Transcription
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. They're 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. Terren Tao won the Fields Medal, the 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?
Ah, I actually don't drink coffee much except in 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 and coffee?
He said that mathematicians are um a means for turning coffee into theorems.
There's a very nerdy follow-up joke to that which is that a co-athetician is a is a is a is a way of turning co thems into fe.
Yeah, that's it's a very 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?
Uh yes. I think I was 10 at the time. Um, so he had a a collaborator in Adelaide, which is the city where I I grew up, George Zakaresh. So he would visit every now and then. At the time, u, I think one of the math professors at the local university introduced me to him. And Erdish was always very good at uh, um, he was known for for um, um, meeting, you know, bright young kids. And so we had a nice conversation. I wish I'd remembered more of it actually. Um, I was just I was too young at the time to realize just sort of how how much of an honor I was really. One thing I remember was that he really treated me like an equal like he you know he didn't condescend as a kid and he later send me a postcard um that uh it was it just had thank you for your nice hospitality. Um he was a math problem um which I didn't solve actually but uh it it did get solved later um by someone else.
Oh, it's amazing. Yeah, he was one of the most prolific mathematicians in in at least modern history and 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 Erdosh number?
Uh yeah, so there's this concept called the Erdish number. Um so Erdish Woodlock in graph theory. Um and so this this concept is inspired by by graph theory. So Erdish himself has a Erdish number of zero. If you've written a paper with Erdish, you get an Erdish number of one. If you've written a paper of someone who's written a paper with Erdish, you um uh you have Erdish number two. So I have an urgent number two for instance and uh I think nowadays people it's common of numbers of four or five people have made similar numbers in other fields as a bacon number. Um so if you've started in a film with Kevin Bacon you have a Kevin Bacon number of one and so forth. And then there's something called the Erdish Bacon number which is the sum of your Erdish number and Bacon number which is usually infinite. Um because you either don't have a chain of papers going to Erdish or you don't have a chain of movies going to to uh to Bacon. But there are half dozen people who have like but combined like seven or eight.
Yes. Yes. I've heard some of random things like that, but uh yeah, he was known in many ways. I I remember hearing from Jim Simons who was my late great mentor and obviously you knew him well. Uh that he had uh he was incredibly uh productive but part of his productivity relied on the use of amphetamines. He used to take some Is that true?
Uh that's what I've heard. Um, apparently one of his friends convinced him to give up empetamines for a month um for a bet or something and Erdish grudgingly did it and then he's he at the end he he just went back and just said you just said mathematics back by one month. So he was I guess hardcore. You don't see that so often nowadays. I think uh I think back then maybe there was there was less of a work life less of a stress and work life balance that you are today.
Now you had work uh related to Erdos, right? The Erdos discrepancy theorem or something. Yeah. What is that? Can you explain that for my audience?
Okay. So, Erdish was famous for posing many many problems. Um, and I I solved a few of them in um over my career. Discrepancy theory is a theory about how irregular sequences can be. Um, so, uh, like if you have a sequence of plus ones and minus ones, um, and if they're random, um, you expect in if you if you pick say 1,000 numbers plus or minus ones at random, you'd expect 500 of them to be pluses, 500 minus 1's. So the discrepancy is defined as the difference between the number of plus ones and number of minus ones. A sequence can have low discrepancy if you view it over the whole sequence. But if you look at subsequences um the this the sequence can have a higher discrepancy. So for example if you take an out outake sequence of plus one - 1 plus one minus one a thousand times. Um the discrepancy over the whole interval is zero because you have 500 plus one 500 minus one it sums to zero. But if you look only over the even numbers you just see plus one plus one plus one plus one or minus one - one - one um then you have a very large discrepancy like 500. Some sequences they can be very well balanced overall but when you restrict to to sub um um subsequences they can have a higher discrepancy. So Erdish was interested in whether you could design a sequence which had um what's called bounded discrepancy um over all what are called homogeneous arithmetic regressions. So could you create a very very long sequence of plus ones and minus ones where if you look at any um finite segments say from one to 100 one to a thousand um the number of plus ones and minus ones only differ by at most two say um but also if you look over the even numbers same thing happens if you look over multiples of three same thing happens so he wondered if it's 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 um I think um someone constructed a sequence of like 1,64 elements or so where the discrepancy was 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 sats over they could show that past that point you had to have a discrepancy of three. Um the the sequences became more and more unbalanced over time but until um um several years ago that was the record. So um the um three was the best lower bound for how much discrepancy these sequences had to have. Um so Erdish asked do these sequences if you continue these sequences on forever and ever do you must you eventually um must the discovery eventually um go to infinity and and this is what I was able to show.
Oh wow. So it does go to in it diverges.
Yes. Yes. Extremely slowly as far as we know like logarithmic or double logithic but it it does it it does go to infinity and I had to use tools from um information theory and number theory.
I've heard that there are people that use some applications of of your work um to detect cheating in the following sense that when a student cheats, of course, our students never cheat, but you know, say they're doing a true false exam or they're, you know, want to mirror something and they want to simulate that they actually got the answers. Maybe they'll put that randomly, true false, true false, true, and it'll be too close. The sum would, you know, plus minus would go to zero. Is that is that relevant?
