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Terence Tao

lines on the record: 252
first: 2025-06-15
most recent: 2025-06-15
sittings or episodes: 1
sources: podcast

Every line below is reproduced as it was said and linked to the record it came from. Nothing here is summarised or generated. Directory · Search · Corrections

“So a lot of mathematicians were involved in the design of lean. So it's designed so that individual lines of code resemble individual lines of argument. You might want to introduce a variable, you want to prove by contradiction. There are various standard things that you can do, and it's written ideally like a one-to-one correspondence. In practice, it isn't because Lean is like explaining a proof to extremely pedantic colleague who will point out, okay, did you really mean this? What happens if this is zero? Okay, how do you justify this?”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“To create new ones. So the idea is not new. These things are called proof assistance. And so they provide languages for which you can create quite complicated, intricate.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“So Wien is a computer language, much like standard languages like Python and C and so forth, except that in most languages, the focus is on using executable code. Lines of code do things. They flip bits or they make a robot move or they deliver you text on the internet or something. So Lean is a language that can also do that. It can also be run as a standard traditional language, but it can also produce certificates. So a software language like Python might do a computation and give you that the answer is seven. It does the sum of people plus fours equal to seven, but lean can produce not just the answer, but a proof that how it got the answer of seven as 3 plus 4 and all the steps involved. So it creates these more complicated objects, not just statements, but statements with proofs attached to them. And every line of code is just a way of piecing together previous state.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“There's a lot of drawing and a lot of bespoke doodles that only make sense to me. And that's the beauty of Blackboard you raise. It's very organic thing. I'm beginning to use more and more computers, partly because AI makes it much easier to do simple coding things. If I wanted to plot a function before which is moderately complicated, I have some iteration or something, I'd have to remember how to set up a Python program and how does a for loop work and debug it and it would take two hours and so forth. And now I can do it in 10, 15 minutes. I'm using more and more computers to do simple explorations.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Mostly pen and paper. Actually, in my office, I have four giant blackboards, and sometimes I just have to write everything I know about the problem on the four blackboards and sit on my couch and just sort of see the whole thing.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“You know how to solve the 10 difficulties separately, then you have to start merging them a few at a time. As a kid, I watched a lot of these Hong Kong action movies from a culture. And one thing is that every time there's a fight scene, so maybe the hero gets swarmed by a hundred bad guy goons or whatever. But it will always be choreographed so that you'd always be only fighting one person at a time and it will defeat that person and move on. Because of that, he could defeat all of them. But whereas if they had fought a bit more intelligently and just swarmed the guy at once, it would make for much, much worse cinema, but they would win.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“It's like trying to solve a computer game where there's unlimited cheat codes available. And so you can set there's a dimension that's large. I'll set it to one. I'll solve the one dimensional problem first. There's a main term and an error term. I'm going to make a spherical cow assumption. I'll assume the error term is zero. And so the way you should solve these problems is not in sort of this iron man mode where you make things maximally difficult. But actually the way you should approach any reasonable mass problem is that if there are 10 things that are making your life difficult, find a version of the problem that turns off nine of the difficulties but only keeps one of them. And so that and then that just so you install nine cheats. Okay, you saw 10 cheats then the game is trivial. Then you saw nine cheats. You saw one form that teaches you how to deal with that particular difficulty. And then you turn that one off and you turn someone else on and then you solve that one.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“A lot of pen and paper. One thing you pick up as a mathematician is sort of a cheating strategically. So the beauty of mathematics is that you get to change the problem and change the rules as you wish. You don't get to do this for any other field. If you're an engineer and someone just built a bridge over this river, you kind of say, I want to build this bridge over here instead. Or I want to put it out of paper instead of steel. But a mathematician, you can do whatever you want.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“To see how to change coordinates in such a way that somehow things in all directions would behave in a reasonably linear fashion. And yeah, my aunt walked in on me while I was doing that and she was asking, what are that? What am I doing doing this?”