Learn. Share. Evolve…

0:00:03 The following is a conversation with Terence Tao,
0:00:07 widely considered to be one of the greatest mathematicians in history,
0:00:10 often referred to as the Mozart of math.
0:00:15 He won the Fields Medal and the Breakthrough Prize in Mathematics
0:00:18 and has contributed groundbreaking work
0:00:22 to a truly astonishing range of fields in mathematics and physics.
0:00:27 This was a huge honor for me, for many reasons,
0:00:32 including the humility and kindness that Terry showed to me
0:00:34 throughout all our interactions.
0:00:35 It means the world.
0:00:38 And now, a quick few-second mention of each sponsor.
0:00:43 Check them out in the description or at lexfriedman.com slash sponsors.
0:00:45 It’s the best way to support this podcast.
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0:00:52 NetSuite for your business, Element for electrolytes,
0:00:54 and AG1 for your health.
0:00:55 Choose Wizen, my friends.
0:00:57 And now, on to the full ad reads.
0:00:58 They’re all here in one place.
0:01:03 I do try to make them interesting by talking about some random things
0:01:04 I’m reading or thinking about.
0:01:07 But if you skip, please still check out the sponsors.
0:01:08 I enjoy their stuff.
0:01:09 Maybe you will, too.
0:01:11 To get in touch with me, for whatever reason,
0:01:13 go to lexfriedman.com slash contact.
0:01:15 All right, let’s go.
0:01:17 This episode is brought to you by Notion,
0:01:20 a note-taking and team collaboration tool.
0:01:24 I use Notion for everything, for personal notes, for planning these podcasts,
0:01:27 for collaborating with other folks,
0:01:30 and for super boosting all of those things with AI,
0:01:34 because Notion does a great job of integrating AI into the whole thing.
0:01:38 You know what’s fascinating is the mechanisms of human memory
0:01:43 before we had widely adopted technologies and tools
0:01:46 for writing and recording stuff,
0:01:47 certainly before the computer.
0:01:50 So you can look at medieval monks, for example,
0:01:55 that would use the now well-studied memory techniques,
0:01:57 like the memory palace,
0:01:58 the spatial memory techniques,
0:01:59 to memorize entire books.
0:02:01 That is certainly the effect of technology,
0:02:03 started by Google Search
0:02:05 and moving to all the other things like Notion,
0:02:08 that we’re offloading more and more and more
0:02:10 of the task of memorization to the computers,
0:02:15 which I think is probably a positive thing
0:02:20 because it frees more of our brain to do deep reasoning,
0:02:24 whether that’s deep dive, focused specialization,
0:02:26 or the journalist type of thinking,
0:02:28 versus memorizing facts.
0:02:31 Although I do think that there’s a kind of
0:02:34 Brackard model that’s formed when you memorize a lot of things,
0:02:39 and from there, from inspiration, arises discovery.
0:02:40 So I don’t know.
0:02:47 It could be a great cost to offloading most of our memorization to the machines.
0:02:50 But it is the way of the world.
0:02:53 Try Notion AI for free when you go to notion.com slash lex.
0:02:56 That’s all lowercase notion.com slash lex
0:02:58 to try the power of Notion AI today.
0:03:01 This episode is also brought to you by Shopify,
0:03:05 a platform designed for anyone to sell anywhere with a great looking online store.
0:03:08 Our future friends has a lot of robots in it.
0:03:10 Looking into that distant future,
0:03:16 you have Amazon warehouses with millions of robots that move packages around.
0:03:22 You have Tesla bots everywhere in the factories and in the home and on the streets and the baristas.
0:03:23 All of that.
0:03:24 That’s our future.
0:03:28 Right now you have something like Shopify that connects a lot of humans in the digital space.
0:03:38 But more and more, there will be an automated, digitized, AI-fueled connection between humans in the physical space.
0:03:43 Like a lot of futures, there’s going to be negative things and there’s going to be positive things.
0:03:48 And like a lot of possible futures, there’s little we could do about stopping it.
0:03:53 All we can do is steer it in the direction that enables human flourishing.
0:04:04 Instead of hiding in fear or fear-mongering, be part of the group of people that are building the best possible trajectory of human civilization.
0:04:10 Anyway, sign up for a $1 per month trial period at shopify.com slash lex.
0:04:11 That’s all lowercase.
0:04:15 Go to shopify.com slash lex to take your business to the next level today.
0:04:22 This episode is also brought to you by NetSuite, an all-in-one cloud business management system.
0:04:26 There’s a lot of messy components to running a business.
0:04:34 And I must ask, and I must wonder, at which point there’s going to be an AI, AGI-like CFO of a company.
0:04:43 An AI agent that handles most, if not all, of the financial responsibilities or all of the things that NetSuite is doing.
0:04:49 At which point will NetSuite increasingly leverage AI for those tasks?
0:05:06 I think probably it 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.
0:05:22 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.
0:05:25 Anyway, that’s our future.
0:05:30 Download the CFO’s Guide to AI and Machine Learning at netsuite.com slash lex.
0:05:32 That’s netsuite.com slash lex.
0:05:39 This episode is also brought to you by Element, my daily zero-sugar and delicious electrolyte mix.
0:05:45 You know, I run along the river often and get to meet some really interesting people.
0:05:50 One of the people I met was preparing for his first ultra-marathon.
0:05:52 I believe he said it was 100 miles.
0:05:59 And that, of course, sparked in me the thought that I need for sure to do one myself.
0:06:10 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.
0:06:16 Of course, the thing that is crazy for me is daily activity for Goggins.
0:06:25 But nevertheless, I think it’s important in the physical domain, the mental domain, and all domains of life to challenge yourself.
0:06:32 And athletic endeavors is one of the most sort of crisp, clear, well-structured way of challenging yourself.
0:06:34 But there’s all kinds of things.
0:06:35 Writing a book.
0:06:40 To be honest, having kids and marriage and relationships and friendships.
0:06:46 All of those, if you take it seriously, if you go all in and do it right, I think that’s a serious challenge.
0:06:50 Because most of us are not prepared for it.
0:06:51 You can learn along the way.
0:06:59 And if you have the rigorous feedback loop of improving, constantly growing as a person, and really doing a great job of the thing,
0:07:04 I think that might as well be an ultra-marathon.
0:07:07 Anyway, get a sample pack for free with any purchase.
0:07:10 Try it at drinkelement.com slash lex.
0:07:15 And finally, this episode is also brought to you by AG1.
0:07:19 An all-in-one daily drink to support better health and peak performance.
0:07:22 I drink it every day.
0:07:28 I’m preparing for a conversation on drugs in the Third Reich.
0:07:34 And funny enough, it’s a kind of way to analyze Hitler’s biography.
0:07:37 It’s to look at what he consumed throughout.
0:07:41 And Norman Oler does a great job of analyzing all of that.
0:07:47 And tells the story of Hitler and the Third Reich in a way that hasn’t really been touched by historians before.
0:07:54 It’s always nice to look at key moments in history through a perspective that’s not often taken.
0:08:00 Anyway, I mention that because I think Hitler had a lot of stomach problems.
0:08:04 And so that was the motivation for getting a doctor.
0:08:08 The doctor that eventually would fill him up with all kinds of drugs.
0:08:15 But the doctor earned Hitler’s trust by giving him probiotics, which is a kind of revolutionary thing at the time.
0:08:20 And so that really helped deal with whatever stomach issues that Hitler was having.
0:08:24 All of that is a reminder that war is waged by humans.
0:08:26 And humans are biological systems.
0:08:31 And biological systems require fuel and supplements and all of that kind of stuff.
0:08:36 And depending on what you put in your body will affect your performance in the short term and the long term.
0:08:40 With meth, that’s true with Hitler.
0:08:43 To his last days in the bunker in Berlin.
0:08:46 All the cocktail of drugs that he was taking.
0:08:49 So, I think I got myself somewhere deep.
0:08:53 I’m not sure how to get out of this.
0:08:57 It deserves a multi-hour conversation versus a few seconds of mention.
0:09:04 But yeah, all of that was sparked by my thinking of AG1 and how much I love it.
0:09:07 I appreciate that you’re listening to this.
0:09:12 And coming along for the wild journey that these ad reads are.
0:09:19 Anyway, AG1 will give you a one-month supply of fish oil when you sign up at drinkag1.com slash Lex.
0:09:22 This is the Lex Friedman podcast.
0:09:28 To support it, please check out our sponsors in the description or at lexfriedman.com slash sponsors.
0:09:32 And now, dear friends, here’s Terrence Tao.
0:09:54 What was the first really difficult research-level math problem that you encountered?
0:09:56 One that gave you pause, maybe?
0:10:02 Well, I mean, in your undergraduate education, you learn about the really hard impossible problems.
0:10:05 Like the Riemann hypothesis, the Trin-Primes conjecture.
0:10:07 You can make problems arbitrarily difficult.
0:10:08 That’s not really a problem.
0:10:10 In fact, there’s even problems that we know to be unsolvable.
0:10:17 What’s really interesting are the problems just on the boundary between what we can do easily and what are hopeless.
0:10:25 But what are problems where existing techniques can do like 90% of the job and then you just need that remaining 10%?
0:10:30 I think as a PhD student, the Kikeya problem certainly caught my eye.
0:10:32 And it just got solved, actually.
0:10:34 It’s a problem I’ve worked on a lot in my early research.
0:10:41 Historically, it came from a little puzzle by the Japanese mathematician Soji Kikeya in like 1918 or so.
0:10:48 So the puzzle is that you have a needle on the plane.
0:10:52 Well, think of like driving on a road or something.
0:10:54 And you want to execute a U-turn.
0:10:55 You want to turn the needle around.
0:10:59 But you want to do it in as little space as possible.
0:11:03 So you want to use this little area in order to turn it around.
0:11:06 But the needle is infinitely maneuverable.
0:11:09 So you can imagine just spinning it around.
0:11:10 It’s a unit needle.
0:11:12 You can spin it around its center.
0:11:15 And I think that gives you a disk of area, I think, pi over 4.
0:11:22 Or you can do a 3-point U-turn, which is what we teach people in their driving schools to do.
0:11:24 And that actually takes area pi over 8.
0:11:27 So it’s a little bit more efficient than a rotation.
0:11:31 And so for a while, people thought that was the most efficient way to turn things around.
0:11:37 But Vesikovic showed that, in fact, you could actually turn the needle around using as little area as you wanted.
0:11:47 So 0.001, there was some really fancy multi-back-and-forth U-turn thing that you could do, that you could turn the needle around.
0:11:50 And in so doing, it would pass through every intermediate direction.
0:11:51 Is this in the two-dimensional plane?
0:11:53 This is in the two-dimensional plane.
0:11:55 So we understand everything in two dimensions.
0:11:57 So the next question is what happens in three dimensions.
0:12:01 So suppose the Hubble Space Telescope is tube in space.
0:12:04 And you want to observe every single star in the universe.
0:12:07 So you want to rotate the telescope to reach every single direction.
0:12:09 And here’s the unrealistic part.
0:12:11 Suppose that space is at a premium, which it totally is not.
0:12:18 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.
0:12:22 How small a volume do you need to do that?
0:12:25 And so you can modify Vesikovic’s construction.
0:12:30 And so if your telescope has zero thickness, then you can use as little volume as you need.
0:12:32 That’s a simple modification of the two-dimensional construction.
0:12:37 But the question is that if your telescope is not zero thickness, but just very, very thin,
0:12:44 some thickness delta, what is the minimum volume needed to be able to see every single direction as a function of delta?
0:12:49 So as delta gets smaller, as your needle gets thinner, the volume should go down.
0:12:50 But how fast does it go down?
0:12:59 And the conjecture was that it goes down very, very slowly, like logarithically, roughly speaking.
0:13:01 And that was proved after a lot of work.
0:13:04 So this seems like a puzzle-wise and interesting.
0:13:08 So it turns out to be surprisingly connected to a lot of problems in partial differential equations,
0:13:12 in number theory, in geometry, combinatorics.
0:13:16 For example, in wave propagation, you splash some water around, you create water waves,
0:13:17 and they travel in various directions.
0:13:22 But waves exhibit both particle and wave type behavior.
0:13:29 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.
0:13:34 And so if you plot it into space and time, it occupies a region which looks like a tube.
0:13:43 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.
0:13:47 Like you can imagine dropping a pebble into a pond and ripples spread out.
0:13:52 But then if you time reverse that scenario, and the equations of wave motion are time reversible,
0:13:59 you can imagine ripples that are converging to a single point, and then a big splash occurs, maybe even a singularity.
0:14:07 And so it’s possible to do that, and geometrically what’s going on is that there’s always sort of light rays.
0:14:12 So like if this wave represents light, for example, you can imagine this wave as a superposition of photons,
0:14:15 all traveling at the speed of light.
0:14:18 They all travel on these light rays, and they’re all focusing at this one point.
0:14:24 So you can have a very dispersed wave, focus into a very concentrated wave at one point in space and time,
0:14:27 but then it defocuses again, and it separates.
0:14:30 But potentially, if the pinjaccio had a negative solution,
0:14:36 so what that means is that there’s a very efficient way to pack tubes pointing in different directions
0:14:40 into a very, very narrow region of a very narrow volume.
0:14:43 Then you would also be able to create waves that start out,
0:14:46 there’ll be some arrangement of waves that start out very, very dispersed,
0:14:49 but they would concentrate not just at a single point,
0:14:55 but there’ll be a lot of concentrations in space and time.
0:15:01 And you could create what’s called a blow-up, where these waves, their amplitude becomes so great
0:15:05 that the laws of physics that they’re governed by are no longer wave equations,
0:15:06 but something more complicated and non-linear.
0:15:11 And so in mathematical physics, we care a lot about whether certain equations
0:15:15 and wave equations are stable or not, whether they can create these singularities.
0:15:19 There’s a famous unsolved problem called the Navier-Stokes regularity problem.
0:15:22 So the Navier-Stokes equations, equations that govern the fluid flow,
0:15:24 or incompressible fluids like water.
0:15:28 The question asks, if you start with a smooth velocity field of water,
0:15:32 can it ever concentrate so much that the velocity becomes infinite at some point?
0:15:33 That’s called a singularity.
0:15:37 We don’t see that in real life.
0:15:40 If you splash around water in a bathtub, it won’t explode on you,
0:15:44 or have water leaving at a speed of light.
0:15:46 But potentially, it is possible.
0:15:55 And in fact, in recent years, the consensus has drifted towards the belief that,
0:16:00 in fact, for certain very special initial configurations of, say, water,
0:16:02 that singularities can form.
0:16:05 But people have not yet been able to actually establish this.
0:16:08 The Clay Foundation has these seven Millennium Prize problems,
0:16:11 has a million-dollar prize for solving one of these problems.
0:16:12 This is one of them.
0:16:14 Of these seven, only one of them has been solved.
0:16:16 At the point, great conjecture.
0:16:22 So, the Kakeha conjecture is not directly, directly related to the Navier-Stokes problem,
0:16:28 but understanding it would help us understand some aspects of things like wave concentration,
0:16:31 which would indirectly probably help us understand the Navier-Stokes problem better.
0:16:33 Can you speak to the Navier-Stokes?
0:16:37 So, the existence and smoothness, like you said, Millennial Prize problem.
0:16:38 Right.
0:16:39 You’ve made a lot of progress on this one.
0:16:43 In 2016, you published a paper, Finite Time Blow-Up,
0:16:46 for an averaged three-dimensional Navier-Stokes equation.
0:16:46 Right.
0:16:51 So, we’re trying to figure out if this thing usually doesn’t blow up.
0:16:52 Right.
0:16:55 But, can we say for sure it never blows up?
0:16:56 Right.
0:16:56 Yeah.
0:16:58 So, yeah, that is literally the million-dollar question.
0:16:59 Yeah.
0:17:03 So, this is what distinguishes mathematicians from pretty much everybody else.
0:17:11 Like, if something holds 99.99% of the time, that’s good enough for most, you know, for most things.
0:17:20 But, mathematicians are one of the few people who really care about whether, like, 100%, really 100% of all situations are covered by, yeah.
0:17:24 So, most fluid, most of the time, water does not blow up.
0:17:28 But, could you design a very special initial state that does this?
0:17:33 And, maybe we should say that this is a set of equations that govern in the field of fluid dynamics.
0:17:34 Yes.
0:17:36 Trying to understand how fluid behaves.
0:17:42 And, it actually turns out to be a really complicated, you know, fluid is an extremely complicated thing to try to model.
0:17:42 Yeah.
0:17:44 So, it has practical importance.
0:17:48 So, this clay price problem concerns what’s called the incompressible Navier-Stokes, which governs things like water.
0:17:51 There’s something called the compressible Navier-Stokes, which governs things like air.
0:17:53 And, that’s particularly important for weather prediction.
0:17:56 Weather prediction, it does a lot of computational fluid dynamics.
0:17:59 A lot of it is actually just trying to solve the Navier-Stokes equations as best they can.
0:18:05 Also, gathering a lot of data so that they can get, they can initialize the equation.
0:18:06 There’s a lot of moving parts.
0:18:08 So, it’s a very important problem, practically.
0:18:12 Why is it difficult to prove general things?
0:18:17 About the set of equations like it not blowing up.
0:18:18 The short answer is Maxwell’s Demon.
0:18:21 So, Maxwell’s Demon is a concept in thermodynamics.
0:18:24 Like, if you have a box of two gases, you know, oxygen and nitrogen.
0:18:27 And, maybe you start with all the oxygen on one side and nitrogen on the other side.
0:18:29 But, there’s no barrier between them.
0:18:30 Then, they will mix.
0:18:32 And, they should stay mixed.
0:18:35 There’s no reason why they should unmix.
0:18:40 But, in principle, because of all the collisions between them, there could be some sort of weird conspiracy.
0:18:50 Like, maybe there’s a microscopic demon called Maxwell’s Demon that will, every time an oxygen and nitrogen atom collide, they will bounce off in such a way that the oxygen sort of drifts onto one side and the nitrogen goes to the other.
0:18:56 And, you could have an extremely improbable configuration emerge, which we never see.
0:19:00 And, statistically, it’s extremely unlikely.
0:19:03 But, mathematically, it’s possible that this can happen.
0:19:05 And, we can’t rule it out.
0:19:09 And, this is a situation that shows up a lot in mathematics.
0:19:11 A basic example is the digits of pi.
0:19:13 3.14159, and so forth.
0:19:16 The digits look like they have no pattern.
0:19:17 And, we believe they have no pattern.
0:19:21 On the long term, you should see as many ones and twos and threes as fours and fives and sixes.
0:19:26 There should be no preference in the digits of pi to favor, let’s say, 7 over 8.
0:19:35 But, maybe there’s some demon in the digits of pi that, like, every time you compute more and more digits, it biases one digit to another.
0:19:39 And, this is a conspiracy that should not happen.
0:19:40 There’s no reason it should happen.
0:19:45 But, there’s no way to prove it with our current technology.
0:19:47 Okay, so, getting back to Navier-Stokes.
0:19:49 A fluid has a certain amount of energy.
0:19:52 And, because the fluid is in motion, the energy gets transported around.
0:19:54 And, water is also viscous.
0:20:02 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.
0:20:08 And, this is what happens when we actually experiment with water.
0:20:11 You splash around, there’s some turbulence and waves and so forth.
0:20:13 But, eventually, it settles down.
0:20:18 And, the lower the amplitude, the smaller the velocity, the more calm it gets.
0:20:26 But, potentially, there is some sort of demon that keeps pushing the energy of the fluid into a smaller and smaller scale.
0:20:27 And, it will move faster and faster.
0:20:31 And, at faster speeds, the effect of viscosity is relatively less.
0:20:41 And, it could happen that it creates some sort of, what’s called a self-similar blow-up scenario, where, you know, the energy of the fluid starts off at some large scale.
0:20:53 And, then, it all sort of transfers the energy into a smaller region of the fluid, which then, at a much faster rate, moves into an even smaller region and so forth.
0:20:59 And, each time it does this, it takes maybe half as long as the previous one.
0:21:07 And, then, you could actually converge to all the energy concentrating in one point in a finite amount of time.
0:21:12 And, that scenario is called finite amount of blow-up.
0:21:14 So, in practice, this doesn’t happen.
0:21:17 So, water is what’s called turbulent.
0:21:23 So, it is true that, if you have a big eddy of water, it will tend to break up into smaller eddies.
0:21:26 But, it won’t transfer all the energy from one big eddy into one smaller eddy.
0:21:28 It will transfer into maybe three or four.
0:21:31 And, then, those ones split up into maybe three or four small eddies of their own.
0:21:37 And, so, the energy gets dispersed to the point where the viscosity can then keep everything under control.
0:21:50 But, if it can somehow concentrate all the energy, keep it all together, and do it fast enough that the viscous effects don’t have enough time to calm everything down, then this blow-up can occur.
0:21:57 So, there are papers who have claimed that, oh, you just need to take into account conservation of energy and just carefully use the viscosity.
0:22:02 And, you can keep everything under control for not just Navier-Stokes, but for many, many types of equations like this.
0:22:10 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 final time blow-up, that velocity stays smooth.
0:22:12 And, it all failed.
0:22:15 There was always some sign error or some subtle mistake, and it couldn’t be salvaged.
0:22:24 So, what I was interested in doing was trying to explain why we were not able to disprove final time blow-up.
0:22:28 I couldn’t do it for the actual equations of fluids, which were too complicated.
0:22:38 But, if I could average the equations of motion of Navier-Stokes, so, basically, if I could turn off certain types of ways in which water interacts, and only keep the ones that I want.
0:22:58 So, in particular, if there’s a fluid, and it could transfer its energy from a large eddy into this small eddy, or this other small eddy, I would turn off the energy channel that would transfer energy to this one, and direct it only into this smaller eddy, while still preserving the law of conservation of energy.
0:22:59 So, you’re trying to make a blow-up.
0:22:59 Yeah.
0:23:06 So, I basically engineer a blow-up by changing volts of physics, which is one thing that mathematicians are allowed to do.
0:23:07 We can change the equation.
0:23:10 How does that help you get closer to the proof of something?
0:23:10 Right.
0:23:13 So, it provides what’s called an obstruction in mathematics.
0:23:26 So, what I did was that, basically, if I turned off certain parts of the equation, which, usually, when you turn off certain interactions, make it less non-linear, it makes it more regular and less likely to blow up.
0:23:35 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.
0:23:51 So, what that means is that, if you wanted to prove global regularity for Navier-Stokes, for the actual equation, you must use some feature of the true equation, which my artificial equation does not satisfy.
0:23:54 So, it rules out certain approaches.
0:24:04 So, the thing about math is, it’s not just about finding, you know, taking a technique that is going to work and applying it, but you need to not take the techniques that don’t work.
0:24:17 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.
0:24:30 So, having these counter-examples for nearby problems kind of rules out, it saves you a lot of time because you’re not wasting energy on things that you now know cannot possibly ever work.
0:24:37 How deeply connected is it to that specific problem of fluid dynamics, or is it some more general intuition you build up about mathematics?
0:24:38 Right, yeah.
0:24:43 So, the key phenomenon that my technique exploits is what’s called supercriticality.
0:24:48 So, in partial differential equations, often these equations are like a tug-of-war between different forces.
0:24:54 So, in Navier-Stokes, there’s the dissipation force coming from viscosity, and it’s very well understood.
0:24:55 It’s linear.
0:24:56 It calms things down.
0:25:00 So, if viscosity was all there was, then nothing bad would ever happen.
0:25:09 But there’s also transport, that energy in one location of space can get transported because the fluid is in motion to other locations.
0:25:13 And that’s a non-linear effect, and that causes all the problems.
0:25:19 So, there are these two competing terms in the Navier-Stokes equation, the dissipation term and the transport term.
0:25:24 If the dissipation term dominates, if it’s large, then basically you get regularity.
0:25:29 And if the transport term dominates, then we don’t know what’s going on.
0:25:30 It’s a very non-linear situation.
0:25:31 It’s unpredictable.
0:25:31 It’s turbulent.
0:25:38 So, sometimes these forces are in balance at small scales, but not in balance at large scales, or vice versa.
0:25:40 So, Navier-Stokes is what’s called supercritical.
0:25:45 So, at smaller and smaller scales, the transport terms are much stronger than the viscosity terms.
0:25:48 So, the viscosity terms are the things that calm things down.
0:25:53 And so, this is why the problem is hard.
0:26:00 In two dimensions, so, the Soviet mathematician, Ladishan Skaya, she, in the 60s, showed in two dimensions there was no blow-up.
0:26:03 And as you mentioned, the Navier-Stokes equation is what’s called critical.
0:26:08 The effect of transport and the effect of viscosity are about the same strength, even at very, very small scales.
0:26:13 And we have a lot of technology to handle critical and also subcritical equations and prove regularity.
0:26:17 But for supercritical equations, it was not clear what was going on.
0:26:21 And I did a lot of work, and then there’s been a lot of follow-up,
0:26:26 showing that for many other types of supercritical equations, you can create all kinds of blow-up examples.
0:26:31 Once the nonlinear effects dominate the linear effects at small scales, you can have all kinds of bad things happen.
0:26:40 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.
0:26:46 I mean, that’s a key qualitative feature that distinguishes some equations for being sort of nice and predictable,
0:26:47 and, you know, like planetary motion.
0:26:53 I mean, there’s certain equations that you can predict for millions of years, or thousands at least.
0:26:54 Again, it’s not really a problem.
0:26:59 But there’s a reason why we can’t predict the weather past two weeks into the future,
0:27:00 because it’s a supercritical equation.
0:27:03 Lots of really strange things are going on at very fine scales.
0:27:12 So, whenever there’s some huge source of nonlinearity, that can create a huge problem for predicting what’s going to happen.
0:27:13 Yeah.
0:27:17 And if nonlinearity is somehow more and more featured and interesting at small scales.
0:27:23 I mean, there’s many equations that are nonlinear, but in many equations, you can approximate things by the bulk.
0:27:29 So, for example, planetary motion, you know, if you wanted to understand the orbit of the Moon or Mars or something,
0:27:35 you don’t really need the microstructure of, like, the seismology of the Moon or, like, exactly how the Mars is distributed.
0:27:39 You just, basically, you can almost approximate these planets by point masses.
0:27:43 And just the aggregate behavior is important.
0:27:50 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.
0:27:54 For supercritical equations, the fine scale information is really important.
0:27:57 So, if we can just linger on the Navier-Stokes equations a little bit.
0:28:12 So, you’ve suggested, maybe you can describe it, that one of the ways to solve it or to negatively resolve it would be to sort of to construct a liquid, a kind of liquid computer.
0:28:12 Right.
0:28:17 And then show that the halting problem from computation theory has consequences for fluid dynamics.
0:28:20 So, show it in that way.
0:28:22 Can you describe this idea?
0:28:22 Right, yeah.
0:28:27 So, this came out of this work of constructing this average equation that blew up.
0:28:33 So, as part of how I had to do this, so, there’s sort of this naive way to do it.
0:28:35 You just keep pushing.
0:28:41 Every time you get energy at one scale, you push it immediately to the next scale as fast as possible.
0:28:44 This is sort of the naive way to force blow up.
0:28:46 It turns out in five and higher dimensions, this works.
0:28:50 But in three dimensions, there was this funny phenomenon that I discovered.
0:28:59 That if you keep, if you change the laws of physics, you just always keep trying to push the energy into smaller, smaller scales.
0:29:04 What happens is that the energy starts getting spread out into many scales at once.
0:29:12 So, you have energy at one scale, you’re pushing it into the next scale, and then as soon as it enters that scale, you also push it to the next scale.
0:29:15 But there’s still some energy left over from the previous scale.
0:29:16 You’re trying to do everything at once.
0:29:19 And this spreads out the energy too much.
0:29:26 And then it turns out that it makes it vulnerable for viscosity to come in and actually just damp out everything.
0:29:30 So, it turns out this directive motion doesn’t actually work.
0:29:34 There was a separate paper by some other authors that actually showed this in three dimensions.
0:29:38 So, what I needed was to program a delay.
0:29:40 So, kind of like airlocks.
0:29:46 So, I needed an equation which would start with a fluid doing something at one scale.
0:29:48 It would push its energy into the next scale.
0:29:54 But it would stay there until all the energy from the larger scale got transferred.
0:29:58 And only after you pushed all the energy in, then you sort of opened the next gate.
