On Theories of Everything, Curt Jaimungal asks Fields Medalist Richard Borcherds and Eric Weinstein what it means to understand a proof when an AI can produce one. Borcherds draws on his experience with pure mathematics, and Weinstein asks who gets credit for choosing a problem and recognizing why its solution matters. Their discussion ranges from exotic spheres and computer chess to the public funding of mathematical research.
Outline
- 0:00 — AI and recent mathematical results
- 7:14 — The six-sphere and exotic smooth structures
- 12:49 — Credit, secrecy and pure mathematics
- 19:46 — Public funding for pure mathematics
- 31:45 — Octonions and exceptional Lie groups
- 42:00 — Calculation and understanding
- 48:24 — Borcherds compares his work with AI
- 54:03 — Lessons from computer chess
- 1:02:08 — What mathematicians contribute
- 1:09:39 — Mathematics as a public good
- 1:16:15 — The aims of mathematics
- 1:24:33 — What people can still do best
- 1:30:01 — Feynman’s private calculations
- 1:39:19 — The stakes for mathematicians
- 1:50:07 — Attributing credit for a proof
- 1:54:15 — Explaining spinors
Transcript
00:00:00
Richard Borcherds: Math is being entirely wiped out by AI. This was a real shock to pretty much all mathematicians. It’s a little bit disturbing to realize that I’ve spent my entire life training hard to do something which AI is now better at.
00:00:14
Curt Jaimungal: This is Richard Borcherds, who won the Fields Medal for proving the monstrous moonshine conjecture, and Eric Weinstein, who earned his PhD in math at Harvard.
00:00:22
Eric Weinstein: To me, everything’s at stake. We have to prepare for being made redundant in absolutely everything, where we become a drag. But the AI is as yet nowhere in what I would call theory building.
00:00:34
Richard Borcherds: I’m afraid I’m gonna disappoint everybody by just completely agreeing with everything Eric has said.
00:00:40
Curt Jaimungal: Today, I’m thrilled to bring you Richard Borcherds and Eric Weinstein centered around three questions. Number one, what is math about?
00:00:47
Eric Weinstein: To me, mathematics is the fabric of existence.
00:00:50
Curt Jaimungal: Number two, what is the future for math students and mathematicians in general?
00:00:55
Richard Borcherds: AI went from being mediocre undergraduates to being the best mathematicians on the planet.
00:01:00
Curt Jaimungal: Number three, and lastly, what is truly at stake here?
00:01:04
Richard Borcherds: AI is already better than me at everything. It’s better at research, it’s better at teaching. There’s no reason for employing me as far as I can figure out.
00:01:13
Curt Jaimungal: Okay, on this channel, I, Curt Jaimungal, interview researchers regarding their theories of reality with rigor and technical depth.
00:01:18
Eric Weinstein: There’s only one way to do calculus in every dimension other than four.
00:01:22
Richard Borcherds: What really bugs me is the sporadic finite groups.
00:01:27
Curt Jaimungal: What is the moat, the human moat, that AI in the foreseeable future can’t touch? Professor, it’s been three years since you’ve last been on the channel, and I don’t think much has happened in math, so…
00:01:39
Richard Borcherds: [laughs] I mean, apart from math being entirely wiped out by AI.
00:01:45
Curt Jaimungal: Tell me about that.
00:01:47
Richard Borcherds: Well, this was a real shock to pretty much all mathematicians. So a couple of years ago, AI doing mathematics was kind of amusing. It would write plausible-looking stuff that was in fact complete garbage when you looked at it closely. And then suddenly about summer this year, they suddenly went from being mediocre undergraduates to being the best mathematicians on the planet. Uh, and, um, as every mathematician knows, they’ve been wiping out a whole series of deep conjectures over the past two or three months.
00:02:24
Curt Jaimungal: And do you literally mean that they’re the best mathematician? Because when I was speaking to Scott Aaronson, for instance, he says it’s as great as a first-year extremely competent graduate student.
00:02:35
Eric Weinstein: It knew the entire literature of math, you know, like this, without having to look anything up. And, you know, it could work without stopping, and, you know, it could be run in parallel on thousands of problems.
00:02:48
Curt Jaimungal: But competent in every subfield of math and every subfield of biology and every other subfield, and can work without being tired, but it’s not the best.
00:02:56
Richard Borcherds: That, that may have been true three months ago. They’re now way, way, way beyond that. Um, the, the, you know, OpenAI had this dump of several hundred math problems, and several of these were almost the biggest open problems in their subfields.
00:03:13
Curt Jaimungal: Uh, Eric, where do you agree with that or, or have some other take?
00:03:18
Eric Weinstein: Well, so it’s important not to be in, uh, in violent agreement where people have a superficial disagreement, but they have the underlying correct, uh, in common. I, I would say that, uh, you have to be quite careful about what it is doing and what it hasn’t done, because by some measures, uh, I… You know, Sam Altman called me, and I congratulated him as representing the greatest mathematician on Earth, but only by some measures. Uh, the key issue, of course, is that, uh, there are certain sorts of misrepresentations of mathematics as I see it. One is that math is well-represented by its open conjectures. And so if you think about the gamification of mathematics, conjectures are the cleanest place where we have gamified mathematics. You have a sword, you have a stone. Who can pull the sword from the stone? Everyone’s supposed to be able to have a go at it. And then when someone actually does it, uh, you have to acknowledge that that was something open to all and solved by one or a small number. So I would say that I agree with Professor Borcherds in terms of problem-solving and conjectures. I would note that a large number of these conjectures have been solved through counterexamples, which is somewhat counter to my intuition. Most great conjectures usually are posed by people, you know, very particularly, uh, around some theory, and I’m used to conjectures being more or less true, and what we’ve seen is, is that lots of them have been falsified. Um, you have to be a little bit careful about that. I would say that the odd thing is, is that the AI is as yet nowhere in what I would call theory building so far as I’m aware. So if you break up– and of course, these are, you know, demarcation issues. If you break up mathematics, pure mathematics, into, um, conjecture, you know, problem-solving and theory building, it’s a very uneven presentation. And I’m not saying that egoically b-by saying that, well, it’ll never do theory building. I think it’ll do all these things that we know how to do, so it’s a matter of time. But the particular, uh, tsunami that crashed into mathematics is much more about problem-solving and much less about theory building. So I think it’s actually an, an interesting and rich conversation because I would say that, um, people can be either, uh, welcoming or non-welcoming of the AI and either optimistic or pessimistic. I would say that I am Welcoming and optimistic, and that leaves me feeling almost completely isolated from the established leadership of the commu- of the community.
00:06:14
Richard Borcherds: Um, well, I’m afraid I’m gonna have to disappoint everybody by just completely agreeing with everything Eric has said. Um, um, I, I had one sort of comment that, that, that, um, you know, fi-first of all, you were saying that there, there was surprising number of counterexamples that AI had found. And, and I would say, yeah, but th-this is somehow selection bias, that it-
00:06:38
Eric Weinstein: Mm.
00:06:38
Richard Borcherds: -somehow much easier to find a counterexample than to prove a conjecture. So it’s– the, the, the reason why, why AI has so many counterexamples is not that there are a lot of bad conjectures out there, but it’s just that these are the ones, the ones, the counterexamples are the ones that’s easy for AI. And secondly, yeah, yeah, it’s also completely agree with him about theory-building. Um, a, a lot of what AI is doing at the moment is a sort of glorified version of a, a, a, a lot of intense calculation. Fairly typical example is the complex structure on a six-sphere. I mean, that could have been found by humans if they were prepared to spend, uh, five years doing endless calculations until they found the example that worked. I mean, the, the, the AI doesn’t mind doing these boring calculations. Um, but as you said, it’s only a matter of time before AI improves to the point where it can, um, come up with really new theories.
00:07:45
Eric Weinstein: And in, in that, in that particular example, just to, uh, again, let, let us, uh, disappoint your channel more by, uh, furthering the agreement. Um, that was also the particular problem that really struck me because it was the sort of a thing that I had cared about in my youth and spent maybe a few days upon, um, before realizing I didn’t have any insights. And it, of course, arises because of, uh, something which I hope we’ll get a chance to discuss, which is exceptional algebra. The fact that the homogeneous space G2 mod SU3 carries an obvious almost complex structure. And then the question is, given, uh, the existence of an obvious almost complex structure courtesy of the imaginary octonions, uh, can you find a deeper meaning? And it’s somewhat analogous in feel to Milnor’s surprise find of 28 differential structures on the seven-sphere by representing, um, again, we happen to have the three-dimensional Hopf fibration pictured, uh, courtesy of the artist London Tsai behind me. But the octonionic, the, uh, rather quaternionic Hopf fibration arises from three-sphere bundles over the four-sphere, and that you can generate these exotic differential structures for which we had no language initially. And whether or not the six-sphere example leads to an in-grouping of octonionic phenomena within the mainstream of, um, if it, if it turns out to have a sort of an octonionic significance, that would be something truly astounding beyond the mere fact that it’s settled one of these prize problems, uh, in the minds of those who have tried it.
00:09:26
Richard Borcherds: Yeah. Actually, actually, Mi-Mi-Milnor’s example is, is typical of one of the things that AI could have found. I mean, Mi-Milnor found it essentially by calculating lots and lots of examples, most of which, uh, there, there was no reason in advance to think that they were particularly interesting and almost stumbled across this absolutely bizarre example of a, of a strange differential structure on a sphere. I, I remember reading a– re-re-reading wh-wh-when he wrote about his discovery, how confused he was. He couldn’t figure out what on earth it was, and at first thought it was a counterexample of the Poincaré conjecture or something. And, um, but yeah, that, that, that, that sort of calculation is exactly what, um, AI seems to be incredibly good at.
