Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI | Lex Fridman Podcast #472

Lex Fridman 0:00 The following is a conversation with Terence Tao, widely considered to be one of the greatest mathematicians in history, often referred to as the Mozart of Math. He won the Fields Medal and the Breakthrough Prize in Mathematics, and has contributed groundbreaking work to a truly astonishing range of fields in mathematics and physics. This was a huge honor for me, for many reasons, including the humility and kindness that Terry showed to me throughout all our interactions. It means the world. This is the Lex Fridman Podcast. To support it, please check out our sponsors in the description or at lexfridman.com/sponsors. And now, dear friends, here's Terence Tao.

Lex Fridman 0:50 What was the first really difficult research-level math problem that you encountered? One that gave you pause maybe?

Terence Tao 0:57 Well, I mean, in your undergraduate, um, education, you learn about the really hard impossible problems, like the Riemann hypothesis, the twin primes conjecture. You can make problems arbitrarily difficult, uh, that's not really a problem. In fact, there's even problems that we know to be unsolvable. What's really interesting, uh, are the problems just at the... on the boundary between what we can do perfectly easily and what are hopeless. Um, but what are problems where, like, existing techniques can do like 90% of the job and then y- you just need that remaining 10%. Um, I think as a PhD student, the Kakeya problem certainly caught my eye, and it just got solved actually. It's a problem I've worked on a lot in my early research. Historically, it came from a little puzzle by the Japanese, uh, mathematician Soichi Kakeya, uh, in like 1918 or so. Um, so, the puzzle is that you, you have, um, a needle, um, in s- on the plane, um, or- or think like- like- uh, like driving on- on- on a road or something, and you want to- to execute a U-turn, you want to turn the needle around, um, but you want to do it in as little space as possible, uh, so you want to use this little area, uh, in order to turn it around. So, um, but the needle is infinitely maneuverable. So you can imagine just spinning it around its, um... uh, it's a unit needle. You can spin it around its center, um, and I think, um, that gives you a disc of- of area, I think, pi over 4. Um, or you can do a three-point U-turn, which is what they- we teach people in- in the driving schools to do, uh, and that actually takes area pi over 8. Um, so it's- it's a little bit more efficient than, um, uh, rotation. And so for a while people thought that was the most efficient, uh, way to turn things around. But Bazhukovich, uh, showed that in fact, uh, you could actually, uh, turn the needle around using as little area as you wanted.

Lex Fridman 2:38 Mm.

Terence Tao 2:38 So .001, there was some really fancy multi, um, uh, um, back and forth U-turn thing that you could- you could do, that- that you could turn the needle around. And in so doing, it would pass through every intermediate direction.

Lex Fridman 2:50 Is this in the two-dimensional plane?

Terence Tao 2:52 This is in the two-dimensional plane. Yeah, so we understand everything in two dimensions. So the next question is, what happens in three dimensions? So suppose like the Hubble Space Telescope is a tube in space, and you want to observe every single star in the universe, so you want to rotate the telescope to reach every single direction. And here's the unrealistic part. Suppose that space is at a premium, which it totally is not.

Lex Fridman 3:13 (laughs) Yeah.

Terence Tao 3:13 Uh, you want to occupy as little volume as possible in order to rotate your- your needle around, in order to see every single star in the sky. Um, how small a volume do you need to do that? And so you can modify Bazhukovich's construction. And so if your telescope has zero thickness, then y- you can use as little volume as you need. That's a- a simple modification of the two-dimensional construction. But the question is that if your telescope is not zero thickness but- but just very, very thin, some thickness delta, what is the minimum volume needed to be able to see every single direction as a function of delta? So as delta gets smaller, as the needle gets- gets thinner, the volume should go down, but- but how fast does it go down? Um, and the conjecture was that it goes down very, very slowly, um, like logarithmically, um, uh, roughly speaking. And that was proved after a lot of work. So this seems like a puzzle. Why is it interesting? So it turns out to be surprisingly connected to a lot of problems in partial differential equations, in number theory, in geometry, combinatorics. For example, in- in wave propagation, if you splash some- some water around, um, you create water waves and they- they travel in various directions. Um, but waves exhibit both w- both particle and wave type behavior. So you can have what's called a wave packet, which is like a- a very localized wave that is localized in space and moving a certain direction in time. And so if you plot it in both space and time, it occupies a region which looks like a tube. And so what can happen is that you can have a wave which initially is- is very dispersed, but it all comes... it all focuses at a single point later in time. Like, you can imagine dropping a- a pebble into a pond and ripples spread out. But then if you time reverse that- that, um, that scenario, and the equations of wave motion are time reversible, you can imagine ripples that are converging, um, to a single point, and then a- a big splash occurs, um, maybe even a singularity. Um, and so it's possible to do that, uh, and geometrically what's going on is that there's also light rays. Um, so like if- if- if this wave represents light, for example, um, you can imagine this wave as a superposition of photons, um, all traveling at the speed of light, they- they all travel on these light rays and they're all focusing at this one point. So you can have a very dispersed wave focus into a very concentrated wave at one point in- in space and time, but then it defocuses again and it's- it separates. But potentially, if the conjecture had a negative solution, so what I mean is that there's- there's a very efficient way to pack, um, tubes pointing in different directions into a very, very narrow region of- of- of... uh, very narrow volume. Then you would also be able to create waves that start out... some... there'll be some arrangement of waves that start out very, very dispersed, but they would concentrate not just at a single point but, um, um, there'll be a- a large, um... uh, there'll be a lot of concentrations in space and time. And, uh, um, and you could create what's called a blowup, where these waves, their amplitude becomes so great that the laws of physics that they're governed by are no longer wave equations, but something more complicated and nonlinear. Um-And so in mathematical physics, we care a lot about whether certain equations in- in- in wave equations are stable or not, uh, whether they can create, um, these singularities.

Terence Tao 6:16 There's a famous unsolved problem called the Navier-Stokes regularity problem. So the Navier-Stokes equations, the equations that govern the fluid flow for incompressible fluids, like water, the question asked, if you start with a smooth velocity field of water, can it ever concentrate so much that, like, the velocity becomes infinite at some point? Uh, that's called a singularity. We don't see that, um, in real life. You know, if you splash around water on- on the bathtub, it won't explode on you, um, or- or have p- have water leaving at the speed of light or anything, but potentially it is possible. Um, and in fact, in recent years, the- the consensus has- has drifted towards the, uh, the- the belief that, uh, that in fact for certain very special con- initial con- uh, configurations of- of, say, water, that singularities can form. But people have not yet been able to, uh, to actually establish this. The Clay Foundation has these seven Millennium Prize problems, has a million dollar prize for solving one of these problems, uh, this is one of them, of these seven, only one of them has been solved, uh, uh, the Poincaré conjecture by Perlmutter. So the Kakeya conjecture is not directly, directly related to the Navier-Stokes problem, but understanding it would help us understand some aspects of things like wave concentration, which would indirectly probably help us, uh, understand the Navier-Stokes problem better.

Lex Fridman 7:32 Can you speak to the Navier-Stokes, so the existence and smoothness, like you said, Millennial Prize problem-

Terence Tao 7:38 Right.

Lex Fridman 7:38 ...you've made a lot of progress on this one. In 2016, you published a paper, Finite Time Blow-Up For An Averaged Three-Dimensional Navier-Stokes Equation.

Terence Tao 7:47 Right. (laughs)

Lex Fridman 7:48 (laughs) So- so we're trying to figure out if this thing usually doesn't blow up-

Terence Tao 7:52 Right.

Lex Fridman 7:53 ...but can we say for sure it never blows up?

Terence Tao 7:56 Right, yeah. So yeah, that is literally the- the- the million dollar question, yeah. So this is what distinguishes mathematicians from pretty much everybody else, like if- if- if something ho- holds 99.99% of the time, um, that's good enough for most, you know, uh, for- for- for- for most things. But mathematicians are one of the few people who really care about whether every, like 100%, really 100% of all, um, situations are covered by- by, um... Yeah, so most fluids, most of the time, um, s- water d- does not blow up, but could you design a s- very special initial state that does this?

Lex Fridman 8:29 And maybe we should say that this is a, this is a set of equations that govern in the field of fluid dynamics-

Terence Tao 8:35 Yes.

Lex Fridman 8:35 ...yeah, trying to understand how a fluid behaves and it's actually turns out to be a really compli-... You know, fluid is-

Terence Tao 8:40 Yeah.

Lex Fridman 8:40 ...extremely complicated thing to try to model.

Terence Tao 8:43 Yeah, so it has practical importance. So this Clay Prize problem concerns what's called the incompressible Navier-Stokes, which governs things like water. There's something called the compressible Navier-Stokes, which governs things like air, and that's particularly important for weather prediction. Weather prediction, it's just a lot of computational fluid dynamics, a lot of it is actually just trying to solve the Navier-Stokes equations as- as best they can. Um, also gathering a lot of data so that they can get, they can ini- initialize the equation. There's a lot of moving parts. So it's very important problem practically.

Lex Fridman 9:09 Why is it difficult to prove general things about the set of equations like it not not blowing up?