0:30:00 And then you push that in as well.
0:30:07 So, by doing that, the energy inches forward scale by scale in such a way that it’s always localized at one scale at a time.
0:30:11 And then it can resist the effects of viscosity because it’s not dispersed.
0:30:18 So, in order to make that happen, I had to construct a rather complicated non-linearity.
0:30:24 And it was basically like, you know, it was constructed like an electronic circuit.
0:30:28 So, I actually thanked my wife for this because she was trained as an electrical engineer.
0:30:34 And, you know, she talked about, you know, she had to design circuits and so forth.
0:30:45 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.
0:30:54 And these diagrams, you can sort of follow up with your eyeballs and say, oh, yeah, the current will build up here and then it will stop and then it will do that.
0:31:00 So, I knew how to build the analog of basic electronic components, you know, like resistors and capacitors and so forth.
0:31:07 And I would stack them together in such a way that I would create something that would open one gate and then there would be a clock.
0:31:10 And then once the clock hits a certain threshold, it would close it.
0:31:13 It would become a Rube Goldberg type machine, but described mathematically.
0:31:15 And this ended up working.
0:31:19 So, what I realized is that if you could pull the same thing off for the actual equations.
0:31:38 So, if the equations of water support a computation, so, like, you can imagine kind of a steampunk, but it’s really waterpunk 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.
0:31:44 But instead of electrons, you can imagine these pulses of water moving at a certain velocity.
0:31:49 And maybe it’s, there are two different configurations corresponding to a bit being up or down.
0:32:03 Probably that if you had two of these moving bodies of water collide, they would come out with some new configuration, which is, which would be something like an AND gate or OR gate, you know, that the output would depend in a very predictable way on the inputs.
0:32:07 And like, you could chain these together and maybe create a Turing machine.
0:32:11 And then you could, you have computers, which are made completely out of water.
0:32:17 And if you have computers, then maybe you can do robotics, you know, hydraulics and so forth.
0:32:25 And so you could create some machine, which is basically a fluid analog, what’s called a von Neumann machine.
0:32:32 So von Neumann proposed, if you want to colonize Mars, the sheer cost of transporting people and machines to Mars is just ridiculous.
0:32:47 But if you could transport one machine to Mars, and this machine had the ability to mine the planet, create some more materials, smelt them, and build more copies of the same machine, then you could colonize the whole planet over time.
0:32:55 So if you could build a fluid machine, which, yeah, so it’s, it’s, it’s a, it’s a, it’s a, it’s a fluid robot.
0:32:56 Okay.
0:32:58 And what it would do, it’s, it’s purpose in life.
0:33:03 It’s programmed so that it would create a smaller version of itself in some sort of cold state.
0:33:04 It wouldn’t start just yet.
0:33:10 Once it’s ready, the big robot conviction of water would transfer all its energy into the smaller configuration and then power down.
0:33:11 Okay.
0:33:12 And then like, like clean itself up.
0:33:18 And then what’s left is this newest state, which would then turn on and do the same thing, but smaller and faster.
0:33:20 And then the equation has a certain scaling symmetry.
0:33:22 Once you do that, it can just keep iterating.
0:33:26 So this in principle would create a blow up for the actual Navier-Stokes.
0:33:29 And this is what I managed to accomplish for this average Navier-Stokes.
0:33:32 So it provided this sort of roadmap to solve the problem.
0:33:39 Now, this is a pipe dream because there are so many things that are missing for this to actually be a reality.
0:33:44 So I, I, I can’t create these basic logic gates.
0:33:48 I don’t, I don’t have these in these special configurations of water.
0:33:59 So, um, I mean, there’s candidates that include vortex rings that might possibly work, but, um, um, but also, you know, analog computing is really nasty, um, compared to digital computing.
0:34:00 I mean, cause there’s always errors.
0:34:04 Um, you have to, you have to do a lot of error correction along the way.
0:34:12 I don’t know how to completely power down the big machine so that it doesn’t interfere with the, the, the writing of a smaller machine, but everything in principle can happen.
0:34:14 Like it doesn’t contradict any of the laws of physics.
0:34:18 Um, so it’s sort of evidence that this thing is possible.
0:34:26 Um, there are other groups who are now pursuing ways to make Navier-Stokes blow up, which are nowhere near as ridiculously complicated as this.
0:34:39 Um, um, they, they actually are pursuing much closer to the direct self-similar model, which can, uh, it, it doesn’t quite work as is, but there could be some simpler scheme than what I just described to make this work.
0:34:46 There is a real leap of genius here to go from Navier-Stokes to this Turing machine.
0:35:03 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, smaller, smaller, and somehow seeing how that could be used.
0:35:06 To say something about a blow up.
0:35:07 I mean, that’s a big leap.
0:35:08 So there’s precedent.
0:35:18 I mean, um, so the, the thing about mathematics is that it’s, it’s really good at, um, spotting connections between what you think of, what you might think of as completely different, um, problems.
0:35:23 Um, but if, if, if the mathematical form is the same, you, you, you, you can, you can, you can draw a connection.
0:35:28 Um, so, um, there’s a lot of work previously on what’s called cellular automator.
0:35:31 Um, the most famous of which is Conway’s Game of Life.
0:35:33 This is infinite discrete grid.
0:35:36 And at any given time, the grid is either occupied by a cell or it’s empty.
0:35:40 And there’s a very simple rule that, uh, tells you how these cells evolve.
0:35:42 So sometimes cells live and sometimes they die.
0:35:51 Um, and there’s, um, you know, um, when I was a, uh, a student, uh, it was a very popular screensaver to actually just have these, these animations going on and they look very chaotic.
0:35:57 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.
0:36:00 Um, so for example, they discovered this thing called a glider.
0:36:05 So a glider is a very tiny configuration of like four or five cells, which evolves and it just moves at a certain direction.
0:36:07 And that’s like this, this vortex rings.
0:36:10 Um, yeah, so this is an analogy.
0:36:19 The Game of Life is kind of like a discrete equation and, and, um, the fluid Navier-Stokes is a continuous equation, but mathematically they have some similar features.
0:36:27 Um, and, um, so over time people discovered more and more interesting things that you could build within the Game of Life.
0:36:28 Game of Life is a very simple system.
0:36:34 It only has like three or four rules, um, to, to do it, but, but you can design all kinds of interesting configurations inside it.
0:36:38 Um, there’s something called a glider gun that does nothing to spit out gliders one at a, one, one at a time.
0:36:47 Um, and then after a lot of effort, people managed to, to create, um, and gates and all gates for gliders.
0:36:55 Like there’s this massive ridiculous structure, which if you, if, if, uh, if, uh, if you have a stream of gliders, um, coming in here and a stream of gliders coming in here,
0:36:57 then you may produce a stream gliders coming out.
0:37:04 If, if, maybe, if both of, of the, um, streams, um, have gliders, then there’ll be an output stream.
0:37:06 But if only one of them does, then nothing comes out.
0:37:08 So they could build something like that.
0:37:17 And once you could build, and, um, these basic gates, then just from software engineering, you can build almost anything.
0:37:19 Um, you can build a Turing machine.
0:37:22 I mean, it’s, it’s like an enormous steampunk type things.
0:37:28 They look ridiculous, but then people also generated self-replicating objects in the game of life.
0:37:36 A massive machine, a boner machine, which over a lot, huge period of time, and it always looked like glider guns inside doing these very steampunk calculations.
0:37:40 It would create another version of itself, which could replicate.
0:37:41 It’s so incredible.
0:37:45 A lot of this was like community crowdsourced by like amateur mathematicians, actually.
0:37:48 Um, so I knew about that, that, that work.
0:37:52 And so that is part of what inspired me to propose the same thing with Navier-Stokes.
0:37:57 Um, which is a much, as I said, analog is much worse than digital.
0:38:03 Like it’s going to be, um, you can’t just directly take the constructions in the game of life and plunk them in.
0:38:05 But again, it just, it shows it’s possible.
0:38:10 You know, there’s a kind of emergence that happens with these cellular automata.
0:38:14 Local rules, maybe it’s similar to fluids.
0:38:15 I don’t know.
0:38:24 But local rules operating at scale can create these incredibly complex dynamic structures.
0:38:28 Do you think any of that is amenable to mathematical analysis?
0:38:33 Do we have the tools to say something profound about that?
0:38:38 The thing is, you can get these emergent, very complicated structures, but only with very carefully prepared initial conditions.
0:38:44 Yeah, so, so these, these, these glider guns and gates and, and software machines, if you just plunk down randomly,
0:38:47 some cells and you, on the left, you will not see any of these.
0:38:57 Um, 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.
0:39:06 Um, but basically through engineering, you know, by, by, by, by, by, by specially designing things in a very special way, you can pick clever constructions.
0:39:15 I wonder if it’s possible to prove the sort of the negative of like, basically prove that only through engineering can you ever create something interesting.
0:39:21 This, this, this is a recurring challenge in mathematics that, um, I call it the dichotomy between structure and randomness.
0:39:24 That most objects that you can generate in mathematics are random.
0:39:26 They look like random, like the digits of pi.
0:39:28 Well, we believe is a good example.
0:39:31 Um, but there’s a very small number of things that have patterns.
0:39:41 Um, but, um, now you can prove something as a pattern by just constructing, you know, like if something has a simple pattern and you have a proof that it, it does something like repeat itself every so often you can do that.
0:39:48 But, um, and you can prove that, that for example, you can, you can prove that most sequences of, of digits have no pattern.
0:39:52 Um, so like if, if you just pick digits randomly, there’s something called the low large numbers.
0:39:55 It tells you, you’re going to get as many ones as, as twos in the long run.
0:40:02 Um, but, um, we have a lot fewer tools to, to, to, to, if I give you a specific pattern.
0:40:06 Like the digits of pi, how can I show that this doesn’t have some weird pattern to it?
0:40:14 Some other work that I have spent a lot of time on is to prove what are called structure theorems or inverse theorems that give tests for when something is, is very structured.
0:40:17 So some functions are, what’s called additive.
0:40:20 Like if you have a function that maps the natural numbers to the natural numbers.
0:40:24 So maybe, um, you know, two maps to four or three maps to six and so forth.
0:40:30 Um, some functions are what’s called additive, which means that if you add, if you add two inputs together, the output gets, gets added as well.
0:40:32 Uh, for example, I’m multiplying by a constant.
0:40:40 If you multiply a number by 10, um, if you, if you, if you, if you multiply a plus b by 10, that’s the same as multiplying a by 10 and b by 10 and then adding them together.
0:40:42 So some, um, functions are additive.
0:40:46 Some functions are kind of additive, but not completely additive.
0:40:53 Um, so for example, if I take a number n, I multiply by the square root of two and I take the integer part of that.
0:40:56 So 10 by square root of two is like 14 point something.
0:40:59 So 10 up to 14, um, 20 up to 28.
0:41:03 Um, so in that case, additivity is true then.
0:41:05 So 10 plus 10 is 20 and 14 plus 40 is 28.
0:41:08 But because of this rounding, uh, sometimes there’s roundoff errors.
0:41:16 And sometimes when you, um, add a plus b, this function doesn’t quite give you the sum of, of the two individual outputs, but the sum plus minus one.
0:41:19 Um, so it’s almost additive, but not quite additive.
0:41:33 Um, so there’s a lot of useful results in mathematics and I’ve worked a lot on developing things like this to the effect that if, if a function exhibits some structure like this, then, um, it’s basically, there’s a reason for why it’s true.
0:41:42 And the reason is because there’s, there’s some other nearby function, which is actually, um, completely structured, which is explaining this sort of partial pattern that you have.
0:41:54 Um, and so if you have these sort of inverse theorems, it, um, it creates this sort of dichotomy that, that either the objects that you study are either have no structure at all, or they are somehow related to something that is structured.
0:41:59 Um, and in either way, in either, um, uh, in either case, you can make progress.
0:42:06 Um, a good example of this is that there’s this old theorem in mathematics called Szemeredi’s theorem, uh, proven in the 1970s.
0:42:09 It concerns trying to find a certain type of pattern in a set of numbers.
0:42:14 The pattern is arithmetic progression, things like three, five, and seven, or, or, or 10, 15, and 20.
0:42:26 And Szemeredi, André, Szemeredi proved that, um, any set of numbers that are sufficiently big, um, what’s called, what’s called positive density, has, um, arithmetic progressions in it of, of any length you wish.
0:42:33 Um, so for example, um, the odd numbers have a set of density one half, um, and they contain arithmetic progressions of any length.
0:42:37 Um, so in that case, it’s obvious because the, the, the odd numbers are really, really structured.
0:42:40 I can just take, uh, 11, 13, 15, 17.
0:42:44 I just, I can, I can easily find arithmetic progressions in, in, in that set.
0:42:48 Um, but, um, Szemeredi’s theorem also applies to random sets.
0:42:56 If I take the set of all numbers and I flip a coin, um, and I, uh, for each number, and I only keep the numbers which, for which I got a heads.
0:43:00 Okay, so I just flip coins, I just randomly take out half the numbers, I keep one half.
0:43:02 So that’s a set that has no, no patterns at all.
0:43:10 But just from random fluctuations, you will still get a lot of, um, um, of arithmetic progressions in that set.
0:43:17 Can you prove that there’s arithmetic progressions of arbitrary length within a random?
0:43:19 Yes, um, have you heard of the infinite monkey theorem?
0:43:23 Usually, mathematicians give boring names to theorists, but occasionally they, they give colorful names.
0:43:32 Yes, the popular version of the infinite monkey theorem is that if you have an infinite number of monkeys in a room with each of a typewriter, they type out, uh, text randomly.
0:43:37 Almost surely one of them is going to generate the entire script of Hamlet or any other finite string of text.
0:43:40 Uh, it will just take some time, quite a lot of time, actually.
0:43:42 But if you have an infinite number, then it happens.
0:43:53 Um, so, um, basically the theorem says that if you take an infinite string of, of digits or whatever, um, eventually any finite pattern you wish will emerge.
0:43:56 Uh, it may take a long time, but it will eventually happen.
0:43:59 Um, in particular, the arithmetic progressions of any length will eventually happen.
0:44:03 Okay, but you need that, you, but you need an extremely long random sequence for this to happen.
0:44:05 I suppose that’s intuitive.
0:44:07 It’s just infinity.
0:44:08 Yeah.
0:44:10 Infinity absorbs a lot of sins.
0:44:11 Yeah.
0:44:13 How are we humans supposed to deal with infinity?
0:44:26 Well, you can think of infinity as, as, as an abstraction of, um, a finite number for which you, you do not have a bound for, um, that, uh, you know, I mean, so nothing in real life is truly infinite.
0:44:35 Um, but, you know, you can, um, you know, you can ask yourself questions like, you know, what if I had as much money as I wanted, you know, or what if I could go as fast as I wanted?
0:44:45 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.
0:44:49 Um, and often the, the mathematics becomes a lot cleaner when you do that.
0:45:01 I mean, in physics, we, we joke about, uh, assuming spherical cows, um, you know, like real world problems have got all kinds of real world effects, but you can idealize, send certain things to infinity, send certain things to zero.
0:45:06 Um, and, um, and the mathematics becomes a lot simpler to work with there.
0:45:16 I wonder how often using infinity, uh, forces us to deviate from, um, the physics of reality.
0:45:16 Yeah.
0:45:18 So there’s a lot of pitfalls.
0:45:30 Um, so, you know, we, we spend a lot of time, you know, undergraduate math classes, teaching analysis, um, and analysis is often about how to take limits and, and, and, and whether, you know, so for example, a plus B is always B plus A.
0:45:34 Um, so when, when you have a finite number of terms and you add them, you can swap them and there’s no, there’s no problem.
0:45:43 But when you have an infinite number of terms, they’re these sort of show games you can play where you can have a series which converges to one value, but you rearrange it and it suddenly converges to another value.
0:45:45 And so you can make mistakes.
0:45:55 You have to know what you’re doing when you allow infinity, um, you have to introduce these epsilons and deltas and, and, and there’s, there’s a certain type of way of reasoning that helps you avoid mistakes.
0:46:06 Um, in more recent years, um, people have started taking results that are true in, in infinite limits and what’s called, and what’s called, and what’s called finalizing them.
0:46:11 Um, so you know that something’s true eventually, but, um, you don’t know when now give me a rate.
0:46:11 Okay.
0:46:18 So if I don’t have an infinite number of monkeys, but, but a large finite number of monkeys, how long do I have to wait for Hamlet to come out?
0:46:21 Um, and, um, that’s a more quantitative question.
0:46:28 Um, and this is something that you can, you can, um, attack by purely finite methods and you can use your finite intuition.
0:46:33 Um, and in this case, it turns out to be exponential in the length of the text that you’re trying to generate.
0:46:38 Um, so, um, and so this is why you never see the monkeys create Hamlet.
0:46:41 You can maybe see them create a four letter word, but nothing that big.
0:46:50 And so I personally find once you finiteize an infinite statement, it’s, it does become much more intuitive and it’s no longer so, so weird.
0:46:56 Um, so even if you’re working with infinity, it’s good to finiteize so that you can have some intuition.
0:46:57 Yeah.
0:47:00 The downside is that the finite groups are just much, much messier.
0:47:07 And, and, uh, yeah, so, so the infinite ones I found first, usually like decades earlier, and then later on people finalize them.
0:47:16 So since we mentioned a lot of math and a lot of physics, uh, what is the difference between mathematics and physics as disciplines, as ways of understanding of seeing the world?
0:47:19 Maybe we can throw an engineering in there.
0:47:22 You mentioned your wife is an engineer, give it new perspective on circuits.
0:47:22 Right.
0:47:28 So this different way of looking at the world, given that you’ve done mathematical physics, you, you’ve, you’ve worn all the hats.
0:47:29 Right.
0:47:33 So I think science in general is interaction between three things.
0:47:35 Um, there’s the real world.
0:47:43 Um, there’s what we observe of the real world, our observations, and then our mental models as to how we think the world works.
0:47:47 Um, so, um, we can’t directly access reality.
0:47:48 Okay.
0:47:52 Uh, all we have are the observations, which are incomplete and they, they have errors.
0:47:59 Um, and, um, there are many, many cases where we would, um, uh, we want to know, for example, what is the weather like tomorrow?
0:48:02 And we don’t yet have the observation and we’d like to, like a prediction.
0:48:08 Um, and then we have these simplified models, sometimes making unrealistic assumptions, you know, spherical cow type things.
0:48:10 Those are the mathematical models.
0:48:12 Mathematics is concerned with the models.
0:48:19 Science collects the observations and it proposes the models that might explain these observations.
0:48:24 What mathematics does is, uh, you, we stay within the model and we ask what are the consequences of that model?
0:48:32 What observations, what, what predictions would the model make of the, of future observations or past observations?
0:48:33 Does it fit observed data?
0:48:35 Um, so there’s definitely a symbiosis.
0:48:48 Um, it’s math, I guess mathematics is, is unusual among other disciplines is that we start from hypotheses, like the axioms of a model and ask what conclusions come up from that model.
0:48:54 Um, in almost any other discipline, uh, you start with the conclusions, you know, I want to do this.
0:48:57 I want to build a bridge, you know, I want to, to make money.
0:48:57 I want to do this.
0:48:58 Okay.
0:49:01 And then you, you, you find the path to get there.
0:49:07 Um, a lot, there’s, there’s a lot less sort of speculation about, you know, suppose I did this, what would happen?
0:49:14 Um, you know, planning and, and, and modeling, um, uh, speculative fiction maybe is one other place.
0:49:16 Uh, but, uh, that’s about it actually.
0:49:20 Most of the things we do in life is conclusions driven, including physics and science.
0:49:22 You know, I mean, they want to know, you know, where is this asteroid going to go?
0:49:24 You know, what, what, what, what is the weather going to be tomorrow?
0:49:31 Um, but, um, but thanks also has this other direction of, of going from the, uh, the axioms.
0:49:32 What do you think?
0:49:36 There is this tension in physics between theory and experiment.
0:49:41 What do you think is the more powerful way of discovering truly novel ideas about reality?
0:49:43 Well, you need both top down and bottom up.
0:49:46 Um, yeah, it’s just a, it’s a, it’s a really interaction between all these things.
0:49:53 So over time, the observations and the theory and the modeling should both get closer to reality.
0:49:59 But initially, and it isn’t, I mean, uh, this is, um, this is, um, this is always the case, you know, they’re, they’re always far apart to begin with.
0:50:04 Um, but you need one to figure out where, where to push the other, you know?
0:50:15 So, um, if your model is predicting anomalies, um, that are not picked up by experiment, that tells experimenters where to look, you know, um, to, to, to, to, to, to find more data to refine the models.
0:50:18 Um, you know, so it, it, it goes, it goes back and forth.
0:50:23 Um, within mathematics itself, there’s, there’s also a theory and experimental component.
0:50:28 It’s just that until very recently, theory has dominated almost completely.
0:50:30 Like 99% of mathematics is theoretical mathematics.
0:50:33 And there’s a very tiny amount of experimental mathematics.
0:50:40 Um, 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.
0:50:45 And so once we had the computers, um, we began to do it a little bit.
0:50:55 Um, although even before, well, like Gauss, for example, he discovered, he conjectured the most basic theorem in, in number theory, which is called the prime number theorem, which predicts how many primes that are up to a million, up to a trillion.
0:50:57 It’s not an obvious question.
0:51:13 And basically what he did was like, he computed, uh, I mean, mostly, um, by himself, but also hired human computers, um, people whose professional job it was to do arithmetic, um, to compute the first hundred thousand primes or something and made tables and made a prediction.
0:51:16 Um, and that was an early example of experimental mathematics.
0:51:23 Um, but until very recently, it was not, um, yeah, I mean, theoretical mathematics was just much more successful.
0:51:30 I mean, because doing complicated mathematical computations is, uh, was just not, not feasible until very recently.
0:51:36 Uh, and even nowadays, you know, even though we have powerful computers, only some mathematical things can be, um, explored numerically.
0:51:38 There’s something called the combinatorial explosion.
0:51:43 If you want us to study, for example, you want to study all possible subsets of numbers one to a thousand.
0:51:45 There’s only one thousand numbers.
0:51:46 How bad could it be?
0:51:56 It turns out the number of different subsets of one to a thousand is two to the power of one thousand, which is way bigger than, than, than any computer can currently, can, can, can any computer ever, or ever, um, enumerate.
0:52:06 Um, so if you have to be, um, there are certain math problems that very quickly become just intractable to attack by direct brute force computation.
0:52:09 Uh, chess is another, um, a famous example.
0:52:14 Uh, the number of chess positions, uh, we can’t get a computer to fully explore.
0:52:28 But now we have AI, um, um, we have tools to explore this space, not with 100% guarantees of success, but with experiment, you know, so, like, um, we can empirically solve chess now.
0:52:37 Uh, for example, uh, we have, we have, uh, very, very good AIs that, that can, you know, they don’t explore every single position in the game tree, but they have found some very good approximation.
0:52:50 Um, and people are using, actually, these chess engines, uh, to make, uh, to do experimental chess, um, that, uh, they’re, they’re revisiting old chess theories about, oh, you know, when you, this type of opening, you know, this is a good, this is a good type of move, this is not.
0:52:57 And they can use these chess engines to actually, uh, refine, uh, in some cases, overturn, um, um, um, conventional wisdom about chess.
0:53:04 And I, I do hope that, uh, that mathematics will, will have a larger experimental component in the future, perhaps powered by AI.
0:53:07 We’ll, of course, talk about that, but in the case of chess.
0:53:17 And there’s a similar thing in mathematics that I don’t believe it’s providing a kind of formal explanation of the different positions.
0:53:20 It’s just saying which position is better or not, that you can intuit as a human being.
0:53:25 And then from that, we humans can construct a theory of the matter.
0:53:29 You’ve mentioned the Plato’s cave allegory.
0:53:30 Mm-hmm.
0:53:37 So, in case people don’t know, it’s where people are observing shadows of reality, not reality itself.
0:53:41 And they believe what they’re observing to be reality.
0:53:50 Is that, in some sense, what mathematicians and maybe all humans are doing, is, um, looking at shadows of reality?
0:53:54 Is it possible for us to truly access reality?
0:53:57 Well, there are these three ontological things.
0:54:02 There’s actual reality, there’s our observations, and our, our models.
0:54:07 Um, and technically they are distinct, and I think they will always be distinct.
0:54:07 Um, right.
0:54:11 But they can get closer, um, over time.
0:54:20 Um, you know, so, um, and the process of getting closer often means that you’re, you have to discard your initial intuitions.
0:54:36 Um, so, um, astronomy provides great examples, you know, like, you know, like, you know, like, you know, an initial model of the world is flat because it looks flat, you know, and, um, and that it’s, and it’s big, you know, and the rest of the universe, the skies is not, you know, like the sun, for example, looks really tiny.
0:55:05 Um, and so, you, you start off with a model which is actually really far from reality, um, but it fits kind of the observations that you have, um, you know, so, you know, so things look good, you know, but over time, as you make more and more observations, bringing it closer to, to, to reality, um, the model gets dragged along with it, you know, and so, over time, we had to realize that the Earth was round, that it spins, it goes around the solar system, solar system goes around the galaxy, and so on and so forth, and the guys about the universe, you know, it’s expanding, um, expansions, it’s self-expanding, accelerating, and in fact,
0:55:11 very recently in this year, I saw this, uh, even the acceleration of the universe itself is, uh, this evidence that is, is non-constant.
0:55:15 And, uh, the explanation behind why that is, is…
0:55:16 It’s catching up.
0:55:18 Um, it’s catching up.
0:55:21 I mean, it’s still, you know, the dark matter, dark energy, this kind of thing.
0:55:21 Yes.
0:55:42 We have, we have a model that sort of explains, that fits the data really well, it just has a few parameters that, um, uh, you have to specify, um, but, so, you know, people say, oh, that’s fudge factors, you know, with, with enough fudge factors, you can, you can explain anything, um, yeah, but, uh, the mathematical point of the model is that, um, you want to have fewer parameters in your model than data points in your observational set.
0:55:48 So, if you have a model with 10 parameters that explains 10 observations, that is a completely useless model.
0:55:49 It’s what’s called overfitted.
0:55:59 But, like, if you have a model with, you know, two parameters, and it explains a trillion observations, which is basically, uh, so, uh, yeah, the, the, the dark matter model, I think, has, like, 14 parameters.
0:56:15 And it explains petabytes of data, um, that, that, that, that the astronomers have, um, you can think of a theory, uh, like, one way to think about, um, uh, physical mathematical theory, uh, theory is, it’s, it’s, it’s a compression of, of the, of the universe, um, and a data compression.
0:56:24 So, you know, you have these petabytes of observations, you’d like to compress it to a model, which you can describe in five pages and specify a certain number of parameters.
0:56:31 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.
0:56:37 In fact, one of the great surprises of our universe and of everything in it is that it’s compressible at all.
0:56:39 That’s the unreasonable effectiveness of mathematics.
0:56:40 Yeah.
0:56:41 Einstein had a quote like that.
0:56:44 The, the most incomprehensible thing about the universe is that it is comprehensible.
0:56:45 Right.
0:56:49 And not just comprehensibly, you can do an equation like E equals mc squared.
0:56:52 There is actually some mathematical possible explanation for that.
0:56:56 Um, so there’s this phenomenon in mathematics called universality.
0:57:01 So many complex systems at the macroscale are coming out of lots of tiny interactions at the macroscale.
0:57:11 And normally because of the common form of explosion, you would think that, uh, the macroscale equations must be like infinitely exponentially more complicated than, than the, uh, the macroscale ones.
0:57:14 And they are, if you want to solve them completely exactly.
0:57:22 Like if you want to model, um, all the atoms in a box of, of air, uh, that’s like Avogadro’s number is humongous, right?
0:57:23 There’s a huge number of particles.
0:57:26 If you actually have to track each one, it would be ridiculous.
0:57:34 But certain laws emerge at the macroscopic scale that almost don’t depend on what’s going on at the macroscale or only depend on a very similar number of parameters.
0:57:44 So if you want to model a gas, um, of, you know, quintillion particles in a box, you just need to know its temperature and pressure and volume and a few parameters, like five or six.
0:57:51 And it models almost everything you need to know about these 10 to 23 or whatever particles.