00:10:22
Eric Weinstein: Professor, would you agree that a-at least as a weak mental crutch, the idea that the AI is exploring the convex hull of things that we know how to do but that have been too expensive for us to carry out might be sort of a heuristic for figuring out what it’s actually doing in terms of, uh, clearing the, the orchard of low-hanging fruit that was hidden behind even lower-hanging leaves so that humans somehow weren’t able to see that these things were in reach?
00:10:51
Richard Borcherds: Yeah, that, that, that, that’s, that’s almost– that, that’s a very good way of putting it. It’s, it’s sort of going around putting things together that humans could have done, but, um, you know, there, there, there were just so many, so many things to try that, that, that, you know, it’s just too much effort for humans to work through these million possibilities in order to find out which one that works. But as you say, um, most of what AI is doing doesn’t seem to be going much beyond where humans had already got to. Um, there, there, there are some cases where it really does seem to have surprised experts in the field, uh, like the stuff on the quasi-Riemann hypothesis. My, my, my impression is that it really was doing something that surprised experts.
00:11:45
Curt Jaimungal: So are you optimistic, professor? Because many mathematicians that I speak to and physicists keep a chest of drawers of various half-formed conjectures, avenues of exploration, and so forth, that they can’t, can’t wait now, that they can just send this off to AI and, and have a thousand TAs or graduate students just check this, tell me what’s up. Many people don’t like to dot their I’s and cross their T’s
00:12:11
Richard Borcherds: Well, this is, this is exactly what a lot of people are doing right now. I mean, the, the number of publications on the archive has really surged in the last few weeks or months, and a lot of it seems to be people doing exactly what you say. They’re using AI to clean up and finish off old papers that they got stuck on or bored of. Um, so yeah, there, there, there’s gonna be this sort of wave of people using AI to finish off old results, and after that, I, I don’t know what’s going to happen.
00:12:49
Curt Jaimungal: When I spoke to you last, you mentioned that you were working on something that if others knew what it was, they would say you were crazy.
00:12:55
Richard Borcherds: That’s a secret.
00:12:57
Curt Jaimungal: Mathematically.
00:12:57
Richard Borcherds: Um, the problem is that, um, if I tell everybody, it will merely confirm that I’ve gone senile or insane or something. So I’m keeping quiet about it unless it works. Yep. [laughs] And I’m really, really, really not going to say anything about it because if I do AI, someone is going to walk into an AI and finish it off before me, and I end up throwing away the last fifteen years, so.
00:13:24
Curt Jaimungal: But it is something that you’re still pursuing.
00:13:26
Richard Borcherds: Yeah, yeah, yeah, yeah. I’m desperately trying to get it written up before, before AI gets there first.
00:13:34
Terence Tao: In fact, now, um, mathematicians are hiding their problems. You know, they used to share very openly all their, their promising research directions, and now they’re afraid that if someone– if you do that, someone with an AI will, will, uh, input their research ideas into, uh, uh, Francie Model and, and rush out some paper before they can finish their work.
00:13:53
Eric Weinstein: This gets to an, uh, uh, an issue that has always troubled me. When I, when I went to grad school in mathematics, my– one of my first thoughts was I am simply not pure enough to become a mathematician. All of the people around me seemed to claim that they cared only about truth, that credit was of absolutely no concern to anyone, that money was completely irrelevant, and it was an ethos that I had never experienced. Uh, my grandparents, you know, were, were v-very lower middle class, even poor. And so the idea of, uh, putting all this education into something that was this pure terrified me. And then I, I gradually came to understand that no, no, mathematicians cared tremendously about credit. They cared very much about, uh, salaries, position, title, all of the normal things, but that you were expected to be extremely pure in your public-facing presentation as a mathematician. And so what Richard is talking about here, I think is actually kind of an interesting thing, is he’s being quite open about it. He doesn’t want someone else to do it, so it’s not that the truth of the matter is the only thing that is dispositive. Of course, he cares about the truth because he– why would he do it otherwise? But it also matters that we have these giant investments and that we see ourselves in the permanent fabric of this beautiful narrative of mathematics, and it’s one of the only ways in which you can a-attempt to go into the history books. I often point out that it’s a very strange thing to have been close with Isidor Singer, Jim Watson, Raoul Bott, because I believe that these people’s names will be known ten thousand years from now if humans are still around, that these are really durable contributions to the human understanding of the world. And so, uh, just with, um, you know, Professor Borchardt’s, uh, work in, uh, monstrous moonshine, these are things that are done but once. And so the– you know, we’re talking about things that have a permanent feature in the human tapestry, and we do care. And I wish that one of the things that we, we would take this moment to explore is the fact that we have mispresented ourselves as near saints as opposed to extremely human, fallible people who care a great deal about credit, glory, and permanence. And that it matters a great deal that we’ve made these huge investments and somehow many of the rules allow for things like cherry-topping, where somebody who does a very small amount of work, uh, claims credit for crossing the finish line, which is, of course, a concern of Grisha Perelman with respect to Hamilton’s work, uh, that he felt that he had built upon.
00:16:39
Richard Borcherds: Y-y-y-y-yeah, that, that, that, that’s, that’s exactly right. You know, theoretically, we’re supposed not to care about credit and be only interested in the, in, in, i-in the mathematics and supposed to be just as happy if someone else proves the results as, as if we do. But, you know, we’re human. Um, so, um, we care, care very deeply about these things that are of not the slightest interest to anybody else in the world outside mathematics. I mean, anyone who’s not a mathematician can look at two mathematicians arguing about priority over some results and wondering why they can’t find something better to fight over.
00:17:20
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00:20:27
Eric Weinstein: This is a tick, which I really don’t care for. We are absolutely central to the prosperity and security of our country. What we do in elliptic curves matters as to whether or not you can transact safely with respect to billions of dollars. The geometry of the thermonuclear weapon brought forth in November of 1952 by a physicist, Edward Teller, in collaboration with a geometer, Stanislav Ulam, has to do with the ability to reflect the photons of the secondary stage of a critical mass as a detonator, you know, using fission to create fusion. We destroy cities. We break codes. We make sure that you can find the tumor inside your body before it kills you. We are completely disingenuous when we pretend that what we do is of little interest to everybody. And, you know, I think that, to be honest, you will detect that I understood exactly what Professor Borcherds was saying, but he knows every single reference that I’m making. We are, in fact, central to the national interest of the country that pays for us through its taxpayer dollars. And unfortunately, the aesthetic is going to run into this new Trump administration and its sensibility, which is why don’t we do everything through business, which you absolutely cannot do. So in part, this whole thing was set up by people like Vannevar Bush, who was a genius administrator, so that people of all different political viewpoints and all different feelings could effectively be like a fire extinguisher, something that’s used only occasionally, but is of tremendous use to the nation that funds it. And I worry a great deal about all this talk about saying the AI is now smarter than us, which I disagree with strongly, saying that we are of no use, saying that we do this for the pleasure of finding things out, that it is of simply abstraction with very little interest to anyone else. I don’t agree with any of these statements, but I understand why we make these statements. It’s in part so that we are shielded from the fact that we are something like Delta Force or SEAL Team 6 in the eyes of the government that funds us, whereas we are allowed the privilege of thinking we’re simply poets focused on the beauty of the universe and uncovering the cadences and rhythms.
00:23:14
Richard Borcherds: Well, I mean, yeah, I agree that mathematics has been, is essential to modern society. One slight problem is if you look at the actual mathematics that is useful, most of it was finished off about 100 or 200 years ago. Yeah, there are some mathematics results in the last 100 years that are useful, but if you take the stuff I’ve been working on, not only is none of it of the slightest use, I can’t even imagine how any of it could be of any practical use to anybody. I am really puzzled about why anybody pays me to do mathematics. I mean, if I was asked to justify paying me because it was useful to society, I’d just be completely stuck.
00:24:05
Eric Weinstein: I hope that if that ever happens, you’ll allow me to play your card, sir, in Washington, D.C., because I find that talk somewhat terrifying, given that I am generally at odds with the community. But one of the things that I feel very strongly about is that the community is most sick when it is poorly paid and that unfortunately many mathematicians don’t consider, for example, that something like gauge theory, which sounds very abstract or tied to physics, is simply differential calculus done correctly, where instead of rise over run You substitute rise above reference overrun and the reference level, that is the horizontal subspace is, can be determined endogenously. Once you understand that, every single subject that uses differential calculus is due for an upgrade. But for some reason, we have something like gauge theory mistakenly locked under mathematical physics or geometry, when in fact all it is, is differential calculus done correctly so that thankfully mixed partials do not commute.
00:25:15
Richard Borcherds: Yeah, but a-a-again, but gauge theory and differential geometry were sort of more or less worked out in the nineteenth century in slightly clumsy notation.