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Even when energy was small. But I developed what's called a gauge transformation. So the equation is kind of like an evolution of heaps of wheat, and they're all bending back and forth. And so there's a lot of motion. But if you imagine stabilizing the flow by attaching little cameras at different points in space, which are trying to move in a way that captures most of the motion, and under this stabilized flow, the flow becomes a lot more linear. I discovered a way to transform the equation to reduce the amount of nonlinear effects. And then I was able to solve the equation. I found this transformation while visiting my aunt in Australia. And I was trying to understand the dynamics of all these fields, and I couldn't do it with pen and paper. And I had not enough facilitative computers to do any computer simulations. So I ended up closing my eyes being on the floor and just imagining myself to actually be this vector field and rolling around to try to”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Feel. And I was interested in the global regularity problem again for this question is it possible for all the energy here to collect at a point. So equation I considered was actually what's called a critical equation where it's actually the behavior at all scales is roughly the same. And I was able barely to show that you couldn't actually force a scenario where all the energy concentrated at one point, that the energy had to disperse a little bit. At the moment, it just a little bit would stay regular. Yeah, this was back in 2000. That was part of why I got interested in Navy Stocks afterwards, actually. Yeah, so I developed some techniques to solve that problem. So part of it was this problem is really nonlinear because of the curvature of the sphere. There was a certain nonlinear effect which was non-perturbative effect. When you sort of looked at it normally, it looked larger than the linear effects of the wave equation. And so it was hard to keep things under control.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“I have worked on some equations. There's something called the wave maps equation or the sigma field model, which is not quite the equation of spacetime gravity itself, but of certain fields that might exist on top of spacetime. So Einstein's equations of relativity just describes space and time itself. But then there's other fields that live on top of that. There's the electromagnetic field, there's young Mills fields. And there's this whole hierarchy of different equations, of which Einstein is considered one of the most nonlinear and difficult, but relatively low on the hierarchy was this thing called the wave maps equation. So it's a wave which at any given point is fixed to be like on a sphere. So think of a bunch of arrows in space and time. And the arrows are pointing in different directions. But they propagate like waves. If you wiggle an arrow, it will propagate and make all the arrows move kind of like sheaves of wheat in the wheat.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“If you propose axioms, then the mathematics lets you follow those axioms to their conclusions. And sometimes you can get quite a long way from initial hypotheses.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Change of perspective is really important, you say, travel broadens the mind. This is intellectual travel, put yourself in the mind of the ancient Greeks or some other person from some other time period. Make hypotheses, spherical cows, whatever, speculate. And this is what mathematicians do and samathists do actually”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Right, yeah. So, modern science is maybe again a victim of its own success is that in order to be more accurate, it has to move further and further away from your initial intuition. And so for someone who hasn't gone through the whole process of science education, it looks more and more suspicious. Because of that. So we need more grounding. I think there are scientists who do excellent outreach. But there's lots of science things that you can do at home, lots of YouTube videos I did a YouTube video recent of Guy Sanderson, and we talked about this earlier, that how the ancient Greeks were able to measure things like the distance of the moon, distance of the Earth, using techniques that you could also replicate yourself. It doesn't all have to be fancy space telescopes and very intimidated mathematics.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Is why analogies are so important. The round earth is not intuitive because we're stuck on it. But round objects in general, we have pretty good intuition. And we have intuition about light works and so forth. And it's actually a good exercise to actually work out how eclipses and phases of the sun and the moon and so forth can be really easily explained by round earth and round moon and models. And you can just take a basketball and a golf ball and a light source and actually do these things yourself. So the intuition is there. But yeah, you have to transfer it.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“That was a leading candidate for many decades. I think it's slowly falling out of fashion because it's not matching experiment.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“What often happens is that when the physicists need something of mathematics, there's often some precursor that the mathematicians worked out earlier. So when Einstein started realizing that space was curbed, he went to some mathematician and asked, is there some theory of curved space that the mathematicians already came up with that could be useful? And he said, oh yeah, I think Riemann came up with something. And so, yeah, Riemann had developed Riemannian geometry, which is precisely a theory of spaces that are curved in various general ways, which turn out to be almost exactly what was needed for Einstein's theory. This is going back to Dublin's unreasonable effectiveness on mathematics. I think the theories that work well, they stay in the universe, tend to also involve the same mathematical objects that work well to solve mathematical problems. Ultimately, they're just both ways of organizing data in useful ways.