Terence Tao 9:17 The short answer is Maxwell's demon.

Lex Fridman 9:19 (laughs) .

Terence Tao 9:19 Uh, so Maxwell's demon is a concept in thermodynamics, like if you have a box with two gases in, o- uh, oxygen and nitrogen, uh, and maybe you start with all the oxygen on one side and nitrogen on the other side, but there's no barrier between them, right? Then they will mix, um, and they should stay mixed, right? There's- there's- there's no reason why they should unmix. But in principle, because of all the collisions between them, there could be some sort of weird conspiracy that- that, um, like maybe there's a microscopic demon called Maxwell's demon that will, um, every time a oxygen and nitrogen atom collide, they will bounce off in such a way that the oxygen sort of drifts onto one side and then nitrogen goes to the other, and, uh, you could have an extremely improbable configuration emerge, uh, which we never see, um, and- and wh- which statistically it's extremely unlikely. But mathematically, it's possible that this can happen and we can't rule it out. Um, and this is a s- situation that shows up a lot in mathematics, um, uh, basically this example is the digits of pi, 3.14159 and so forth, the digits look like they have no pattern, and we believe they have no pattern. On the long term, you should see as many ones and twos and threes as fours and fives and sixes, uh, that there should be no preference in the digits of pi to favor, let's say, seven over eight. Um, but maybe there's some demon in the digits of pi that- that, like, every time you compute more and more digits, it sort of biases one digit to another, um, and this is a conspiracy that should not happen. There's no reason it- it should happen, but, um, there's- there's- there's no way to prove it, uh, with our current technology. Okay, so getting back to Navier-Stokes, a fluid has a certain amount of energy, and because the fluid is in motion, the energy gets transported around, and, uh, water is also viscous. So if the energy is spread out over many different locations, the natural viscosity of the fluid will just damp out the energy and it will- it will- it will go to zero. Um, and this is what happens, um, in, um, uh, when we actually experiment with water, like it, you splash around, there's- there's some turbulence and- and waves and so forth, but eventually it- it settles down and- and- and the- the lower the amplitude, the- the smaller the velocity, the- the more calm it gets. Um, but potentially, there is some sort of demon that keeps pushing the, uh, the energy of the fluid into a smaller and smaller scale and it will move faster and faster. And at faster speeds, the effect of viscosity is relatively less, and so it could happen that- that it- it creates a s- some sort of, um, um, what's called a self-similar blob scenario where, you know, um, the energy of the fluid sta- starts off at some, um, large scale and then it all sort of, um, transfers its energy into a smaller, um, region of- of- of the fluid, which then at a much faster rate, um, moves into, um, an even s- smaller, uh, region and so forth. Um, and- and each time it does this, uh, it takes maybe half as- as- as long as- as- as the previous one, and then s- uh, you could- you could- you could actually, uh, converge to- to all the energy, um, concentrating in one point in a- a finite amount of time. Um, and that- that- that's, uh, that scenario is called finite time blow-up. Um, so in practice, this doesn't happen.Um, so water is what's called turbulent. Um, so it is true that, um, if you have a big eddy of water, it will tend to break up into smaller eddies, but it won't transfer all the s- the energy from one big eddy into one small eddy. It will transfer it into maybe three or four. And then those ones split up into maybe three or four small eddies of their own. And so the energy gets dispersed to the point where the viscosity can, can then k- keep everything under control. Um, but if it can somehow, um, concentrate, um, all the energy, keep it all together, um, and do it fast enough that, that the viscous effects, uh, don't have enough time to calm everything down, then this blowup can occur. So there are papers who had claimed that, "Oh, you just need to take into account conservation of energy and just carefully use the viscosity and you can keep everything under control for not just in Navier-Stokes, but for many, many types of equations like this." And so like in, in the past, there had been many attempts to try to obtain what's called global regularity for Navier-Stokes, which is the opposite of final time blowup, that velocity stays smooth, and it all failed. There was always some sign error or some subtle mistake and, and it c- it couldn't be salvaged. Um, so what I was interested in doing was trying to explain why we were not able to disprove, um, final time blowup. I couldn't do it for the actual equations of fluids, which were too complicated, but if I could average the equations of motion of Navier-Stokes, so physically if, if, um, if I could turn off certain types of, of ways in which water interacts and only keep the ones that I want. Um, so in particular, um, if, um, if it's a fluid and it could transfer its energy from a large eddy into this small eddy or this other small eddy, I would turn off the energy channel that would transfer energy to this, this one and, and direct it only into, um, this smaller eddy while still preserving the law of conservation of energy.

Lex Fridman 13:58 So you're trying to make it blow up?

Terence Tao 14:00 Yeah. Yeah, so I, I, I basically engineer, um, a blowup by changing the laws of physics, which is one thing that mathematicians are allowed to do. We can change the equation.

Lex Fridman 14:08 How does that help you get closer to the proof of something?

Terence Tao 14:10 Right. So it provides what's called an ob- obstruction in mathematics. Um, so, so what I did was that, uh, basically if I turned off the, um, certain parts of the equation so that... which usually when you turn off certain interactions make it less nonlinear, it makes it more regular and less likely to blow up. But I found that by turning off a very well designed set of, of, of, of, of, of interactions, I could force all the energy to blow up, uh, in finite time. So what that means is that if you wanted to prove, um, global regularity for Navier-Stokes, um, for the actual equation, you had... you must use some feature of the true equation which, which my artificial equation, um, does not satisfy. So it, it rules out certain, um, certain approaches. So, um, the thing about math is, is it's not just about finding or taking a technique that is gonna work and applying it, but you, you need to not take the techniques that don't work. Um, and for the problems that are really hard, often there are dozens of ways that y- you might think might apply to solve the problem, but, uh, it's only after a lot of experience that you realize there's no way that these methods are going to work. So having these counterexamples for nearby problems, um, kind of rules out, um, uh... it, it saves you a lot of time because you, you, you're not wasting, um, energy on, on things that you now know cannot possibly ever work.

Lex Fridman 15:30 How deeply connected is it to that specific problem of fluid dynamics, or is it some more general intuition you build up about mathematics?

Terence Tao 15:38 Right. Yeah. So the key phenomenon that, uh, was, uh, my technique exploits is what's called supercriticality. So in partial differential equations, often these equations are like a tug of war between different forces. So in Navier-Stokes there's the dissipation, um, force coming from viscosity and, and it's very well understood. It's linear, it calms things down. If, if viscosity was all there was, then, then nothing bad would ever happen. Um, but there's also transport, um, that, that energy from in one location of space can get transported because the fluid is in motion to, to other locations. Um, and that's a nonlinear effect and that causes all the, all the problems. Um, so there are these two competing terms in the Navier-Stokes equation, uh, the dissipation term and the transport term. If the dissipation term dominates, if it's, if it's large then basically you get regularity and if, um, if the transport term dominates then, uh, then we don't know what's going on. It's a very nonlinear situation. It's unpredictable. It's turbulent. So sometimes these forces are in balance at small scales but not in balance at large scales or, or vice versa. Um, so Navier-Stokes is what's called supercritical. So at, at smaller and smaller scales the transport terms are much stronger than the viscosity terms. So the viscosity terms are the things that calm things down, um, and so this is, um, um, this is why the problem is hard. In two dimensions, so the Soviet mathematician, uh, Ledochinskaya, she, in the '60s showed in two dimensions there was no blowup. And in two dimensions, the Navier-Stokes equations is what's called critical. The effect of transport and the effect of viscosity are about the same strength even at very, very small scales. And we have a lot of technology to handle critical and also subcritical equations and prove, um, regularity. But for supercritical equations it was not clear what was going on, and I did a lot of work and then there's been a lot of follow-up showing that for many other types of supercritical equations you can create all kinds of blowup examples. Once the nonlinear effects dominate the linear effects at small scales you can have all kinds of bad things happen. So this is sort of one of the main insights of this, this line of work is that supercriticality versus criticality and subcriticality, this, this makes a big difference. I mean, the, that's a key qualitative feature that distinguishes some equations from being sort of nice and predictable and, and, you know, like, like planetary motion and... I mean, there, there's certain equations that, that you can predict for millions of years and, uh, or, or thousands at least. Again, it's not really a problem. But, but there's a reason why we can't predict the weather past two weeks into the future because it's a supercritical equation. Lots of really strange things are going on at, at very fine scales.

Lex Fridman 18:04 So whenever there is some huge source of nonlinearity-

Terence Tao 18:09 Yeah.

Lex Fridman 18:09 ... that can create a huge problem for predicting what's gonna happen?

Terence Tao 18:13 Yeah. And if the nonlinearity is somehow more and more featured and interesting at, at small scales.Um, I mean there's- there's many equations that are nonlinear, but, um, in many- in- in many equations you can approximate things by the bulk. Um, so for example, planetary motion. You know, if you wanted to, uh, understand the orbit of th- the moon or Mars or something, you don't really need the microstructure of, uh, like the seismology of the moon or- or like exactly how the mass is distributed, um, you just... basically, you- you can almost approximate these planets by point masses. Uh, and you... it's just the aggregate behavior is important. Um, but if you want to model a fluid, um, like- like the weather, you can't just say, "In Los Angeles, the temperature is this, the wind speed is this." For supercritical equations, the finest-grain information is- is really important.