0:58:05 Um, so we, we have, um, we, we don’t understand universality anywhere near as we would like mathematically, but there are much simpler toy models where we do, um, have a good understanding of why universality occurs.
0:58:14 Um, um, most basic one is, is the central limit theorem that explains why the bell curve shows up everywhere in nature, that so many things are distributed by, that was called a Gaussian distribution.
0:58:18 The famous bell curve, uh, there’s not even a meme with this curve.
0:58:22 And even the meme applies broadly, the universality to the meme.
0:58:26 Yes, you can go meta, uh, if you like, but there are many, many processes.
0:58:33 For example, you can, you can take lots and lots of independent, um, random variables and average them together, um, uh, in, in various ways.
0:58:40 You can take a simple average or more complicated average, and we can prove in various cases that, that these, these bell curves, these Gaussians emerge.
0:58:42 And it is a satisfying, satisfying explanation.
0:58:44 Um, sometimes they don’t.
0:58:51 Um, so, so if you have many different inputs and they will correlate it in some systemic way, then you can get something very far from a bell curve show up.
0:58:54 Uh, and this is also important to know when a situation fails.
0:58:58 So universality is not a 100% reliable thing to rely on.
0:59:03 Um, that, um, um, that the global financial crisis was a famous example of this.
0:59:14 Uh, people thought that, uh, um, mortgage defaults, um, um, had this sort of, um, Gaussian type behavior that, that if you, if you ask, if a population of, of, uh, you know,
0:59:22 100,000 Americans with mortgages, that’s what, what proportion of the world default in the mortgages, um, if everything was decorrelated, it would be a nice bell curve.
0:59:26 And, and, and like, you can, you can, you can, you can manage risk with options and derivatives and so forth.
0:59:29 And, um, and it is a very beautiful theory.
0:59:37 Um, but if there are systemic shocks in the economy, uh, that can push everybody to default at the same time, uh, that’s very non-Gaussing behavior.
0:59:42 Um, and, uh, this wasn’t fully accounted for in, uh, 2008.
0:59:48 Um, now I think there’s some more awareness that this is a systemic risk is actually a much bigger issue.
0:59:53 And, uh, just because the model is pretty, uh, and nice, uh, it may not match reality.
0:59:59 And so, so the mathematics of working out what models do is really important.
1:00:07 Um, but, um, also the, the science of validating when the models fit reality and when they don’t, um, I mean, that you need both.
1:00:20 Um, and, but mathematics can help because it can, for example, these central limit theorems, it tells you that if you have certain axioms, like, like, like, uh, non-correlation, that if all the inputs were not correlated to each other, um, then you have this constant behavior.
1:00:24 If things are fine, it tells you where to look for weaknesses in the model.
1:00:40 So if you have a mathematical understanding of central limit theorem and someone proposes to use these Gaussian copulas or whatever to model, um, default risk, um, if you’re mathematically, um, trained, you would say, okay, but what are the systemic correlation between all your inputs?
1:00:45 And so then, then you can ask the economists, you know, how, how, how much risk is that?
1:00:47 Um, and then you can, you can, you can go look for that.
1:00:51 So there’s always this, this, this synergy between science and mathematics.
1:00:54 A little bit on the topic of universality.
1:01:03 You’re known and celebrated for working across an incredible breadth of mathematics reminiscent of Hilbert a century ago.
1:01:11 In fact, the great Fields Medal winning mathematician, Tim Gowers has said that you are the closest thing we get to Hilbert.
1:01:15 He’s a colleague of yours.
1:01:16 Good friend.
1:01:21 But anyway, so you are known for this ability to go both deep and broad in mathematics.
1:01:30 So you’re the perfect person to ask, do you think there are threads that connect all the disparate areas of mathematics?
1:01:35 Is there a kind of deep underlying structure, uh, to all of mathematics?
1:01:49 There’s certainly a lot of connecting threads, um, and a lot of the progress of mathematics has, can be represented by taking by stories of two fields of mathematics that were previously not connected and finding connections.
1:01:53 Um, an ancient example is, um, geometry and number theory.
1:01:57 You know, so, so, so in the times of ancient Greeks, these were considered different subjects.
1:01:59 Um, I mean, mathematicians worked on both.
1:02:04 You know, you could, uh, work both on geometry most famously, but also on numbers.
1:02:08 Um, but they were not really considered related.
1:02:15 Um, I mean, a little bit like, you know, you could say that, that this length was five times this length because you could take five copies of this length and so forth.
1:02:26 But it wasn’t until Descartes who really realized that, uh, who developed, what we now call analytic geometry, that you can, you can parametrize the plane, a geometric object by, um, by two real numbers.
1:02:31 Every point can be, and so geometric problems can be turned into, into problems about numbers.
1:02:36 Um, and today this feels almost trivial.
1:02:39 Like there’s, there’s, there’s, there’s no content to list.
1:02:45 Like, of course, uh, you, you know, um, the plane is X, X, and Y, and of course that’s what we teach and it’s internalized.
1:02:51 Um, but it was an important development that these, these two fields are, uh, will unify.
1:02:55 Um, and this process has just gone on throughout mathematics over and over again.
1:03:00 Algebra and geometry were separated and now we have a spirit algebraic geometry that connects them and over and over again.
1:03:04 And that’s certainly the type of mathematics that I enjoy the most.
1:03:07 So I think there’s sort of different styles to being a mathematician.
1:03:13 I think hedgehogs and fox, a fox knows many things a little bit, but a hedgehog knows one thing very, very well.
1:03:17 Um, and in mathematics, there’s definitely both hedgehogs and foxes.
1:03:20 Um, and then there’s people who are kind of, uh, who can play both roles.
1:03:27 Um, and I think ideal collaboration between mathematicians involves very, you need some diversity.
1:03:31 Like, um, a fox working with many hedgehogs or vice versa.
1:03:35 So, yeah, but I identify mostly as a fox, uh, certainly.
1:03:48 I, I, I like, uh, arbitrage somehow, you know, like, like, um, learning how one field works, learning the tricks of that wheel and then going to another field, which people don’t think is related, but I can, I can adapt the tricks.
1:03:51 So see the connections between the fields.
1:03:52 Yeah.
1:03:55 So there are other mathematicians who are far deeper than I am.
1:03:57 Like, they’re really, they’re really hedgehogs.
1:04:04 You know, they, they, they know everything about one field and they’re much faster and, and, and more effective in that field, but I can, I can give them these extra tools.
1:04:10 I mean, you’ve said that you can be both the hedgehog and the, and the fox, depending on the context and depending on the collaboration.
1:04:17 So what can you, if it’s at all possible, speak to the difference between those two ways of thinking about a problem?
1:04:25 Say you’re encountering a new problem, you know, searching for the connections versus like very singular focus.
1:04:30 I’m much more comfortable with, with the, uh, the, uh, the fox paradigm.
1:04:35 So, um, yeah, I, I like looking for analogies, narratives.
1:04:37 Um, I, I spend a lot of time.
1:04:41 If there’s a result, I see it in one field and I like the result.
1:04:43 It’s a cool result, but I don’t like the proof.
1:04:47 Like it uses types of mathematics that I’m not super familiar with.
1:04:51 Um, I often try to reprove it myself using the tools that I favor.
1:04:53 Um, often my proof is worse.
1:05:00 Um, but, um, by the exercise of doing so, um, I can say, oh, now I can see what the other proof was trying.
1:05:07 Um, and from that, I can get some understanding of, of the tools that are used in, in that field.
1:05:13 So it’s very exploratory, very doing crazy things in crazy fields and like reinventing the wheel a lot.
1:05:13 Yeah.
1:05:23 Whereas so the hedgehog style is, uh, I think much more scholarly, you know, you, you, you very knowledge-based, you, you, you, you, you stay up to speed on like all the developments in this field.
1:05:29 You, you know, all the history, um, you have a very good understanding of, of exactly the strengths and weaknesses of, of each particular technique.
1:05:37 Um, yeah, uh, I think you, you’d rely a lot more on sort of calculation than sort of trying to find narratives.
1:05:43 Um, so yeah, I mean, I could do that too, but, uh, there are other people who are extremely good at that.
1:05:52 Let’s step back and, uh, uh, maybe look at the, the, a bit of a romanticized version of mathematics.
1:06:03 So, uh, I think you’ve said that early on in your life, uh, math was more like a puzzle solving activity when you were, uh, young.
1:06:10 When did you first encounter a problem or proof where you realized math can have a kind of elegance and beauty to it?
1:06:14 That’s a good question.
1:06:20 Um, when I came to graduate school, uh, in Princeton, um, so John Conway was there at the time.
1:06:21 He passed away a few years ago.
1:06:27 But, uh, I remember one of the very first research talks I went to was a talk by Conway on what he called extreme proof.
1:06:33 So Conway had just had this, this amazing way of thinking about all kinds of things in a, in a way that you wouldn’t normally think of.
1:06:42 So, um, he thought of proofs themselves as occupying some sort of space, you know, so, so, um, if you want to prove something, let’s say that there’s infinitely many primes.
1:06:45 Okay, there will be different proofs, but you could, you could rank them in different axes.
1:06:50 Like some proofs are elegant, some proofs are long, some proofs are, uh, um, are elementary and so forth.
1:06:52 Um, and so this is cloud.
1:06:55 So the space of all proofs itself has some sort of shape.
1:07:00 Um, and so he was interested in, in extreme points of this shape.
1:07:08 Like out of all, all these proofs, what is one of those, the shortest at the expense of everything, everything else or, or the most elementary or, or whatever.
1:07:12 Um, and so he gave some examples of well-known theorems.
1:07:16 And then he would give what he thought was, was the extreme proof, um, in these different aspects.
1:07:38 Um, um, I, I just found that really eyeopening, um, that, that, um, you know, it’s, it’s not just getting a proof for a result was interesting, but, but once you have that proof, you know, trying to, to, uh, to optimize it in various ways, um, that, that proof, um, uh, proofing itself had some craftsmanship to it.
1:07:41 Um, it’s something for my writing style.
1:07:49 Um, that, you know, like when you do your, your math assignments and as your undergraduate, your homework and so forth, you’re sort of encouraged to just write down any proof that works.
1:07:50 Okay.
1:07:50 Okay.
1:07:53 And they hand it in, they get a, get a, get a, as long as it gets a tick mark, you, you move on.
1:08:01 Um, but if you want your, your, your results to actually be influential and be read by people, um, it can’t just be correct.
1:08:09 It should also, um, be a pleasure to read, you know, um, motivated, um, be adaptable to, to generalize to other things.
1:08:12 Um, it’s the same in many other disciplines, like, like coding.
1:08:14 And there’s a, uh, there’s a lot of analogies between math and coding.
1:08:16 I like analogies if you haven’t noticed.
1:08:28 Um, but, um, you know, like you can code something spaghettical that works for a certain task and it’s quick and dirty and it works, but, uh, there’s lots of good principles for, for, um, writing a code.
1:08:32 Well, so that other people can use it, build upon it and so on and has fewer bugs and whatever.
1:08:37 Um, and there’s similar things with mathematics, so.
1:08:45 Yeah, the, first of all, there’s so many beautiful things there and Kama is one of the great minds, uh, in mathematics ever and computer science.
1:08:49 Uh, just even considering the space of proofs.
1:08:49 Yeah.
1:08:54 And saying, okay, what does this space look like and what are the extremes?
1:09:01 Uh, like you mentioned, coding is an analogy is interesting because there’s also this activity called, uh, code golf.
1:09:02 Oh yeah, yeah, yeah.
1:09:11 Which I also find beautiful and fun, uh, where people use different programming languages to try to write the shortest possible program that accomplishes a particular task.
1:09:11 Yeah.
1:09:13 And I believe there’s even competitions on this.
1:09:14 Yeah, yeah, yeah, yeah.
1:09:25 And, uh, it’s also a nice way to stress test, not just the, sort of the programs or in this case, the proofs,
1:09:31 but also the different languages, maybe that’s a different notation or whatever to use to, to accomplish a different task.
1:09:31 Yeah, you learn a lot.
1:09:42 I mean, it may seem like a frivolous exercise, but it can generate all these insights, which if you didn’t have this artificial, um, objective to, to, to pursue, you, you might not see.
1:09:47 What do you use the most beautiful or elegant equation in mathematics?
1:09:53 I mean, one of the things that people often look to in, in beauty is the simplicity.
1:10:04 So if you look at E equals MC squared, so when, when a few concepts come together, that’s why the Euler identity is often considered, uh, the most beautiful equation in mathematics.
1:10:08 Do you, do you find beauty in that one, in the Euler identity?
1:10:08 Yeah.
1:10:16 Well, as I said, I mean, what I find most appealing is, is connections between different things that you, um, so the, if you, uh, if the pi i equals minus one.
1:10:19 Um, so yeah, people are, oh, these are all the fundamental constants.
1:10:19 Okay.
1:10:21 That, that, that’s, I mean, that’s cute.
1:10:37 Um, but, but to me, so the exponential function was, or to measure exponential growth, you know, so the compound interest or decay, you know, anything which is continuously growing, continuously decreasing growth and decay or dilation or contraction is modeled by the exponential function.
1:10:42 Um, whereas pi, uh, comes around from circles and rotation, right?
1:10:45 If you want to rotate a needle, for example, a hundred degrees, uh, you need to rotate by pi radians.
1:10:53 And i, complex numbers, represents the swapping between you and imaginary axes, so a 90 degree rotation, so a change in direction.
1:10:58 So the exponential function represents growth and decay in the direction that you already are.
1:11:09 Um, when you stick an i in the exponential, now it’s, it’s, instead of motion in the same direction as your current position, it’s motion as a right angle as your current position, so rotation.
1:11:17 Um, and then, so, e to pi equals minus one tells you that if you rotate for a time pi, you end up at the other direction.
1:11:25 So it unifies geometry through dilation and exponential growth, or dynamics, through this act of, of complexification, rotation by, by, by, by i.
1:11:28 So it, it, it connects together all these tools, mathematics, you know, yeah.
1:11:36 That thing was geometry and complex and complex and, um, the complex numbers, they’re all considered almost, yeah, they’re all next door neighbors in mathematics because of this identity.
1:11:46 Do you, do you think the thing you mentioned is cute, the, the, the, the collision of notations from these disparate fields, um, is just a frivolous side effect?
1:11:53 Or do you think there is legitimate, like, value in one notation, all the, our old friends come together in the night?
1:11:56 Right, well, it’s, it’s, it’s confirmation that you have the right concepts.
1:12:11 Um, so, when you first study anything, um, you, you have to measure things and give them names, um, and initially, sometimes, you’re, because your, your model is, again, too far off from reality, you give the wrong things the best names.
1:12:14 And you only find out later what’s, what’s really important.
1:12:15 Physicists can do this sometimes.
1:12:17 I mean, but it turns out, okay.
1:12:22 So, actually, with physics, so, e equals mc squared, okay, so, uh, one of the, the big things was the e, right?
1:12:31 So, when, when Aristotle first came up with his laws of, of motion and then, and then, um, Galileo and Newton and so forth, you know, they saw the things they could, they could measure.
1:12:34 They could measure mass and acceleration and force and so forth.
1:12:39 And so, Newtonian mechanics, for example, you know, I think it was MA was the famous, uh, Newton’s second law of motion.
1:12:41 So, those were the, the primary objects.
1:12:43 So, they gave them the central billing in the theory.
1:12:50 It was only later, after people started analyzing these equations, that there always seemed to be these quantities that were conserved.
1:12:52 Um, so, in particular, momentum and energy.
1:12:57 Um, uh, and it’s not obvious that things happen in energy.
1:13:01 Like, it’s not something you can directly measure the same way you can measure mass and, and, and velocity and so forth.
1:13:04 But over time, people realized that this was actually a really fundamental concept.
1:13:14 Hamilton, eventually, in the 19th century, reformulated Newton’s laws of physics into what’s called Hamiltonian mechanics, where the energy, which is now called the Hamiltonian, was the dominant object.
1:13:21 Once you know how to measure the Hamiltonian of any system, you can describe completely the dynamics, like what happens to all the states.
1:13:25 Like, it’s, um, it, it really was a central actor, which was not obvious initially.
1:13:33 Um, and this, uh, helped actually, uh, this change of perspective really helped when quantum mechanics came along.
1:13:46 Uh, because, um, the early physicists who studied quantum mechanics, they had a lot of trouble trying to adapt their Newtonian thinking, because, you know, everything was a particle and so forth to, to, to quantum mechanics.
1:13:55 Um, and, um, and, um, but, um, but again, once you specify it, you specify the entire dynamics, you know, and it’s really, really hard to, to give an answer to that.
1:14:09 Um, but it turns out that the Hamiltonian, which was so, um, secretly behind the scenes in classical mechanics, also is the key, uh, object in, um, um, um, in quantum mechanics, that there’s, there’s also an object called Hamiltonian.
1:14:10 It’s a different type of object.
1:14:12 It’s what’s called an operator rather than, than a function.
1:14:16 But, um, and, um, but again, once you specify it, you specify the entire dynamics.
1:14:22 So, there’s something called Schrodinger’s equation that tells you exactly how quantum systems evolve once you have the Hamiltonian.
1:14:29 So, side by side, they look completely different objects, you know, like, so one involves particles, one involves waves, and so forth.
1:14:35 But with this centrality, you could start actually transferring a lot of intuition and facts from classical mechanics to quantum mechanics.
1:14:39 So, for example, in classical mechanics, there’s this thing called Noether’s theorem.
1:14:43 Every time there’s a symmetry in a physical system, there is a conservation law.
1:14:46 So, the laws of physics are translation invariant.
1:14:52 Like, if I move 10 steps to the left, I experience the same laws of physics as if I was here, and that corresponds to conservation momentum.
1:14:57 Um, if I turn around by, by some angle, again, I experience the same laws of physics.
1:14:59 Uh, this corresponds to the conservation angle of momentum.
1:15:03 If I wait for 10 minutes, um, I still have the same laws of physics.
1:15:04 Um, so there’s time transition invariance.
1:15:06 This corresponds to the law of conservation of energy.
1:15:11 Um, so there’s this fundamental connection between symmetry and conservation.
1:15:16 Um, and that’s also true in quantum mechanics, even though the equations are completely different.
1:15:19 But because they’re both coming from the Hamiltonian, the Hamiltonian controls everything.
1:15:23 Um, every time the Hamiltonian has a symmetry, the equations will have a conservation law.
1:15:31 Um, so it’s, it’s, it’s, it’s, it’s, it’s, once you have the right language, it actually makes things, um, a lot, a lot cleaner.
1:15:37 One of the points why we can’t unify quantum mechanics and general relativity yet, we haven’t figured out what the fundamental objects are.
1:15:42 Like, for example, we have to give up the notion of space and time being these almost Euclidean type spaces.
1:15:49 And it has to be, um, you know, and, you know, we kind of know that at very tiny scales, uh, um, there’s going to be quantum fluctuations.
1:15:56 There’s a space, space, time foam, um, and trying to, to use Cartesian coordinates X, Y, Z is going to be, it’s, it’s, it’s, it’s, it’s a non-starter.
1:16:05 But we don’t know how to, what to replace it with, um, we don’t actually have the mathematical, um, um, concepts.
1:16:08 The analog or Hamiltonian that sort of organized everything.
1:16:18 Does your gut say that there is a theory of everything, so this is even possible to unify, to find this language that unifies general relativity and quantum mechanics?
1:16:19 I believe so.
1:16:24 I mean, the history of physics has been that of unification, much like mathematics, um, over the years.
1:16:28 You know, electricity and magnetism were separate theories, and then Maxwell unified them.
1:16:33 You know, Newton unified the motions of the heavens for the motions of objects on the earth and so forth.
1:16:35 So it should happen.
1:16:41 It’s just that the, um, uh, again, to go back to this model of the observations and theory.
1:16:44 Part of our problem is that physics is a victim of its own success.
1:16:51 That our two big theories of, of, of physics, general relativity and quantum mechanics are so, are so good now.
1:17:10 So together, they cover 99.9% of sort of all the observations we can make, um, and you have to, like, either go to extremely insane particle accelerations or, or the early universe or, or things that are really hard to measure, um, in order to get any deviation from either of these two theories to the point where you can actually figure out how to, how to combine them together.
1:17:18 Um, but I have faith that we, you know, we’ve, we’ve, we’ve been doing this for centuries and we’ve made progress before and there’s no reason why we should stop.
1:17:23 Do you think you will be a mathematician that develops a theory of everything?
1:17:35 What often happens is that when the physicists need, uh, um, some theory of mathematics, there’s often some precursor that the mathematicians, um, worked out earlier.
1:17:45 So when Einstein started realizing that space was curved, he went to some mathematician and asked, you know, is there, is there some theory of curved space that the mathematicians already came up with that could be useful?
1:17:48 And he said, oh yeah, there’s a, I think, uh, Riemann came up with something.
1:18:00 Um, and so yeah, Riemann had developed Riemannian geometry, um, which is precisely, um, you know, a theory of spaces that are curved in, in various general ways, which turned out to be almost exactly what was needed, um, for Einstein’s theory.
1:18:03 This is going back to Wiggins’ unreasonable effectiveness on mathematics.
1:18:11 I think the theories that work well if they explain the universe tend to also involve the same mathematical objects that work well to solve mathematical problems.
1:18:14 Ultimately, they’re just sort of both ways of organizing data.
1:18:16 Um, in, in, in, in, in useful ways.
1:18:22 It just feels like you might need to go to some weird land that’s very hard to, to intuit.
1:18:24 Like, you know, you have like string theory.
1:18:27 Yeah, that, that’s, that was, that was a leading candidate for many decades.
1:18:32 I think it’s slowly pulling out of fashion because it’s, it’s not matching experiment.
1:18:37 So one of the big challenges, of course, like you said, is experiment is very tough.
1:18:37 Yes.
1:18:41 Because of the, how effective both theories are.
1:18:49 But the other is like, just, you know, you’re talking about, you’re not just deviating from space-time.
1:18:51 You’re going into like some crazy number of dimensions.
1:18:52 Yeah.
1:18:58 You’re doing all kinds of weird stuff that, to us, we’ve gone so far from this flat earth that we started at.
1:19:09 And now we’re just, it’s, it’s very hard to use our limited descendants of, uh, uh, cognition to intuit what that reality really is like.
1:19:16 This is why analogies are so important, you know, I mean, so yeah, the round earth is not intuitive because we’re stuck on it.
1:19:23 Um, but you know, but you, you, you, you know, but round objects in general, we have pretty good intuition over, uh, and we have intuition about light works and so forth.
1:19:35 And like, it’s, it’s actually a good exercise to actually work out how eclipses and phases of, of the sun and the moon and so forth that can be really easily explained by, by, by, by round earth and round moon, you know, um, and models.
1:19:42 Um, and, and you can just take, you know, a basketball and a golf ball and a, and a light source and actually do these things yourself.
1:19:46 Um, so the intuition is there, um, but yeah, you have to transfer it.
1:19:54 That is a big leap intellectual for us to go to from flat to round earth because, you know, our life is mostly lived in flat land.
1:19:55 Yeah.
1:19:56 To load that information.
1:19:57 And we’re all like, take it for granted.
1:20:03 We take so many things for granted because science has established a lot of evidence for this kind of thing.
1:20:06 But, you know, we’re in a round rock.
1:20:07 Yeah.
1:20:09 Flying through space.
1:20:09 Yeah.
1:20:10 Yeah.
1:20:11 That’s a big leap.
1:20:15 And you have to take a chain of those leaps the more and more and more we progress.
1:20:15 Right.
1:20:15 Yeah.
1:20:24 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.
1:20:30 And so, um, for someone who hasn’t gone through the whole process of science education, it looks more and more suspicious because of that.
1:20:33 So, you know, we, we, we need, we need more grounding.
1:20:41 I mean, I think, um, I mean, you know, there are, there are scientists who do excellent outreach, um, but there’s, there’s, there’s, there’s, there’s, there’s, there’s lots of science things that you can do at home.
1:20:42 I mean, there’s lots of YouTube videos.
1:20:50 I did a YouTube video recently with Grant Sanderson that we talked about earlier that, uh, you know, how the ancient Greeks were able to measure things like the distance to the moon, distance to the earth.
1:20:54 And, you know, using techniques that you could, you could also replicate yourself.
1:21:00 Um, it doesn’t all have to be like fancy space telescopes and, and very intimidating mathematics.
1:21:01 Yeah.
1:21:02 That’s, uh, I highly recommend that.
1:21:06 I believe you give a lecture and you also did an incredible video with Grant.
1:21:14 It’s a beautiful experience to try to put yourself in the mind of a person from that time shrouded in mystery.
1:21:14 Right.
1:21:19 You know, you’re like on this planet, you don’t know the shape of it, the size of it.
1:21:24 You see some stars, you see some, you see some things and you try to like localize yourself in this world.
1:21:25 Yeah.
1:21:25 Yeah.
1:21:28 And try to make some kind of general statements about distance to places.
1:21:30 Change of perspective is really important.
1:21:31 You say travel burdens the mind.
1:21:36 This is intellectual travel, you know, put yourself in the mind of the ancient Greeks or, or some other.
1:21:42 Put some, some other time period, make hypotheses, spherical cows, whatever, you know, speculate.
1:21:47 Um, and you know, this is, this is what mathematicians do and some artists do actually.
1:21:52 It’s just incredible that given the extreme constraints, you could still say very powerful things.
1:21:54 That’s why it’s inspiring.
1:22:01 Looking back in history, how much can be figured out when you don’t have much to figure out stuff with.
1:22:05 If you propose axioms, then the mathematics lets you follow those axioms to their conclusions.
1:22:09 And sometimes you can get quite a lot, quite a long way from, you know, initial hypotheses.
1:22:12 If we stay in the land of the weird, you mentioned general relativity.
1:22:18 You’ve, uh, you’ve contributed, uh, to the mathematical understanding of Einstein’s field equations.
1:22:19 Can you explain this work?
1:22:30 And, uh, from a sort of mathematical standpoint, uh, what, what aspects of general relativity are intriguing to you, challenging to you?
1:22:32 I have worked on some equations.
1:22:44 There’s something called the, the wave maps equation, or the sigma field model, which is not quite the equation of space-time gravity itself, but of certain fields that might exist on top of space-time.
1:22:51 Um, so, uh, Einstein’s equations of relativity just describes space and time itself, um, but then there’s other fields that live on top of that.
1:23:01 Uh, there’s the electromagnetic field, um, there’s, uh, things called Yang-Mills fields, and there’s this whole hierarchy of different equations, of which Einstein is considered one of the most nonlinear and difficult.
1:23:05 But, uh, relatively low on the hierarchy was this thing called the wave maps equation.
1:23:10 So, it’s a wave which, at any given point, uh, is fixed to be, like, on a sphere.
1:23:17 Um, so, uh, I can think of a bunch of arrows in space and time, and, and, and, and, and, yeah, so it’s pointing in, in different directions.
1:23:19 Um, but they propagate like waves.
1:23:26 If, if, if you wiggle an arrow, it will propagate and create and make all the arrows move, kind of like, uh, sheaves of wheat in the wheat field.
1:23:34 And I was interested in the global regularity problem, again, for this question, like, is it possible for, for all the energy here to, to, to, to collect at a point?
1:23:40 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.
1:23:48 Um, and I was able barely to show that, um, that you couldn’t actually force a scenario where all the energy concentrated at one point.
1:23:53 That the energy had to disperse a little bit, and the moment it disperse a little bit, it would, it would, it would, it would stay regular.
1:23:55 Yeah, this was back in 2000.
1:23:58 That was part of why I got interested in Larry Stokes afterwards, actually.
1:24:02 Yeah, so, I developed some techniques to, um, to solve that problem.
1:24:07 So, part of it is, it was, um, this problem is really non-linear, uh, because of the curvature of the sphere.
1:24:12 Um, there’s, there was a certain non-linear effect, which was a non-perturbative effect.
1:24:17 It was, when you sort of looked at it normally, it looked larger than the linear effects of the wave equation.
1:24:22 Um, and so, it was hard to, to keep things under control, even when the energy was small.
1:24:24 But I developed what’s called a gauge transformation.
1:24:30 So, the equation is kind of like an evolution of, of, of sheaves of wheat, and they’re all bending back and forth.
1:24:32 And so, there’s a lot of motion.
