a certificate of trust. So I can do trust the mathematics. So there's so many interesting questions. There's one you're known for being a great collaborator. So what is the right way to approach solving a difficult problem in mathematics when you're collaborating? Are you doing a divide and conquer type of thing or are you focused in a particular part and your brain storming? There's always a brain storming 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. It's not like an engineering project where some other theory has been established for decades, and its implementation is the main difficulty. You have to figure out even what is the right path. So this is what I said about cheating first, you know, it's like to go back to the bridge building analogy. So first, assume you have an infinite budget and unlimited amounts of workforce and so forth. Now can you build this bridge? Okay, now have infinite budget, but only finite workforce now can you do that and so forth. So I mean, of course, no engineer can actually do this because they have fixed requirements. Yes, there's this sort of jam sessions at the beginning where you try all kinds of crazy things and you make all these assumptions that aren't realistic, but you plan to fix later. And you try to see if there's even some skeleton and I'm going to push them, might work. And then hopefully that breaks up the problem into smaller subproblems, which you don't know how to do. But then you focus on the subproblems. And sometimes different collaborators are better at working on certain things. So one of my themes I'm known for is a theme of Ben Green, which is called the Green Talfeerum. It's a statement that the primes contain earthquake precautions of any event. So it was a modification of the theme of similarity. And the way we collaborated was that Ben had already proven a similar result for progressions of then three. He showed that sets like the primes contain loss and loss of questions of plan three, even subsets of the primes, certain subsets do. But his techniques only worked for them through questions. They didn't work for longer progressions. But I had these techniques coming from a gothic theory, which is something that I had been playing with and I knew better than I'd been at the time. And so if I could justify certain randomness properties of some set relating to the primes, there's a certain technical condition, which if I could have it. If Ben could supply me this fact, I could conclude the theorem. But what I asked was a really difficult question in number theory, which he said, no, there's no way we can prove this. So he said, can you prove your part of the theorem using a weak hypothesis that I have a chance to prove it? And he proposed something which he could prove that it was too weak for me. I can't use this. So there was this conversation going back and forth. It's still different cheats, too. Yeah, I want to cheat more. He wants to cheat less. But eventually we found a property which A he could prove and B I could use. And then we could prove out here. And so there's all kinds of dynamics. I mean, every collaboration has some story. There's no two of the same. And then on the flip side of that, like you mentioned, would lean programming. Now that's almost like a different story because 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. So it makes everything compatible and yeah, and trustable. 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. And math paper, basically, is a blueprint, in a sense. It is taking a difficult statement, like a big theorem and breaking it up into me, are 100 little limits. 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 much detail as possible. And to try to make each step kind of self-contained, and depending on only a very specific number of previous statements that I've been proven, so that each node of this blueprint graph that gets generated can be tackled independently of all the others. And you don't even need to know how the whole thing works. So it's like a modern supply chain. And if you want to create an iPhone or some other complicated object, no one person can build a single object. But you can have specialists 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 there's a really exciting possibility, because you can have, if you can find problems that could be broken down in this way, then you could have thousands of contributors, right? Yes, yes, yes, yes. 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 is experimental. I think the platform that lean and other software tools, so a GitHub and things like that, allow, they will allow experimental mathematics to be to scale up to a much greater degree than we can do now. So right now, if you wanted to do any mathematical exploration 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 says one mathematician coding lots and lots of Python, whatever. And because coding is such an error for an activity, it's not practical to allow other people to collaborate with you on writing modules for your code, because if one of the modules has a bug in it, the whole thing is unreliable. So you get these spook spaghetti code written by non-professional programmers with my mathematicians. And they're clunky and slow. And so because of that, it's hard to really mass-produce experimental results. But 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 Theory's Project. Basically, we generated about 22 million little problems at abstract algebra. We should back up and tell you what the project is. Okay, so abstract algebra studies operations like multiplication and addition and the abstract properties. Okay, so multiplication, for example, is commutative. X times Y is always Y times X is for numbers. And it's also associative. X times Y times Z is the same as X times Y times Z. So these operations obey some laws that don't obey others. For example, X times X is not always equal to X. So that laws are not always true. So given any operation, it obeys some laws and not others. And so we generated about 4,000 of these possible laws of algebra that certain operations can satisfy. And our question is, which laws imply which other ones? So for example, does commutativity imply associativity? And the answer is no, because it turns out you can describe an operation which obeys the cognitive law, but it doesn't obey the associative law. So by producing an example, you can show that commutatively does not imply associativity. But some of the laws do imply other laws by substitution and so forth. And you can write down some algebraic proofs. So we look at all the pairs between these 4,000 laws and this up to 22 million of these pairs. And for each pair, we ask, does this law imply this law? If so, give a proof, if not, give a count example. So 22 million problems, each one of which you could give to an undergraduate algebra student. And they had a decent chance of solving the problem. Although there are a few, at least 22 million, there are like 100 or so that are really quite hard, but a lot are easy. And the project was just to work out to determine the entire graph, like which ones imply which other ones? That's an incredible project, by the way. Such a good idea, such a good test that the very thing we've been talking about at a scale that's remarkable. Yeah, so it would not have been feasible. I mean, the state of the art in the literature was like 15 equations and sort of highly implied. That's the limit of what a human repentant people can do. So you need to scale that up. So you need a crowdsource, but you also need to trust all the-- I mean, no one person can check 22 million of these proofs. You need to be computerized. And so it only became possible with Lean. We were hoping to use a lot of AI as well. So the project is almost complete. So at least 20 million, all but two have been settled. And all those two, we have a pen and paper proof of the two. And we're formalizing-- in fact, as this morning, I was working on it and finishing it. So we're almost done on this. It's incredible. It's-- yeah, the fact is how many people were able to get-- Bar 50, which in mathematics is considered a huge number. It's a huge number. Yeah, crazy. Yeah. So we're going to have a paper 50 all this, having a big appendix of food contributor, 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 organize the contributions by level of expertise to the people of the contributors? No, OK. I'm asking you a lot of pot head questions here. But I'm imagining a bunch of humans, and maybe in the future, some AI's. Can there be like an elo rating type of situation? Or a gamification of this? The beauty of all these lean projects, is that automatically, you get all this data. So like, every week, I uploaded this guitar and could have tracks who contributed what. So you could generate statistics from any later point in time. You could say, oh, this person contributed this many lines of code, or whatever, I mean. These are very crude metrics. I would definitely not want this to become part of your tenure review or something. But I mean, I think already in enterprise computing, people do use some of these metrics as part of the assessment of performance of an employee. Again, this is the direction which is a bit scary for academics to go down where we don't like metrics so much. And yet, academics use metrics. They just use old ones. Number of papers. Yeah, it's true that way. I mean, it feels like this is a metric while flawed is going in the more in the right direction, right? Yeah, it's interesting. At least it's a very interesting metric. Yeah, I think it's interesting to study. I mean, I think you can do studies of whether these are better predictors. There's this problem called good-hard slow. If a statistic is actually used to incentivize performance, it becomes gained. And then it is no longer a useful measure. Oh, humans always get-- Yeah, I know. I mean, it's mainly rational. So what we've done for this project is self-report. So there are actually standard categories from the sciences of what types of contributions people give. So there's this concept and validation and resources and coding and so forth. So there's a standard list of crawl also categories. And we just ask each contributor to this big matrix of all the authors and all the categories just to tick the boxes where they think they're contributed. And just give a rough idea. Also, you did some coding and you provided some compute, but you didn't do any for pen and paper for-- 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 their sciences of lead author and second author and so forth. Which we're proud of. We make all the authors equal status, but it doesn't quite scale to this size. So a decade ago, I was involved in these things called polymath projects. It was the 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. And this was a huge bottom neck, actually. But still, we had projects that were 10 authors or so. But we had decided at the time not to try to decide who did what, but to have a single pseudonym. So we created this fictional character called DHJ Polymath. In the spillover board by Kiwaki, this is the pseudonym for a famous group of mathematicians in the 20th century. And so the paper was altered on the pseudonym. So none of us got the author credit. This actually turned out to be not so great for a couple of reasons. So one is that if you actually wanted to be considered for 10 year old or whatever, you could not use this paper as you submitted as only publications because it didn't have the form of author credit. But the other thing that we've reckoned as much later is that when people refer to these projects, they naturally refer to the most famous person who was involved in the project. Oh, so this was Tim Gowas' playoff project. This was Tim Gowas' playoff project and not mention the other 19 or whatever people that were involved. - Oh, yeah. - So we're trying something different this time around where we have everyone's an author, but we will have an appendix with his matrix and we'll see how that works. - I mean, 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 the lead programming language just a few years ago and you're saying that this might be the future of mathematics. And so it's also exciting that you're embracing one of the greatest mathematicians in the world embracing what seems like the paving of the future of mathematics. 