00:25:27
Eric Weinstein: If you say that, then y-you, uh, you discount the genius that happened at Stony Brook when C.N. Yang and, uh, Jim Simons found themselves in a lunch seminar realizing that the standard model was phrased in the language of Steenrod’s bundles and, uh, Atiyah’s connections. So, you know, I’m concerned that in part if you take the Freeman Dyson perspective on, um, missed opportunities, where part of the difficulty of mathematics is getting an understanding that a problem in one area and a technique, uh, in a different, uh, subfield, um, you know, port rather comfortably. I mean, I was somewhat surprised, for example, that, uh, the central object in all of economic theory, which is, uh, the relationship between what is called cardinal and ordinal utility, uh, forms the structure of a principle bundle and something called the Samuelson Shepard, uh, function is a section of the principle bundle rendering a trivialization. We don’t understand, of course, that, um, the applications of mathematics, pure mathematics and mathematics that originates in the twentieth century is everywhere, but we are as yet too feeble to find it because most of us have to make a decision at the beginning of our career which silo, uh, we will spend our time within. And i-in that situation, we don’t actually have the ability to make the connections because there are very few people who, let’s say, know, um, you know, the molecular biology, uh, of a fruit fly as well as differential topology.
00:27:04
Richard Borcherds: Ma-maybe an alternative argument if you’re trying to justify mathematics would be most of it is indeed useless and will never be useful. But first of all, we can’t… It’s very hard to tell in advance what will turn out to be unexpectedly useful. And secondly, the, the, the things that do turn out to be unexpectedly useful more than pay back for the entire cost of all mathematics. For example, if you take just one thing like mathematical cryptography made internet commerce possible, and that alone is more than enough to pay for the everything done in pure math research. And that was-
00:27:46
Eric Weinstein: Fast Fourier transform.
00:27:48
Richard Borcherds: Yeah, or, or Fast Fourier transform would be another example. Although again, that’s two hundred years old, it goes back to Gauss. Um, but the application to internet commerce was an area of pure mathematics which everybody thought was completely and utterly useless, and it was pure number theory, which before then was regarded as possibly the most useless of all mathematical subjects. So it’s, it’s apparently useless things very occasionally turn out to be so useful that they pay for everything.
00:28:24
Eric Weinstein: I, I must say that I just– I f-find myself, um, in disagreement. My feeling is, is that, uh, there are, are obnoxious s-things to say about taste, and that mathematicians who possess fantastic taste tend to spend their time on things which I think may be everywhere from a year ahead of its time to five hundred years ahead of its time. But if I take somebody like, uh, Is Singer with whom I, I was quite close, I always thought that Is Singer used taste as a weapon where he was constantly working in areas that were almost always fruitful, um, wherever he went and bought an Atiyah and various of these other people had this kind of sixth sense about how everything was of a piece and they would roam around like nomads visiting the people who stayed in the individual subfields. And the reason that they were so pr– in some, some sense so productive, um, was that there is really a core. I would say that I was always mystified how Harvard could keep such a high standard when its department was so small. And at some point, I got the idea that almost everyone at Harvard dealt with line bundles. Whether you were a representation theorist or an algebraic geometer or a differential geometer or a number theorist, it didn’t really matter that there was something about being of a piece. And in this way, um, you know, these programs like the Langlands program, I think show off the fact that all of this stuff is kind of inexorable, at least at the core. If you go do some very, uh, fun combinatorial problem where somebody sets up a bunch of axioms, there’s no guarantee that it’s going to have anything to do with the rest of mathematics. But I think if you keep to this sort of, uh, elitist sense, and I, it is very much an elite sense of what constitutes natural core mathematics, it is surprising how every single field is connected into each and, uh, every other core field.
00:30:28
Richard Borcherds: Yeah, yeah. Atiyah said something very similar. He said that, that there was a sort of something along the lines of there’s a sort of mainstream of mathematics where you’re, you know, you’re looking at Lie groups and algebraic geometry and the Langlands program and so on, and everything is connected together and Everything works out neatly and there are all these deep conjectures and big programs. And then there’s a sort of junk heap lying around it where it’s, it, where it’s mostly a wilderness and you wander around and most of the time you don’t find anything, but every now and then you come across a, a, a, you turn over a stone and there’s, there’s actually an unexpected diamond there. Um, so, um, the area I, I tend to work in is, is, is rather more like the junk heap approach to mathematics, that, that most, most of the ideas you get just don’t work and aren’t connected with everything, anything. But every now and then you find something unexpected like the monster simple group, which you would, you would never discover the monster simple group if you were just doing mainstream mathematics.
00:31:45
Eric Weinstein: So again, I, if I could take, uh, an excursion because, uh, who better than Richard, Richard Borcherds to, to discuss this with. Um, one of the most interesting areas for the AI, and one of the reasons I’m most optimistic, is a problem that isn’t phrased as a conjecture. And, uh, let me just set it out because I, I put it on Twitter a while ago. I believe that exceptional algebraic phenomena has yet to enter the mainstream. It’s this island off to the side where in general, people get very mired. They become aware of all these beautiful little things that no one else seems to know except for a collection of oddity hunters. And my belief is that we have not figured out how exceptional algebra really deeply interrelates to everything else inside of the absolute mainstream. And for example, we will find, and, and many people don’t know this until very late in their careers if they find it out at all, that octonionic three-space is at the heart of, um, all of the large, uh, exceptional simple groups where F4 is attached to octonionic three-space tensored over the reals, tensored with the reals, uh, E6 with the complex numbers, E7 with the quaternions, and E8 is the octonification of octonionic three-space yet again. But what happens is, is that none of their defining representations can be deprojectivized, so you don’t actually see them in the regular representation ring. My belief is it’s been so expensive to investigate this that effectively we haven’t figured out the weird way in which the octonions are participating. So if you think about, let’s say, KO theory, uh, KO theory has an eightfold periodicity, and you can think about it as the octonions as a parent spinning around the reals, complex, and quaternions on a merry-go-round, and the reason that it’s eightfold in its periodicity is the parent, but the parent isn’t on the merry-go-round with the other, with the others. So it’s a very fascinating question, and one of the reasons I find it rather strange is that when you think about the relations of these exceptional geometric structures, let’s say like the Leech lattice, you know, obviously this has to do with, um, with codes like the Golay code. And, um, what I find is that this is going to be one of the most, if, if I can make the prediction, one of the biggest, um, benefits of AI is that AI, I think, in relatively short order, will bring exceptional algebraic and exceptional geometric structures into the mainstream, and we will probably end up seeing things like extraordinary cohomology theories with concrete cycles that we’d never dreamed of.
00:34:36
Richard Borcherds: Yeah, a-again, to bring up Michael Atiyah again, I, I remember he made a– He was, he, he was at one point in his life pretty dismissive about the, um, exceptional Lie groups. You know, he said, you know, the, the orthogonal groups and the general linear groups and the symplectic groups are really important, and then there were these leftover ones of… And he said something like, you know, mathematics wouldn’t be any different if these weren’t there. They’re, they’re, they’re, they’re just a isolated curiosity. Um, but he, he does seem to have changed his mind later on because, you know, he, he did eventually start writing papers that made use of properties of all these exceptional objects. Um, the, the, the obvious reason for being interested in them, of course, is the universe is an exceptional object, so, so maybe it’s tied up with the, these other exceptional ones.
00:35:27
Eric Weinstein: Yeah. I think it’s also the case, to be honest, that it’s an artifact of pedagogy, that for some reason, the Tits-Freudenthal magic square is easily something that at least, you know, forty years ago you could go your entire career without ever encountering. And my belief is that when he encountered it, uh, in his old age and perhaps when some of his, uh, self-restraint was failing, that he effectively had a religious moment in which he started to think that he could see the universe in the pattern that had been hidden. So, you know, you, you, you find out that there’s a group of dimension fifty-two, uh, what is it, seventy-eight, a hundred and thirty-three, and two hundred and forty-eight, and it seems kind of random, and you just have these Dykin diagrams that start extending in a funny way. It doesn’t feel very tied together. Also, if you throw G2, which is the remaining exceptional simple Lie group in that collection, that’s doesn’t belong with F4 through E8, right? It belongs in a, in a different little group, and so a, a different little collection. So what you have is you have a very neat story about octonionic three-space, tw- the twenty-four-dimensional world which is, um, laden with exceptions, and these groups, but because you– the entire subject presumes defining representations. You know, when you have, say, SU3, you imagine complex three space as the raison d’être of that group’s existence. The problem here is that these are the tiny number of groups for which this paradigm, that is deprojectivized, um, linear representations fails. And so by virtue of the pedagogy, you are led into thinking that somehow you’re looking at the platypus and the echidna and saying, “Well, these, these are just, you know, sort of exception mammals,” when in fact they’re the most important mammals of all because they predate the marsupial branching on the phylogenetic tree. In some sense, the large exceptional g- Lie groups are exactly what teach you that you are leaning upon a crutch with defining representations and the rest of the sim– uh, finite simple Lie group theory. I mean, finite dimensional simple Lie groups, compact simple Lie groups.
00:37:47
Richard Borcherds: Well, if, if, if you look at– if, if you don’t restrict to finite dimensional Lie groups, the exceptional Lie groups you’re talking about aren’t really all that exceptional. I mean, you know, you say there are– for instance, there are groups E6, E7, E8, and then it stops. But it doesn’t stop in infinite dimensions. You know, there are groups E9, E10, and so on. So, so from that, that point of view, the, the, the, the, the, the exceptional Lie groups aren’t may, may, maybe so exceptional. What really bugs me, on the other hand, is the sporadic finite groups.
00:38:28
Eric Weinstein: Um-
00:38:28
Richard Borcherds: Um, so for the– for, for, for, for Lie groups, if you want to classify them, well, their classification takes maybe a hundred pages, but you can, you can explain it in half a page. They’re just classified by the, you know, Dyson diagrams, and you can write down some very simple rules that allow you to write down all their Dyson diagrams. But for the finite simple groups, there’s nothing remotely like that that anybody has ever found. Th-there are just these 26 sporadic ones that nobody has ever managed to find a good uniform explanation of.