Lex Fridman 18:54 If we can just linger on the Navier–Stokes, uh, equations a little bit. So you've suggested... maybe you can describe it, that one of the ways to, uh, solve it, or to negatively resolve it, would be to... sort of to construct a liquid, a kind of liquid computer-

Terence Tao 19:13 Right.

Lex Fridman 19:13 ... and then show that the halting problem from computation theory has, uh, consequences for fluid dynamics. So, uh-

Terence Tao 19:19 Right.

Lex Fridman 19:19 ... show it in that way. Can you describe this? This idea?

Terence Tao 19:22 Right. Yeah. So this came out of- of this work of constructing th- this- this av- this average equation that- that blew up. Um, so one, um... as- as part of how I had to- to do this... so there's sort of this naïve way to do it. You ju- y- you just keep tr- pushing, um, um... every time you- you get energy at one scale, you- you push it immediately to the next scale as- as fast as possible. This is sort of the naïve way to- to- to- to force blowup. Um, it turns out in five and higher dimensions, this works. Um, but in three dimensions, there was this funny phenomenon that I discovered, that if you- if you keep... if- if you change the laws of physics, you just always keep trying to push, um, the energy into smaller and smaller scales. Um, what happens is that the energy s- starts getting spread out into multi- ma- many scales at once, um, so that you have- you have energy at one scale, you're pushing it i- into the next scale, and then, um, as soon as it enters that scale, you also push it to the next scale, but there's still some energy left over from the previous scale. Um, you're trying to do everything at once, um, and this spreads out the energy too much, um, and then it turns out that- that, um, it makes it vulnerable for viscosity to come in and actually just damp out everything. So, um... so it turns out this- this direct approach doesn't- doesn't actually work. There was a separate paper by some other authors that actually showed this, um, in three dimensions. Um, so what I needed was to program a delay, um, so kind of like airlocks. Uh, so, um, I- I needed an equation which would start with a- a fluid doing something at one scale, it would push this energy into the next scale, but it would s- stay there until all the energy from the- from the larger scale got transferred. And only after you pushed all the energy in, then you sort of opened the next gate, uh, and- and then you- you push that in as well. So, um, by- by doing that, it kind of... the energy inches forward scale by scale in such a way that it's always, um, localized at one scale at a time, um, and then it can resist the effects of viscosity because it's- it's not dispersed. Um, so in order to make that happen, um... yeah, I had to construct a rather complicated non-linearity, um, and it was basically like, um... you know, like it was constructing like an e- electronic circuit. So I- I actually thank my wife for this because she was trained as a electrical engineer, um, and, um, you know, she- she talked about, um, uh... you know, she had to design circuits and so forth, and, you know, if you- if you want a circuit that does a certain thing, like maybe, uh, have a light that- that flashes on and then turns off and then on and then off, you can build it from s- from more primitive components, you know, capacitors and resistors and so forth, and you have to board a diagram, and you, um... and these diagrams, you can- you can sort of follow with your eyeballs and say, "Oh, yeah. The- the- the current will- will board up here and then it will stop and then it will do that." So I knew how to build the analog of basic electronic components, you know, like resistors and capacitors and so forth, and- and I would- I would stack them together, um, in such- in such a way that- that I would create something that would open one gate and then there'll be a clock that would... and then once the clock hits a certain threshold, it would close it. Uh, become a Rube Goldberg type machine, but described mathematically. And this ended up working. So what I realized is that if you could pull the same thing off for the actual equations... so if the equations of water support a computation. So, um, like if... you can imagine kind of a steampunk, but it's really water punk, uh, type of thing where, um... you know, so modern computers are electronic, you know, they- they- they're powered by- by electrons p- passing through very tiny wires and interacting with oth- other electrons and- and so forth, but instead of electrons, you- you can imagine these pulses of- of water moving at a certain velocity, and maybe it's... they're two different configurations corresponding to a bit being up or down. Probably if you had two of these moving bodies of water collide, uh, they would come out with some new configuration which is... which would be something like an AND gate or OR gate, you know, that... uh, it would... it... the- the output would depend in a very predictable way on- on the inputs. And, like, you could chain these together and maybe create a Turing machine, and- and then you could... you have computers, uh, which are made completely out of water. Um, and if you have computers, then maybe you can do robotics, uh, you know, so... I mean, you know, hydraulics and so forth. Um, and so you could create some machine which is basically a fluid analog of what's called a Von Neumann machine. Uh, so Von Neumann proposed... if you want to colonize Mars, the sheer cost of transporting people and machines to Mars is just ridiculous, but if you could transport one machine to Mars and this machine had the ability to mine the planet, create some more materials, smelt them, and build more copies of the same machine, um, then you could colonize a whole planet, um, over time. Um, so, uh, if you could build a fluid machine which, uh... yeah. So it's- it's- it's a- it's a robo... it's a fluid robot, okay, and what it would do... it- its purpose in life, it's programmed so that it would create a smaller version of itself in some sort of cold state. It- it wouldn't start just yet. Once it's ready, the big robot configured with water would transfer a- all its energy into the s- the smaller configuration and then power down, okay, and then, like- like... uh, clean itself up, and then what's left is this newest state which would then turn on and do the same thing, but smaller and faster.... and then the equation has a certain scaling symmetry. O- once you do that, it can just keep iterating. So this, in principle, would create a blow up, uh, for the actual Navier-Stokes, and this is what I managed to accomplish for this average Navier-Stokes. So it provided this sort of roadmap to solve the problem. Now, this is, uh, a, a pipe dream because, uh, there are so many things that are missing for this to actually be a reality. Um, so, um, I, I, I can't create these ba- basic logic gates. Um, I, I don't, I don't have these in these special configurations of water. Um, I mean, there's candidates. There's things called vortex rings that might possibly work, but, um, um, but also, you know, analog computing is really nasty, um, compared to digital computing. I mean, 'cause there's always errors. Um, you'll, you have to, you have to do a lot of error correction along the way. I don't know how to completely power down the big machine so that it doesn't interfere with the, the running of the smaller machine. But everything in principle can happen. Like, it doesn't contradict any of the laws of physics. Um, so it's sort of evidence that this thing is possible. Um, there are other groups who are n- now pursuing ways to make Navier-Stokes blow up which are nowhere near as ridiculously complicated as this. Um, um, they, they actually are, are pursuing much closer to the, 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.

Lex Fridman 25:40 There is a real leap of genius here to go from Navier-Stokes to this Turing machine. So it goes from what, the self-similar blob scenario that you're trying to get the smaller and smaller blob-

Terence Tao 25:53 Mm-hmm.

Lex Fridman 25:53 ... to now having a liquid Turing machine get smaller-

Terence Tao 25:57 Right.

Lex Fridman 25:57 ... and smaller and smaller, and somehow seeing how that could be used to say something about a blow up. I mean, that's a big leap.

Terence Tao 26:08 So there's precedent. I mean, um, so the, the thing about mathematics is that it's, it's really good at, um, spotting connections between what you think of, what you might think of as completely different, um, problems. Um, but if, if the mathematical form is the same, you, you could, you, you can, you can draw a connection. Um, so, um,

Terence Tao 26:26 there's a lot of work previously on what are called c- cellular automata.

Lex Fridman 26:28 Mm-hmm.

Terence Tao 26:29 Um, the most famous of which is Conway's Game of Life. There's this infinite discrete grid, and any given time, the grid is either occupied by a cell or it's empty. And there's a very simple rule that, uh, that tells you how these cells evolve. So s- sometimes cells live, and sometimes they, they die. Um, and there's a, um, you know, um, when I was a, a, a student, uh, it was a very popular screensaver to actually just have these, these animations l- going. And, and they look very chaotic. In, in fact they look a little bit like turbulent flow sometimes. But at some point, people discovered more and more interesting structures within this Game of Life. Um, so for example they discovered this thing called a glider. So a glider is a v- very tiny configuration of like four or five cells, which evolves, and it just moves at a certain direction. And that's like this, this vortex rings, this analog. Um, yeah, so this is an a- analogy. The Game of Life is kinda like a discrete equation and, and, um, the fluid Navier-Stokes is, is a continuous equation. But mathematically, they have some similar features. Um, and, um, so over time people discovered more and more interesting things that you could build within the Game of Life. The Game of Life is a very simple system. It only has like three or four rules, um, to, to do it. But, but you can design all kinds of interesting configurations inside it. Um, there's something ca- called a glider gun that does nothing but spit out gliders one at a t- one, one at a time. Um, and then after a lot of effort people managed to, to create, um, AND gates and OR gates for gliders. Like there's this massive ridiculous structure which if you, if a, if a, if you have a stream of gliders, um, coming in here and a stream of gliders coming in here, then you may produce extreme gliders coming out if, if maybe, maybe if both of, of the, um, streams, um, have gliders then there'll be an- an ou- output stream. But if only one of them does, then nothing comes out.