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 IMO problems. 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 problems? - Yeah, the difficulty increases exponentially with the number of steps involved in the proof. It's a commentary explosion. - So I think of large language models is that they make mistakes. And so if your proof has got 20 steps and your high school level has a 10% failure rate at each step of going in the wrong direction. It's extremely unlikely to actually reach the end. - Actually, just to take a small tangent here how hard is the problem of mapping from natural language to the formal program? - Oh, yeah, it's extremely hard actually. Natural language, you know, it's very fault-tolerant like you can make a few minor grammatical errors and speak in the second language you can get some idea over what you're saying. But formal language, if you get one little thing wrong, I do that the whole thing is nonsense. Even formal to formal, it's very hard. There are different incompatible for our prefaces and languages. There's lean, but also cock and Isabelle and so forth. And I keep even converting from a formal language to formal language. It's an unsolved problem. - That is fascinating. - Okay, so but once you have an informal language, they're using their RL train model. So something akin to alpha zero 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 the system and what do you think is the gap? - Yeah, we talked early about things that are amazing over time become kind of normalized. So now somehow it's of course geometry is a silver 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, the approach doesn't scale currently. Three days of Google's service, server time to sort of one high school math formula. This is not a scalable plus spec. Especially with the exponential increase in as the complexity increases. Which mentioned that they got a silver model performance. - The equivalent of, I mean, the equivalent of a silver model. - So first of all, they took way more time than was a lot of, and they had this assistance where the humans started helped by formalizing. But also they're giving us those formats for the solution, which I guess is formally verified. So I guess that's fair. There will be a proposal at some point to actually have an AI Math Olympiad where at the same time as the human contestants get the actual little bit problems. AI will also be given the same problems at the same time period. And the outputs will have to be graded by the same judges. And which means that it will have to be written in natural language rather than formal language. - I hope that happens. I hope that this is what happens. I hope that's what happens. - It won't happen this IMO. The performance is not good enough in the time period. But there are smaller competitions. There are competitions where the answer is a number rather than a long-form proof. And that's actually a lot better at problems where there's a specific numerical answer. 'Cause it's easy to enforce learning on it. Yeah, you've got the right answer, you've got the wrong answer. It's a very clear signal. But a long-form proof either has to be formal and then the lean can give it thumbs up, thumbs down. Or it's informal. But then you need a human to create it. To tell, and if you're trying to do billions of reinforcements learning runs, you can't hire enough humans to grade those. It's a very hard enough for the last time you want us to do reinforcements learning on just the regular text that people get. But now if you hire people not just give thumbs up, thumbs down, but actually check the output mathematically. Yeah, that's too expensive. So if we explore this possible future, what is the thing that humans do that's most special in mathematics? So that you could see AI not cracking for a while. So inventing new theories. So coming up with new conjectures versus proving the conjectures, building new abstractions, new representations, maybe an AI-terrestrial with seeing new connections between disparate fields. I think the nature of what mathematicians do over time has changed a lot. So a thousand years ago, mathematicians had to compute the date of Easter and then we'd be complicated calculations. But it's all automated centuries. Maybe we don't need that anymore. They used to navigate to do spherical navigation, spherical trigonometry to navigate how to get from the old board to the new. So I think it's very complicated calculations. Again, we'd been automated. Even a lot of undergraduate mathematics, even before AI, like both from alpha, for example, is not a language monopoly. It can solve a lot of undergraduate level math tasks. So on the computational side, verifying routine things, like having a problem and say here's a problem in partial differential equations. Could you still be using any of the 20 standard techniques? And they have the ASUP trial or 20 and here that 100 different permutations and disease map results. And that type of thing, I think, it worked very well. Type of scaling to, once you solve one problem to make the AI attack 100 adjacent problems. The things that humans do still, so-so, where the AI really struggles right now is knowing when it's made a wrong turn. You can say, "Oh, I'm gonna solve this problem. "I'm gonna split up this problem into these two cases. "I'm gonna try this technique." And sometimes, if you're lucky, it's a simple problem, it's the right technique and you solve the problem. And sometimes it will have a problem. It would propose an approach which is just complete nonsense. But like, it looks like a proof. So this is one annoying thing about LLM generated mathematics. So we've had human-generated mathematics as a very low quality. Submissions who don't have the formal training and so forth. But if a human proof is bad, you can tell it's bad pretty quickly. It makes really basic mistakes. But the AI-generated proofs, they can look superficially flawless. And that's partly because that's what the reinforcement learning has, like you train them to do. To make things, to produce text that looks like what is correct, which for many applications is good enough. So the AI was often really subtle and then when you spot them, they're really stupid. Like, no human would have vacuumed that mistake. Yeah, it's actually really frustrating in the programming context because I program a lot. And yeah, when a human makes low quality code, there's something called code smell, right? You can tell, you can tell. They're immediately like, okay, they're signs. But with AI-generated code and then you're right, eventually you find an obvious dumb thing that just looks like good code. Yeah, so. It's very tricky to and frustrating for some reason to yeah, so yeah, so the sense of smell Yes, this is this is this is one thing a humans have And there's a metaphorical mathematical smell that This we it is not clear how to get there has to do with you that eventually um, I mean so the way um Alpha zero and so forth to make progress and go and chest and so forth is is in some sense they have developed a sense of smell will go and chest positions you know that that this position is good for white this good for black They can't initiate why um, but just having that that sense of smell lets them strategize So if a eyes gained that ability to sort of assess the viability of certain proof strategies says so you can say I'm going to try to to break up this problem into two small subtasks and then you can say oh this looks good The 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 made the problem worse because each of the two sub problems is actually harder than your original problem Which is actually what normally happens if you try a random thing to try you normally you actually it's very easy to transform a problem into eating harder problem Very rarely do problem transport is a simpler problem um Yeah, so if they can pick up a sense of smell then they could maybe start competing with A human though with my position 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. 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 Mm-hmm do the code smut like you're Yes A professor child. 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 parts of systems that are component enough to understand complex instructions Um, they can work at massive scale But also unreliable Like it's it's a interesting uh A bit unreliable in subtle ways was we was providing sufficiently good output It's an interesting combination um, you know, I mean you have we you have like graduate students who work with who I kind of like this, but not as 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 use um, I mean i'm so Tim Kahler like you you can imagine Yaki's for saw like in 2000 He was envisioning what mathematics would look like in in like two and a half decades And asking yeah, he he wrote in his in his article like a hypothetical composition between a mathematical assistant of the future And himself, you know, if you find a silver problem and they would have to have a composition Sometimes the human would propose an idea and the AI would would Evaluate it and sometimes the air would propose an idea um and And sometimes the competition was required and air would just go and say okay. I've checked the 100 cases needed here or The first you said this is for all and I've checked it but in after 100 and it looks good so far or hang on. There's a problem that it equals 46 And 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 that go to both on both sides calculations could pose on both sides I've had conversations with AI where I say I can 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 try to prompt it Okay, so here's the problem I suggest using this tool and then it'll find is this lovely argument using a completely different tool Which eventually goes you know into the weeds and say no, no, no try using this again Am I start using this and then you'll go back to the tool that I wanted to do before um and like you have to keep railroading it Onto the path you want and I could eventually force it To give the proof I wanted um but it was like hurting cats um like and we might have personal effort I had to take to not just sort of prompt it But I also check it output because it I've a lot of what it looked like it's gonna work I know there's a problem on 917 and basically arguing with it Um like it was more exhausting than doing it on assisted. So like it but that's the currency to be hard. I wonder if there's There's a phase shift that happens towards no longer feels like hurting cats and Maybe you'll surprise us how quickly that comes you know, I believe so um so in formalization I mentioned before that it takes 10 times longer to formalize a proof that about it by hand With these modern AI tools is And also just better tooling it's the the lean um Developers are doing a great job Adding more more features and making it use a friend of you. It's going from 9 to 8 to 7. Okay, no big deal but one day you'll drop all the one um and that's the phase shift Because suddenly um it makes sense When you write a paper to write it in lean first Or through a conversation with AI which is generating lean on the fire with you and it becomes natural for journals to accept