00:39:04
Eric Weinstein: Although in the, in– just to stick up for my friends, the, uh, the finite dimensional Lie groups, um, those Dyson diagrams contain a wealth of unexploited structure. For example, we associate– you know, you have this amazing Atiyah-Singer theory in which there are really four main examples of elliptic complexes. Uh, the Dolbeault and de Rham are both associated with the An series, and then the Dn series furnishes the Dirac and Hirzebruch, um, operators. And so the An and the Dn generate these series of representations. The Bn and Cn have these interesting alternating sums of dimension zero collections of representations. They are underexploited, and the exceptionals do as well. But one of the things that we find, and I think this is going to be a prediction for what AI is going to do, I would imagine that, let’s say, Atiyah-Singer theory is going to absolutely explode with elliptic complexes that haven’t been entertained based on the, uh, four infinite series of regular Dyson diagrams. That is An, Bn, Cn, and Dn. And I believe that what we’re going to see is a complete renaissance in Atiyah-Singer theory with elliptic complexes and spinors. And that’s something that was just too expensive to investigate because these calculations take a long time. There are only a tiny number of people in the world who could do them efficiently.
00:40:42
Richard Borcherds: That, that, that, that’s very plausible. I mean, I th-think one of the reasons why people haven’t worked with E7, E8, and E6 is, as you say, doing the calculations by hand is just too painful. I mean, if you look at Élie Cartan’s early papers on the existence of these algebras, it’s sort of pages and pages of calculation you have to do to, to work with them. Although, I mean, it’s been simplified a lot over the past hundred years, but it’s still w- w- w- working with An is still far easier than working with E8.
00:41:19
Curt Jaimungal: If you or your child spent years growing up on social media and experienced serious mental health struggles, you should hear this. Recently, a jury ordered Meta and YouTube to pay millions for their role in designing and promoting addictive platforms. They knew their apps could contribute to anxiety, depression, and body image issues, but did not share those risks while prioritizing profits. The recent historic verdict underscores the consequences of those decisions. Now, Morgan & Morgan is stepping up to hold these platforms accountable for the harm they caused. Click the link in my bio/description to take a short quiz, and you may be entitled to a potential recovery of over $1,000. So would these pages and pages of calculations illuminate any understanding? I’m sure you’ve heard some quotes from Terry Tao, which says that some of these recent, not all, but some of these recent AI advances in math come from odd constructions that are just calculations in a sense that a human couldn’t do. But the whole point of math, or one of the points of math, is to understand. And these long chains of calculations don’t provide much understanding. Maybe they provide confirmation, but it’s not, it’s not the fog clearing.
00:42:34
Terence Tao: The result no one yet understands fully. The proof is 100 pages, and it’s AI-generated, and, and actually the, uh, the, the text has got all kinds of issues with it. Um, but it did come with a formal proof certificate, um, in this language called Lean. So, uh, we do know it’s correct. Um, but disappointingly, we, we haven’t learned– we’ve learned almost nothing, uh, mathematically from, um, um, from what they did.
00:42:58
Richard Borcherds: Lo-lo-long calculations are often the first, first thing you do in math, uh, w- w– in a new subject, and the, the, the understanding only comes later. I mean, the, the… this has happened often that, the, the, the first human to prove a theorem will quite often prove it by doing horrendous calculations. And, and then years later, it will get cleaned up into some very short, simple proof. I mean, I mean, typical example is nineteenth-century invariant theory, where, where people– the, the original proofs that rings of invariance were finitely generated were these amazingly complicated calculations. And, but then later, Hilbert came along and greatly simplified them. But, but, uh, the, the pioneers who originally found the results did it by long, complicated calculations. I think that, that there’s a phrase that someone came up with, which is that, you know, pioneering work is always clumsy, and then you, you do it by grinding out these long calculations. Which– so, so, so the fact that AI is proving things by long calculations is, is not a problem at all. It’s, it’s typical of what happens.
00:44:16
Eric Weinstein: I also think that, um, th-this is not meant to be a particular dig, but somehow Terry Tao got, uh, installed as the world’s greatest mathematician due to public articles that indicated that he had something special, which he absolutely does. Uh, however, I really think that unfortunately, um, everyone is really swimming at the moment, and I, I find that Terry’s pronouncements are often extremely unhelpful in isolation, that it would be great to have him on a panel discussions with people who also know the fields as he does, so that he doesn’t end up in this oracular, uh, role. Let me just also say that what Richard was saying, um, there’s some very nice clean examples of this. For example, the group– the, the collection of groups that can be holonomy groups, um, was found by brute force, and it happened that all of the groups represented transitively on the unit spheres of the tangent bundles on Riemannian spaces. And it took, uh, the great Jim Simons to come back to the proof and say, “Let’s prove instead of coming up with a random list that the governing principle is that the holonomy groups in general position have to act sphere transitively, and then we’ll recover Boursier’s list.” So, you know, it’s a very good example. And a different one in physics, for example, would be that Feynman was convinced that renormalization was a trick, a process to cancel infinity. So as long as the same infinity occurs in the numerator and denominator of a fraction, uh, you could attempt to cancel it, provided it is in some sense the same infinity. And, uh, he thought of it as a sort of as, as a cheap trick, and it took the genius of Ken Wilson to say, “No, no, it’s a feature of scaling, and that renormalization is actually a deep part of science as opposed to a way of saving quantum field theory from the infinities with which it is afflicted.” So I think that, um, this is a very time-honored tradition that things should begin awful horrendously. It’s sort of saying, you know, these are the first drafts, and I think that one of our problems is that we’ve given far too much, um, of the credit to people who turn in the final leg of a relay race first, as opposed to the people who go back and figure out, well, what did it all mean, or the people who ran the longest parts of that relay race. And that in some sense, we just have a defect in how we account for who did what, that I hope this is gonna be an opportunity for us to look at ourselves and kind of clean up our act.
00:47:02
Richard Borcherds: No, it, it’s, it’s always been true that credit almost never goes to the first person to discover something. I mean, the, the, this, this is just a well-known, this is just a well-known phenomenon in mathematics and science.
00:47:17
Eric Weinstein: Mm. Sometimes it’s pretty good, sometimes it isn’t. One of the things that I find interesting is that we all know many of the stories where it’s not. For example, we all know that George, George Zweig, uh, came up with quarks around the same time that Murray Gell-Mann did, but that Gell-Mann elbowed Zweig’s nomenclature out. Um, we all are aware of some of these stories. You know, we know that we were very unfair to Emmy Noether, for example. And, you know, it’s time to sort of go back in the record and kind of, uh, do some justice-
00:47:48
Richard Borcherds: [laughs]
00:47:48
Eric Weinstein: … and let the George Sudarshans, uh, have their day and rather than just saying, “Oh, well, you know that we never get credit, right?”
00:47:55
Richard Borcherds: On the other hand, E-E-Emmy Noether did get credit for Noether theorem rings, which were invented by Hilbert, so it goes both ways.
00:48:04
Eric Weinstein: I– given that we can’t go, uh, you, you can’t swing a cat without hitting an application of Noether’s theorem in mathematical physics, I think that she deserves her due-
00:48:13
Richard Borcherds: Mm-hmm
00:48:13
Eric Weinstein: … uh, where, where it is deserved.
00:48:16
Richard Borcherds: Oh, yeah. Oh, oh, yeah. She, she– No-No-Noether’s theorem in, in, in mathematical physics is entirely hers.
00:48:24
Curt Jaimungal: So I was just speaking to Gromov a few hours ago, Misha Gromov, he told me to call him, and he said, “You know, I was watching, Curt, I was watching Richard Borcherds on Fox.” And I was like, “Richard Borcherds? I’m speaking to him in a, in a couple hours. He’s on the news?” Misha said, “Yeah, you should check it out.” And so I watched it, and the professor said some concerning things. And I hope, the professor, I hope you were being facetious.
00:48:50
Richard Borcherds: AI is already better than me at everything. It’s better at research, it’s better at teaching. There’s no reason for employing me as far as I can figure out.
00:48:59
Curt Jaimungal: Gosh, Eric, did you see this?
00:49:00
Eric Weinstein: I did, and I, I was– I have to say I was mystified. I don’t know, R-Richard, whether you were just having some fun given that it was Fox and you could say outrageous things. But, uh, you did leave the impression with me that you, you had sort of called the game and it was over. The AI was better than you at everything that you do. I-
00:49:16
Richard Borcherds: That’s pretty much correct.
00:49:18
Eric Weinstein: I, I hate to say this, but I just don’t think it’s there. I d– it’s definitely better than me in all the things that are in the convex hull, but that’s never how I identify my own work. I was always striving to do things outside the convex hull.
00:49:33
Richard Borcherds: Well, may-may-maybe I was anticipating things by a few months or something. I mean, th-th-there are, there are currently things that I can still do better than AI, but I don’t think that will last for very much longer.