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“I believe so. I mean, the history of physics has been that of unification, much like mathematics over the years. Electricity and magnetism was separate theories and then Maxwell unified them. Newton unified the motions of the heavens for the motions of objects on the earth and so forth. So it should happen. It's just that, again, to go back to this model of the observations and theory, part of our problem is that physics is a victim of no success. two big theories of physics general relativity and quantum mechanics are so good now that together they cover 99.9 percent of sort of all the observations we can make and you have to like either go to extremely insane particle accelerations or or the early universe or things that are really hard to measure 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 but i have faith that we you know we've”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“One of the poems 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, you know, and we kind of know that at very tiny scales, there's going to be quantum fluctuations of space-time foam. And trying to use Cartesian coordinates XYZ is a non-starter. But we don't know how to replace it with. We don't actually have the mathematical concepts, the analog of the Hamiltonian that sort of organized everything.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Of physics. So there's time transition invariance. This corresponds to the whole consortium of energy. So this fundamental connection between symmetry and conservation. And that's also true in quantum mechanics, even though the equations are completely different. But because they're both coming from the Hamiltonian, the Hamiltonian controls everything. Every time the Hamiltonian has a symmetry, the equations will have a conservation law. So once you have the right language, it actually makes sense.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“There's also an object in the Hamiltonian. It's a different type of object, it's what's called an operator rather than a function. But again, once you specify it, you specify the entire dynamics. So there's something called Schroenz equation that tells you exactly how quantum systems evolve once you have a Hamiltonian. So side by side, they look completely different objects. 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 classical mechanics, there's this thing called Notice theorem. Every time there's a symmetry in a physical system, there was a conservation law. So the laws of physics are translation invariant. Like if I move 10 steps to the left, I experience the same laws of physics as if I was here. And that corresponds to conservation momentum. If I turn around by some angle, again, I experience the same laws of physics. This corresponds to the conservation angle of momentum. If I wait for 10 minutes, I still have the same laws.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Was the dominant object. Once you know how to measure the Hamiltonian of any system, you can describe completely the dynamics, like what happens to all the states. It really was a central actor which was not obvious initially. And this helped actually this change of perspective really helped when quantum mechanics came along because the early physicists who studied quantum mechanics, they had a lot of trouble trying to adapt their Newtonian thinking because everything was a particle and so forth to quantum mechanics because everything because it was a wave. It just looked really, really weird. You ask, what is the quantum version of ethics MA? And it's really, really hard to give an answer to that. But it turns out that the Hamiltonian, which was so secretly behind the scenes in classical mechanics, also is the key object in quantum mechanics that this”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“So, actually, physics equals z squared. So one of the big things was the E. So when Aristotle first came up with his flaws of motion and then Galileo and Newton and so forth, they saw the things they could measure. They could measure mass and acceleration and force and so forth. So Newtonian mechanics, for example, ethical MA was famous Newton's second law of motion. So those were the primary objects. So they gave them the central billing in the theory. It was only later after people started analyzing these equations that always seemed to be these quantities that were conserved. So in particular momentum and energy. And it's not obvious that things happen energy. It's not something you can directly measure the same way you can measure mass and velocity and so forth. But over time, people realize that this was actually a really fundamental concept. Hamilton eventually in 19th century reformulated Newton's laws of physics into what's called Hamiltonian mechanics, where the energy, which is now called the Hamiltonian,”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Well, it's confirmation that you have the right concepts. So when you first study anything, you have to measure things and give them names. And initially, sometimes your model is, again, too far off from reality, you give the wrong things the best names. You only find out later what's really important.