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

Terence Tao is widely considered to be one of the greatest mathematicians in history. He won the Fields Medal and the Breakthrough Prize in Mathematics, and has contributed to a wide range of fields from fluid dynamics with Navier-Stokes equations to mathematical physics & quantum mechanics, prime numbers & analytics number theory, harmonic analysis, compressed sensing, random matrix theory, combinatorics, and progress on many of the hardest problems in the history of mathematics. Thank you for listening ❤ Check out our sponsors: https://lexfridman.com/sponsors/ep472-sc See below for timestamps, transcript, and to give feedback, submit questions, contact Lex, etc. Transcript: https://lexfridman.com/terence-tao-transcript CONTACT LEX: Feedback – give feedback to Lex: https://lexfridman.com/survey AMA – submit questions, videos or call-in: https://lexfridman.com/ama Hiring – join our team: https://lexfridman.com/hiring Other – other ways to get in touch: https://lexfridman.com/contact EPISODE LINKS: Terence’s Blog: https://terrytao.wordpress.com/ Terence’s YouTube: https://www.youtube.com/@TerenceTao27 Terence’s Books: https://amzn.to/43H9Aiq SPONSORS: To support this podcast, check out our sponsors & get discounts: Notion: Note-taking and team collaboration. Go to https://notion.com/lex https://lexfridman.com/s/notion-ep472-sc Shopify: Sell stuff online. Go to https://shopify.com/lex https://lexfridman.com/s/shopify-ep472-sc NetSuite: Business management software. Go to http://netsuite.com/lex https://lexfridman.com/s/netsuite-ep472-sc LMNT: Zero-sugar electrolyte drink mix. Go to https://drinkLMNT.com/lex https://lexfridman.com/s/lmnt-ep472-sc AG1: All-in-one daily nutrition drink. Go to https://drinkag1.com/lex https://lexfridman.com/s/ag1-ep472-sc OUTLINE: (00:00) – Introduction (00:36) – Sponsors, Comments, and Reflections (09:49) – First hard problem (15:16) – Navier–Stokes singularity (35:25) – Game of life (42:00) – Infinity (47:07) – Math vs Physics (53:26) – Nature of reality (1:16:08) – Theory of everything (1:22:09) – General relativity (1:25:37) – Solving difficult problems (1:29:00) – AI-assisted theorem proving (1:41:50) – Lean programming language (1:51:50) – DeepMind’s AlphaProof (1:56:45) – Human mathematicians vs AI (2:06:37) – AI winning the Fields Medal (2:13:47) – Grigori Perelman (2:26:29) – Twin Prime Conjecture (2:43:04) – Collatz conjecture (2:49:50) – P = NP (2:52:43) – Fields Medal (3:00:18) – Andrew Wiles and Fermat’s Last Theorem (3:04:15) – Productivity (3:06:54) – Advice for young people (3:15:17) – The greatest mathematician of all time PODCAST LINKS: – Podcast Website: https://lexfridman.com/podcast – Apple Podcasts: https://apple.co/2lwqZIr – Spotify: https://spoti.fi/2nEwCF8 – RSS: https://lexfridman.com/feed/podcast/ – Podcast Playlist: https://www.youtube.com/playlist?list=PLrAXtmErZgOdP_8GztsuKi9nrraNbKKp4 – Clips Channel: https://www.youtube.com/lexclips

Speaker names marked ? are inferred from speaking time and the show’s known hosts — they are not voice-verified.

• 0:00Lex Fridman?

  The following is a conversation with Terrence Tao, widely considered to be one of the greatest mathematicians in history, often referred to as the Mozart of math. He won the Fields Medal and the Breakthrough Prize in Mathematics, and has contributed groundbreaking work to a truly astonishing range of fields in mathematics and physics. This was a huge honor for me, for many reasons, including the humility and kindness that Terry showed to me throughout all our interactions. It means the world. And now, a quick, few second mention of each sponsor. Check them out in the description or at lexfridman.com/sponsors. It's the best way to support this podcast.

• 0:46Lex Fridman?

  We've got Notion for teamwork, Shopify for selling stuff online, NetSuite for your business, Element for electrolytes, and the Age One for your health. Choose wisely, my friends. And now, onto the full ad reads. They're all here in one place. I do try to make them interesting by talking about some random things I'm reading or thinking about. But if you skip, please still check out the sponsors. I enjoy their stuff, maybe you will too. To get in touch with me, for whatever reason, go to lexfridman.com/contact. All right, let's go. This episode is brought to you by Notion, a note-taking and team collaboration tool. I use Notion for everything, for personal notes, for planning these podcasts, for collaborating with other folks, and for super boosting all of those things with AI,

• 1:30Lex Fridman?

  because Notion does a great job of integrating AI into the whole thing. You know what's fascinating is the mechanisms of human memory before we had widely adopted technologies and tools for writing and recording stuff, certainly before the computer. So you can look at medieval monks, for example, that would use the now well-studied memory techniques, like the memory palace, the spatial memory techniques to memorize entire books. That is certainly the effect of technology, started by Google Search and moving to all the other things like Notion, that we're offloading more and more and more of the task of memorization to the computers, which I think is probably a positive thing,

• 2:15Lex Fridman?

  because it frees more of our brain to do deep reasoning, whether that's deep dive, focused specialization, or the journalist type of thinking versus memorizing facts.

• 2:29Lex Fridman?

  Although I do think that there's a kind of black car model that's formed when you memorize a lot of things. And from there, from inspiration arises discovery. So I don't know, it could be a great cost to offloading most of our memorization to the machines. But it is the way of the world. Try Notion AI for free when you go to notion.com/lex. That's all lowercase notion.com/lex to try the power of Notion AI today. This episode is also brought to you by Shopify, a platform designed for anyone to sell anywhere with a great looking online store. Our future friends has a lot of robots in it. Looking into that distant future, you have Amazon warehouses with millions of

• 3:13Lex Fridman?

  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. But 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 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,

• 3:58Lex Fridman?

  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 lowercase. Go to Shopify.com/lex to take your business to the next level today. This episode is also brought to you by NetSuite, an all-in-one cloud business management system.

• 4:23Lex Fridman?

  There's a lot of messy components to running a business. And I must ask, and I must wonder, at which point there's going to be an AI AGI like CFO of a company? An AI agent that handles most, if not all, of the financial responsibilities or all of the things that NetSuite is doing. At which point will NetSuite increasingly leverage AI for those tasks? I think probably 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.

• 5:07Lex Fridman?

  I suspect that running a company is a lot more difficult than people realize, but there's a lot of sort of paperwork type stuff that could be automated, could be digitized, could be summarized, integrated, and used as a foundation for the said humans to make decisions. Anyway, that's our future. Download the CFO's Guide to AI and Machine Learning at netsuite.com/lex. That's netsuite.com/lex. This episode is also brought to you by Element, my daily zero sugar and delicious electrolyte mix. You know, I run along the river often and get to meet some really interesting people. One of the people I met was preparing for his first ultra marathon.

• 5:50Lex Fridman?

  I believe he said it was 100 miles. And that, of course, sparked in me the thought that I need for sure to do one myself. Some time ago now, I was planning to do something with David Goggins, and I think that's still on the sort of to-do list between the two of us, to do some crazy physical feat. Of course, the thing that is crazy for me is a daily activity for Goggins. 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.

• 6:33Lex Fridman?

  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. Because most of us are not prepared for it. You can learn along the way. And if you have the rigorous feedback loop of improving constantly and growing as a person and really doing a great job of the thing, I think that might as well be an ultra matter thought. Anyway, get a sample pack for free with any purchase. Try it at drinkelement.com/lex. And finally, this episode is also brought to you by AG1, an all-in-one daily drink

• 7:18Lex Fridman?

  to support better health and peak performance. I drink it every day. I'm preparing for a conversation on drugs in the Third Reich. And funny enough, it's a kind of way to analyze Hitler's biography is to look at what he consumed throughout. And Norman Oller does a great job of analyzing all of that and tells the story of Hitler and the Third Reich in a way that hasn't really been touched by historians before. It's always nice to look at key moments in history through a perspective that's not often taken. Anyway, I mention that because I think Hitler had a lot of stomach problems.

• 8:01Lex Fridman?

  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 war is waged by humans and humans are biological systems and biological systems require fuel and supplements and all of that kind of stuff. And depending on what you put in your body will affect your performance in the short term and the long term. With meth, that's true with Hitler, to his last days in the bunker in Berlin, all the cocktail of drugs that he was taking, so.

• 8:48Lex Fridman?

  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 the Lex Fridman Podcast. To support it, please check out our sponsors in the description or at lexfridman.com/sponsors. And now, dear friends, here's Terrence Tao.

• 9:49Lex Fridman?

  What was the first really difficult research-level math problem that you encountered? One that gave you pause, maybe?

• 9:57Speaker 01

  Well, I mean, in your undergraduate education, you learn about the really hard and possible problems, like the Riemann hypothesis, the twin primes conjecture. You can make problems arbitrarily difficult. That's not really a problem. In fact, there's even problems that we know to be unsolvable. What's really interesting are the problems just on the boundary between what we can do 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 Kakeya 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, Soichiro Kakeya, in like 1918 or so.

• 10:42Speaker 01

  So the puzzle is that you have a needle on the plane. We'll think like driving on a road or something. And you wanted to execute a U-turn. You want to turn the needle around. But you want to do it in as little space as possible. So you want to use as little area in order to turn it around. So, but the needle is infinitely maneuverable.

• 11:07Speaker 01

  So you can imagine just spinning it around its, it's the unit needle. You can spin it around its center. And I think that gives you a disk of area, I think, pi over four. Or you can do a three-point U-turn, 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 Vesicovic 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 U-turn thing that you could do, that you could turn a needle around. And in so doing, it would pass through every intermediate direction. Is this in the two-dimensional plane?

• 11:52Speaker 01

  This is in the two-dimensional plane. Yeah, so we understand everything in two dimensions. So the next question is what happens in three dimensions? So suppose like the Hubble Space Telescope is tube in space. And you want to observe every single star in the universe. So you want to rotate the telescope to reach every single direction. And here's the unrealistic part. Suppose that space is at a premium, which it totally is not. You want to occupy as little volume as possible in order to rotate your needle around in order to see every single star in the sky. How small a volume do you need to do that? And so you can modify Vesicovic's 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 but just very, very thin,

• 12:37Speaker 01

  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 your needle gets thinner, the volume should go down. But but how fast does it go down? And the conjecture was that it goes down very, very slowly, like logarithmically, roughly speaking. And that was proved after a lot of work. So this seems like a puzzle. Why is it interesting? So it turns out to be surprisingly connected to a lot of problems in partial differential equations, in number theory, in geometry, combinatorics. For example, in wave propagation, you splash some water around, you create water waves and they travel in various directions. But waves exhibit both particle and wave type behavior.

• 13:22Speaker 01

  So you can have what's called a wave packet, which is like a very localized wave that is localized in space and moving a certain direction in time. And so if you plot it in both space and time, it occupies a region which looks like a tube. And so what can happen is that you can have a wave which initially is very dispersed, but it all focuses at a single point later in time. Like you can imagine dropping a pebble into a pond and it will 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 geometrically what's going on is that there's always sort of light rays.

• 14:07Speaker 01

  So like if this wave represents light, for example, you can imagine this wave as a superposition of photons all traveling at the speed of light. They all travel on these light rays and they're all focusing at this one point. So you can have a very dispersed wave focus into a very concentrated wave at one point in space and time, but then it defocuses again and it separates. But potentially, if the Binjaki had a negative solution, so what I mean is that there's a very efficient way to pack tubes pointing in different directions into a very, very narrow region of 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,

• 14:49Speaker 01

  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, their amplitude becomes so great that the laws of physics that they're governed by are no longer wave equations, but something more complicated and nonlinear. And so in mathematical physics, we care a lot about whether certain equations and wave equations are stable or not, whether they can create these singularities. There's a famous unsolved problem called the Navier-Stokes regularity problem. So the Navier-Stokes equations, equations that govern the fluid flow or compressible fluids like water. The question asks, if you start with a smooth velocity field of water, can it ever concentrate so much that like the velocity becomes infinite at some point? That's called a singularity.

• 15:34Speaker 01

  We don't see that in real life. You know, if you splash around water on a bathtub, it won't explode on you. Or have water leaving at the speed of light. But potentially it is possible.

• 15:48Speaker 01

  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, has a million dollar prize for solving one of these problems. This is one of them. Of these seven, only one of them has been solved. The point great conjecture, like Perlmutter. So the Kekhane 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.

• 16:32Lex Fridman?

  Can you speak to the Navier-Stokes? So the existence and smoothness, like you said, millennial prize problem. You've made a lot of progress on this one. In 2016, you published a paper, Finite Time Blow Up for an Averaged Three-Dimensional Navier-Stokes Equation.

• 16:46Speaker 01

  Right.

• 16:47Lex Fridman?

  So we're trying to figure out if this thing usually doesn't blow up.

• 16:52Speaker 01

  Right.

• 16:52Lex Fridman?

  But can we say for sure it never blows up?

• 16:56Speaker 01

  Right. Yeah. So yeah, that is literally the million dollar question. Yeah. So this is what distinguishes mathematicians from pretty much everybody else. Like if something holds 9.99% of the time, that's good enough for most, you know, for most things. But mathematicians are one of the few people who really care about whether like 100%, really 100% of all situations are covered by, yeah. So most fluids, most of the time, water does not blow up. But could you design a very special initial state that does this?

• 17:28Lex Fridman?

  And maybe we should say that this is a set of equations that govern in the field of fluid dynamics, trying to understand how fluid behaves. And it's actually turns out to be a really complicated, you know, fluid is extremely complicated thing to try to model.

• 17:42Speaker 01

  Yeah. So it has practical importance. So this Clay Prize problem concerns what's called the incompressible Navier-Stokes, which governs things like water. There's something called the compressible Navier-Stokes, which governs things like air. And that's particularly important for weather prediction. Weather prediction, it does a lot of computation of fluid dynamics. A lot of it is actually just trying to solve the Navier-Stokes equations as best they can. Also gathering a lot of data so that they can get, they can initialize the equation. There's a lot of moving parts. So it's a very important problem practically.

• 18:09Lex Fridman?

  Why is it difficult to prove general things?

• 18:13Lex Fridman?

  About the set of equations like it not not blowing up.

• 18:17Speaker 01

  Short answer is Maxwell's demon. So Maxwell's demon is a concept in thermodynamics. Like if you have a box of two gases, oxygen and nitrogen, and maybe you start with all the oxygen on one side and nitrogen on the other side, but there's no barrier between them, then they will mix. And they should stay mixed. There's no reason why they should unmix. But in principle, because of all the collisions between them, there could be some sort of weird conspiracy that that like maybe there's a microscopic demon called Maxwell's demon. That will every time oxygen and nitrogen add them collide, they will bounce off in such a way that the oxygen sort of drifts onto one side and then nitrogen goes to the other. And you could have an extremely improbable configuration emerge, which we never see. And and we statistically is extremely unlikely.

• 19:00Speaker 01

  But mathematically, it's possible that this can happen and we can't rule it out. And this is a situation that shows up a lot in mathematics. A basic example is the digits of pi, 3.14159 and so forth. The digits look like they have no pattern and we believe they have no pattern. On the long term, you should see as many ones and twos and threes as fours and fives and sixes. There should be no preference in the digits of pi to favor, let's say, seven over eight. But maybe there is some demon in the digits of pi that that like every time you compute more and 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 there's no way to prove it with our current technology.

• 19:46Speaker 01

  Okay, so getting back to Navier-Stokes, a fluid has a certain amount of energy. And because the fluid is in motion, the energy gets transported around. And water is also viscous. So if the energy is spread out over many different locations, the natural viscosity of the fluid will just damp out the energy and it will go to zero. And this is what happens in. When we actually experiment with water, 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 it will move faster and faster, and at faster speeds, the effectiveness of custody is relatively less.

• 20:31Speaker 01

  And so it could happen that it creates some sort of, what's called a self-similar blob scenario, where the energy of the fluid starts off at some large scale, and then it all sort of transfers its energy into a smaller region of the fluid, which then at a much faster rate moves into an even smaller region and so forth. And each time it does this, it takes maybe half as long as the previous one. And then you could actually converge to all the energy concentrating in one point in a finite amount of time. And that scenario is called finite time blow up. So in practice, this doesn't happen.

• 21:15Speaker 01

  So water is what's called turbulent. So it is true that if you have a big eddy of water, it will tend to break up into smaller eddies, but it won't transfer all this energy from one big eddy into one smaller eddy. It will transfer into maybe three or four. And then those ones split up into maybe three or four small eddies of their own. And so the energy gets dispersed to the point where the viscosity can then keep everything under control. But if it can somehow concentrate all the energy, keep it all together, and do it fast enough that the viscous effects don't have enough time to calm everything down, then this blob can occur. So there are papers who had claimed that, oh, you just need to take into account conservation of energy and just carefully use the viscosity and you can keep everything under control

• 21:59Speaker 01

  for, not just Navier-Stokes, but for many, many types of equations like this. And so in the past, there have been many attempts to try to obtain what's called global regularity for Navier-Stokes, which is the opposite of finite time blow up, that velocity stays 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 time blow up. I couldn't do it for the actual equations of fluids, which were too complicated. But if I could average the equations of motion of Navier-Stokes, so physically, 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 it's a fluid and it could transfer its energy

• 22:44Speaker 01

  from a large eddy into this small eddy or this other small eddy, I would turn off the energy channel that would transfer energy to this one and direct it only into this smaller eddy, while still preserving the law of conservation of energy.

• 22:58Lex Fridman?

  So you're trying to make a blow up?

• 22:59Speaker 01

  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.

• 23:07Lex Fridman?

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

• 23:10Speaker 01

  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 up in finite time. So what that means is that if you wanted to prove global 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.

• 23:55Speaker 01

  So the thing about math is it's not just about finding, taking a technique that is going to work and applying it, but you need to not take the techniques that don't work. And for the problems that are really hard, often there are dozens of ways that you might think might apply to solve the problem. But it's only after a lot of experience that you realize there's no way that these methods are going to work. So having these counterexamples for nearby problems kind of 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.

• 24:30Lex Fridman?

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

• 24:37Speaker 01

  Right, yeah, so the key phenomenon that my technique exploits is what's called supercriticality.

• 24:44Speaker 01

  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 in one location of space can get transported because the fluid is in motion to other locations. And that's a nonlinear effect, and that causes all the problems. 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.