1:24:41 Um, but like, if you imagine, like, stabilizing the flow by attaching little cameras at different points in space, which are trying to move in a way that captures most of the motion.
1:24:45 Um, and under this sort of stabilized flow, the flow becomes a lot more linear.
1:24:52 I discovered a way to transform the, the equation to reduce the amount of non-linear effects.
1:24:55 Um, and then I was able to, to, to, to solve the equation.
1:25:03 I found this transformation while visiting my aunt in Australia, and I was trying to understand the dynamics of all these fields, and I, I couldn’t do it with pen and paper.
1:25:07 Um, and I had not enough facility for computers to do any computer simulations.
1:25:21 So, I ended up closing my eyes, being on, on the floor, and just imagining myself to actually be the specter field, and rolling around to try to, to see how to change coordinates in such a way that somehow things in all directions would behave in a reasonably linear fashion.
1:25:27 And, uh, yeah, my aunt walked in on me while I was doing that, and she was asking, what am I, what am I doing, doing this?
1:25:29 It’s complicated, is the answer.
1:25:32 Yeah, yeah, and, you know, she said, okay, fine, you know, you’re a young man, I don’t ask questions.
1:25:39 I, I, I have to ask about the, you know, um, how do you approach solving difficult problems?
1:25:51 What, if it’s possible to go inside your mind when you’re thinking, are you visualizing in your mind the mathematical objects?
1:25:52 Symbols, maybe?
1:25:56 What are you visualizing in your mind usually when you’re thinking?
1:25:57 Um, a lot of pen and paper.
1:26:02 One thing you pick up as a mathematician is sort of, uh, I call it cheating strategically.
1:26:10 Um, so, uh, the, the beauty of mathematics is that, is that you get to change the rule, change the problem, change the rules as you wish.
1:26:13 Uh, like this, you don’t get to do this for any other field.
1:26:20 Like, you know, if, if you’re an engineer and someone says, build a bridge over this river, you can’t say, I want to build this bridge over here instead, or I want to build it out of paper instead of steel.
1:26:23 Um, but, um, you can, you can, you can do whatever you want.
1:26:31 Um, it’s, it’s like trying to solve a computer game where you can, there’s unlimited cheat codes available.
1:26:37 Uh, and so, you know, you, you, you can, you can set this, there’s a dimension that’s large.
1:26:38 I’ll set it to one.
1:26:39 I’d solve the one-dimensional problem first.
1:26:41 There’s a main term and an error term.
1:26:43 I’m going to make a spherical car assumption.
1:26:44 I’ll assume the error term is zero.
1:26:49 And so the way you should solve these problems is, is not in sort of this Ironman mode where
1:26:50 you make things maximally difficult.
1:26:56 Um, but actually the way you should, you should approach any reasonable math problem is that
1:27:00 you, if there are 10 things that are making life difficult, find a version of the problem
1:27:02 that turns off nine of the difficulties, but only keeps one of them.
1:27:08 Um, and so that, um, and then that just figured, so you, you, you, you install nine cheats.
1:27:09 Okay.
1:27:12 If you saw 10 cheats, then, then the game is trivial, but you saw nine cheats, you solve
1:27:15 one problem that, that, that, that, that, that teaches you how to deal with that particular
1:27:16 difficulty.
1:27:19 And then you turn that one off and you turn someone else, something else on, and then you
1:27:20 solve that one.
1:27:24 And after you, you know how to solve the 10 problems, 10 difficulties separately, then you
1:27:26 have to start merging them a few at a time.
1:27:32 Um, I, I, as a kid, I watched a lot of these Hong Kong action movies, um, from my
1:27:33 culture.
1:27:37 Um, and, uh, one thing is that every time it’s a fight scene, you know, so maybe the
1:27:43 hero gets swarmed by a hundred bad guy goons or whatever, but it will always be choreographed
1:27:46 so that you’d always be only fighting one person at a time and then it would defeat that person
1:27:47 and move on.
1:27:50 And, and because of that, they could, they could defeat all of them.
1:27:50 Right.
1:27:55 But whereas if they had fought a bit more intelligently and just swarmed the guy at once, uh, it would
1:28:00 make for much, uh, much worse choreo, uh, cinema, but, uh, but they would win.
1:28:04 Are you usually a pen and paper?
1:28:07 Are you working, uh, with computer and LaTeX?
1:28:09 I’m mostly pen and paper actually.
1:28:11 So in my office, I have four giant blackboards.
1:28:16 Um, and sometimes I just have to write everything I know about the problem on the four blackboards
1:28:19 and then sit my couch and just sort of see the whole thing.
1:28:23 Is it all symbols like notation or is there some drawings?
1:28:27 Oh, there’s a lot of drawing and a lot of bespoke doodles that, uh, only makes sense to
1:28:27 me.
1:28:32 Um, I mean, and, and, and that’s the beauty of blackboards you raise and it’s, it’s, it’s,
1:28:33 it’s very organic thing.
1:28:38 Um, I’m beginning to use more and more computers, um, partly because AI makes it much easier
1:28:43 to do simple coding things that, you know, if I wanted to plot a function before, which
1:28:46 is moderately complicated as an iteration or something, you know, I’d have to, to remember
1:28:50 how to set up a Python program and, and, and, and, and, and how does a for loop work and,
1:28:53 and, and debug it and it would take two hours and so forth.
1:28:58 And, and now I can do it in 10, 15 minutes as much, um, yeah, I’m, I’m using more and
1:29:00 more, uh, computers to do simple explorations.
1:29:03 Let’s talk about AI a little bit if we could.
1:29:09 So, um, maybe a good entry point is just talking about computer assisted proofs in general.
1:29:18 Can you describe the lean formal proof programming language and how it can help as a proof assistant
1:29:24 and maybe how you started using it and how, uh, it has helped you.
1:29:31 So, um, lean is a computer language, um, much like sort of standard languages like Python and C and so forth.
1:29:36 Except that in most languages, the focus is on producing executable code.
1:29:41 Lines of code do things, you know, they, they flip bits or they make a robot move or, or they, they deliver
1:29:43 your text on the internet or something.
1:29:46 Um, so lean is a language that can also do that.
1:29:51 Uh, it can also be run as a standard, uh, traditional language, but it can also produce
1:29:52 certificates.
1:29:56 So a software language like Python might do a computation and give you the answer is seven.
1:29:56 Okay.
1:30:01 Then it does the sum of three plus four is equal to seven, but, uh, lean can produce not just
1:30:05 the answer, but, but a proof that, uh, how it got the, the answer of seven as three plus
1:30:11 four, uh, and all the steps involved in, in, in, in, in, um, so it’s, so it creates these
1:30:14 more complicated objects, not just statements, but statements with proofs attached to them.
1:30:20 Um, and, um, every line of code is just a way of piecing together previous statements
1:30:21 to, to create new ones.
1:30:23 So the idea is not new.
1:30:24 These things are called proof assistance.
1:30:29 And so they provide languages for which you can create quite complicated, um, intricate,
1:30:30 um, mathematical proofs.
1:30:37 And, um, they produce these certificates that give a 100% guarantee that your arguments are
1:30:37 correct.
1:30:42 If you trust the compiler of lean, but they made the compiler really small and you can,
1:30:43 there are several different compilers available for the same.
1:30:49 Can you give people some intuition about the, the difference between writing on pen and paper
1:30:52 versus using lean programming language?
1:30:55 How hard is it to formalize a statement?
1:30:59 So lean, a lot of mathematicians were involved in the design of lean.
1:31:05 So it’s, it’s designed so that, um, individual lines of code resemble individual lines of
1:31:05 mathematical argument.
1:31:07 Like you might want to introduce a variable.
1:31:08 You want to, want to prove that contradiction.
1:31:13 You want your, um, um, there are various standard things that you can do and, and it’s, it’s
1:31:13 written.
1:31:17 So ideally it should like a one-to-one correspondence in practice.
1:31:22 It isn’t because lean is like explaining a proof to extremely pedantic colleague who will,
1:31:24 will point out, okay, did you really mean this?
1:31:26 Like what, what happens if this is zero?
1:31:26 Okay.
1:31:28 Um, did you, how do you justify this?
1:31:34 Um, so lean has a lot of automation in it, um, to try to, to, uh, to be less annoying.
1:31:38 Um, so for example, um, every mathematical object has to come of a type.
1:31:45 Like if I, if I talk about X, is X a real number or, um, a natural number or, or a function
1:31:45 or something?
1:31:50 Um, if you write things informally, um, it’s often in some context.
1:31:56 You say, you know, um, clearly X is equal to, uh, let X be the sum of Y and Z and Y and
1:31:57 Z were already real numbers.
1:31:58 So X should also be a real number.
1:32:00 Um, so lean can do a lot of that.
1:32:05 Um, but every so often it says, wait a minute, uh, can you tell me more about what this object
1:32:07 is, uh, what, what type of object it is?
1:32:12 You have to think more, um, at a philosophical level, well, not just sort of computations
1:32:16 that you’re doing, but sort of what each object actually, um, is in some sense.
1:32:22 Is he using something like LLMs to do, uh, the type inference or like you mentioned with
1:32:22 a real number?
1:32:26 It’s, it’s using much more traditional, what’s called good old fashioned AI.
1:32:29 You can represent all these things as trees and there’s always algorithm to match one tree
1:32:30 to another tree.
1:32:36 So it’s actually doable to figure out if something is, uh, a real number or a natural number.
1:32:39 Every object sort of comes with a history of where it came from and you can, you can kind
1:32:39 of trace.
1:32:40 Oh, I see.
1:32:41 Um, yeah.
1:32:43 So it’s, it’s, it’s designed for reliability.
1:32:47 So, uh, modern AIs are not used in, it’s a disjoint technology.
1:32:50 People are beginning to use AIs on top of lean.
1:32:55 So when a mathematician tries to program, um, improvement in lean, um, often there’s
1:32:59 a step, okay, now I want to use, um, the fundamental calculus, say, okay, to do the next step.
1:33:05 So the lean developers have built this, this massive project called Metholib, a collection
1:33:08 of tens of thousands of useful facts about methodical objects.
1:33:12 And somewhere in there is the fundamental calculus, but you need to find it.
1:33:15 So a lot, the bottleneck now is actually lemma search.
1:33:20 You know, there’s a tool that, that you know is in there somewhere and you need to find
1:33:20 it.
1:33:24 Um, and so you can, there are various search engines specialized for Metholib that you can
1:33:24 do.
1:33:28 Um, but there’s now these large language models that you can say, okay, um, I need the fundamental
1:33:29 calculus at this point.
1:33:34 And it was like, okay, um, uh, for example, um, when I code, I have GitHub Copilot installed
1:33:39 as a plugin to my IDE and it scans my text and it sees what I need.
1:33:41 It says, you know, I might even type it.
1:33:43 Now I need to use the fundamental calculus.
1:33:43 Okay.
1:33:45 And then it might suggest, okay, try this.
1:33:48 And like maybe 25% of the time it works exactly.
1:33:49 And then another.
1:33:53 10, 50% of the time it doesn’t quite work, but it’s close enough that I can say, oh yeah,
1:33:55 if I just change it here and here, it will work.
1:33:57 And then like half the time it gives me complete rubbish.
1:34:02 Um, so, but people are beginning to use AIs a little bit on top.
1:34:09 Um, mostly on the level of basically fancy autocomplete, um, that, uh, you can type half of one line
1:34:10 of a proof and it will find, it will tell you.
1:34:10 Yeah.
1:34:16 But, but, but a fancy, especially fancy with the sort of capital letter F is, uh, uh, remove
1:34:17 some of the friction.
1:34:18 Yeah.
1:34:22 What a mathematician might feel when they move from pen and paper to formalizing.
1:34:23 Yes.
1:34:23 Yeah.
1:34:27 So right now I estimate that the effort, time and effort taken to formalize a proof is about
1:34:29 10 times the amount taken to, to write it out.
1:34:30 Yeah.
1:34:35 So it’s doable, but, uh, you don’t, it’s, it’s annoying.
1:34:38 But doesn’t it like kill the whole vibe of being a mathematician?
1:34:39 Yeah.
1:34:42 So, I mean, having a pedantic coworker, right?
1:34:42 Yeah.
1:34:44 If that was the only aspect of it.
1:34:44 Okay.
1:34:46 But, um, okay.
1:34:49 So there’s some, there’s some cases where it’s actually more pleasant to do things formally.
1:34:54 So there was a theorem I formalized and there was a certain constant 12, um, that, that
1:34:56 came out at, um, in, in the final statement.
1:34:57 And so this 12 had to be carried all through the proof.
1:35:00 Um, and like everything had to be checked.
1:35:03 I did go through all the, all these other numbers that had to be consistent with this
1:35:04 final number 12.
1:35:07 And then, so we wrote a paper through this theorem with this number 12.
1:35:10 And then a few weeks later, someone said, oh, we can actually improve this 12 to an 11
1:35:12 by reworking some of these steps.
1:35:16 And when this happens with pen and paper, um, like every time you change a parameter, you
1:35:20 have to check line by line that every single line of your proof still works.
1:35:23 And there can be subtle things that you didn’t quite realize some properties on number
1:35:25 12 that you didn’t even realize that you were taking advantage of.
1:35:27 So a proof can break down at a subtle place.
1:35:31 Um, so we had formalized the proof with this constant 12.
1:35:35 And then when this, this new paper came out, uh, we said, oh, let’s, uh, so that took like
1:35:39 three weeks to formalize, uh, and like 20 people to formalize this, this, this original proof.
1:35:44 I said, oh, but now let’s, let’s, um, uh, uh, let’s update the 12 to 11.
1:35:49 And what you can do with lean, so you just, in your headline theorem, you change the 12 to 11,
1:35:54 you run the compiler and like of the thousands of lines of code you have, 90% of them still
1:35:55 work.
1:35:57 And there’s a couple that are lined in red.
1:36:01 Now I can’t justify these steps, but it immediately isolates which steps you need to change, but you
1:36:03 can skip over everything, which, which works just fine.
1:36:09 Um, and if you program things correctly, um, with good programming practices, most of your
1:36:09 lines will not be read.
1:36:14 Um, and there’ll just be a few places where you, I mean, if you don’t hard code your constants,
1:36:18 but you sort of, uh, um, um, you use smart tactics and so forth.
1:36:22 Uh, you can, you can, you can localize, um, the things you need to change to, to a very
1:36:24 small, um, period of time.
1:36:28 So it’s like within a day or two, we had updated our proof because this is very quick process.
1:36:30 You, um, you make a change.
1:36:33 There are 10 things now that don’t work for each one.
1:36:36 You make a change and now there’s five more things that don’t work, but, but the process
1:36:39 converges much more smoothly than with pen and paper.
1:36:40 So that’s for writing.
1:36:41 Are you able to read it?
1:36:46 Like if somebody else has a proof, are you able to like, how, what’s, what’s the, uh, versus
1:36:47 paper?
1:36:47 Yeah.
1:36:52 So the proofs are longer, but each individual piece is easier to read.
1:36:58 So, um, if you take a math paper and you jump to page 27 and you look at paragraph six and
1:37:04 you have a line of, of, of text or math, I often can’t read it immediately because it assumes
1:37:08 various definitions, which I had to go back and, and maybe on 10 pages earlier, this was
1:37:12 defined and this, um, the proof is scattered all over the place and you basically are forced
1:37:13 to read fairly sequentially.
1:37:19 Um, it’s, it’s not like say a novel where like, you know, in theory, you could open up
1:37:20 a novel halfway through and start reading.
1:37:24 There’s a lot of context, but when a proof and lean, if you put your cursor on a line of
1:37:29 code, every single object there, you can hover over it and it would say what it is, where it
1:37:30 came from, where the stuff is justified.
1:37:34 You can trace things back much easier than sort of flipping through a math paper.
1:37:39 So one thing that lean really enables is actually collaborating on proofs at a really atomic
1:37:41 scale that you really couldn’t do in the past.
1:37:45 So traditionally a pen and paper, um, when you want to collaborate with another mathematician,
1:37:50 um, either you do it at a blackboard where you, um, you can really interact, but if you’re
1:37:54 doing it sort of by email or something, um, basically, yeah, you have to segment it.
1:37:58 So I’m going to, I’m going to finish section three, you do section four, but, uh, you can’t
1:38:02 really sort of work on the same thing, uh, collaborative at the same time.
1:38:06 But with lean, you can be trying to formalize some portion of the proof and say, oh, I got
1:38:07 stuck at line 67 here.
1:38:10 I need to prove this thing, but it doesn’t quite work.
1:38:12 Here’s the, like the three lines of code I’m having trouble with.
1:38:16 Um, but because all the context is there, someone else can say, oh, okay, I recognize what you
1:38:17 need to do.
1:38:22 You need to apply this trick or this tool and you can do extremely atomic level conversations.
1:38:27 So because of lean, I can collaborate, you know, with dozens of people across the world.
1:38:29 Most of them I don’t, have never met in person.
1:38:34 Um, and I may not know actually even whether they’re, um, how reliable they are in, in,
1:38:38 in their, um, um, in, in the proof they give me, but lean gives me a certificate of, of,
1:38:39 of trust.
1:38:42 Um, so I can do, I can do trustless mathematics.
1:38:44 So there’s so many interesting questions.
1:38:49 There’s one, you’re, you’re known for being a great collaborator.
1:38:56 So what is the right way to approach solving a difficult problem in mathematics when you’re
1:38:57 collaborating?
1:39:02 Are you doing a divide and conquer type of thing or are you brains, are you focused on
1:39:05 a particular part and you’re brainstorming?
1:39:07 There’s always a brainstorming process first.
1:39:07 Yeah.
1:39:12 So math research projects sort of by their nature, when you start, you don’t really know how to
1:39:13 do the problem.
1:39:17 Um, it’s not like an engineering project where somehow the theory has been established for
1:39:20 decades and it’s implementation is the main difficulty.
1:39:22 You have to figure out even what is the right path.
1:39:27 So, so this is what I said about, about cheating first, you know, um, it’s like, um, to go
1:39:31 back to the bridge building analogy, you know, so first assume you have infinite budget and
1:39:34 like unlimited amounts of, of, of, of workforce and so forth.
1:39:35 Now can you, can you build this bridge?
1:39:35 Okay.
1:39:36 Okay.
1:39:38 Now have infinite budget, but only finite workforce.
1:39:39 All right.
1:39:39 Now can you do that?
1:39:40 And so what?
1:39:45 Um, so, uh, I mean, of course, no, no engineer can actually do this.
1:39:47 Like I said, you have fixed requirements.
1:39:47 Yes.
1:39:52 There’s this sort of jam sessions or at the beginning where you try all kinds of crazy things and
1:39:55 you, you, you make all these assumptions that are unrealistic, but you plan to fix later.
1:40:01 Um, and you try to see if there’s even some skeleton of an approach that might work.
1:40:06 Um, and then hopefully that breaks up the problem into smaller sub problems, which you don’t know
1:40:09 how to do, but then you, uh, you focus on, on the sub ones.
1:40:13 And sometimes different collaborators are better at, at working on, on certain things.
1:40:18 Um, so one of my theorems I’m known for is a theorem of Ben Green, which is now called
1:40:19 the Green Tau theorem.
1:40:23 Um, it’s a statement that the primes contain algorithmic progressions of any length.
1:40:25 So it was a modification of this theorem already.
1:40:30 And the way we collaborated was that Ben had already proven a similar result for progressions
1:40:31 of length three.
1:40:35 Um, he showed that sets like the primes contain lots and lots of progressions of length three.
1:40:39 Um, even, and even, um, subsets of the prime, certain subsets do.
1:40:43 Um, but his techniques only worked for, um, for length three progressions.
1:40:44 They didn’t work for longer progressions.
1:40:49 Um, but I had these techniques coming from a gothic theory, which is something that I had
1:40:52 been playing with and, and, uh, I knew better than Ben at the time.
1:40:58 Um, and so, um, if I could justify certain randomness properties of some set relating to
1:41:03 the primes, like there’s, there’s a certain technical condition, which if I could have it,
1:41:06 if Ben could supply me this fact, I could give, I could conclude the theorem.
1:41:12 But I, what I asked was a really difficult question in number theory, which, um, he said,
1:41:13 no, there’s no way we can prove this.
1:41:17 Can you, so he said, can you prove your part of the theorem using a weaker hypothesis that
1:41:18 I have a chance to prove it?
1:41:21 And he proposed something which he could prove, but it was too weak for me.
1:41:23 Uh, I can’t use this.
1:41:28 Um, so there’s this, there was this conversation going back and forth, um, it’s sort of a
1:41:29 different cheats too.
1:41:29 Yeah.
1:41:30 Yeah.
1:41:31 I want to cheat more.
1:41:31 He wants to cheat less.
1:41:32 Yeah.
1:41:37 Uh, but eventually we found a, a, a, a, a, a property, which a, he could prove in B I could
1:41:37 use.
1:41:39 Um, and then we, we could prove our view.
1:41:44 Uh, and, um, yeah, so there’s, there’s, there’s a, there are all kinds of dynamics, you
1:41:44 know?
1:41:49 I mean, it’s, it’s, it’s every, every, um, collaboration has a, has a, has some story.
1:41:50 It’s no two of the same.
1:41:55 And then on the flip side of that, like you mentioned with lean programming, now that’s
1:42:00 almost like a different story because you can do, you can create, I think you’ve mentioned
1:42:07 a kind of a blueprint for a problem and then you can really do a divide and conquer with lean
1:42:13 where you’re working on separate parts and they’re using the computer system proof
1:42:16 checker essentially to make sure that everything is correct along the way.
1:42:19 So it makes everything compatible and, uh, yeah, and trustable.
1:42:20 Um, yeah.
1:42:26 So currently only a few mathematical projects can be cut up in this way at the current state
1:42:26 of the art.
1:42:30 Most of the lean activity is on formalizing proofs that have already been proven by humans.
1:42:34 A math paper basically is a boop, a blueprint in a sense.
1:42:38 It is taking a difficult statement, like big theorem and breaking up into me a hundred little
1:42:45 lemurs, um, but often not all written with enough detail that each one can be sort of directly
1:42:45 formalized.
1:42:52 A blueprint is like a really pedantically written version of a paper where every step is explained
1:42:54 as, as much detail as, as, as possible.
1:43:00 And to try and make each step kind of self-contained, um, and, or depending on only a very specific
1:43:05 number of previous statements that have been proven so that each node of this blueprint
1:43:08 graph that gets generated can be tackled independently of all the others.
1:43:10 And you don’t even need to know how the whole thing works.
1:43:14 Um, so it’s like a modern supply chain, you know, like if you want to create an iPhone or
1:43:20 some other complicated object, um, no one person can, can build a single object, but you can
1:43:24 have specialists who, who just, if they’re given some widgets from some other company, they
1:43:26 can combine them together to form a slightly bigger widget.
1:43:31 I think that’s a really exciting possibility because you can have, if you can find problems
1:43:37 that could be broken down this way, then you can have, you know, thousands of contributors,
1:43:37 right?
1:43:37 Yes.
1:43:38 They’ll be completely distributed.
1:43:42 So I told you before about the split between theoretical and experimental mathematics.
1:43:45 And right now, most mathematics is theoretical and only a tiny bit is experimental.
1:43:50 I think the platform that lean and other software tools, uh, so, um, GitHub and things like
1:43:56 that, um, allow, uh, they will allow experimental mathematics to be, to scale up, um, to a much
1:43:57 greater degree than we can do now.
1:44:04 So right now, if you want to, um, um, do any mathematical exploration, uh, of some mathematical
1:44:06 pattern or something, you need some code to write out the pattern.
1:44:10 And I mean, sometimes there are some computer algebra packages that help, but often it’s
1:44:13 just one mathematician coding lots and lots of Python or whatever.
1:44:19 And because coding is such an error prone activity, it’s not practical to allow other people
1:44:23 to collaborate with you on writing a module for your code, because if one of the modules has
1:44:25 a bug in it, the whole thing is unreliable.
1:44:33 Um, so it’s, these are, uh, so you get these bespoke, uh, spaghetti code that written by non-professional
1:44:37 programmers, but mathematicians, you know, and they’re clunky and, and, and slow.
1:44:42 And, um, um, and so because of that, it’s, it’s, it’s hard to, to really mass produce
1:44:43 experimental results.
1:44:50 Um, but, um, yeah, but I think with lean, I mean, so I’m already starting some projects
1:44:54 where we are not just experimenting with data, but experimenting with proofs.
1:44:56 So I have this project called the Equational Theories Project.
1:45:00 Basically, we generated about 22 million little problems in abstract algebra.
1:45:02 Maybe I should back up and tell you what, what the project is.
1:45:03 Okay.
1:45:07 So abstract algebra studies operations like multiplication and addition and their abstract properties.
1:45:08 Okay.
1:45:10 So multiplication, for example, is commutative.
1:45:12 X times Y is always Y times X, at least for numbers.
1:45:14 Um, and it’s also associative.
1:45:17 X times Y times Z is the same as X times Y times Z.
1:45:23 Um, so, um, these operations obey some laws that don’t obey others.
1:45:25 For example, X times X is not always equal to X.
1:45:26 So that law is not always true.
1:45:30 So given any, any operation, it obeys some laws and not others.
1:45:36 Um, and so we generated about 4,000 of these possible laws of algebra that certain operations
1:45:36 can satisfy.
1:45:39 And our question is, which laws imply which other ones?
1:45:43 Um, so for example, does commutativity imply associativity?
1:45:47 And the answer is no, because it turns out you can describe an operation which obeys the
1:45:49 commutative law, but doesn’t obey the associative law.
1:45:53 So by producing an example, you can, you can show that commutativity does not imply associativity.
1:45:57 But some other laws do imply other laws by substitution and so forth.
1:45:59 Uh, and you can write down some, some algebraic proof.
1:46:04 So we look at all the pairs between these 4,000 laws and there’s over 22, 22 million of these
1:46:04 pairs.
1:46:07 And for each pair, we ask, does this law imply this law?
1:46:10 If so, give a, give, uh, give a proof.
1:46:11 If not, give a counterexample.
1:46:17 Um, so 22 million problems, each one of which you could give to like an undergraduate
1:46:19 algebra student and they had a decent chance of solving the problem.
1:46:23 Although there are a few, at least 22 million, there are like a hundred or so that are really
1:46:24 quite hard.
1:46:24 Okay.
1:46:25 But a lot are easy.
1:46:30 And the project was just to, to work out, to determine the entire graph, like, like which
1:46:30 ones imply which other ones.
1:46:32 That’s an incredible project, by the way.
1:46:33 Such a good idea.
1:46:37 Such a good test of the very thing we’ve been talking about on a scale that’s remarkable.
1:46:37 Yeah.
1:46:39 So it would not have been feasible.
1:46:43 You know, I mean, the state of the art in the literature was like, you know, 15 equations
1:46:46 and sort of how they imply, that’s sort of at the limit of what a human with pen and paper
1:46:46 can do.
1:46:48 So, so you need to scale it up.
1:46:54 So you need to crowdsource, but you also need to trust all the, um, you know, I, I mean,
1:46:57 no one person can check 22 million of these proofs.
1:46:59 You need to be computerized.
1:47:02 And so it only became possible with, with lean.
1:47:05 Um, we were hoping to use a lot of AI as well.
1:47:07 Um, so the project is almost complete.
1:47:10 Um, so all these 22 million, all but two had been settled.
1:47:15 Um, and, uh, well, actually, and of those two, uh, we have a pen and paper proof of the
1:47:17 two, uh, and we were formalizing it.
1:47:22 In fact, I was, this morning I was working on, um, so we’re almost done on this.
1:47:23 Um, it’s incredible.
1:47:23 Yeah.
1:47:26 How many people were able to get, uh,
1:47:30 which in mathematics is considered a huge number.
1:47:31 It’s a huge number.
1:47:32 That’s crazy.
1:47:32 Yeah.
1:47:37 So we’re going to have a paper of 50 authors, uh, and a big appendix of food contributor.
1:47:37 What?
1:47:41 Here’s an interesting question, not to maybe speak even more generally about it.
1:47:48 When you have this pool of people, is there a way to, uh, organize the contributions by level
1:47:50 of expertise of the people, of the contributors?
1:47:51 Now, okay.
1:47:56 Uh, I’m asking a lot of pothead questions here, but I’m imagining.
1:47:59 A bunch of humans, and maybe in the future, some AIs.