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“As pi comes around from circles and rotation. If you want to rotate a needle, for example, 180 degrees, you need to rotate by pi radians. And I, complex numbers, represents this opportunity to imagine axes of a 90 degree rotation. So a change in direction. So the x-matri function represents growth and decay in the direction that you already are. When you stick an i in the exponential now, it's instead of motion in the same direction as your current position, the motion at right angles to your current position. So rotation. And then e to pi a goes 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 dynamics through this act of complexification, rotation by i. So it connects together all these tools, mathematics, the dynamics, geometry and complex and complex and the complex numbers.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Yeah, well, as I said, I mean, what I find most appealing is connections between different things. So if you e to the pi i equals minus one. So yeah, people use all the fundamental constants. Okay, that's cute. But to me, so the exponential function was introduced by Euler, to measure exponential growth. So compound interest or decay, anything which is continuously growing, continuously decreasing growth and decay or dilation or contraction is modeled by the exponential function.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Yeah, you learn a lot. I mean, it may seem like a frivolous exercise, but it can generate all these insights, which if you didn't have this artificial objective to pursue, you might not see.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“But you can code something spaghetti that works for a certain task and it's quick and dirty and it works. But there's lots of good principles for writing code well so that other people can use it, build upon it, and so on and has fewer bugs and whatever. And there's similar things with mathematics.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“And so he gave some examples of well known theorems, and then he would give what he thought was the extreme proof in these different aspects. I just found that really eye-opening a proof was interesting. But once you have that proof, trying to optimize it in various ways, that proofing itself had some craftsmanship to it. Something for my writing style that, you know, like when you do your math assignments as an undergraduate, your homework and so forth, you're sort of encouraged to just write down any proof that works. And they hand it in and get as long as it gets a tick mark, you move on. But if you want your results to actually be influential and be read by people,”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“It's a good question. When I came to graduate school in Princeton, so John Conway was there at the time. He passed away a few years ago. But I remember one of the very first research talks I went to was a talk by Conway on what he called extreme proof. So Conway had just had this amazing way of thinking about all kinds of things in a way that you would normally think of. He thought of proofs themselves as occupying some sort of space. So if you want to prove something, let's say that there's infamy primes. You always have different proofs, but you could rank them in different axes. Like some proofs are elegant, some proofs are long, some proofs are elementary and so forth. And so this is cloud, so this space of all proofs itself has some sort of shape. And so he was interested in extreme points of this shape. Out of all these proofs, what is one of those, the shortest at the expense of everything else, or the most elementary or whatever?”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Often my proof is worse. But by the exercise of doing so, I can say, oh, now I can see what the other proof was trying to do. And from that, I can get some understanding of the tools that are used in that field. So it's very exploratory, doing crazy things and crazy fields and reinventing the wheel a lot. Whereas some of the hedgehog style is, I think, much more scholarly. You're very knowledge-based. You stay up to speed on all the developments in this field. You know all the history. You have a very good understanding of exactly the strengths and weaknesses of each particular technique. Yeah, I think you'd rely a lot more on sort of calculation than sort of trying to find narratives. So yeah, I mean, I could do that too, but there are other people who are extremely good at that.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“I'm much more comfortable with the Fox paradigm. I like looking for analogies, narratives. I spend a lot of time. If there's a result, I see it in one field, and I like the result, it's the cool result, but I don't like the proof. It uses... Types of mathematics that I'm not super familiar with. I often try to reprove it myself using the tools that I favor.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“In the fields. There are other mathematicians who are far deeper than I am who are really hedgehogs. They know everything about one field and they're much faster and more effective in that field. But I can give them these extra tools.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“You need some diversity, a fox working with many hedgehogs or vice versa. But I identify mostly as a fox, certainly. I like arbitrage somehow, like learning how one field works, learning the tricks of that field, and then going to another field, which people don't think is related, but I can adapt the tricks.