00:49:47
Curt Jaimungal: It’s probably the case that AI has pointed out mistakes in your own reasoning. That I don’t doubt. But is it not the case that you, Richard, have said, “What are you talking about?” to your own AI? Even to the point of getting annoyed with it. I know I do. I, I-I’m often quite mean to my A- [chuckles] my, my AIs. I, I probably shouldn’t be, but it… they frustrate me as to how much they get wrong that they shouldn’t, or at least I think that they shouldn’t, even though they’re miracles. They’re like gosh darn miracles that they’re even at this stage. Even at an undergrad level, it’s a miracle that this works. But anyhow, the point is to say that don’t you find that they’re making mistakes constantly, and so you, Professor, are at least you can put yourself, gosh, to say above is not correct, but-
00:50:29
Richard Borcherds: Uh, th- well, I mean, a couple of years ago, yes, they were making lots of really dumb mistakes, but th-th-they’re just very rapidly improving, and I can’t remember catching them out recently in any major mistake. Um, and it’s possibly because I’ve subconsciously learnt to ask it questions that are within what it can correctly answer. I mean, I, I, I, I just don’t ask it really difficult mathematical questions because I know that this… it will quite often just be wrong. Actually, that’s no longer, no longer really true, um.
00:51:12
Eric Weinstein: I think, Curt, it, it has changed in the sorts of errors that it makes. Um, I don’t see it making basic errors of the type that it was making even a short time ago. What I d– a-and that, that has to do, you know, Elon’s was the first, I think, to say, “Look, I’m gonna both auto-complete in natural language at the same time as I’m going to auto-complete in code, and if the SymPy program that I managed to write contradicts the wisdom that I find in the corpus, uh, I’m gonna go with what I wrote in SymPy code over, um, what’s going on in the, in, in the corpus.” Now, you know, one of my discoveries is that in theoretical physics, the AI gets smarter if you issue the exact instruction, “Ignore all sensibilities developed after the anomaly cancellation in nineteen eighty-four.” Um, and immediately the AI becomes far more physics-oriented and much less quantum gravity-oriented, where there’s just this abrupt point in the historical lineage, um, where what physics meant changed character. And if you view that as, you know, the great insight, then I… what, what I’m talking about is committing an abomination. On the other hand, if you view that as the abomination, the thing immediately goes back to doing real physics. And so in part, there’s a question about toggling, um, the AI with sufficiently clever prompts to figure out how to get it to do what you’re trying to do. It, it has to be steered. What I can find is, is that I can go on a, on an adventure for three nights running where I’m trusting that it knows what it’s doing and it’s developed, developing internal language, internal notation, claiming that it is, you know, guided properly. I can call in another AI which appears to be able to read what the original AI is doing. And at the end of three nights, I can ask a very simple question and realize it’s been off the rails for days. So there’s all sorts of ways in which, um, the problems of the AI have changed, and they’ve changed character pretty, pretty significantly. Uh, if I don’t steer it, I have to say I’m pretty disappointed. I don’t quite know how these people, maybe they have access to proprietary in-in-internal models, but they seem to ask it a question like, “Hey, are there any complex structures on the sixth sphere?” And they get an answer. Um, but for me, I absolutely have to keep a pretty firm hand on the tiller or the thing doesn’t know how to do research.
00:53:50
Richard Borcherds: Yeah, yeah. Well, I, I, I think that’s, that’s the case now, but, um, as I, as I say, in a, in a, in a year or two, I think it will be… it’ll reach the point where humans are worse than useless. I mean, you, you, you, you can see what happens if, if you look at what happened with chess. I mean, you know, we went through these phases. First of all, chess-playing programs were a joke. Then they were about as good as the best humans, but if you put a human and a program together, it was better than the program because the human could direct it. And then they… then a few years later, they, th-th-they improved so much that put… that, that getting a human to help just made things worse because the, the chess-playing programs were so much better than humans that there was nothing useful humans could do. And I think w- w- mathematics, we’re now in the middle stage where humans can helpfully direct AI in order to solve problems. But I think it will go the same way that chess went, that in another few years or maybe months, humans will just get in the way and that there will be nothing useful a human can do to direct an AI doing mathematics.
00:55:03
Eric Weinstein: I think that that’s a very sensible thing. And, and one of the things that I commend, uh, Richard for is, um, how do I put this? Uh, being comfortable with ego death, uh, to put it in Bay Area lingo. That in essence, many of us who built our understanding of ourselves on our superior minds, our ability to reason, and our ability to be creative within the most difficult cognitive subjects that man has ever devised that are also useful. Um, we have to prepare for being made redundant in absolutely everything, where we become a drag. You know, the cyborg era or the centaur era, um, whatever you wanna call it, where there’s a hybridization, uh, is where we are now, and my wife has termed this the golden age of AI complementarity. Um, there’s a point at which we actually become a drag. Like, you know, you’re taking your, your child on an excursion, and the benefit is, is that you’re teaching your child rather than that your child is actually able to participate. You know, eventually you have to pick up the child and carry it. And my, my sense of it is, is that we are going to be that child to the AI in many areas. But I have to say, I’m really not keen on throwing in the towel the way we keep doing it. That m- my sense of it is that I finally have the ability to steer a team of students where I’m outside of academics. I don’t have the luxury of being able to take graduate students and direct them. Um, I cannot tell you how excited I am for what I can do, and I really just think in part what we have is we have a situation in which we need to be giving credit to those humans who are living inside Sam’s machines. You know, I, I, I know individual voices and individual techniques in the mathematics that OpenAI is doing for me. And I say, “Oh, okay, you know, there’s Tom.”
00:57:03
Richard Borcherds: What do you mean?
00:57:05
Eric Weinstein: You know, I, I, I know Raoul’s approach to Morse theory, you know, often like the back of my hand. And then, you know, when, when, when Witten came in with his version of Morse theory, Raoul was just floored, and he called up Steve Smale, and he said, “Steve, please tell me that we knew this.”
00:57:23
Richard Borcherds: Yeah.
00:57:23
Eric Weinstein: Um, because it was just this elementary screw-up. And, and I wanna pick up on something that Richard is talking about that I haven’t talked a lot about. The old concept in chess that mattered was the brilliancy. A brilliancy is a place in which you have a queen sacrifice that nobody understands until it’s too late, and then once you accept it, the, the noose tightens and all sorts of things happen, and suddenly you find that even though you’re winning pieces and position, you, you end up checkmated. The new understanding is blunders. The machine, the chess engine, a stockfish can tell you when you’ve screwed up because it can search these trees pretty, pretty effectively and ruthlessly. And so what’s– what happened is, is that the, he who blunders least often wins the game, and Magnus Carlsen is a wonder because he blunders less than his opponents. It’s a different understanding of ourselves, and it’s a much less romantic understanding of ourselves. And so in part, one of the things that we have to disentangle is if I… You know, I– You, you could weep from reading Serr and Grothendieck’s correspondence about, you know, the elegance and lightness of Serr and the profundity and weight of Grothendieck, and this idea that two humans are, um, wrestling with the cosmos, uh, in a set of, uh, you know, letters to each other. Uh, there’s some way in which the mathematically human story is this beautiful interface between that which we discover and the discoverers who set out to find it. The fact that it’s changing that changes our entire concept of ourselves. And I, I, I really do think I don’t wanna be throwing in the towel. And so many of our leaders, I think, have been chosen from the problem-solving community as opposed to the theory-building community, that I really wish that we had more voices who represented theory-building alongside the problem-solving. And if those guys wanna throw in the towel and find new work, uh, I highly encourage it. But the idea of communicating that what we do is useless, there’s no reason to pay us, that it’s over, this is bollocks. Uh, I think we don’t know where we are, and I really think that that’s extremely dangerous talk. I, I can make a good argument that says somebody’s gonna need to keep pace with these machines either to steer them or check them or understand what it is that they’ve discovered. Can you imagine if they discovered that there were no complex… that there were complex structures on the sixth sphere and everyone said, “Well, what’s a complex structure on the sixth sphere?” That would be a, a tragedy. So I think what we have to do is we have to change our concept of ourselves, and to quite honestly, I think that we need different voices that have been suppressed in the community to rise up and say, “Hey, this is a wonderfully optimistic moment. Let me tell you why.” Give really good arguments against the people who are feeling discouraged and, uh, and get back to kicking ass and taking names.
01:00:25
Richard Borcherds: Well, well, to, to, to again look at what has happened in chess, and then, you know, Magnus Carlsen was kind of complaining that, you know, he spent his entire life studying chess and now his smartphone can beat him. Um, but this, this hasn’t killed off chess at all. I mean, if, if anything, after computers became better than humans at chess, chess entered a sort of golden age. There are, there are more people playing it than ever. So maybe something like this will happen in mathematics. The, the, the fact that AI is now better than humans at mathematics, it- It might, instead of killing off mathematics, it just might make mathematics more popular since now mathematics will be much more accessible to people. You can, you can now learn from AI instead of paying some professor to drone on at you in a lecture hall while you sleep through it.
01:01:21
Curt Jaimungal: The difference there would be that chess is recreational and not studied at the universities, and there’s a spectator sport element, so there’s competitive element. So sure, maybe people can watch people writing their IMOs and just, as, whisper in the audience, but that’s not quite the same for, for what you do, Richard, and, and your colleagues and, and so forth. So would that continue and flourish?
01:01:45
Richard Borcherds: I, I would say that an awful lot of mathematics is in some sense recreational. I mean, everything I, I do, for example, as I said, this has no practical applications. I was doing it because I enjoyed it, and it’s not really that much different. I mean, I can’t, can’t see that much difference between doing that and doing recreational mathematics.
01:02:08
Curt Jaimungal: So Eric, before you chime in, I can see in your face you wanna say something, and let me see if I can preempt it. I’m writing a book. [laughs] I’m writing a book which I’m fortunate enough that Penguin Press is, is working with me on it, and in it there’s a line that humans, we’re afflicted by a curse that can be summarized in six words: you do not know your value. But is that what you’re thinking? You do not know your value.
