Terry Tao: "LLMs Are Simpler Than You Think – The Real Mystery Is Why They Work!"

Every time you enter a password or.

Buy something online or send any kind of encrypted WhatsApp message, you're betting your security on a pattern in prime numbers. A pattern no mathematician on earth has.

Ever been able to prove.

There are the atoms of multiplication. They're supposed to be random, unpredictable, and our entire digital security infrastructure assumes that they are. But here's the thing, we don't actually know that. We've tested it with supercomputers on trillions of cases. But mathematical proof, it's still elusive. Today, the man sitting across from me has solved more legendary math problems than almost any human alive. Terence Tao won the Fields Medal, that Nobel Prize of math, and he's tackled questions that have stumped the greatest minds for centuries. And he just told me there could be an undiscovered pattern hiding in prime numbers.

A pattern that, if it exists, could break the encryption protecting every financial transaction you'll ever do.

We're going to talk about the beauty of numbers, why AI keeps getting the math wrong, what it was like to.

Meet the legendary Paul erdos as a 10 year old, and whether or not mathematics is invented or discovered. Let's go deep into the impossible with the Mozart of math.

First question I always ask a mathematician is, how do you like your coffee?

I actually don't drink coffee much, except on social occasions. Actually black, no sugar. Okay.

So the reason I asked that maybe you'll recognize as Erdos, I believe said. What did he say about mathematicians in coffee?

He said that mathematicians are a means for turning coffee into theorems. There's a very nerdy follow up joke to that, which is that a co mathematician is a way of turning co theorems into feet.

Into feet.

Yeah, it's a very inside joke.

That's right, that's a dad joke. Plus a mathematician joke that's really, really bad but really good at the same time. Well, the reason I bring up Erdos, of course, you actually met him when you were a kid, didn't you?

Yes, I think I was 10 at the time. So he had a collaborator in Adelaide, which is the city where I grew up, George Zegeres. So he would visit every now and then at the time. I think one of the math professors at the local university introduced me to him and Erdos was always very good at, he was known for meeting bright young kids and so we had a nice conversation. I wish I'd remembered more of it actually. I was too young at the time to realize just sort of how much of an honor it was really the one Thing I remember was that he really treated me like an equal, like he didn't condescend as a kid. And he later sent me a postcard that it just said, thank you for your nice hospitality. Here's a math problem which I didn't solve actually, but it did get soul plated by someone else.

Oh, that's amazing. Yeah. He was one of the most prolific mathematicians in at least modern history, maybe in all time history. And famously there's a relationship between the number of authors you have to go through before you're related to him. Right. What is your Erds number?

Yeah, so there's this concept called the Erds number. So Erds worked a lot in graph theory, and so this concept is inspired by graph theory. So Erds himself has an erds number of 0. If you've written a paper with Erds, you get an erdish number of 1. If you've written a paper of someone who's written a paper with Erds, you have an Erds number two. So I have an Erdish number two, for instance. And I think nowadays people, it's common to have Erdish numbers of four or five. People have made similar numbers in other fields.

There's a Bacon number. So if you've starred in a film with Kevin Bacon, you have a Kevin Bacon number of one and so forth. And then there's something called the Erds Bacon number, which is the sum of your Erds number and Bacon number, which is usually infinite because you either don't have a chain of papers going to Erds or you don't have a chain of movies going to Bacon. But there are a half dozen people who have like the combinable, like seven or eight.

Yes, yes, I've heard of random things like that, but yeah, he was known in many ways. I remember hearing from Jim Simons, who is my late great mentor, and obviously you knew him well, that he had. He was incredibly productive, but part of his productivity relied on the use of amphetamines. He used to take some. Is that true?

That's what I've heard. Apparently one of his friends convinced him to give up amphetamines for a month for a bet or something. And Erdos grudgingly did it. And then at the end he just went back and just said, you just set mathematics back by one month. So he was, I guess, hardcore. You don't see that so often nowadays. I think back then maybe there was less of a stress on work, life balance than you are today now.

You had work related to ERDs, right, the ERDs discrepancy theorem or something.

Yeah.

What is that? Can you explain that for my audience?

Okay, so ERDS was famous for posing many, many problems, and I solved a few of them over my career. Discrepancy theory is a theory about how irregular sequences can be. So, like, if you have a sequence of plus ones and minus ones, and if they're random, if you pick, say, 1,000 numbers plus or minus ones at random, you'd expect 500 of them to be plus ones, 500 min ones. So the discrepancy is defined as the difference between the number of plus ones and the number of minus ones. A sequence can have low discrepancy if you view it over the whole sequence, but if you look at subsequences, the sequence can have a higher discrepancy. So, for example, if you take an outing sequence of plus one, minus one, plus one, minus one a thousand times, the discrepancy over the whole interval is zero because you have 500 plus 1, 500 minus 1, it sums to zero. But if you look only over the even numbers, you just see plus 1, plus 1, plus 1, plus one, or minus one, minus one, minus one, then you have a very large discrepancy. Like 500.

Some sequences, they can be very well balanced overall, but when you restrict to subsequences, they can have a higher discrepancy. So ERDS was interested in whether you could design a sequence which had what's called bounded discrepancy over all what are called homogeneous arithmetic regressions. So could you create a very, very long sequence of plus 1s and minus 1s, where if you look at any finite segments, say, from 1 to 101 to 1000, the number of plus ones and minus ones only differ by at most 2. But also if you look over the even numbers, same thing happens. If you look over the three, same thing happens. So they wondered if it was possible to make an extremely uniformly distributed sequence that was always balanced no matter how you looked. And you can do it for quite a while. I think someone constructed a sequence of like 1164 elements or so, where the discrepancy was never bigger than plus or minus two.

So extremely well balanced. So there are some extremely uniform sequences, but using a really huge supercomputer and something called a SAT silver, they could show that past that point you had to have a discrepancy of 3. The sequence became more and more unbalanced over time. But until several years ago, that was the record. So three was the best lower bound for how much discrepancy these sequences had to have. So Erlich asked, do these sequences, if you continue these sequences on forever and ever, must the discrepancy eventually go to infinity? And this is what I was able to show.

Oh, wow. So it does go to infinity.

Divergence. Yes, yes, extremely slowly, as far as we know, logarithmic or double logarithmic. But it does go infinity. And I had to use tools from information theory and number theory.

I've heard that there are people that use some applications of your work to detect cheating in the following sense, that when a student cheats, of course, our students never cheat, but say they're doing a true false exam or they want to mirror something and they want to simulate that they actually got the answers. Maybe they'll put down randomly true, false, true, false, true, and it'll be too close. The sum would plus minus would go to zero. Is that relevance?

It is connected, yeah. So there are other statistics patterns that random sequences have and artificial sequences don't. I don't think my low discrepancy work directly based on that. But there are other patterns. The most famous is called Benfit's law, which is a very unintuitive law that, roughly speaking, 30% of all numbers in the world start with 1, which sounds very weird because numbers can start with 1, 2, 3, or up to 9. But you can take, for example, take all the countries in the world, take their population, and there's about 100 odd countries in the world, about a third of them. The population would start with one. China, for instance.