It's it is connected. Yeah. So, there are other statistical patterns that um random sequences have and artificial sequences um don't. uh I don't think my law discrepancy work directly based on that but there are other patterns there the most famous is called Benford's law which is very unintuitive law that uh roughly speaking 30% of all numbers in the world start with one which sounds very weird because uh you know numbers can start with one two three or up to nine but you can take for example take all the countries in the world take their population and there's about about 100 odd countries in the world about a third of them the population will start with one um China for instance um but um or you can take um the the the net wealth of of um of several millionaires and billionaires or whatever and you also find that most of them start with one or birthdays. Yeah, the pattern is quite universal. Um, but whereas if you if you pick numbers randomly, if you like fudge your accounting books and you you um uh when you pick numbers random um artificially, they don't necessarily obey that.
They're uniform. So you highlight that or you try to make them you think that uniform is correct.
So so humans are actually really quite bad at at creating truly random patterns. Um, and so yeah, you can distinguish natural patterns from from human generated ones.
Interesting. So, one thing that I've, you know, been to talk to you about for a long time are kind of the the limits of mathematical induction. So, you mentioned that you start with this small number and then you kind of add on to it. And I I do want to hearken to uh the work of of Jim Simons. He's most famously known well for being a multi-billionaire um establishing philanthropies that support money research and and and hopefully uh other very highly competent scientists. But um but one of his lesserk 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 I should I should have you explain what a minimal surface is. But as I understand it is you can sort of think of it physically if you had some 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 in zero or one 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, right? And he showed it didn't work. I would have stopped at two. Right. So most mathematical induction, you know, seems to continue to infinity. But you've 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?
Yeah. So induction means different things. I think the philosophers and philosophy of science induction reverse is something slightly different where you you take facts that you observe from small examples and you um you induce from that what what things um um a prediction for what will happen for larger cases and um it's 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 um so there's a precise um principle of mathematical induction. But 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 um some number n you it uh you can you you know for sure that it implies then um the same thing for n plus one then it implies it's true for all for all numbers. Um the analogy often given is just a row of dominoes. Um so if each each domino um represents one case of 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 um whenever you can prove it, it it tips over the next domino, then um no matter how long the um the string of dominoes is, you you can you can knock over every single um uh domino chain. But it's really important that your arguments are 100% watertight. You know, if a string of dominoes and like the 97th domino um doesn't tip over the 98th, you know, it then it stops there. So it's a principle that only works in the water mathematics which is one of the few places where you really can have 100% guarantees.
So Simon's Yeah. So he discovered what's called the Simon's cone. Okay, you're pushing a little bit because this this is geometry is not completely my my area of mathematics but yeah so um minimal surfaces are most famously are two vision surface like soap films. Um 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 or um at one dimension minimal services are very boring. just straight lines but you can consider them threedimensional surfaces in fourdimensional space which already is hard to visualize but mathematically you can consider it and five and six. Weirdly sometimes um 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 it gets you some intuition um which can help guide you with with the problems that you do care about. Yeah. So um it turns out yeah that in uh up below eight dimensions I think here all minimal surfaces are smooth. You you you can't tie a soap bubble and create an any kind of knot. Um
yeah because there there's always some way to uh pull it apart and reduce the the surface tension. Yeah. Starting in eight dimensions he discovered a very surprising uh fact that that singularities do can actually form. Yeah. that that there is there is this uh yeah it looks like a cone except in much higher dimensions and there's no way to to modify the cone to make it to reduce the surface area. If if you made a cone if you try to arrange soap into a cone in three dimensions you could just remove the the um do what's called a surgery. You 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 we we need to understand high dimensional geometry is much um much better um than than we're used to um and a lot of our old intuition is actually um which you get from low dimensional geometry is actually completely false in in high dimensions. So just to give you one example like if you inscribe a a a circle inside a square it occupies a fair you know a pretty large chunk of of of of the of of the um 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 like if you take a thousand dimensional cube and you inscribe a thousand dimensional ball inside it, it's actually incredibly tiny. It's like 0.00001%. Like um balls become extremely poor space filling.
Yeah. They're nowhere near space filling um in high dimensions. And this is important when you look at clouds of data and uh you know like if if you have some you're taking a thousand measurements and that's like a thousand data points but but there's some errors in them. you know, do you measure the the root mean square error which is like trying to place uh this uh your your measurement inside some ball or do you measure the worst error worst of the 10,000 errors which is like placing the question is do you want your arrow bars in high dimensions to be like a ball or a cube and it starts making a difference
right they I mean there 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 um you know proof by um ridiculum. Can you talk about what what's your favorite type of mathematical proof and when you're on to it, you just get so excited to finish the front?
Proof by contradiction. I think Hardy had a great quote that um in chess um a chess player may offer, you know, a pawn or a bishop. Um but a mathematician offers the entire game, you know. So, uh it was say
sacrifice. Yeah.