Lex Fridman 28:07 Mm-hmm.

Terence Tao 28:08 So they could build something like that. And once you could build an, um, these basic gates then just from software engineering you can build almost anything. Um, uh, you can build a Turing machine. I mean, it, it's, it's like an enormous steampunk type things. They look ridiculous, but then people also generated self-replicating objects in the Game of Life, a massive machine, a Bonomo machine, which over a lo- huge period of time and it always looked like glider guns inside doing these very steampunk calculations. It would create another version of itself which could replicate-

Lex Fridman 28:40 That's so incredible.

Terence Tao 28:42 A lot of this was like community crowdsourced by, uh, like amateur mathematicians actually. Um, so I knew about that, that, that work, and so th- that is part of what inspired me to propose the same thing for Navier-Stokes. Um, that really just... analog is much... I should say analog is much worse than digital. Like, uh, it's gonna be, um... you can't just directly take the constructions in the Game of Life and plunk them in, but again, it just, it shows it's possible.

Lex Fridman 29:06 You know, there's a kinda emergence that happens with these cellular automata local rules.

Terence Tao 29:13 Mm-hmm.

Lex Fridman 29:13 Maybe it's similar to fluids, I don't know, but local rules operating at scale can create these incredibly complex dynamic structures. Do you think any of that is amenable to mathematical analysis? Do we have the tools to say something profound about that?

Terence Tao 29:34 The thing is you can get this emergent very complicated structures but only with very carefully prepared initial conditions, yeah? So, so these, these, these glider guns and, and gates and, and software machines, if you just plunk down randomly some cells and you run the Game of Life you will not see any of these. Um, and th- that's the analogous situation with Navier-Stokes again, you know, that, that with, with typical initial conditions you will n- you will not have any of this weird computation going on. Um, but basically through engineering, you know, by, by, by s- specially designing things in a very special way you can make clever constructions.

Lex Fridman 30:07 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-

Terence Tao 30:14 Yeah. Yeah, yeah, yeah.

Lex Fridman 30:15 ... something interesting.

Terence Tao 30:16 This, this is a recurring challenge in mathematics that, um-... I call it the dichotomy between structure and randomness, that most objects that you can generate in mathematics are random. Uh, they look like ran- like, the digits of pi, well, we believe is a good example. Um, but there's a very small number of things that have patterns. Um, but, um, now, you can prove something has a pattern by just constructing, you know, like if, if something has a simple pattern and you have a proof that it, it does something like repeat itself every so often, you can do that. But, um, and you c- you can prove that, that, for example, you can, you can prove that most sequences of, of digits have no pattern. Um, so like if, if you just pick digits randomly, there's something called the law of large numbers that tells you that you're going to get as many oneses as, as twos in the long run. Um, but, um, we have a lot fewer tools to, to, to... If I give you a specific pattern like the digits of pi, how can I show that this doesn't have some weird pattern to it? Some other work that I spend 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. So some functions are what's called additive, like if you have a function that maps the natural numbers to the natural numbers, so maybe, um, you know, two maps to four, three maps to six and so forth. Um, some functions are what's, what's called additive, which means that if you add, if you add two inputs together, the output gets, gets added as well. Uh, for example, uh, multiplying by a constant, if you multiply a number by 10, um, if you, i- i- if you multiply A plus B by 10, that's the same as multiplying A by 10 and B by 10 and then adding them together, so some, um, functions are additive. Some functions are kind of additive but not completely additive. Um, so for example, if I take a number N, I multiply by the square root of two and I take the integer part of that, so 10 by the square root of two is like 14 point something, so 10 will map to 14. Um, 20 will map to 28. Um, so in that case, a- additivity is true then, so 10 plus 10 is 20 and 14 plus 14 is 28, but because of this rounding, uh, sometimes there's round off errors and, and sometimes when you, um, add A plus B, this function doesn't quite give you the sum of, of the two individual outputs, but the sum plus or minus one. Um, so it's almost additive, but not quite additive. Um, so there's a lot of useful results in mathematics and I've worked a lot on developing things like this, to the effect that if, if a function exhibits some structure like this, then, um, it's basically, th- there's a reason for why it's true and the reason is because there's, there's some other nearby function which is actually, um, com- completely structured which is explaining this sort of partial pattern that you have. Um, and so if you have these sort of inverse theorems, it, um, it creates this sort of dichotomy that, that either the objects that you study are either have no structure at all or they are somehow related to something that is structured. Um, and in either way, in either, um, uh, in either case you can make progress. Um,

Terence Tao 33:01 a good example of this is that there's this old theorem in mathematics called Szemerédi's theorem, uh, proven in the 1970s. It concerns trying to find a certain type of pattern in a set of numbers and the, the pattern is arithmetic progression, things like 3, 5, and 7, or, or, or 10, 15, and 20. And Szemerédi, Endre Szemerédi proved that, um, any set of, 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. Um, so for example, um, the odd numbers have a set of density one half, um, and they contain arithmetic progressions of any length. Um, so in that case it's obvious because the, the od- the odd numbers are really, really structured. I can just take, uh, uh, 11, 13, 15, 17, I can just, I can, I can easily find arithmetic progressions in, in, in that set. Um, but, um, Szemerédi's theorem also applies to random sets. If, if I take the set of all numbers and I flip a coin, um, and I, uh, for each number and I only keep th- the numbers which, for which I got a heads, okay, so I just flip coins, I just randomly take out half the numbers, I keep one half, so that's a set that has no, no patterns at all, but just from random fluctuations you will still get a lot of, um, um, of arithmetic pro- progressions in that set.

Lex Fridman 34:10 Can you prove that there's arithmetic progressions of arbitrary length within a random-

Terence Tao 34:17 Yes. Um, have you heard of the infinite monkey theorem? Usually mathematicians give boring names to theorems, but occasionally-

Lex Fridman 34:23 Yeah.

Terence Tao 34:23 ... they, they give colorful names.

Lex Fridman 34:24 Yes.

Terence Tao 34:24 The popular version of the infinite monkey theorem is that if you have an infinite number of monkeys in a room with each a typewriter, they type out, uh, text randomly. Almost surely one of them is going to generate the entire score of Hamlet or any other finite string of text. Uh, it will just take some time, uh, quite a lot of time actually. But if you have an infinite number, then it happens. Um, so, um, basically the theorem says that if you take an infinite string of, of digits or whatever, um, eventually any finite pattern you wish will emerge. Uh, it may take a long time, but it will eventually happen. Um, in particular, arithmetic progressions of any length will eventually happen. Okay, but you need a, b- but you need an extremely long random sequence for this to happen.

Lex Fridman 35:04 I suppose that's intuitive, it's just infinity.

Terence Tao 35:08 Yeah, infinity absorbs a lot of sins.

Lex Fridman 35:11 Yeah. How are we humans supposed to deal with infinity?

Terence Tao 35:15 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 full, um, that, uh, you know, I mean, so nothing in real life is truly infinite, um, but, you know, you can, um, you know, you can ask these sort of questions like, "What if I had as much money as I wanted?" You know, or, "What if I could go as fast as I wanted?" And a way in which mathematicians formalize that is, mathematics has found a formalism to idealize instead of something being extremely large or extremely small, to actually be exactly infinite or zero, um, and often the, the mathematics becomes a, a lot cleaner when you do that. I mean, in, 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 s- send some things to infinity, send so- some things to zero, um, and, uh, and the mathematics becomes a lot simpler to work with then.

Lex Fridman 36:06 I wonder how often using infinity, uh, forces us to deviate from, um, the physics of reality.

Terence Tao 36:17 Yeah, so there's, there's a lot of pitfalls, um, so, you know, we, we spend a lot of time in, uh, undergraduate math classes teaching analysis, um, and analysis is often about how to take limits and, and, and whether you... You know, so for example, A plus B is always B plus A.Um, so when, when you have a finite number of terms and you add them, you can swap them and there's n- there's no problem. But when you have an infinite number of terms, there are these sort of show games you can play where you can have a series which converges to one value, but you rearrange it and suddenly it converges to another value, and so you can make mistakes. You have to know what you're doing when you allow infinity. Um, you have to introduce these epsilons and deltas and, and, and there's, there's a certain type of way of reasoning that helps you avoid mistakes. Um, in more recent years, um, uh, people have started taking results that are true in, in infinite limits and try to

Lex Fridman 37:05 (laughs)

Terence Tao 37:05 ... what's ca- what's called finitizing them. Um, so you know that something's true eventually, but, um, you don't know when. Now, give me a rate. Okay, so such that if I have, don't have an infinite number of monkeys but, but a large finite number of monkeys, how long do you have to wait for Hamlet to come out? Um, and, uh, that's a more qua- quantitative question. Um, and this is something that you can, you can, um, attack by purely finite methods and you can use your finite intuition. Um, and i- in this case it turns out to be exponential in the length of the text that you're, you're trying to generate. Um, so if... um, and so this is why you never see the monkeys create Hamlet. You can maybe see them create a four-letter word, but nothing that big. And so I personally find once you finitize an infinite statement, it's, it does become much more intuitive a- and it's no longer so, so weird. Um...