Or you can take the net wealth of several millionaires and billionaires or whatever, and you also find that Muslims start with one, or birthdays. The pattern is quite universal. But whereas if you pick numbers randomly, if you fudge your accounting books, when you pick numbers artificially, they don't necessarily.

Obey that they're uniform. So you violate that or you try to make them, you think they're uniform.

So humans are actually really quite bad at creating truly random patterns. And so, yeah, you can distinguish natural patterns from human generated ones.

Interesting. So one thing that I've been dying to talk to you about for a long time are kind of the limits of mathematical induction. So you mentioned that you start with a small number and then you kind of add on to it. And I do want to hearken to the work of Jim Simons. He's most famously known for being a multi billionaire establishing philanthropies. That's a important. My Research and hopefully other very highly competent scientists. But one of his lesser known things that was actually very important in, at least in my understanding of how mathematical induction works or violates, is his work on minimal surfaces, where he showed something really fascinating.

So I should have you explain what a minimal surface is. But as I understand it, you can sort of think of it physically. If you had some shape, say a coat hanger, and you made it into a loop, and then you wanted to attach it to another loop using a soap bubble, the shape that would. Would obtain would be called a minimal surface. Is that correct? Okay. And then he showed that there are such minimal surfaces and 0 or 1 dimension, or it was known that that was true. And then he showed it in dimension two. It exists in dimension three and dimension four and dimension six, seven, and then he got to eight.

Actually, it didn't work. And I would have stopped at 2. Right. So most mathematical induction, you know, seems to continue to infinity. But you already told me one thing that doesn't continue to infinity, as you might naively expect. What are the limits of mathematical induction? Maybe define what it is first. What is mathematical induction?

Right. Yeah. So induction, these different things, I think the philosophers and philosophy of science. Induction reverses something slightly different where you take facts that you observe from small examples and you induce from that a prediction for what will happen for larger cases. And it's a very basic procedure in the scientific method because you do experiments and then you extrapolate from the experiments. Mathematical induction is a more precise form of reasoning where. So there's a precise principle of mathematical induction that if you have a statement that you want to be true for all natural numbers, 1, 2, 3, 4, and so forth, and you know it's true for one, and whenever you know it's true for sum over n, you know for sure that it implies the same thing for n +1, then it implies it's true for all numbers. The analogy often given is just a row of dominoes.

So if each domino represents one case of what you're trying to prove, and you can prove the first case, you knock the first domino over, and you know that each domino, whenever you can prove it, it tips over the next domino. Then no matter how long the string of dominoes is, you can knock over every single domino chain. But it's really important that you're arguing with the 100% watertight. If it's dominoes and the 97th domino doesn't tip over the 98th, then it stops there. So it's a principle that only Works in the world of mathematics, which is one of the few places where you really can have 100% guarantees. So he discovered what's called the Simon's cone. You pushed him a little bit because this geometry is not in my area of mathematics, but. Yeah.

So minimal surfaces most famously are two dimensional surfaces like soap films. But in mathematics, there's nothing stopping you from considering the same notion in other dimensions. In one dimension, it's just like rubber bands. One dimension minimal surfaces are very boring. They're just straight lines. But you can consider them three dimensional surfaces in four dimensional space, which already is hard to visualize, but mathematically you can consider it. And 5 and 6. Weirdly, sometimes problems become easier in higher dimensions.

So even if you care about the physical world and you only care about two and three dimensions, sometimes it makes sense as a mathematician to first study higher dimensions. It gets you some intuition which can help guide you with the problems that you do care about. Yeah. So it turns out that up below eight dimensions, I think here, all minimal surfaces are smooth. You can't tie a soap bubble and create any kind of knot. Yeah. Because there's always some way to pull it apart and reduce the surface tension. Yeah.

Starting in eight dimensions, he discovered a very surprising fact that singularities do can actually form that there is this. Yeah. It looks like a cone, except in much higher dimensions. And there was no way to modify the cone to make it to reduce the surface area. If you made a cone. If you try to arrange soap into a cone in three dimensions, you could just remove the. Do what's called a surgery. You remove the vertex of the cone and replace it by two rounded nubs.

Okay. And that would reduce the surface tension. Interesting. You can't do that in higher dimensions. So nowadays because of data science, actually we need to understand high dimensional geometries much better than we're used to. And a lot of our old intuition is actually, which you get from low dimensional geometry is actually completely false in high dimensions. So just to give you one example, if you inscribe a circle inside a square, it occupies a pretty large chunk of the square, maybe like 75% or something. And if you inscribe a ball inside a cube, it's still pretty big.

I think about half the volume of a cube. But if you take a thousand dimensional cube and you inscribe a thousand dimensional ball inside it, it's like incredibly tiny. It's like 0.0001%. Balls become extremely poor space filling. Yeah, they're nowhere near space filling in high Dimensions. And this is important when you look at clouds of data, and if you have some taking 1,000 measurements, and that's like 1,000 data points, but there's some errors in them. Do you. You measure the root mean square error, which is like trying to place your measurement inside some ball, or do you measure the worst of the 1,000 errors, which is like placing.

The question is, do you want your error bars in high dimensions to be like a ball or a cube and it starts making a difference? Right.

There's significant differences between that approximation. So you mentioned that technique of going to higher dimensions to solve problems in lower dimensions. That's one of the many tools that mathematicians use. Others include proof by Raduchio and Absurdium. Can you talk about what's your favorite type of mathematical proof? When you're onto it, you just get so excited to finish the film.

Proof by Contradiction. I think Hardy had a great quote that in chess, a chess player may offer a pawn or a bishop, but a mathematician offers the entire game.

The sacrifice.

Yeah. So he says, okay, we want to prove this conclusion. I will give you that the conclusion is false. I will just let you run with it. And. But you do that and I will show that it gives you a contradiction. It actually is a technique. So on the one hand, it is very unintuitive.

The undergraduate students that we teach, they struggle a lot with the notion of proof of contradiction. On the other hand, it is a concept that I have seen primary school students teach each other. So in recess, you might see kids play the game of who can name the largest number. So they say, okay, 1,000 and then a million, a billion, a billion billion. And they'll go on like this. But at some point, someone will realize, one of the kids will realize that no matter what number the other kid says they can just say that number plus one. They have proven that there was no largest number in the natural numbers. And this is a proof of contradiction, because if anyone ever did claim a natural number years largest natural number, you just add one and you have contradicted them.

So it is actually a very intuitive proof technique, but you have to teach it the right way. And sometimes case conditions help. Type of mathematics that I the type of proof arguments that I like the best are ones that make unexpected connections between different areas of mathematics, like, say, between discrete mathematics and continuous mathematics. We talk about low dimensions and high dimensions. You can have a problem which has to do with combinatorics and nothing to do with the real world, but you find that there is some physical model of It. And you can use ideas from physics, of course, as physicists do this all the time. They also have correspondences which. Which are really quite amazing.