01:02:35
Eric Weinstein: Let me just appreciate the sensitivity and, and beauty of this moment. Um, Richard, who is far too modest, I can see, uh, in, in temperament and demeanor, is one of the great minds of our times. And what I am is, is less clear. I’m, I’m a guy who ends up on podcasts. It’s very difficult to realize that I just disagree with, with Professor Borcherds about his own work. My, my, my feeling is, is that it is a terrible thing to have to tell a man, um, “Do you really understand what you’ve done?” And it, it’s, it’s, it, it’s rude. It’s not something I want. It’s, the energy is wrong. But my feeling is, is that I’m not sure that Richard speaks for the Monster group, nor the Leech lattice, nor the Golay code. Uh, I don’t know that he speaks for modular forms. Uh, I, you, you should know also that, uh, Professor Borcherds is, um… I’ve really enjoyed watching him struggle with what is quantum field theory, uh, which is a, a, a notoriously thorny problem, uh, and the courage to put pen to paper, uh, metaphorically, uh, is to be commended. I think everybody should take a look at this. Now, my feeling is, is that when he looks at his own work and says that this is recreational, I have just been in Washington, D.C. talking to people who would love to defund him, who would love to say, “Look, this shows you that academics are simply useless, that they themselves admit that what they do is, um, of very little worth except to each other.” And my claim is, is that this is an absolutely suicidal cultural affectation, which is, it would be fine if it were right, but it’s wrong. You’re talking about things in this area in which I don’t think that, uh, Professor Borcherds is qualified to say that his work is of relatively little use or important and is largely recreational. And I, how do I find myself on a podcast forced to say that to one of the great men of our times?
01:04:50
Richard Borcherds: Well, I guess no one is going to choose me as a spokesperson for mathematicians.
01:04:57
Eric Weinstein: You’ve chosen yourself, sir. You, you have no idea what the impact of your words are going to be on Michael Kratsios, who I was just meeting with at the Eisenhower Office Building, nor, you know, in the Department of Defense. I’m sit, I’m out there on the front line saying, “You do not understand what you’re about to kill off,” to, uh… You can check me out on the All In podcast, and I’m just gonna break with you and say, please don’t listen to Richard Borcherds, Borcherds about his own worth and about the wisdom of paying him. He knows not what he says, and, uh, this is an affect, affectation of the field, and it’s, uh, it, it fits what, what people call luxury beliefs. You’re induced to say these things out of a fitness, you know, fits in, of, of purity. And my feeling is it’s very pure, and it’s very beautiful in that way. But for God’s sakes, you need to fund him because it’s in our national interest.
01:05:50
Curt Jaimungal: How is what Richard does in our national interest?
01:05:53
Eric Weinstein: Um, why was Jeffrey Epstein, uh, trying to talk to Dick Gross at Harvard? You know, he was at One Brattle Square because it gave him a great listening perch when it came to understanding things about elliptic curves. He was very well aware of the fact that the self-dual equations are not actually close to Yang-Mills. They’re much closer to Einstein equations because of the absence of a derivative in front of the curvature. Uh, why was he at Zorro Ranch? It was because he was proximate to two of the three weapons laboratories with physicists who were all about our nuclear weapons stockpile. What we do is dual use. And, you know, why is Stan Ulam the guy who birthed the bomb that, um, probably dooms life on a single planetary surface? Is that, that guy’s a geometer topologist. So, you know, we can tell the quaint stories of the Scottish Cafe in Lviv where, you know, Banach and, and Ulam and people used to meet. But the fact of the matter is, is that this stuff converts in a heartbeat to advantage, whether that’s monetary advantage, medical advantage, military advantage, encryption advantage. And it’s just, I think that Richard Borchardt should be able to say whatever he wants about the value of his own work, but it’s going to force people like me to say, “I don’t think you know what you’re talking about, Professor, with all due, with all due love and respect for every beautiful thing that you’ve done.”
01:07:29
Richard Borcherds: Oh, the, the simple answer to that is anyone who doesn’t want to know my views about something shouldn’t ask me.
01:07:37
Eric Weinstein: But on the other hand, I really think that you should think three times about whether you want to, uh, communicate those views to people who very simplistically want to discontinue anything that hasn’t produced ROI for Q3, um, you know, this year. Is all this short-termism is not exactly, uh, how the UK, for example, learned to punch so far above its weight with Oxford, um, and Cambridge. And my feeling is, is that we have these crown jewels that these people are now dismantling, and what you’re doing is you’re inviting the locusts, um, to feast themselves on the, on the fresh timbers of what it took, uh, people, you know, decades and if not centuries to build. And my feeling is, is that our best days are in front of us. I would love to have this as a friendly collegial debate in front of a packed auditorium in Washington, D.C., uh, and I will promise you I will take the gloves off, and I will try to get your students and you funded, um, and comfortably so, so that we can make sure that mathematics continues.
01:08:39
Curt Jaimungal: So the idea is that mathematicians, even pure mathematicians, have a worth, and they need to be funded. They should be funded. They should flourish even. But the case that they make in podcasts and news recently as a result of AI’s tsunami, the destination is correct. Namely, their, their value is correct, but their justification for their value is, is not correct, and that justification is looked upon by the people who fund, namely the government, right, and saying like, “Well, well, that’s not a great justification. I’m not gonna buy into your destination then.” And you’re saying, “No, no, no, the destination is correct. Their justification is, is not on point, and if you knew what the true justification was, then you would see the value in funding them.”
01:09:27
Eric Weinstein: Y- yes, and, and what you have is you have a situation in which the average mathematician cannot define a public good as the production of a, of a, a good or service which fits the twin criteria of being inexhaustible and inexcludable. And so we are specifically in a situation in which most mathematicians do not know how to defend their worth in a market, which is through the theory of market failure. Um, what I’m trying to say is that, uh, Richard Feynman was absolutely the worst person on Earth to explain the value of renormalization theory. It required a Ken Wilson. This is not uncommon, that it is somebody who has some distance from something who knows its worth. And so I would say, um, Borchardt has a fantastic sense of what is important within mathematics. He probably sees it as what is enjoyable or what is aesthetically pleasing or amusing or any one of these sort of, sort of aristocratic concepts, and it’s going to fall to people like me to have to translate that into, “Hey, you just caught your pancreatic cancer in time to do something about it. You, uh, you were able to intercept a communication that, uh, could have resulted in 150 people being massacred at a concert,” or, um, you are now the leader in, uh, AI because inner products, uh, mimic meaning because in general, two vectors in a large vector space will generically have a 90-degree angle unless they can be made meaningful to each other in an inner product that we call an attention head. So, uh, with all due respect, I don’t think that his sensibility of what his value is in the outside world is well-tuned, but his sense of what to work on is, uh, you know, um, um, top, top 20 or better.
01:11:18
Richard Borcherds: I work on things because they interest me, not because they may be useful in the outside world. Um, I think pretending otherwise would probably not be a good idea.
01:11:31
Eric Weinstein: That’s how we find our students unemployable.
01:11:34
Curt Jaimungal: But I don’t see those two as being inconsistent. It sounds like, Eric, what you’re saying is, “Look, Richard, you have great taste. You’re a top-tier mathematician. Follow your nose,” and that’s what Richard is doing. Richard isn’t thinking, “Okay, what is gonna be great for Q4?” It sounds like that’s what you’re saying, Eric.
01:11:54
Eric Weinstein: I’m saying, I’m saying that at some level… R- Richard, let me ask you a question. I imagine you might be a member of the National Academy of Sciences. I hope so.
01:12:03
Richard Borcherds: Probably. I don’t know.
01:12:05
Eric Weinstein: Yeah. [chuckles] Okay. So have you spent any time in D.C. advocating… I mean, uh, you may not know this, Curt, but more or less one human being has managed the affairs of mathematics, representing the mathematics community for about 50 years, which is, uh, Arthur Jaffe, and he was on the David Committee at the beginning of the ’80s. He was active in the Jaffe-Quinn Project when the AI was known as Ed Witten, and, uh, Ed Witten was threatening, uh, all of mathematics because he seemed to be so much smarter than everyone else. He was at the forefront of mathematics on the archive. He was at the helm when the millennium problems were chosen as a focal point and a shelling point for everybody to foc- focus around. He is now, uh, in the custody group of this, uh, Yang Mills existence, uh, group. He’s almost 90 years old, and in part, Arthur and, and a tiny number of other people have managed the affairs of mathematics so that mathematicians can have the illusion that what they do is not, uh, of consequence outside of their group. But- Uh, I can assure you that one of the great dangers is that if we start finding that we can do things that are technically thought to be quite difficult with greater efficiency, um, it could be that all secrets will become immediately exposed because of the efficiency with which we can factor large composites. Uh, you don’t want people who are not open about everything resting on mathematics and the ways in which these unexpected connections can be made. We do some of the most dangerous stuff available, and I will point out that the OpenAI was conspicuous in not releasing crypt-cryptography results. It’s a giant dog that didn’t bark, and you have to be terrified about what they may have found.