And maybe they'll offer expedite refereeing you know that 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 the literature and not worry so much about the correctness Because it that's been certified um Papers are getting longer and longer in mathematics and it's harder and harder to get good your refereeing for The really long ones unless they really important It is actually an issue which and the formalization is coming in 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 going to see much more Like math label grow right potentially exponentially. Yeah. Yeah. It's uh, it's uh, it's a virtuous Cycle okay. I mean one phase shift or this type that happened in the past was the adoption of late tech So so late tech is this type of language that all mathematicians use now So in the past people is all kinds of word processes and typewriters and whatever But at some point late tech became easier to use than or other competitors And that people just switch you know within a few years like it was just a dramatic patient It's a wild out there question, but what What year how far away are we from a AI system being a collaborator on a proof that wins the fields model so that level Okay, um Well, it depends on level collaboration. Yeah, no like it deserves to be to get the fields model like So I have an already like I can imagine if was we're a metal-witting paper having some AI systems in writing it You know, uh, just you know like the all-complete alone is already I use it like it speeds up my own writing Um, like you know you you can have a theorem and you have a proof of three cases And I write down the proof of first case and the autocomplete just suggested that now these have a proof of second case could work And like it was exactly correct. That was great. Save me like five ten minutes of All the typing but in that case the AI system doesn't get the fields model. No I was talking 20 years 50 years a hundred years. What do you think? Okay. Sorry. I gave a prediction in print So by 2026, which is now next year. Um, there will be Math collaborations with AI. So I'm not feels metal-witting, but but like actual research level published ideas that in our generation by AI maybe not the ideas, but at least some of the computations Um Forifications. Yeah, I mean that already happened. That's really happened. Yeah, there are there are problems that were solved By a complicated process conversing with AI to propose things and the human goes and tries it and then contact doesn't work But that yeah, if I pose a different idea 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 Mathematical human earth edition and an AI involved um, but no, uh, it's hard to sort of disentangle credit um I mean these tools they they do not uh replicate all the skills needed to mathematics But they can replicate sort of some non-trivial percentage of them you know 30 40 percent so they can fill in gaps um, you know, so coding is is a is a good example, you know, so I um Um, it's annoying for me to code and Python. I'm not I'm not a native um no professional programmer um But um the with AI that the if the fiction cost of doing it is it's much reduced So it fills in that gap for me um It's getting quite good at literature review um, I mean it's still a problem with um hallucinating you know the references that don't exist Um but this I think is a civil problem is 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 know um you should in a few years get the point where you you have a Alarm that you need and uh say anyone proven this number before and they will do Basically a fancy web search AI systems. Yeah. Yeah. They're these six papers where something similar has happened And I mean you can ask you right now and it'll give you six papers of which maybe one is legitimate and relevant one exists But it's not relevant and for our hallucinate um it has a non-zero success rate right now, but uh it's there's so much garbage so much the signal noise ratio is so poor that it's most helpful when you already somewhat know the relationship and you just need to be prompted to be reminded of a paper that was at least consciously in your memory or it's just helping you discover new you were not even aware of but it is the correct citation yeah that's yeah that it can sometimes do but but when it does it's buried in a list of options to which the other 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 quite a correctly yeah yeah it does it breaks you out of the silos of yeah yeah yeah yeah you know yeah now there's a big hump to overcome right now I mean it's it's like self-driving cars right yeah the safety margin it has to be really high yeah to be to be feasible so yeah so there's a last mile problem but with a lot of AI applications that you know they can develop tools that work 20% 80% of the time but it's still not good enough and in fact even worse than good some ways I mean another way of asking the fields matter of question is what year do you think you'll wake up and be like real surprise you read the headline the news or something happened that AI did like you know real break through something it doesn't you know like fields metal it even hypothesis it could be like really just this alpha zero moment would go that 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 thought was unrelated oh interesting generating a conjecture that's a beautiful conjecture yeah and and actually has a real time so being correct and then meaningful because that's actually kind of doable I suppose but the word of the data is yeah yeah no that would be truly amazing you come on a struggle a lot I mean so a version of this is I mean the physicists have a dream of getting the AI to discover new laws of physics you know the the dreams you just feed it all this data okay and and it says he was a new pattern that we didn't see before but it actually even struggled the current state of the art even struggles to discover all laws of physics from the data I mean or if it does there's a big concern with contamination that they did it only because like it's somewhere in this training data it is someone new you know boils law whatever ball of you trying to do we construct part of it so we don't have the right type of training data for this yeah so for laws of physics like we don't have like a million different universes we have a million different balls of nature and like a lot of what we're missing in math is actually the negative space of so we have published things of things that people have been able to prove and conjectures that end up being verified or we kind of examples produced but we don't have data 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 said oh but we should like you change our claim to modify it in this way to actually make it more plausible there's a trial and error process which is a real integral part of human mathematical discovery which we don't record because it's embarrassing we make mistakes and we only like to publish our wins and the AI has no access to the data to train on I sometimes joke that basically you know I didn't get an AI has to go through a grad school and actually you know go to grad courses do the assignments go to office hours make mistakes get advice on how to correct the mistakes and learn from that let me ask you if I may about Gregory Proman you mentioned that you try to be careful in your work and not let a problem completely consume you just you've really fallen love with the problem and it really cannot rest until you solve it but you also hasted to add that sometimes this approach actually can be very successful and the example you gave is Gregory Proman who proved the Poincare conjecture and did so by working alone for seven years with basically little contact with the outside world can you explain this one millennial prize problem has been solved Poincare conjecture and maybe speak to the journey that Gregory Proman's been on all right so it's a question about curved spaces that's a good example so I think I think it was a 2D surface interested me around you could maybe be a tourist with a hole in it or kind of many holes and there are many different topologies a priori that the surface could have even if you assume that it's bounded and smooth and so forth so we have figured out how to classify surfaces as a first approximation everything is to tell my son called the genus how many holes it has so the sphere has a genus zero a donut has genus one and so forth and one way you can tell the surfaces apart probably the sphere has which is quite simply connected if you take any closed loop on the sphere like a big close to a rope you can contract it to a point and while staying on the surface and the sphere has this property but a tourist doesn't then if you're on a sure us and you take a rope that goes around say the outer diameter of course there's no way it can't get through the hole there's no way to contract a point so it turns out that the this the sphere is the only surface with this property of contractability I mean up to like continuous deformations of the sphere so some things that are what are called topologically equivalent of the sphere so point where you ask the same question higher dimensions so this it becomes hard to visualize because surface you can think of as embedded in three dimensions but a curved free space we don't have good intuition of 40 space to to to live it and then they're also 3D space that can't even fit into four dimensions you need five or six or or higher but anyway mathematically you can still pause 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 point great conjecture weirdly in higher dimensions four and five it was actually easier so 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 the sphere but three was really hard so people try many approaches there's sort of commentary approaches where you chop up the the surface into little triangles or tetrahedron you you just try to argue based on how the faces interact each other there were algebraic approaches there's various algebraic objects like things called the fundamental group that you can attach to these homology and homology and and and all these very fancy tools they also didn't quite work but but your Hamilton's proposed a partial differential equations approach so you take you take so the problem is that you so you have this object which is so secret is a sphere but it's given to you in a really in a in a real way so like I think of a ball that's being kind of crumpled up and twisted and it's not obvious that is the ball but like if you have some sort of surface which is which is a deformed sphere you could you could for kind of think of it as a surface of a balloon you could try to inflate it you blow it up and naturally as you fill the air the wrinkles were sort of smooth out and it will turn into a nice round sphere unless of course it was a toy or something in which case it would get stuck at some point like if you inflate a toy or there would be a point in the middle when the inner ring shrinks to zero you get a singularity and you can't blow up any further you can't blow any further so if you created this flow which is called witty flow which is a way of taking an arbitrary surface or space and smoothing it out to make it rounder and rounder to make it look like a sphere and he wanted to show that either this process would give you a sphere or it would create a singularity I can very much like how PDs either have global regularity or finite and blow up basically it's almost exactly the same thing it's all connected and so and and he showed that for two dimensions two dimensional surfaces if you start with some connection no singularity is ever formed you never manage a trouble and you could flow and it will give you a sphere and so he got a new proof of the two dimensional result but by the