So, yeah, those feel like magic to me.

Yeah. I mean, the most famous one, at least to my physicist mind, is the proof that by contradiction, that square root of 2 is irrational. So that's Euclid's original proof, or what does trace back to before him?

Euclid or Pythagorean, I think. Pythagoreans, maybe.

Pythagoreans, yeah. Euclid proved the prime, basically by the same sort of idea as the infinity plus one. Right?

Yeah, yeah, yeah. No, we have a lot to thank Euclid for, actually. I mean, he wasn't the first to write down many of the theorems, like the Pythagorean theorem, for example. I think the Babylonians had a version, the Chinese had a version, but really, he was the one who introduced this notion of proof that complex facts about mathematics you could deduce from simpler axioms. And it was extremely influential way of thinking which you hadn't seen before.

So the square root operation, just as a notion, has always fascinated me, you know, for. For one thing, it seems to occur in physics, you know, quite regularly. And I'll get into some examples that pique my curiosity and eventually I do want to tie this to Wigner's famous statement about the unreasonable effectiveness of mathematics to the physical world. And we'll talk about that in just a bit, which also tangentially involves the square root. But the square root, in physics, at least, for example, in classical mechanics, you can construct things, operators that involve the position of momentum, called Poisson Brack brackets. And as soon as you take them from the classical world to the quantum world, instead of commuting being equal to zero, they become equal not to zero, but times a fundamental constant, times the square root of negative one. And it's just so baffling to me that once you introduce the concept of a square root and imaginary number, then so much mathematics is open to physicists. And I wonder, you know, is there like, could we, an intelligent alien, you know, who knew all of mathematics, could they have taught us this? Or is there something special about the square root operation? In LLMs, they use LU decomposition and we have spinors that would have spinner representation in our square roots.

Is there something special about the square root? Or like, in other words, why don't we say cube roots or fifth roots or 100th roots? Why is the square root something that's. That's so prevalent in physics, for example.

Yeah. So we experience the world as in a continuum Euclidean space. And the notion of numbers that are most natural to us from our spatial intuition are the real numbers. So real numbers have lots of wonderful properties. The algebra of real numbers works really well for things like addition is commutative, X plus Y is Y plus X, and X times Y is Y times X, multiplication is commutative, and so forth. But they have one flaw, which is that not every polynomial equation has roots. So if you take the equation X squared plus one equals zero in the real numbers, X squared plus one is never zero because X squared is always positive. So it's not what's called algebraically complete, but it's very close to being complete.

So if you take a polynomial which is odd degree, like a cubic x cubed plus 3x plus 1, it must have a root, because a cubic polynomial or an odd degree polynomial, when you make X very, very big, it becomes very large and positive. And when you make X very, very large and negative, it becomes negative. And because the reals are continuous, polynomials are continuous. Therefore, at some point in between, you must hit zero to get from negative to positive. Sort of a half of all the polynomials in the world, you can solve them in the reals and half you can't. So, with the benefit of hindsight, this really suggests that you should make the real twice as big in order to get this really useful property of algebraic completeness. And so, as it turns out, there are these numbers called the complex numbers, which are twice as big as they were. So the real numbers are one dimensional and the complex numbers are two dimensional, and they have wonderful, wonderful properties.

Lots of very nice geometric structure, nice algebraic structure ultimately coming from this algebraic completeness. And so we also know from algebra that the way to make something twice as big, a number system twice as big is to add a square root that you didn't have before. If you want to make a system three times as big, you should add a cube root that you didn't have previously. So once you know that you're looking for a new number system that's twice as big as what you started with, it's very natural to look for, to throw in a square root of a number that doesn't currently have a square root, such as minus one. So that's kind of the, in retrospect.

You might have predicted.

Yeah, I mean, this is not historically how complex numbers were discovered, but this could be sort of one explanation.

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Right. And then there's a whole other class of numbers, transcendental numbers, where you have. They don't solve polynomial equations, correct?

Yeah, yeah.

We don't have roots of polynomials.

We've learned that the notion of number is very flexible. I mean, people get upset when they learn that. But it feels simpler to have one notion of everything taught in school and then not have that change. People were very upset when the notion of a planet got changed 10, 15 years ago. And we occasionally do that with math, too. The number one used to be prime about 100 years ago.

Oh, really? I mean, I still consider it prime, but I still consider prime a planet.

Yeah, yeah. But it's because mathematicians and science, in any field of study, when you first study a subject, you don't really know what concepts are the most fundamental and important and which ones are not. So you make a guess based on your experience. So maybe you think that numbers that don't have any smaller factor are important, so you call them prime numbers. Or maybe the stars that move in the sky are important, so you call them planets. But over time, you realize that actually there are slightly better definitions that have better properties. Okay, so I can't speak to the astronomers why the new notion of a planet is better. But, for example, what we've learned is that one of the really important properties of primes is what's called the fundamental theorem of arithmetic, that any number can be broken up into primes in exactly one way other than rearranging the factors.

So 12 is 2 times 3 times 2 or 2 times 2 times 3. But other than interchanging the order, that's the only way to break up a number of primes. Just like there's only one way to break up a chemical compound into atoms. So primes are like the atoms of multiplication. But if you made one prime, then you would have to give up the fundamental theorem of arithmetic, because now 12 is also 1 times 2 times 2 times 3. And you just add too many exceptions to this really important factor in number theory. So we made a decision to therefore redefine prime numbers when it was okay.

But just like with Pluto, there's no consequence to life on Earth. It's more sort of.

Yeah, it's a human convention, but we update our human conventions to match reality better over time.

I've done work in prime pairs. Is that right?

Yes. Yeah. So primes are one of the oldest subjects in mathematics. Euclid had the first theorem almost ever, and it was about prime numbers more than 2,000 years ago. And so it's very frustrating and annoying that even the most basic questions about primes we still cannot answer definitively. We have good guesses like so for almost all questions of primes, we can predict the answer, but we cannot get the 100% mathematical standard of proof for many of them. And one of the most basic questions, which is at least 300 years old, is called the Trin prime conjecture that there should be. So Euclid show there's infinitely many primes.

The primes never end. You can always find primes bigger than any number you wish. But we cannot find, we cannot say the same yet for prime twins. So these are pairs of primes that differ by. By the closest they can, which is two, for example, 11 and 13. Well, okay, two and three are closer, but after two, all primes are odd, so the closest you can get is two. And so we can observe that every so often the primes, they don't seem to obey a pattern. Sometimes the prime gaps are large, sometimes they're small, but every so often they come close to each other and you get a twin, and they seem to occur infinitely often.

We can find trillions and trillions, at least by computer, but we have never been able to prove that they go on forever. We have this prediction that the primes behave basically like a random sequence of numbers and random sequences. If you have a random sequence of the same density as the primes, they will hit form twins infinitely often. But the primes are not random. We believe they're what's called pseudorandom, that they have no obvious pattern besides the ones that we can obviously see, such as them being odd. So, I mean, it's a very likely hypothesis, but we can't prove it.