• 25:29Speaker 01

  It's a very nonlinear situation, it's unpredictable, it's turbulent. So sometimes these forces are in balance at small scales, but not in balance at large scales, or 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 the things that calm things down. And so this is why the problem is hard. In two dimensions, so the Soviet mathematician Ladashin Skaya, she in the sixties showed in two dimensions, there was no blow up. And in two dimensions, the Navier-Stokes equation is what's called critical. The effect of transport and the effect of viscosity are about the same strength, even at very, very small scales. And we have a lot of technology to handle critical and also subcritical equations and prove regularity.

• 26:14Speaker 01

  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's a key qualitative feature that distinguishes some equations from being sort of nice and predictable, and 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,

• 26:59Speaker 01

  because it's a supercritical equation. Lots of really strange things are going on at very fine scales.

• 27:04Lex Fridman?

  So whenever there is some huge source of nonlinearity...

• 27:09Lex Fridman?

  That can create a huge problem for predicting what's gonna happen.

• 27:12Speaker 01

  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, if you want 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.

• 27:37Speaker 01

  You just basically, you can almost approximate these planets by point masses. And just the aggregate behavior is important. But if you want to model a fluid, like the weather, you can't just say in Los Angeles, the temperature is this, the wind speed is this. For supercritical equations, the fine scale information is really important.

• 27:54Lex Fridman?

  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

• 28:04Lex Fridman?

  solve it or to negatively resolve it would be to sort of to construct a liquid, a kind of liquid computer.

• 28:12Speaker 01

  Right.

• 28:12Lex Fridman?

  And then show that the halting problem from computation theory has consequences for fluid dynamics. So show it in that way. Can you describe this?

• 28:21Speaker 01

  Right, yeah, so this came out of this work of constructing this average equation that blew up. So as part of how I had to do this, so this sort of naive way to do it, you just keep pushing. Every time you get energy at one scale, you push it immediately to the next scale as fast as possible. This is sort of the naive way to force blow up. It turns out in five and higher dimensions, this works. But in three dimensions, there was this funny phenomenon that I discovered that if you keep, if you change laws of physics, you just always keep trying to push

• 28:58Speaker 01

  the energy into smaller and smaller scales. What happens is that the energy starts getting spread out into many scales at once. So you have an energy at one scale, you're pushing it into the next scale. And then as soon as it enters that scale, you also push it to the next scale. But there's still some energy left over from the previous scale. You're trying to do everything at once. And this spreads out the energy too much. And then it turns out that it makes it vulnerable for viscosity to come in and actually just damp out everything. So it turns out this directed波 doesn't actually work. There was a separate paper by some other authors that actually showed this in three dimensions. So what I needed was to program a delay. So kind of like airlocks.

• 29:40Speaker 01

  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 largest scale got transferred. And only after you pushed all the energy in, then you sort of open the next gate and then you push that in as well. So by doing that, it kind of, the energy inches forward scale by scale in such a way that it's always localized at one scale at a time. And then it can resist the effects of viscosity because it's not dispersed. So in order to make that happen, yeah, I had to construct a rather complicated nonlinearity. And it was basically like, you know, like it was constructed like an electronic circuit.

• 30:24Speaker 01

  So I actually thank my wife for this because she was trained as an electrical 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 then off, you can build it from more primitive components, you know, capacitors and resistors and so forth. And you have to build a diagram and these diagrams, you can sort of follow it with your eyeballs and say, oh yeah, the current will blow up here and then it will stop and then it will do that. So I knew how to build the analog of basic electronic components, you know, like resistors and capacitors and so forth. And I would stack them together in such a way that I would create something that would open one gate and then there'd be a clock that would, and then once the clock hits

• 31:09Speaker 01

  a certain threshold, it would close it. It's kind of a Rube Goldberg type machine, but described mathematically. And this ended up working. So what I realized is that if you could pull the same thing off for the actual equations. So if The equations of water support a computation. So like you can imagine kind of a steampunk, 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 at a certain velocity. And maybe it's the two different configurations corresponding to a bit being up or down. Probably if you had two of these moving bodies of water collide,

• 31:53Speaker 01

  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 in a very predictable way on the inputs. And like you could chain these together and maybe create a Turing machine. And then you have computers, which are made completely out of water. And if you have computers, then maybe you can do robotics, you know, hydraulics and so forth. And so you could create some machine, which is basically a fluid analog, what's called a von Neumann machine. So von Neumann proposed, if you want to colonize Mars, the sheer cost of transporting people and machines to Mars is just ridiculous. But if you could transport one machine to Mars,

• 32:35Speaker 01

  and this machine had the ability to mine the planet, create some more materials, smelt them, and build more copies of the same machine, then you could colonize a whole planet over time. So if you could build a fluid machine, which, yeah, so it's a fluid robot. And what it would do, its 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, the water would transfer all its energy into the smaller configuration and then power down, okay, and then like clean itself up. And then what's left is this newer state, which would then turn on and do the same thing, but smaller and faster. And then the equation has a certain scaling symmetry.

• 33:20Speaker 01

  Once you do that, it can just keep iterating. So this in principle would create a blow up for the actual Navier-Stokes. And this is what I managed to accomplish for this average Navier-Stokes. So it provided this sort of roadmap. To solve the problem. Now, this is a pipe dream because there are so many things that are missing for this to actually be a reality. So I can't create these basic logic gates. I don't have these in these special connotations of water. I mean, there's candidates, there's things called vortex rings that might possibly work, but but also, you know, analog computing is really nasty compared to digital computing. I mean, because there's always errors. You have

• 34:01Speaker 01

  to do a lot of error correction along the way. I don't know how to completely power down the big machine so that it doesn't interfere with the running of the smaller machine. But everything in principle can happen, 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 is, but there could be some simpler scheme than what I just described to make this work.

• 34:40Lex Fridman?

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

• 35:07Speaker 01

  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 is the same, you can you can you can draw a connection. So there's a lot of previously on what is called cellular automata, the most famous of which is Conway's game of life. There's this infinite discrete grid, and at any given time, the grid is either occupied by a cell or it's empty. And there's a very simple rule that tells you how these cells evolve. So sometimes cells live and sometimes they die. And this, you know, when I was a student, it was a very popular screensaver to actually just have these, these animations going on and they look very chaotic.

• 35:51Speaker 01

  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 this thing called a glider. So a glider is a very tiny configuration of like four or five cells, which evolves and it just moves at a certain direction. And that's like this vortex rings.

• 36:09Speaker 01

  Yeah, so this is an analogy. The game of life is kind of a discrete equation and the fluid Navier-Sogis is a continuous equation, but mathematically they have some similar features.

• 36:21Speaker 01

  And so over time, people discovered more and more interesting things you could build within the game of life. The game of life is a very simple system. It only has like three or four rules to do it. But you can design all kinds of interesting configurations inside it. There's something called a glider gun that does nothing to spit out gliders one at a time.

• 36:40Speaker 01

  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, if, if, 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 Turing machine. I mean, it's like an enormous steampunk type things. They look ridiculous.

• 37:24Speaker 01

  But then people also generated self replicating objects in the game of life. A massive machine, a quantum machine, which over a huge period of time, and there were always glider guns inside doing these very steampunk calculations. It would create another version of itself. Which could replicate.

• 37:40Lex Fridman?

  It's so incredible.

• 37:41Speaker 01

  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 to Navier-Stokes. Which is a much, as I say, analog is much worse than digital. Like it's gonna be, you can't just directly take the constructions from the game of life and plonk them in. But again, it just, it shows it's possible.

• 38:05Lex Fridman?

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

• 38:33Speaker 01

  The thing is, you can get this emergent, very complicated structures, but only with very carefully prepared initial conditions. Yeah, so, so these, these, these glider guns and gates and, and sort of machines, if you just plonk down randomly some cells and you, and like, you'll not see any of these. And that's the analogous situation of Navier-Stokes again, you know, that, that with, with typical initial conditions, you will not, you will not have any of this weird computation going on. But basically through engineering, you know, by, by, by specially designing things in a very special way, you can pick clever constructions.

• 39:07Lex Fridman?

  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

• 39:14Speaker 01

  Yeah, yeah, yeah.

• 39:14Lex Fridman?

  Something interesting.

• 39:15Speaker 01

  This, this is a 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, like the digits of pi, well, we believe is 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, it does something like repeat itself every so often, you can do that. But, and you can prove that, that, for example, you can, you can prove that most sequences of, of digits have no pattern. So like if you just pick digits randomly, there's something called the low large numbers that tells you, you're gonna get as many ones as, as twos in the long run. But.

• 39:59Speaker 01

  We have a lot fewer 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 spend a lot of time on is to prove or call structure theorems or inverse theorems that give tests for when something is very structured. So some functions are what's called additive, like if you have a function that maps natural numbers to natural numbers. So maybe, you know, two maps to four, three maps to six and so forth. Some functions what's called additive, which means that if you add, if you add two inputs together, the output gets added as well. For example, multiply by a constant. If you multiply a number by 10, if you multiply a plus b by 10, that's the same as multiplying a by 10 and b by 10 and then adding them together. So some functions additive.

• 40:43Speaker 01

  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 point something, so 10 up to 14, 20 up to 28.

• 41:01Speaker 01

  So in that case, additivity is true then, so 10 plus 10 is 20 and 14 plus 14 is 28. But because of this rounding, sometimes there's round off 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 developing things like this to the effect that if a function exhibits 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.

• 41:43Speaker 01

  And so if you have these inverse theorems, it creates this sort of dichotomy that either

• 41:49Speaker 01

  The objects that you study are either have no structure at all, or they are somehow related to something that is structured. And in either way, in either, in either case, you can make progress. A good example of this is that there's this old theorem in mathematics called Zermatt's theorem, proven in the 1970s. It concerns trying to find a certain type of pattern in a set of numbers. The patterns have make progression, things like three, five and seven, or 10, 15 and 20. And Zermatt, Andrei Zermatt, proved that any set of numbers that is sufficiently big, was called positive density, has arithmetic progressions in it of any length you wish. So for example, the odd numbers have a density one half, and they contain arithmetic progressions of any length.

• 42:34Speaker 01

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

• 43:10Lex Fridman?

  Can you prove that there's arithmetic progressions of arbitrary length within a random?

• 43:17Speaker 01

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

• 43:23Lex Fridman?

  Yes.

• 43:24Speaker 01

  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 square 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 theorem says that if you take an infinite string of digits or whatever, eventually, any finite pattern you wish will emerge. It may take a long time, but it will eventually happen. In particular, arithmetic progressions of any length will eventually happen. Okay, but you need, but you need an extremely long random sequence for this to happen.

• 44:04Lex Fridman?

  I suppose that's intuitive. It's just infinity.

• 44:08Speaker 01

  Yeah, infinity absorbs a lot of sins.

• 44:10Lex Fridman?

  Yeah. How are we humans supposed to deal with infinity?

• 44:14Speaker 01

  Well, you can think of infinity as as an abstraction of a finite number for which you you do not have a bound for. That, you know, I mean, so nothing in real life is truly infinite. But, you know, you can, you know, you can ask yourself questions like, what if I had as much money as I wanted? You know, 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 the mathematics becomes a lot cleaner when you do that. I mean, in physics, we joke about assuming spherical cows. You know, like reward problems have got all kinds of reward effects,

• 44:57Speaker 01

  but you can idealize send something to infinity, send something to zero. And and the mathematics becomes a lot simpler to work with it.

• 45:06Lex Fridman?

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

• 45:16Speaker 01

  Yeah, so there's a lot of pitfalls. So, you know, we we spend a lot of time in undergraduate math classes, teaching analysis. And analysis is often about how to take limits and and and whether you know. So, for example, A plus B is always B plus A. So when you have a finite number of terms, you add them, you can swap them, and there's no problem. But when you have infinite number of terms, there are these sort of show games you can play where you can have a series which converges to one value, but you rearrange it, and it suddenly converges to another value. And so you can make mistakes. You have to know what you're doing when you allow infinity. You have to introduce these eps and deltas, and and there's a certain type of way of reasoning that helps you avoid mistakes.

• 45:58Speaker 01

  In more recent years, people have started taking results that are true in infinite limits and what's called finalizing them. So you know that something's true eventually, but you don't know when. Now give me a rate. Okay, so if I don't have an infinite number of monkeys, but a large finite number of monkeys, how long do I have to wait for hamlet to come out? And that's a more quantitative question. And this is something that you can attack by purely finite methods and you can use your finite intuition. And in this case, it turns out to be exponential in the length of the text that you're trying to generate. So and so this is why you never see the monkeys create hamlet. You can maybe see them create a four-letter word, but nothing that big.

• 46:42Speaker 01

  And so I personally find, once you finalize an infinite statement, it does become much more intuitive and it's no longer so weird.

• 46:50Lex Fridman?

  So even if you're working with infinity, it's good to finalize so that you can have some intuition.

• 46:57Speaker 01

  Yeah, the downside is that the finalized proofs are just much, much messier. And yeah, so the infinite ones are found first, usually like decades earlier, and then later on, people finalize them.

• 47:07Lex Fridman?

  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 in 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.

• 47:29Speaker 01

  Right, so I think science in general is interaction between three things. There's the real world, there's what we observe of the real world, our observations, and then our mental models as to how we think the world works. So we can't directly access reality. Okay, all we have are the 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? And we don't yet have the observation and we'd like to make a prediction. And then we have these simplified models, sometimes making unrealistic assumptions, you know, spherical cow type things. Those are the mathematical models.

• 48:10Speaker 01

  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 we ask what are the consequences of that model? What observations, what predictions would the model make of the of future observations? Or past observations to test it fit observed data? So there's definitely symbiosis.

• 48:38Speaker 01

  It's math, 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.

• 48:50Speaker 01

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

• 49:03Speaker 01

  A lot, there's a lot less sort of speculation about, suppose I did this, what would happen? You know, planning and modeling.

• 49:12Speaker 01

  Speculative fiction maybe is is one other place, but that's about it. Actually, most of the things we do in life is conclusions driven, including physics and science. I mean, they want to know, you know, where is this asteroid going to go? You know, what was, what is the weather going to be tomorrow? But mathematics also has this other direction of going from the the axioms.

• 49:32Lex Fridman?

  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?

• 49:41Speaker 01

  Well, you need both, top down and bottom up. Yeah, it's a really interaction machine, all these things. So over time, the observations and the theory and the modeling should both get closer to reality. But initially, and it is, I mean, this is, this is always the case, you know, they're always far apart to begin with. But you need one to figure out where to push the other, you know? So if your model is predicting anomalies that are not picked by experiment, that tells experimenters where to look, you know, to to to to find more data to refine the models. You know, so it goes, it goes back and forth.

• 50:20Speaker 01

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

• 50:35Speaker 01

  I mean, people do do it, you know, like if they want to study prime numbers or whatever, they can just generate large data sets. And so, once we had the computers, we began to do it a little bit. Although even before, well, like Gauss, for example, he discovered he conjectured the most basic theorem in number theory, it's called the prime number theorem, which predicts how many primes that 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's an early example of experimental mathematics.

• 51:18Speaker 01

  But until very recently, it was not, yeah, I mean, theoretical mathematics was just much more successful. I mean, 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 something called the commutator explosion. If you want to study, for example, Zermatt's theorem, you 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 any computer can currently can enumerate, any computer ever will ever enumerate.

• 51:58Speaker 01

  So you have to, there are certain math problems that very quickly become just intractable to attack by direct brute force computation. Chess is another famous example. The number of chess positions, we can't get a computer to fully explore. But now we have AI. We have tools to explore this space, not with 100% guarantees of success, but with experiment. You know, so like we can empirically solve chess now. For example, we have, we have very, very good AIs that can, 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 make, to do experimental chess.

• 52:43Speaker 01

  That they're revisiting old chess theories about, oh, you know, when you, this type of opening, this is a good type of move, this is not. And they can use these chess engines to actually refine, in some cases overturn conventional wisdom about chess. And I do hope that mathematics will have a larger experimental component in the future, perhaps powered by AI.

• 53:04Lex Fridman?

  Well, of course, talk about that. But in the case of chess, and there's a similar thing in mathematics, I don't believe it's providing a kind of formal explanation of the different positions. It's just saying which position is better or not, that you can intuit as a human being. And then from that, we humans can construct a theory of the matter. You've mentioned the Plato's Cave allegory. So in case people don't know, it's where people are observing shadows of reality, not reality itself. And they believe what they're observing to be reality. Is that in some sense what mathematicians and maybe all humans are doing?

• 53:45Lex Fridman?

  Is looking at shadows of reality? Is it possible for us to truly access reality?

• 53:55Speaker 01

  Well, there are these three ontological things. There's actual reality, there's observations and our models. And technically, they are distinct, and I think they will always be distinct. But they can get closer over time.

• 54:13Speaker 01

  You know, so and the process of getting closer often means that you you have to discard your initial intuitions. So astronomy provides great examples, you know, like, you know, like, you know, an initial model of the world is that it's flat because it looks flat, you know, and and that it's and it's big, you know, and the rest of the universe, the skies is not, you know, like the sun, for example, looks really tiny. 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. You know, so, you know, so things look good, you know, but over time, as you make more and more observations, bring it closer to to reality, the model gets dragged along with it. And so over time, we had to realize that the earth was round, that it

• 54:54Speaker 01

  spins, it goes around the solar system, solar system goes on the galaxy, and so on and so forth. And the entire universe is expanding. The expansions is self-expanding, accelerating. And in fact, very recently, in this year, so there's even the acceleration of the universe itself is this evidence that is non-constant.

• 55:12Lex Fridman?

  And the explanation behind why that is.

• 55:15Speaker 01

  It's catching up.

• 55:16Lex Fridman?

  It's catching up. I mean, it's still, you know, the dark matter, dark energy, this this kind of thing.

• 55:21Speaker 01

  Yes. We have 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 sudge factors, you know, with with enough sudge factors, you can explain anything. Yes, but the mathematical point of the model is that you want to have fewer parameters in your model than data points in your observational set. So if you have a model with 10 parameters that explains 10 observations, that is a completely useless model. It's what's called overfitted. But like, if you have a model with, you know, two parameters and it explains a trillion observations, which is basically, so, yeah, the the dark matter model, I think it has like 14 parameters, and it explains petabytes of data that that the astronomers have.

• 56:05Speaker 01

  You can think of a theory, like one way to think about a physical mathematical theory is it's 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 if it can fit to reasonable accuracy, you know, almost all of your observations. I mean, the more compression that you make, the better your theory.

• 56:31Lex Fridman?