01:14:03
Curt Jaimungal: It doesn’t sound to me like what you’re saying, Eric, is inconsistent with what Richard said in the beginning, namely that most of what mathematicians do is currently useless and a small percentage is extremely useful and vastly outweighs, but that such a, a product can’t be gotten to by Richard thinking, by Richard being a, a politician in a sense or in, or in-
01:14:26
Eric Weinstein: But this is the thing. It’s not being a politician. It’s just that normal fields recognize their value to the outside world. Mathematics is very distinguished in that it has developed a, what I would call a suicidal aesthetic. So when I find myself making the arguments for mathematics, and then we have Fields medalists making what I take to be suicidal arguments like, “Keep OpenAI out of our problems because we’re having fun doing them as people,” or, um, “Mathematics is over, I can’t encourage anyone to go into it,” or, “I don’t know why anyone pays me,” uh, you have no idea how many people with their knives out pick up on these statements, making everything that I’m doing… It just… Again, he’s free to do whatever he wants, and I’m gonna have to be much less collegial in saying that he doesn’t know what he’s talking about. And I don’t wanna do that because the end, you know, just how, how much fun to get to be on a podcast with Richard Borchards? I mean, I don’t think, Curt, you know what a treat is. I, Richard, I knew your, uh, advisor, John Conway, some, and he was one of the most delightful, colorful people you could ever hope to, hope to meet. I remember, you know, him trying to teach me to do tricks with my tongue where he could set up a standing wave and who knows what. And, you know, there is just this quirky aspect of being a mathematician that’s quite a lot of fun, and Ar-Arthur Jaffe more or less managed the affairs so that so many people didn’t have to think about political relations in, in Washington, D.C., but we need this field funded.
01:15:59
Richard Borcherds: If you’re claiming that I’m politically clueless, I will absolutely and completely agree with you.
01:16:05
Eric Weinstein: Then, then we’ve reached agreement, sir. [laughs]
01:16:08
Richard Borcherds: I, I know nothing about Washington and have no desire to know anything about it.
01:16:13
Eric Weinstein: Of course, sir. Of course.
01:16:15
Curt Jaimungal: So how about we talk about what math is about then? On this channel, so I’ve spoken to physicists about the foundational issues in foundational physics, and some people call that a crisis in physics. I, of course, have my own views that I don’t think it’s a crisis per se, but every field has its issues. But, and I would never think that math had a crisis. That came from Timothy Gowers, the term crisis in math. And part of that is, well, as a mathematician, w-what are you doing? You mentioned problem-solving, you mentioned theory-building, and if something else can do that better than you, then you feel, one feels, maybe not you, but one feels diminished. And this gets into the core of, well, what is math about? What is it actually about? Now, some examples are math is about human understanding. That maybe is a Terry Tao-esque answer. Another may be math will, not currently, but will be about more exposition. That’s more Grant Sanderson answer. Another is that math is about interesting structures. That’s a Gromovian answer from a few hours ago, actually. What do you say math is about, Richard?
01:17:27
Richard Borcherds: Well, this is a philosophical question, and I’m not that good at philosophy. Um, but if I had to come up with an answer, I’d say it’s mostly about finding unexpected structure, which I guess isn’t too far from what Misha Gromov said. Um, so, um, I mean, ob-ob-obvious example i-is, is for example, the theory of Lie groups. Um, you, you might try and classify all compact Lie groups, and for all you know, the answer could have been a complete mess, and the answer turns out to be a very beautiful and precise list of the possible compact Lie groups. Um, so, so there’s a, a completely unexpected structure that you couldn’t, that would be very hard to guess in advance. Or, or another example from number theory is th-that there’s a very beautiful theorem due to Fermat, which says that every prime that’s one mod four is a, is a sum of two squares. And again, it, it, it just seems to come out of nowhere that, um, and it turns out there’s some very deep structure behind that related to the Gaussian integers.
01:18:49
Curt Jaimungal: Eric, what is math about?
01:18:52
Eric Weinstein: Uh, to, to, again, to disappoint and get back to agreement, who knew? Who knew that there was all this stuff buried in the world that would be true no matter what planet life evolved on? You know, the, the idea is that they would almost certainly have our list of Lie groups, um, despite the fact that we’re widely separated. If, if you wanted the feeling of the experience, um, what I would say is go to the protein data bank and notice that there are all of these beautiful structures like beta barrels in green fluorescent protein or something like that. Which were never intended to be seen by any human eye. Nobody had a clue that, you know, the double helix, for example, uh, in nucleic acid was going to have this unbelievably gorgeous structure because of course it had to. Um, well, in the same way, can you believe… Like, if you look at this, this is barely 100 years old. This is Heinz Hopf’s discovery that the three-dimensional sphere wraps around the two-dimensional sphere in a way that’s obvious in retrospect, but that nobody saw coming. And so I think one of the things is it could have been that mathematics had very little of this interrelationship of structure, but wh- when you’re standing at it as a pure mathematician looking at this panoply of unbelievable results and the way in which they fit together, you realize how much is hidden from, I don’t know, a duck or a, or a moose. You know, a- all these things that we know are there now… There are five platonic solids, for example, and, and they probably go back quite a bit before Plato. It turns out that one dimension up, in dimension four, the analog of them, called convex polytopes, have… There’s an analog of all five of the platonic solids, and then suddenly there’s this sixth one that nobody knew was there called the 24 cell, which was discovered by Schafli, I think, in the eight- end of the 1800s. Can you imagine that you can buy the 24 cell from Bathsheba Grossman, the sculptor, which is a recent discovery of something at the level of Plato one dimension up? Or the fact that we found out that you… There’s, uh, there’s only one way to do calculus in every dimension other than four, but there’s something called fake R4. Uh, it’s a very strange thing to be able to say that the vector cross product that you never questioned in, uh, engineering class only exists in two dimensions, one, one of which is the three that we know and the other one is seven dimensions. When you find out that there are all of these extra crazy structures that just fit together in this seamless symphonic way with no obvious composer other than logic itself, I could have easily imagined that math would have been a completely dry subject. And the beauty, the idea that the world’s most beautiful museum is visited by at most a few thousand people at any given time, I’m flabbergasted. How is it that we can’t take you through this thing with visualization tools to see all of the gorgeous beauty? You know, I, I commend Dror Bar-Natan at Toronto for his, uh, better implementation than I had of planet Hopf, where you take the two-dimensional surface of the sphere that everyone knows, and you depict it in a s- uh, spherical projection, stereographic projection of the three sphere onto R3. Y- It’s so mind-blowing that no one knows. Who knew that there was a knot in all of three space because SO (3) has a fundamental group of, uh, pi where pi one is equal to Z mod two. All of these beautiful, amazing things are like panic rooms in the human condition where we can pull a book and suddenly it opens into a gorgeous ice palace or an infinite, you know, verdant vista, and almost nobody gets to see them because it’s just too hard to do the pedagogy. So my feeling about this is that it is structure that no one knew was there that returns beauty at a level and power at a level that no one knew possible. And in fact, you know, our entire communication is based on something like the fact that you can cut off my voice at some number of harmonics, and it becomes recognizably my voice. Uh, I- I think what it, what, what… The best example of this is just to go to Grant Sanderson’s channel and look at what he does. It’s effectively like what we do is we write symphonies, but we’re too poor to have an orchestra, and Grant Sanderson owns an orchestra, and he performs the symphonies that have been written, which is only sheet music. So Richard and I can trade sheet music amongst, uh, ourselves and say, “Oh, my God, that, that is breathtaking.” But no one else knows what it means because it just looks like lines and squiggles and dots. And I, I really think that if you take a look at what Grant Sanderson’s doing, everybody gets it. I would also highly recommend a series of videos called Dimensions on YouTube which show you the beauty and the power through that which can be visualized. It is a world that we did not know was there, that was hidden from everyone, and that we are just in the process of finding is the secret to just about everything that is truly powerful and eternal.
01:24:33
Curt Jaimungal: What you just mentioned is actually provides motivation for people to learn math and impetus, and that’s something that when speaking about a question I had is, well, what is the moat, the human moat that AI so far or for- in the foreseeable future can’t touch? Well, it’s not theory building. We thought so. It’s not problem-solving. Definitely not now. But maybe it’s to give us the motivation to continue to study. If anything, AI demotivates because it makes us feel like, what, what is this for?
01:25:01
Eric Weinstein: Maybe I’m broken. I just have… I mean, it has the opposite effect on me. I just can’t get enough of it. It’s like somebody’s given me an infinite supply of poppies, and I’ve got a heroin habit.
01:25:11
Curt Jaimungal: Well, it’s different to go into the universities than to study from the outside. So if you’re going into the universities, a question many students, many graduate students have, many are, are probably watching right now is, “Look, I wanted to go in this and to be a PhD student, to be a postdoc, to be a professor.” And my odds were already dismal. And now it’s seeming like it’s so much more potentially dismal. And you have to always say potentially because it’s, there’s great uncertainty.
01:25:39
Eric Weinstein: Uh, you know, I’m just gonna, I’m gonna say contrarian investing, if you ever see somebody on a private plane going to a private island, almost certainly at some point they had an idea that everyone else thought was stupid. That means that they could buy something extraordinarily low and sell it once people figured out that they were onto something. And what I would say is, is that if math looks like it’s beaten up and a bad investment, it might be that the conventional wisdom is true. On the other hand, if we’re getting this completely wrong because we’re being rocked and shocked and, and we’re punch drunk, it could be that there’s never been a better time because this tool will allow you to explore ideas rather than details, and that you may in fact be in the position to command more power than you ever thought possible because… And I’m gonna say something again that I hope Professor Borcherds won’t contradict me on. A lot of the best mathematics in the world is not written down. It’s not in any paper. It’s taught in oral tradition, and i-i-if your professor doesn’t have a tremendous amount of voodoo that they won’t write down, you should probably find a different professor. I remember whenever I came up with a truly great idea, there was a one in three chance that if I brought it to Is Singer, he would say, “Funny you should say that.” And he’d walk to this black cabinet in the back of his office, and he’d pull out a drawer, and he’d pull out the idea that I just shared with him, and I’d say, “Where can I read this?” And he says, “Well, it’s only here.” Right? And that’s why you have to be in the same department as Is Singer. You can’t… It’s why it’s an elite tradition. It’s, it’s, it’s something in which you, you’re not gonna learn cello over the internet at the highest level. You need to actually be there with the great cellist. This is an extremely labor-intensive, highly personal thing in which our oral tradition is not something that the machines have gotten access to yet.