Pseudorandomness, meaning that it could be derived from some algorithm, but not in all cases or something. What's pseudorandom versus random distinction?

Random means not deterministic. There is no single. So the primes, if you forgotten what the primes were and you have to regenerate them by a computer program, you would generate exactly the same set, whereas if you are generating a set by rolling dice or flipping coins, you would get a different set. Pseudorandom are sets which are either random or deterministic, but statistically, they are indistinguishable from random noise. So, for example, a random number should just. If you have a random sequence of numbers, there should be just as many numbers that end in 5 or end in 7 as end in 6. Like, the digits should be equally distributed. Now, the params, they're not perfectly pseudorandom because they do have certain patterns, like they tend to be odd, for instance.

But there's ways of excluding those sort of obvious biases. And once you exclude them, it's expected that there's no tests that can distinguish them from random numbers. This is important for cryptography, actually, because there are many cryptosystems, like the ones we use to encrypt web traffic, cryptocurrency, financial transactions, where data like sensitive data, like passwords or credit card numbers are encrypted using mathematical routines that rely implicitly on primes having no pattern. And so they use primes in various mathematical ways to mix up these numbers. And we believe if by doing so, the data that we actually send looks indistinguishable from random noise and conveys no information about your personal data. And we really hope that that's true. So one reason why it's important for mathematicians to actually study prime numbers is that we occasionally get a shock that. I mean, it hasn't really happened in number theory in decades at least.

But there could be really unusual undiscovered patterns in the prime numbers that we weren't previously aware of, and if they existed, that they could present a vulnerability to crypto systems. There have been a few other cryptosystems where similar patterns have been discovered, I think not for priors, very elliptic curves and other things where people actually had to migrate to a different crypto system because of these weaknesses.

Yeah. So that brings up kind of an inversion, maybe contradiction of what Wigner said. He commented on the unreasonable effectiveness of mathematics in the physical sciences or in the natural world. But what you said just made me think about the kind of inverse of that, which is the unreasonable effectiveness of physics in the mathematical world. In other words, you mentioned cryptography, and it said that quantum computers can perhaps factor and break these previously considered to be uncorrect. So, yeah, my question is, what is it about quantum computers that could then illuminate or elucidate things in number theory and pure mathematics from the physical world to the you know, the quantum world to the mathematical world. Do you see that as, you know, sort of a viable topic?

So quantum computers are a fascinating topic. Yeah, they interface with maths in various ways.

So.

So one is actually just the actual software engineering of creating good quantum algorithms. So it requires a very different type of software mindset. So classical computers, we have this sequential way of thinking where you just sort of, you have these bits of memory and you flip them and if you do this, you do that. And we have decades of experience. For quantum computer, the state is not a bunch of 01 bits, but it's a wave function. And the operations you assign to them, you have to multiple them. You're only allowed to multiply them by matrices, really, really large matrices. Except that your basic operations, your matrices are mostly the identity and you only change a few corner bits at a time.

But you want to couple them together in a very efficient way so that you can do really complicated operations. So quantum computers are both exponentially more powerful than classical computers, but also exponentially more limited. So because they can handle superposition of quantum states simultaneously, in principle, there's this exponential speed up and in. And for certain applications like factoring and I think quantum chemistry.

Lagrangians.

Right. They are, at least in principle, very, very powerful. But quantum mechanics is also very restrictive. The number of things you can do to quantum state, you can only do linear operations and only do time reversible operations.

Non destructive.

Yeah. So this requires you to develop the theories of reversible computing. Error correction is also much, much more annoy. So yeah, so there's software challenges. Maybe once quantum computers become a reality, they will be used to do large scale computations or type that we haven't done before. Whether they have a practical impact on the actual theory of mathematics. I don't know of any examples off the top of my head, but certainly classical complexity theory has been very influential. Historically, mathematicians only cared about whether something was true or false or provable or disprovable.

And with the advent of computers, people also started asking questions of how computable is an object? Like, so if they could prove that something exists, could you go further and actually say, is there an algorithm to compute? And is the algorithm exponential time or polynomial time? So a much finer grained notion of truth actually than just is true or it's false. But how easy is it to actually compute? And, and that has led to very productive mathematics. Sometimes just the effort to not just show something exists, but actually find it creates new techniques. Complexity theory has offered, given a Much more nuanced understanding of how true a statement is. And yeah, this has led to a better understanding of, you just proved some that something is true, but you may not have any insight. So what was the key ingredient that made it work? Or if you had two different proofs, which proof is better? But maybe one proof leads to a faster algorithm than the other. And so you can say, oh, that proof actually is stronger, it's more efficient. So it indirectly sort of provides much more insight into the proofs that mathematicians want.

And has AI actually enabled new discoveries in mathematics or new proofs that otherwise would not have existed?

Slowly it's beginning to, I mean, by itself. So the big weakness of these AIs right now is that they can begin to produce output that looks like, say, a human mathematician reasoning their way through a problem. But it's not grounded, it, it's probabilistic. They often make mistakes. And much like if I were to get a student to solve a problem on a blackboard and they're nervous and they just say the first thing comes to mind, they might get it right or they might get it wrong, but if it's a weak student and they don't have sort of fundamental knowledge of what they're actually doing, once they go off the rails, they can go really off the rails. And this is something which is a fundamental problem with the current large language model. But if you use them as a component of a more rigorous and grounded reasoning system. So if you converse with large language models, for them to make suggestions, but you understand the output and you can verify it.

So then people have had some success talking about their math problems to a large language model. Large language model will produce some suggestions, some of which the human expert can dismiss as not viable. Some of us would be thinking, oh, I thought of that already. But one or two is, oh, that's actually something we're. Which I should have come up with myself, but I just didn't realize. So one thing where AI models are already beginning to be useful is in literature review type tasks where there is a class of problem. And in the literature, there are maybe say a dozen ways already to attack this problem and the human working on the problem. Maybe you can remember six of them, but you forgot the other six don't come to your mind.

And you can use these Nike models to prompt to, to remind you of the missing six. They may also hallucinate three more that don't exist. So you can't trust them, supervise them. Yeah, you have to verify there is hope in the future. So this separate math technology to automate to have the software that can automatically verify certain types of proofs. Right. And so the hope is that if you force the large language models to only output in some language that you can verify and to filter out the hallucinations, has it been able to reproduce.

A wiles proof from Matzlaus theorem or your work and now your Stokes? I mean, has it been able to actually just simply reproduce what a natural intelligence person like you do?

This is issue. It can, but often because of what's called contamination. So if a result is like taught in textbooks software, then it is implicitly in the training data that these AIs train on. And so they're basically memorizing the same way again that a student at the board may just repeat from memory a proof that they saw in a textbook. So AI has basically read all the textbooks in the world. It's hard to discern when that happens, whether it was training data or whether they really sort of thought it up. If you ask the AI to explain their chain of thought, they often give complete nonsense. It's clear that they just didn't know.