Lex Fridman 37:51 So even if you're working with infinity, it's good to finitize so that you can have some intuition.

Terence Tao 37:57 Yeah. The downside is that the finitized proofs are just much, much messier.

Lex Fridman 38:00 Yeah.

Terence Tao 38:00 And, and, uh, yeah, so the, so the infinite ones are found first usually, like decades earlier, uh, and then later on people finitized

Terence Tao 38:07 them.

Lex Fridman 38:07 So since we mentioned a lot of the math and a lot of physics-

Terence Tao 38:10 Mm-hmm.

Lex Fridman 38:10 ... uh, what to you is the difference between mathematics and physics as disciplines, as ways of understanding, of seeing the world? Maybe we can throw in engineering in there. You mentioned your wife is an engineer, give a new perspective on circuits.

Terence Tao 38:23 Right.

Lex Fridman 38:23 So there's a different way of looking at the world given that you've done mathematical physics, so you, you've, you've worn all the hats.

Terence Tao 38:30 Right. So I think science in general is an interaction between three things. Um, there's the real world, um, there's what we observe of the real world, our observations, and then our mental models as to h- how we think the world w- works. Um, so, um, we can't directly access reality, okay? Uh, all we have are the, uh, observations, which are incomplete and they, they have errors, um, and, um, there are many, many cases where we would, um, uh, we want to know, for example, what is the weather like tomorrow and we don't yet ha- have the observation and we'd like to make a prediction. Um, and then we have these simplified models, sometimes making unrealistic assumptions, you know, spherical cow type things. Those are the mathematical models.

Lex Fridman 39:10 Mm-hmm.

Terence Tao 39:11 Mathematics is concerned with the models. Science collects the observations and it proposes the models that might explain these observations. What mathematics does i- uh, we, we stay within the model and we ask, "What are the consequences of that model? What observations, what, what predictions would the model make of the, of future observations? Uh, or past observations, does it fit observed data?" Um, so there, there's definitely a symbiosis. Um, it's... matha- 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 m- model. Um, in almost any other discipline, uh, you start with the conclusions. You know, "I want to do this. I want to build a bridge," you know, "I want, I want to, to make money. I want to do this." Okay, and then you, you, you find the paths to get there. Um, a lot... there's, uh, there's a lot less sort of speculation about, "Suppose I did this, what would happen?" Um, you know, planning and, and, and modeling. Um, uh, speculative fiction maybe is, is one other place, uh, but, uh, that's about it actually. Most of the things we do in life is conclusions driven, including physics and sci... you know, I mean, they want to know, you know, where is this asteroid gonna go, you know, or what, what, "What is the weather gonna be tomorrow?" Um, but, um, mathematics also has this other direction of, of going from the, uh, the axioms.

Lex Fridman 40:32 W- what do you think... there is this tension in physics between theory and experiment.

Terence Tao 40:36 Mm-hmm.

Lex Fridman 40:37 What do you think is a more powerful way of discovering truly novel ideas about reality?

Terence Tao 40:42 Well, you need both, top-down and bottom-up. Um, yeah, it's just a, it's, it's really an interaction between all these things. So over time, the observations and the theory and the modeling should g- both g- get closer to reality. But initially, and it is, I mean, uh, this is, um, this is always the case, you know, they're, they're always far apart to begin with. Um, but you need one to figure out, uh, where, where to push the other, you know? So, um, if your model is predicting anomalies, um, that are not picked up by experiment, that tells experimenters where to look, you know, um, to, to, to, to, to find more data to refine the models. Um, you know, so it, it, it goes, it goes back and forth. Um, within mathematics itself, there's, there's also a theory and experimental component. It's just that until very recently, theory has dominated almost completely. Like 99% of mathematics is theoretical mathematics, and there's a very tiny amount of experimental mathematics. Um, I mean, people do do it, you know. Like if they want to study prime numbers or whatever, they can just generate large datasets and with a com-... so once we had computers, um, we began to do it a little bit. Um, although even before... well, like Gauss for example, he discovered or he conjectured the most basic theorem in, in number theory, it's called the prime number theorem, which predicts how many primes that... up to a million, up to a trillion. It's not an obvious question. And basically what he did was that he computed, uh, I mean, mostly u- um, by himself, but also hired human computers, um, people who, whose professional job it was to do arithmetic, um, to compute the first 100,000 primes or something and made tables and made a prediction. Um, that was an early example of experimental mathematics. Um, but until very recently, it was not, um... yeah, I mean, theoretical mathematics was just much more successful. I mean, of course, doing complicated mathematical computations is, uh, was just not, not feasible, uh, until very recently.... and even nowadays, you know, even though- though we have powerful computers, only some mathematical things can be, um, explored numerically. There's something called the combinatorial explosion. If you want us to study, for example, Zermelo's theorem, we want to study all possible subsets of the numbers one to 1,000, there's only 1,000 numbers. How bad could it be? It turns out the number of different subsets of- of one to 1,000 is two to the power of 1,000, which is way bigger than- than- than any computer can currently can- can enu- in fact, any computer ever will ever, um, enumerate. Um, so if you have to be, um... There are certain math problems that very quickly become just in- intractable to attack by direct brute force computation. Uh, chess is another, um, uh, famous example. Uh, it's the number of chess positions, uh, we can't get a c- computer to fully explore. But now we have AI, um, um, we have tools to explore the space not with 100% guarantees of s- success, but with experiment, you know. So like, um, we can empirically solve chess now, right? For example, uh, we have, we have, uh, very, very good AIs that they can... You know, they don't explore every single position in- in the game tree, but they have found some very good approximation. Um, and people are using, actually, these chess engines, uh, to make... uh, to do experimental chess. Um, that, uh, they're- they're revisiting old chess theories about, oh, you know, when you... this type of opening, you know, that- that this is good... this is a good type of move, this is not, and then using chess engines to actually, uh, refine, and- and in some cases overturn, um, um, conventional wisdom about chess. And I- I do hope that, uh, that mathematics will- will have a larger experimental component in the future, perhaps powered by AI.

Lex Fridman 44:05 We'll, of course, ta- talk about that, but in the case of chess, and there's a similar thing in mathematics, the... I don't believe it's providing a kind of formal explanation of the different positions.

Terence Tao 44:17 No.

Lex Fridman 44:17 It's just saying which position is better or not, and then you can intuit as a human being. And then from that, we humans can construct-

Terence Tao 44:24 Yes.

Lex Fridman 44:24 ... a theory of the matter.

Terence Tao 44:27 Right.

Lex Fridman 44:27 You've mentioned the Plato's cave allegory.

Terence Tao 44:30 Mm-hmm.

Lex Fridman 44:30 So, in case people don't know, it's where people are observing shadows of reality, not reality itself, and they believe what they're observing to be reality. Is that, in some sense, what mathematicians and maybe all humans are doing, is, um, looking at shadows of reality? Do... Is it possible for us to truly access reality?

Terence Tao 44:55 Well, there are these three ontological things. There's actual reality, there's our observations, and our- our models. Um, and- and technically, they are distinct, and I think they will always be distinct, um...

Lex Fridman 45:08 Right.

Terence Tao 45:08 ... but they can get closer, um, over time. Um, you know, so, um... And the process of getting closer often means that you- you have to discard your initial intuitions, um, so, um, a- astronomy pro- provides great examples, you know, like, you know, like, you'll... An initial model of the world is that it's flat because it's- it looks flat, you know, uh, and, um, and that it's... and it's big, you know, and the rest of the universe, the skies is not... You know, like, the sun, for example, looks really tiny-

Lex Fridman 45:37 Mm-hmm.

Terence Tao 45:38 ... 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 for... you know. But- but over time, as you make more and more observations, bringing it closer to- to- to reality, um, the model gets dragged along with it, you know. And so, over time, we had to realize that the Earth was round, that it spins, it goes around the solar system, the solar system goes around the galaxy, and so on and so forth, and the- the guy's part of the universe is expanding. Um, the expansions are self-expanding, accelerating, and in fact, very recently, in- in this year or so, there's a... even the explosion of the universe itself is, uh, there's evidence now that it's- it's non-constant.

Lex Fridman 46:12 And, uh, the explanation behind why that is, is-

Terence Tao 46:16 It's catching up. Um... (laughs)

Lex Fridman 46:18 (laughs) it's catching up. I mean, it's still, you know, the dark matter, dark energy-

Terence Tao 46:20 Yeah.

Lex Fridman 46:21 ... this- this kind of thing.

Terence Tao 46:21 Mm-hmm. Yes. We have a- 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. So, if you have a model with 10 parameters that explains 10 out- 10 observations, that is a completely useless model. It's what's called over-fitted. But, like, if you have a model with- with, you know, two parameters and it explains a- a trillion observations, which is basically... uh... so, uh, yeah, the- the- the dark matter model, I think, has like 14 parameters and it explains petabytes of data, um, that- that- that- that the astronomers have. Um, you can think of a- of a theory, uh, like one way to think about, um, uh, a physical mathematical theory... a- a theory is- it's- it's a compression of- of the universe, um, and, uh, uh, data compression. So, you know, you have these petabytes of observations, you like to compress it to a model which you can describe in five pages and specify a certain number of parameters, and if it can fit to reasonable accuracy, you know, a- almost all of- of the observations, that... I mean, the more compression that you make, the better your theory.