way that's a beautiful explanation of a ratio flow and its application in this context how difficult is the mathematics here but for the 2D case is it yeah these are quite sophisticated equations on par with the Einstein equations slightly simpler but yeah but they were considered hard nonlinear equations to solve and there's lots of special tricks in 2D that that that helped but in 3D the problem was that this equation was actually super critical it has the same problem as Navier Stokes as you blow up maybe the curvature could get concentrated in finite smaller smaller regions and it it looked more and more nonlinear and things just look worse and worse and that we all kinds of singularities that showed up some singularities there's these things called neck pinches where where the surface sort of behaves like a like a barbell and it pinches at a point some some singularities are simple enough that you can sort of see what you do next you just make a snip and then you can turn one surface into two and you built them separately but those those are the kind of the prospect that's from really nasty like nonsense singularities showed up that you couldn't see how to resolve in any way that you couldn't do any surgery too so you need to classify all the singularities like what are all the possible things can go wrong so what problem did first of all he he made the problem he turned the problem a super critical problem to a critical problem I said before about how the invention of the of energy a Hamiltonian like really clarified Newtonian mechanics so he introduced something which is now comparements reduced volume and performance entropy he introduced new quantities kind of like energy that looked the same at a every single scale and turn the problem into a critical one where the non-linearities actually suddenly looked a lot less scary than they did before and then he had to solve you still had to analyze the singularities of this critical problem and that itself was a problem similar to this way of balancing it worked on actually so 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 resolve the point where he can actually so quite like a lot of really ambitious steps and like nothing that a locked language model today for example could I mean at best I could imagine proposing this idea as one of hundreds of different things to try but the other night and I would be complete dead ends but you don't only find out after months of work he must have had some sense that this was the right track to pursue because it takes years to get from A to B so you've done like you said actually you see even strictly mathematically but more broadly in terms of the process he's done similarly difficult things what can you infer from the process he was going through because 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 AI doesn't know when it's failing what happens to you you're sitting in your office when you realize the thing you did the last few days maybe weeks yeah is a failure well for me I switch to a different problem I'm a fox I'm not a hedgehog but you'll generally that is a break that you can take is just step away and look at it yeah I'm a problem yeah you can modify the problem too I mean yeah you can ask them cheat if there's a specific thing that's blocking you at that this some bad case keeps showing up at that for which your tool doesn't work you can just assume by fear this bad case doesn't occur so you do some magical thinking but strategically okay for the point to see if the rest of the argument goes through if there's multiple problems with with with your approach then maybe you just give up okay but if this is the only problem that but everything else checks out then it's still worth biting so yeah you have to do some some so forward reconnaissance sometimes that's true and that is sometimes productive to assume like okay we'll figure it out oh yeah yeah um sometimes actually it's even productive to make mistakes so um one of the I mean um there's a project which actually we wanted some prizes for you to do the job before other people um we worked on this PD problem again actually this blow off regularity type problem um and it was considered very hard um jump again um it was another few as my first you 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 anything fit and we were excited we were planning celebrationary to we'll get together and have champagne or something um and we started writing it up um and one of what one of us not me I keep it another quarter said oh um in this in this lemma here we we have to estimate these 13 terms that that show up in this expansion and we it's made 12 of them but in our notes I can't find it for the estimation of 13th can you can someone supply that and I said sure look at this and I like you it's oh yeah we didn't cover that we completely omitted this term and this turn 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 uh and all different permutations and there was always this one thing one turn 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 and was already um we stuck at this which we tried increasingly desperate things and at crazy things um and after two years we found that the picture is like somewhat different but quite a bit from our initial um strategy which did actually didn't generate these problem editors and and actually solve the problem so we we solved the problem after two years but if we hadn't had that initial full-storm of nearly solving the problem we would have given up by month two or something and worked on an easier problem um yeah if we had known it would take two years not sure we would have started the project yeah sometimes actually having the incorrect you know it's like Columbus traveling in New Orleans and the incorrect version of the measurement of the size of the earth um and he thought he was going to find a new trade route in India or at least that was how he sold it in his prospectus I mean it could be that he actually secretly knew but just on the psychological element do you have like emotional or like self-doubt or just overwhelmed you most like that you know because this stuff it feels like math it's so engrossing that like it can break you when you like invest so much yourself on the problem and then it turns out wrong you could start to a similar way chest 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 is as a job you know you you have a problem and if it doesn't work out you you will you call 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 about that go diseases where where we have just latch on to the one problem and they spend years and years thinking about nothing but that one problem and um you know maybe they're career sufferers and so forth they say oh but how could this big win this will you know once I once I finish this problem hour I will make up for all the years of off off lost opportunity and then that's that's yeah I mean occasionally occasionally it works but yeah I really don't recommend it for people who have got the right fortitude yeah so 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 well uh when we do crowd proposals uh we kind of say we will we will study this set of problems but even though we don't promise definitely by five years I will supply a proof of all these things you know um you promise to make some progress or discover some interesting phenomena and maybe you don't solve the problem but you find some related problem that you can say something new about 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 readman hypothesis go luck conjecture twin prime that's again so I mean the promised I could readman about this is those are so far out of reach I think so yeah there's no even viable strategy like even if I actually all my all the cheats that I know of like it this is still no way to give 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 so the natural numbers have two basic operations attach some addition and multiplication um so if you want to generate the national numbers you can do one or two things you can just start with one and add one to itself over and over again and that generates you the national numbers so additively they're very easy to generate one two three five or you can take the prime number if you want to generate multiplicatively you can take all the prime numbers two three five seven and more plan more together um and together they because you all the the national numbers except maybe for one so there are these two separate ways of thinking about the national numbers we added to point of view and a more significant point of view um and separately they're not so bad um so like any question about that national was only was addition is rather easy to solve and any question that only was multiplication is rather easy to solve um but what has been frustrating is that you combine the two together um and suddenly you get an extremely rich i mean we know that there are statements in numbers that are actually as understandable there are certain polynomials in some number variables in us is the solution in the national numbers and the answer depends on on a on a understandable statement um like like whether um the axioms of mathematics are consistent or not um but um yeah but even the simplest problems that combine something more multiplicative such as the primes with some additives such as chipsing by two uh separately we understand both from well but if you ask when you shift the prime by two do you can you get up how often can you get another prime we it's been amazingly hot to relate the two and we should say that the twin prime conjectures just that it posits that there are infinitely many pairs of prime numbers that differ by two it's now the interesting thing is that you have been very successful at pushing forward the field and 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 she's mind-blowing you can prove something like that right yeah so what if realized because of this this type of of research is that there's different patterns have different levels of uh indestructibility um so so what makes the twin prime problem hard is that you can take all the primes in the world you know three five seven eleven so both there are some twins in there eleven and thirteen is a twin prime perfect in process so well but you could easily if you wanted to redact the primes to get rid of to get rid of the um these twins like the twins they show up and in the infinite many of the world. them, but that you recently spassed, there's not, I mean, initially, it's quite a few, but once you got to the millions, trillions, they become rarer and rarer. And you could actually just, you know, if someone was given access to the database of privacy, just edit it out a few, a few privacy on there, they could make the trend package at your false by just removing like 0.0 or 1% of the privacy or something, just well chosen to do this. And so you could present a censored database of the prize, which passes all of the statistical tests of the primes, you know, if it obeys things like the problem of theorem and other sex of the primes, but doesn't contain any trend primes anymore. And this is a real obstacle to the trend prime conjecture. It means that any proof strategy to actually find trend primes in the actual primes must fail when applied to these slightly edited primes. And so it must be some very subtle, delicate feature of the primes that you can't just get from like, like, I could get statistical analysis. Okay, so that's all. Yeah. On the other hand, I think progression has turned out to be much more robust. Like, you can take the primes and you can eliminate 99% of the primes, actually, you know, and you can take any 90% and you want. And it turns out, and it's another thing we prove is that you still get as many progressions. As many progressions