Yeah, I mean I've found that even we tried with my student Evan Watson, we just gave it the information about the orbit of Mercury over the past 3,000 years, which JPL up the road here has access to and could predict. And then we said, well, if you observe this in this planet, you know, basically could you first discover this anomalous procession of the perihelion and Mercury and then could you predict it, you know, and it was just completely unable. It required us. We had to first discretize everything, make everything Euclidean, which then totally ruins it. Right. So I've proposed and I want to get your take on it. Kind of a joke, I call it the Keating test, but it's basically the Turing Test. We'll know when it's true, when actual AI can come up with new and unknown, heretofore unknown, you know, predictions that can be verified by humans like you.

Yeah, I think that's a very promising use case of AI neural networks in general. They're designed to make patterns, to detect correlations and things. So there have been a few examples in mathematics where for example, in knot theory, a neural network, not a fancy LLM so much, but like a more old school neural network was used to detect correlations between different types of knot invariants. That was not believed people did not suspect existed before. And initially this type of correlation was just sort of this black box relationship. So Knots have these loops in space which some can be untangled and some can't be. They come with all these numbers. They're called knot invariants.

And so the neural network found that by feeding a database of like 1,000,000 knots, that there was one not invariant called the signature, which could be predicted with really high accuracy from a whole bunch of other invariants called hyperbolic invariants. But this neural network was this black box. You just fed in these 20 numbers as your hyperbolic invariants, and it was spit out. The signature should be plus three, and like 90% of the time it was correct. But once they had this black box, they could analyze it. They could say, okay, suppose I change this input. I modified this hyperbole volume order. How much does this change the output? And so it's like a box of 20 dials that they could play with.

And basically by running experiments, they could see that three of these inputs were actually really important and the other 17 were very peripheral. And by doing those types of analysis, they actually got some insight as to what the relationship was, and they could actually make a formal mathematical prediction, which they could then prove. So once you have these neural network models, you could actually probe them. So in your astronomy example, maybe a neural network might not be able to tell you exactly what the new law of physics would have to be, but it can say, well, I can at least predict the orbit of Mercury over the next thousand years, and here's my model. And then you can just try to tweak it. Now, suppose I changed the period of Mercury or the mass, whatever, what happens to it? And maybe you can work out laws of nature experimentally. It gives you a new paradigm to access reality in traditional experiment or theory.

Yeah, I've used that example. There's a lot of AI doom people that think we're, you know, AI is going to run amok and turn us all into paperclips and all sorts of nonsense. But, you know, because it seems to have this feature that you mentioned that it's sort of averaging over all of human knowledge. And so it'll have errors and I'll have mistakes, but it's bounded by the amount of human knowledge that's used in some level in its training set.

But then there's something magical about it.

And I wonder, the mathematics, I mean, I'm in the presence of greatness, right? So the mathematics, though, aren't that. I mean, it's matrix multiplication at a massive scale, high dimensions and huge volumes. But is it really that complicated intrinsically?

So the mathematics to Train and run a large language model or any other modern AI is not that complicated. Yeah, so an undergraduate math major would have all the prerequisites. Basically you need to know how matrix multiplication works and a little bit of calculus. But the area where we don't have a good mathematical theory is how to evaluate how to predict the performance, performance of these models. So the mystery is not so much how they run. We know how to make a large language model and how to train it and how to run it. But what is surprising is that it works really well for certain tasks and it doesn't work well for others. And we don't know in advance.

We don't have good rules, even heuristic rules of thumb for predicting which tasks are good, which ones are not. We can only just make empirical experiments. Part of the reason is that the data that you train on, data on one level is just strings of zeros and ones. And mathematically we understand sort of very, very random data, so completely random zeros and ones. We have the mathematical probability theory which explains we can analyze this fiction very well. And then we have very, very structured types of data, like a sequence which is all ones or all zeros or just alternating 10, 10 in a very periodic fashion. Very structured data we understand very well. But the TYP data, that is natural data, like English text, so you can digitize that as strings of zeros and ones, but very specific zeros and ones, but not so specific that they're completely predictable, but they still seem to be somewhat predictable.

And. Yeah, so we don't have good mathematics for partially structured objects. This analogy in physics, actually. So in physics we have continuum mechanics, which is the one where everything's sort of averaged out and, and we have a good theory there. And then we have atomic level physics where you just have, you can look at individual molecules and particles, but at the meso scale there's lots of intermediate structures like cells, for example, biological cells.

Emergent.

It's emergent. Yeah, it's emergent. And we don't have good mathematics for this. You know, I mean, in principle you could break down atoms, but you can't possibly analyze.

Yeah, it's not mathematically impossible, but in practice it might be physically impossible. We mentioned, you know, inevitably when we talk about LLMs, you know, the middle L is language. We'll get to my friend Galileo I brought, I want you to get your impression on one of his math books and treatises. But he said that the book of knowledge of nature, the universe, is written in the language of mathematics. And it kind of was echoed later by Wigner and so forth, as we already discussed. But is it really a language? Yes, it has a vocabulary and it has a syntax, but in the same text, Shakespeare and math, if they're truly at root, some probably proto, er, language or something like that, then they should have more combinations or similarities, I would think. But again, I want your opinion. Do you think of math as a language or is it much more than that?

Well, certainly when mathematicians talk to each other or to other scientists, I mean, they have to use math as a language. I think the difference between mathematical language and natural language is that mathematical language sort of has evolved over time to describe, it's to describe the underlying mathematics as efficiently as possible. Natural language is not always about efficiency. I mean, you also want to convey nuance and emotion and art or just express frustration or whatever. So it isn't driven purely by efficiency, but mathematics pretty much is, partly because over time we try to do more and more ambitious mathematical tasks. And if we didn't optimize our math language in this fashion, we will not be able to do these more complicated tasks. And the same is true in the sciences. We keep updating our laws of nature so that we can make more complex predictions.

When you optimize a language for efficiency, you're basically just trying to compress a description of the universe into as minimal and elegant a form as possible. And so when you're doing that, you are somehow getting to the essence of, of how the universe actually works. So presumably the universe does operate by some laws of nature, which maybe you don't know yet, but we'd like to believe that these are simple predictable laws and it isn't just some big chaotic. There isn't some agent that's just making things up as they go along. And the whole history of science has been sort of validating that belief. The naturalistic viewpoint to philosophy and mathematics has been trying to do the same thing thing to mathematical theories, trying to find the most elegant minimal inputs that would explain lots and lots of mathematical phenomena. So maybe that's why they sort of converge over time. And this is why Bugner also observed that the types of mathematical language and formalism that is good for mathematics, for example, the language of curved space to describe all kinds of geometries, happens to coincide quite well with the language that it would describe.

So, you know, universe, like Einstein's use of that same language to describe spacetime, figure of space.