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

• 56:40Speaker 01

  Yeah, Einstein had a quote like that. The most incomprehensible thing about the universe is that it is comprehensible.

• 56:44Lex Fridman?

  Right. And not just comprehensible, you can do an equation like E equals mc squared.

• 56:49Speaker 01

  There is actually some mathematical possible explanation for that. So there's this phenomenon in mathematics called universality. So many complex systems at the macro scale are coming out of lots of tiny interactions at the macro scale. And normally because of the Compton explosion, you would think that the macro scale equations must be 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, that's like Avogadro's number is humongous. There's a huge number of particles. If you actually have to track each one, it would be ridiculous. But certain laws emerge at the microscopic scale that almost don't depend on what's

• 57:31Speaker 01

  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 its temperature and pressure and volume and a few parameters, like five or six, and it models almost everything you need to know about these 10 to 23 or whatever particles. So we have.

• 57:56Speaker 01

  We don't understand universality anywhere near as we would like mathematically, but there are much simpler toy models where we do have a good understanding of why universality occurs. 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 what's called a Gaussian distribution, a famous bell curve. There's now even a meme with this curve. And even the meme applies broadly, the universality to the meme. Yes, you can call meta if you like, but there are many, many processes. For example, you can take lots of independent random variables and average them together in various ways. You can take a simple average or more complicated average. And we can prove in various cases that these bell curves, these Gaussians emerge.

• 58:40Speaker 01

  And it is a satisfying explanation. Sometimes they don't. So if you have many different inputs and they're all correlated in some systemic way, then you can get something very far from a bell curve show up. And this is also important to know when a system fails. So universality is not a 100% reliable thing to rely on. That the global financial crisis was a famous example of this. People thought that mortgage defaults had this sort of Gaussian type behavior, that if you ask a population of, you know, 100,000 Americans with mortgages, ask what proportion of them would default on their mortgages. If everything was decorrelated, it would be a nice bell curve.

• 59:23Speaker 01

  And you can manage risk with 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 to default at the same time, that's very non-Gaussian behavior. And this wasn't fully accounted for in 2008. Now I think there's some more awareness that this systemic risk is actually a much bigger issue. And just because the model is pretty and nice, it may not match reality.

• 59:55Speaker 01

  The mathematics of working out what models do is really important. But also the science of validating when the models fit reality and when they don't. I mean, you need both. And but mathematics can help because it can. For example, the central lembertium, 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 this Gaussian behavior. Things are fine. It tells you where to look for weaknesses in the model. So if you have a mathematical understanding of central lembertium, and someone proposes to use these Gaussian copulas, whatever, to model default risk. If you're mathematically trained, you would say, OK, but what if there's systemic correlation between all your inputs?

• 1:00:40Speaker 01

  And so then you can ask the economists, you know, how much risk is that? And then you can go look for that. So there's always this synergy between science and mathematics.

• 1:00:52Lex Fridman?

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

• 1:01:35Speaker 01

  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 fields of mathematics that were previously not connected and finding connections. An ancient example is geometry and number theory. And so in the times of the ancient Greeks, these were considered different subjects. I mean, mathematicians worked on both. You know, 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 know, you could say that this length was five times this length because you could take five copies of this length and so forth. But it wasn't until Descartes, who really realized that, who developed analytic geometry,

• 1:02:20Speaker 01

  that you can you can parameterize the plane, a geometric object, by two real numbers. Every point can be, and so geometric problems can be turned into into problems about numbers.

• 1:02:33Speaker 01

  And today, this feels almost trivial, like there's no content to this. Like, of course, you know, a plane is X, X and Y, because that's what we teach and it's internalized. But it was an important development that these two fields were unified. And this process has just gone on throughout mathematics over and over again. Algebra and geometry were separated, and now we have this 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 foxes. A 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.

• 1:03:17Speaker 01

  And then there's people who are kind of, who can play both roles. And I think ideal collaboration between mathematicians involves very, you need some diversity. Like a fox working with many hedgehogs or vice versa. So, yeah, but I identify mostly as a fox. Certainly, I like arbitrage somehow, you know, like learning how one field works, learning the tricks of that, and then going to another field, which people don't think is related, but I can I can adapt the tricks.

• 1:03:49Lex Fridman?

  So see the connections between the fields.

• 1:03:51Speaker 01

  Yeah, so there are other mathematicians who are far deeper than I am. Like, they're really hedgehogs, you know, 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.

• 1:04:04Lex Fridman?

  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?

• 1:04:26Speaker 01

  I'm much more comfortable with the the fox paradigm, yeah. So.

• 1:04:32Speaker 01

  Yeah, I like looking for analogies, narratives. I spent a lot of time, if there's a result, I see it in one field and I like the result. It's a cool result, but I don't like the proof. Like it uses types of mathematics that I'm not super familiar with. I often try to reprove it myself using the tools that I favor. Often my proof is worse, but by the exercise of doing so, I can say, oh, now I can see what the other proof was trying to do. And from that, I can get some understanding of the tools that are used in that field. So it's very exploratory, very doing crazy things in crazy fields and like reinventing the wheel a lot.

• 1:05:13Speaker 01

  Yeah, whereas the hedgehog style is, I think, much more scholarly, you know, you're very knowledge based. You stay up to speed on like all the developments in this field. You know all the history. You have a very good understanding of exactly the strengths and weaknesses of each particular technique. Yeah, I think you'd 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 people who are extremely good at that.

• 1:05:43Lex Fridman?

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

• 1:06:13Speaker 01

  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 had this amazing way of thinking about all kinds of things in a way that you wouldn't normally think of. So he thought of proofs themselves as occupying some sort of space. So if you want to prove something, let's say that there's infinitely many primes, okay? You have all different proofs, but you could rank them in different axes. Like some proofs are elegant, some proofs are long, some proofs are elementary and so forth. And so this is cloud, so the space of all proofs itself has some sort of shape.

• 1:06:56Speaker 01

  And so he was interested in in extreme points of this shape like out of all these proofs, what is one of the shortest at the expense of everything else or or the most elementary or or whatever. And so he gave some examples of well-known theorems and then he would give what he thought was was the extreme proof in these different aspects. I just found that really eye opening that, you know, it's not just getting a proof for was interesting, but but once you have that proof, you know, trying to to to optimize it in various ways, that that proof, proofing itself had some craftsmanship to it. It's something for my

• 1:07:40Speaker 01

  writing style that, you know, like when you do your math assignments and as an undergraduate, your homework and so forth, you you're sort of encouraged to just write down any proof that works. Okay. And they hand it in, get it, as long as it gets a tick mark, you you move on. But if you want 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 to other things. It's the same in many other disciplines like like coding. There's a lot of analogies between math and coding. I like analogies, if you haven't noticed. But, you know, like, you can code something spagetic that works

• 1:08:21Speaker 01

  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, build upon it, and so on, and has fewer bugs and whatever. And there's similar things with mathematics. So.

• 1:08:37Lex Fridman?

  Yeah, the first of all, there's so many beautiful things there. And Conway is one of the great minds in mathematics ever and computer science. Just even considering the space of proofs.

• 1:08:49Speaker 01

  Yeah.

• 1:08:50Lex Fridman?

  And saying, okay, what does this space look like? And what are the extremes? Like you mentioned coding as an analogy is interesting, because there's also this activity called code golf.

• 1:09:01Speaker 01

  Oh, yeah, yeah, yeah.

• 1:09:02Lex Fridman?

  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.

• 1:09:11Speaker 01

  Yeah.

• 1:09:12Lex Fridman?

  Then I believe there's even competitions on this.

• 1:09:13Speaker 01

  Yeah, yeah, yeah.

• 1:09:14Lex Fridman?

  And it's also a nice way to stress test, not just the.

• 1:09:22Lex Fridman?

  Sort of the programs or in this case the proofs, but also the different languages. Maybe that's a different notation or whatever to use to accomplish a different task.

• 1:09:31Speaker 01

  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.

• 1:09:42Lex Fridman?

  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 mc squared. So when a few concepts come together, that's why the Euler identity is often considered the most beautiful equation in mathematics. Do you find beauty in that one, in the Euler identity?

• 1:10:08Speaker 01

  Yeah, well, as I said, I mean, what I find most appealing is connections between different things that you. So the if you eat the pi i equals minus one. So yeah, people always use all the fundamental constants. Okay, that that's I mean, that's cute. But but to me, so the exponential function was introduced by Euler to measure exponential growth. And so I think compound interest or decay or anything which is continuously growing, continuously decreasing, growth and decay or dilation or contraction is modeled by the exponential function. Whereas pi comes around from circles and rotation. If you want to rotate a needle, for example, hundred degrees, you need to rotate by pi radians. And i, complex numbers, represents this, what we would imagine axis of a ninety degree rotation. So a change in direction.

• 1:10:53Speaker 01

  So the exponential function represents growth and decay in the direction that you already are. When you stick an i in the exponential, now it's instead of motion in the same direction as your composition, it's motion as right angles to your composition, so rotation. And then so if the pi i equals minus one tells you that if you rotate for time pi, you end up at the other direction. So it unifies geometry through dilation and exponential growth or dynamics through this act of of complexification, rotation by i. So it connects together all these two's mathematics. Yeah, yeah, that I was jumping complex and complex and the complex numbers, they're all considered almost, yeah, they're all next door neighbors in mathematics because of this identity.

• 1:11:37Lex Fridman?

  Do you think the thing you mentioned is cute, the the collisional notations from these disparate fields? It's just a frivolous side effect? Or do you think there is legitimate like value in one notation? All the our old friends come together.

• 1:11:53Speaker 01

  Right, well, it's it's it's confirmation that you have the right concepts. So when you first study anything, you you have to measure things and give them names. And initially, sometimes you're because your your model is again too far off from reality. You give the wrong things the best names and you only find out later what's what's really important.

• 1:12:14Lex Fridman?

  Physicists can do this sometimes. I mean, but it turns out, okay.

• 1:12:17Speaker 01

  So actually, physics, so E equals M squared. Okay, so one of the the big things was the E. So when when Aristotle first came up with his laws of of motion and then and then Galileo and Newton and so forth. You know, they saw the things they could they could measure. They could measure mass and acceleration and force and so forth. And so Newtonian mechanics, for example, I think was M A was the famous Newton's second law of motion. So those were the the primary objects. So they gave them the central billing in the theory. It was only later, after people started analyzing these equations, that there always seemed to be these quantities that were conserved. So in particular, momentum and energy.

• 1:12:54Speaker 01

  And it's not obvious that things happen in energy. Like, it's not something you can directly measure the same way you can measure mass and and velocity. So both, but over time, people realized that this was actually a really fundamental concept. Hamilton, eventually in nineteenth century, reformulated Newton's laws of physics into what's called Hamiltonian mechanics. Where the energy, which is now called the Hamiltonian, was the dominant object. Once you know how to measure the Hamiltonian of any system, you can describe completely that the dynamics, like what happens to it to all the states. Like it's it really was a central actor, which was not obvious initially.

• 1:13:28Speaker 01

  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.

• 1:13:41Speaker 01

  Adapted Newtonian thinking, because everything was a particle and so forth, to quantum mechanics. Because everything was a wave, but it just looked really, really weird. Like, what is the quantum vision if it was MA? 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 a Hamiltonian. It's a different type of object, it's what's called an operator rather than a function. But again, once you specify it, you specify the entire dynamics. So there's some called Schrödinger's equation that tells you exactly how quantum systems evolve once you have a Hamiltonian.

• 1:14:22Speaker 01

  So side by side, they look completely different objects, you know, like one involves particles, one involves waves and so forth. But with this centrality, you could start actually transferring a lot of intuition and facts from classical mechanics to quantum mechanics. So, for example, in classical mechanics, there's this thing called notice theorem. Every time there's a symmetry in a physical system, there is 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 if 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 conservation angular momentum. If I wait for 10 minutes, I still have the same laws of physics. So there's time transition invariance, this corresponds to the law of conservation of energy.

• 1:15:07Speaker 01

  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 has symmetry, the equations will have a conservation law. So it's, it's, it's, it's, once you have the right language, it actually makes things a lot cleaner.

• 1:15:32Speaker 01

  One of the points why we can't unify quantum mechanics and general relativity yet. We haven't figured out what the fundamental optics are like. For example, we have to give up the notion of space and time being these almost creating type spaces. And it has to be, you know, and you know, we kind of know that at very tiny scales. This can be quantum fluctuations, this is space time foam. And trying to use Cartesian coordinates X, Y, Z is going to be, it's just, it's a nonstarter. But we don't know how to, what to replace it with. We don't actually have the mathematical concepts, the analog of the Hamiltonian that sort of organized everything.

• 1:16:08Lex Fridman?

  Does your guts 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?

• 1:16:19Speaker 01

  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 were separate theories. And then Maxwell unified them, you know, Newton unified 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 observations and theory. Part of our problem is that physics is a victim of no success. That of two big theories of physics, general relativity and quantum mechanics are so, 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 accelerations or, or the early universe or, or things that are really hard to measure.

• 1:17:03Speaker 01

  In order to get any deviation from either of these two theories to the point where you can actually figure out how to combine them together. But I have faith that we, you know, we've, we've been doing this for centuries. We've made progress before. There's no reason why we should stop.

• 1:17:18Lex Fridman?

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

• 1:17:24Speaker 01

  What often happens is that. When the physicists need some theory of mathematics, there's often some precursor that the mathematicians worked out earlier. So when Einstein started realizing that space was curved, he went to some mathematician and asked, is there some theory of curved space that the mathematicians already came up with that could be useful? And he said, oh, yeah, I think Riemann came up with something. And so, yeah, Riemann had developed Riemannian geometry, which is precisely a theory of spaces that are curved in various general ways, which turned out to be almost exactly what was needed for Einstein's theory. This is going back to witness unreasonable effectiveness of mathematics. I think the theories that work well if they explain the universe tend to also involve

• 1:18:08Speaker 01

  the same mathematical objects that work well to solve mathematical problems. Ultimately, they're just both ways of organizing data in useful ways.

• 1:18:16Lex Fridman?

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

• 1:18:24Speaker 01

  Yeah, that was a leading candidate for many decades. It's, I think, is slowly falling out of fashion because it's not matching experiment.

• 1:18:32Lex Fridman?

  So one of the big challenges, of course, like you said, is experiment is very tough.

• 1:18:37Speaker 01

  Yes.

• 1:18:37Lex Fridman?

  Because of the how effective both theories are. But the other is like, just, you know, you're talking about, you're not just deviating from spacetime, you're going into like some crazy number of dimensions, you're doing all kinds of weird stuff that to us, we've gone so far from this flat earth that we started at, like you mentioned.

• 1:18:59Speaker 01

  Yeah, yeah, yeah.

• 1:18:59Lex Fridman?

  And now we're just, it's very hard to use our limited abiding descendants of cognition to intuit what that reality really is like.

• 1:19:09Speaker 01

  This is why analogies are so important. I mean, so, yeah, the round earth is not intuitive because we're stuck on it. But, you know, but round objects in general, we have pretty good intuition over. And we've been interested about light works and so forth. And like, it's actually a good exercise to actually work out how eclipses and phases of the sun and the moon and so forth can be really easily explained by round earth and round moon. And models and and you can just take, you know, a basketball and a golf ball and and and a light source and actually do these things yourself. So the intuition is there, but you have to transfer it.

• 1:19:47Lex Fridman?

  That is a big leap in to lecture for us to go from flat to round earth because, you know, our life is mostly lived in flat land.

• 1:19:54Speaker 01

  Yeah.

• 1:19:55Lex Fridman?

  To load that information and we're all like, take it for granted. We take so many things for granted because science has established a lot of evidence for this kind of thing. But, you know, we're in a round rock.

• 1:20:07Speaker 01

  Yeah.

• 1:20:08Lex Fridman?

  Flying through space.

• 1:20:09Speaker 01

  Yeah, yeah.

• 1:20:10Lex Fridman?

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

• 1:20:15Speaker 01

  Right, yeah. So modern science is maybe, again, a victim of its own success is that, you know, in order to be more accurate, it has to to move further and further away from your initial intuition. And so for someone who hasn't gone through the whole process of science education, it looks more and more suspicious because of that. So, you know, we need, we need more grounding. I mean, I think, I mean, you know, there are, there are scientists who do excellent outreach. But there's, there's, there's, there's lots of science things that you can do at home. I have 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.

• 1:21:00Lex Fridman?

  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 mystery.

• 1:21:14Speaker 01

  Right.

• 1:21:15Lex Fridman?

  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.

• 1:21:25Speaker 01

  Yeah, yeah.

• 1:21:25Lex Fridman?

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

• 1:21:29Speaker 01

  Change of perspective is really important. You say travel broadens the mind. This is intellectual travel, you know, put yourself in the mind of the ancient Greeks or or some other person, some other time period, make hypotheses, spherical cows, whatever, speculate. And, you know, this is. This is what mathematicians do and some artists do, actually.

• 1:21:48Lex Fridman?

  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

• 1:21:58Speaker 01

  Right.

• 1:21:58Lex Fridman?

  when you don't have much to figure out stuff with.

• 1:22:01Speaker 01

  If you propose axioms, then the mathematics lets you follow those axioms to their conclusions. And sometimes you can get quite a long way from initial hypotheses.

• 1:22:09Lex Fridman?

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

• 1:22:30Speaker 01

  I have worked on some equations, something called the wave maps equation, or the sigma field model, which is not quite the equation of spacetime gravity itself, but of certain fields that might exist on top of spacetime. So Einstein's equation of relativity just describes space and time itself. But then there's other fields that live on top of that. There's the electromagnetic field, there's things called Yang-Mills fields. And there's this whole hierarchy of different equations, of which Einstein considered one of the most nonlinear and difficult. But relatively low in the hierarchy was this thing called the wave maps equation. So it's a wave which, at any given point, is fixed to be like on a sphere.

• 1:23:11Speaker 01

  So I can think of a bunch of arrows in space and time, and the arrows are just 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 sheaves of wheat in a wheat field. And I was interested in the global regularity problem again for this question. Like, is it possible for all the energy here to collect at a point? So the equation I considered was actually what's called a critical equation, where it's actually the behavior at all scales is roughly the same. And I was able barely to show that that you couldn't actually force a scenario where all the energy concentrated at one point, that at the end you had to dismiss a little bit. At the moment, it just a little bit, it would stay regular. Yeah, this was back in 2000.