1:48:06 Can there be, like, an ELO rating type of situation where, like, a gamification of this?
1:48:10 The beauty of, of these lean projects is that automatically you get all this data, you know?
1:48:14 So, like, like, everything’s uploaded to this GitHub, and GitHub tracks who contributed what.
1:48:20 Um, so you could generate statistics from, at any, at any later point in time, you could say,
1:48:23 oh, this person contributed this many lines of code or whatever.
1:48:25 I mean, these are very crude metrics.
1:48:28 Um, I would, I would definitely not want this to become, like, you know, part of your 10-year
1:48:29 review or something.
1:48:36 Um, but, um, I mean, I think already in, in, in enterprise computing, right, people do use
1:48:41 some of these metrics as part of, of the assessment of, of, uh, performance of, of an employee.
1:48:45 Um, again, this is the direction which is a bit scary for academics to go down.
1:48:48 We, we, we, we don’t like metrics so much.
1:48:55 And yet, academics use metrics, they just use old ones, number of papers.
1:48:59 Yeah, yeah, it’s true, it’s true that, yeah, I mean, um.
1:49:04 It feels like this is a metric while flawed is, is going in the, more in the right direction,
1:49:04 right?
1:49:05 Yeah.
1:49:08 It’s an interesting, I mean, at least it’s a very interesting metric.
1:49:10 Yeah, I think it’s interesting to study.
1:49:13 I mean, I think you can, you can do studies of, of, of whether these are better predictors.
1:49:15 Um, there’s this problem called Goodhart’s Law.
1:49:19 If a statistic is actually used to incentivize performance, it becomes gamed.
1:49:21 Um, and then it is no longer a useful measure.
1:49:25 Oh, humans, always, yeah, yeah, no, I mean, it’s, it’s, it’s rational.
1:49:28 So what we’ve done for this project is, is self-report.
1:49:34 So, um, there are actually these standard categories, um, from the sciences of what types of contributions
1:49:34 people give.
1:49:40 So there’s, there’s a concept and validation and resources and, and, and, and, and coding and so forth.
1:49:43 Um, so we, we, we, there’s a standard list of 12 or so categories.
1:49:48 Um, and we just ask each contributor to this big matrix of all the, of all the authors and
1:49:51 all the categories just to tick the boxes where they think that they contributed.
1:49:57 Um, and just give a rough idea, you know, like, oh, so you did some coding and, and, uh, and
1:50:01 you provided some compute, but you didn’t do any of the pen and paper verification or whatever.
1:50:03 And I think that that works out.
1:50:06 Traditionally, mathematicians just order alphabetically by surname.
1:50:10 So we don’t have this tradition as in the sciences of, you know, lead author and second
1:50:14 author and so forth, like, which we’re proud of, you know, we make all the authors equal
1:50:17 status, but it doesn’t quite scale to this size.
1:50:21 So a decade ago, I was involved in these things called polymath projects.
1:50:24 It was the crowdsourcing mathematics, but without the lean component.
1:50:29 So it was limited by, you needed a human moderator to actually check that all the contributions coming
1:50:29 in were actually valid.
1:50:32 Um, and this was a huge bottleneck actually.
1:50:39 Um, but still we had projects that were, you know, 10 authors or so, but we had decided
1:50:44 at the time, um, not to try to decide who did what, um, but to have a single pseudonym.
1:50:50 Um, so we created this fictional character called DHJ polymath in the spirit of Bobaki.
1:50:55 Bobaki is the pseudonym for a famous group of mathematicians in the 20th century.
1:50:58 But, um, and so the paper was also authored on the pseudonym.
1:50:59 So none of us got the author credit.
1:51:03 Um, this actually turned out to be not so great for a couple of reasons.
1:51:08 So, so one is that if you actually wanted to be considered for 10 years or whatever, you
1:51:14 could not use this paper in your, uh, uh, as you’re submitted as one of your publications
1:51:16 because it was, you didn’t have the formal author credit.
1:51:23 Um, um, but the other thing that we’ve recognized much later is that when people referred to
1:51:27 these projects, they naturally refer to the most famous person who was involved in the
1:51:27 project.
1:51:28 Oh yeah.
1:51:29 So this was Tim Gower’s polymath project.
1:51:34 This was Terrence Towers’ polymath project and not mentioned the, the other 19 or whatever
1:51:35 people that were involved.
1:51:36 Ah, yeah.
1:51:40 So we’re trying something different this time around where we have, everyone’s an author,
1:51:44 um, but we will have an appendix with this matrix and we’ll see how that works.
1:51:47 I mean, uh, so both projects are incredible.
1:51:52 Just the fact that you’re involved in such huge collaborations, but I think I saw a talk from
1:51:56 Kevin Buzzard about, uh, the lean programming language is a few years ago and he was saying
1:51:59 that, uh, this might be the future of mathematics.
1:52:05 And so it’s also exciting that you’re embracing, uh, one of the greatest mathematicians in the
1:52:10 world embracing this, what seems like the paving of the future of mathematics.
1:52:18 Um, so I have to ask you here about the integration of AI into this whole process.
1:52:24 So DeepMind’s alpha proof was trained using reinforcement learning, um, both failed and
1:52:27 successful formal lean proofs of IMO problems.
1:52:31 So this is sort of high level high school.
1:52:32 Oh, very high level.
1:52:32 Yes.
1:52:35 Very high level, high school level mathematics problems.
1:52:36 What do you think about the system?
1:52:41 And maybe what is the gap between this system that is able to prove the high school level
1:52:46 problems versus gradual level, uh, problems?
1:52:46 Yeah.
1:52:52 The difficulty increases exponentially with the, the number of steps involved in the proof is
1:52:53 a commentorial explosion.
1:52:57 So the thing of large language models is, is that they make mistakes.
1:53:02 And so if a proof has got 20 steps and your offline board has a 10% failure rate, um, at
1:53:07 each step, um, of, of going in the wrong direction, like, uh, it’s, it’s just extremely unlikely
1:53:09 to actually, uh, reach the end.
1:53:16 Actually, uh, just to take a small tangent here is how hard is the problem of mapping from natural
1:53:18 language to the formal program?
1:53:20 Well, yeah, it’s extremely hard.
1:53:23 Actually, um, natural language, you know, it’s very fault tolerant.
1:53:27 Um, like you can make a few minor grammatical errors and a speaker in the second language
1:53:28 can get some idea of what you’re saying.
1:53:31 Um, yeah, but, but formal language, yeah.
1:53:35 You know, if you get one little thing wrong, um, I think that the whole thing is, is, is,
1:53:36 is nonsense.
1:53:39 Um, even formal to formal is, is, is very hard.
1:53:42 There are different incompatible, um, uh, proof of system languages.
1:53:45 Uh, there’s lean, but also cock and Isabel and so forth.
1:53:48 And actually even converting from a formal language to formal language, um,
1:53:51 is, uh, it’s, uh, it’s an unsolved, it’s an unsolved problem.
1:53:52 That is fascinating.
1:53:53 Okay.
1:54:02 So, uh, but once you have an informal language, they’re using, um, their RL train model.
1:54:07 So something, something akin to alpha zero that they use to go to then try to come up with
1:54:07 proofs.
1:54:08 They also have a model.
1:54:11 I believe it’s a separate model for geometric problems.
1:54:14 So what impresses you about the system?
1:54:17 And, um, what do you think is the gap?
1:54:18 Yeah.
1:54:21 We talked earlier about things that are amazing over time, become kind of normalized.
1:54:26 Um, so yeah, now somehow it’s, of course, geometry is a silver book problem.
1:54:26 Right.
1:54:27 That’s true.
1:54:27 That’s true.
1:54:29 I mean, it’s still beautiful.
1:54:29 Yeah.
1:54:29 Yeah.
1:54:31 No, it’s, it’s, it’s, it’s a great work.
1:54:32 It shows what’s possible.
1:54:35 I mean, um, it’s, it, um, the approach doesn’t scale currently.
1:54:36 Yeah.
1:54:41 Three days of Google’s servers, server time to sort of one, uh, high school math from there.
1:54:46 This, this is not a scalable, uh, prospect, um, especially with the exponential increase in,
1:54:51 um, as, as the complexity, um, increases, which mentioned that they got a silver metal performance.
1:54:57 The equivalent of, I mean, so first of all, they took way more time than was, uh, allotted.
1:55:01 Um, and they had this assistance where, where the humans started helped by, by formalizing.
1:55:07 Um, but, uh, also they’re giving us those full marks for the solution, which I guess is formally
1:55:08 verified.
1:55:09 So I guess that that’s, that’s fair.
1:55:16 Um, uh, um, there, there are efforts, there was, there will be a proposal at some point to actually
1:55:22 have an, an AI math Olympiad where at the same time as the human contestants get the, the actual
1:55:28 Olympiad, um, problems, AIs will also be given the same problems with the same time period.
1:55:31 Um, and the outputs will have to be graded by the same judges.
1:55:36 Um, um, um, and which means that will have to be written in natural language rather than
1:55:37 formal language.
1:55:38 Oh, I hope that happens.
1:55:40 I hope that this IMO happens.
1:55:41 I hope, I hope next one.
1:55:42 It won’t happen this IMO.
1:55:45 The performance is not good enough in, in, in the time period.
1:55:51 And, and, uh, um, but there are smaller competitions, uh, there are competitions where the, the answer
1:55:54 is a, is a number rather than a long form proof.
1:56:00 Um, and that’s, that’s, um, AI is actually a lot better at, um, problems where there’s a
1:56:01 specific numerical answer.
1:56:06 Um, cause it’s, it’s, it’s easy to, to, to, uh, to reinforce, to reinforce some learning
1:56:06 on it.
1:56:06 Yeah.
1:56:07 You got the right answer.
1:56:08 You got the wrong answer.
1:56:12 Uh, it is, it’s, it’s a very clear signal, but a long form proof either has to be formal
1:56:16 and then the lean can give it thumbs up, thumbs down, or it’s informal.
1:56:22 Um, but then you need a human to grade it to tell, uh, and if you’re trying to do a billions
1:56:27 of, of reinforcement learning, um, you know, um, um, runs, you’re not, you can’t hire enough
1:56:29 humans to, uh, to grade those.
1:56:33 Um, I mean, it’s already hard enough for, for the last language to do reinforcement learning
1:56:38 on, on just the regular text that people get, but now we actually hire people, not just
1:56:42 give thumbs up, thumbs down, but actually check the, the output mathematically.
1:56:43 Yeah.
1:56:44 Uh, that’s too expensive.
1:56:51 So if we, uh, just explore this possible future, what, what, what is the thing that humans do
1:56:54 that’s most special in, uh, in mathematics?
1:56:59 So that you could see AI, uh, not cracking for a while.
1:57:05 Well, so inventing new theories, so coming up with new conjectures versus, uh, proving the
1:57:13 conjectures, building new abstractions, new representations, maybe, uh, an AI turner style
1:57:16 with, uh, seeing new connections between disparate fields.
1:57:17 That’s a good question.
1:57:21 Um, I think the nature of what mathematicians do over time has changed a lot.
1:57:26 Um, you know, um, so a thousand years ago, mathematicians had to compute the date of Easter,
1:57:31 uh, and those really complicated, uh, calculations, you know, but it’s all automated, been automated
1:57:32 centuries.
1:57:33 Uh, we don’t need that anymore.
1:57:37 So, you know, they used to navigate, to do spherical navigation, spherical trigonometry
1:57:42 to navigate how to get from, from, um, the old world to the new or something, a very complicated
1:57:42 calculation.
1:57:47 Again, we’d been automated, um, you know, even a lot of undergraduate mathematics, even before
1:57:52 AI, um, like Wolfram Alpha, for example, uh, it’s, it’s not a language model, but it can
1:57:54 solve a lot of undergraduate level math tasks.
1:58:00 So on the computational side, verifying routine things like having a problem and, um, and say,
1:58:02 here’s a problem in partial differential equations.
1:58:04 Could you solve it using any of the 20 standard techniques?
1:58:07 Um, and they say, yes, I’ve tried all 20.
1:58:10 I hear that 100 different permutations and, and here’s my results.
1:58:13 Um, and that type of thing, I think it will work very well.
1:58:20 Um, type of scaling to once you solve one problem to, to make the AI attack a hundred adjacent
1:58:20 problems.
1:58:29 Um, the things that humans do still, so, so where the AI really struggles right now, um, is knowing
1:58:33 when it’s made a wrong turn, um, and it can say, oh, I’m going to solve this problem.
1:58:37 I’m going to split up this problem into, um, into these two cases.
1:58:38 I’m going to try this technique.
1:58:43 And, um, sometimes if you’re lucky and it’s a simple problem, it’s the right technique and
1:58:43 you solve the problem.
1:58:46 And sometimes it will get, it will have a problem.
1:58:49 It would, it would propose an approach which is just complete nonsense.
1:58:52 Um, and, but like, it looks like a proof.
1:58:56 Um, so this is one annoying thing about LLM generated mathematics.
1:59:03 So, um, yeah, we, we, we’ve had human generated mathematics as very low quality, um, uh, like,
1:59:06 you know, submissions for people who don’t have the formal training and so forth.
1:59:09 But if a human proof is bad, you can tell it’s bad pretty quickly.
1:59:15 It makes really basic mistakes, but the AI generator proofs, they can look superficially
1:59:16 flawless.
1:59:19 Uh, and it’s partly because that’s what the reinforcement learning has like you train them
1:59:25 to do, uh, to, to make things, to, to produce text that looks like, um, uh, what is correct,
1:59:26 which for many applications is good enough.
1:59:30 Um, uh, so the errors often really subtle.
1:59:34 And then when you spot them that they’re really stupid, um, like, you know, like no
1:59:35 human would have actually made that mistake.
1:59:35 Yeah.
1:59:40 It’s actually really frustrating in the programming context because I program a lot and yeah, when
1:59:45 a human makes when a low quality code, there’s something called code smell, right?
1:59:51 You can tell, you can tell immediately like there’s signs, but with, with AI generate code
1:59:53 and then you’re right.
1:59:59 Eventually you find an obvious dumb thing that just looks like good code.
1:59:59 Yeah.
2:00:04 So, um, it’s very tricky to, and frustrating for some reason to have to work.
2:00:04 Yeah.
2:00:08 So the sense of smell, this is, this is, this is one thing that humans have.
2:00:16 Um, and there’s a metaphorical mathematical smell that, uh, this is not clear how to get
2:00:17 the AI to duplicate that.
2:00:24 Eventually, um, I mean, so the, the, the way, um, alpha zero and so forth to make progress
2:00:28 on go and chess and so forth is, is in some sense they have developed a sense of smell
2:00:31 for go and chess positions, you know, that, that this position is good for white.
2:00:32 It’s good for black.
2:00:34 Um, they can’t enunciate why.
2:00:39 Um, but just having that, that sense of smell lets them strategize.
2:00:45 So if AI’s gained that ability to sort of assess a viability of certain proof strategies, so
2:00:50 you can say, uh, I’m going to try to, to break up this problem into two small subtasks and
2:00:52 they can say, oh, this looks good.
2:00:55 The two tasks look like they’re simpler tasks than, than your main task.
2:00:57 And they still got a good chance of being true.
2:00:58 Um, so this is good to try.
2:01:02 Or no, if you’ve, you’ve made the problem worse because each of the two sub problems
2:01:05 is actually harder than your original problem, which is actually what normally happens if
2:01:07 you try a random, uh, thing to try.
2:01:11 Normally you actually, it’s very easy to transform a problem into an even harder problem.
2:01:14 Very rarely do you transform into a simpler problem.
2:01:21 Um, yeah, so if they can pick up a sense of smell, then they could maybe start competing
2:01:23 with, uh, uh, human level mathematicians.
2:01:27 So, so this is a hard question, but not competing, but collaborating.
2:01:27 Yeah.
2:01:28 If, okay.
2:01:29 Hypothetical.
2:01:36 If I gave you an oracle that was able to do some aspect of what you do and you could just
2:01:36 collaborate with it.
2:01:36 Yeah.
2:01:37 Yeah.
2:01:37 Yeah.
2:01:40 What would that oracle, what would you like that oracle to be able to do?
2:01:44 Would you like it to, uh, maybe be a verifier, like check?
2:01:45 Mm-hmm.
2:01:51 Do the codes, like you’re, yes, uh, uh, Professor Tao, this is the correct, this is a good,
2:01:54 this is a promising fruitful direction.
2:01:54 Yeah.
2:01:54 Yeah.
2:01:55 Yeah.
2:02:01 Or, or would you like it to, uh, generate possible proofs and then you see which one is
2:02:02 the right one.
2:02:08 Um, or would you like it to maybe generate different representation, different, totally different
2:02:10 ways of seeing this problem?
2:02:10 Yeah.
2:02:11 I think all of the above.
2:02:15 Um, a lot of it is, we don’t know how to use these tools because, because it,
2:02:21 it’s a paradigm that it’s not, um, yeah, we have not had in the past assistants that are
2:02:28 competent enough to understand complex instructions, um, that can work at massive scale, but are
2:02:29 also unreliable.
2:02:36 Uh, it’s, it’s an interesting, uh, a bit unreliable in subtle ways whilst we, whilst providing sufficiently
2:02:37 good output.
2:02:39 Um, it’s, um, it’s an interesting combination.
2:02:43 Um, you know, I mean, you have, you have like graduate students that you work with who
2:02:45 are kind of like this, but not at scale.
2:02:51 Um, you know, and, and, and we had previous software tools that, um, can work at scale,
2:02:52 but, but very narrow.
2:02:58 Um, so we have to figure out how to, how to use, um, I mean, um, so Tim Goward, actually,
2:03:03 you mentioned, he actually foresaw like in, in 2000, he was envisioning what mathematics
2:03:06 would look like in, in actually two and a half decades.
2:03:14 And yeah, he, he wrote in his, in his article, like a hypothetical conversation between a mathematical
2:03:18 assistant of the future, um, and himself, you know, trying to solve a problem and they
2:03:22 would have to have a conversation that sometimes the human would propose an idea and the AI
2:03:24 would, would, uh, evaluate it.
2:03:29 Uh, and sometimes the AI would propose an idea, um, and, uh, and sometimes a competition
2:03:32 was required and the AI would just go and say, okay, I’ve, I’ve checked the 100 cases
2:03:37 needed here, or, um, uh, the first, uh, you, you said this is true for all N, I’ve checked
2:03:42 the N up to 100, um, and it looks good so far, or hang on, there’s a problem at N equals 46.
2:03:47 And so just a free form conversation where you don’t know in advance where things are going
2:03:52 to go, but just based on, on, I think ideas could propose on both sides, calculations could
2:03:52 propose on both sides.
2:03:57 I’ve had conversations with AI where I say, okay, let’s, we’re going to collaborate to solve
2:03:58 this math problem.
2:03:59 And it’s a problem that I already know a solution to.
2:04:01 So I, I try to prompt it.
2:04:01 Okay.
2:04:02 So here’s the problem.
2:04:06 I suggest using this tool and it will find this, this lovely argument using a completely
2:04:10 different tool, which eventually goes into the weeds and say, no, no, no, try using this.
2:04:10 Okay.
2:04:14 And it might start using this and then it’ll go back to the tool that I wanted to do before.
2:04:18 Um, and like you have to keep railroading it, um, onto the path you want.
2:04:21 And I could eventually force it to give the proof I wanted.
2:04:26 Um, but it was like herding cats, um, like, and the amount of personal effort I had to
2:04:31 take to not just sort of prompt it, but also check its output because it, after a lot of
2:04:32 what it looked like, it was going to work.
2:04:36 I know there’s a problem on 917 and basically arguing with it.
2:04:40 Um, like it was more exhausting than doing it, uh, unassisted.
2:04:43 So like, but that’s the current state of the art.
2:04:49 I wonder if there’s, there’s a phase shift that happens to where it’s no longer feels
2:04:53 like herding cats and maybe it’ll surprise us how quickly that comes.
2:04:55 I believe so.
2:04:59 Um, so in formalization, I mentioned before that it takes 10 times longer to formalize
2:05:04 a proof than to write it by hand with these modern AI tools and also just better tooling
2:05:10 that the lean, um, um, developers are doing a great job adding more and more features and
2:05:11 making it user friendly.
2:05:13 It’s going on from nine to eight to seven.
2:05:14 Okay.
2:05:14 No big deal.
2:05:17 But one day it will drop below one.
2:05:24 Um, and that’s the phase shift because suddenly, um, it makes sense when you write a paper to,
2:05:29 to write it in lean first, uh, or through a conversation with AI, which is generally, um,
2:05:30 on the fly with you.
2:05:35 And it becomes natural for journals to accept, uh, you know, maybe they’ll offer an expedite
2:05:40 refereeing, you know, that if, if a paper has already been formalized in lean, um,
2:05:44 they’ll just ask the referee to comment on, on the significance of the results and how
2:05:48 it connects to literature and not worry so much about the correctness, um, because that’s
2:05:49 been certified.
2:05:53 Um, papers are getting longer and longer in mathematics and like it’s harder and harder
2:05:57 to get good refereeing for, um, the really long ones, unless they’re really important.
2:06:01 Uh, it is actually an issue which, and the formalization is coming in at just the right
2:06:03 time for this to be.
2:06:07 And the easier and easier it gets because of the tooling and all the other factors, then
2:06:11 you’re going to see much more like math lib will grow potentially exponentially.
2:06:15 It’s a, it’s a, it’s a, it’s a virtuous, uh, cycle.
2:06:15 Okay.
2:06:19 I mean, one phase shift of this type that happened in the past was, uh, the adoption of LaTeX.
2:06:22 So, so LaTeX is this typesetting language that all musicians use now.
2:06:26 So in the past, people use all kinds of word processors and typewriters and whatever.
2:06:31 But at some point LaTeX became easier to use than all other competitors.
2:06:36 And like people would switch, you know, within a few years, like it was just a dramatic, um,
2:06:47 it’s a wild out there question, but what, what year, how far away are we from a, uh, AI system
2:06:52 being a collaborator on a proof that wins the Fields Medal?
2:06:53 So that level.
2:06:55 Okay.
2:06:57 Um, well, it depends on the level of collaboration.
2:07:00 I mean, no, like it deserves to be, to get the Fields Medal.
2:07:02 Like, so half and half.
2:07:05 Already, like I can imagine if it was for a metal winning paper,
2:07:09 having some AI systems in writing it, you know, uh, just, you know,
2:07:13 like the autocomplete alone is already, I, I use it, like it speeds up my, my own writing.
2:07:18 Um, um, like, you know, you, you, you can have a theorem and you have a proof and the proof
2:07:22 has three cases and I, I write down the proof of the first case and the autocomplete just
2:07:24 suggests that now here’s how the proof of the second case could work.
2:07:26 And like, it was exactly correct.
2:07:26 That was great.
2:07:29 Saved me like five, 10 minutes of, uh, of, of typing.
2:07:32 But in that case, the AI system doesn’t get the Fields Medal.
2:07:33 No.
2:07:40 Are we talking 20 years, 50 years, a hundred years?
2:07:41 What do you think?
2:07:41 Okay.
2:07:47 Uh, so I, I gave a prediction in print, but so by 2026, which is now next year, um, there
2:07:52 will be math collaborations, you know, where the AI, so not Fields Medal winning, but, but
2:07:53 like actual research level math papers.
2:07:57 Like published ideas that are in part generated by AI.
2:08:02 Um, maybe not the ideas, but at least, uh, some of the computations, um, um, um, the
2:08:03 verifications.
2:08:03 Yeah.
2:08:03 I mean,
2:08:04 that, that already happened.
2:08:05 That’s already happened.
2:08:05 Yeah.
2:08:12 There are, there are problems that were solved, uh, by a complicated process, conversing with
2:08:13 AI to propose things.
2:08:16 And then the human goes and tries it and it, and then kind of comes like, doesn’t work.
2:08:19 Um, but it was a different idea.
2:08:22 Um, it, it’s, it’s hard to disentangle exactly.
2:08:28 Um, there are certainly math results, which could only have been accomplished because there
2:08:30 was a math, math, human mathematician and an AI involved.
2:08:35 Um, but, uh, it’s hard to sort of disentangle credit.
2:08:43 Um, I mean, these tools, they, they do not, uh, replicate all the skills needed to do mathematics,
2:08:47 but they can replicate sort of some non-trivial percentage of them, you know, 30, 40%.
2:08:49 So they can fill in gaps.
2:08:56 Um, you know, so, uh, coding is, is, is, is a, is a good example, you know, so I, I, um, um,
2:08:57 it’s annoying for me to, to code in Python.
2:09:01 I’m not, I’m not a native, um, no professional, um, programmer.
2:09:08 Um, but, um, the, with AI that the, the, the, the friction cost of, of doing it is, is, is
2:09:08 much reduced.
2:09:10 Uh, so it, it fills in that gap for me.
2:09:14 Um, AI is getting quite good at literature review.
2:09:18 Um, I mean, there’s still a problem with, um, hallucinating, you know, the references that
2:09:19 don’t exist.
2:09:22 Um, but this, I think is a silverware problem.
2:09:27 Uh, if you train in the right way and so forth, you can, you can, and, um, and verify, um,
2:09:33 you know, using the internet, um, you, you know, um, you should in a few years get the
2:09:37 point where you, you have a, a lemma that you need and, uh, say, has anyone proven this
2:09:38 lemma before?
2:09:43 And we will do basically a fancy web search AI system and say, yeah, yeah, there are these
2:09:45 six papers where something similar has happened.
2:09:49 And I mean, you can ask you right now and it will give you six papers of which maybe one
2:09:51 is legitimate and relevant.
2:09:56 one exists, but it’s not relevant and for a hallucinated, um, it has a non-zero success
2:10:01 rate right now, but, uh, it’s, there’s so much garbage, uh, so much, the signal to noise
2:10:07 ratio is so poor that it’s, it’s, um, it’s most helpful when you already somewhat know the
2:10:07 literature.
2:10:13 Um, and you just need to be prompted to be reminded of a paper that was really subconsciously
2:10:13 in your memory.
2:10:17 Or it’s just helping you discover new, you were not even aware of, but is the correct
2:10:18 citation.
2:10:19 Yeah.
2:10:24 That’s, yeah, that it can sometimes do, but, but when it does, it’s, it’s buried in, in a
2:10:25 list of options to which the other.
2:10:26 That are bad.
2:10:26 Yeah.
2:10:30 I mean, being able to automatically generate a related work section that is correct.
2:10:31 Yeah.
2:10:36 That’s actually a beautiful thing that might be another phase shift because it assigns credit
2:10:37 correctly.
2:10:37 Yeah.
2:10:38 It does.
2:10:40 It breaks you out of the silos of.
2:10:40 Yeah.
2:10:40 Yeah.
2:10:40 Yeah.
2:10:40 Yeah.
2:10:44 No, I mean, yeah, no, there’s a big hump to overcome right now.
2:10:49 I mean, it’s, it’s, it’s like self-driving cars, you know, the safety margin has to be really
2:10:52 high for it to be, um, uh, to be feasible.
2:10:56 So yeah, so there’s a last mile problem, um, with a lot of AI applications.
2:11:03 Um, that, uh, you know, they can develop tools that work 20%, 80% of the time, but it’s still
2:11:04 not good enough.
2:11:07 Um, and in fact, even worse than good in some ways.
2:11:13 I mean, another way of asking the feels metal question is what year do you think you’ll wake
2:11:15 up and be like real surprised?
2:11:22 You read the headline, the news or something happened that AI did like, you know, real breakthrough
2:11:23 something.
2:11:25 It doesn’t, you know, like feels metal, even hypothesis.
2:11:31 It could be like really just this alpha zero moment would go that kind of thing.
2:11:31 Right.
2:11:39 Um, yeah, this, this decade, I can, I can see it like making a conjecture between two unrelated
2:11:41 two, two things that people thought was unrelated.
2:11:42 Oh, interesting.
2:11:43 Generating a conjecture.
2:11:45 That’s a beautiful conjecture.
2:11:45 Yeah.
2:11:49 And, and actually has a real chance of being correct and, and, and meaningful.
2:11:55 And, um, because that’s actually kind of doable, I suppose, but the word of the data is, it’s
2:11:57 for, yeah, yeah, no, that would be truly amazing.
2:12:00 Um, the current models struggle a lot.