Yeah, exactly. So one of the questions I love to ask mathematicians that have been on from Jim Simons and Steven Trogatz and many others is whether or not you believe that math is invented or discovered. So there's four options. You could say invented, discovered, both, or neither. So where do you come down on this classic division?

Definitely both. So I mean, we, I think there is an innate mathematical structure which we are trying to discover. But in order to do that, we have to invent mathematical language. And initially it's not a very good language. We are focusing on the wrong things. But over time, as I said, to try to make our language more efficient and more powerful, it sort of naturally converges to the ideal Platonic ideal of mathematics. And that certainly feels like discovery, but it's done through human beings. So, yeah, it's both invention and discovery.

Yeah, that's what Jim Simon's telling. When we look at the future of education, you're not only a Fields medalist, a mathematician and father and everything else that you do, but you're a teacher and you're an educator. Talk to me about your vision for the future. What's your philosophy of teaching?

Yeah, so it needs to evolve quite a bit for many reasons. So the world has become infinitely more complex and unstable and unpredictable. And now with AI, humans used to have a monopoly on cognitive tasks and now AI. So one of the problems with AI actually, I mean, the way the subject develops, it's not so much that they overtake human research level mathematics or any other discipline in the near future, but already undergraduate level mathematics, for instance, many of the homework assignments that we assign right now, they can be done by AI. So we have to reinvent the way we teach. So one thing that will become more important is students will need to have much more training in how to validate information that they see. So in the past we had a small number of authoritative source information or textbooks and your teacher or something. And you didn't have social media and the Internet and all kinds of information.

Now AI of all information of really variable quality. On the other hand, in the past when you had information that was low quality in content, it was also low quality in presentation. So you could tell that a really well produced textbook would likely have more accurate content than something written in crayon or something. But now our ability to produce high quality presentation has far output paced our ability to produce high quality content. So you cannot have YouTube videos or textbooks that look flawless. And now AI generated output, but have got lots of fundamental mistakes. So yeah, we need to encourage critical thinking. I already see teachers experimenting with things like here is a question that I would have assigned, but I've given it to ChatGPT and this is the answer that they give.

It's really wrong. Please critique it and correct it.

Interesting.

And these are, I think, more of the skills, more interactive. So not treating knowledge as a passive thing to be acquired by an authority, but something that you always have to.

Question and struggle with. Interesting.

Yeah.

That kind of reminds me John Preskill at Caltech talking about quantum computing and quantum supremacy and so forth. And one of the ways to overcome some of the issues with error correction in quantum computing is just throw more qubits at the problem. And I wonder, will we throw more AIs at the problem? You know, this kind of flipped it through natural, you know, human brains that an AI to prove what's wrong. But will we be at a place where AI could police itself? And so what would it take to trust them?

It's good to make them more reliable, but I think, well, maybe if we use a very different architecture from the current AIs. So by nature they are inherently unreliable, but we have ways to use unreliable tools. Random number generators are the most unreliable device technology we have, but they're extremely useful for all kinds of things like cryptography. I think as long as you pair these AIs with good verification and you only use the AIs to the extent that you can verify the outputs and no further, then they can be a great tool. I see them more as complementing human scientists and mathematicians. So because there are so few human scientists in the world and we only have so much, much time to work on research, we tend to focus on sort of high value, high priority, isolated problems. But in mathematics and the sciences, there are millions and millions. There's a long tail of lots and lots of less well known problems which should require some attention.

And they're not the most difficult or important, but it'll be good to have someone or something look at them. And so I think AI, actually their best use case is, is not to target them on the most high profile problems, but actually on the millions of medium difficulty problems. And they may fail and they may only do they only solve 10% of these million problems, but that's 100,000 problems solved. So scale is the big advantage. You cannot scale a graduate student this way. Okay, not legally. No, not legally or ethically, but AI, I think that's where the big, the real value lies.

What's your highest priority task right now?

Well, research wise, what I'm interested in most nowadays is new workflows to modernize mathematics and make it more collaborative, more accessible to the public, and to integrate in these new tools like AI. The way we've done mathematics has not changed fundamentally in centuries. You may see I have blackboards in my office. We still work with pen and paper. We use computers on little bit, but not so much. And our collaborations are still very small. We work with two, three people in the sciences, of course, thousands, thousands, in large part because we don't know how to incorporate contributions in the general public. There's a barrier to entry.

First of all, a lot of what we do is very technical, but we need to synthesize proofs where every single step has to be verified. So if we had thousands of people, you had to verify a thousand little components. It wasn't feasible to verify recently. Also, because of all these factors, we don't collaborate as much with the other sciences as we ought to, especially the new sciences, which are so data driven and connected with reward in new ways, like social network analysis or whatever. So that is, I think, the direction in which my research is going into it. It's almost more the sociology of mathematics actually than the technique. And more recently I've been interested in trying to secure funding for mathematical research that has become very interesting, unstable in recent years.

We'll talk about that in a bit. So ritual friend Sergio Kleinerman asked me a question related to what you just brought up, sociology of science. And he wondered how it was stressful for you to be reputed the best mathematician on earth, a Fields medalist, a very young and extremely successful mathematician. Did that affect you? Was that challenge for you with that mantle, that weight on your shoulders? Perhaps, or maybe not.

Okay, I do remember the year that was not 2006. My life did change in many ways. So suddenly I got invitations like embassies, and I would meet with people who I would not only meet, and I got asked to be on all these committees. Suddenly my opinion was sought after. So that was a sea change. I mean, I was only son of these already, but so that took them getting used to. But I think one thing that helps ground mathematicians a little bit is that as a pure mathematician, your main task is you have these problems you want to solve and you need proof theorems that solve these problems. And your proof has to be correct and every step has to be validated.

And it doesn't matter how famous you are or how much of a reputation you have, you can't just say, I've proven something, trust me, okay? You have to supply the details, you have to supply the proofs. And if you don't have the proof, you don't up the proof. So I think this naturally provides some check on just sort of how high your ego can go death from these awards. Because, I mean, there are countless problems that I would love to solve, the Trueheim conjecture we talked about, but the hundreds of problems that I would love to solve, and I just know I don't know how to solve. And so I know more problems I can't solve than the problems I have solved. So I think that. So that keeps you somewhat obvious.

What about the. Get out. The old trope that, you know, mathematicians do their best work by age 30. We're you're 50s now, you and I. What, what do you make of. Of that statement? Jim Simons used to tell me he didn't really believe it. He thought that actually a clock starts, you know, at a certain moment, and then you have 10 years or 20 years to do. And he, he did stop at age 30, but that was because he worked at Pensley for 10 years, not because he hit an arbitrary age.

Right.

You've heard this trope.

Do you make of it? Yeah. So different mathematicians have had different career tracks. So I definitely had more stereotypical. And I had skipped a childhood. I accelerated, I skipped several grades. And so, yeah, I did all of my work when I was younger. But there are other mathematicians who started quite late. They didn't become interested in mathematics until college when they switched became quite good.