• 1:23:55Speaker 01

  That was part of why I got interested in Neri's talks afterwards, actually. Yeah, so I developed some techniques to solve that problem. So part of it, 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 effect. 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 heaps of 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

• 1:24:37Speaker 01

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

• 1:25:21Speaker 01

  And yeah, my aunt walked in on me while I was doing that and she was asking, what am I doing, doing this?

• 1:25:27Lex Fridman?

  It's complicated, is the answer.

• 1:25:29Speaker 01

  Yeah, and okay, fine, you know, you're a young man, I don't ask questions.

• 1:25:34Lex Fridman?

  I have to ask about the, you know, how do you approach solving difficult problems? What, if it's possible?

• 1:25:44Lex Fridman?

  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?

• 1:25:56Speaker 01

  A lot of pen and paper. One thing you pick up as a mathematician is sort of, I call it cheating strategically. So the beauty of mathematics is that you get to change the rules, change the problem, change the rules as you wish. You don't get to do this for any other field. Like, you know, if you're an engineer and someone says, build a bridge over this river, you can say, I want to build this bridge over here instead, or I want to build it out of paper instead of steel. But a mathematician, you can do whatever you want.

• 1:26:26Speaker 01

  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. There's a dimension that's too large. I'll set it to one. I'd solve the one dimension problem first. So there's a main term and an error term. I'm going to make a spherical cow or something. I've seen the error term is zero. And so the way you should solve these problems is not in sort of this Iron Man mode where you make things maximally difficult. But actually, the way you should, you should approach any reasonable math problem is that you, if there are 10 things that are making your life difficult, find a version of the problem that turns off nine of the difficulties and only keeps one of them.

• 1:27:04Speaker 01

  And solve that. And then that just, so you install nine cheats. Okay, you install 10 cheats, then the game is trivial. But you install nine cheats, you solve one problem, that teaches you how to deal with that particular difficulty. And then you turn that one off and you turn someone else, something else on. And then you solve that one. And after you know how to solve the 10 problems, 10 difficulties separately, then you have to start merging them a few at a time.

• 1:27:28Speaker 01

  I, as a kid, I watched a lot of these Hong Kong action movies. From a culture. And one thing is that every time it's a fight scene, you know, so maybe the hero gets swarmed by 100 bad guy goons or whatever. But it'll always be choreographed so that you'd always be only fighting one person at a time. And then you would defeat that person and move on. And because of that, he could defeat all of them. But whereas if they had fought a bit more intelligently and just swarmed the guy at once, it would make for much, much worse choreo cinema. But they would win.

• 1:28:01Lex Fridman?

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

• 1:28:07Speaker 01

  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 four blackboards and then sit on my couch and just sort of see the whole thing.

• 1:28:19Lex Fridman?

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

• 1:28:23Speaker 01

  Oh, there's a lot of drawing and a lot of bespoke doodles that only make sense to me. I mean, and the beauty of a blackboard is you erase and it's a very organic thing. I'm beginning to use more and more computers, partly because AI makes it much easier to do simple coding things. You know, if I wanted to plot a function before, which is moderately complicated, has some iteration or something, you know, I'd have to remember how to set up a Python program and how does a for 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.

• 1:29:01Lex Fridman?

  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? And how it can help as a proof assistant? And maybe how you started using it and how it has helped you?

• 1:29:25Speaker 01

  So Lean is a computer language, much like sort of standard languages like Python and C and so forth. Except that in most languages, the focus is on producing executable code. Lines of code do things, you know, they flip bits or they make a robot move or they deliver you text on the internet 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 language like Python might do a computation and give you that the answer is seven. Okay, that it does the sum of three plus four is equal to seven. But Lean can produce not just the answer, but a proof that how it got the answer of seven as three plus four and all the steps involved in.

• 1:30:09Speaker 01

  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 assistants. And so they provide languages for which you can create quite complicated, intricate mathematical proofs. And they produce these certificates that give a 100% guarantee that your arguments are correct if you trust the compiler of Lean. But they made the compiler really small. And you can, there are several different compilers available for the same.

• 1:30:43Lex Fridman?

  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 a statement?

• 1:30:55Speaker 01

  So Lean, a lot of mathematicians were involved in the design of Lean. So it's designed so that individual lines of code resemble individual lines of a mathematical argument. You might want to introduce a variable, you want to prove a contradiction. There are various standard things that you can do and it's written so ideally it should be like a one-time correspondence. In practice, it isn't because Lean is like explaining a proof to an extremely pedantic colleague who will point out, okay, did you really mean this? 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 be less annoying. So, for example, every mathematical object has to come with a type.

• 1:31:38Speaker 01

  Like, if I talk about x, is x a real number or a natural number or a function or something? If you write things informally, it's often in a certain context. You say, you know, clearly x is equal to, let x be the sum of y and z, and y and z were already real numbers, so x should also be a real number. 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.

• 1:32:16Lex Fridman?

  Is it using something like LLMs to do the type inference? Or like you mentioned with the real number?

• 1:32:22Speaker 01

  It's using much more traditional, what's called good old fashioned AI. You can represent all these things as trees, and there's always algorithms to match one tree to another tree.

• 1:32:30Lex Fridman?

  So it's actually doable to figure out if something is a... A real number or a natural number.

• 1:32:36Speaker 01

  Yeah, every object sort of comes with a history of where it came from and you can you can kind of trace.

• 1:32:39Lex Fridman?

  Oh, I see.

• 1:32:41Speaker 01

  Yeah, so it's it's designed for reliability. So modern AIs are not used in it's a disjointed technology. People are beginning to use AIs 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 calculus, say, okay, to do the next step. So the Lean developers have built this this massive project called Mathlib, a collection of tens of thousands of useful facts about mathematical objects. And somewhere in there is the fundamental calculus, but you need to find it. So a lot of the bottleneck now is actually LeMA 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 Mathlib that you can do.

• 1:33:24Speaker 01

  But there's now these large language models that you can say, I need the fundamental calculus at this point. And it was like, okay, for example, when I code, I have GitHub Copilot installed as a plugin to my IDE and it scans my text and it sees what I need, says, you know, I might even type. Now I need to use the fundamental calculus. Okay, and then it might suggest, okay, try this. And like maybe 25% of the time it works exactly. And then another 10, 50% of the time it doesn't quite work, but it it's close enough that I can say, oh, yeah, if I just change it here and here, it will work. And then like half the time it gives me complete rubbish.

• 1:33:59Speaker 01

  So but people are beginning to use AIs a little bit on top. Yeah, mostly on the level of basically fancy autocomplete. That you can type half of one line of a proof and it will find it will tell you.

• 1:34:10Lex Fridman?

  Yeah, but but a fancy, especially fancy with the sort of capital letter F is

• 1:34:16Lex Fridman?

  removes some of the friction mathematician might feel when they move from pen and paper to formalizing.

• 1:34:23Speaker 01

  Yes, yeah, so right now I estimate that the effort time and effort taken to formalize a proof is about 10 times the amount taken to to write it out. Yeah, so it's doable, but you don't it's it's annoying.

• 1:34:35Lex Fridman?

  But doesn't it like kill the whole vibe of being a mathematician?

• 1:34:39Speaker 01

  Yeah, so I mean.

• 1:34:40Lex Fridman?

  Having a pedantic coworker.

• 1:34:42Speaker 01

  Right, yeah, if if that was the only aspect of it, okay, but. Okay, there's some cases where it's actually more pleasant to do this formally. So there's a theorem I formalized and there was a certain constant 12 that came out in the final statement. And so this 12 had to be carried all through the proof. And like everything had to be checked that it goes, all these other numbers had to be consistent with this final number 12. And then so we wrote a paper through this theorem with this number 12. And then a few weeks later, someone said, oh, we can actually improve this 12 to an 11 by reworking some of these steps. And when this happens with pen and paper, like every time you change a parameter, you have to check line by line that every single line of your proof still works. And there can be subtle things that you didn't quite realize. Some properties of number 12 that you didn't even realize that you were taking advantage of. So a proof can break down at a subtle place.

• 1:35:29Speaker 01

  So we had formalized the proof with this constant 12. And then when this new paper came out, we said, oh, let's, so that took like three weeks to formalize. And like 20 people to formalize this original proof. I said, oh, but now let's, let's, let's update the 12 to 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 of 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 immediately 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.

• 1:36:10Speaker 01

  And there'll just be a few places where you, I mean, if you don't hard code your constants, but you sort of,

• 1:36:17Speaker 01

  you use smart tactics and so forth, 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. Of course, this is very quick process.

• 1:36:30Speaker 01

  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.

• 1:36:39Lex Fridman?

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

• 1:36:47Speaker 01

  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 twenty seven and you look at paragraph six and you have a line of text of math, I often can't read it immediately because it assumes various definitions, which I have to go back and maybe on ten 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 and start reading. There's a lot of context. But with Lean, if you put your cursor on a line of code, every single object there, you can hover over it and it will say what it is, where it came from, where it's justified.

• 1:37:31Speaker 01

  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, a 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. Say, I'm gonna, I'm gonna 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 sixty seven here. I need to prove this thing, but it doesn't quite work. Here's like the three lines of code I'm having trouble with. But because all the context is there, someone else can say, oh, okay, I recognize what you need to do.

• 1:38:17Speaker 01

  You need to apply this trick or this tool. And you can do extremely atomic level conversations. So because of Lean, I can collaborate, you know, with dozens of people across the world, most of whom I don't have never met in person. And I may not know actually even whether they're, how reliable they are in their, in the proof they give me. But Lean gives me a certificate of trust. So I can do, I can do trustless mathematics.

• 1:38:42Lex Fridman?

  So there's so many interesting questions. There's, so one, you're, you're known for being a great collaborator. So what is the right way to approach?

• 1:38:54Lex Fridman?

  Solving a difficult problem in mathematics when you're collaborating. Are you doing a divide and conquer type of thing? Or are you focusing on a particular part and you're brainstorming?

• 1:39:05Speaker 01

  There's always a brainstorming process first. Yeah, so math research projects by their nature, when you start, you don't really know how to do the problem. It's not like an engineering project where somehow the theory has been established for decades and its implementation is the main difficulty. You have to figure out even what is the right path. So this is what I said about cheating first. It's like, to go back to the bridge building analogy. First, assume you have infinite budget and like 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. Like I say, they have fixed requirements.

• 1:39:47Speaker 01

  Yes, there's this sort of jam sessions always at the beginning where you try all kinds of crazy things and you make all these assumptions that are unrealistic, but you plan to fix later.

• 1:39:57Speaker 01

  And you try to see if there's even some skeleton of an approach that might work. And then hopefully that breaks up the problem into smaller sub-problems, which you don't know how to do, but then you focus on the sub ones. And sometimes different collaborators are better at working on certain things.

• 1:40:15Speaker 01

  So one of my theorems I'm known for is a theorem Ben Green, it's called the green tower theorem. It's a statement that the primes contain arithmetic progressions of N event. So it was a modification of this theorem as I'm already. And the way we collaborated was that Ben had already proven a similar result for progressions of N three. He showed that just like the primes contain lossless questions of N three, and even subsets of the primes, certain subsets do. But his techniques only worked for N three progressions. They didn't work for longer progressions. But I had these techniques coming from a godic theory, which is something that I had been playing with and I knew better than Ben at the time. And so if I could justify certain randomness properties of some set relating to primes,

• 1:40:58Speaker 01

  like there's a certain technical condition which if I could. Habit. 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 weaker hypothesis that I have a chance to prove it? And he proposed something which he could prove, but it was too weak for me. I can't use this. So there was this conversation going back and forth.

• 1:41:27Lex Fridman?

  Different cheats too.

• 1:41:29Speaker 01

  Yeah, 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 that theorem. Yeah, so there's all kinds of dynamics. You know, I mean, every collaboration has some story. No two are the same.

• 1:41:50Lex Fridman?

  And then on the flip side of that, like you mentioned with lean programming. Now, that's almost like a different story because you can do, you can create, I think you've mentioned a kind of a blueprint

• 1:42:02Speaker 01

  Right.

• 1:42:02Lex Fridman?

  for a problem. And then you can really do a divide and conquer with lean, where you're working on separate parts.

• 1:42:10Speaker 01

  Right.

• 1:42:10Lex Fridman?

  And they're using the computer system proof checker essentially.

• 1:42:14Speaker 01

  Yeah.

• 1:42:14Lex Fridman?

  To make sure that everything is correct along the way.

• 1:42:16Speaker 01

  Yeah, 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. A math paper basically is a blueprint in a sense. It is taking a difficult statement, like a big theorem and breaking it up into a hundred little lemmas.

• 1:42:41Speaker 01

  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 you're trying to make each step kind of self-contained. And or depending on only a very specific number of previous statements that have been proven. So that each node of this blueprint graph that gets generated can be tackled independently of all the others. And you don't even need to know how the whole thing works. So it's like a modern supply chain, you know, like if you want to create an iPhone or some other complicated object. No one person can build up a single object, but you can have specialists to just, if they're given some widgets from some other company, they can combine them together to form a slightly bigger widget.

• 1:43:27Lex Fridman?

  I think that's a really exciting possibility because you can have. If you can find problems that could be

• 1:43:33Speaker 01

  Right.

• 1:43:33Lex Fridman?

  broken down this way, then you could have, you know, thousands of contributors, right?

• 1:43:37Speaker 01

  Yes, yes, yes. So I told you before about the split between theoretical and experimental mathematics. And right now, most mathematics is theoretical and only a tiny bit is experimental. I think the platform that Lean and other software tools, so 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 want to

• 1:44:01Speaker 01

  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's just one mathematician coding lots and lots of Python or whatever. And because coding is such an error-prone activity, it's not practical to allow other people to collaborate with you on writing modules for your code. Because if one of the modules has a bug in it, the whole thing is unreliable. So it's, these are, so you get these bespoke spaghetti code written by not professional programmers, but mathematicians, you know, and they're clunky and slow.

• 1:44:39Speaker 01

  And so because of that, it's hard to really mass produce experimental results.

• 1:44:45Speaker 01

  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 Equation of Theories project. Basically, we generated about 22 million little problems in abstract algebra. Maybe I should back up and tell you what the project is. Okay, so abstract algebra studies operations like multiplication and addition and their abstract properties. So multiplication, for example, is commutative. X times y is always y times x, at least 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 law is not always true. So given any operation, it obeys some laws and not others.

• 1:45:32Speaker 01

  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?

• 1:45:44Speaker 01

  And the answer is no, because it turns out you can describe an operation which obeys the commutative law but doesn't obey the associative law. So by producing an example, you can you can show that commutativity does not imply associativity. But some other laws do imply other laws by substitution and so forth, and you can write down some algebraic proof. So we look at all the pairs between these 4,000 laws and there's about 22 million of these pairs. And for each pair, we ask, does this law imply this law? If so, give a give a give a proof. If not, give a counterexample. So 22 million problems, each one of which you could give to like an undergraduate algebra student, and they had a decent chance of solving the problem. Although there are a few, of these 22 million, there are like a hundred or so that are really quite hard, okay, but a lot are easy.

• 1:46:25Speaker 01

  And the project was just to to work out, to determine the entire graph, like which ones imply which other ones.

• 1:46:31Lex Fridman?

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

• 1:46:37Speaker 01

  Yeah, so it would not have been feasible. I mean, the state of the art in the literature was like, you know, 15 equations and sort of how they implied. That's sort of at the limit of what a human with a pen and paper can do. So you need to scale that up, so you need to crowdsource, but you also need to trust all the, I mean, no one person can check 22 million of these proofs. You need it 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 of these 22 million, all but two have been settled.

• 1:47:10Lex Fridman?

  Wow.

• 1:47:11Speaker 01

  And, well, actually, and of those two, we have a pen and paper proof of the two, and we were formalizing it. In fact, this morning, I was working on finishing it. So we're almost done on this.

• 1:47:22Lex Fridman?

  It's incredible.

• 1:47:23Speaker 01

  It's, yeah.

• 1:47:24Lex Fridman?

  How many people were able to?

• 1:47:26Speaker 01

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

• 1:47:30Lex Fridman?

  It's a huge number, that's crazy.

• 1:47:32Speaker 01

  Yeah, so we're going to have a paper of 50 of this, and a big appendix of food contributor what?

• 1:47:38Lex Fridman?

  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 the level of expertise of the people, of the contributors? Now, okay, I'm asking a lot of pothead questions here, but I'm imagining a bunch of humans and maybe in the future some AIs.

• 1:47:59Speaker 01

  Yeah.

• 1:48:00Lex Fridman?

  Can there be like an Elo rating type of situation where like a gamification of this?

• 1:48:06Speaker 01

  The beauty of these lean projects is that automatically you get all this data. Yeah. So like, like everything's, we upload it to this GitHub and GitHub tracks who contributed what. So you could generate statistics from at 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, I would definitely not want this to become like, you know, part of your tenure review or something.

• 1:48:31Speaker 01

  But I mean, I think already in enterprise computing, right? People do use some of these metrics as part of the assessment of performance of an employee. Again, this is a direction which is a bit scary for academics to go down. We, we don't like metrics so much.

• 1:48:48Lex Fridman?

  And yet academics use metrics, they just use old ones, number of papers.

• 1:48:56Speaker 01

  Yeah, it's true. It's true that, I mean.

• 1:48:59Lex Fridman?

  It feels like this is a metric while flawed is going in the more in the right direction, right?

• 1:49:05Speaker 01

  Yeah.

• 1:49:05Lex Fridman?

  It's interesting. At least it's a very interesting metric.

• 1:49:08Speaker 01

  Yeah, I think it's interesting to study. I mean, I think you can do studies of whether these are better predictors. There's this problem called Goodhart's law. If a statistic is actually used to incentivize performance, it becomes gamed and then it is no longer a useful measure.

• 1:49:22Lex Fridman?

  Oh, humans, always.

• 1:49:23Speaker 01

  Yeah, yeah, no, I mean, it's rational. So what we've done for this project is self-report. So there are actually these 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 we, there's a standard list of role or so categories. And we just ask each contributor to, there's a big matrix of all the, of all the authors and all the categories, just to tick the boxes where they think that they contributed. And just to give a rough idea, you know, like, oh, so you did some coding and, and you provided some compute, but you didn't do any independent paper verification or whatever. And I think that that works out. Traditionally, mathematicians just order alphabetically by surname.

• 1:50:06Speaker 01

  So we don't have this tradition, as in the sciences of, you know, lead author and second author and so forth. Like, which we're proud of. You know, we make all the authors equal status, but it doesn't quite scale to this size. So a decade ago, I was involved in these things called polymath projects. It was 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 bottleneck, actually. But still, we had projects that were, you know, 10 authors or so. But we had decided at the time not to try to decide who did what, but to have a single pseudonym. So we created this fictional character called DHJ polymath.