are much, you know, they're all coaches of arbitrary length. Yes. Yes. That's crazy. Yeah. So 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 an infinite monkey type phenomenon. For any fixed length of your set, you don't get arbitrary, that's progressions. You only get quite short progressions. But you're saying twin primes not an infinite monkey of phenomena. I mean, it's a very subtle monkey. It's still an infinite monkey phenomenon. Right. Yeah. If the primes were really genuinely random, if the primes were generated by monkeys, then yes. In fact, the infinite monkey theorem would all, 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. So the reason why we care about the trim and how conjecture is, it's a test case for whether we can genuinely, completely 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, at least with 100% probability, or probably 10 to 100% as you go out further and further. Yeah. So the primes we believe that the random, the reason why primes are indestructible is that regardless of whether it looks random or looks structured, like periodic, in both cases, the arithmetic regression appear, but for different reasons. And this is basically all the ways in which the thing, there are many proofs of, of these sort of arithmetic regression 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. But in twin primes, if, if the primes are random, then you're happy, you win. If the primes are structured, they could be structured in, in a specific way that eliminates the twins. And we can't rule out that one conspiracy. And yeah, you were able to make a zanish term progress on the K2 version. Right. Yeah. So the one thing about conspiracies is that any one conspiracy theory is really how to disprove. That, you know, if you believe the word is what by lizards is, here's some evidence that it's not my man. This is what that episode's taught about lizards. Yeah. You may have encountered this kind of phenomenon. Yeah. So like, like, I'm a pure, like, there's, there's almost no way to it. Definitely, without a conspiracy. And the same is true in mathematics, but a conspiracy is taught solely devoted to learning twin primes. You know, like it would, you have to also infiltrate other areas of mathematics, 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 conspiracy rule out other conspiracies. So, you know, if the word is one, this is the kind also be one by the ins. Right. Right. So one unreasonable thing is, it's hard to disprove, but more than one, there are, there are tools. So yeah. So for example, we, we know there's simply many primes that are no two, which are, so there is something that pairs up right which differ by at most 246. Actually, it is, it's the code. So there's a bound, yes, on the right. So, like the twin primes, the thing called cousin primes that differ by by four, this thing called sexy primes that differ by six. What are sexy primes? Primes that differ by six. The name is much less. Of course, it was much less exciting than the name suggests. 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 requires too much energy somehow in this conspiracy space. How do you do the bound part? How do you, how do you develop a bound for the difference between the primes? Okay. So there's an infinite number of, so it's ultimately based on what's called the pigeonhole principle. So the pigeonhole principle, it's a statement that if you have a number of pigeons and they all have to go over the pigeonholes and you have more pigeons than pigeonholes, then one of the pigeonholes has to have at least two pigeons. So that 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, two of them have to be at most 10 apart, because you can divide up the numbers from one to a hundred into one hundred pigeonholes. Let's say they have 101 numbers, 101 numbers, then two of them have to be a distance less than 10 apart because two of them have to belong to the same pigeonhole. So it's a basic, basic feature of a basic principle in mathematics. So it doesn't quite work with the primes directly because the primes get sparser and sparser as you go out, that a few and a few numbers are prived. But it turns out that there's a way to assign weights to numbers. So there are numbers that are kind of almost prived, but they don't have no factors at all other than themselves in one. They have very few factors. And it turns out that we understand almost primes a lot better than primes. And so for example, it was known for a long time that they were trying to almost prived. This has been worked out. So almost primes are something we cannot understand. So you can actually restrict the attention to a suitable set of almost primes. And whereas the primes are very spars overall relative to the almost primes actually are much less spars. They make you can set up a set of almost primes where the primes of density like say 1%. And that gives you a shot at proving by applying also a pigeonhole principle that there's persa primes that are just only 100% apart. But in order to determine how conjecture you need to get the density of primes is up to up to 50%. Once you get up to 50%, you would get trim primes. But unfortunately there are barriers. We know that no matter what kind of goods that are almost primes you pick, the density of primes can never get up off 50%. It's called the parody barrier. And I would love to find, yes, so one of my long-term dreams is to find a way to breach that barrier. Because it would open up not only trim primes conjecture, the go-back conjecture, and many other problems in number theory are commonly blocked. Because our current techniques would require improving going beyond this theoretical parody barrier. It's like pulling past the speed of light. Yeah, so we should say a twin primes conjecture. One of the biggest problems in the history of mathematics, go-back conjecture also. They feel like extra neighbors. Is there been days when you felt you saw the path? Oh, yeah. Sometimes you try something and it works super well. You again, again the sense of mathematical smell we talked about earlier. You learn from experience when things are going too well. Because there are certain difficulties that you sort of have to encounter. I think the way of calling my put it is that if you are on the streets of New York and you put in a blindfold and you put in a car and after some hours the blindfold is off and then you're in Beijing. That was too easy. There was no ocean being crossed. Even if you don't know exactly what was done, you're suspecting that something wasn't right. But is that still in the back of your head? Do you return to the prime numbers every once in a while to see? Yeah, when I have nothing better to do, which is less than that time. It's busy with so many things these days. When I free time, and I'm too frustrated to work on my view of research projects, I also don't want to do my ministry of sub-order. I don't want to do some errands for my family. I can play with these things for fun. Usually you get nowhere. Yeah, you have to just say, "Okay, fine. Once again, nothing happened. I will move on." Very occasionally, one of these problems I actually solved. Sometimes, as you say, you think you solved it, and then you're forward for maybe 15 minutes, and then you think, "I should check this because this is too easy. It could be true, and usually is." What's your gut say about when these problems would be solved when prime and go back to prime? I think we'll keep getting more partial results. It doesn't need at least one, this parity barrier is the biggest remaining obstacle. There are simpler versions of the conjecture where we are getting really close. I think we will, in 10 years, we will have many more, much closer results. We may not have the whole thing. Trin times is somewhere close. Reemun hypothesis, I have no clue. It has happened by accident. The Reemun hypothesis is more general conjecture about the distribution of prime numbers. Right. Yeah, it states that viewed more applicatively. For questions only involving multiplication, no addition. The primes really do behave well. as randomly as you could hope. So there's a phenomenon in probably called square root cancellation that you know, like if you want to poll, say America, above on some issue, and you ask one or two voters, and you may have sampled a bad sample, and then you get a really imprecise measurement of the full average. But if you sample more and more people, the accuracy gets better and better, and it actually improves like the square root of the number of people you sample. So yeah, if you sample 1,000 people, you can get like a two feet percent margin of error. So in the same sense, if you measure the primes in a certain multiplicative sense, there's a certain type of statistic you can measure and it's called the remunzator function, and it fluctuates up and down. But in some sense, as you keep averaging more and more, if you sample and 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 women hypothesis is a very elegant way that captures this. But as with many other ways in mathematics, we have very few tools to show that something really genuine behaves like Bidi random. And this is not just a little bit random, but it's asking that behaves as random, as a dukely random set, this square root cancellation. And we know here, because of things related to the parody property, most of us usual techniques cannot hope to settle this question. The proof has to come out of left fuel, yeah, but what that is, no one has any serious proposal. And there's various ways to sort of, as I said, you can modify the primes a little bit, and you can destroy the remuner hypothesis. So, it has to be very delicate. You kind of apply something that has huge margins of error. It has to be just barely work. And there's like all these pitfalls, like dodge very adeptly. The prime numbers are just fascinating, yeah. What to use most mysterious about the prime numbers? So, like, conjectually, 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 obvious patterns, they behave very randomly. And just assuming that they behave, so there's something called the cream of random model for the primes, that after a certain point, primes just behave like a random set. And there's various flight modifications as a model, but this has been a very good model. It matches the numerics. It tells us what to predict. Like, I can tell you of complete certainty, the trim back into this true. The random model gives overwhelming odds of this true. I just can't prove it. Most of our mathematics is optimized for solving things with patterns in them. And the primes have this anti-pattern as doom, almost everything, really. But we can't prove that. Yeah, I guess it's not mysterious at the price of the event. It's kind of randomly because there's no reason for them to be, to have any kind of secret pattern. But what is mysterious is what is the mechanism that really forces the randomness to happen? And this is just absent. Another incredibly surprisingly difficult problem is the collage conjecture. Oh, yes. Simple to state, beautiful to visualize in a simplicity, and yet extremely difficult to solve. And yet you have been able to make progress. Paul Radar said about the collage conjecture that mathematics may not be ready for such problems. Others have stated that it is an extraordinarily difficult problem completely out of reach. This is in 2010, out of reach of present day mathematics and yet, you have made some progress. Why is it so difficult to make? Can you actually even explain what it is? Oh, yeah. So it's a problem that you can explain. It helps with some visual aids, but yeah. So you take any natural number, like say 13. 