Yeah. So, okay, we want to prove this is a conclusion. I will give you that the conclusion is false. Okay. I will just let you um uh run with it. Okay. And but you do that and I will show that it's gives you contradiction. It actually is a technique. Um so um it on the one hand it it is very unintuitive. Um the undergraduate students that that we teach they struggle a lot with the notion of of proof of contradiction. On the other hand it is a concept that I have seen primary school students teach each other. So um in recess you might see um 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 will go on like this but at some point someone will realize that 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's no largest number in in the natural numbers and this is a proof of contradiction because if anyone ever did claim to have natural number largest natural numbers they add one and you have contradicted them. Well, it is actually a very intuitive proof technique, but uh but you have to you have to teach it the right way and sometimes kids can teach themselves.
Type of mathematics that I that 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 you can um have a problem which is has to do with commentary nothing to do with the real world but you find that there's 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 uh which are really quite uh quite amazing. So yeah those feel like magic to me.
Mhm. Yeah. I mean, the most famous one at least to my physicist mind is the proof that by contradiction that square root of two is irrational. So that's was the Uklid's original proof or what does trace back to before him? Uklid or Pythagor the Pythagoreans. Maybe Pythagoreans. Yeah. Uclid prove the prime basically by the same sort of idea as the infinity plus one, right?
Yeah. Yeah. Yeah. Um, no, we have a lot to thank Uklid for actually. I mean, he wasn't the first to write down many of of of the theorems like the Pythagoran theorem, for example. I think the Babylonians had a version, the Chinese had a version. Um but really he was the one who introduced this this notion of proof that um complex facts about mathematics you could you could deduce from from simpler axioms um 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 piqu my curiosity. Um and eventually I do want to tie this to Wignner's famous statement about the unreasonable effectiveness of mathematics to the physical world and and we'll talk about that in just a bit which also tangentially involves the square root. the square root in physics at least for example in classical mechanics you can construct things operators that involve the position of momentum called pluson brackets and as soon as you take them from the classical world to the quantum world um instead of commuting being equal to zero they become equal not to zero but times a fundamental constant times the square root of negative 1 and it's just so baffling to me that once you introduce the concept of a square root and imaginary number um 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 mathematic could they have taught us this or is there something special about the square root operation in LLMs they use LU decomposition in we have spinners they have a spinner representation there are square roots is there something special about the square root or like in other words why don't we say with cube roots or fifth roots or hundth root why is the square root something that's that's so prevalent in physics for example
yeah so um so We experience the world as in a continuum ukidian space and the notion of numbers that are most natural to us from our spatial intuition are the real numbers. Right? So real numbers have lots of wonderful properties. Um the algebra of real numbers works really well. You know things like uh um addition is is commutive. X plus y is y + x and x * y is y * x. Multiplication is commutive and so forth. But they have one flaw which is that not every polinomial equation has roots. Um so uh if you take um the equation x2 + 1= 0 in in the real numbers x2 + 1 is is never zero because x2 is always positive. So it's what's called it's not what's called algebraically complete but it's very close to being complete. Um so if you take a polinomial which is odd degree like a cubic x cub + 3x + 1 it must have a root because a a cubic polinomial or an odd degree polinomial when you make x very very big it becomes very large and positive and when you make x very very um large and negative it becomes negative and because um the reals are continuous um polinomials are continuous um therefore at some point in between you must hit zero to get from negative to positive sort of sort of for half of all the polinomials in the world um the um um you can solve them in the reals and half you can't. So um in with with the benefit of of hindsight this really suggests that you should make the real sort of twice as big in order to get this really useful property of algebraic completeness. And so um as it turns out there these numbers called the complex numbers which um are twice as big as so the real numbers are onedimensional 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. Um and so um 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 you didn't have previously. Uh so once you know that you're looking for a new number system that's twice as big as 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
how 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 polinomial equations. Correct. Is
Yeah. Yeah. Uh we have roots of polinomial.
We we've learned that the notion of number is very flexible. Um I mean people get upset when when they they learn that that what what you know it it feels simpler to have you know one notion of everything taught in school and then not have that changed. You know, like people were very upset when the notion of a planet got changed like 1015 years ago. And uh we occasionally do that with with math too. Like the number one used to be prime about 100 years ago.
Oh, really? Oh, I I mean I still consider it prime, but I still consider a planet. Okay.
Uh yeah. Yeah. But it um it's it's because um in all mathematicians and scient any any in any study when you first study a subject um you don't really know what concepts are the most fundamental and important and which ones are not. So you make a guess um based on 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 that the the 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 um the new notion of a planet is is better, but um but for example, one of the 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 other than rearranging the factors. So 12 is 2 * 3 * um two or or 2 * 2 * 3 but other than interchanging the um 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 mlication. But if you made one prime, then you wouldn't you would lose you would have to give up the fundamental theorem of arithmetic because now 12 is also 1 * 2 * 2 * 3. Um and you just add too many exceptions to this really important um factor in number theory. So we made a decision to therefore redefine what it was. Okay. But just like with Pluto, there's no there's no consequence to life on Earth. It's it's more sort of
Yeah. It's it's it's a human convention. Um, but it's it's but we we update our human conventions to match reality better over time.
You I've done work in prime pairs. Is that right?