• 1:50:49Speaker 01

  In the spirit of Boba Key, Boba Key is the pseudonym for a famous group of mathematicians of the 20th century. But, and so the paper was authored 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 tenure or whatever, you could not use this paper in your, as you submitted as one of your publications, because it was, you didn't have the formal author credit. But the other thing that we've recognized much later is that when people referred to these projects, they naturally referred to the most famous person who was involved in the project. Oh, so this was Tim Gowers' polymath project, this was Terence Thou's polymath project,

• 1:51:32Speaker 01

  and not mentioned the other 19 or whatever people that were involved.

• 1:51:35Lex Fridman?

  Ah, yeah.

• 1:51:36Speaker 01

  So we're trying something different this time around, where we have, everyone's an author, but we will have an appendix with this matrix, and we'll see how that works.

• 1:51:45Lex Fridman?

  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 Lean programming languages 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 this, what seems like the paving of the future of mathematics. So I have to ask you here about.

• 1:52:15Lex Fridman?

  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.

• 1:52:27Speaker 01

  Yes.

• 1:52:27Lex Fridman?

  So this is sort of high level, high school.

• 1:52:31Speaker 01

  Oh, very high level, yes.

• 1:52:32Lex Fridman?

  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?

• 1:52:46Speaker 01

  Yeah, the difficulty increases exponentially with the number of steps involved in the proof. It's a cometary explosion. So I think of large language models is that they make mistakes. And so if a proof has got 20 steps and your large language model has a 10% failure rate at each step of going in the wrong direction, like it's just extremely unlikely to actually reach the end.

• 1:53:09Lex Fridman?

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

• 1:53:18Speaker 01

  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 a speaker in the second language can get some idea of what you're saying. Yeah, but formal language, yeah, if you get one little thing wrong, I think that the whole thing is nonsense. Even formal to formal is very hard. There are different incompatible proof systems and languages. There's Lean, but also COCH and Isabelle and so forth. And actually, even converting from a formal language to a formal language is an unsolved, basically unsolved problem.

• 1:53:51Lex Fridman?

  That is fascinating. Okay, so, but once you have an informal language, they're using their RL-trained model, so something akin to AlphaZero that they used to go to then try to come up with proofs. They also have a model, I believe it's a separate model for geometric problems. So what impresses you about this system? And?

• 1:54:15Speaker 01

  What do you think is the gap? Yeah, we talked earlier about things that are amazing over time become kind of normalized. So yeah, now somehow it's, of course, geometry is a silver bullet problem. Right, that's true, that's true. I mean, it's still beautiful. Yeah, I know. It's a great work that shows what's possible. I mean, it's, the approach doesn't scale currently. Yeah, three days of Google's server time to solve one high school math problem. This is not a scalable prospect, especially with the exponential increase in, as the complexity increases. We should mention that they got a silver medal performance. The equivalent of, I mean. Equivalent of a silver medal. So first of all, they took way more time than was allotted.

• 1:54:57Speaker 01

  And they had this assistance where the humans started helped by formalizing. But also they're giving us those four marks for the solution, which I guess is formally verified. So I guess that's fair. There are efforts, there will be a proposal at some point to actually have an AI Math Olympiad where at the same time as the human contestants get the, the actual problems, the AIs will also be given the same problems, the same time period, and the outputs will have to be graded by the same judges. Which means that will have to be written in natural language rather than formal language. I hope that happens. I hope this IMO happens. I hope, I hope next one.

• 1:55:41Speaker 01

  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, that's, AIs are actually a lot better at problems where there's a specific numerical answer, because it's easy to reinforce, to reinforce on learning on it. Yeah, you got the right answer, you got the wrong answer. It's a very clear signal. But a long form proof either has to be formal, and then the lean can give a 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 reinforcement learning, you know, runs,

• 1:56:26Speaker 01

  you're not, you can't hire enough humans to create those. I mean, it's already hard enough for the last language models to do reinforcement 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.

• 1:56:45Lex Fridman?

  So if we just explore this possible future, what is the thing that humans do that's most special in mathematics? So that you could see AI not cracking for a while. So inventing new theories to coming up with new conjectures versus proving the conjectures, building new abstractions, new representations, maybe an AI turnstile with seeing new connections between disparate fields.

• 1:57:17Speaker 01

  It's a good question. I think the nature of what mathematicians do over time has changed a lot. You know, so a thousand years ago, mathematicians had to compute the date of Easter and there's really complicated calculations, you know, but it's all automated, been automated centuries. We don't need that anymore. You know, they used to navigate to do spherical navigation, spherical trigonometry to navigate how to get from the old world to the new or something very complicated calculation. Again, we've been automated.

• 1:57:45Speaker 01

  You know, even a lot of undergraduate mathematics, even before AI, like Wolfram Alpha, for example, is not a language model, but it can solve a lot of undergraduate level math tasks. So on the computational side, verifying routine things like having a problem and say, here's a problem in partial differential equations. Could you solve it using any of the 20 standard techniques? And they have, yes, I've tried all 20. I hear the 100 different permutations and these are my results. And that type of thing, I think it will work 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 where the AI really struggles right now is knowing when it's made a wrong turn.

• 1:58:32Speaker 01

  And it can say, oh, I'm gonna solve this problem. I'm gonna split up this one 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 get, it will have a problem. It would propose an approach which is just complete nonsense. And 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 very low quality.

• 1:59:03Speaker 01

  Like, you know, submissions where we 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 it's partly because that's what the reinforcement learning has actually trained them to do, to make them to produce text that looks like what is correct, which for many applications is good enough.

• 1:59:28Speaker 01

  So the errors are often really subtle. And then when you spot them, they're really stupid. Like, you know, like no human would have actually made that mistake.

• 1:59:35Lex Fridman?

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

• 1:59:59Speaker 01

  Yeah, so.

• 2:00:00Lex Fridman?

  It's very tricky to, and frustrating for some reason to.

• 2:00:03Speaker 01

  Yeah, so, yeah, so the sense of smell. This is, this is one thing that humans have. And there's a metaphorical mathematical smell that this is not clear how to get the AI to do that. Eventually, I mean, so the way.

• 2:00:22Speaker 01

  Alpha zero and so forth, they make progress on Go and chess and so forth. This is in some sense, they have developed a sense of smell for Go and chess positions. That this position is good for white, it's good for black. They can't initiate why, but just having that that sense of smell lets them strategize. So if AIs gain that ability to sort of a sense of viability of certain proof strategies. So you can say, I'm going to try to break up this problem into two small subtasks. And they 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. So this is good to try. Or no, you've you've you made the problem worse because each of the two subproblems is actually harder than your original problem. Which is actually what normally happens if you try a random thing to try.

• 2:01:07Speaker 01

  Normally, it's very easy to transform a problem into an even harder problem. Very rarely do you transform a simpler problem.

• 2:01:16Speaker 01

  Yeah, so if they can pick up a sense of smell, then they could maybe start competing with a human level methodology.

• 2:01:24Lex Fridman?

  So this is a hard question, but not competing, but collaborating.

• 2:01:27Speaker 01

  Yeah.

• 2:01:27Lex Fridman?

  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.

• 2:01:36Speaker 01

  Yeah, yeah, yeah.

• 2:01:37Lex Fridman?

  What would that oracle, what would you like that oracle to be able to do? Would you like it to maybe be a verifier, like check? Do the codes smell like you're? Yes, Professor Tao, this is the correct, this is a good, this is a promising fruitful direction.

• 2:01:54Speaker 01

  Yeah, yeah, yeah.

• 2:01:55Lex Fridman?

  Or or would you like it to generate possible proofs? And then you see which one is the right one? Or would you like it to maybe generate different representation, different, totally different ways of seeing this problem?

• 2:02:10Speaker 01

  I think all of the above. A lot of it is, we don't know how to use these tools because it's a paradigm that is not.

• 2:02:19Speaker 01

  Yeah, we have not had in the past assistants that are competent enough to understand complex instructions that can work at massive scale, but are also unreliable. It's an interesting, a bit unreliable in subtle ways whilst we whilst providing sufficiently good output. It's an interesting combination. You know, I mean, you have you have graduate students you work with who kind of like this, but not at scale.

• 2:02:47Speaker 01

  Yeah, and and we had previous software tools that can work at scale, but but very narrow.

• 2:02:54Speaker 01

  So we have to figure out how to how to use. I mean, so Tim Coward, actually, you mentioned, he actually foresaw like in two thousand, he was envisioning what mathematics would look like in in actually two and a half decades.

• 2:03:08Speaker 01

  That's funny. Yeah, he he wrote in his in his article, like a hypothetical conversation between a mathematical assistant of the future and himself, you know, trying to solve a problem. And they would have a conversation. Sometimes the human would propose an idea and the AI would would evaluate it. And sometimes the AI would propose an idea. And and sometimes the competition was required and AI would just go and say, okay, I've checked the hundred cases needed here. Or the first, you said this is true for all N, I've checked the N up to one hundred, and it looks good so far. Or hang on, there's a problem at N equals forty six. And so that's a free form conversation where you don't know in advance where things are going to go, but just based on, I think ideas are good proposed on both sides, calculations could propose on both sides.

• 2:03:53Speaker 01

  I've had conversations with AI where I say, okay, let's, we're going to collaborate to solve this math problem. And it's a problem that I already know a solution to. So I try to prompt it. Okay, so here's the problem. I suggest using this tool. And then it'll find this lovely argument using a completely different tool, which eventually goes into the weeds and say, no, no, no, try using this. Okay, and it might start using this and then it'll go back to the tool that I wanted to do before. And you have to keep railroading it onto the path you want. And I could eventually force it to give the proof I wanted. But it was like herding cats. And the amount of personal effort I had to take to not just to prompt it, but also check its output, because a lot of what it looked like is going to work, and I know there's a problem on nine seventeen. And basically arguing with it.

• 2:04:37Speaker 01

  Like, it was more exhausting than doing it unassisted. So like, but that's the current state of the art. I wonder if there's.

• 2:04:45Lex Fridman?

  There's a phase shift that happens to where it's no longer feels like herding cats. And maybe you'll surprise us how quickly that comes.

• 2:04:53Speaker 01

  I I believe so. So in formalization, I mentioned before that it takes 10 times longer to formalize a proof that I write at my hand. With these modern AI tools, and also just better tooling, the lean developers are doing a great job adding more and more features and making it user-friendly. It's going from nine to eight to seven. Okay, no big deal. But one day it will drop to a one, and that's a phase shift. Because suddenly, it makes sense when you write a paper to write it in lean first, or through a conversation with AI, which is generally on the file with you. And it becomes natural for journals to accept, maybe they'll offer expedite refereeing.

• 2:05:36Speaker 01

  If a paper has already been formalized in lean, they'll just ask the referee to comment on the significance of the results and how it connects to literature and not worry so much about the correctness, because it has been certified. Papers are getting longer and longer in mathematics, and it's harder and harder to get good refereeing for the really long ones, unless they're really important. It is actually an issue, which the formalization is coming at just the right time for this to be.

• 2:06:03Lex Fridman?

  And the easier and easier it gets because of the tooling and all the other factors, then you're going to see much more like math lib will grow potentially exponentially. It's a virtuous cycle. Okay.

• 2:06:15Speaker 01

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

• 2:06:37Lex Fridman?

  It's a wild out there question, but what year, how far away are we from?

• 2:06:45Lex Fridman?

  A AI system being a collaborator on a proof that wins the Fields Medal. So that level.

• 2:06:54Speaker 01

  Okay, well, it depends on the level of collaboration.

• 2:06:57Lex Fridman?

  No, like it deserves to be to get the Fields Medal, like so.

• 2:07:02Speaker 01

  Already like I can imagine if it was a medal winning paper, having some AI systems in writing it, you know, just, you know, like the autocomplete alone is already, I use it, like it speeds up my own writing. Like, you know, you can have a theorem, you have a proof and the proof has three cases. And I write down the proof of first case and the autocomplete just suggests that now here's how the proof of second case could work. And like he was exactly correct. That was great. Saved me like five, 10 minutes of typing.

• 2:07:30Lex Fridman?

  But in that case, the AI system doesn't get the Fields Medal.

• 2:07:32Speaker 01

  No.

• 2:07:37Lex Fridman?

  Are we talking 20 years, 50 years, 100 years? What do you think?

• 2:07:41Speaker 01

  Okay, so I gave a prediction in print about 2026, which is now next year. There will be math collaborations with AI. So not Fields Medal winning, but but like actual research level.

• 2:07:54Lex Fridman?

  Like published ideas that in part generate about AI.

• 2:07:57Speaker 01

  Maybe not the ideas, but at least some of the computations.

• 2:08:02Speaker 01

  The verifications, yeah, I mean.

• 2:08:04Lex Fridman?

  Has that already happened?

• 2:08:04Speaker 01

  Has it already happened? Yeah, there are, there are problems that were solved

• 2:08:10Speaker 01

  by a complicated process conversing with AI to propose things and the human goes and tries it. And then contract doesn't work, but it was a different idea. It's hard to disentangle exactly.

• 2:08:24Speaker 01

  There are certainly math results which could only have been accomplished because there was a math, a human mathematician and an AI involved.

• 2:08:32Speaker 01

  But it's hard to disentangle credit.

• 2:08:37Speaker 01

  I mean, these tools, they do not replicate all the skills needed to do mathematics, but they can replicate sort of some non-trivial percentage of them, thirty, forty percent. So they can fill in gaps. You know, so coding is a good example, you know. So I, it's annoying for me to code in Python. I'm not, I'm not a native, you know, professional programmer. But with AI, the friction cost of doing it is much reduced. So it fills in that gap for me.

• 2:09:12Speaker 01

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

• 2:09:30Speaker 01

  You know, you should, in a few years, get to the point where you have a dilemma that you need. And say, has anyone proven this dilemma before? And we will do basically a fancy web search AI system and say, yeah, there are these six papers where something similar has happened. And I mean, you can ask it right now and it will give you six papers of which maybe one is legitimate and relevant, one exists but is not relevant and four are hallucinated. It has a non-zero success rate right now, but it's, there's so much garbage, so much, the signal to noise ratio is so poor that it's, it's most helpful when you're already somewhat knowledgeable and you just need to be prompted to be reminded of a paper that was already subconsciously in your memory.

• 2:10:13Lex Fridman?

  Versus helping you discover new you were not even aware of, but is the correct citation.

• 2:10:18Speaker 01

  Yeah, that's, yeah, that it can sometimes do, but, but when it does, it's buried in a list of options which the other.

• 2:10:25Lex Fridman?

  That are bad.

• 2:10:26Speaker 01

  Yeah.

• 2:10:26Lex Fridman?

  I mean, being able to automatically generate a related work section that is correct.

• 2:10:31Speaker 01

  Yeah.

• 2:10:32Lex Fridman?

  That's actually a beautiful thing that might be another face shift because it assigns credit correctly.

• 2:10:37Speaker 01

  Yeah.

• 2:10:37Lex Fridman?

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

• 2:10:40Speaker 01

  Yeah, yeah, yeah.

• 2:10:41Lex Fridman?

  Thought, you know.

• 2:10:41Speaker 01

  Yeah, no, there's a big hump to overcome right now. I mean, it's like self-driving cars.

• 2:10:47Lex Fridman?

  Right.

• 2:10:48Speaker 01

  The safety margin has to be really high.

• 2:10:50Lex Fridman?

  Yeah.

• 2:10:50Speaker 01

  For it to be feasible. So, yeah, so there's a last mile problem 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 in some ways.

• 2:11:07Lex Fridman?

  I mean, another way of asking the fields medal question is, what year do you think you wake up and be like real surprised? You read the headline, the news, or something happened that AI did, like, you know, real breakthrough, something. It doesn't, you know, like fields medal, even hypothesis, it could be like really just this alpha zero moment would go, that kind of thing.

• 2:11:31Speaker 01

  Right, right. Yeah, this decade, I can, I can see it like making a conjecture between two unrelated, two, two things that people thought was unrelated.

• 2:11:42Lex Fridman?

  Oh, interesting, generating a conjecture, that's a beautiful conjecture.

• 2:11:45Speaker 01

  Yeah, and, and actually has a real chance of being correct and, and meaningful.

• 2:11:50Lex Fridman?

  Because that's actually kind of doable, I suppose, but the, what are the data is? Yeah.

• 2:11:55Speaker 01

  Yeah.

• 2:11:55Lex Fridman?

  No, that would be truly amazing.

• 2:11:57Speaker 01

  Um, the current models struggle a lot. I mean, so a version of this is, I mean, the physicists have a dream of getting the AIs to discover new laws of physics.

• 2:12:08Speaker 01

  You know, the, the dream is you just feed it all this data, okay? Uh, uh, and, and this is, here is a new patent that we didn't see before. But it actually even struggled, the current state of the art even struggles to discover old laws of physics from the data. I mean, uh, or if it does, uh, there's a big concern of contamination, that it did it only because like somewhere in its training data, it already somehow knew, um, you know, Boyle's law, whatever, you're trying to, to reconstruct.

• 2:12:34Speaker 01

  Part of it is that we don't have the right type of training data for this. Yeah, so for laws of physics, like we, we don't have like a million different universes with a million different balls of nature.

• 2:12:45Speaker 01

  And, um, like.

• 2:12:48Speaker 01

  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 ended up being verified or where the counter examples produced. But we don't have data on on things that were proposed and they're kind of a good thing to try. But then people quickly realized that it was the wrong conjecture. And then they they said, oh, but we should actually 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 and we only like to publish our wins. And the AI has no access to this data to train on.

• 2:13:32Speaker 01

  I sometimes joke that basically AI has to go through 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.

• 2:13:47Lex Fridman?

  Let me ask you, if I may, about Gregori Perlman. You mentioned that you try to be careful in your work and not let a problem completely consume you. Just you've really fallen in love with the problem and really cannot rest until you solve it. But you also hasted to add that sometimes this approach actually can be very successful. An example you gave is Gregori Perlman, who proved the Poindexter conjecture and did so by working alone for seven years with basically little contact with the outside world. Can you explain this one Millennium Prize problem that's been solved, Poindexter conjecture? And maybe speak to the journey that Gregori Perlman's been on.