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Problems about numbers. And today this feels almost trivial. There's no content to this. Like, of course, a plane is X and Y, because that's what we teach. And it's internalized. But it was an important development that these two fields were unified. And this process has just gone on throughout mathematics over and over again. Algebra and geometry were separated and now we have this fluid algebraic geometry that connects them and over and over again. And that's certainly the type of mathematics that I enjoy the most. So I think there's sort of different styles to being a mathematician. I think hedgehogs and fox, Fox knows many things a little bit, but a hedgehog knows one thing very, very well. And in mathematics, there's definitely both hedgehogs and foxes. And then there's people who are kind of who can play both roles. And I think an ideal collaboration between mathematicians involves”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“There's certainly a lot of connecting threads and a lot of the progress of mathematics can be represented by taking stories of two fields of mathematics that were previously not connected and finding connections. An ancient example is geometry and number theory. So in the times of ancient Greeks, these were considered different subjects. I mean, mathematicians worked on both, you know, Euclid worked both on geometry most famously, but also on numbers. But they were not really considered related, a little bit like you could say that this length was five times less length because you could take five copies of this length and so forth. But it wasn't until Descartes, you really realized that you developed what I call analytic geometry. You can parameterize the plane, a geometric object by two real numbers. Every point can be. And so geometric problems can be turned into...”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Non correlation that if all the inputs were not correlated to each other, then you have these classic behaviors that things are fine. It tells you where to look for weaknesses in the model. So if you have a mathematical understanding of central limit theorem and someone proposes If you're mathematically trained, you would say, okay, but what are this systemic correlation between all your inputs? And so then you can ask the economists, how much of a risk is that? And then you can go look for that. So there's always this synergy between science and mathematics.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“If everything was decorated, it would be an asteroid bell curve, and you can manage risk with options and derivatives and so forth. And it is a very beautiful theory. But if there are systemic shocks in the economy that can push everybody to default at the same time, that's very non-Gaussian behavior. And this wasn't fully accounted for in 2008. Now I think there's some more awareness this is a systemic risk is actually a much bigger issue. And just because the model is pretty and nice, it may not match reality. So the mathematics of working out what models do is really important. But also the size of validating when the models fit reality and when they don't, I mean, you need both. But my facts can help because, for example, the central limit theorems, it told you that if you have certain axioms like”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Yes, you can call Meta if you like. But there are many, many processes, for example, you can take lots and lots of independent random variables and average them together in various ways. You can take a simple average or more complicated average. And we can prove in various cases that these bell curves, these Gaussians, emerge. And it is a satisfying explanation. Sometimes they don't. So if you have many different inputs and they're all correlated in some systemic way, then you can get something very far from a Bocave Schwab. And this is also important to know when the system fails. So universality is not a 100% reliable thing to rely on at that global financial crisis was a famous example of this. People thought that mortgage defaults had this sort of Gaussian type behavior that if you ask if a population of 100,000 Americans with mortgages, what proportion of them would default in their mortgages.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“And it models almost everything you need to know about these 10 to 23 or whatever particles. We don't understand universality anywhere new as we would like mathematically, but there are much simpler toy models where we do have a good understanding of why universality occurs. Most basic one is the central limit theorem that explains why the Bell curve shows up everywhere in nature, that so many things are distributed by what's called a Gaussian distribution, a famous bell curve. There's now even a meme with this curve.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Is actually some mathematical possible explanation for that. So there's this phenomenon in mathematics called universality. So many complex systems at the macro scale are coming out of lots of tiny interactions at the macro scale. And normally because of the common form of explosion, you would think that the macro scale equations must be infinitely exponentially more complicated than the macroscale ones. And they are, if you want to solve them completely exactly. Like if you want to model all the atoms in a box of air. Abagajo's number is humongous. There's a huge number of particles. If you actually have to track each one, it'll be ridiculous. But certain laws emerge at the macroscopic scale that almost don't depend on what's going on at the macro scale or only depend on a very small number of parameters. So if you want to model a gasillion particles in a box, you just need to know as temperature and pressure and volume a few parameters.