Yes. Yeah. So primes are one of the oldest subjects in mathematics is we Uklid had the first theorem almost ever um um and it was about prime numbers more than 2,000 years ago. And so it's very um frustrating and annoying that even the most basic questions about primes we still cannot answer definitively. um we can we have good guesses uh like so for almost all question primes we can predict the answer um but we cannot get the 100% mathematical standard of proof for for many of them and one of the most basic questions which is at least 300 years old is called the twin prime conjecture that um there should be um so you could show there's infinitely many primes the primes never end you can always find primes bigger than any number you wish but um we cannot find we cannot say the same yet for prime twins so these are pairs are primes that differ by the closest they can, which is two. Uh, for example, 11 and 13. Um, well, okay, two and three are closer, but um, but after two, all primes are odd. So, the the closest you can get is is two. And so, we can observe that every so often, you know, the primes they they don't seem to obey a pattern. Sometimes, um, the prime gaps are large, sometimes they're small, but every so often they they come close to each other and you get a twin. And they seem to occur infinitely often, you know, as we we can find trillions and trillions of these by computer. But we have never been able to prove that they go on forever. We have this prediction that the primes behave like basically like a random sequence of numbers. Um and um random sequences with the if if you have a random sequence of of the same density as the primes, they will hit um um form twins infinitely often. But the primes are not random. We believe they're what's called pseudo random. um that they have no obvious pattern besides the ones that we can we can obviously see such as them being um uh being odd you know so I mean it's it's a very likely hypothesis but we can't prove it
pseudo randomness meaning that it could be derived from some algorithm but not in all cases or something pseudo random versus random distinction
um random means not deterministic that that there's no single like so the primes if you forgotten what the primes were and you had to regenerate them by a computer program you would generate exactly the same set whereas if you were generating um set by rolling dice or flipping coins you would get a different set. Pseudo random sets which are either random or deterministic but statistically they are indistinguishable from random noise. So um for example um a random number should just if you have a random sequence of numbers there should be just as many numbers that end in five as or end in seven and six like the the digits should be equally distributed. Now the primes they're not perfectly suited random because they they do avoid certain they do have certain patterns like they tend to be odd um for instance but there's ways of excluding those sort of obvious biases and once you exclude them um um there should it's expected there's no no tests that can distinguish them from random um um numbers. This is important for cryptography actually because there are many crypto systems like the ones we use to encrypt web traffic um cryptocurrency um you know financial transactions where data like sensitive data like a passwords or or um or credit card numbers are encrypted using mathematical um 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 by doing so the data that we actually send looks indistinguishable from random noise and conveys no information about um your your personal data. And we really hope that that's that's true. I mean um so one reason why it's important for mathematicians to actually study prime numbers is that we occasionally get a shock that um I mean it hasn't really happened in number theory in in c in decades at least but but but there could be really unusual undiscovered patterns in the prime numbers that we weren't previously aware of. And if they existed, they could present a a vulnerability to crypto crypto systems. There have been a few other crypto systems where similar patterns have been discovered. I think not for prize where 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 what kind of an inversion maybe contradiction of uh of what Wner said. He commented on the unreasonable effectiveness of mathematics and the physical sciences or 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 um quantum computers can perhaps factor and break these um 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 in 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 uh uh topic?
So uh quantum computers are a fascinating topic. Yeah, they interface with maths in various ways. Um so one one is actually just the actual uh software engineering of of creating good quantum algorithms. Um so it requires a very different type of software mindset you know. So classical computers we have this sort of 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 and and we have decades of experience with um yeah of a quantum computer the state is is not a bunch of of 01 bits but it's it's a wave function and the operations you you assign to them you have to multiply them but you're only allowed to multiply them by matrices really really large matrices um except that your basic operations um um your matrices are mostly the identity and there's only you only change a few corner bits at a time but you want to couple them together in a in a very efficient way so that you can do really complicated um operations. Um yes so quantum computers are both exponentially more powerful than than classical computers but also exponentially more limited.
Yes. Um so because they can handle um superposition quantum state simultaneously in principle there's this exponential speed up um and in in for certain applications like factoring um and I think quantum chemistry grounds
right um they are at least in principle very very um uh very powerful but quantum mechanics is also very restrictive the number of things you can do to quantum state um you can only do linear operations and only do um time reversible operations
non-destructive yeah
yeah so um um Yeah. So this has um yeah, so this requires you to develop fields of reversible computing. Error correction is also much much more annoying um in um uh and and so yeah so so there's there's software challenges. Maybe once quantum computers become a reality they will be used to do large scale computations of a 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. Um but certainly classical complexity theory um has been very influential. Historically, mathematicians only cared about something whether something was true or false um or provable or disprovable. Um and with the advent of computers, people also started asking questions of how comput how how computable is is an object like? So if they if they could prove that something exists um could you go further and actually say you know is there an algorithm to comput and is the algorithm exponential time or or polinomial time. So a much finer grained notion of of truth actually than than just it's true or it's false. But how easy is it to actually um compute
um and it um that has led to very productive mathematics. I mean sometimes just the the effort to um not just show something exists but actually um find it um creates um yeah creates new techniques. Yeah, complexity theory has offered as has sort of um given a much more nuanced um understanding of of of how true a statement is or how um and yeah, this has led to to um a better understanding of if you just prove something is is true and you don't but uh uh you may not have any insight. So what was what was the key ingredient that made it work? Or if you had two different proofs, which one 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 is stronger. It's more efficient. It's
so it yeah it it it indirectly sort of um provides much more insight is into the um into the proofs that mathematicians want.