Lex Fridman 47:32 In fact, one of the great surprises of our universe and of everything in it is that it's compressible at all.

Terence Tao 47:38 Yeah.

Lex Fridman 47:38 It's the unreasonable effectiveness of mathematics.

Terence Tao 47:40 Yeah. Einstein had a quote like that. The- the most incomprehensible thing about the universe is that it is comprehensible.

Lex Fridman 47:45 Right. And not just comprehensively. You can do an equation like E equals MC squared.

Terence Tao 47:49 There is actually a- some mathematical possible explanation for that. Um, so there's this phenomenon in mathematics called universality. So, many complex systems at the macroscale are coming out of lots of tiny interactions at- at the macroscale, and normally, because of the combinatorial explosion, you would think that, uh, the macroscale equations must be like infinitely, exponentially more complicated than- than the, uh, the microscale ones. And they are, if you want to solve them completely exactly. Like if you want to model, um, all the atoms in a box of- of air, uh, that's like Avogadro's number is humongous, right? There's a huge number of particles. If you actually had to track each one, it would be ridiculous. But certain laws emerge at the microscopic scale that almost don't depend on what's going on at the macroscale or only depend on a very small number of parameters. So, if you want to model a gas, um, of, you know-... quintillion particles in, in a box. You just need to know its temperature and pressure and volume and a few parameters, like five or six, and it models almost everything you, you need to know about these 10 to the 23 or whatever particles. 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 univers- universality occurs. Um, uh, most basic one is, is the central limit theorem. That explains why the bell curve shows up everywhere in nature, that so many things are distributed by, uh, what's called a Gaussian distribution, uh, the famous bell curve. Uh, there's now even a meme with this curve.

Lex Fridman 49:18 And even the meme applies broadly.

Terence Tao 49:21 Yeah.

Lex Fridman 49:21 There's a universality to the meme.

Terence Tao 49:23 Yeah, yes, you can call meta-

Lex Fridman 49:24 (laughs)

Terence Tao 49:24 ... uh, if you like. But there are many, many processes. For example, you c- you can take lots and lots of independent, um, random variables and average them together, um, uh, in, in various ways. You basically... you could take a simple average or more complicated average, and we can prove in various cases that, that these, these bell curves, these Gaussians emerge, and it is a satisfying, satisfying explanation. Um, sometimes they don't. Um, so, so if you have many different inputs and they're all correlated in some s- systemic way, then y- you can get something very far from a bell curve to show up. Uh, and this is also important to know when this fails. So universality is not a 100% reliable thing to rely on. In fact, um, um, the, the global financial crisis was a, a famous example of this. Uh, people thought that, uh, um, mortgage de- defaults, um, um, had this sort of, um, Gaussian-type behavior that, that if you, if you ask... if a population of, of, of, uh, you know, 100,000 Americans with mortgages, ask what, what proportion of them would default on their mortgages, um, if everything was de-correlated, it would be a nice bell curve and, and like you can, you can, you can manage risk with options and derivatives and so forth and, um, and it is a very beautiful theory. Um, but if there are systemic shocks in the economy (laughs) , uh, that can push everybody to default at the same time, uh, that's very non-Gaussian behavior. Um, and, uh, this wasn't fully accounted for (laughs) in, uh, 2008. Uh, now I think there's some more awareness that this is a... systemic risk is actually a, a much bigger issue. And, uh, just because the model is pretty, uh, and nice, uh, it may not match reality. And so, so the mathematics of working out what models do is really important, um, but, um, also the, the, the science of validating when the models, uh, fit reality and when they don't, um, I mean, that... you need both. Um, and... but mathematics can help because it c- it can, uh... for example, these central limit theorems, it, it tells you that, that if you have certain axioms like, like, like, uh, non-correlation, that if all the inputs were not correlated to each other, um, then you have this Gaussian behavior so things are fine. It, it tells you where to look for weaknesses in the model. So if you have a mathematical understanding of central limit theorem and someone proposes to use these Gaussian copulas or whatever to mo- to model, um, default risk, um, if you're mathematically, um, trained, you would say, "Okay, but what are the systemic correlation between all your inputs?" And so then the... then you can ask the economists, you know, "How, how, how much of a risk is that?" Um, and then you can, you can, you can go look for that. So there's always this, this, this synergy between science and, and mathematics.

Lex Fridman 51:52 A little bit on the topic of universality...

Terence Tao 51:54 Mm-hmm.

Lex Fridman 51:56 You're known and celebrated for working across an incredible breadth of mathematics, reminiscent of Hilbert a century ago. In fact, the great Fields Medal-winning mathematician, Tim Gowers, has said that you are the closest thing we get to Hilbert.

Terence Tao 52:12 Ah. (laughs)

Lex Fridman 52:12 (laughs) He's a colleague of yours.

Terence Tao 52:15 Oh, yeah. Good friend.

Lex Fridman 52:16 But anyway, so y- yeah, you are known for this ability to go both deep and broad in mathematics, so you're the perfect person to ask, do you think there are threads that connect all the disparate area- areas of mathematics? Is there a kind of deep underlying structure, uh, to all of mathematics?

Terence Tao 52:36 There's certainly a lot of connecting threads, um, and a lot of the progress of mathematics has... can be represented by taking... by stories of two fields of mathematics that were previously not connected and finding connections. Um, an ancient example is, um, geometry and number theory. You know, so, so in the times of the ancient Greeks, these were considered different subjects. Um, I mean, mathematicians worked on both. You know, you could, uh, work both on, on geometry most famously, but also on numbers. Um, but they were not really considered related. Um, I mean, a little bit, like, you know, you, you could say that, that, that this length was five times this length because you could take five copies of this length and so forth. But it wasn't until Descartes who really realized that... uh, who developed what I call analytic geometry, that you can, you can parametrize the plane, a geometric object, by, um, by two real numbers, right? Every point can be... and so geometric problems can be turned into, into problems about numbers. Um, and the, the... today, this feels almost trivial, like the... there's like... there's, there's, there's no content to this. Like, of course, uh, you, you know, um, a plane is X, X and Y, you know, 'cause that's what we teach and it's internalized. Um, but it was an important development that these, these two fields, uh, uh, were unified. Um, and this process has just gone on throughout mathematics over and over again, that algebra and geometry were separated, and now we have this root algebraic geometry that connects them and, and over and over again. And that's certainly the type of mathematics that I enjoy the most. So I think there's sort of different styles to being a mathematician. I think hedgehogs and foxes, a fox knows many things a little bit, but a, a hedgehog knows one thing very, very well. Um, and in mathematics, there's definitely both hedgehogs and foxes, um, and then there's people who are kind of... uh, who can play both roles. Um, and I think i- ideal collaboration between mathematicians in- involves very... you need some diversity, like, um, a f- a fox working with many h- hedgehogs or vice versa. So, yeah, but, but I identify mostly a- as a fox, uh, certainly. I, I, I like, uh, arbitrage somehow, you know, like, um, learning how one field works, learning the tricks of that world, and then going to another field which...... people don't think of it as related, but I can, I can adapt the tricks.

Lex Fridman 54:49 So see the connections-

Terence Tao 54:51 Yeah.

Lex Fridman 54:51 ... between the fields.

Terence Tao 54:52 Yeah. So there are other mathematicians who are far deeper than I am, like who, they're, who have, they're really hedgehogs. You know, they, they've, they, they know everything about one field and they're, they're much faster and, and more effective in that field. But I can, I can give them these extra tools.

Lex Fridman 55:05 I mean, you've said that you can be both a hedgehog and a, and, and the fox, depending on the context-

Terence Tao 55:09 Yeah.

Lex Fridman 55:09 ... depending on the collaboration. So what... Can you, if it's at all possible, speak to the difference between those two ways of thinking about a problem? Say you're encountering a new problem, you know, searching for the connections versus, like, very singular focus?

Terence Tao 55:26 I'm much more comfortable with, with the, uh, the, uh, the fox paradigm. Yeah, so, um, yeah, I- I like looking for analogies, narratives. Um, I, I spend a lot of time, if, if there's a result, I see it in, in one field and I like the result, it's a cool result, but I don't like the proof. Like, it uses-

Lex Fridman 55:45 (laughs)

Terence Tao 55:45 ... types of mathematics that I'm not super familiar with.

Lex Fridman 55:47 Yes.