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“So you have these petabytes of observations, you'd like to compress it to a model which you can describe in five pages and specify a certain number of parameters. And if it can fit to reasonable accuracy almost all of your observations, I mean, the more compression that you make, the better your theory.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“This kind of thing, we have a model that sort of explains, that fits the data really well. It just has a few parameters that you have to specify. So people say, oh, that's fudge factors with enough fud factors, you explain anything. But the mathematical point of the model is that you want to have fewer parameters in your model than data points in your observational set. So if you have a model with 10 parameters that explains 10 up 10 observations, that is a completely useless model. It's what's called overfitted. But if you have a model with two parameters and it explains a trillion observations, which is basically, so yeah, the dark matter model, I think it has like 14 parameters and it explains petabytes of data that the astronomers have. You can think of a theory. One way to think about a physical method is it's a compression of the universe and data compression.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“The model gets dragged along with it. And so over time, we had to realize that the earth was round, that it spins, it goes around this whole system, this whole system goes around the galaxy, and so on and so forth. And the guy's about the universe and it's expanding. Expansion is self-expanding, accelerating. And in fact, very recently in this year, we also saw this, even the version of the universe itself, this evidence now is non-constant.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“There are these three ontological things. There's actual reality, there's observations and our models. And technically they are distinct, and I think they will always be distinct, but they can get closer over time. And the process of getting closer often means that you have to discard your initial intuitions. astronomy provides great examples like an initial model of the world is that it's flat because it looks flat and that it's and it's big you know and the rest of the universe is the skies is not you know like the sun for example looks really tiny um and so you start off with a model which is actually really far from reality um but it fits kind of the observations that you have um you know so you know so things look good you know but over time as you make more and more observations bring it closer to reality”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Positions, we can't get a computer to fully explore. But now we have AI. We have tools to explore this space not with 100% guarantees of success, but with experiment. We can empirically solve chess now. For example, we have very, very good AIs that don't explore every single position in the game tree, but they have found some very good approximation. And people are using actually these chest engines to do experimental chess that they're revisiting old chess theories about, oh, you know, when you do this type of opening, this is a good type of move, this is not. And then you can use these chess engines to actually refine, in some cases, overturn commercial wisdom about chess. And I do hope that mathematics. Perhaps powered by A”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Experimental mathematics. But until very recently, it was not, I mean, theoretical mathematics was just much more successful. I mean, because doing complicated mathematical computations was just not feasible until very recently. And even nowadays, even though we have powerful computers, only some mathematical things can be explored numerically. There's some called the combinatorial explosion. If you want to study, for example, Zamari's theorem, you want to study all possible subsets of numbers 1 to 1000, there's only 1,000 numbers. How bad could it be? It turns out the number of different subsets of 1 to 1000 is 2 to the power 1000, which is way bigger than any computer can currently, any computer ever, or ever enumerate. So you have to be, there are certain math problems that very quickly become just intractable to attack by direct brute force computation. Chess is another famous example. The number of chess”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Within Mepetic itself, there's also a theory and experimental component. It's just that until very recently, theory has dominated almost completely, like 99% of mathematics is theoretical mathematics. There's a very tiny amount of experimental mathematics. People do do it. If they want to study prime numbers or whatever, they can just generate large data sets. So once we had computers, we began to do it a little bit. Although even before, well, like Gauss, for example, he discovered who he conjectured the most basic theorem in number theory, which is 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, I mean, mostly by himself, but also hired human computers, people whose professional job it was to do arithmetic, to compute the first hundred thousand fribs or something and made tables and made a prediction. That's an early example of”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Well, you need both top down and bottom up. It's a really interaction between all these things. So over time, the observations and the theory and the modeling should both get closer to reality. But initially, and this is always the case, they're always far apart to begin with. But you need one to figure out where to push the other. If your model is predicting anomalies that are not picked up by experiment, that tells experimenters where to look. to find more data, to find the models. So it goes back and forth.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source