Has AI actually enabled new discoveries in mathematics or new proofs that otherwise would not have existed?
slowly it's beginning to um I mean by itself um so the um the big weakness um 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 um it's not grounded um that um it's it's it is it's proistic okay um they often make mistakes and um much like you know if if I were to get a student to solve a problem on a blackboard and they're nervous and they just say the person comes to they might get it right or they might get it wrong. Um but if it's a weak student and they don't have sort of um fundamental knowledge of of what they're actually doing, once they go off the rails, they can go really um off the rails. And and this is something which is a fundamental problem with the current large language models. But if you use them as a component of um a more um rigorous and grounded reasoning system. So if you if you converse with a a large language mod to for them to make suggestions but you ver you understand with the output and you can verify it. Um so um then people have had some success talking about their math problems a large language model will produce will produce some suggestions some of which the human expert can dismiss as as not viable. um some of us would be thinking oh I thought of that already but one or two is oh that's actually something which um I should have known I should have come up with myself but I just didn't um realize so one thing where um AI models are already beginning to be useful is in like 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 you 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 um and you can use the these night models to prompt you to remind you of of the of the missing six. They may also hallucinate three more that don't exist. So you you do need you can't trust them
supervise them.
Yeah. You have to you have to verify. There is hope in the future, you know. So this separate method technology to to auto to have there's software that can automatically verify certain types of proofs,
right? Um and so the hope is that if you force the large language models to only to only output in some verified um some language that you can verify and to filter out the hallucinations has it been able to reproduce a wild you know proof format/ theorem or your work navier stokes I mean has it been able to actually just just simply quote unquote reproduce what I natural intelligence person like you
um there's this issue um it can but but uh often because of what's called contamination um so um if if a result is like taught in textbooks software then it would 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 reproduce from memory a proof that they saw in a textbook and so AI has basically read all the textbooks in the world it's hard to to discern when that happens whether it was training data or whether they they really sort of thought it up if you ask the AI to explain their their chain of thought um they often give complete nonsense like it it um it's clear that it they just didn't know.
Yeah. I mean, I found that even we tried with my student Evan Watson, we tried to uh we just gave it um the um information about the orbit of Mercury over the past 3,000 years, which JPL up the road here has 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 uklidian 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 keing test but it's basically the touring test you will know when it's true when actual AI can come up with new and unknown here to for unknown you know predictions that can be verified by humans like you
yeah yeah know I think I think that's a very promising use case of of AI I mean um yeah I mean you know neural networks in general I mean they're designed to make patterns to to detect correlations and things. Yeah. So there have been a few examples in mathematics where for example in in not theory um um a neural network not a fancy LM so much but like a more old school neuronet network um was used to detect correlations between different types of not invariance that was not um believed people did not suspect existed before.
Yeah. And then once once I mean and initially this type of correlation was um was just sort of this blackbox relationship. So um so so knots have um are these loops in space which um you know some are some can be untangled and some some can't be um they come with all these numbers they're called not invariants um and so um the neural network found that that by by feeding a database of like 1 million knots um 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. No wow.
Um uh so but this this this neon what was this black box you just fed in these 20 numbers as your hyper in variance and it was sped 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 suppose I changed this input um I I just um I modified this hyperbolic volume or whatever how much does this change the output and so they they it's like a box of 20 dials that they could they could they could play with and basically by running um experiments they could see that there three of these inputs were actually really important and the other 17 were were were very peripheral and by doing those kind types of analysis they 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 you know once you have these neural network models of of um he could actually um probe them. So in in your astronomy example, maybe um your a new network might not be able to to tell you exactly what the um the new law of physics would have to be, but it can say, well, I can at least predict um the orbit of Mercury over the next thousand years and and and here's my my model. And then and then you can you can just try to treat it. Now suppose I I changed my my the the period of of Mercury or or the uh 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 than traditional experiment or theory.
Yeah, I've used that example to kind of 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 paper clips 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 and so it'll have errors and it'll have mistakes, but it's bounded by the amount of human knowledge that's used. In some levels, it's 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 and 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. So 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 of these models. So, the mystery is not so much how they run. Okay, we can, 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 of even heuristic rules of thumb for predicting which tasks are good, which ones and not. We can only just make empirical experiments.
Part of the reason is that the data that we train on, like so, we, so data on one level is just strings of zeros and ones, and mathematically, we understand sort of very, very random data. So like if complete noise, completely random zeros and ones, we have the mathematical probability theory which explains, we can analyze this 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 1 0 1 0 in a very periodic fashion. Very structured data, we understand very well.
But the type of data that is natural data, like English text, you know, so you can digitize that as strings of zeros and ones, but very specific zeros and ones, right? 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 of physics, actually. So in physics, we have continuum mechanics, which is one where everything is sort of averaged out 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.
Virgin: It's emergent.
Yeah, it's emergent. Um, and we don't have good mathematics for this. Um, you know, I mean, in principle, you could break into atoms, but you can't possibly analyze.