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 would become 40. Of course, 13 times 3 is 39 and add one to your 40. So it's a simple process. For odd numbers and even numbers, they're both very easy operations. And then you put together that's still reasonably simple. But then you ask what happens when you iterate it. You take the output that you just got and feed it back in. So 13 becomes 40. 40 is now even divided by two is 20. 20 is still even divided by 10 to 10. 5 and then 5 times 3 plus 1 is 16. And then 8, 4, 2, 1. So, and then for 1, it goes 1, 4, 2, 1, 4, 2, 1. It cycles forever. So this sequence I just described, 13, 40, 20, 10s or both, these are also called hairstone sequences. Because there's an over-stimplified model of hairstone formation, which is not actually quite correct, but it's some I've taught to high school students as a first approximation, is that a little nugget of ice gets a nice crystal forms in cloud. And it goes up and down because of the wind. And sometimes it's cold, it gets a bit more mass. And maybe it melts a little bit. And this process is going up and down, creates this of partially melted ice, which I mentioned because it's hairstone. And eventually it falls down to the earth. So the conjecture is that no matter how high you start up, you take a number, which is in the millions or billions, this process that goes up if you're hard and down, if you're even, eventually goes down to the earth all the time. No matter where you start, it was very simple. I'll generally end up at 1. And you might climb for a while. Right. Yeah, so it's-- Yeah, if you plot it, these sequences, they look like Brownian motion. They look like the stock market. They just go up and down in a seemingly random pattern. And in fact, usually that's what happens. If you plug in a random number, you can actually prove at least initially that it would look like random walk. And that's actually a random walk with a downward drift. It's like if you're always gambling on a roulette at the casino with odds slightly weighted against you. So sometimes you win, sometimes you lose. But over in the long run, you lose a bit more than you win. And so normally your wallet will go to 0 if you just keep playing over and over again. So statistically, it makes sense. Yes. So the result that I proved roughly speaking asserts that statistically, like 9%, 10% of all inputs would drift down to maybe not all the way to 1, but to be much, much smaller than what you started. So it's like if I told you that if you go to a casino, most of the time, you end up-- if you keep playing up long enough, you end up with a smaller amount of any wallet when you start. That's kind of like the result that I proved. So why is that result, like can you continue down the thread to prove the full conjecture? Well, the problem is that I used arguments from probability theory. And there's always this exceptional event. So in probability, we have this low large numbers, which tells you things like if you play a casino with a game at a casino with a losing expectation, over time you are guaranteed almost surely with probability as close to 100% as you wish, you're guaranteed to lose money. But there's always this exceptional outlier. It is mathematically possible that even in the game it's also not in your favor. You could just keep winning slightly more than you lose. Very much like how in Navier Stokes, there could be most of the time your waves can disperse. There could be just one outlier choice of initial conditions that would lead you to blow up. And there could be one outlier choice of a special number that you stick in. That shoes of infinity were all other numbers crushed to earth, crushed to one. And in fact, there's some mathematicians who've Alex Contorovich, for instance, who've proposed that actually these collapse iterations are like these similar automata. Actually, if you look at what they happen on in binary, they do actually look a little bit like these game of life type patency, and in analogy to how the game of life can create these massive self-opticating 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 to encode is to create a version of a cell which is larger. heavier than air machine encoded in a number that flies forever. So Conway, in fact, worked on this problem as well. So Conway, so similar, in fact, that was more on inspirations for the Navier Stokes project. Conway studied generalizations of the collapse problem where instead of, more than where three and adding one or dividing by two, you have more complicated branching 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 complicate enough, you can actually encode two ring machines and you can actually make these problems undecidable and do things like this. In fact, he met a programming language for these kind of fractional linear transformations. He had a fact track on four track. And he showed that you couldn't program-- it was too incomplete. You could make a program that if your number you inserted in was encoded as a prime, it would sink to zero. It would go down otherwise it would go up and things like that. So the general cluster problems is really as complicated as all the mathematics. Some of the mystery of the cellular terminal that we talked about having a mathematical framework to say anything about cellular terminal, maybe the same kind of framework is required-- Yeah, clocks and gesture. Yeah, if you want to do it not statistically, but you really want 100% of all inputs to 2 to 4 to Earth. Yeah, so what might be feasible is-- yeah, still saying 99%, you know, going to go to one. But like everything, yeah, that books hard. What would you say is out of these within reach, famous problems is the hardest problem we have today. Is there even a hypothesis? We want to-- there. Pecos MPs is a good one because that's a metaphor. If you solve that in the positive sense that you can find a Pecos MP 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. There's ripple effects. Pecos MP has more ripple effects than basically any other. Right. If the readman hypothesis is disproven, that would be a big mental shock to the number theorists, but it would have follow-on effects for cryptography. Because a lot of cryptography uses number theory, uses number theory constructions, evolving primes and so forth. And it relies very much on the intuition that number theorists have built over many, many years of what operations evolving primes behave randomly and what ones don't. And in particular, our encryption methods are designed to turn text information on it into text which is indistinguishable from random noise. So, enhance, we believe to be almost impossible to crack at least mathematically. But if something as core to our belief that we want to hypothesis is wrong, it means that there are actual patterns of the primes that we are not aware of. And if there's one, there's probably going more. And suddenly a lot of our crypto systems are in doubt. But then, how do you then say stuff about the primes? Yeah. That you go towards the collection and conjecture again. Because you do you want it to be random, right? You want it to be random. So more broadly, I'm just looking for more tools, more ways to show that things are random. How do you prove a conspiracy doesn't happen, right? Is there any chance to you that P equals NP? Is there some, can you imagine a possible universe? It is possible. I mean, this is various scenarios. I mean, if there's one way, it is technically possible, but in fact, it's never actually implementable. The evidence is sort of slightly pushing in favor of no, but we'd probably be as low as an NP. I mean, it seems like it's one of those cases, you know, similar to Roman hypothesis. I think the evidence is leaning pretty heavily on the no. Certainly more on the no than on the yes. The funny thing about P equals NP is that we have also a lot more obstructions than we do for almost any other problem. So while there's evidence, we also have a lot of results ruling out many, many types of approaches to the problem. This is the one thing that the computer science is actually very good at. It's actually saying that certain approaches cannot work. No go theorems. It could be understandable. We don't, yeah, we don't know. There's a funny story I read that when you won the field's medal, somebody from the internet wrote you and asked, you know, what are you going to do now that you won this procedure award? And then you just quickly, very humbly said that, you know, this shiny medal is not going to solve any of the problems I'll currently working on. It's just, first of all, it's funny to me that you would answer an email in that context. And second of all, it just shows your humility. But anyway, maybe you could speak to the field's medal, but it's another way for me to ask about Gregoria Perlman. What do you think about him famously declining the field's medal in the millennial 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. Well, he's, he's somewhat of an outlier, even among mathematicians who tend to have someone idealistic views. I've never met him. I think I'll be interested in me the one day, but I'd never have the chance. I know people who met him. He's always had strong views about certain things. I mean, it's not like he was completely isolated from the math community. I mean, he would give talks and papers and so forth. But at some point, he just decided not to engage with the rest of the community. He was dissolution of something. I don't know. And he decided to, to, to, to peace out and collect mushrooms in St. Petersburg or something. And that's, that's fine. You know, and you can, you can do that. I mean, that's another sort of flip side. I mean, we are not a lot of problems that we solve. You know, some of them do have practical application. That's, that's great. But like, if you stop thinking about a problem, you know, so he hasn't published since in this field, but that's fine. There's many, many other people who have done so as well. Yeah. So I guess one thing I didn't realize initially with the field's metal is that it sort of makes you part of the establishment. You know, so, you know, most mathematicians, you know, there's, you just, career mathematicians, you know, you just focus on publishing your next paper, maybe getting one, just to promote a one, one rank, you know, and, and starting a few projects, maybe having to take some students or something. Yeah. But then suddenly people want your opinion on things and you have to think a little bit about, you know, things that you might just sort of foolishly say because, you know, no one's going to listen to you. So it's more important now. Is it constraining to you? Are you able to still have fun and be a rebel and try crazy stuff and play with the idea? I have a lot less free time than I had previously. I mean, mostly by choice. I mean, I can always see I have the option to sort of decline. So I declare a lot of things. I could decline even more. Or I could acquire a repetition of things so unreliable that you would always been asking anymore. I love the different algorithms here. This is great. It's always an option. But, you know, there are things that are like, I mean, so I mean, I don't spend as much time as I do as a postdoc, you know, just just working on the one part of the time or fooling around. I still do that a little bit, but yeah, as you're advancing your career, some of the more soft skills. So math somehow front notes all the technical skills to the early stages of a career. So, yeah, so it's, as a postdoc, as a publisher of Paris, you incentivize to basically focus on proving very technical themes, to sort of prove yourself as well as prove the theorems. But then as you get more senior, you have to start mentoring and giving interviews and trying to shape direction of the field both research wise and, you know, sometimes you have to, you know, to present with administrative