Terence Tao 55:47 Um, I often try to reprove it myself using the tools that I, I favor. Um, often my proof is worse, um, but, um, by the exercise of doing so, um, I can say, "Oh, now I can see what the other proof was trying to do." Um, and from that, I can get some understanding of, of the tools that are used in, in that field. So it's very exploratory, very doing crazy things in crazy fields and, and, like, yeah, reinventing the wheel a lot. Yeah, whereas, so the hedgehog style is, uh, I think much more scholarly. You know, you, you, you're very knowledge-based. You, you, you, you stay up to speed on, like, all the developments in this field. You, you know all the history. Um, you have a very good understanding of, of exactly the strengths and weaknesses of, of each particular, uh, technique. Um, yeah, uh, I think you, you'd rely a lot more on sort of calculation than sort of trying to find narratives. Um, so yeah, I mean, I can do that too, but, uh, there are other people who are extremely good at that. (laughs)

Lex Fridman 56:44 Let's step back and, uh, uh, maybe look at the, the, a bit of a romanticized version of mathematics.

Terence Tao 56:52 Mm-hmm.

Lex Fridman 56:53 So, uh, I think you've said that early on in your life, uh, math was more like a puzzle-solving activity-

Terence Tao 57:01 Mm-hmm.

Lex Fridman 57:01 ... when you were-

Terence Tao 57:02 Mm-hmm.

Lex Fridman 57:02 ... uh, young. When did you first encounter a problem or proof where you realized math can have a kind of elegance and beauty to it?

Terence Tao 57:14 That's a good question. Um, when I came to graduate school, uh, in Princeton, um, so John Conway was there at the time. He, he passed away a, a few years ago. But, uh, I remember one of the very first research talks I, I went to was a talk by Conway on what he called extreme proof. So Conway just had this, this amazing way of, of thinking about all kinds of things in a, in a way that you wouldn't normally think of. So, um, he thought of proofs themselves as occupying some sort of space. You know, so, so, um, if you want to prove something, let- let's say that there's infinitely many primes, okay? There, you have all your different proofs. But you could, you could rank them in different axes, like some proofs are elegant and some proofs are long, some proofs are, are, um, are elementary and so forth. Um, and so there's this cloud, uh, so the space of all proofs i- itself has some sort of shape. Um, and so he, he was interested in, in extreme points of the shape. Like, out of all, all these proofs, which one is the shortest at the ex- at the expense of everyth- everything else? Or, or the most elementary or, or whatever. Um, and so he gave some examples of well-known theorems and then he would give what he thought was, was the extreme proof, um, in these different aspects. Um, um, I, I just found that really eye-opening, 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, tr- trying to, to, uh, to optimize it in various ways, um, that, that proof, um, uh, proofing itself had some craftsmanship to it. Um, it, it certainly informed my writing style, um, that, you know, like when you do your, your math assignments in, as a undergraduate, your homework and so forth, you, you're sort of encouraged to just write down any proof that works, okay? And hand it in and get a, get, as long as it gets a tick mark, you, you move on. Um, but if you want your, your results to actually be influential and be read by people, um, it can't just be correct. It should also, um, be a pleasure to read, you know, um, motivated, um, be adaptable to, to generalize to other, um, things. Um, it's the same in, in many other disciplines, like, like coding. There's an, there's a, uh, there's a lot of analogies between math and coding. I like analogies, if you haven't noticed.

Lex Fridman 59:17 (laughs)

Terence Tao 59:17 Um, but, um-

Lex Fridman 59:19 Yeah.

Terence Tao 59:19 ... you know, like you can code something, spaghetti code that works for a certain task and it's quick and, and dirty and it works, but, uh, there's lots of good principles for, for, um, writing, uh, code well so that other people can use it, build upon it and so then it has fewer bugs and whatever.

Lex Fridman 59:33 Mm-hmm.

Terence Tao 59:34 Um, and there's a, uh, similar things with, with math- mathematics. So...

Lex Fridman 59:37 Yeah, the (laughs) ... for, first of all there's so many beautiful things there and, and Conway is one of the great minds, uh, in mathematics ever, and computer science. Uh, just even considering the space of proofs-

Terence Tao 59:50 Yeah.

Lex Fridman 59:50 ... and saying, "Okay, what does this space look like? And what are the extremes?" Uh, like you mentioned, coding as an analogy is interesting because there's also this activity called, uh, Code Golf-

Terence Tao 1:00:02 Oh, yeah, yeah, yeah.

Lex Fridman 1:00:02 ... 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.

Terence Tao 1:00:12 Yeah.

Lex Fridman 1:00:12 And I believe there's even competitions on this.

Terence Tao 1:00:14 Yeah, yeah, yeah. Yeah, yeah.

Lex Fridman 1:00:15 And, uh, it's also a nice way to stress test not just the, s- sort of the programs, or in this case, the proofs, but also the different languages. Maybe that's a different notation or whatever to use to, to accomplish a different task.

Terence Tao 1:00:31 Yeah. Yeah, you learn a lot. I mean, it, it may seem like a frivolous exercise, but i- it can generate all these insights which, if you didn't have this artificial, um, objective to, to p- to pursue, you, you might not see.

Lex Fridman 1:00:43 What to you is the most beautiful or elegant equation in mathematics? I mean, one of the things that people often look to in, in beauty is the simplicity. So if you look at E equals MC squared.

Terence Tao 1:00:56 Mm-hmm.

Lex Fridman 1:00:56 So when, when a few concepts come together, that's why the Euler identity-

Terence Tao 1:01:01 Mm.

Lex Fridman 1:01:01 ... is often considered...... uh, the most beautiful equation in mathematics.

Terence Tao 1:01:05 Mm-hmm. Mm-hmm.

Lex Fridman 1:01:05 Do, do you find beauty in that one, in the Euler identity?

Terence Tao 1:01:08 Yeah. Well, as I said, I mean, what I find most appealing is, is connections between different things that, that, um... So the- if you, uh, e to the pi i equals minus one. Um, so yeah, people are like, "Oh, this uses all the fundamental constants." Okay. That- that- that's... I mean, that's cute. Um, but, but to me-

Lex Fridman 1:01:24 (laughs)

Terence Tao 1:01:24 So the-

Lex Fridman 1:01:25 Yeah.

Terence Tao 1:01:25 ... exponential function was introduced by Euler to measure exponential growth, you know, so comp- compound interest or decay or anything which is continuously growing, continuously decreasing. Growth and decay or dilation or contraction is modeled by the exponential function. Um, whereas pi, uh, comes around from circles and rotation, right? If you want to rotate a needle, for example, 180 degrees, uh, you need to rotate it by pi radians. And i, complex numbers, represents this whole thing, we want to imagine the axis of a 90-degree rotation, so a change in direction. So the exponential function represents growth and decay in the direction that you already are. Um, when you stick an i in the exponential, it- it- it now, it's- it's, uh, instead of motion in the same direction as your current position, it's motion as right angles to your current position. So rotation. Um, and then so, e to the- e to the pi i equals minus one tells you that if you rotate for time pi, you end up at the other direction. So it unifies geometry through dilation and exponential growth, you know, dynamics through this act of, of classificion, rotation by, by, by i. So it, it connects together all these tools in mathematics, you know. Yeah. That language, geometry, and complex and complex and, um, the complex numbers, they're all considered almost... You know, they're all next door neighbors in mathematics because of this iden- identity.

Lex Fridman 1:02:37 Do, do you think the thing you mentioned is cute, the, the, the collision of notations from these disparate fields, um, is just a frivolous side effect? Or do you think there is legitimate, like, value in-

Terence Tao 1:02:49 Um-

Lex Fridman 1:02:49 ... notation? All the, our old friends come together (laughs) in the night.

Terence Tao 1:02:53 Right. Well, it's, it's, it's confirmation that you have the right concepts. Um, so when you first study anything, um, you, you have to measure things and give them names. Um, and initially, sometimes your, because your, your model is, again, too far off from reality, you give the wrong things the best names, and you only find out later what's, what's really important.

Lex Fridman 1:03:14 Physicists can do this sometimes. I mean, but it turns out okay.