Yeah, it's not mathematically impossible, but in practice, it might be physically impossible. You 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 a book I want you to get your impression on, one of his math books and treatises. Uh, but, but he said, you know, that the book of knowledge of nature, the universe is written in the language of mathematics. And it kind of, you know, was echoed later, you know, 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. And you know, but in the same text, you know, Shakespeare and math, if they're truly, you know, at root some proto-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, you know, to describe the underlying mathematics as efficiently as possible. Natural language is not always about efficiency. Um, you know, I mean, you also want to convey nuance and emotion and art, or just express frustration or whatever. Um, so it isn't driven purely by efficiency. Um, but mathematics pretty much is, partly because we over time, we try to do more and more ambitious mathematical tasks, and if we didn't optimize our math language in this fashion, it, uh, we would not be able to do these more complicated tasks. And the same is true in the sciences. You know, 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 how the universe actually works, you know. So presumably, the universe does operate by some laws of nature, which maybe you don't know yet. We'd like to believe that there are 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. You know, this naturalistic philosophy and mathematics has been trying to do the same 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 why we now also observe 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 describes the universe, like Einstein's use of that same language to describe spacetime.
Yeah, exactly. Um, so one of the questions I love to ask mathematicians that have been on, from Jim Simons and Steven Strogatz and many others, is whether or not you believe that math is invented or discovered. So there are four options: you could say invented, discovered, both, or neither. So where do you come down on this classic, classic debate?
Um, definitely both. So, I mean, we, I think there is an innate mathematical structure which we are trying to discover. And but in order to do that, we had to invent mathematical language. And initially, it's not a very good language. We are focusing on the wrong things. Um, but over time, as I said, to try to make our language more efficient and more powerful, it sort of naturally converges to the, um, ideal Platonic ideal of mathematics. And that certainly feels like discovery. Um, but it's done through human means. So, yeah, it's both invention and discovery.
Yeah, that's what Jim Simons told me. Uh, when we look at the future of education, you're not only a Fields Medalist, a mathematician, a 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, um, it needs to evolve quite a bit for many reasons. Yeah. So the world has become infinitely more complex and unstable and unpredictable. Um, and now, you know, and now with AI, you know, humans used to have sort of a monopoly on cognitive tasks like, you know, and now AI. So one of the problems with AI, actually, I mean, the way the subject develops is it's not so much that they will overtake human, like 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 teach, we assign right now, they can be done by AI.
Yeah.
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, you know. So in the past, we had like a small number of authoritative sources of information or textbooks and your teacher or something, and we didn't have social media and the internet and all kinds of information now. And now AI, 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, you know. So you could tell that like 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 outpaced our ability to produce high-quality content. So you cannot have YouTube videos or textbooks that look flawless, okay, and now AI-generated output, but have got lots of fundamental mistakes.
So, yeah, we need to encourage, you know, critical thinking, you know. 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 wrong. Please critique it and correct it. Interesting. Um, and these are, I think, more of the skills, more interactive, you know, 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 of 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, um, I wonder, will we throw more AIs at the problem? You know, this kind of flipped it. You throw natural, you know, human brains that have 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. You know, random number generators are the most unreliable device technology we have, but they're extremely useful for all kinds of things like cryptography.
Yeah, I think as long as you pair these AIs with good verification and you only use the AI to the extent that you can verify the outputs and no further, then they can be a great tool. Um, I see them more as complementing human scientists and mathematicians. Um, so, because there are so few human scientists and we don't only have so 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 not to target them on the most high-profile problems, but actually on the millions of medium-difficulty problems. And, you know, they may fail, and they may only solve 10% of these million problems, but that's 100,000 problems solved. So scale is their big advantage. You know, you cannot scale a graduate student this way.
Okay. Um, but
No, not legally or ethically. Okay. 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. Now, we even see our blackboards in my office, you know, we still work with pen and paper. We use computers a little bit, but not so much. And our collaborations are still very small. We work with two, three people. In the sciences, of course, you know, thousands, thousands, in large part because we don't know how to incorporate contributions from 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, we had to verify a thousand little components. It wasn't feasible until very recently. Also, because of all these factors, it's we don't collaborate as much with the other sciences as we ought to, especially the new sciences which are so data-driven and connect the real world in new ways, you know, with social network analysis or whatever. So that is, I think, the direction in which my research is going. 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 metamathematical research that has become very unstable in recent years.
We'll talk about that in a bit. So, retro 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, you know, reputed the best mathematician on Earth, the Fields Medalist, a very young and extremely successful mathematician. Did that affect you? Was that a challenge for you, that mantle, that weight on your shoulders, perhaps, or maybe not?
Okay. I do remember the year of the Fields Medal, 2006. My life did change in many ways. Like, so suddenly I got invitations, like embassies, and I would meet with people who I would not normally meet. And I got asked on all these committees. You know, suddenly my opinion was sought after. So that was a sea change. Um, I mean, I was, I mean, some of you already, but so that took some getting used to. But I think one thing that helps ground mathematicians a little bit is that, you know, I mean, as a pure mathematician, your main task is you have these problems you want to solve, and you want to prove theorems that solve these problems. And your proof has to be correct, and every step has to be validated. It doesn't matter how famous you are or how much reputation you have. You can't just say, I've proven something. Trust me. Okay. You have to supply the details. And if you don't have the proof, you don't have the proof. So I think this naturally provides some check on just how high your ego can go, just from these awards, because, you know, I mean, there are countless problems that I would love to solve, and, you know, the Trinham 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 keeps you somewhat honest.