2:12:04 I mean, so, um, a version of this is, um, I mean, the physicists have a dream of getting
2:12:06 the AIs to discover new laws of physics.
2:12:10 Um, uh, you know, the, the, the dream is you just feed it all this data.
2:12:11 Okay.
2:12:15 Uh, and, and, and this is a, here, here is a new patent that we didn’t see before, but it
2:12:18 actually even struggled with the current state of the art, even struggles to discover old
2:12:20 laws of physics, um, from the data.
2:12:25 I mean, uh, or if it does, uh, there’s a big concern of contamination that it did it only
2:12:29 because it’s like somewhere in its training data, it already somehow knew, um, you know,
2:12:32 Boyle’s law or whatever you’re trying to, to, to reconstruct.
2:12:37 Um, part of it is that we don’t have the right type of training data for this.
2:12:41 Um, yeah, so for laws of physics, like we, we don’t have like a million different universes
2:12:42 with a million different laws of nature.
2:12:50 Um, and, um, like a lot of what we’re missing in math is actually the negative space.
2:12:55 So we have published things of things that people have been able to prove, um, and conjectures
2:13:00 that ended up being verified, um, or maybe counterexamples produced, but, um, we don’t
2:13:05 have data on, on things that were proposed and they’re kind of a good thing to try, but then
2:13:09 people quickly realized that it was the wrong conjecture and then they, they said, oh, but
2:13:13 we, we should actually change, um, our claim to modify it in this way to actually make it
2:13:14 more plausible.
2:13:20 Um, there’s, there’s a trial and error process, which is a real integral part of human mathematical
2:13:23 discovery, which we don’t record because it’s embarrassing.
2:13:26 Uh, we make mistakes and, and we only like to publish our, our wins.
2:13:31 Um, and, uh, the AI has no access to this data to train on.
2:13:38 Um, I sometimes joke that basically AI has to go through, um, a grad school and actually,
2:13:44 you know, go to grad courses, do the assignments, go to office hours, make mistakes, um, get advice
2:13:46 on how to correct the mistakes and learn from that.
2:13:52 Let me, uh, ask you, if I may, about, uh, Grigori Perlman.
2:13:57 You mentioned that you try to be careful in your work and not let a problem completely consume
2:13:58 you.
2:14:02 Just, you’ve really fallen in love with the problem and really cannot rest until you solve
2:14:03 it.
2:14:07 But you also hasted to add that sometimes this approach actually can be very successful.
2:14:14 An example you gave is Grigori Perlman who proved the Poincare conjecture and did so by
2:14:19 working alone for seven years with basically little contact with the outside world.
2:14:26 Can you explain this one millennial prize problem that’s been solved, Poincare conjecture, and
2:14:30 maybe speak to the journey that Grigori Perlman’s been on?
2:14:34 All right, so it’s, it’s a question about curved spaces.
2:14:35 Earth is a good example.
2:14:36 So Earth, you can think of as a 2D surface.
2:14:40 In just being round, you know, it could maybe be a torus with a hole in it or kind of many
2:14:40 holes.
2:14:46 And there are many different topologies, a priori, that, that a surface could have, um, even if
2:14:49 you assume that it’s, it’s bounded and, and, and, and smooth and so forth.
2:14:54 So we’ve, we have figured out how to classify surfaces as a first approximation, uh, everything’s
2:14:56 determined by something called the genus, how many holes it has.
2:14:59 So a sphere has genus zero, a donut has genus one and so forth.
2:15:03 And one way you can tell the surfaces apart, probably the sphere has, which is called simply
2:15:04 connected.
2:15:09 If you take any closed loop on the sphere, like a big closed loop of rope, you can contract
2:15:11 it to a point and while staying on the surface.
2:15:14 And the sphere has this property, but a torus doesn’t.
2:15:18 And if you’re on a torus and you take a rope that goes around, say the, the outer diameter
2:15:21 torus, there’s no way it can’t get through the hole.
2:15:23 There’s no way to, to contract it to a point.
2:15:29 So it turns out that the, the, the sphere is the only surface with this property of contract
2:15:31 ability, I mean, up to like continuous deformations of the sphere.
2:15:35 So, um, so things that I want to call topologically, um, equivalent of the sphere.
2:15:38 So Poincare asked the same question, higher dimensions.
2:15:43 Um, so this, it becomes hard to visualize, uh, because, um, surface you can think of as embedded
2:15:47 in three dimensions, but as a curved free space, we don’t have good intuition of
2:15:49 four D space to, to, to, to limit.
2:15:52 And then there are also three D spaces that can’t even fit into four dimensions.
2:15:54 You need five or six or, or higher.
2:15:59 But anyway, uh, mathematically you can still pose this question that if you have a bounded
2:16:03 three dimensional space now, which is also has this simply connected property that every
2:16:04 loop can be contracted.
2:16:06 Can you turn it into a three dimensional version of the sphere?
2:16:08 And so this is the Poincare conjecture.
2:16:11 Weirdly in higher dimensions, four and five, it was actually easier.
2:16:14 So, uh, it was solved first in higher dimensions.
2:16:16 There’s somehow more room to do the deformation.
2:16:19 It’s easier to, to, to move things around to the sphere.
2:16:21 But three was really hard.
2:16:23 So people tried many approaches.
2:16:27 There’s sort of commentary approaches where you chop up the, the surface into little triangles
2:16:31 or tetrahedra and you, you just try to argue based on how the faces interact each other.
2:16:35 Um, there were, um, algebraic approaches.
2:16:38 Uh, there’s, there’s various algebraic objects, uh, like things called the fundamental group
2:16:43 that you can attach to these homology and cohomology and, and, and, and all these very
2:16:44 fancy tools.
2:16:45 Um, they also didn’t quite work.
2:16:51 Um, but Richard Hamilton’s proposed a, um, partial differential equations approach.
2:16:56 So you take, um, you take, so the problem is that you’re, so you have this object, which
2:17:02 is sort of secretly is a sphere, but it’s given to you in a, in a, in a, in a weird way.
2:17:05 So it’s like, I think of a ball that’s been kind of crumpled up and twisted.
2:17:06 And it’s not obvious that it’s a ball.
2:17:12 Um, but, um, like if you, if you have some sort of surface, which is, which is a deformed
2:17:18 sphere, you could, um, uh, you could, for example, think of it as the surface of a balloon.
2:17:19 You could try to inflate it.
2:17:20 You blow it up.
2:17:25 Um, and naturally as you fill it with air, um, the, the wrinkles will sort of smooth out
2:17:28 and it will turn into, um, um, a nice round sphere.
2:17:31 Um, uh, unless of course it was a torus or something, in which case it would get stuck
2:17:32 at some point.
2:17:35 Like if you inflate a torus, there would, there’d be a point in the middle.
2:17:38 When the inner ring shrinks to zero, you get, you get a singularity and you can’t
2:17:39 blow up any further.
2:17:40 Uh, you can’t flow any further.
2:17:45 So he created this flow, which is now called Ritchie flow, which is a way of taking an
2:17:49 arbitrary surface or, or space and smoothing it out to make it rounder and rounder, to
2:17:50 make it look like a sphere.
2:17:56 And he wanted to show that either, uh, this process would give you a sphere or it would
2:17:56 create a singularity.
2:18:00 Um, I can very much like how PDEs, either they have global regularity or finite and
2:18:01 blow up.
2:18:03 I can basically, it’s almost exactly the same thing.
2:18:04 It’s all connected.
2:18:10 Um, and so, and, and he showed that for two dimensions, two dimensional surfaces, um, uh, if you start
2:18:13 with something connected, no singularity is ever formed.
2:18:16 Um, you, you never ran into trouble and you could flow and it will give you a sphere.
2:18:19 And it, so he got a new proof of the two dimensional result.
2:18:23 Well, by the way, that’s a beautiful explanation of Ritchie flow and its application in this context.
2:18:25 How difficult is the mathematics here?
2:18:26 Like for the 2D case?
2:18:27 Yeah.
2:18:27 Yeah.
2:18:32 These are quite sophisticated equations on par with the Einstein equations, slightly simpler,
2:18:38 but, um, um, yeah, but, but they were considered hard nonlinear equations to solve.
2:18:41 Um, and there’s lots of special tricks in 2D that, that, that helped.
2:18:46 But in 3D, the problem was that, uh, this equation was actually supercritical.
2:18:47 So it has the same problems as Navier-Stokes.
2:18:52 As you blow up, um, maybe the curvature could get concentrated in finer and smaller, smaller
2:18:52 regions.
2:18:57 And it, um, it looked more and more nonlinear and things just look worse and worse.
2:19:00 And there could be all kinds of singularities that showed up.
2:19:05 Um, some singularities, um, like if, uh, there’s these things called neck pinches where, where,
2:19:11 where the, uh, the surface sort of behaves like a, like a, like a barbell and it, it pinches
2:19:11 at a point.
2:19:14 Some, some singularities are simple enough that you can sort of see what to do next.
2:19:17 You just make a snip and then you can turn one surface into two and evolve them separately.
2:19:22 But there was, there was a, the, the prospect that there’s some really nasty, like knotted
2:19:28 singularities showed up that you, you couldn’t see how to, um, resolve in any way, uh, that
2:19:29 you couldn’t do any surgery to.
2:19:34 Um, so you need to classify all the singularities, like what are all the possible ways that things
2:19:34 can go wrong?
2:19:40 Um, so what Perlman did was, first of all, he, he made the problem, he turned the problem
2:19:41 from a supercritical problem to a critical problem.
2:19:47 Um, I said before about how, um, the invention of the, of, of energy, the Hamiltonian, like
2:19:50 really clarified, um, Newtonian mechanics.
2:19:54 Um, uh, so he introduced, uh, something which is now called Perlman’s reduced volume and
2:19:55 Perlman’s entropy.
2:20:00 Um, he introduced new quantities, kind of like energy that looked the same at every single
2:20:04 scale and turned the problem into a critical one where the nonlinearities actually suddenly
2:20:06 looked a lot less scary than they did before.
2:20:10 Um, and then he had to solve, he still had to analyze the singularities of this critical
2:20:10 problem.
2:20:14 Uh, and that itself was a problem similar to this wavemaps thing I worked on, actually.
2:20:19 Um, so on the, on the level of difficulty of that, so he managed to classify all the singularities
2:20:22 of this problem and show how to apply surgery to each of these.
2:20:31 So, um, quite, uh, like a lot of really ambitious steps, um, and like, like nothing that a large
2:20:37 language model today, for example, could, I mean, um, at best, uh, I could imagine a model
2:20:41 proposing this idea as one of hundreds of different things to try.
2:20:45 Um, but the other 99 would be complete dead ends, but you’d only find out after months
2:20:52 of work, he must’ve had some sense that this was the right track to pursue because it takes
2:20:53 years to get them from A to B.
2:20:58 So you’ve done, like you said, actually, you see, even strictly mathematically, but more
2:21:05 broadly in terms of the process, you’ve done similarly difficult things.
2:21:08 What, what can you infer from the process he was going through?
2:21:09 Cause he was doing it alone.
2:21:12 What are some low points in a process like that?
2:21:17 When you start to like, you’ve mentioned hardship, like, uh, AI doesn’t know when it’s
2:21:18 failing.
2:21:19 What happens to you?
2:21:24 You’re sitting in your office when you realize the thing you did the last few
2:21:27 days, maybe weeks is a failure.
2:21:28 Well, for me, I switched to a different problem.
2:21:32 Uh, so, uh, I’m, I’m, I’m, I’m a fox.
2:21:32 I’m not a hedgehog.
2:21:36 But you legitimately, that is a break that you can take is, is to step away and look at
2:21:37 a different problem.
2:21:37 Yeah.
2:21:39 You can modify the problem too.
2:21:43 Um, I mean, um, yeah, you can ask them if, if there’s a specific thing that’s blocking
2:21:49 you at that, just, um, some bad case keeps showing up that, that, that for which your
2:21:53 tool doesn’t work, you can just assume by fiat this, this bad case doesn’t occur.
2:21:58 So you, you do some magical thinking, um, for the, you know, but, but strategically,
2:21:58 okay.
2:22:02 For the point to see if the rest of the argument goes through, um, if there’s multiple problems,
2:22:05 uh, with, with, with your approach, then maybe you just give up.
2:22:05 Okay.
2:22:09 But if this is the only problem that, you know, then everything else checks out, then it’s
2:22:10 still worth fighting.
2:22:17 Um, so yeah, you have to do some, some, so forward reconnaissance sometimes to, uh, you
2:22:17 know.
2:22:20 And that is sometimes productive to assume like, okay, we’ll figure it out.
2:22:21 Oh yeah.
2:22:21 Yeah.
2:22:22 Eventually.
2:22:24 Um, sometimes actually it’s, it’s even productive to make mistakes.
2:22:31 So, um, one of the, I mean, um, there was a project which actually, uh, we won some prizes
2:22:36 for actually, but, uh, before other people, um, we worked on this PD problem again, actually
2:22:37 this blow off regularity type problem.
2:22:39 Um, and it was considered very hard.
2:22:45 Um, Jean Bougain, um, uh, who was another field’s methodist who worked on a special case
2:22:47 of this, but he could not solve the general case.
2:22:51 Um, and we worked on this problem for two months and we found, we thought we solved it.
2:22:56 We, we had this, this cute argument that if anything fit and we were excited, uh, we were
2:22:59 planning celebration to all get together and have champagne or something.
2:23:02 Um, and we started writing it up.
2:23:06 Um, and one of, one of us, not me actually, but another co-author said, oh,
2:23:11 um, in this, in this lemma here, we, um, we have to estimate these 13 terms that, that
2:23:15 show up in this expansion and we estimate 12 of them, but in our notes, I can’t find the
2:23:16 estimation of the 13th.
2:23:17 Can you, can someone supply that?
2:23:19 And I said, sure, I’ll look at this.
2:23:21 And like you said, yeah, we didn’t cover that.
2:23:22 We completely omitted this term.
2:23:25 And this term turned out to be worse than the other 12 terms put together.
2:23:27 Um, in fact, we could not estimate this term.
2:23:30 Um, and we tried for a few more months and all different permutations.
2:23:34 And there was always this one thing, one term that we could not control.
2:23:38 Um, and so like, um, this was very frustrating.
2:23:44 Um, but because we had already invested months and months of evidence already, um, we stuck
2:23:47 at this, which we tried increasingly desperate things and crazy things.
2:23:52 Um, and after two years, we found that approach is somewhat different, but quite a bit from
2:23:57 our initial, um, strategy, which did actually didn’t generate these problematic terms and, and
2:23:58 actually solve the problem.
2:24:03 So we, we solved the problem after two years, but if we hadn’t had that initial full storm
2:24:07 of nearly solving the problem, we would have given up by month two or something and worked
2:24:08 on an easier problem.
2:24:13 Um, yeah, if we had known it would take two years, not sure we would have started the project.
2:24:14 Yeah.
2:24:18 Sometimes actually having the incorrect, you know, it’s, it’s like Columbus traveling to the
2:24:22 new world, the incorrect version of a measurement of the size of the earth.
2:24:27 Um, he thought he was going to find a new trade route to India, uh, or at least that was how
2:24:28 he sold it in his prospectus.
2:24:35 I mean, it could be that he secretly knew, but just on a psychological element, do you have
2:24:41 like emotional or like self doubt that just overwhelms you moments like that?
2:24:44 You know, cause this stuff, it feels like math.
2:24:51 It’s, it’s so engrossing that like it can break you when you like invest so much yourself
2:24:53 on the problem and then it turns out wrong.
2:24:58 You could start to similar way chess has broken some people.
2:24:58 Yeah.
2:25:04 Um, I, I think different mathematicians have different levels of emotional investment in
2:25:05 what they do.
2:25:08 I mean, I think for some people, it’s just a job, you know, you, you have a problem and
2:25:10 if it doesn’t work out, you, you, you go on the next one.
2:25:12 Um, yeah.
2:25:18 So the fact that you can always move on to another problem, um, it reduces the emotional
2:25:18 connection.
2:25:23 I mean, there are cases, you know, so there are certain problems that are what are called
2:25:27 mathematical diseases where, where, where, where just latch onto that one problem and
2:25:30 they spend years and years thinking about nothing but that one problem.
2:25:34 And, um, you know, maybe the, the career suffers and so forth.
2:25:36 You say, oh, but I’ll get this big win.
2:25:42 This will, you know, once I, once I finish this problem, I will make up for all the years
2:25:44 of, of, of, of lost opportunity.
2:25:51 And that’s, that’s, I mean, occasionally, occasionally it works, but I, I, um, I really don’t recommend
2:25:53 it for people who have the right fortitude.
2:25:54 Yeah.
2:25:57 So I, I, I’ve never been super invested in any one problem.
2:26:01 Um, one thing that helps is that we don’t need to call our problems in advance.
2:26:07 Uh, um, well, uh, when we do grant proposals, uh, we sort of say we will, we will study this
2:26:12 set of problems, but even though we don’t promise definitely by five years, I will supply a proof
2:26:13 of all these things.
2:26:18 You know, um, you promise to make some progress or discover some interesting phenomena.
2:26:23 Uh, and maybe you don’t solve the problem, but you find some related problem that you can
2:26:23 say something new about.
2:26:26 Uh, and that’s, that’s a much more feasible task.
2:26:29 But I’m sure for you, there’s problems like this.
2:26:36 You have, you have, um, made so much progress towards the hardest problems in the history of
2:26:37 mathematics.
2:26:41 So is there, is there a problem that just haunts you?
2:26:46 It sits there in the dark corners, you know, twin prime conjecture, Riemann hypothesis,
2:26:47 global conjecture.
2:26:52 Twin prime, that sounds, well, again, so, I mean, the problem is like Riemann hypothesis,
2:26:53 those are so far out of reach.
2:26:55 Do you think so?
2:26:57 Yeah, there’s no even viable strategy.
2:27:03 Like, even if I activate all my, all the cheats that I know of in this problem, like it, there’s
2:27:04 just still no way to get made to be.
2:27:12 Um, like it’s, it’s, um, I think it needs a breakthrough in another area of mathematics to
2:27:12 happen first.
2:27:17 And for someone to recognize that it would be a useful thing to transport into this problem.
2:27:22 So we, we should maybe step back for a little bit and just talk about prime numbers.
2:27:22 Okay.
2:27:25 So they’re often referred to as the atoms of mathematics.
2:27:30 Can you just speak to the structure that these, uh, atoms provide?
2:27:34 The natural numbers have two basic operations attached to them, addition and multiplication.
2:27:38 Um, so if you want to generate the natural numbers, you can do one of two things.
2:27:41 You can just start with one and add one to itself over and over again.
2:27:42 And that generates you the natural numbers.
2:27:46 So additively, they’re very easy to generate one, two, three, four, five, or you can take
2:27:47 the prime number.
2:27:49 If you want to generate multiplicatively, you can take all the prime numbers, two, three,
2:27:51 five, seven, and multiply them all together.
2:27:55 Um, and together, they, they, they gives you all the, the, the natural numbers, except maybe
2:27:56 for one.
2:28:00 So there are these two separate ways of thinking about the natural numbers from an additive
2:28:02 point of view and a more multiplicative point of view.
2:28:05 Um, and separately, they’re not so bad.
2:28:10 Um, so like any question about that natural numbers that only was addition is relatively
2:28:10 easy to solve.
2:28:14 And any question that only was multiplication is relatively easy to solve.
2:28:17 Um, but what has been frustrating is that you combine the two together.
2:28:23 Um, and suddenly you get the extremely rich, I mean, we know that there are statements in
2:28:25 number theory that are actually as undecidable.
2:28:27 There are certain polynomials in some number of variables.
2:28:29 Is there a solution in the natural numbers?
2:28:31 And the answer depends on, on an undecidable statement.
2:28:36 Um, like, like whether, um, the axioms of mathematics are consistent or not.
2:28:43 Um, but, um, yeah, but even the, the simplest problems that combine something more applicative
2:28:48 such as the primes with something additive such as shifting by two, uh, separately, we understand
2:28:52 both of them well, but if you ask, when you shift the prime by two, do you, can you get
2:28:54 a, how often can you get another prime?
2:28:58 We, it’s been amazingly hard to relate the two.
2:29:04 And we should say that the twin prime conjecture is just that it posits that there are infinitely
2:29:06 many pairs of prime numbers that differ by two.
2:29:13 Now, the interesting thing is that you have been very successful at pushing forward the
2:29:18 field and answering these complicated questions, uh, of this variety.
2:29:23 Like you mentioned the green tile theorem, it proves that prime numbers contain arithmetic
2:29:24 progressions of any length.
2:29:24 Right.
2:29:27 It’s just mind blowing that you can prove something like that.
2:29:27 Right.
2:29:28 Yeah.
2:29:33 So what we’ve realized because of this, this, this type of research is that there’s different
2:29:36 patterns have different levels of, uh, interstructibility.
2:29:41 Um, so, so what makes the twin prime conjecture hard is that you can take all the primes in
2:29:44 the world, you know, three, five, seven, 11, so forth.
2:29:46 There are some twins in there.
2:29:51 11 and 13 is a twin prime, pair of twin primes and so forth, but you could easily, if you
2:29:56 wanted to, um, redact the primes to get rid of, to get rid of the, um, these twins.
2:30:00 Like the twins, they show up and there are infinitely many of them, but they’re actually reasonably
2:30:01 sparse.
2:30:04 Um, not, there’s, there’s not, I mean, initially there’s quite a few, but once you got to the
2:30:07 millions, trillions, they become rarer and rarer.
2:30:12 And you could actually just, you know, if, if, if someone was given access to the database
2:30:15 of primes, you just edit out a few, a few primes here and there, they could make the twin
2:30:20 prime conjecture false by just removing like 0.01% of the primes or something, um, just
2:30:23 well, well chosen to, to, um, to do this.
2:30:30 And so you could present a censored database of the primes, which passes all of the statistical
2:30:34 tests of the primes, you know, that it obeys things like the polynomial theorem and other
2:30:37 things about the primes, but it doesn’t contain any trim primes anymore.
2:30:40 Um, and this is a real obstacle for the twin prime conjecture.
2:30:48 It means that any proof strategy to actually find twin primes in the actual primes must fail
2:30:51 when applied to these slightly edited primes.
2:30:57 And so it must be some very, um, subtle, delicate feature of the primes that you can’t just get
2:31:00 from like, like aggregate statistical analysis.
2:31:01 Okay.
2:31:02 So that’s all.
2:31:02 Yeah.
2:31:06 On the other hand, I think progressions has turned out to be much more robust.
2:31:10 um, like you can take the primes and you can eliminate 99% of the primes actually, you
2:31:13 know, and you can take, take any 99% you want.
2:31:17 And, uh, it turns out, and another thing we proved is that you still get as make progressions.
2:31:22 Um, as make progressions are much, you know, they’re like cockroaches of arbitrary length.
2:31:23 Yes, yes.
2:31:24 That’s crazy.
2:31:25 Yeah.
2:31:30 I mean, so, so, uh, for, for people who don’t know arithmetic progressions is a sequence of
2:31:31 numbers that differ by some fixed amount.
2:31:32 Yeah.
2:31:34 But it’s again, like it’s, it’s an infinite monkey type phenomenon.
2:31:38 For any fixed length of your set, you don’t get arbitrary lengths of progressions.
2:31:40 You only get quite short progressions.
2:31:43 But you’re saying twin primes is not an infinite monkey phenomenon.
2:31:47 I mean, it’s a very subtle, it’s still an infinite monkey phenomenon.
2:31:47 Right.
2:31:48 Yeah.
2:31:53 If the primes were really genuinely random, if the primes were generated by monkeys, um,
2:31:56 then yes, in fact, the infinite monkey theorem would.
2:32:02 Oh, but you’re saying that twin prime is, it doesn’t, you can’t use the same tools.
2:32:04 Like the, it doesn’t appear random almost.
2:32:05 Well, we don’t know.
2:32:09 Uh, yeah, we, we, we, we believe the primes behave like a random set.
2:32:14 And so the reason why we care about the twin prime conjecture is it’s a test case for whether
2:32:19 we can genuinely confidently say with, with 0% chance of error that the primes behave like
2:32:20 a random set.
2:32:20 Okay.
2:32:23 Random, yeah, random versions of the primes we know contain twins.
2:32:29 Um, at least we’re, we’re, we’re 100% probably, uh, or probably tending to 100% as you go
2:32:30 out further and further.
2:32:32 Um, yeah.
2:32:36 So the primes we believe that they’re random, um, the reason why ethnic progressions are
2:32:41 indestructible is that regardless of whether you’re saying it looks random or looks, um,
2:32:46 structured, like periodic, in both cases, um, ethnic progressions appear, but for different
2:32:47 reasons.
2:32:51 Um, and this is basically all the ways in which the thing, uh, there are many proofs
2:32:55 of, of these sort of ethnic progression epithereums, and they’re all proven by some sort of dichotomy
2:32:57 where your set is either structured or random.
2:33:00 And in both cases you can say something and then you put the two together.
2:33:06 Um, but in twin primes, if, if the primes are random, then you’re happy, you win.
2:33:11 If your primes are structured, they can be structured in a specific way that eliminates the
2:33:12 twin, the twins.
2:33:15 Uh, and we can’t rule out that one conspiracy.
2:33:20 And yet you were able to make a, as I understand, progress on the K-tuple version.
2:33:21 Right.
2:33:21 Yeah.
2:33:25 So, um, the, the one funny thing about conspiracies is that any one conspiracy theory is really
2:33:26 hard to disprove.
2:33:27 Uh-huh.
2:33:30 That, you know, if you believe the world is run by lizards, you say, here’s some evidence
2:33:34 that, that it, it, not run by lizards, but that, that evidence was planted by lizards.
2:33:38 So, um, you may have encountered, uh, uh, this kind of phenomenon.
2:33:38 Yeah.
2:33:44 So, like, like, um, a pure, like, there’s, there’s almost no way to, um, definitively rule out
2:33:44 a conspiracy.
2:33:49 And the same is true in mathematics, that a conspiracy is solely devoted to eliminating
2:33:50 twin primes.
2:33:53 You know, like, you would, you would have to also infiltrate other areas of mathematics
2:33:56 to sort of, but, but like, it could be made consistent, at least as far as we know.
2:34:02 But there’s a weird phenomenon that you can make one, um, uh, one conspiracy rule out other
2:34:03 conspiracies.
2:34:07 So, you know, if the, if the world is, is run by lizards, they can’t also be run by aliens.
2:34:08 Right.
2:34:09 Right.
2:34:12 So one unreasonable thing is, is, is, is, is hard to disprove, but, but more than one,
2:34:13 there are, there are tools.
2:34:15 Um, so, yeah.
2:34:19 So, for example, we, we know there’s infinitely many primes that are, um, uh, no two, which
2:34:24 are, um, so the infinite pairs of primes which differ by at most, uh, um, 246 actually
2:34:26 is, is, is, is, is, is, is the current.
2:34:27 So there’s like a bound.
2:34:27 Yes.
2:34:28 On the.
2:34:28 Right.
2:34:33 So like there’s twin primes, there’s things called cousin primes that differ by, by four.
2:34:35 Um, there’s things called sexy primes that differ by six.
2:34:37 Uh, what are sexy primes?
2:34:38 Primes that differ by six.
2:34:42 The name, the name is much less, it costs as much less exciting than the name suggests.
2:34:42 Got it.
2:34:48 Um, so you can make a conspiracy rule out one of these, but like once you have like 50
2:34:50 of them, it turns out that you can’t rule out all of them at once.
2:34:54 It just, it requires too much energy somehow in this conspiracy space.
2:34:56 How do you do the bound part?
2:35:02 How do you, how do you develop a bound for the difference between the primes that there’s
2:35:03 an infinite number of?
2:35:05 So it’s ultimately based on, uh, what’s called the pigeonhole principle.
2:35:09 Um, so the pigeonhole principle, uh, it’s a statement that if you have a number of pigeons
2:35:14 and they all have to go into pigeonholes and you have more pigeons than pigeonholes, then
2:35:16 one of the pigeonholes has to have at least two pigeons in.
2:35:17 So there has to be two pigeons that are close together.
2:35:22 So for instance, if you have a hundred numbers and they all range from one to a thousand,
2:35:27 um, two of them have to be at most 10 apart because you can divide up the numbers from one
2:35:29 to a hundred into 100 pigeonholes.