My advisor, when I was In Princeton, my PhD advisor, Eli Stein, I would meet with him every week and I would discuss the problems that he'd assigned me to work on. And I'd spent hours trying all kinds of crazy things and I'll report all these things I tried didn't work. I tried this. It didn't work. All this energy and time. And he would just sort of look at what I wrote in blackboard and just think for a few seconds and said, you know, the difficulty you're having is exactly the same difficulty that so and so had in this paper. So he goes, and you push out this one paper, a preprint, to read this. This will solve your problem.

So there was a different way of doing mathematics. I couldn't see how he put. Because I would go home and read it and it would solve my problem. I would then hit another structure next week. But I spent hours on these problems. And he just thought about it for 10 seconds. He just knew from experience what to do. It's the wisdom.

Just wisdom? Yeah, wisdom. And so I think as you get older, you Find different ways to do mathematics which it may not be as flashy in terms of more brute force than what it does as a fast bench. It can be more productive. I can now pull the symmetric off my own graduate students and so it's like quite satisfying. You see full circle.

You could do second order, you could say. When my advisor told me reading about grand advisor, let me ask you a question related to pedagogy. So it's obvious from what we've already talked about with Wigner that math is really important for physics. Do you believe that there's an experimental or physics minimum amount of knowledge that a mathematician should have? I've asked this a theoretical physicist that's much more closely related to experimental physics. But do you believe that there's a certain amount of connection to the real world that a mathematician can benefit from?

Oh, definitely. I think one great thing about mathematics is that there are so many ways to approach mathematics. So you can be a very visual mathematician and so you see pictures, you can be a very symbolic mathematician and you just view it as a game of manipulating numbers or symbols. Or you can be a very physics oriented mathematician and you always use physical analogies and you use insights from various soft fields of physics to help you. I mean so there's some very direct connections. If you study caucho differential equations then very naturally you should know some physics because physics has so many examples of great differential equations. And having intuition about say how fluids work or how waves work can really to relate really helps me. And I think just in general, just the more you know in other areas, I mean sometimes I find ventures from thinking economic terms like if you want to prove that X is less than Y, one way to think about it is that if you own Y amounts of stuff then can you buy X.

And sometimes if you don't have a, you know, sometimes you can't do it directly, but maybe you can trade in Y +Z and then use Z to buy X. So like if you put yourself in the mindset of you have some bizarre and you there are certain merchants where you can trade X for Y but you want to negotiate, you want to get a good price for these things and you don't want to trade X Y from Z if it's a bad deal. That kind of mindset can actually be very helpful in seeing sort of the right route how to get from X to Y. Sometimes you can think some types of methodology you can think of as games. So in analysis, the voxel, the statements which say things like for every Epsilon. There's a delta such as blah blah blah. It's called epsilon Delta type truths. And undergraduates are often very.

I hate those because even yeah they're quite complicated in terms of games. If you're used to games like chess and so forth, if your opponent moves here, how do you counter that route? And so if you think every time someone gives you an episode Sloan, you need to find a delta to counter it. And if you think in these sort of game theoretic terms, sometimes that can provide you a useful mindset. So yeah, you can use intuition, biology, social sciences, every academic discipline has theory. Yeah.

That reminds me of this book that I've been wanting to show you and we did take a look at it before we started recording. So this is called the Compaso gmf. So it's. The English version is Galileo's. It's by Galileo. Galileo. The operations of the geometric and military compass. This is not for finding north and south but instead it's for finding really doing a calculation.

So it's really an early version of a slider book. So this is the 1649 2nd edition. The 1601 1st edition as several times our salary. He's at the University of California so I didn't afford be able to afford that. But what's so amazing, in addition to Galileo's actual signature, which we can zoom in on there, I don't know how in about this paper seven years after he died, but he had a stockpile. He was a minor celebrity now he never left Italy, he never got outside.

Of Italy, just the preserve for 40 years.

It is, it is. Isn't it beautiful? I find it like a treasure. I'll bring it up in just a bit. But here's an example of it. So it had segments, it had. It had. Was made of metal and it had indications on it. It could do angles and so forth, but it could also do calculations.

And one of the calculations kind of funny to think about is he goes in, in this. I think I mark it in this post it note. Want to take a look at that page, Terry? He talks about. It's basically an instruction manual. So nowadays we get the device, we get an iPhone. It doesn't come with an instruction manual or anything day and you're expected to be able to use it. So at some point he starts talking about, you know, comparing lengths of lines. But I think on this page here he goes rule for monetary exchange.

So you just mentioned this. You want to read that. That would be cool.

By the means of the same arithmetic Lines we can change every kind of currency to every other in a very easy and speedy way. We first set up instrument taking leftwise the price and money you want to exchange and fitting this crosswise the price of money in which the king is made. Utter's got an example of this. Everything is clearly understood. Suppose you wish to exchange Florentine gold scutti into vintage. Since the price of value of de cat is 6 lira, 430 is necessary. We work out gold currency sodi given a scooter's price of 160 soi the price of tickets 124. I'm so glad we ought to do the same.

What?

Because I get into the fun? Exactly. I think it's so funny because you know, nowadays the scooty is worth nothing. I mean it might be worth a dollar, dollars or whatever, but if Galileo had just put away a couple first editions of this book for his heirs, they'd be worth billions of dollars. But we mentioned this notion of currency conversion and my friend Eric Weinstein and I know, long known the same have worked on gauge theory applied to it. So what do you make of this? How interesting.

Okay, yeah, currency exchange actually is a very good example. So gauge theory has this reputation of being this really obstruse area of physics in mathematics. But it comes down to many quantities in the real world are scalar, but they don't have a natural unit. So currency is one example. If I have a certain amount of wealth, I can measure it in dollars or euro or whatever. And so you can refer to a number, but it is not actually itself a number. Wealth is not a number, but you can measure it by numbers. Gauge theory is about quantities which can be measured, measured by numbers or by coordinates xyz but there's a choice of which units to use or which axes to use.

And maybe if you're a different location on the earth, you may have to use different units. And so as you go from one country to the next, your units may change it. So you need some way to convert as you go from one location to the next. Similarly so in the real world, the electromagnetic fields, which in high school we teach that these fields are vectors, there's some triple of numbers at every point we call E and one for B. But actually they're not numbers, they're directions in some abstract space. And so as you go from, as you move from one place to another, these numbers will change in a certain way. So gauge theory is just about how to manage these convergence. And one day you may Decide that I'm going to price my currency here not in pounds but in lira.

And so that doesn't change how wealthy you are, but it does change the gauge. And so there's a mathematics of how this gauge works and which things are gauge invariant, which things are not so poor curvature. If you go around in a loop and you follow and you just transport whatever you, your vector or whatever it is along with your gauge, sometimes you end up back where you start and sometimes you don't. There's a correction and the correction is, and the correction doesn't matter actually what units of currency or what's your Gaussian variant. So for example, if you have a certain amount of dollars and you travel to Europe and you go to the Euros, then you go back to the US and put back into dollars because of exchange rate piece and so forth, you might not have exactly the same one. So in a sense that is some curvature. It's not exactly curvature, but it's a bit like curvature in the currency bundle of the world currency actually is a nice metaphor. Yeah, okay.