• 2:14:31Speaker 01

  All right, so it's a question about curved spaces. Earth is a good example. So I think it was a 2D surface. In this round, you could maybe be a torus with a hole in it or can have many holes. And there are many different topologies a priori that that a surface could have. Even if you assume that it's bounded and and smooth and so forth. So we have figured out how to classify surfaces. As a first approximation, everything is determined by some of the genus, how many holes it has. So a sphere has genus zero, a donut has genus one and so forth. And one way you can tell these surfaces apart, probably the sphere has, which is called simply connected. If you take any closed loop on the sphere, like a big closed rope, you can contract it to a point and while staying on the surface. And the sphere has this property, but a torus doesn't.

• 2:15:14Speaker 01

  And if you're on a torus and you take a rope that goes around, say, the the outer diameter of a torus, there is no way, it can't get through the hole. There is no way to contract it to 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 things that I want to call topologically equivalent to the sphere. So Poincaré asked the same question in higher dimensions. So this it becomes hard to visualize because a surface you can think of as embedded in three dimensions, but as a curved three space, we don't have good intuition of four D space to to to live in. And then there are also three D spaces that can't even fit into four dimensions. You need five or six or or higher. But anyway, mathematically, you can still pose this question that if you have a bounded

• 2:15:59Speaker 01

  three dimensional space now, which is also has this simply connected property, that every loop can be contracted, can you turn it into a three dimensional version of the sphere? And so this is the Poincaré conjecture. Weirdly, in higher dimensions, four and five, it was actually easier. So 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 a sphere. But three was really hard. So people tried many approaches. There's sort of commentary approaches where you chop up the the surface into little triangles or tetrahedra and you you just try to argue based on how the faces interact each other. There were algebraic approaches. There's there's various algebraic objects, like things called the fundamental group that you can attach to these homology and cohomology and and and all these very fancy tools.

• 2:16:44Speaker 01

  They also didn't quite work. But Richard 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 sort of secretly is a sphere, but it's given to you in a really in a weird way. So it's like I think of a ball that's been kind of crumpled up and twisted and it's not obvious that it's a ball. But like if you have some sort of surface, which is, which is a deformed sphere, you could, you could, for example, think of it as the surface of a balloon. You could try to inflate it, you blow it up. And naturally, as you fill it with air, the wrinkles will sort of smooth out and it will turn into a nice round sphere.

• 2:17:29Speaker 01

  Unless, of course, it was a torus or something, in which case it would get stuck at some point. Like if you inflated a torus, there would be a point in the middle. When the inner ring shrinks to zero, you get a singularity and you can't blow up any further. You can't flow any further. So he created this flow, which is now called Ricci flow, which is a way of taking an arbitrary surface or space and smoothing it out to make it rounder and rounder, to make it look like a sphere. And he wanted to show that either this process would give you a sphere or it would create a singularity. Like you're very much like how PDEs either have global regularity or finite and blow up. Okay, it basically is almost exactly the same thing. It's all connected.

• 2:18:06Speaker 01

  And so, and he showed that for two dimensions, two dimensional surfaces, if you started something connected, no singularity is ever formed. You never ran into trouble and you could flow and it will give you a sphere. And so he got a new proof of the two dimensional result.

• 2:18:20Lex Fridman?

  Well, by the way, that's a beautiful explanation of Ricci flow and its application in this context. How difficult is the mathematics here, like for the 2D case?

• 2:18:26Speaker 01

  Yeah, these are quite sophisticated equations on par with the Einstein equations. Slightly simpler, but yeah, but they were considered hard non-linear equations to solve. And there's lots of special tricks in 2D that helped. But in 3D, the problem was that this equation was actually super critical. The same problem as Navier-Stokes. As you blow up, maybe the curvature could get concentrated in finite, smaller, smaller regions. And it looked more and more nonlinear and things just looked worse and worse. And there could be all kinds of singularities that showed up. Some singularities, like there's these things called neck pinches, where the surface sort of behaves like a barbell and it pinches at a point.

• 2:19:12Speaker 01

  Some singularities are simple enough that you can sort of see what to do next. You just make a snip and then you can turn one surface into two and evolve them separately. But there was the prospect that some really nasty, like knotted singularities showed up that you couldn't see how to resolve in any way, that you couldn't do any surgery to. So you need to classify all the singularities, like what are all the possibilities that things can go wrong? So what Perman did was first of all, he made the problem, he turned the problem from a supercritical problem to a critical problem. I said before about how the invention of the of of energy, the Hamiltonian, that really clarified Newtonian mechanics. So he introduced something which is now called Perman's reduced volume and Perman's entropy.

• 2:19:56Speaker 01

  He introduced new quantities, kind of like energy, that looked the same at every single scale, and turned the problem into a critical one, where the nonlinear is actually suddenly looked a lot less scary than they did before. And then he had to solve, he still had to analyze the singularities of this critical problem. And that itself was a problem similar to this wave荡thing I worked on, actually. So on the on the level of difficulty of that, so he managed to classify all the singularities of this problem and show how to apply surgery to each of these, and through that was able to to resolve the point grid conjecture. So quite, like, a lot of really ambitious steps, and like, like, nothing that a large language model today, for example, could. I mean, at best, I could imagine proposing this idea as one of hundreds of different

• 2:20:40Speaker 01

  things to try. But the other ninety-nine would be complete dead ends, but you'd only find out after months of of work. He must have had some sense that this was the right track to pursue, because, you know, it takes years to get from from A to B.

• 2:20:54Lex Fridman?

  So you've done, like you said, actually, you see, even strictly mathematically, but more broadly, in terms of the process, you've done similarly difficult things. What what can you infer from the process he was going through, 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 what it's failing. What happens to you, you're sitting in your office, when you realize the thing you did for the last few days, maybe weeks, is a failure?

• 2:21:27Speaker 01

  Well, for me, I switch to a different problem. So I'm a fox, I'm not a hedgehog.

• 2:21:32Lex Fridman?

  But you legitimately, that is a break that you can take, is is to step away and look at it from a problem.

• 2:21:37Speaker 01

  Yeah. You can modify the problem, too. I mean, you can add some cheat, if if there's a specific thing that's blocking you that this.

• 2:21:46Speaker 01

  Some bad case keeps showing up for which your tool doesn't work. You can just assume by fiat 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 your approach, then maybe you just give up. But if this is the only problem, then everything else checks out, then it's still worth fighting.

• 2:22:12Speaker 01

  So yeah, you have to do some sort of forward reconnaissance sometimes.

• 2:22:17Speaker 01

  And that is sometimes productive to assume like, okay, we'll figure it out. Oh, yeah, yeah, yeah. Sometimes, actually, it's even productive to make mistakes. So one of the, I mean, there was a project which actually we won some prizes for. I have four other people. We worked on this PDE problem, again, actually this blow off regularity type problem. And it was considered very hard.

• 2:22:41Speaker 01

  Jean Bourguin, who was another fierce methodist who worked on this on a special case of this, but he could not solve the general case. And we worked on this problem for two months and we found, we thought we solved it. We had this acute argument that everything fit and we were excited. We were planning celebrationary to all get together and have champagne or something. And we started writing it up and one of us, not me, but another co-author said, oh, in this, in this dilemma here,

• 2:23:09Speaker 01

  we have to estimate these 13 terms that show up in this expansion. And we estimate 12 of them, but in our notes, I can't find the estimation of 13, can you? Can someone supply that? And I said, sure, look at this and argue, yeah, we didn't cover, we completely omitted this term. And this term turned out to be worse than the other 12 terms put together. In fact, we could not estimate this term. And we tried for a few more months and all different permutations. And there was always this one thing, one term that we could not control. And so like.

• 2:23:37Speaker 01

  This was very frustrating. But because we had already invested months and months of effort in this already, we stuck at this, we tried increasingly desperate things and crazy things. And after two years, we found an approach which is somewhat different, but quite a bit from our initial strategy, which did actually didn't generate these problematic terms and actually solve the problem. So we solved the problem after two years. But if we hadn't had that initial false dawn of nearly solving the problem, we would have given up by month two or something and worked on an easier problem.

• 2:24:10Speaker 01

  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 the new world, they had an incorrect version of the measurement of the size of the earth. He thought he was going to find a new trade route to India. Or at least that was how he saw it in his prospectus. I mean, it could be that he secretly knew, but.

• 2:24:31Lex Fridman?

  Just on the psychological element, do you have like emotional or

• 2:24:38Lex Fridman?

  like self doubt that just overwhelms you most like that? You know, because this stuff, it feels like math is so engrossing that like it can break you when you like invest so much yourself in the problem. And then it turns out wrong, you could start to.

• 2:24:56Lex Fridman?

  Similar way chess has broken some people.

• 2:24:58Speaker 01

  Yeah, I, I think different mathematicians have different levels of emotional investment in what they do. I mean, I think for some people, it's as a job, you know, you, you have a problem and if it doesn't work out, you will go on the next one.

• 2:25:12Speaker 01

  Yeah, so the fact that you can always move on to another problem, it reduces the emotional connection. I mean, there are cases, you know, so there are certain problems that are what I call macro diseases, where we're just latch on to that one problem and they spend years and years thinking about nothing but that one problem. And, you know, maybe their career suffers and so forth. They say, oh, but I could this big win, this will, you know, once I, once I finish this problem, I will make up for all the years of, of, of lost opportunity. And that's, that's, I mean, occasionally, occasionally it works, but I, I really don't recommend it for people who have the right fortitude. Yeah, so I, I've never been super invested in any one problem.

• 2:25:59Speaker 01

  One thing that helps is that we don't need to call up problems in advance. Well, when we do graph proposals, we sound and say, 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.

• 2:26:15Speaker 01

  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.

• 2:26:26Lex Fridman?

  But I'm sure for you, there's problems like this. You have you have

• 2:26:32Lex Fridman?

  made so much progress towards the hardest problems in the history of mathematics. So is there is there a problem that just haunts you? It sits there in the dark corners, you know, twin prime conjecture, Riemann hypothesis, Globach conjecture.

• 2:26:48Speaker 01

  Twin prime, that sounds, well, again, so I mean, the problems like the Riemann hypothesis, those are so far out of reach.

• 2:26:54Lex Fridman?

  You think so?

• 2:26:55Speaker 01

  Yeah, there's no even viable strategy, like even if I activate all my all the cheats that I know of in this problem. Like it, there's just still no way to get me to be. Like it's it's

• 2:27:08Speaker 01

  I think it needs a breakthrough in another area of mathematics to happen first. And for someone to recognize that it would be a useful thing to transport into this problem.

• 2:27:18Lex Fridman?

  So we should maybe step back for a little bit and just talk about prime numbers.

• 2:27:22Speaker 01

  Okay.

• 2:27:22Lex Fridman?

  So they're often referred to as the atoms of mathematics. Can you just speak to the structure that these atoms provide?

• 2:27:30Speaker 01

  The natural numbers have two basic operations, addition and multiplication. So if you want to generate the natural numbers, you can do one of two things. You can just start with one and add one to itself over and over again, and that generates you the natural numbers. So additively, they're very easy to generate, one, two, three, four, 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 multiply them all together. And together, that gives you all the natural numbers, except maybe for one. So there are these two separate ways of thinking about the natural numbers. From an additive point of view and a particular point of view. And separately, they're not so bad. So like any question about that a natural was only was addition is relatively easy to solve. And any question that only was multiplication is relatively easy to solve.

• 2:28:15Speaker 01

  But what has been frustrating is that you combine the two together. And suddenly you get this extremely rich. I mean, we know that there are statements in number theory that are actually as undecided. There are certain polynomials in some number variables. There's a solution in the natural numbers. And the answer depends on an undecided statement, like whether the axioms of mathematics are consistent or not. But yeah, but even the simplest problems that combine something more implicative, such as the primes with something additive, such as shifting by two. Separately, we understand both of them well. But if you ask, when you shift the prime by two, do you, can you get a, how often can you get another prime? We, it's been amazingly hard to relate the two.

• 2:28:59Lex Fridman?

  And we should say that the twin prime conjecture is just that. It posits that there are infinitely many pairs of prime numbers that differ by two. Now, the interesting thing is that you have been very successful at pushing forward the field in answering these complicated questions of this variety. Like you mentioned, the green tile theorem. It proves that prime numbers contain arithmetic progressions of any length.

• 2:29:24Speaker 01

  Right.

• 2:29:25Lex Fridman?

  Which is mind blowing that you could prove something like that.

• 2:29:27Speaker 01

  Right. Yeah, so what we've realized because of this type of research is that this different patterns have different levels of indestructibility. So, so what makes the twin prime problem hard is that if you take all the primes in the world, you know, three, five, seven, 11, so forth. There are some twins in there, 11 and 13 is a twin prime, pair of twin primes and so forth. But you could easily, if you wanted to, redact the primes to get rid of, to get rid of the, these twins. Like the twins, they show up and there are infinitely many of them, but they're actually reasonably sparse. Not, there's, there's not, I mean, initially there's quite a few, but once you go to the millions, trillions, they become rarer and rarer. And you could actually just, you know, if, if someone was given access to the database of primes,

• 2:30:12Speaker 01

  you just edit out a few, a few primes here and there. They could make the twin prime conjecture false by just removing like point zero one percent of the primes or something. Just well, well chosen to, to, to do this. And so you could present a censored database of the primes, which passes all of the statistical tests of the primes. You know, it'll base things like the prime number theorem and other things about the primes, but doesn't contain any twin primes anymore. And this is a real obstacle for the twin prime conjecture. It means that any proof strategy to actually find twin primes in the actual primes must fail when applied to these slightly edited primes. And so it must be some very subtle, delicate feature of the primes

• 2:30:56Speaker 01

  that you can't just get from like aggregate statistical analysis.

• 2:31:01Lex Fridman?

  Okay, so that's out.

• 2:31:02Speaker 01

  Yeah. On the other hand, arithmetic progressions has turned out to be much more robust. Like, you can take the primes and you can eliminate 99% of the primes, actually. And you can take any 90% if you want. And it turns out, and another thing we proved is that you still get arithmetic progressions. Arithmetic progressions are much, you know, they're like cockroaches.

• 2:31:21Lex Fridman?

  Of arbitrary length, though.

• 2:31:22Speaker 01

  Yes, yes.

• 2:31:23Lex Fridman?

  That's crazy. I mean, so for people who don't know, arithmetic progressions is a sequence of numbers that differ by some fixed amount.

• 2:31:31Speaker 01

  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 arithmetic progressions. You only get quite short progressions.

• 2:31:40Lex Fridman?

  But you're saying twin primes is not an infinite monkey phenomenon? I mean, it's a very subtle one. It's still an infinite monkey phenomenon.

• 2:31:47Speaker 01

  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.

• 2:31:56Lex Fridman?

  Oh, but you're saying that twin prime is, it doesn't, you can't use the same tools? Like, it doesn't appear random almost?

• 2:32:04Speaker 01

  Well, we don't know. Yeah, we believe the primes behave like a random set. And so the reason why we care about the twin prime conjecture is it's a test case for whether we can genuinely confidently say with 0% chance of error that the primes behave like a random set. Okay, random, yeah, random versions of the primes we know contain twins. At least with 100% probability, or probably tending to 100% as you go out further and further.

• 2:32:32Speaker 01

  Yeah, so the primes we believe are the random. The reason why arithmetic progressions are indestructible is that regardless of whether your set looks random or looks structured, like periodic, in both cases, arithmetic progressions appear, but for different reasons. And this is basically all the ways in which the theorem, there are many proofs of these sort of arithmetic progression theorem, and they're all proven by some sort of dichotomy where your set is either structured or random, and in both cases, you can say something, and then you put the two together. But in twin primes, if the primes are random, then you're happy, you win. But if your primes are structured, they could be structured in a specific way that eliminates the twins. And we can't rule out that one conspiracy.

• 2:33:15Lex Fridman?

  And yet, you were able to make, as I understand, progress on the K-tuple version.

• 2:33:20Speaker 01

  Right, yeah. So the one funny thing about conspiracies is that any one conspiracy theory is really hard to disprove. That, you know, if you believe the world is run by lizards, you say, here's some evidence that it's not random, this is what, that evidence was planted by the lizards.

• 2:33:34Lex Fridman?

  Yeah.

• 2:33:35Speaker 01

  You may have encountered this kind of phenomenon. So like, a pure, like, there's almost no way to definitively rule out a conspiracy. And the same is true in mathematics, that a conspiracy is solely devoted to living twin primes. You know, you have to also infiltrate other areas of mathematics, 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 world is run by lizards, it can also be run by aliens.

• 2:34:07Lex Fridman?

  Right.

• 2:34:09Speaker 01

  So one unreasonable thing is hard to disprove, but more than one, there are tools. So, yeah, so for example, we know there's infinitely many primes that are no two, which are, so the infinite pair of primes which differ by at most 246, actually, is the current record.

• 2:34:26Lex Fridman?

  So there's like a bound.

• 2:34:27Speaker 01

  Yes. Right. So like, there's twin primes, there's this thing called cousin primes that differ by four, this thing called sexy primes that differ by six.

• 2:34:36Lex Fridman?

  What are sexy primes?

• 2:34:37Speaker 01

  Primes that differ by six. The name is much, it's much as exciting than the name suggests.

• 2:34:44Speaker 01

  So you can make a conspiracy rule out one of these, but like, once you have like fifty 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.

• 2:34:55Lex Fridman?

  How do you do the bound part? How do you, how do you develop a bound for the different team deposit?

• 2:35:00Speaker 01

  Okay, so.

• 2:35:01Lex Fridman?

  That there's an infinite number of.

• 2:35:03Speaker 01

  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 into pigeonholes, and you have more pigeons than pigeonholes, then one of the pigeonholes has to have at least two pigeons in it. So there has to be two pigeons that are close together. So for instance, if you have a hundred numbers and they all range from one to a thousand, two of them have to be at most ten apart. Because you can divide up the numbers from one to a hundred into one hundred pigeonholes. Let's say they are hundred, they are hundred and one numbers, they are hundred and one numbers, then two of them have to be a distance less than ten apart, because two of them have to belong to the same pigeonhole. So it's a basic, um, basic feature of a basic principle in mathematics.

• 2:35:44Speaker 01

  So it doesn't quite work with the primes already, because the primes get sparser and sparser as you go out. That a few and few numbers are prime. But it turns out that there's a way to assign weights to the, to, to numbers. So there are numbers that are kind of almost prime, but they're not, they, they don't have no factors at all other than themselves and one. They have very few factors. And it turns out that we understand almost primes a lot better than we understand primes. And so, for example, it was known for a long time that there were twin almost primes. This has been worked out. So almost primes are things we can't understand. So you can actually restrict attention to a suitable set of almost primes. And, uh, whereas the primes are very sparse

• 2:36:28Speaker 01

  overall, relative to the almost primes, they're actually much less sparse. They may, you can set up a set of almost primes where the primes have density like, say, one percent. And that gives you a shot at proving, by applying some sort of pigeonhole principle, that, that there's pairs of primes that are just only a hundred, a hundred apart. But in order to prove the twin prime conjecture, you need to get the density of primes of the almost primes up to, up to a threshold of fifty percent.