things. And it's kind of the right social contract because you need to work in the trenches to see what can help mathematicians. The other side of the establishment sort of the really positive thing is that you get to be a light that's an inspiration to a lot of young mathematicians and young people that are just interested in mathematics. It's like, yeah, it's just how the human mind works. This is where I would probably say that I like to feel as metal 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 want to give sort of respect to somebody like Gregorio Proman, who is critical of awards in his mind. Those are his principles and any human that's able for their principles to do the thing their most humans would not be able to do is beautiful to see. Some recognition is necessary and important, but yeah, it's also important to not let these things take over your life. And like only be concerned about getting the next big award or whatever. I mean, yeah, yeah. So again, you see these people try to only solve like a really big math problems and not work on things that are less sexy, you wish, but actually, it's still interesting and it's instructive. As you say, like, the way the human mind works, we understand things better when they're attached to humans. And also, if they're attached to a small number of humans, the way our humans mind is wide, we can comprehend the relationship between 10 or 20 people. But once you get beyond like 100 people, there's a limit. I pick this name for it, beyond which it just becomes the other. And so we have to simplify the whole massive, you know, 9.9% of humanity becomes the other. And often these models are incorrect in this course as all kinds of problems. But so, yeah, so to humanize a subject, you know, if you identify a small number of people, I say, these are representative people of the subject, you know, role models, for example, that has some role. But it can also be, yeah, too much of it can be harmful because it's, I'll be the first to say that my own career trough, for this not that of a typical mathematician. I, the very accelerated education, I skipped a lot of classes. I think I was very fortunate mentoring opportunities. And I think I was at the right time, just because someone doesn't have my trajectory, you know, it doesn't mean that they can't be good mathematicians, I mean, they're very different style. And we need people with different style. And, you know, even if, and sometimes too much focus is given on the, on the person who does the last step to complete a project in mathematics or elsewhere, that's really taken, you know, centuries or decades with lots and lots of, building a lot of previous work. But that's a story that's difficult to tell, if you're not an expert, because, you know, it's easy to just say one person did this one thing, you know, it makes for much simpler history. I think on the whole, it is a hugely positive thing to talk about Steve Jobs as a representative of Apple. When I personally know, and of course, I've already knows the incredible design, the incredible engineering teams, just the individual humans than those teams are not a team. their 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. Steve Jobs. Yeah. Yeah. As a starting point, you know, yeah. As a first approximation. That's how you can read some biographies and then look into which deeper first approximation. Yeah. That's right. So you were a person to Andrew Wiles at that time. Oh yeah. Professor there. It's a funny moment. How history is just all interconnected. And at that time, he announced that he proved from us last year. Oh yeah. What did you think maybe looking back now with more context about that moment in math history? Yeah. So I was a graduate student at the time. I mean, I vaguely remember, you know, there was press attention and we all had the same we had pigeonholes in the same mail group, you know, so we all put you on mail and like suddenly Andrew Wiles' mail box exploded to be overflowing. That's a good metric. Yeah. You know, so, yeah, we all talked about it at T and so forth. I mean, we didn't understand most of us sort of understand the proof. We understand sort of high-level details. Like there's an ongoing project to formalize it in lean, Kevin Bonsett is actually. Yeah. Can we take that small tangent? Is it how difficult does that? Because as I understand it from us, the proof from us last theorem has like super complicated objects. Yeah. Yeah. It's really difficult to formalize now. Yeah. I guess you're right. The objects that they use, you can define them. So they've been defined in lean. Okay. So just defining what they are can be done. That's really not trivial, but it's been done there. But there's a lot of really basic facts about these objects that have taken decades to prove and that they're in all these different math papers. And so a lot of lots of these have reformalized as well. Kevin's, Kevin Bonsett's goal, actually, he has five-year grant to formalize philosophy. And his aim is that he doesn't think he could be able to get all the way down to the basic axioms. But you want to formalize it to the point where the only things that he needs to rely on as black boxes are things that were known by 1980 to number theories at the time. And then some other person or some other work would have he done to get from there. So it's a different area mathematics than the type of mathematics I'm used to. In analysis, which is kind of my area, the objects we study are kind of much closer to the ground. We study things like prime numbers and functions and things that are within scope of a high school math education to at least define. But then this is very advanced algebraic side of number theory, where people have been building structures and structures for quite a while. And it's a very sturdy structure. It's been very, at the base, at least it's extremely what developed in the textbooks and so forth. But it does get to the point where if you haven't taken these years of study and you want to ask about what is going on at level six of this tower. You have to spend quite a bit of time before they can even get to the point where you can see when you recognize. What inspires you about his journey that was similar as we talked about seven years, mostly working in secret. Yeah, that is a romantic, it kind of fits with the sort of the romantic image that people have of mathematicians, to the extent that they think of things that are at all as these kind of eccentric wizards or something. So that sort of accentuated that perspective. I mean, it is a great achievement. His style of solving problems is so different from my own. But which is great. I mean, we need people like that. He's bigger, like what in terms of like you like the collaborative. I like moving on from a problem if it's giving too much difficulty. But you need the people who have the tenacity and the fearlessness. I've collaborated with people like that where I want to give up because the first approach that we tried to work and the second one didn't approach, they convinced and they have the third, fourth, and the fifth approach works. And I have to eat my words. Okay, I didn't think this was going to work, but yes, you were right along. And we should say for people that 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 very high quality. So there's something to be said about being able to jump from top of your topic. Yeah, it works for me. I mean, also people who are very productive and they take focus very deeply on, yeah. I think everyone has to find their own workflow. 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 mathematics. And so we have a certain curriculum and so forth. I mean, maybe like if you do math competitions or something, you get a slightly different experience. But I think many people, they don't find their native math language until very late or usually too late. So they stop doing mathematics and they have a bad experience with a teacher who's trying to teach them one way to do mathematics that they don't like it. My theory is that humans don't come, evolution has not given us a math center or a brain directly. We have a vision center and a language center and some other centers, which have evolution as home, but we don't have innate sense of mathematics. But our other centers are sophisticated enough that different people, we can repurpose other areas or a brain to do mathematics. So some people have figured out how to use the visual center to do mathematics. So they think very visually when they do mathematics. Some people have repurposed their language center and they think very symbolically. Some people, if they are very competitive and they like gaming, there's a part of your brain, it's very good at solving puzzles and games and that can be repurposed. But like when I talk to my mathematicians, they don't quite think that I can tell that they're using some of different styles of thinking. I mean, not just joint, but they may prefer visual. I don't like you prefer visual so much, I need also visual aids myself. My effect provides the common language, so we can still talk to each other even if we are thinking in different ways. But you can tell there's a different set of subsystems being used in the thinking process. They take it from past, they're very quick at things that I struggle with in vice versa and yet they still get to the same goal. That's beautiful. But I mean, the way we educate, unless you have a personalized tutor or something, I mean, education, sort of just financial skill has to be mass-produced. You have to teach to 30 kids. They have 30 different styles. You can't teach 30 different ways. On that topic, what advice would you give to students, young students who are struggling with math, but are interested in it and would like to get better? Is there something in this complicated educational context? What would you. Yeah, it's a tricky problem. One nice thing is that there are now lots of sources for my faculty in Richmond outside the classroom. So in my day, there are already math competitions. And they're also like popular math books in the library. Now you have YouTube, there are forums devoted to solving math puzzles. And math shows up in other places. For example, there are hobbyists who play poker for fun. And they are for very specific reasons. I'm interested in very specific probability questions. And they actually. There's a community of amateur probabilists in poker, in chess, in baseball. I mean, there's this method of a lot of the place. And I'm hoping actually with these new sort of tools for lean and so forth, that actually we can incorporate the broader public into math research projects. This is almost. It doesn't happen at all currently. So in the sciences, there's some scope for citizen science. Astronomers, the amateurs who we discover comments, and there's biologists, there are people who could identify butterflies and so forth. And in math, there are small number of activities where amateur mathematicians can discover new primes and so forth. But previously, because we have to verify every single contribution, like most mathematical research projects, it would not help to have input from the general public. And it would just be time consuming, because just error-checking and everything. But 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 have already contributed to some of these formalizing projects, who contributed to math with them. 