Yeah. It's surprising that you get from different symmetry laws and so forth that you get properties that are unexpected and things even emerge where they wouldn't be expected. And one sort of commonality. A fellow Fields medalist, I believe the only physicist to win IT is Edward Whitman on the Institute for Advanced Study. And of course he's known for contributions to stream theory. He's all or 60 years older than us. So fellow Fields medalist Edward Witten Institute for Advanced Study is the first physicist mega, the only physicist winifieldsmo. He's worked extensively in quantum gravity and string theory, et cetera.

What do you make of the current status of the IT and the mathematical nature of it that seems to only be able to solve things in very high dimensional spaces for which we have no evidence. Where's your outlook as an outsider perhaps?

So it's, I mean physics as you know, of course history of we had to reevaluate our conception of the universe and the nature of reality several times already. So it was the Copernican revolution that leads one big Earth is the center of the universe. You know, this is Einstein works relatively that space time had to be curved. And then of course there's quantum mechanics that reality should be gripped in by wave functions and quantum fields in a way that this theory is now a victim's own success. Right. Because we can now explain 99.9% of all observable phenomena by these theories, except that at really tiny Scales or the origin of the universe, it doesn't work and the mathematics is inconsistent and so we have to replace it by something. So in particular, the idea that space time is a smooth manifold does not seem to be compatible with con operations that below pine cleanse. So we need something else to replace it.

The problem is that there's infinitely many candidates for what to replace it with, despite mathematics being unreasonably effective. Ask you the right mathematics, you can't just conk in any biabago theory. And I'd hope that this will explain. So yeah, for many decades string theory was the leading contender. It's a very elegant theory, I'm not an expert and understanding it has not quite withdrawed up the expectations of providing at least not a unique canonical theory that would fit the data. Maybe Pomercy is too flexible, gives you too many possible ideas.

And that brings up a question I've been meaning to ask you. In mathematics there's Godel's incompleteness theorem, which sets a bound on what's possible to extract from a given system of axioms and possibly bound what's possible to prove. As I understand in physics we don't have that. Right. We don't have any proof. We can't prove that gravity is always 9.8 meters per second. Right. So it's provisional and subject to new data.

Right. So, and that's part of the beauty of it. But the closest we seem to have is what Popper, you know, suggested as a, as the sinequan non is a the definition of good science is that it's falsifiable. Do you think? I've often joked that physicists have mathematician envy. Yeah. A lot of people say, you know, sociology has physics envy. But I think that because we can't prove stuff. So is there always going to be this limit to, you know, what is capable of being asked of, of a physical theory? Because we can't, as I said, we can't even prove we.

You can prove one plus one equals two. It takes what a ballock organic be as n takes 200 pages. But we can't prove anything in physics. Where does that leave us in the epistemological search for truth?

I think you just always have to keep separate the real world and our models of the real world. So I mean, physics has provided us with mathia models which are with which you can prove things. So relativity, for example, Eisenhower's equations are a completely precise mathematical equation. And you can get stuff by initial conditions. Still, if you specify the initial conditions of Spacetime, there is one mathematical solution unless this in and you can prove theorems about that. And so the models you can both file. So by and I mean they are on the status of a mathematical construct. Where the physics comes in is how that model interfaces with with reality.

So even if it doesn't quite match, even if it's technically falsified by experiment, doesn't actually mean that the theory is destroyed. Newtonian gravity is still a very useful theory even with technically accurate. It's good enough for modeling planets and comets and so forth. As long as you don't conflate your model with the reality, you can have both your mathematical cake needed to where Very good.

Just as we were wrapping up, Terry grabbed the chalk and gave us a lightning talk about how he helped to crack a brutal image analysis problem that was vexing physicians trying to get the best quality images of their MRI machines. Terry and his colleagues cracked this mystery using what he calls compressed sensing, using math to reconstruct physical images from far less data than ever before. The result, MRIs that run up to 10 times faster. Enjoy your infrared treat.

I was talking to some statisticians and engineers about an image acquisition problem which they had converted into this sort of math puzzle about how to solve a certain system of linear equations. And they were reporting some results which were amazing that they were able to reconstruct an image using much fewer measurements than traditional imaging. And they were hoping to use this for medical particle imaging. And I talked to them and I solved their little linear algebra problem. In fact, I first was trying to disprove it because I couldn't believe how good the results were. But on trying to do that, I figured out how it worked. And this technique, we published it and it became very widespread. In fact, nowadays most of the big manufacturers of medical MRI machines, they use our technology methods, which is now called compressed sensing, to speed up MRI scans by like a factor of 10 or so.

Yeah, you sort of never know. I mean, there's a lot of work I do, for example these days is how to tell if given some sequence of numbers, whether it has patterns or whether it's structured or whether it's random. And what kind of tests can you apply and which tests are sort of better than others in various ways. As you said, there could be ways you could use this to detect fraud maybe, or filter out noise and try to get better signal acquisition algorithms. It's a whole ecosystem. I think in order for the more applied scientists and engineers to get the ambient ideas from the literature in order to solve their problems. They, they need the people from the more basic sciences to ask questions more in a curiosity driven way. And maybe things that we do directly have a practical impact, but there's this unreasonable effectiveness.

If you don't have these people asking these questions. The people downstream who are actually trying to make practical application things a reality, they can spend a lot more time and maybe a lot more money trying to invest. So to give one example, Shannon developed this theory of communication complexity over a century ago. Just theoretically, if you could only send a certain number of bits of messages per second, how much information can you send and what's the best way to compress this data? And there's this whole practical theory that was developed actually long before the digital revolution. Later, when we needed, when everyone had cell phones and we needed to transmit huge amounts of data simultaneously and we wanted to make sure that cell phones didn't interfere with each other. All this mathematical work was really important. It may not have directly told you how to build the phones, but it did things like it provided the theoretical limit. It was called the Shannon bound, like exactly how much information you could cram into a certain amount of spectrum.

Spectrum.

And so because of that, you could plan, you could buy, purchase a certain amount of spectrum and you would know sort of theoretically how much information you could communicate from that. And you can do budgets, budgeting and planning. There's still lots of engineering that needs to be done. But mathematics can tell you what's possible. So yeah, you need this basic science and it's much cheaper to do that when it's still mathematics and you do it by pen and paper rather than deploy a billion dollars and realize that it doesn't have the capacity that you need or has too much.

Right, yeah. In which case it's wasteful. Awesome. Okay.

I know. If you enjoyed this conversation with Terry, you're going to want to catch part.

One of our interview.

We talked about the dramatic cuts that terry faced at UCLA thanks to the Trump administration's policies in middle of 2025. And you'll also want to check out my recent conversation with Stephen Wolfram, one of the deepest thinking mathematicians of all time. Don't forget to like, comment and subscribe.