What about the, you know, the old trope that mathematicians do their best work by age 30? You're in your 50s now, you and I. What do you make 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 it. And he did stop at age 30, but that was because he worked at Polymath for 10 years, not because he hit an arbitrary age. All right. You've heard this chart. What do you make of it?
Yeah. So, um, different mathematicians have different career tracks. So I definitely was stereotypical. I had, I skipped, I skipped a child at accelerated, I skipped several grades. Um, and so, yeah, I did all of my work when I was younger. Um, but there are other mathematicians who started quite late. They were, they didn't become interested in mathematics until college, when they switched and became quite good. My advisor when I was in Princeton, my PhD advisor, I would meet with him every week, and I would discuss the problems that he'd assigned me to work on, and I'd spend hours trying all kinds of crazy things, and I'd report all these things I tried didn't work. I tried this, it didn't work. Just all this energy and time. And he would just sort of look at what I wrote on the blackboard and just think for a few seconds and say, you know, the difficulty you're having is exactly the same difficulty that so-and-so had in this paper. So if you go to the filing cabinet and you pull out this one paper, a preprint, read this, this will solve your problem. So, yeah, there was a different way of doing mathematics. I didn't, I couldn't see how he put it because I would go home and read it, and it would solve my problem. I would then hit another structure the next week. But I spent hours on these problems, and he just thought about it for 10 seconds and just knew from experience what to do.
It's the wisdom. Just
Wisdom. Yeah, wisdom. So, um, I think as you get older, you approach, you find different ways to do mathematics, which, it may not be as flashy in terms of more brute force than it would be as the first, but actually more, it can be more productive. I mean, yeah, I can now pull the syntax. You can say what my advisor told me. Rudy, about your advisor. Let me ask you a question related to pedagogy. So it's obvious, you know, 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 to 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. Um, I think one of mathematics' strengths is that there are so many ways to approach mathematics. So, you can be a very visual mathematician, and 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 are some very direct connections. If you study ordinary differential equations, then very naturally, you should know some physics because physics has so many examples of great differential equations. And having intuition about how fluids work or how waves work really helps me. And I think just in general, just the more you know in other areas, you know, I mean, I've sometimes I've done thinking in economic terms, like if you want to prove X less than Y, one way to think about it is that if you own Y amounts of stuff, 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 for Z and then use Z to buy X. So like, if you have, if you put yourself in a mindset of you got some bizarre and you can, there's certain merchants where you can trade X for Y, but you want to negotiate when you want to get a good price for these things, and you don't want to trade X, Y, and 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 pick some types of methods you can think of as games. So in analysis, there are walks of statements which say things like, for every epsilon there's a delta such that blah blah blah. It's called epsilon-delta type proofs, and undergraduates are often very, um, hate those because, yeah, it's they're quite complicated. In terms of games, like if you're used to games like chess, and so, you know, so like if your opponent moves here, how do you counter that move? And so if you think, every time someone gives you epsilon, you need a kind of delta to counter it. And you think in these sort of game-theoretic terms, sometimes that can provide you a useful mindset. So yeah, you can use interest in biology, social sciences, you know, every academic discipline has to do it.
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 Compasso Geometrico. So it's the English version is Galileo's. It's by Galileo Galilei, 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 slide rule. So this is the 1649 second edition. The 1601 first edition is several times our salaries at the University of California. So I wasn't 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 this paper is seven years after he died, but he had a stockpile. He was a minor celebrity. He never left Italy. He never got outside of Italy, just to observe for 40 years.
It is, isn't it beautiful? I find it like a treasure. I'll bring that 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 this, I think I marked it in this post-it note. Want to take a look at that page, Terry? Um, he talks about it's basically an instruction manual. So nowadays, we get the device, we get an iPhone that doesn't come with an instruction manual, right? And you're expected to be able to use it. So at some point, he starts talking about, you know, comparing links 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 the instrument, taking leftwise the price and money you want to exchange, and fitting this crosswise to the price of money which the king has been made. We illustrate this by example, and then everything is clearly understood. Suppose you wish to exchange Florentine gold scudi in the tax, since the price of value is 6 for is necessary. We work out called Kodi, given that scudi's price is 160, so the price of tickets is 124." I'm so glad we ought to do the same.
Exactly. I think it's so funny because, you know, nowadays the scudi is worth nothing. I mean, it might be worth a couple 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. Uh, but we mentioned this notion of currency conversion, and you know, my friend Eric Weinstein and I know the same have worked on gauge theory applied to it. So what do you make of this understanding that's okay, yeah, currency exchange actually is a very good example.