• 2:36:51Speaker 01

  Once you get up to fifty percent, you will get twin primes. But unfortunately, there are barriers.

• 2:36:56Speaker 01

  We know that, that no matter what kind of good set of almost primes you pick, the density of primes can never get above fifty percent. It's called the parity 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 to twin prime conjecture, the Gobak conjecture and many other problems in number theory are currently blocked. Because our current techniques would require going beyond this theoretical parity barrier. It's like, it's like going past the speed of light.

• 2:37:23Lex Fridman?

  Yeah, so we should say a twin prime conjecture. One of the biggest problems in the history of mathematics, Gobak conjecture also. They feel like next door neighbors. Is there been days when you felt you saw the path?

• 2:37:36Speaker 01

  Oh, yeah. Um, yeah, sometimes you try something and it works super well. You, 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. Um, I think the way a colleague might put it is that, you know, like if you are on the streets of New York and you're put in a blindfold and you're in a car and, and after some hours, the pipe was off and you're in Beijing. Um, you know, I mean, that was too easy somehow, like, like there was no ocean being crossed. Even if you don't know exactly what, how, what, what was done, you're suspecting that there's something wasn't right.

• 2:38:20Lex Fridman?

  But is that still in the back of your head to, do you return to these, to the prime? Do you return to the prime numbers every once in a while to see?

• 2:38:29Speaker 01

  Yeah, when I have nothing better to do, which is less and less than ten times, which is I get busy with so many things these days. But yeah, when I have free time and I'm not, and I'm too frustrated to, to work on my sort of real research projects and I also don't want to do my administrative support. I don't want to do some errands for my family. I can play with these, these things for fun and usually you get nowhere. Yeah, you have to learn to just say, okay, fine. Once again, nothing happened, I will, I will move on. Yeah, very occasionally, one of these problems I actually solved, or sometimes, as you say, you think you solved it and then you're euphoric for maybe 15 minutes and then you think I should check this because this is too easy, too good to be true and it usually is.

• 2:39:10Lex Fridman?

  What's your gut say about when these problems would be solved, twin prime and Gobak?

• 2:39:16Speaker 01

  Prime, I think we'll keep getting, keep getting more partial results. It does need at least one, this parity barrier is the biggest remaining obstacle. There are simpler versions of conjecture where we are getting really close. So I think we will, in 10 years, we will have many more, much closer results. We may not have the whole thing. Yeah, so twin primes is somewhat close. Riemann hypothesis, I have no, I mean, it has to happen by accident, I think.

• 2:39:46Lex Fridman?

  So the Riemann hypothesis is a kind of more general conjecture about the distribution of prime numbers.

• 2:39:52Speaker 01

  Right, yeah, it's, it's, it's, they sort of viewed modulcatively, like for questions only involving multiplication, no addition. The primes really do behave as randomly as you could hope. So there's a phenomenon in probability called square root cancellation that, you know, if you want to poll, say, America upon some issue and you, you ask one or two voters and you may have sampled a bad sample and then you get, you get a really imprecise measurement of the full average. But if you sample more and more people, the accuracy gets better and better. And it actually improves like the square root of the number of people you sample. So yeah, if you sample a thousand people, you can get like a two, three percent margin of error.

• 2:40:32Speaker 01

  So in the same sense, if you measure the primes in a certain applicative sense, there's a certain type of statistic you can measure. And it's called the Riemann's data function and it fluctuates up and down. But in some sense, as you keep averaging more and more, if you sample more and more, the fluctuation should go down as if they were random. And there's a very precise way to quantify that. And the Riemann hypothesis is a very elegant way that captures this. But as with many other ways in mathematics, we have very few tools to show that something really genuinely behaves like really random. And this is actually not just a little bit random, but it's asking that it behaves as random as a actually random set. This, this, this square root cancellation.

• 2:41:11Speaker 01

  And we know because of things related to the parity problem, actually, that most of us's usual techniques cannot hope to settle this question. The proof has to come out of left field. Yeah, but what that is, yeah, no one has any serious proposal.

• 2:41:30Speaker 01

  Yeah, and there's various ways to sort of, as I said, you can modify the primes a little bit, and you can destroy the Riemann hypothesis. So like, it has to be very delicate. You can't apply something that has huge margins of error. It has to just barely work. And like, there's like all these pitfalls that you like dodge very adeptly.

• 2:41:50Lex Fridman?

  The prime numbers is just fascinating.

• 2:41:51Speaker 01

  Yeah, yeah, yeah.

• 2:41:52Lex Fridman?

  What to you is most mysterious about the prime numbers?

• 2:41:59Speaker 01

  That's a good question. So like conjecturally, we have a good model of them. I mean, like, as I said, I mean, they have certain patterns, like the primes are usually odd, for instance. But apart from these obvious patterns, they behave very randomly. And just assuming that they behave, so there's something called the Kriemer Random Model of the primes, that after a certain point, primes just behave like a random set. And there's various slight modifications to this model. But this has been a very good model. It matches the numerics. It tells us what to predict. Like, I can tell you with complete certainty the Trinbik conjecture is true. The random model gives overwhelming odds that it's true. I just can't prove it. Most of our mathematics is optimized for solving things with patterns in them.

• 2:42:39Speaker 01

  And the primes have this anti-pattern, as do almost everything, really. But we can't prove that. Yeah, I guess it's not mysterious that the primes be random. It's kind of random because there's no reason for them to be,

• 2:42:54Speaker 01

  to have any kind of secret pattern. But what is mysterious is what is the mechanism that really forces the randomness to happen? This is just absent.

• 2:43:04Lex Fridman?

  Another incredibly surprisingly difficult problem is the Collatz conjecture.

• 2:43:09Speaker 01

  Oh, yes.

• 2:43:09Lex Fridman?

  Simple to state, beautiful to visualize, in its simplicity, and yet extremely difficult to solve. And yet you have been able to make progress. Paul Eredar said about the Colossus 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?

• 2:43:41Speaker 01

  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, because 13 times 3 is 39, add one, you get 40. So it's a simple process. For odd numbers and even numbers, they're both very easy operations. And then you put it together, it's still reasonably simple. But then you ask what happens when you innovate it. You take the output that you just got and feed it back in. So 13 becomes 40, 40 is now even,

• 2:44:21Speaker 01

  divide by two is 20, 20 is still even, divide by 10, 10, 5, and then 5 times 3 plus 1, 16, and then 8, 4, 2, 1. So and then from 1 it goes 1, 4, 2, 1, 4, 2, 1, it cycles forever. So the sequence I just described, 13, 40, 20, 10, so these are also called hailstone sequences, because there's an oversimplified model of hailstone formation, which is not actually quite correct, but it's still somewhat taught to high school students as a first approximation, is that a little nugget of ice gets an ice crystal, it forms and clouded, it goes up and down because of the wind, and sometimes when it's cold, it acquires a bit more mass, and maybe it melts a little bit. And this process of going up and down

• 2:45:04Speaker 01

  creates this sort of partially melted ice, which eventually becomes hailstone, and eventually it falls off the earth. So the conjecture is that no matter how high you start up, like you take a number which is in the millions or billions, you have this process that goes up if you're odd and down if you're even, it eventually goes down to earth all the time.

• 2:45:22Lex Fridman?

  No matter where you start with this very simple algorithm, you end up at 1.

• 2:45:26Speaker 01

  Right.

• 2:45:27Lex Fridman?

  And you might climb for a while.

• 2:45:28Speaker 01

  Right. Yeah, so it's

• 2:45:29Lex Fridman?

  Come down.

• 2:45:30Speaker 01

  Yeah, if you plotted these sequences, they look like brownie in 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, that if you plug in a random number, you can actually prove, at least initially, that it would look like random walk. And it's actually a random walk with a downward drift. It's like if you're always gambling on roulette at the casino with odds slightly weighted against you. 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 hit, will go to zero if you just keep playing over and over again.

• 2:46:06Lex Fridman?

  So statistically, it makes sense.

• 2:46:08Speaker 01

  Yes, so the result that I proved, roughly speaking, is that statistically, like ninety percent of all inputs would would drift down to, maybe not all the way to one, but to be much, much smaller than what you started. So it's like if I told you that if you go to a casino, most of the time you end up, if you keep playing for long enough, you end up with a smaller amount in your wallet than when you started. That's kind of like the result that I proved.

• 2:46:34Lex Fridman?

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

• 2:46:42Speaker 01

  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, or almost surely, with probability as close to one hundred percent as you wish, you're guaranteed to lose money. But there's always this exceptional outlier. Like, it is mathematically possible that even in the game, if the odds are not in your favor, you could just keep winning slightly more than you lose. Very much like how in Navier-Stokes, there could be, you know, 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.

• 2:47:28Speaker 01

  And there could be one outlier choice of a special number that you stick in that shoots off infinity, while all other numbers crash to earth, crash to one. In fact, there's some mathematicians who've, Alex Kontorovich, for instance, who've proposed that that actually these collapse iterations are like these similar automata. Actually, if you look at what happened in binary, they do actually look a little bit like like these game of life type patterns. And in an analogy to how the game of life can create these massive self-fabricating objects and so forth, possibly you could create some sort of heavier than a flying machine, a number which is actually encoding this machine, which is just, whose job it

• 2:48:12Speaker 01

  is to encode, is to create a version of itself which is larger.

• 2:48:17Lex Fridman?

  Heavier than air machine encoded in a number

• 2:48:20Speaker 01

  Yeah.

• 2:48:20Lex Fridman?

  That flies forever.

• 2:48:22Speaker 01

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

• 2:48:25Lex Fridman?

  Oh, wow.

• 2:48:25Speaker 01

  So Conway, so similar, in fact, that was one of my inspirations for the Neve Stokes project. Conway studied generalizations of the collapse problem where instead of multiplying by three and adding one or dividing by two, you have a more complicated branch of rules. But instead of having two cases, maybe you have seventeen cases and then you go up and down. And he showed that once your iteration gets complicated enough, you can actually encode Turing machines and you can actually make these problems undecidable and do things like this. In fact, he invented a programming language for these kind of fractional linear transformations. He called a fact track as a play on a fort track. And he showed that you can program, if it was Turing complete,

• 2:49:06Speaker 01

  you could, you could, you could, 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.

• 2:49:22Lex Fridman?

  Some of the mystery of the cellular automata that we talked about, having a mathematical framework to say anything about cellular automata. Maybe the same kind of framework is required.

• 2:49:33Speaker 01

  Yeah, yeah.

• 2:49:34Lex Fridman?

  For Glocks's conjecture.

• 2:49:35Speaker 01

  Yeah, if you want to do it not statistically, but you really want one hundred percent of all inputs to for the earth. Yeah, so what might be feasible is statistically ninety-nine percent, you know, go to one. But like everything, you know, that looks hard.

• 2:49:50Lex Fridman?

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

• 2:49:59Speaker 01

  It's up there. P goes NP is a good one because like that's a meta problem. Like if you solve that in the, in the positive sense, that you can find a P goes NP algorithm, then potentially this solves a lot of other problems as well.

• 2:50:13Lex Fridman?

  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. P goes on P has more ripple effects than basically any other.

• 2:50:23Speaker 01

  Right. If the Riemann hypothesis is disproven, that'd be a big mental shock to the number theorists. But it would have follow-on effects for cryptography.

• 2:50:36Speaker 01

  Because a lot of cryptography uses number theory, uses number theory constructions involving primes and so forth. And it relies very much on the intuition that number theories are built over many, many years of what operations involving primes behave randomly and what ones don't. And in particular, encryption methods are designed to turn textual information on it into text which is indistinguishable from random noise. So and hence, we believe to be almost impossible to crack, at least mathematically.

• 2:51:10Speaker 01

  But if something as core to our beliefs as the Riemann hypothesis is wrong, it means that there are actual patterns of the primes that we're not aware of. And if there's one, there's probably going more. And suddenly, a lot of our crypto systems are in doubt.

• 2:51:27Lex Fridman?

  Yeah.

• 2:51:29Lex Fridman?

  But then how do you then say stuff about the primes?

• 2:51:33Speaker 01

  Yeah.

• 2:51:33Lex Fridman?

  Like you're going towards the Kolax conjecture again. Because you want it to be random, right?

• 2:51:41Speaker 01

  Yes.

• 2:51:41Lex Fridman?

  You want it to be random.

• 2:51:42Speaker 01

  So more broadly, I'm just looking for more tools, more ways to show that that

• 2:51:46Lex Fridman?

  Yeah.

• 2:51:46Speaker 01

  that things are random. How do you prove a conspiracy doesn't happen?

• 2:51:49Lex Fridman?

  Right. Is there any chance to you that P equals NP? Is there some, can you imagine a possible universe?

• 2:51:56Speaker 01

  It is possible. I mean, there's various scenarios. I mean, there's one where it is technically possible, but in fact, it's never actually implementable. The evidence is sort of slightly pushing in favor of no, that we'd probably P is not equal to NP.

• 2:52:10Lex Fridman?

  I mean, it seems like it's one of those cases similar to Riemann hypothesis. I think the evidence is leaning pretty heavily on the no.

• 2:52:19Speaker 01

  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 is 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 scientists have actually been very good at. It's actually saying that certain approaches cannot work, no-go theorems. It could be undecidedable. We don't, yeah, we don't know.

• 2:52:43Lex Fridman?

  There's a funny story I read that when you won the Fields Medal, somebody from the internet wrote you.

• 2:52:50Lex Fridman?

  And asked, you know, what are you going to do now that you've won this prestigious 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'm currently working on. So I'm going to keep working on them. It's just, first of all, it's funny to me that you would answer an email in that context. And second of all, it, um, it just shows your humility. But anyway, uh, maybe you could speak to the Fields Medal, but it's another way for me to ask about Gregor Perlman, what do you think about him famously declining the Fields 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

• 2:53:34Lex Fridman?

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

• 2:53:39Speaker 01

  Yeah, he's somewhat of an outlier, even among mathematicians who tend to, uh, to have somewhat idealistic views. I've never met him. I think I'd be interested to meet him one day, but I'd never had the chance. I know people who've met him, but he's always had strong views about certain things. Um, I mean, it's not like he was completely isolated from the math community. I mean, he would, he would give talks and write papers and so forth. But at some point, he just decided not to engage with the rest of the community. He was disillusioned or something. Um, I don't know. Um, and he decided to, to, uh, to peace out, uh, and, you know, collect mushrooms in St. Petersburg or something. And that's, that's fine, you know, and you can, you can do that. Um, I mean, that's another sort

• 2:54:21Speaker 01

  of flip side. I mean, we are not, a lot of our problems that we solve, you know, they, some of them do have practical application. That's, that's great. But, uh, like if you stop thinking about a problem, you know, so he's, he hasn't published since in this field, but that's fine. There's many, many other people who've done so as well. Um, yeah. So I guess one thing I didn't realize initially with the Fields Medal is that it sort of makes you part of the establishment. You know, so You know, most mathematicians, you know, there's just career mathematicians, you know, you just focus on publishing a next paper, maybe getting one to promote one rank, you know, and starting a few projects, maybe taking some students or something. Yeah, but then suddenly people want your opinion on things and you have to think a

• 2:55:04Speaker 01

  little bit about the, you know, things that you might just foolishly say because you know no one's going to listen to you. It's more important now.

• 2:55:11Lex Fridman?

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

• 2:55:18Speaker 01

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

• 2:55:36Lex Fridman?

  I love the different algorithms here. This is great.

• 2:55:40Speaker 01

  This is, it's always an option.

• 2:55:43Speaker 01

  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 one problem at a time or fooling around. I still do that a little bit, but yeah, as you advance in your career, some of the more soft skills, so math somehow front loads all the technical skills to the early stages of your career. So, yeah, so it's as a postdoc, it's publish or perish. You're incentivized to basically focus on proving very technical theorems to prove yourself as well as prove the theorems. But then as, as you get more senior, you have to start, you know, mentoring and giving interviews and, and trying to shape direction of the field, both research wise and, you

• 2:56:28Speaker 01

  know, sometimes you have to, you know, do various administrative things. And it's kind of the right social contract because you, you need to work in the trenches to see what can help mathematicians.

• 2:56:39Lex Fridman?

  The other side of the establishment, sort of the really positive thing is that you get to be a light that's an inspiration to a lot of young mathematicians or young people that are just interested in mathematics. It's like, it's just how the human mind works. This is where I would probably say that I like the field medal that it does inspire a lot of young people somehow. I don't, this is just how human brains work.

• 2:57:05Speaker 01

  Yeah.

• 2:57:05Lex Fridman?

  At the same time, I also want to give sort of respect to somebody like Gregory Perlman, who. Is critical of awards in his mind, those are his principles. And any human that's able for their principles to like, do the thing that most humans would not be able to do. It's beautiful to see.

• 2:57:24Speaker 01

  Some recognition is necessarily important, but yeah, it's, it's also important to not let these things take over your life.

• 2:57:31Lex Fridman?

  Yeah.

• 2:57:32Speaker 01

  And like only be concerned about getting the next big award or whatever. I mean, yeah, so again, you see these people try to only solve like a really big math problems and not work on on things that are less sexy, if you wish, but but but actually still interesting and instructive. As you say, like the way the human mind works, it's, we understand things better when they're attached to humans. And also, if they're attached to a small number of humans, like the way our human mind is wired, we can comprehend the relationships between ten or twenty people. But once you get beyond like a hundred people, I this is a limit. I think there's a name for it.

• 2:58:14Speaker 01

  Beyond which it just becomes the other. And so we have, you have to simplify the poem as of, you know, ninety nine point nine percent of humanity becomes the other. And often these models are incorrect and this causes 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. 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 path is not that of a typical mathematician. I the very accelerated education, I skipped a lot of classes.

• 2:58:57Speaker 01

  I think I was very fortunate mentoring opportunities. And I think I was at the right place at the right time. Just because someone doesn't have my trajectory, it doesn't mean that they can't be good mathematicians. I mean, they may be mathematicians in a very different style. And we need people with a different style.

• 2:59:16Speaker 01

  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, 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 it's easy to just say, one person did this one thing, you know, it makes for a much simpler history.