Terence Tao 1:03:18 So actually, physics, so E equals MC squared. Okay. So, uh, one of the, the big things was the E, right? So when, when Aristotle first came up with his laws of, of motion and then, and then, um, Galileo and Newton and so forth, you know, they saw the things that they could measure. They could measure mass and acceleration and force and so forth. And so Newtonian mechanics, for example, you know, F equals MA was the famous, uh, Newton's second law of motion. So those were the, the primary objects, so they gave them the central billing in the theory. It was only later after people started analyzing these equations that there always seemed to be these quantities that were conserved. Um, so, um, in particular, momentum and energy. Um, uh, and it's not obvious that things have an energy. Like, it's not something you can directly measure the same way you can measure mass and, and, and velocity and so forth. But over time, people realized that this was actually a really fundamental concept. Hamilton eventually, in the 19th century, reformulated Newton's laws of physics into what is called Hamiltonian mechanics, where the energy, which is now called the Hamiltonian, was the dominant object. Once you know how to measure the Hamiltonian of any system, you can describe completely the, the dynamics. Like, well, what happens to, to all, all the states? Like, it, um, it, it really was a central actor, which was not obvious initially. Um, and this, uh, helped actually, uh, this change of perspective really helped when quantum mechanics came along, uh, because I- um... The early physicists who studied quantum mechanics, they had a lot of trouble trying to adapt their N- Newtonian thinking because you know, everything was a particle and so forth, to, to, to quantum mechanics, you know, because now everything, because it was a wave, it just looked really, really weird. Um, like he asked, "What is the quantum version of F equals MA?" And it's really, really hard to, to give an answer to that. Um, but it turns out that the Hamiltonian, which was so, um, secretly behind the scenes in classical mechanics, also is the key, uh, object in, um, um, in quantum mechanics, that this- there's also an object called a Hamiltonian. It's a dif- different type of object. It's what's called an operator rather than, than a function, but, um, and, um, but again, once you specify it, you specify the entire dynamics. So this thing called Schoenflies equation that tells you exactly how quantum systems evolve once you have a Hamiltonian. So side by side, they look completely different objects, you know. Like, so one involves particles, one involves waves and so forth. But with this centrality, you could start actually transferring a lot of intuition and facts from classical mechanics to quantum mechanics. So for example, in, in classical mechanics, there's this thing called Noether's theorem. Every time there's a symmetry in a physical system, there is a conservation law. So the laws of physics are translation invariant. Like, if I move 10 steps to the left, I experience the same laws of physics as if I was here, and that corresponds to conservation of mome- momentum. Um, if I turn around by n- by some angle, again I experience the same laws of physics. Uh, this corresponds to conservation of angular momentum. If I wait for 10 minutes, um, I still have the same law, laws of physics, um, so there's time translation variance. This corresponds to the law of conservation of energy. Um, so there's this fundamental connection between symmetry and conservation, um, and that's also true in quantum mechanics, even though the equations are completely different, but because they're both coming from the Hamiltonian, the Hamiltonian controls everything. Um, every time the Hamiltonian has symmetry, the equations will, will have a conservation law. Um, so it's, it's, it's, it's, it's... Once you have the right language, it actually makes things, um, uh, a lot, a lot cleaner. One of the problems why we can't unify quantum mechanics and general relativity yet, we haven't figured out what the fundamental objects are. Like, for example, we have to give up the notion of space and time being these almost Euclidean type spaces, and it has to be s- um, you know, and, you know, we kind of know that at very tiny scales, uh, um, there's going to be quantum fluctuations, there's a space, spacetime foam, um, and trying to, to use Cartesian coordinates X, Y, Z is going to be, it's, it's just, it's, it's a nonstarter. But we don't know how to, what to replace it with, um, we don't actually have the mathematical, we, um...... um, concepts, uh, the analog of a Hamiltonian that sort of organized everything.

Lex Fridman 1:07:09 Does your gut say that there is a theory of everything, so this is po- even possible to unify, to find this language that unifies general relativity and quantum mechanics?

Terence Tao 1:07:19 I believe so. I mean, the history of physics has been that unification, much like mathematics, um, over the years. You know, electricity and magnetism were, were separate theories, and then Maxwell unified them. You know, Newton unified the, the motions of the heavens with the motions on, of, of objects on the earth and so forth. So it should happen. Uh, it's just that the, um, uh... again, to go back to this model of the, of the observations and, and, and theory, part of our problem is that physics is a victim of its own success. That, uh, two big theories of, of, of physics, general relativity and quantum mechanics, are so, are so good now that together they cover 99.9% of sort of all the observations we can make. Um, and you have to, like, either go to extremely insane particle accelerations or, or the early universe or, or, or things that are really hard to measure, um, in order to get any deviation from either of these two theories to the point where you can actually figure out how to, how to c- c- combine them together. 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. There, there's no reason why we should stop.

Lex Fridman 1:08:18 Do you think it will be a mathematician that develops a theory of everything?

Terence Tao 1:08:24 What often happens is that when the physicists need, uh, um, some theory of math- mathematics, there's often some precursor that the mathematicians, um, worked out earlier. So when Einstein started realizing that space was curved, he went to some mathematician and asked, you know, "Is there, is there some theory of curved space that mathematicians already came up with that could be useful?" And he said, "Oh, yeah. There's a... I think, uh, Riemann came up with something." Um, and so, yeah, Riemann had developed, you know, Riemannian geometry, um, which is precisely, um, you know, a, a, a, a theory of spaces that are curved in, in various general ways, which turned out to be almost exactly what was needed, um, for Einstein's theory. This is going back to, to Wigner's unreasonable effectiveness of mathematics. I think the theories that work well to explain the universe tend to also involve the same mathematical objects that work well to solve mathematical problems.

Lex Fridman 1:09:11 Mm-hmm.

Terence Tao 1:09:12 Ultimately, they just have both ways of organizing data, um, in, in, in, in useful ways.

Lex Fridman 1:09:17 It just feels like you might need to go to some weird land that's very hard to, to intuit, like-

Terence Tao 1:09:23 Yeah, yeah.

Lex Fridman 1:09:23 ... you know, you have, like, string theory.

Terence Tao 1:09:25 Yeah. That, that's, that was, that was a leading candidate for many decades. It's, I think it's slowly falling out of fashion because it's, it's not matching experiment, uh...

Lex Fridman 1:09:33 So one of the big challenges, of course, like you said, is experiment is very tough-

Terence Tao 1:09:38 Yes.

Lex Fridman 1:09:38 ... because of the how effective-

Terence Tao 1:09:40 Yeah.

Lex Fridman 1:09:40 ... both theories are. But the other is, like, just, you know, you're talking about you're not just deviating from spacetime. You're going into, like, some crazy number of dimensions.

Terence Tao 1:09:52 Yeah. Yeah.

Lex Fridman 1:09:52 You're doing all kinds of weird stuff that to us... We've gone so far from this flat Earth that we started at-

Terence Tao 1:09:58 Yes.

Lex Fridman 1:09:58 ... like you mentioned. (laughs)

Terence Tao 1:09:59 Yeah, yeah, yeah, yeah.

Lex Fridman 1:10:00 And now we're just... it's, it's very hard to use our limited ape descendants of, uh, uh, cognition to intuit what that reality really is like.

Terence Tao 1:10:10 This is why analogies are so important, you know. I mean, so yeah, when... the round Earth is not intuitive because we're, we're stuck on it, um, but, you know, but y- y- y- you know, but round objects in general, we have pretty good intuition, a little bit. Uh, and we have intuition about light works and so forth. And, like, it's, it's actually a good exercise to actually work out how eclipses and phases of, of the sun and the moon and so forth can be really easily explained by, by, by, by round Earth and round moon, you know, um, and models. Um, and, and i- and you can just take, you know, a basketball and a golf ball and, and a, and, and a light source and actually do these things yourself. Um, so the intuition is there, um, but yeah, you have to transfer it.

Lex Fridman 1:10:47 That is a big leap intellectually for us to go from flat to round Earth-

Terence Tao 1:10:51 (laughs)

Lex Fridman 1:10:51 ... because, you know, our life is mostly lived in flat land.

Terence Tao 1:10:55 Yeah.

Lex Fridman 1:10:55 To load that information, and we all, like, take it for granted. We take so many things for granted because science has established a lot of evidence-

Terence Tao 1:11:03 Mm-hmm.

Lex Fridman 1:11:03 ... for this kind of thing, but, you know, we're on a r- r- ro- round rock (laughs) -

Terence Tao 1:11:08 Yeah.

Lex Fridman 1:11:08 ... flying through space.

Terence Tao 1:11:10 Yeah, yeah. Yeah.

Lex Fridman 1:11:10 And it's a big leap. And you have to take a chain of those leaps the more and more and more we progress.

Terence Tao 1:11:15 Right. Yeah, so modern science is maybe again a victim of its own success, is that, you know, in order to be more accurate, it has to, to move further and further away from your initial intuition. And so, um, for someone who hasn't gone through the whole process of science education, it looks more and more sus- suspicious-

Lex Fridman 1:11:30 Yeah.

Terence Tao 1:11:30 ... because of that. So, you know, we, we, we need, we need more grounding. I mean, I, I think, um... I mean, you know, there are, there are scientists who do excellent outreach, um, but there's, there's, there's, there's, there's lots of science things that you can do at home. I, uh, there's lots of YouTube videos. I did a YouTube video recently of Grant Sandeson. We talked about this earlier, that, uh, you know, how, how the ancient Greeks were able to measure things like the distance to the moon, distance to the earth and, you know, using techniques that you c- you could also replicate yourself. Um, it doesn't all have to be, like, fancy space telescopes and, and really int- intimidating mathematics.

Lex Fridman 1:12:01 Yeah. That's, uh... I highly recommend that. I believe you give a lecture, and you also did an incredible video with Grant. It's a beautiful experience to try to put yourself in the mind of a person from that time-

Terence Tao 1:12:12 Mm-hmm.

Lex Fridman 1:12:13 ... shrouded in mystery.

Terence Tao 1:12:15 Right.

Lex Fridman 1:12:15 You know, you're, like, on this planet. You don't know the shape of it, the size of it. You see some stars, you see some, you see some things, and you try to, like, localize yourself in this world-

Terence Tao 1:12:25 Yeah, yeah.

Lex Fridman 1:12:25 ... and try to make some kind of general statements about distance to places.

Terence Tao 1:12:29 Change of perspective is really important. They say travel broadens the mind. Uh, this is intellectual travel, you know. Put yourself