2:35:33 Let’s, let’s say if you have a hundred, if you have 101 numbers, 101 numbers, then two
2:35:37 of them have to be, uh, distance less than 10 apart because two of them have to belong to
2:35:37 the same pigeonhole.
2:35:43 So it’s a basic, um, basic feature of, uh, a basic principle in mathematics.
2:35:48 Um, so it doesn’t quite work with the primes already because the primes get sparser and sparser
2:35:51 as you go out, that, that fewer and fewer numbers are prime.
2:35:56 But it turns out that there’s a way to assign weights to the, to, to numbers.
2:36:00 Like, um, so there are numbers that are kind of almost prime, uh, but they’re not, they,
2:36:04 they don’t have no factors at all other than themselves in one, but they have very few
2:36:05 factors.
2:36:09 Um, and it turns out that we understand almost primes a lot better than we can assign
2:36:09 primes.
2:36:14 Um, and so, for example, it was known for a long time that there were twin almost primes.
2:36:15 This has been worked out.
2:36:17 So almost primes are something we can’t understand.
2:36:22 So you can actually restrict attention to a suitable set of almost primes.
2:36:30 And, uh, whereas the primes are very sparse overall, uh, relative to the almost primes, they
2:36:31 actually are much less sparse.
2:36:35 They may, um, you can set up a set of almost primes where the primes are density like, say,
2:36:35 one percent.
2:36:41 Um, and that gives you a shot at proving by applying some sort of original principle that,
2:36:43 that there’s pairs of primes that are just only a hundred, a hundred apart.
2:36:47 But in order to prove the twin prime conjecture, you need to get the density of primes in something
2:36:49 almost up to, up to a threshold of 50%.
2:36:52 Um, once you get up to 50%, you will get twin primes.
2:36:54 But, uh, unfortunately there are barriers.
2:37:00 Um, we know that, that no matter what kind of good set of almost primes you pick, the density
2:37:01 of primes can never get above 50%.
2:37:03 It’s called the parity barrier.
2:37:05 Um, and I would love to find, yeah.
2:37:09 So one of my long-term dreams is to find a way to breach that barrier because it would
2:37:13 open up not only the twin prime conjecture, the Goldbach conjecture, and many other problems
2:37:18 in number theory are currently blocked because our current techniques would require going beyond
2:37:21 this theoretical, um, parity barrier.
2:37:23 It’s like, it’s like, it’s like pulling past the speed of light.
2:37:23 Yeah.
2:37:27 So we should say a twin prime conjecture, one of the biggest problems in the history of
2:37:32 mathematics, Goldbach conjecture also, um, they feel like next door neighbors.
2:37:36 Uh, is there been days when you felt you saw the path?
2:37:37 Oh yeah.
2:37:39 Um, um, yeah.
2:37:42 Uh, sometimes you try something and it works super well.
2:37:48 Um, you, you, again, again, the sense of mathematical smell, uh, we talked about earlier, uh, you learn
2:37:53 from experience when things are going too well because there are certain difficulties that
2:37:54 you sort of have to encounter.
2:38:01 Um, um, I think the way a colleague might put it is that, um, you know, like if, if you are
2:38:06 on the streets of New York and you put in a blindfold and you put in a car and, and, uh, after some
2:38:11 hours, um, you, the blindfold is off and you’re in Beijing, um, you know, I mean, that was too
2:38:12 easy somehow.
2:38:14 Like, like there was no ocean being crossed.
2:38:19 Um, even if you don’t know exactly what, how, what, what was done, uh, you’re suspecting that
2:38:20 there’s something that wasn’t right.
2:38:26 But is that still in the back of your head to, do you return to these, to the prime, do you return
2:38:29 to the prime numbers every once in a while to see?
2:38:29 Yeah.
2:38:33 When I have nothing better to do, which is less and less than I have, which is, I get busy
2:38:37 with so many things these days, but yeah, when I have free time and I’m not, and I’m too
2:38:40 frustrated to, to work on my sort of real research projects.
2:38:44 And I also don’t want to do my administrative stuff, but I don’t want to do some errands
2:38:44 for my family.
2:38:48 Um, I can play with these, these things, um, for fun.
2:38:50 Uh, and usually you get nowhere.
2:38:50 Yeah.
2:38:52 You have to learn to just say, okay, fine.
2:38:54 I, once again, nothing happened.
2:38:55 I will, I will move on.
2:39:01 Um, yeah, very occasionally one of these problems I actually solved, uh, well, sometimes as you
2:39:05 say, you think you solved it and then you’re euphoric for, uh, maybe 15 minutes.
2:39:09 And then you think I should check this because this is too easy to be true.
2:39:10 And it usually is.
2:39:16 What’s your gut say about when these problems would be, uh, solved twin prime and go back?
2:39:16 Prime.
2:39:19 I think we’ll keep getting, keep getting more partial results.
2:39:23 Um, it doesn’t need at least one.
2:39:26 This parity barrier is, is the biggest remaining obstacle.
2:39:31 Um, there are simpler versions of the conjecture where we are getting really close.
2:39:38 Um, so I think we will, in 10 years, we will have many more, much closer results.
2:39:39 We may not have the whole thing.
2:39:40 Um, yeah.
2:39:42 So twin primes is somewhat close.
2:39:46 Riemann hypothesis, I have no, I mean, it has to happen by accident.
2:39:51 I think, uh, so the Riemann hypothesis is a kind of more general conjecture about the distribution
2:39:52 of prime numbers.
2:39:52 Right.
2:39:53 Yeah.
2:39:56 It’s, it’s, it’s, it’s sort of viewed more applicatively, like for, for questions only
2:40:01 involving multiplication, no addition, the primes really do behave as randomly as, as you
2:40:01 could hope.
2:40:07 So there’s a phenomenon in probability called square root cancellation that, um, you know,
2:40:13 like if you want to poll say America upon some issue, um, and you, you ask one or two voters
2:40:17 and you may have sampled a bad sample and then you get, you get a really imprecise, um, measurement
2:40:22 of the, of the full average, but if you sample more and more people, the accuracy gets better
2:40:22 and better.
2:40:27 And the accuracy improves like the square root of the number of people you, uh, you sampled.
2:40:31 So yeah, if you sample, um, a thousand people, you can get like a two, three percent margin
2:40:31 of error.
2:40:36 So in the same sense, if you measure the primes in a certain multiplicative sense, there’s a
2:40:40 certain type of statistic you can measure and it’s called the Riemann’s data function and
2:40:41 it fluctuates up and down.
2:40:46 But in some sense, um, as you keep averaging more and more, if you sample more and more,
2:40:48 the fluctuations should go down as if they were random.
2:40:50 And there’s a very precise way to quantify that.
2:40:54 And the Riemann hypothesis is a very elegant way that captures this.
2:40:58 But, um, as with many other ways in mathematics, we have,
2:41:02 very few tools to show that something really genuinely behaves like really random.
2:41:06 And this is actually not just a little bit random, but it’s, it’s asking that it behaves
2:41:08 as random as it actually random set.
2:41:10 This, this, this square root cancellation.
2:41:15 And we know because of things related to the parity problem, actually, that most of us,
2:41:18 usual techniques cannot hope to settle this question.
2:41:21 Um, the proof has to come out of left field.
2:41:28 Um, yeah, but, uh, what that is, yeah, no one has any serious proposal.
2:41:32 Um, yeah, and, and there’s, there’s various ways to sort of, as I said, you can modify the
2:41:35 primes a little bit and you can destroy the Riemann hypothesis.
2:41:38 Um, so like, it has to be very delicate.
2:41:41 You can’t apply something that has huge margins of error.
2:41:43 It has to just barely work.
2:41:49 Um, and like, um, there’s like all these pits, pitfalls that you like dodge very adeptly.
2:41:51 The prime numbers is just fascinating.
2:41:52 Yeah, yeah, yeah.
2:41:57 What, what to you is, um, most mysterious about the prime numbers?
2:42:00 That’s a good question.
2:42:03 So like conjecturally, we have a good model of them.
2:42:06 I mean, like, as I said, I mean, they have certain patterns, like the primes are usually
2:42:07 odd, for instance.
2:42:11 But apart from these sort of obvious patterns, they behave very randomly and just assuming
2:42:12 that they behave.
2:42:16 So there’s something called the Kramer random model of the primes, that, that, that after
2:42:18 a certain point, primes just behave like a random set.
2:42:22 Um, and there’s various slight modifications to this model, but this has been a very good
2:42:22 model.
2:42:24 It matches the numerics.
2:42:26 It tells us what to predict.
2:42:28 Like I can tell you with complete certainty, the true and prime conjecture is true.
2:42:31 Uh, the random model gives overwhelming odds it is true.
2:42:32 I just can’t prove it.
2:42:37 Most of our mathematics is optimized for solving things with patterns.
2:42:46 Um, and the primes have this anti-patent, um, as do almost everything really, but we can’t
2:42:46 prove that.
2:42:47 Yeah.
2:42:50 I guess it’s not mysterious that the primes be random, it’s kind of random because there’s
2:42:57 sort of no reason for them to be, um, uh, to have any kind of secret pattern, but what is
2:43:01 mysterious is what is the mechanism that really forces the randomness to happen.
2:43:03 Uh, and this is just absent.
2:43:09 Another incredibly surprisingly difficult problem is the collage conjecture.
2:43:09 Oh, yes.
2:43:17 Simple to state, beautiful to visualize in its simplicity, and yet extremely, uh, difficult
2:43:18 to solve.
2:43:20 And yet you have been able to make progress.
2:43:26 Uh, uh, Paul Urdar said about the collage conjecture that mathematics may not be ready for
2:43:27 such problems.
2:43:32 Others have stated that it is an extraordinarily difficult problem, completely out of reach.
2:43:35 This is in 2010, out of reach of present day mathematics.
2:43:37 And yet you have made some progress.
2:43:39 Why is it so difficult to make?
2:43:41 Can you actually even explain what it is?
2:43:41 Oh, yeah.
2:43:41 Yeah.
2:43:43 So it’s, it’s, it’s a problem that you can explain.
2:43:49 Um, yeah, it, um, it helps with some, um, visual aids, but yeah.
2:43:53 So you take any natural number, like say 13 and you apply the following procedure to it.
2:43:58 So if it’s even, you divide it by two and if it’s odd, you multiply it by three and add
2:43:59 one.
2:44:01 So even numbers get smaller, odd numbers get bigger.
2:44:04 So 13, uh, would become 40 because 13 times three is 39.
2:44:05 Add one, you get 40.
2:44:09 So it’s a simple process for odd numbers and even numbers.
2:44:10 They’re both very easy operations.
2:44:11 And then you put it together.
2:44:13 It’s still reasonably simple.
2:44:16 Um, but then you ask what happens when you iterate it.
2:44:18 You take the output that you just got and feed it back in.
2:44:20 So 13 becomes 40.
2:44:22 40 is now even divided by two is 20.
2:44:27 20 is still even divided by two, 10, five, and then five times three plus one is 16.
2:44:29 And then eight, four, two, one.
2:44:33 So, uh, and then from one, it goes one, four, two, one, four, two, one.
2:44:33 It cycles forever.
2:44:39 So the sequence I just described, um, you know, 13, 40, 20, 10, so both, uh, these are also
2:44:44 called hailstone sequences because there’s an oversimplified model of, of hailstone formation,
2:44:47 you know, which is not actually quite correct, but it’s still somehow taught to high school
2:44:53 students as a first approximation is that, um, like a little nugget of ice gets, gets a nice
2:44:53 crystal.
2:44:57 It forms in a cloud and it goes up and down because of the wind.
2:45:02 And sometimes when it’s cold, it requires a bit more mass and maybe it melts a little bit.
2:45:06 And this process of going up and down creates this sort of partially melted ice, which eventually
2:45:07 causes hailstone.
2:45:09 And eventually it falls down to the earth.
2:45:14 So the conjecture is that no matter how high you start up, like you take a number, which
2:45:18 is in the millions or billions, you go, this process that goes up if you’re odd and down,
2:45:22 if you’re even, it eventually goes down to earth all the time.
2:45:27 No matter where you start with this very simple algorithm, you end up at one and you
2:45:28 might climb for a while.
2:45:28 Right.
2:45:29 Yeah.
2:45:30 So it’s now, yeah.
2:45:33 If you plot it, um, these sequences, they look like Brownian motion.
2:45:37 Um, they look like the stock market, you know, they just go up and down in a, in a seemingly
2:45:38 random pattern.
2:45:43 And in fact, usually that’s what happens that if you plug in a random number, you can actually
2:45:46 prove that at least initially that it would look like, um, a random walk.
2:45:49 Um, and that’s actually a random walk with a downward drift.
2:45:55 Um, it’s like, if you’re always gambling on a roulette at the casino with odds slightly
2:45:55 weighted against you.
2:46:00 So sometimes you, you win, sometimes you lose, but over in the long run, you lose a bit more
2:46:01 than you win.
2:46:04 Um, and so normally your wallet will hit, will go to zero.
2:46:06 Um, if you just keep playing over and over again.
2:46:08 So statistically it makes sense.
2:46:09 Yes.
2:46:16 So, so the result that I, I proved roughly speaking is that, that statistically like 90% of all inputs
2:46:21 would, would drift down to maybe not all the way to one, but to be much, much smaller
2:46:22 than what you started.
2:46:27 So it’s, it’s like, if I told you that if you go to a casino, most of the time you end
2:46:31 up, if you keep playing up long enough, you end up with a smaller amount in your wallet
2:46:31 than when you started.
2:46:34 Um, that’s kind of like the, what the result that I proved.
2:46:36 So why is that result?
2:46:41 Like, can you continue down that thread to prove the full conjecture?
2:46:46 Well, the, the problem is that, um, my, I used arguments from probability theory.
2:46:48 Um, and there’s always this exceptional event.
2:46:53 So, you know, so in probability we have this, this low, large numbers, um, which tells you
2:46:58 things like if you play a casino with a, um, a game at a casino with a losing, um, expectation
2:47:04 over time, you are guaranteed or almost surely with probably probability as close to 100% as
2:47:05 you wish, you’re guaranteed to lose money.
2:47:08 But there’s always this exceptional outlier.
2:47:13 Like it is mathematically possible that even in the game is, is the odds are not in your
2:47:13 favor.
2:47:18 You could just keep winning slightly more often than you lose very much like how in Navier
2:47:22 Stokes, it could be, you know, um, most of the time, um, your waves can disperse.
2:47:27 There could be just one outlier choice of initial conditions that would lead you to blow up.
2:47:34 And there could be one outlier choice of, um, um, a special number that you stick in that
2:47:38 shoots off infinity while all other numbers crash to earth, uh, crash to one.
2:47:44 Um, in fact, um, there’s some mathematicians, um, who’ve, uh, Alex Kontorovic, for instance,
2:47:50 who’ve proposed that, um, that actually, um, these Kaldats, uh, iterations are like these
2:47:51 cellular automator.
2:47:55 Um, um, yeah, actually, if you look at what happened in binary, they do actually look a little
2:47:57 bit like, like these game of life type patterns.
2:48:03 Um, and in an analogy to how the game of life can create these, these massive, like self-applicating
2:48:07 objects and so forth, possibly you could create some sort of heavier than air flying machine,
2:48:13 a number, which is actually encoding this machine, which is just whose job it is to encode
2:48:16 is to create a version of a cell, which is, which is larger.
2:48:22 Heavier than air machine encoded in a number that flies forever.
2:48:22 Yeah.
2:48:25 So Conway, in fact, worked on, worked on this problem as well.
2:48:25 Oh, wow.
2:48:30 So Conway, um, so similar, in fact, that was one of my inspirations for the Navi, Navi Stokes
2:48:35 project, that Conway studied generalizations of the Kaldats problem, where instead of
2:48:39 multiplying by three and adding one or dividing by two, you have more complicated branching
2:48:43 rules, but, but instead of having two cases, maybe you have 17 cases and then you go up
2:48:43 and down.
2:48:49 And he showed that once your iteration gets complicated enough, you can actually encode
2:48:52 Turing machines and you can actually make these problems undecidable and do things like
2:48:52 this.
2:48:58 In fact, he invented a programming language for, uh, these kind of fractional linear transformations.
2:49:06 And he showed that you can, um, you can, um, you can program if it was too incomplete, you
2:49:10 could, you could, you could, uh, um, you could make a program that if, if your number you insert
2:49:14 in was encoded as a prime, it would sink to zero, it would go down, otherwise it would go
2:49:16 up, uh, and things like that.
2:49:22 Um, so the general class of problems is, is really, uh, as complicated as all the mathematics.
2:49:27 Some of the mystery of the cellular automata that we talked about, uh, having a mathematical
2:49:33 framework to say anything about cellular automata, maybe the same kind of framework is required.
2:49:33 Yeah.
2:49:34 Yeah.
2:49:34 Yeah.
2:49:40 If you want to do it, not statistically, but you really want 100%, 100% of all inputs
2:49:41 to, to, to, for the earth.
2:49:41 Yeah.
2:49:48 So what might be feasible is, is, is assisting 99%, you know, go to one, but like everything,
2:49:50 you know, uh, that looks hard.
2:49:56 What would you say is out of these within reach famous problems is the hardest problem we have
2:49:57 today?
2:49:58 Is the Riemann hypothesis?
2:50:00 Riemann is up there.
2:50:05 Um, P equals NP is, is a good one because like, uh, that’s, that’s, that’s a meta problem.
2:50:10 Like if you solve that in the, um, in the positive sense that you can find a P equals NP algorithm,
2:50:13 then potentially this solves a lot of other problems as well.
2:50:17 And we should mention some of the conjectures we’ve been talking about, you know, a lot of
2:50:19 stuff is built on top of them now.
2:50:20 There’s ripple effects.
2:50:23 P equals NP has more ripple effects than basically any other.
2:50:23 Right.
2:50:30 If the Riemann hypothesis is disproven, um, that’d be a big mental shock to the number of
2:50:35 theorists, uh, but it would have follow on effects for, um, cryptography.
2:50:41 Um, because a lot of cryptography uses number theory, um, uses number theory constructions
2:50:42 involving primes and so forth.
2:50:47 And, um, it relies very much on the intuition that number of those are built over many, many
2:50:51 years of what operations involving primes behave randomly and what ones don’t.
2:50:58 Um, and in particular, um, encryption, um, methods are designed to turn text with information
2:51:02 on it into text, which is indistinguishable from, um, from random noise.
2:51:08 So, um, and hence we believe to be almost impossible to crack, um, at least mathematically.
2:51:16 Um, but, uh, if something as core to our beliefs as the Riemann hypothesis is wrong, it means that
2:51:20 there are, there are actual patterns of the primes that we’re not aware of.
2:51:25 And if there’s one, there’s probably going to be more, um, and suddenly a lot of our crypto
2:51:26 systems are in doubt.
2:51:27 Yeah.
2:51:32 But then how do you then say stuff about the primes?
2:51:33 Yeah.
2:51:37 Like you’re going towards the, uh, collect conjecture again.
2:51:41 Um, because if I, I, you, you want it to be random, right?
2:51:42 You want it to be random.
2:51:43 Yeah.
2:51:46 So more broadly, I’m just looking for more tools, more ways to show that, that, that things
2:51:47 are random.
2:51:49 How do you prove a conspiracy doesn’t happen?
2:51:49 Right.
2:51:53 Is there any chance to you that P equals NP?
2:51:56 Is there some, can you imagine a possible universe?
2:51:57 It is possible.
2:52:00 I mean, there’s, there’s various, uh, scenarios.
2:52:04 I mean, there’s, there’s one where it is technically possible, but in fact, it’s never
2:52:05 actually implementable.
2:52:10 The evidence is sort of slightly pushing in favor of no, that we’d probably P is not equal
2:52:10 to NP.
2:52:14 I mean, it seems like it’s one of those cases seem more similar to Riemann hypothesis that
2:52:19 I think the evidence is leaning pretty heavily on the no.
2:52:21 Certainly more on the no than on the yes.
2:52:25 The funny thing about P equals NP is that we have also a lot more obstructions than we do
2:52:26 for almost any other problem.
2:52:31 Um, so while there’s evidence, uh, we also have a lot of results ruling out many, many
2:52:33 types of approaches to the problem.
2:52:36 Uh, this is the one thing that the computer scientists have actually been very good at.
2:52:39 It’s actually saying that, that certain approaches cannot work.
2:52:40 No go theorems.
2:52:41 It could be undecidable.
2:52:42 We don’t, yeah, we don’t know.
2:52:47 There’s a funny story I read that when you won the Fields Medal, somebody from the internet
2:52:54 wrote you and asked, uh, you know, what are you going to do now that you’ve won this prestigious
2:52:54 award?
2:53:00 And then you just quickly, very humbly said that, you know, this, a shiny medal is not
2:53:01 going to solve any of the problems I’m currently working on.
2:53:04 So I’m just, I’m going to keep working on them.
2:53:08 It’s just, first of all, it’s funny to me that you would answer an email in that context.
2:53:14 And second of all, it, um, it just shows your humility, but anyway, uh, maybe you could speak
2:53:21 to the Fields Medal, but it’s another way for me to ask, uh, about, uh, Gregorio Perlman.
2:53:26 What do you think about him famously declining the Fields Medal and the millennial prize, which
2:53:30 came with a $1 million of prize money?
2:53:32 He stated that I’m not interested in money or fame.
2:53:35 The prize is completely irrelevant for me.
2:53:39 If the proof is correct, then no other recognition is needed.
2:53:40 Yeah.
2:53:45 No, he’s, he’s somewhat of an outlier, um, even among mathematicians who tend to, uh, to
2:53:47 have, uh, somewhat idealistic views.
2:53:48 Um, I’ve never met him.
2:53:51 I think I’d be interested to meet him one day, but I never had the chance.
2:53:54 I know people who met him, but he’s always had strong views about certain things.
2:53:58 Um, you know, I mean, it’s, it’s not like he was completely isolated from the math community.
2:54:01 I mean, he would, he would give talks and write papers and so forth.
2:54:04 Um, but at some point he just decided not to engage with the rest of the community.
2:54:07 There was, he was disillusioned or something.
2:54:08 Um, I don’t know.
2:54:15 Um, and he decided to, to, uh, uh, to peace out, uh, and, you know, collect mushrooms in
2:54:15 St. Petersburg or something.
2:54:17 And that’s, that’s fine.
2:54:19 You know, you can, you can do that.
2:54:21 Um, I mean, that’s another sort of flip side.
2:54:25 I mean, we are not, a lot of problems that we solve, you know, they, some of them do have
2:54:27 practical application and that’s, that’s great.
2:54:33 But, uh, like if you stop thinking about a problem that you, you know, so he’s, he hasn’t published
2:54:35 since in this field, but that’s fine.
2:54:37 There’s many, many other people who’ve done so as well.
2:54:39 Um, yeah.
2:54:43 So I guess one thing I didn’t realize initially with the Fields Medal is that it, it sort of
2:54:45 makes you part of the establishment.
2:54:50 Um, you know, so, you know, most mathematicians, you know, there’s, uh, just career mathematicians,
2:54:54 you know, you just focus on publishing your next paper, maybe getting one, just to promote
2:54:59 one, one rank, you know, and, and starting a few projects, maybe taking some students or
2:54:59 something.
2:55:00 Yeah.
2:55:04 But then suddenly people want your opinion on things and, uh, you have to think a little
2:55:07 bit about, uh, you know, things that you might just so foolishly say because you know
2:55:08 no one’s going to listen to you.
2:55:10 Uh, it’s more important now.
2:55:12 Is it constraining to you?
2:55:14 Are you able to still have fun and be a rebel?
2:55:18 And try crazy stuff and play with ideas.
2:55:22 I have a lot less free time than I had previously.
2:55:24 Um, I mean, mostly by choice.
2:55:28 I mean, I, I, I can always see I have the option to sort of, uh, decline.
2:55:29 So I decline a lot of things.
2:55:33 I, I think I could decline even more, um, or I could acquire a reputation of being so
2:55:35 unreliable that people don’t even ask anymore.
2:55:39 Uh, this is, I love the different algorithms here.
2:55:41 This is, it’s always an option.
2:55:44 Um, but you know, um,
2:55:49 There are things that are like, I mean, so I mean, I, I, I don’t spend as much time
2:55:53 as I do as a postdoc, you know, just, just working on one problem at a time or, um, fooling
2:55:54 around.
2:55:59 I still do that a little bit, but yeah, as you’re advancing your career, some of the
2:56:03 more soft skills, so math somehow front loads all the technical skills to the early stages
2:56:03 of your career.
2:56:05 Um, so, um, yeah.
2:56:09 So it’s, uh, as a postdoc is published or perish, you’re, you’re, you’re, you’re incentivized
2:56:14 to basically focus on, on proving very technical theorems to sort of prove yourself, um, as well
2:56:15 as proof the theorems.
2:56:22 Um, but then as, as you get more senior, you have to start, you know, mentoring and, and, and, and
2:56:27 giving interviews, uh, and, uh, and trying to shape, um, direction of the field, both research
2:56:27 wise.
2:56:32 And, and, and, you know, uh, sometimes you have to, uh, uh, you know, do various administrative
2:56:32 things.
2:56:37 And it’s kind of the right social contract because you, you need to, to work in the trenches
2:56:39 to see what can help mathematicians.
2:56:43 The other side of the establishment sort of the, the, the really positive thing is that,
2:56:48 um, you get to be a light that’s an inspiration to a lot of young mathematicians or young people
2:56:50 that are just interested in mathematics.
2:56:53 It’s like, yeah, it’s just how the human mind works.
2:57:01 This is where I would probably, uh, say that I like the Fields Medal, that it does inspire
2:57:03 a lot of young people somehow.
2:57:05 I don’t, this is just how human brains work.
2:57:10 At the same time, I also want to give sort of respect to somebody like Gregorio Perlman,
2:57:14 who is critical of awards in his mind.
2:57:19 Those are his principles and any human that’s able for their principles to like do the thing
2:57:22 that most humans would not be able to do.
2:57:24 It’s beautiful to see.
2:57:26 Some recognition is, is necessarily important.
2:57:31 And, uh, but yeah, it’s, it’s also important to not let these things take over your life
2:57:36 and like only be concerned about, uh, getting the next big award or whatever.
2:57:42 Um, I mean, yeah, so again, you see these people try to only solve like a really big math problems
2:57:48 and not work on, on, on things that are less, uh, sexy, if you wish, but, but, but actually
2:57:50 still interesting and instructive.
2:57:55 As you say, like the way the human mind works, it’s, um, we understand things better when they’re
2:57:56 attached to humans.
2:58:01 Um, and also, uh, if they’re attached to a small number of humans, like I said, there’s,
2:58:04 there’s the way our human mind is, is, is wired.
2:58:09 We can comprehend the relationships between the 10 or 20 people, you know, but once you
2:58:12 get beyond like a hundred people, I, this is, there’s a, there’s a limit.
2:58:13 I figured there’s a name for it.
2:58:16 Um, beyond which, uh, it just becomes the other.
2:58:22 Um, and, uh, so we have, you have to simplify the pole mass of, you know, 99.9% of humanity
2:58:22 becomes the other.
2:58:27 Um, and, uh, and often these models are, are, are incorrect and this causes all kinds of
2:58:28 problems.
2:58:33 But, um, so yeah, so to humanize a subject, you know, if you identify a small number of
2:58:37 people and say, you know, these are representative people of the subject, you know, role models,
2:58:45 for example, um, that has some role, um, but it can also be, um, uh, yeah, too much of it
2:58:51 can be harmful because it’s, I’ll be the first to say that my own career path is not that of
2:58:52 a typical mathematician.
2:58:56 Um, I, the very accelerated education, I skipped a lot of classes.
2:59:01 Um, I think I was, had very fortunate mentoring opportunities, um, and I think I was at the
2:59:07 right place at the right time just because someone does, doesn’t have my, um, trajectory, you
2:59:09 know, doesn’t mean that they can’t be good mathematicians.
2:59:11 I mean, they, they, they, they, they, they, they, they, they just in, in a very different
2:59:14 style, uh, and we need people with a different style.
2:59:21 Um, and, you know, even if, and sometimes too much focus is given on the, on the person who does
2:59:26 the last step to complete, um, a project in mathematics or elsewhere, that’s, that’s really
2:59:30 taken, you know, centuries or decades with lots and lots of putting on lots of previous work.