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, as a very first step, opens it up to the programming community too. Yes, the people who are already comfortable with programming. It seems like programming is somehow maybe just the feeling, but it feels more accessible to folks than math. Math is seen as this like extreme, especially in modern mathematics. It's seen as this extremely difficult to enter area and programming is not, so that could be just an entry point. You can execute and get results. You can print out the world pretty quickly. If programming was taught as an almost entirely theoretical subject where you just taught the computer science, the theory of functions and routines and so forth, and outside of some very specialized homework assignments, you would not like your program like on the weekend for fun. Or there would be as consider as hot as math. Yeah, so this is it. There are communities of non-matheticians where they're deploying math for some very specific purpose, you know, like optimising their poker game. And for them, then math becomes fun for them. What advice would you give in general to young people, how to pick a career, how to find themselves? That's a tough, tough, tough question. Yeah, so there's a lot of certainty now in the world, you know, I mean, there was this period after the war where, at least in the West, you know, if you came from a good demographic, you know, like you, there was a very stable path through it to a good creator. You go to college, you get an education, you pick one profession, and you stick to it. It's becoming much more think of the past. So I think you just have to be adaptable and flexible. I think people will have to get to the goals that are transferable, you know, like learning one specific program in language or one specific subject, mathematics or something, it's, it's, that is still this lot of super transferable skill, but sort of knowing how to reason with abstract concepts or how to problem solve and things go wrong, these are things which I think we will still need, even as our tools get better, you know, you would be working with AI and so forth. But actually, you're an interesting case study. I mean, you're like 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, but you've learned lean, that's not, that's a non trivial thing to learn. Like that's a, yeah, that's a, for a lot of people, that's extremely uncomfortable leap to take, right? Yeah, a lot of mathematicians. Plus, I've always been interested in new ways to do mathematics. I feel like a lot of the ways we do things right now are inefficient. I spend, many of my colleagues, you 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 and why can't we search and get a quick response and so on. 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 Schwarzer had just formalized one of his new theorems and there's some other developments in computer assisted proof that look quite interesting. And I'd say, oh, we should, we should, we should want to workshop on this. This is a pretty good idea. And then I was a bit too enthusiastic about this idea. So I got more than told to actually write it. So I did with a bunch of other people, colonization, Jordan Allen Bergen, and I'm part of other people. And it was, it wasn't, nice success, we brought together a bunch of mathematicians and computer scientists and other people and and we got up speed and stability out. And it was really interesting developments that most mathematicians didn't know what was going on. That lots of nice proofs of concept, you know, it's just so hints of of what was going to happen. This was just before chat GBD, but there was even then there was one talk about language models and the potential capability of those in the future. So that got me excited about the subject. So I started giving talks about this is some of which, more of us just start looking at. Now that I mentioned this conference and then chat GBD came out and suddenly AI was everywhere. And so I got interviewed a lot about this topic. And in particular, the interaction between AI and formal proof of assistance and I said, yeah, they should be combined. This is this is this is 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, 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'm a wander mathematician, just trust me, you know, when I say that this is going to change mathematics and I'm not doing it any when I don't do any of it myself. So I thought I had to actually justify it. A lot of what I get into actually, I don't quite see an advice as how much time I'm going to spend on it. And it's only after I'm sort of waist deep in a project that I realized 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 beginner, right? Or have some of the sort of challenges that a beginner would, right? And yeah, new concepts, new ways of thinking. Also, you know, sucking at a thing that others, I think, I think in that target, you know, you could be a field as a matter of winning mathematician and undergrad knows something better. Yeah. I think mathematics inherently, I mean, mathematics is so huge these days that nobody knows all of modern mathematics. And inevitably, we make mistakes and, you know, you can't cover up your mistakes with just sort of provider or and I mean, because people will ask for your proofs and if you don't have the proofs, you know, the proofs. I don't love math. Yeah. So it does keep us honest. I mean, you can still, it's not a perfect panacea, but I think we do have more of a culture of admitting error than because we're forced to all the time. Big ridiculous question. I'm sorry for it once again. Who is the greatest mathematician of all time? Maybe one who's no longer with us? Who are the candidates? The Euler, Gauss, Newton, Ramanogen, Hilbert. So first of all, I say, I mentioned before, like there's some time dependence. But on the day, yeah, like if you, if you, if you poke cumulatively over time, for example, you could like, like, like, like, this is one of the pretenders. And then maybe some unnamed anonymous mathematicians before that, you know, whoever came up with the concept of numbers, you know, you know, do mathematicians today still feel the impact of Hilbert? Oh, yeah, directly of 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. A hard to solve. The statement of the open problems. Yeah, I mean, you know, this is bystander effect in everywhere. Like if no one says you should do X. And I just sort of mills around, winning for someone else to do something. And like nothing gets done. So and like it's the one thing that actually you have to teach undergraduate mathematics is that you always try something. So you see a lot of paralysis in an undergraduate trying a math problem. If they recognize that there's a certain technique that can be applied, they will try it. But there are problems for which they see none of their standard techniques obviously applies. And the common reaction is then just paralysis. I don't know what to do. I think there's a quote from the Simpsons. I've tried nothing and I'm all that ideas. So you know, like the next step then is to try anything like no matter how stupid. And in fact, almost the stupid of the better. Which, you know, I think it was almost guaranteed to fail. But the way it fails is going to be instructive. Like it fails because you're not at all taking to account this hypothesis. Oh, this hypothesis must be useful. That's a clue. I think he also suggested somewhere this fascinating approach, which really stuck with me. As they're using it, it really works. I think you said it's called structured procrastination. No, yes. It's when you really don't want to do a thing. The imagine a thing you don't want to do more. Yes. That's worse than that. And then in that way, you procrastinate by not doing the thing that's worse. Yeah. It's a nice hack. It actually works. Yeah. Yeah. With anything, psychology is really important. You talk to athletes like my phone rannes and so forth. And they talk about what's the most important thing? Is it the training veteran or the diet and so on? So much of it is psychology. You know, just tricking yourself to think in the form of responsible. So you're motivated to do it. Is there something our human mind will never be able to comprehend? Well, I sort of, I guess, the mathematician. I mean, yeah, it's a bit of a reduction. There must be some, it's a bit large number that you can't understand. That's the first thing I came to mind. So that, but even broadly, is there, is there something about our mind that 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, for example, if I didn't even have pen and paper, like if I had no technology whatsoever, okay? So I'm not allowed blackboard pen and paper. You're already much more limited than you would be. Incredibly limited. Even language, the English language is a technology. It's one that's been very internalized. So you're right. They're really, the formulation of the problem is incorrect because there really is no longer a just a solo human or already augmented in extremely complicated, intricate ways, right? Yeah, yeah. So we're already like a collective intelligence. Yes, yes. So humanity plural has much more intelligence in principle on his good days than the individual humans put together. It can all have less. Yeah, so yeah, the mathematical community plural is incredibly super intelligent, entity that no single human mathematician can come close to to duplicating. You see it a little bit on these questions analysis sites. So this mathematical flow, which is the math version of stackable flow. And like, sometimes you get like this very quick responses to very difficult questions from the community. And it's a pleasure to watch, I can't wait to hear that. expert. I'm a fan spectator of that of that site just seeing the brilliance of the different people there. The depth of knowledge that some people have and the willingness to engage in the rigor and the nuance of the particular questions, it's pretty cool to watch. It's almost just fun to watch. What gives you hope about this whole thing we have going on, human civilization? I think the younger generation is always really creative and enthusiastic and inventive. It's a pleasure working with young students. The progress of science tells us that the problems that used to be really difficult can become extremely trivial to solve. Navigation, just knowing where you work on the planet, was this horrendous problem people, people who died and lost fortunes because they couldn't navigate. We have devices in our pockets that do this automatically for us. It's a completely solved problem. Things that are seen unfeasible for us now could be just homely exercises for them. 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. Because in the next 200 years, just imagine showing up in 200 years. Yeah. Well, already plenty has happened. If you could go back in time and talk to you if your teenage self was something. I mean, just the internet and now AI, I mean, again, they've been internalized. Of course, I can understand a voice and give reason why, slightly incorrect answers to any question, but yeah, this was mind blowing even two years ago. And in the moment, it's hilarious to watch on the internet and so on. The drama, 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, it's 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 they infrastructure and the culture is healthy, the community of humans can be so much more intelligent and mature and rational than the individuals within it. Well, one place I can always count on rationality is the comment section of your blog, which I'm a fan of. There's a lot of really smart people there. And thank you, of course, for putting those ideas out on the blog. And I can't tell you how honored I am that you would spend your time with me today. I was looking forward to this for a long time. Terry, I'm a huge fan. You inspire me. You inspire millions of people. Thank you so much for talking. 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 lexfreedman.com/sponsors. And now, let me leave you 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. [BLANK_AUDIO]