Yes. So, so gauge theory has this reputation of being, it's a really abstruse area of physics and mathematics, which, but it comes down to many quantities in the real world are scalar, but they don't have a natural unit. So, yeah, so currency is one example. So, you know, 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, it's not a number, well, it's not a number, but you can measure it by numbers. Gauge theory is about quantities which can be 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 at 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 unit just may change. So you need some way to convert as you go from one location to the next. Similarly, so, you know, in the electromagnetic fields, which as we teach in high school, these fields are vectors, it's you know, there's some triple numbers at every point, E and B. But actually, they're not numbers, they are directions in some abstract space. And so 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 conversions. And one day you may decide that I'm going to price my currency here, not in pounds, but in euros. 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, for example, curvature. If you go around in a loop and you follow and you just transport whatever your vector is along your gauge, sometimes you end up back where you started, and you don't, there's a correction. And the correction doesn't matter actually what units of currency or what your gauge is. It's gauge invariant. So, for example, if you have a certain amount of dollars and you travel to Europe and you convert to euros, then you show back to the US and put back into dollars, because of exchange fees, and so you might not have exactly seen one. So in a sense, that is some curvature in it's not exactly curvature, but it's a bit like curvature in the currency bundle of the world. Yes. So actually currency actually is a nice metaphor. Yeah.
Okay. So, yeah, it's surprising that you get from different symmetry laws and so forth that you get properties that are unexpected. And then things emerge where they wouldn't be expected. And one sort of commonality of fellow Fields Medalists, I believe the only physicist to win it is Edward Witten, of the Institute for Advanced Study. And of course, he's known for contributions to string theory. He's 60 years older than us. So fellow Fields Medalist Edward Witten, Institute for Advanced Study, was the first physicist, maybe the only physicist, to win a Fields Medal. He's worked extensively on quantum gravity and string theory, etc. What do you make of the current status of 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, in the history of science, we've had to re-evaluate our conception of the universe and the nature of reality several times already. There was the Copernican revolution that deduced Earth is the center of the universe. You know, there's Einstein's Earth relativity that spacetime had to be curved. And then, of course, there's quantum mechanics, that reality is actually should be represented by wave functions and quantum fields. In a way that this is now a victim of its own success, because we can now explain like all 99.9% of all observable phenomena by these theories, except that at really tiny scales or the original universe, it doesn't work. Um, and the mathematics is inconsistent. Um, and so we have to replace it by something. Um, so in particular, the idea that spacetime is a smooth manifold does not seem to be compatible with observations at the Planck length. So we need something else to replace it. The problem is that there are infinitely many candidates for what to replace it with, despite mathematics being unreasonably effective. Has to be the right mathematics. If I had just conjured any theory and I'd hope this will work. So, yeah, for many decades, string theory was the leading contender. Um, it's a very elegant theory. I mean, I'm not an expert, but my understanding is it has not quite up the patience of providing at least not a unique, canonical theory that would fit the data. Maybe the problem is too flexible, gives you too many possible answers.
And that brings up, you know, a question I've been meaning to ask you. In mathematics, there's Gödel's incompleteness theorems, which set 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 m/s², 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 the sine qua non as a 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 what is capable of being asked of a physical theory, because we can't, as I said, we can't prove. We can prove 1 + 1 = 2, it takes what, a balok identity and 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 keep separate the real world and our models of the real world. So, I mean, physics has provided us with mathematical models which you can prove things with. Relativity, for example, Einstein's equations are completely precise mathematical equations, and you can start with initial conditions, still very, you can specify the initial conditions of spacetime, you can, there is one methodical solution, and the system doesn't, okay, but you can prove about that. And so the models, you can both, and I mean, they have the status of a mathematical construct. Where the physics comes in is how that model interfaces with reality. So, you know, even if it doesn't quite match, you know, even if it's technically also by experiment, it doesn't actually mean that the theory is destroyed. Newtonian gravity is still a very useful theory, technically accurate, it's good enough for modeling planets and comets. And so I think as much as you don't conflate your model with the reality, you can have both your mechanical cake and eat it too, 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 out 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. You're in for a treat.
Um, 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, like 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're hoping to use this for medical imaging. And I talked to them, and I solved their 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. And 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 a 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. You know, as you said, you know, there could be ways you could use this to take forward, 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 need the people from the more basic sciences to ask questions more in a curiosity-driven way. And maybe things that we do, if they don't directly have practical impact. But this, yeah, this unreasonable effectiveness, you know, like if you don't have these people asking these questions, the people downstream who are actually trying to make things practical applications a reality, they can waste, it can, they can spend a lot more time and maybe a lot more money trying to invest, like to give one example, Shannon developed this 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 could, it did things like it provided the theoretical limit, was called the Shannon bound, like exactly how much information you could cram into a certain amount of spectrum. And so because of that, you could plan, you could purchase a certain amount of spectrum, and you would know theoretically how much information you could communicate from that, and you can do budgeting and planning. And yeah, and 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 you're still doing mathematics by pen and paper rather than deploy a billion dollars and realize that it doesn't have the capacity that you need or it has too much.
In which case it's wasteful and inefficient. 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 the middle of 2025. And you'll also want to check out my recent conversation with Steven Wolfram, one of the deepest thinking mathematicians of all time. Don't forget to like, comment, and subscribe.