2:59:34 Um, but that’s a story that’s difficult to tell, um, if you’re not an expert because, you
2:59:38 know, it’s easier to just say one person did this one thing, you know, it makes for a much
2:59:39 simpler history.
2:59:47 I think on the whole, it, um, is a hugely positive thing to, to talk about Steve Jobs as a representative
2:59:53 of Apple when I personally know, and of course, everybody knows the incredible design, the
2:59:58 incredible engineering teams, just the individual humans on those teams.
3:00:01 They’re not a team, they’re individual humans on a team.
3:00:06 And there’s a lot of brilliance there, but it’s just a nice shorthand, like a very, like
3:00:06 pie.
3:00:07 Yeah.
3:00:08 Steve Jobs.
3:00:08 Yeah.
3:00:08 Yeah.
3:00:13 As a starting point, you see, you know, as a first approximation, that’s how you.
3:00:16 And then read some biographies and then look into much deeper first approximation.
3:00:17 Yeah.
3:00:17 That’s right.
3:00:21 Uh, so you mentioned you were a Princeton to, um, Andrew Wiles at that time.
3:00:22 Oh yeah.
3:00:23 He’s a professor there.
3:00:26 It’s a funny moment how history is just all interconnected.
3:00:29 And at that time he announced that he proved the Fermat’s last theorem.
3:00:36 What did you think, maybe looking back now with more context about that moment in math history?
3:00:37 Yeah.
3:00:38 So I was a graduate student at the time.
3:00:43 I mean, I, I vaguely remember, you know, there was press attention and, uh, um, we all
3:00:47 had the same, um, uh, we had pigeonholes in the same mailroom, you know, so we were all
3:00:51 pitching out mail and like suddenly Andrew Wiles’ mailbox exploded to be overflowing.
3:00:53 That’s a good, that’s a good metric.
3:00:54 Yeah.
3:00:58 Um, you know, so yeah, we, we all talked about it at, at tea and so forth.
3:01:01 I mean, we, we didn’t understand, most of us sort of understand the proof.
3:01:04 Um, we understand sort of high level details.
3:01:07 Um, like there’s an ongoing project to formalize it in lean, right?
3:01:08 Kevin Buzzard is actually.
3:01:08 Yeah.
3:01:10 Can we take that small tangent?
3:01:12 Is it, is it, how difficult is that?
3:01:17 Cause as, as I understand the Fermat’s last, the, the proof for, uh, Fermat’s last theorem
3:01:19 has like super complicated objects.
3:01:20 Yeah.
3:01:22 It’s really difficult to formalize now.
3:01:22 Yeah.
3:01:23 I guess, yeah, you’re right.
3:01:26 The, the objects that they use, um, you can define them.
3:01:28 Uh, so they’ve been defined in lean.
3:01:28 Okay.
3:01:31 So, so just defining what they are can be done.
3:01:33 Uh, that’s really not trivial, but it’s been done.
3:01:39 But there’s a lot of really basic facts about, um, these objects that have taken decades
3:01:41 to prove that they’re, they’re in all these different math papers.
3:01:44 And so lots of these have to be formalized as well.
3:01:51 Um, Kevin’s, uh, Kevin Buzzard’s goal, actually, he has a five-year grant to formalize Fermat’s
3:01:55 class theorem and his aim is that he doesn’t think he will be able to get all the way down
3:02:00 to the basic axioms, but he wants to formalize it to the point where the only things that he
3:02:05 needs to rely on as black boxes are things that were known by 1980 to, um, to number theories
3:02:06 at the time.
3:02:11 Um, and then some other person or some other work would have to be done to, to, to, to get
3:02:11 from there.
3:02:16 Um, so it’s, it’s a different area of mathematics than, um, the type of mathematics I’m used
3:02:22 to, um, um, in analysis, which is kind of my area, um, the objects we study are kind of
3:02:23 much closer to the ground.
3:02:28 We study, I study things like prime numbers and, and, and functions and, and things that
3:02:35 are within scope of a high school, um, uh, math education to at least, uh, define, um, yeah.
3:02:39 But then there’s this very advanced algebraic side of number theory where people have been
3:02:41 building structures upon structures for, for quite a while.
3:02:44 Um, and it’s, it’s a very sturdy structure.
3:02:48 There’s a, it’s, it’s, it’s been, it’s been very, um, at the base, at least it’s extremely
3:02:50 well-developed with textbooks and so forth.
3:02:56 But, um, um, it does get to the point where, um, if you’re, if you haven’t taken these years
3:03:00 of study and you want to ask about what, what is going on at, um, like level six of this
3:03:04 tower, you have to spend quite a bit of time before they can even get to the point where
3:03:05 you can see, you see something you recognize.
3:03:13 What inspires you about his journey that we similar, as we talked about seven years, mostly
3:03:14 working in secret.
3:03:15 Yeah.
3:03:17 Uh, yes, that is a romantic, uh, yeah.
3:03:22 So it kind of fits with the sort of the, the romantic image that I think people have of
3:03:26 mathematicians to the extent that they think of anything at all as these kind of eccentric,
3:03:28 uh, you know, wizards or something.
3:03:34 Um, so that certainly kind of, uh, uh, accentuated that perspective.
3:03:36 You know, I mean, it’s, it is a great achievement.
3:03:42 His style of solving problems is so different from my own, um, but which is great.
3:03:43 I mean, we, we need people like that.
3:03:44 Can you speak to it?
3:03:48 Like what, uh, in, in terms of like the, you like the collaborative.
3:03:52 I like moving on from a problem if it’s giving too much difficulty.
3:03:57 Um, but you need the people who have the tenacity and the fearlessness.
3:04:01 Um, you know, I’ve, I’ve collaborated with, with people like that where I want to give
3:04:05 up, uh, because the first approach that we tried didn’t work and the second one didn’t
3:04:09 approach, but they’re convinced and they have the third, fourth and the fifth of what works.
3:04:12 Um, and I’d have to eat my words.
3:04:12 Okay.
3:04:15 I didn’t think this was going to work, but yes, you were right all along.
3:04:20 And we should say for people who don’t know, not only are you known for the brilliance of
3:04:25 your work, but the incredible productivity, just the number of papers, which are all of very
3:04:25 high quality.
3:04:30 So there’s something to be said about being able to jump from topic to topic.
3:04:31 Yeah.
3:04:31 It works for me.
3:04:32 Yeah.
3:04:35 I mean, there are also people who are very productive and they, they focus very deeply
3:04:35 on.
3:04:36 Yeah.
3:04:38 I think everyone has to find their own workflow.
3:04:43 Um, like one thing, which is a shame in mathematics is that we have mathematics.
3:04:46 There’s sort of a one size fits all approach to teaching, teaching mathematics.
3:04:50 Um, and, you know, so we have a certain curriculum and so forth.
3:04:54 I mean, you know, maybe like if you do math competitions or something, you get a slightly different
3:04:54 experience.
3:05:01 But, um, I think many people, um, they don’t find their, their native math language, uh, until
3:05:03 very late or usually too late.
3:05:08 So they, they, they, they stop doing mathematics and they have a bad experience with a teacher who’s
3:05:10 trying to teach them one way to do mathematics that they don’t like it.
3:05:18 Um, my theory is that, um, humans don’t come, evolution has not given us a math center of
3:05:18 our brain directly.
3:05:24 We have a vision center and a language center and some other centers, um, which have evolution
3:05:26 as honed, but we, it doesn’t, we don’t have an innate sense of mathematics.
3:05:35 Um, but our other centers are sophisticated enough that different people, uh, we, we, we can repurpose
3:05:38 other areas of our brain to do mathematics.
3:05:41 So some people have figured out how to use the visual center to do mathematics.
3:05:43 And so they think, think very visually when they do mathematics.
3:05:47 Some people have repurposed their, their language center and they think very symbolically.
3:05:52 Um, you know, um, some people like if, if they are very competitive and they, they’re like
3:05:57 gaming, there’s a type of, there’s a part of your brain that’s very good at, at, at, uh,
3:06:01 at solving puzzles and games and, and, and, and that can be repurposed.
3:06:07 But like when I talk to other mathematicians, you know, they don’t quite think that I can
3:06:10 tell that they’re using some of the different styles of, of thinking than I am.
3:06:14 I mean, not, not disjoint, but they, they may prefer visual.
3:06:16 Like I’m, I, I, I don’t actually prefer visual so much.
3:06:18 I need lots of visual aids myself.
3:06:23 Um, you know, mathematics provides a common language so we can still talk to each other,
3:06:25 even if we are thinking in different ways.
3:06:31 But you can tell there’s a different set of subsystems being used in the thinking process.
3:06:33 Like they, they take different paths.
3:06:35 They’re very quick at things that I struggle with and vice versa.
3:06:38 Um, and yet they still get to the same goal.
3:06:44 Um, and yeah, but I mean, the way we educate, unless you have like a personalized tutor or
3:06:48 something, I mean, education sort of just by initial scale has to be mass produced.
3:06:52 You know, you have to teach the 30 kids, you know, they have 30 different styles.
3:06:54 You can’t, you can’t teach 30 different ways.
3:07:00 On that topic, what advice would you give to students, uh, young students who are struggling
3:07:04 with math and, but are interested in it and would like to get better?
3:07:06 Is there something in this?
3:07:06 Yeah.
3:07:10 Um, in this complicated educational context, what, what would you, yeah, it’s a tricky
3:07:10 problem.
3:07:15 One nice thing is that there are now lots of sources for my faculty enrichment outside the
3:07:15 classroom.
3:07:19 Um, so in, in, in my day, there already, there are math competitions.
3:07:22 Um, and you know, they’re also like popular math books in the library.
3:07:26 Um, yeah, but, but now you have, you know, YouTube, uh, there are, there are forums just
3:07:32 devoted to solving, you know, math puzzles and, um, and math shows up in, in other places,
3:07:36 you know, like, um, for example, there, there are hobbyists who play poker, uh, for fun.
3:07:42 Uh, and, um, um, they, they, they, you know, they are for very specific reasons, are interested
3:07:43 in very specific probability questions.
3:07:50 Um, and, and, uh, they actually, you know, there’s a community of amateur probabilists in,
3:07:53 in, in, in poker, um, in chess and baseball.
3:07:58 I mean, there’s, there’s, there’s, uh, yeah, um, there’s math all over the place.
3:08:03 Um, and I’m, I’m, I’m hoping actually with, uh, with these new sort of tools for lean and
3:08:08 so forth, that actually we can incorporate the broader public into math research projects.
3:08:12 Um, like this is almost, it doesn’t happen at all currently.
3:08:17 So in the sciences, there’s some scope for citizen science, like astronomers, uh, they’re
3:08:21 amateurs who would discover comets and there’s biologists, they’re people who could identify
3:08:26 butterflies and so forth, um, and in method, um, there are a small number of activities
3:08:30 where, um, amateur mathematicians can like discover new primes and so forth.
3:08:36 But, but previously, because we have to verify every single contribution, um, like most mathematical
3:08:40 research projects, it would not help to have input from the general public.
3:08:45 In fact, it would, it would just be, be time consuming because just error checking and everything.
3:08:50 Um, but you know, one thing about these formalization projects is that they are bringing together
3:08:52 more, bringing in more people.
3:08:56 So I’m sure that high school students have already contributed to some of these formalizing
3:08:57 projects who contributed to MathLib.
3:09:02 Um, you know, you don’t need to be a PhD holder to just work on one atomic thing.
3:09:08 There’s something about the formalization here that also, as a very first step, opens it up
3:09:10 to the programming community too.
3:09:11 Yes.
3:09:13 The people who are already comfortable with programming.
3:09:18 It seems like programming is somehow maybe just the feeling, but it feels more accessible
3:09:20 to folks than math.
3:09:26 Math is seen as this like extreme, especially modern mathematics seen as this extremely difficult
3:09:29 to enter area and programming is not.
3:09:30 So that could be just an entry point.
3:09:33 You can execute code and you can get results, you know, you can print a whole world pretty
3:09:34 quickly.
3:09:41 Um, you know, like if, uh, if programming was taught as an almost entirely theoretical subject
3:09:47 where you just taught the computer science, the theory of functions and, and, and, and, and
3:09:50 routines and so forth and, and outside of some, some very specialized homework assignments,
3:09:56 you’re not actually programmed like on the weekend for fun or yeah, that would be as considered
3:09:56 as hard as math.
3:10:04 Um, yeah, so as I said, you know, there are communities of non-mathematicians where they’re
3:10:08 deploying math for some very specific purpose, you know, like, like optimizing their poker game
3:10:12 and, and for them, then math becomes fun for them.
3:10:16 Uh, what advice would you give in general to young people, how to pick a career, how
3:10:17 to find themselves?
3:10:20 Like that’s a tough, tough, tough question.
3:10:20 Yeah.
3:10:25 So, um, there’s a lot of certainty now in the world, you know, I mean, I, there was this period
3:10:30 after the war where, uh, at least in the West, you know, if you came from a good demographic,
3:10:35 you, uh, you know, like you, there was a very stable path to it, to a good career.
3:10:40 You go to college, you get an education, you pick one profession and you stick to it.
3:10:42 It’s becoming much more of a thing of the past.
3:10:46 So I think you just have to be adaptable and flexible.
3:10:50 I think people will have to get skills that are transferable, you know, like, like learning
3:10:53 one specific programming language or one specific subject of mathematics or something.
3:10:58 It’s, it’s, it’s, that itself is not a super transferable skill, but sort of knowing how
3:11:05 to, um, reason with, with abstract concepts or how to problem solve and things go wrong.
3:11:10 So anyway, these are things which I think we will still need, even as our tools get better
3:11:13 and, you know, you’ll, you’ll be working with AI as well and so forth.
3:11:15 But actually you’re an interesting case study.
3:11:22 I mean, you’re like one of the great living mathematicians, right?
3:11:26 And then you had a way of doing things and then all of a sudden you start learning.
3:11:31 I mean, first of all, you kept learning new fields, but you learn lean.
3:11:33 That’s not, that’s a non-trivial thing to learn.
3:11:38 Like that’s a, yeah, that’s a, for a lot of people, that’s an extremely uncomfortable
3:11:39 leap to take, right?
3:11:40 Yeah.
3:11:41 A lot of mathematicians.
3:11:44 First of all, I’ve always been interested in new ways due to mathematics.
3:11:49 I, I, I feel like a lot of the ways we do things right now are inefficient.
3:11:55 Um, I, I, I spent, I, me and my colleagues, we spend a lot of time doing very routine computations
3:11:58 or doing things that other mathematicians would instantly know how to do.
3:12:02 And we don’t know how to do them and why can’t we search and get a quick response and
3:12:02 so on.
3:12:07 So that’s why I’ve always been interested in exploring new workflows.
3:12:13 About four or five years ago, I was on a committee where we had to ask for ideas for interesting
3:12:14 workshops to run at a math institute.
3:12:19 And at the time, Peter Schultz had just, uh, formalized one of his, his, um, new theorems.
3:12:25 And, um, there’s some other developments in computer assisted proof that look quite interesting.
3:12:29 And I said, oh, we should, we should, uh, um, we should run a workshop on this.
3:12:29 This is a pretty good idea.
3:12:33 Um, and then I was a bit too enthusiastic about this idea.
3:12:36 So I, I got voluntold to actually run it.
3:12:41 Um, so I did with a bunch of other people, Kevin Buzzard and Jordan Ellenberg and a bunch
3:12:42 of other people.
3:12:45 Um, and it was, it was a, a, a, a nice success.
3:12:49 We brought together a bunch of mathematicians and computer scientists and other people.
3:12:51 And, and we got up to speed on the state of the yard.
3:12:57 Um, and it was really interesting, um, developments that, but most mathematicians didn’t know what
3:12:57 was going on.
3:13:02 Um, um, that lots of nice proofs of concept, you know, just sort of hints of, of what was
3:13:02 going to happen.
3:13:06 This was just before chat GBD, but there was even then there was one talk about language
3:13:09 models and the potential, um, capability of those in the future.
3:13:12 So that got me excited about the subject.
3:13:16 So I started giving talks, um, about this is something we should, more of us should start
3:13:22 looking at, um, now that I arranged the runner’s conference and then chat GPT came out and like
3:13:23 suddenly AI was everywhere.
3:13:28 And so, uh, I got interviewed a lot, um, about, about this topic.
3:13:33 Um, and in particular, um, the interaction between AI and formal proof assistants.
3:13:34 And I said, yeah, they should be combined.
3:13:38 This, this is, this is, um, this is perfect synergy to happen here.
3:13:42 And at some point I realized that I have to actually do not just talk the talk, but walk
3:13:45 the walk, you know, like, you know, I don’t work in machine learning and I don’t work
3:13:49 in proof formalization and there’s a limit to how much I can just rely on authority and
3:13:51 saying, you know, I, I, I’m a, I’m a, I’m a mathematician.
3:13:52 Just trust me.
3:13:55 You know, when I say that this is going to change mathematics and I’m not doing it any, and I
3:13:56 don’t do any of it myself.
3:14:02 So I felt like I had to actually, uh, uh, justify it.
3:14:07 You know, a lot of what I get into actually, um, I don’t quite see an advice as how much
3:14:08 time I’m going to spend on it.
3:14:14 And it’s only after I’m sort of waist deep in, in, in, in a project that I, I realized by
3:14:14 that point I’m committed.
3:14:19 Well, that’s deeply admirable that you’re willing to go into the fray, be in some small
3:14:21 way, beginner, right?
3:14:26 Or have some of the sort of challenges that a beginner would, right?
3:14:29 It’s new, new concepts, new ways of thinking.
3:14:36 Also, you know, sucking at a thing that others, I think, I think in that talk, you know, you
3:14:40 could be a field metal winning mathematician and an undergrad knows something better.
3:14:41 Yeah.
3:14:47 Um, I think mathematics inherently, I mean, mathematics is so huge these days that nobody
3:14:48 knows all of modern mathematics.
3:14:55 Um, and inevitably we make mistakes and, um, you know, uh, you can’t cover up your mistakes
3:14:57 with just sort of bravado.
3:15:01 And, and, uh, I mean, because people will ask for your proofs and if you don’t have the
3:15:02 proofs, you don’t have the proofs.
3:15:03 Um, I don’t love math.
3:15:04 Yeah.
3:15:06 So it does keep us honest.
3:15:11 I mean, not, I mean, you can still, uh, it’s not a perfect, uh, panacea, but I think, uh,
3:15:16 uh, we do have more of a culture of admitting error than, cause we’re forced to all the time.
3:15:18 Big, ridiculous question.
3:15:19 I’m sorry for it.
3:15:23 Once again, who is the greatest mathematician of all time?
3:15:26 Maybe one who’s no longer with us.
3:15:28 Uh, who are the candidates?
3:15:32 Euler, Gauss, Newton, Ramanujan, Hilbert.
3:15:35 So first of all, as I mentioned before, like there’s, there’s some time dependence.
3:15:37 On the day.
3:15:37 Yeah.
3:15:41 Like, like if you, if you, if you, if you pop cumulatively over time, for example, Euclid
3:15:44 like, like sort of like is, is, is one of the leading contenders.
3:15:50 Um, and then maybe some unnamed anonymous mathematicians before that, um, you know, whoever came up with
3:15:55 the concept of numbers, you know, um, do mathematicians today still feel the impact of
3:16:00 Hilbert just directly of what everything that’s happened in the 20th century.
3:16:00 Yeah.
3:16:00 Yeah.
3:16:01 Hilbert spaces.
3:16:05 We have lots of things that are named after him, uh, of course, just the arrangement of
3:16:07 mathematics and just the introduction of certain concepts.
3:16:10 I mean, 23 problems have been extremely influential.
3:16:16 There’s some strange power to the declaring which problems are hard to solve.
3:16:18 The statement of the open problems.
3:16:19 Yeah.
3:16:22 I mean, you know, this is bystander effect everywhere.
3:16:27 Like if, if no one says you should do X, everyone just sort of mills around waiting for somebody
3:16:30 else to, to, uh, to do something and, and like nothing gets done.
3:16:35 Um, so, and, and like, like it’s the point of, one thing that actually, uh, you have to
3:16:39 teach undergraduates in mathematics is that you should always try something.
3:16:45 So, um, you see a lot of paralysis, um, in an undergraduate trying a math problem.
3:16:49 If they recognize that there’s a certain technique that, that can be applied, they will try it.
3:16:53 But there are problems for which they see none of their standard techniques obviously applies.
3:16:56 And the common reaction is then just paralysis.
3:16:58 I don’t know what to do.
3:17:01 I, oh, um, I think there’s a quote from the Simpsons.
3:17:03 I’ve tried nothing and I’m all out of ideas.
3:17:11 Um, so, you know, like the next step then is to try anything, like no matter how stupid, um, and in fact,
3:17:12 it’s almost the stupid of the better.
3:17:18 Um, which, you know, I’m, I think we’re just almost guaranteed to fail, but the way it fails
3:17:19 is going to be instructive.
3:17:23 Um, like it fails because you, you, you’re not at all taking into account this hypothesis.
3:17:24 Oh, this hypothesis must be useful.
3:17:25 That’s a clue.
3:17:30 I think you also suggested somewhere this, this fascinating approach, which really stuck with
3:17:32 me as they’re using it.
3:17:32 It really works.
3:17:35 I think you said it’s called structured procrastination.
3:17:36 No, yes.
3:17:38 It’s when you really don’t want to do a thing.
3:17:41 Do you imagine a thing you don’t want to do more?
3:17:42 Yes, yes, yes.
3:17:43 That’s worse than that.
3:17:47 And then in that way you procrastinate by not doing the thing that’s worse.
3:17:48 Yeah, yeah.
3:17:50 That’s a nice, it’s a nice hack.
3:17:50 It actually works.
3:17:52 Yeah, yeah.
3:17:57 There’s, um, I mean, with anything like, you know, I mean, like you’ve, um, psychology is
3:17:58 really important.
3:18:02 Like you, you, you, you talk to athletes like marathon runners and so forth and, you know,
3:18:05 and they talk about what’s the most important thing is that the training regimen or the diet
3:18:06 and so forth.
3:18:11 So much of it is like your psychology, um, you know, just tricking yourself to, to think that
3:18:12 the problem is feasible.
3:18:14 Um, so that you can be motivated to do it.
3:18:19 Is there something our human mind will never be able to comprehend?
3:18:23 Well, I sort of, I guess a mathematician, I mean, you know, it’s my induction.
3:18:28 I, it’s really, there must be some, it’s a really large number that you can’t understand.
3:18:30 That was the first thing that came to mind.
3:18:36 So that, but even broadly, is there, are we, is there something about our mind that we’re
3:18:40 going to be limited even with the help of mathematics?
3:18:41 Well, okay.
3:18:44 I mean, it’s like, how much augmentation are you willing?
3:18:49 Like, for example, if I didn’t even have pen and paper, um, like if I had no technology
3:18:50 whatsoever, okay.
3:18:51 So I’ve not allowed blackboard, pen and paper.
3:18:55 You’re already much more limited than you would be.
3:18:56 Incredibly limited.
3:18:57 Even language.
3:18:58 The English language is a technology.
3:19:02 Uh, it’s, uh, it’s one that’s been very internalized.
3:19:03 So you’re right.
3:19:07 They’re really, the, the, the, the formulation of the problem is incorrect because there really
3:19:10 is no longer a, just a solo human.
3:19:17 We’re already augmented in extremely complicated, intricate ways, right?
3:19:17 Yeah.
3:19:17 Yeah.
3:19:19 So we’re already like a collective intelligence.
3:19:20 Yes.
3:19:20 Yeah.
3:19:21 I guess.
3:19:26 So humanity plural has much more intelligence in principle on his good days.
3:19:29 than, than the individual humans put together.
3:19:30 Uh, it can all have less.
3:19:30 Okay.
3:19:32 But, um, um, yeah.
3:19:36 So yeah, math, math, math, math, math, math, the math community plural is, is, is incredibly
3:19:43 super intelligent, uh, entity, um, that, uh, no single human mathematician can, can come
3:19:44 close to, to, to replicating.
3:19:47 You see it a little bit on these like question analysis sites.
3:19:50 Um, uh, so this math overflow, which is the math version of stack overflow.
3:19:55 And like, sometimes you get like this very quick responses to very difficult questions from
3:19:56 the community.
3:20:00 Um, and it’s, it’s, it’s, it’s a pleasure to watch actually as a, as an expert.
3:20:06 I’m a fan spectator of that, uh, of that site, just seeing the brilliance of the different
3:20:12 people there, um, the depth of knowledge that some people have and the willingness to engage
3:20:15 in the, in the rigor and the nuance of the particular question.
3:20:16 It’s pretty cool to watch.
3:20:17 It’s fun.
3:20:18 It’s almost like just fun to watch.
3:20:23 Uh, what gives you hope about this whole thing we have going on human civilization?
3:20:29 I think, uh, yeah, uh, the, uh, the younger generation is always like, like really creative
3:20:30 and enthusiastic and, and inventive.
3:20:36 Um, it’s a pleasure working with, with, with, uh, with, uh, with, uh, with young students.
3:20:43 Um, you know, the, uh, the progress of science tells us that the problems that used to be really
3:20:48 difficult can become extremely, you know, can become like trivial to solve, you know,
3:20:54 I mean, like it was like navigation, you know, just, just knowing where you were on the planet
3:20:55 was this horrendous problem.
3:21:00 People, people died, um, you know, uh, or lost fortunes because they couldn’t navigate,
3:21:03 you know, and we have devices in our pockets that do this automatically for us, like it’s
3:21:06 a completely solved problem, you know?
3:21:10 So things that are seem unfeasible for us now could be maybe just sort of homework exercises
3:21:11 for things.
3:21:16 Yeah, one of the things I find really sad about the finiteness of life is that I won’t
3:21:21 get to see all the cool things we create as a civilization, you know, that, cause it, in
3:21:26 the next hundred years, 200 years, just imagine showing, showing up in 200 years.
3:21:26 Yeah.
3:21:30 Well, already plenty has happened, you know, like if, if you could go back in time and talk
3:21:32 to your, your teenage self or something, you know what I mean?
3:21:33 Yeah.
3:21:39 And just the internet and, and now AI, I mean, again, they’ve been into, they’re beginning
3:21:42 to be internalized and say, yeah, of course, uh, and AI can understand our voice.
3:21:47 And, and give reasonable, you know, slightly incorrect answers to, to any question, but
3:21:49 you know, this was mind blowing even two years ago.
3:21:55 And in the moment, it’s hilarious to watch on the internet and so on, the, the drama, uh,
3:21:57 people take everything for granted very quickly.
3:22:01 And then they, we humans seem to entertain ourselves with drama.
3:22:06 Well, out of anything that’s created, somebody needs to take one opinion and another person
3:22:08 needs to take an opposite opinion and argue with each other about it.
3:22:13 But when you look at the arc of things, I mean, it’s just even in progress of robotics.
3:22:18 Just to take a step back and be like, wow, this is beautiful that we humans are able to create
3:22:18 this.
3:22:19 Yeah.
3:22:23 When the infrastructure and the culture is, is healthy, you know, the community of humans
3:22:29 can be so much more intelligent and mature and, and, and rational than the individuals
3:22:30 within it.
3:22:35 Well, one place I can always count on rationality is the comment section of your blog, which
3:22:36 I’m a big fan of.
3:22:38 There’s a lot of really smart people there.
3:22:43 And thank you, uh, of course, for, uh, for putting those ideas out on the blog.
3:22:50 And it’s, I can’t tell you how, uh, honored I am that you would spend your time with me today.
3:22:52 I was looking forward to this for a long time.
3:22:54 Terry, I’m a huge fan.
3:22:57 Um, you inspire me, you inspire millions of people.
3:22:58 Thank you so much for talking.
3:22:58 Thank you.
3:22:58 It was a pleasure.
3:23:02 Thanks for listening to this conversation with Terrence Tao.
3:23:07 To support this podcast, please check out our sponsors in the description or at lexfreedman.com
3:23:08 slash sponsors.
3:23:13 And now let me leave you with some words from Galileo Galilei.
3:23:19 Mathematics is the language with which God has written the universe.
3:23:24 Thank you for listening and hope to see you next time.
3:23:41 Thanks for listening and hope to see you next time.