• 2:59:40Lex Fridman?

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

• 3:00:06Speaker 01

  Yeah.

• 3:00:07Lex Fridman?

  Steve Jobs.

• 3:00:08Speaker 01

  Yeah, yeah. As a, as a starting point, you see, as a first approximation.

• 3:00:12Lex Fridman?

  And then read some biographies and then look into much deeper first approximation.

• 3:00:16Speaker 01

  Yeah.

• 3:00:17Lex Fridman?

  That's right. So you mentioned you were a person to Andrew Wiles at that time.

• 3:00:21Speaker 01

  Oh, yeah.

• 3:00:22Lex Fridman?

  Professor there. It's a funny moment how history is just all interconnected. And at that time, he announced that he proved the Fermat's Last Theorem. What did you think, maybe looking back now with more context about that moment in math history?

• 3:00:37Speaker 01

  Yes, 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 mailbox, you know, so we all picked up mail and like suddenly Andrew Wiles' mailbox exploded to be overflowing.

• 3:00:52Lex Fridman?

  That's a good, that's a good metric.

• 3:00:53Speaker 01

  Yeah. You know, so, yeah, we all talked about it at T and so forth. I mean, we, 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, right? Kevin Buzzard is actually.

• 3:01:08Lex Fridman?

  Yeah. Can, can we take that small tangent? Is it, is it, how difficult is that? Because as, as I understand the Fermat's Last, the, the proof for Fermat's Last Theorem has like super complicated objects.

• 3:01:20Speaker 01

  Yeah.

• 3:01:20Lex Fridman?

  Really difficult to formalize, no?

• 3:01:22Speaker 01

  Yeah, I guess, yeah, you're right. The, the objects that they use, you can define them. So they've been defined in Lean. Okay. So, 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 lots of these have to be formalized as well. Kevin Buzzard's goal, actually, he has a five-year grant to formalize famous last few. And his aim is that he doesn't think he will be able to get all the way down to the basic axioms. But he wants to formalize it to the point where the only things that he needs to rely on is black boxes. Are things that were known by nineteen eighty to to number theories at the time.

• 3:02:07Speaker 01

  And then some other person or some other work would have he done to to get from there. So it's a different area of mathematics than the type of mathematics I'm used to. In analysis, which is kind of my area, the objects we study are kind of much closer to the ground. We study things like prime numbers and and functions and things that are within scope of a high school math education to at least define. Yeah, but then this is very advanced algebraic side of number theory where people have been building structures upon structures for quite a while. And it's a very sturdy structure. It's been very

• 3:02:47Speaker 01

  at the base, at least it's extremely well developed with 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 like 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 you see something you recognize.

• 3:03:06Lex Fridman?

  What inspires you about his journey that was similar as we talked about? Seven years, mostly working in secret.

• 3:03:15Speaker 01

  Yeah, that is a romantic. So it kind of fits with sort of the the romantic image that I think people have of mathematicians to the extent they think of them at all as these kind of eccentric wizards or something. So that certainly kind of accentuated that perspective. You know, I mean, it's 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.

• 3:03:43Lex Fridman?

  Can you speak to it? Like what in terms of like you like the collaborative?

• 3:03:49Speaker 01

  I like moving on from a problem if it's giving too much difficulty.

• 3:03:54Speaker 01

  But you need the people who have the tenacity and the fearlessness. And I've collaborated with with people like that where I want to give up. Because the first approach that we tried didn't work and the second one didn't approach. They're convinced and they have the third, fourth, and the fifth approach works.

• 3:04:11Speaker 01

  And I'd have to eat my words. Okay, I didn't think this was gonna work, but yes, you were right all along.

• 3:04:16Lex Fridman?

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

• 3:04:31Speaker 01

  Yeah, it works for me. Yeah, I mean, but there are also people who are very productive and they 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.

• 3:04:48Speaker 01

  And so we have a certain curriculum and so forth. I mean, you know, maybe like if you do math competitions or something, you get a slightly different experience. But I think many people, they don't find their native math language until very late or usually too late. So they they stop doing mathematics and they have a bad experience with a teacher who's trying to teach them one way to do mathematics that they don't like it.

• 3:05:12Speaker 01

  My theory is that humans don't come, evolution has not given us a math center of a brain directly. We have a vision center and a language center and some other centers which have, evolution has honed. But we, it doesn't, we don't have an innate sense of mathematics.

• 3:05:28Speaker 01

  But our other centers are sophisticated enough that different people, we can repurpose other areas of our brain to do mathematics. So some people have figured out how to use the visual center to do mathematics. And so they think very visually when they do mathematics. Some people have repurposed their their language center and they think very symbolically. You know, some people like if they are very competitive and they like gaming, there's a type, there's this part of your brain that's very good at solving puzzles and games and that can be repurposed.

• 3:06:03Speaker 01

  Like when I talk about the mathematicians, you know, they don't quite think that I can tell that they're using some other different styles of thinking than I am. I mean, not not disjoint, but they they may prefer visual. I I I don't actually prefer visual so much. I need lots of visual aids myself. Um, you know, mathematics provides a common language, so we can still talk to each other, even if we are thinking in in different ways.

• 3:06:25Lex Fridman?

  But you can tell there's a different set of subsystems being used in the thinking process.

• 3:06:31Speaker 01

  Yeah, like they they take different paths. They're very quick at things that I struggle with and vice versa. Um, and yet they still get to the same goal.

• 3:06:38Lex Fridman?

  Yeah, that's beautiful.

• 3:06:40Speaker 01

  And yeah, but I mean, the way we educate, unless you have like a personalized tutor or something, I mean, education sort of just by nature scale has to be mass produced. You know, you have to teach the thirty kids. You know, they have thirty different styles. You can't you can't teach thirty different ways.

• 3:06:54Lex Fridman?

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

• 3:07:06Speaker 01

  Yeah.

• 3:07:07Lex Fridman?

  In this complicated educational context, what what would you?

• 3:07:09Speaker 01

  Yeah, it's a tricky problem. One nice thing is that there are now lots of sources for mathematical enrichment outside the classroom. Um, so in in in my day, there are already there are math competitions. Um, and you know, there are also like popular math books in the library. Um, you know, but but now you have, you know, YouTube, uh, there are there are forums just devoted to solving, you know, math puzzles and um, and math shows up in in other places, you know, like, um, for example, there there are hobbyists who play poker for fun. Uh, and um, they they, you know, they are for very specific reasons are interested in very specific probability questions.

• 3:07:44Lex Fridman?

  Yes.

• 3:07:45Speaker 01

  And and they actually, you know, this community of amateur probabilists in in in poker, um, in chess, in baseball. I mean, there's there's there's, uh, yeah. There's math all over 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. Like this is almost, it doesn't happen at all currently. So in the sciences, there is some scope for citizen science, like astronomers, amateurs who discover comets and there's biologists, there are people who could identify butterflies and so forth. And in math, there are smaller activities where amateur mathematicians can like discover new primes and so forth.

• 3:08:30Speaker 01

  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. In fact, 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 contribute to Mathlib. You know, you don't need to be a PhD holder to just work on one atomic thing.

• 3:09:03Lex Fridman?

  There's something about the formalization here that also, as a very first step, opens it up to the programming community too. The people who are already comfortable with program. It seems like programming is somehow maybe just a feeling, but it feels more accessible to folks than math. Math is seen as this like extreme, especially modern mathematics, seen as this extremely difficult to enter area and programming is not. So that could be just an entry point.

• 3:09:30Speaker 01

  You can execute code and you can get results, you know, you can print out the world pretty quickly. Yeah, like if, if programming was taught as an almost entirely theoretical subject, where you just talk the computer science, the theory of functions and routines and so forth. And outside of some very specialized homework assignments, you would not actually program like on the weekend for fun. Or yeah, they would be as considered as hard as math.

• 3:09:59Speaker 01

  Yeah, so as I said, there are communities of non mathematicians where they're deploying math for some very specific purpose, you know, like optimizing their poker game. And for them, then that becomes fun for them.

• 3:10:12Lex Fridman?

  What advice would you give in general to young people how to pick a career? How to find themselves?

• 3:10:17Speaker 01

  Yeah, that's a tough, tough, tough question. Yeah, so there's a lot of uncertainty 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, you know, like you, there was a very stable path to a good career. You go to college, you get an education, you pick one profession and you stick to it. It's becoming much more a thing of the past. So I think you just have to be adaptable and flexible. I think people have to get skills that are transferable, you know, like learning one specific programming language or one specific subject of mathematics or something. It's, that itself is not a super transferable skill, but sort of knowing how to reason with

• 3:11:02Speaker 01

  abstract concepts or how to problem solve when things were wrong or something. These are things which I think we will still need, even as our tools get better and, you know, you'll be working with AI sports and so forth.

• 3:11:13Lex Fridman?

  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.

• 3:11:29Speaker 01

  Yeah.

• 3:11:30Lex Fridman?

  But you learned lean. That's, that's a non trivial thing to learn. Like that's a, that's a, for a lot of people, that's an extremely uncomfortable leap to take, right?

• 3:11:39Speaker 01

  Yeah.

• 3:11:40Lex Fridman?

  A lot of mathematicians.

• 3:11:42Speaker 01

  First of all, I've always been interested in new ways to do mathematics. I, I feel like a lot of the ways we do things right now are inefficient. Like me and my colleagues, we spend a lot of time doing very routine computations or doing things that other mathematicians would instantly know how to do and we don't know how to do them. And why can't we search and get a quick response and stuff? 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 Scholz had just formalized one of his new theorems. And there were some other developments in computer assisted proof that.

• 3:12:24Speaker 01

  Look quite interesting and I said, oh, we should, we should, um, we should run a workshop on this. This is a pretty good idea. Um, and then I was a bit too enthusiastic about this idea. So I, I got voluntold to actually run it. Um, so I did with a bunch of other people. Kevin Bazet and Jordan Ellenberg and a bunch of other people. And it was, it was a nice success. We brought together a bunch of mathematicians and computer scientists and other people. And, and we got up to speed and stayed the yard. Um, and it was really interesting developments that, that most mathematicians didn't know was going on. That lots of nice proofs of concept, you know, it's just sort of hints of, of what was going to happen. This is just before chat G P T, but there was even then there was one talk about language models and the potential capability of those in the future.

• 3:13:10Speaker 01

  So that got me excited about the subject. So I started giving talks about this is something we should, more of us should start looking at. Now that I ran this conference and then chat G P T came out and like suddenly A I was everywhere. And so, uh, I got interviewed a lot, um, about, about this topic. Um, and in particular, um, the interaction between A I and formal proof systems. And I said, yeah, that they should be combined. This, this is, this is, um, this is 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 saying, oh, I, I'm a, I'm a one of a mathematician.

• 3:13:51Speaker 01

  Just trust me, you know, when I say that this is going to change mathematics and I'm not doing it any, and I don't do it myself. So I felt like I had to actually, uh, uh, justify it. Yeah, there's a lot of what I get into actually, um, I don't quite see in advice as how much time I'm going to spend on it. And it's only after I'm sort of waist deep in, in, in, in a project that I, I realized by that point I'm committed.

• 3:14:15Lex Fridman?

  Well, that's deeply admirable that you're willing to go into the fray, be in some small way a beginner. Right? Or have some of the sort of challenges that a beginner would, right?

• 3:14:26Speaker 01

  Yeah.

• 3:14:27Lex Fridman?

  New concepts, new ways of thinking, also, you know, sucking at a thing that others. I think, I think in that talk, you could be a Fields Medal winning mathematician and undergrad knows something better.

• 3:14:41Speaker 01

  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 bravado. And, I mean, because people will ask for your proofs, and if you don't have the proofs, you don't have the proofs.

• 3:15:02Lex Fridman?

  I don't love math.

• 3:15:03Speaker 01

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

• 3:15:17Lex Fridman?

  Big ridiculous question, I'm sorry for it once again. Who is the greatest mathematician of all time? Maybe one who's no longer with us. Who are the candidates? Euler, Gauss, Newton, Ramanujan, Hilbert?

• 3:15:32Speaker 01

  So first of all, as I mentioned before, like there's some time dependence.

• 3:15:36Lex Fridman?

  On the day.

• 3:15:37Speaker 01

  Yeah, like if you, if you, if you pop cumulatively over time, for example, Euclid, like, like, sort of, like, is, is one of the league contenders. And then maybe some unnamed, anonymous mathematicians before that, you know, whoever came up with the concept of numbers, you know?

• 3:15:53Lex Fridman?

  Do mathematicians today still feel the impact of Hilbert?

• 3:15:56Speaker 01

  Just. Oh, yeah.

• 3:15:57Lex Fridman?

  Directly of everything that's happened in the 20th century?

• 3:16:00Speaker 01

  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.

• 3:16:11Lex Fridman?

  There's some strange power to the declaring which problems

• 3:16:15Speaker 01

  Yeah.

• 3:16:16Lex Fridman?

  are hard to solve, the statement of the open problems.

• 3:16:19Speaker 01

  Yeah, I mean, this is bystander effect everywhere. Like, if no one says you should do X, everyone just mills around waiting for somebody else to do something. And like, nothing gets done. So, and like, it's the point of one thing that actually you have to teach undergraduates in mathematics is that you should 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.

• 3:17:00Speaker 01

  Or I think there's a quote from The Simpsons, I've tried nothing and I'm all out of ideas.

• 3:17:05Speaker 01

  So, you know, like, the next step then is to try anything, like, no matter how stupid. And in fact, almost the stupid are the better. Which, you know, I'm one, a technique which is almost guaranteed to fail, but the way it fails is going to be instructive. Like, it fails because you're not at all taking into account this hypothesis. Oh, this hypothesis must be useful. That's a clue.

• 3:17:25Lex Fridman?

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

• 3:17:35Speaker 01

  No, yes.

• 3:17:37Lex Fridman?

  It's when you really don't want to do a thing, they imagine a thing you don't want to do more.

• 3:17:41Speaker 01

  Yes.

• 3:17:42Lex Fridman?

  That's worse than that. And then in that way, you procrastinate by not doing the thing that's worse.

• 3:17:48Speaker 01

  Yeah, yeah.

• 3:17:48Lex Fridman?

  It's a nice, it's a nice hack. It actually works.

• 3:17:51Speaker 01

  Yeah, yeah. I mean, with anything, like, you know, I mean, like, psychology is really important. Like, you talk to athletes, like, marathon runners and so forth. And, you know, and they talk about what's the most important thing, is it the training regimen or the diet and so on? Actually, so much of it is actually psychology. You know, just tricking yourself to think that the problem is feasible, so that you're motivated to do it.

• 3:18:15Lex Fridman?

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

• 3:18:20Speaker 01

  Well, I sort of, I guess a mathematician, I mean, yeah, it's a reduction. There must be some, there must be a large number of I can understand. That's the first thing that came to mind.

• 3:18:30Lex Fridman?

  So that, but even broadly, is there, are we, is there something about our mind that's we're going to be limited, even with the help of mathematics?

• 3:18:41Speaker 01

  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've not allowed blackboard, pen and paper.

• 3:18:52Lex Fridman?

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

• 3:18:55Speaker 01

  Incredibly limited, even language, the English language is a technology.

• 3:19:00Speaker 01

  It's a, it's one that's been very internalized.

• 3:19:03Lex Fridman?

  So you're right, they're really, the formulation of the problem isn't correct because there really is no longer a, just a solo human, we're already augmented in extremely complicated, intricate ways, right?

• 3:19:17Speaker 01

  Yeah, yeah.

• 3:19:17Lex Fridman?

  So we're already like a collective intelligence.

• 3:19:20Speaker 01

  Yes, yeah, I guess, so humanity plural has much more intelligence in principle on his good days than, than the individual humans put together. It cannot have less, okay, but, yeah, so, yeah, the mathematical community plural is incredibly super intelligent entity that no single human mathematician can, can come close to, to replicating. You see it a little bit on these like question analysis sites. So this math overflow, which is the math version of stack overflow. And like, sometimes you get like this very quick responses to very difficult questions from the community. And it is, it's a pleasure to watch, actually, as an expert.

• 3:20:00Lex Fridman?

  I'm a fan spectator of that, of that site, just seeing the brilliance of the different people there, the depth of knowledge that people have, and the willingness to engage in the, in the rigor and the nuance of the particular question is pretty cool to watch. It's fun, it's almost like just fun to watch. What gives you hope about this whole thing we have going on, human civilization?

• 3:20:24Speaker 01

  I think, yeah, the younger generation is always like, like really creative and enthusiastic and inventive.

• 3:20:32Speaker 01

  It's a pleasure working with, with, with, with young students.

• 3:20:38Speaker 01

  You know, the the progress of science tells us that the problems that used to be really difficult can become extremely, you know, can become like trivial to solve, you know. I mean, like it was like navigation, you know, just knowing where you were on the planet. Was this horrendous problem? People died, you know, or lost fortunes because they couldn't navigate, you know, and we have devices in our pockets that do this automatically for us, like it's a completely solved problem, you know, so things that are seem unfeasible for us now could be maybe just sort of homework exercises for them.

• 3:21:12Lex Fridman?

  Yeah, one of the things I find really sad about the finiteness of life is that I won't get to see all the cool things we create as a civilization, you know, that because in the next hundred years, 200 years, just imagine showing showing up in 200 years.

• 3:21:26Speaker 01

  Yeah, well, already plenty has happened, you know, like if you could go back in time and talk to your teenage self or something, you know, I mean, yeah, just the internet and our AI, I mean, again, they've been they're beginning to be internalized and says, yeah, of course, an AI can understand our voice and and give reasonable, you know, slightly incorrect answers to any question, but you know, this was mind blowing even two years ago.

• 3:21:50Lex Fridman?

  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, just even in progress of robotics, just to take a step back and be like, wow, this is beautiful, the way humans are able to create this.

• 3:22:19Speaker 01

  Yeah, when the 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.

• 3:22:30Lex Fridman?

  Well, one place I can always count on rationality is the comment section of your blog, which I'm a big fan of. There's a lot of really smart people there. And thank you, of course, for for putting those ideas out on the blog. And it's 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 to me.

• 3:22:58Speaker 01

  Thank you. It was a pleasure.

• 3:23:00Lex Fridman?

  Thanks for listening to this conversation with Terrence Tao. To support this podcast, please check out our sponsors in the description or at lexfridman.com/sponsors. And now, let me leave you with some words from Galileo Galilei. "Mathematics is the language with which God has written the universe." Thank you for listening and hope to see you next time.