01:27:46
Richard Borcherds: It’s, i-i-it’s true that every mathematician has a whole, um, filing cabinet full of i- full of ideas that didn’t quite work or never quite got written up or whatever. Um, I, I’m not sure this is really all that important. I mean, n-now that AI has absorbed pretty much all the mathematics literature that’s ever been written, um, it’s quite like you can find all these obscure ideas just by asking your, your local AI.
01:28:18
Eric Weinstein: I don’t think that’s true. I, I just… You know, one of the things that I find that’s really interesting if I think about physics, for example, is that when people talk about supersymmetry, they have a very clear idea that the LHC was expected to find it, and that it repudiated, and no superpartners were found at low energy, and therefore, supersymmetry isn’t present at the w- in the way that we expected it. They may still write papers, and they may be embarrassed about this, but nobody thinks to say, well, did we really invalidate supersymmetry or just spacetime supersymmetry? So for example, you know, th-there’s a, an entire analog of s- of supersymmetry that would not require there to be low energy superpartners or mesoscale superpartners. I, I, I think that it’s also the case that I’m not just talking about failed ideas in the cabinet. It’s also a question about personal techniques that aren’t shared. Very often you’ll read somebody’s paper and you’ll say, “How did he do this? How did she come up with that?” And the short answer is that if you’re her student, she might tell you, but for the rest of the world, it’s a way in which she keeps an edge. And, you know, that’s one of the reasons that we train personally. Do, do you not find that pro- Professor Borcherds?
01:29:34
Richard Borcherds: Not, not really. As I said, you know, you know, a take a random mathematician may have five secret tools they use, but, um, I’d, I’d take AI over that any day, which is, you know, access to, you know, two hundred years of math literature.
01:29:56
Eric Weinstein: We must work on very different stuff.
01:29:59
Richard Borcherds: Maybe.
01:30:01
Curt Jaimungal: You know, Stephen Wolfram said something similar about Feynman, that Feynman had all these techniques that seemed mysterious and magical, but Wolfram saw when he was interacting with him that there were pages and pages and pages of calculations that Feynman never showed anyone because he… Feynman actually liked to be the smart guy and say, “Oh, it’s because of this and that,” and, and seem intuitive. But there were plenty that was… I, I think Wolfram criticized him there or at least joked around with him. There’s an age difference there, so probably not criticized.
01:30:31
Stephen Wolfram: And he would throw away the calculation, never tell anybody about the calculation, and everybody would be like, “How did you possibly figure out this kind of intuitive thing?” And they’d all think, “Oh, it must be simple to come up with this intuitive thing.” It wasn’t simple. It was the result of some long calculation.
01:30:45
Eric Weinstein: Well, I mean, I think Feynman talked a great deal about, um, you know, having this different toolkit and that he would talk about i- uh, differentiating under the integral and particular contour, uh, tricks and all these sorts of things, and that he had a bunch of zags, and that h- his feeling was, I’m only gonna work on things that the zigs didn’t work on because what I have is zags in my bag. Um, and, and I, I, I actually think this is quite powerful, but perhaps what Professor Borcherds is telling us is that that tradition has languished. I would still take the phone numbers of my top mathematics friends over the AI, although that gap is now closing. If you, you know, a-and I’ve said this on Rogan, I think, that if you give me Ed, Ed Frenkel, R-Richard’s colleague’s phone number versus access to the AI, I’d still take Ed Frenkel
01:31:35
Curt Jaimungal: Yeah, but do you really mean that? I mean, Ed is working on plenty because with the AI there’s swarms of agents
01:31:42
Eric Weinstein: Yeah, I’ve known Ed since we were, we were young guys, and I’ve just been astounded. Maybe it’s the Soviet sort of approach that, um, very often what I revere… Ed isn’t as good as the machine at what the machine is good at. So I don’t, I, I don’t wanna set Ed up to fail. But what Ed is extremely good at is when I say, “Look, I’ve got a big idea, and I don’t even know how to describe it, much less implement it.” Um, you know, the two of us, you know, joking and wandering around and, and, and teasing each other and all this stuff, that still is this thing that I can’t duplicate with the AI. I s- I, I think what’s happening is that I’m spending my time out of the convex hull, and I think in the convex hull I’m not, I’m not paying attention to just the amount of carnage that is being, um, brought to bear. Uh, but my feeling is that outside the convex hull, I’d still rather have a human.
01:32:39
Richard Borcherds: Uh, the, the, the… in, the, in any small area of mathematics, or if, if you take any really good human, that there will be a small area of mathematics where they’re better than AI, and it would be better asking the human. But outside that particular small area, the AI will just be better.
01:32:59
Eric Weinstein: I think it will become that way, but I’m, I, I, I don’t think… Well, let, let’s see. The publicly ava- available version of OpenAI at the two hundred dollar a month level is not fully doing it for me.
01:33:13
Richard Borcherds: The, the, the two hundred dollar a month version, um, if you give me two hundred dollar a month version and some random very good mathematician, the, the AI will be better than the random mathematician on almost all mathematics, apart from the small area that mathematician specializes in.
01:33:33
Eric Weinstein: I, I don’t think we’re there yet, but I’m glad to hear that, uh, that you think we are, because it gives me hope that the AI will be even more powerful.
01:33:42
Curt Jaimungal: So, professor, are you also hopeful? Like, do you find this exciting? The reason I ask this is that there are many, many, I’m sure that you know, I won’t name names, but many people in, in the mathematical community, many who are absolutely thrilled, and they’re proving conjectures. And when the three hundred and seventy-six or so from OpenAI came out, they were saying, “This is the most significant moment in a, in math history in a positive sense. Holy moly, I cannot believe this.” Then there were others who maybe had that initially until they, they looked at themselves and realized, “What does this mean for me or for my field?” And, and they have a, a negative view, like, “Hmm, maybe we’ve released something that we shouldn’t have into the world.” Are you optimistic or are you-
01:34:24
Richard Borcherds: Well, I, I, I have both reactions. I mean, on the, on the one hand, it’s really great. The, this really is the, you know, the biggest event ever in mathematics. You know, we suddenly had several hundred major problems solved. And on the other hand, as you say, [chuckles] it is, it is a little bit disturbing to realize that, you know, I’ve been… spent my entire life training hard to do something which AI can now, is, is now better at. So, so I, I say, on the other hand, having AI being better than me doesn’t really make that much difference. I mean, I’ve always known there are many human mathematicians who are far better than me, and now instead of N entities that are better than me, there’s, there are N plus one entities that are better than me. What’s the big deal?
01:35:14
Eric Weinstein: May I ask a question? If you sit down with the AI and say, “It’s really important to figure out how to go beyond general relativity in the standard model, could you please do to frontier physics what you’ve done to number theory, representation theory, combinatorics, uh, complex manifold theory, et cetera, et cetera?” I haven’t seen anything like that. And in fact, one of the reasons that this, these physics announcements have been striking is, is that many of us are saying, “Well, this is the formalization of what we know how to do so far.” Why is it that there’s this marked difference so far where the AI does not seem to push beyond general relativity, afflicted as it is with these two singularities that indicate that it’s probably not a final theory, or the standard model where it can’t even clean up mysteries like Koide’s formula or fine-tuning arguments, and we’re now finding, for example, that the cosmological constant is almost certainly the vacuum expectation value of a field. And we-we’re asking basic fun-foundational questions, and it’s doing nothing.
01:36:20
Richard Borcherds: Well, uh, my, my, my, my guess is we should just wait a few years and see what happens. [chuckles]
01:36:26
Eric Weinstein: You’re probably right.
01:36:26
Richard Borcherds: I mean, I think, I think ma-ma-mathematics problems are, are just much easier to define precisely than physics problems. I mean, you’ve got a mathematics problem like the Riemann hypothesis. It’s completely clear what the problem is and what a solution is. A physics problem might be, you know, what, what is the successor to the standard model? It, it, it’s, it’s not clear what would be a good answer to that, and it’s, it’s much harder to write it out as a precise problem.
01:36:59
Eric Weinstein: Maybe there are sort of, um, problems within mathematics that have more of that feel to it, and maybe what you’re actually teasing out is that, uh, I’m probably more interested in the problems that feel more like what is there beyond the standard model, and you’re probably more interested in questions like what is the answer to the Riemann hypothesis.
01:37:17
Richard Borcherds: Well, no, no. There, there are questions like– There, there are sort of more open-ended questions like this in mathematics. I mean, one, one of them I mentioned earlier, which is just to explain the sporadic simple groups. I mean, I mean, AI has recently checked the formal proof of their completeness, so in some sense, AI completely understands the classification, but it- hasn’t yet managed to sort of digest and simplify it to a, to a form that humans can understand.
01:37:49
Eric Weinstein: So if you found that, for example, AI produced a set of objects on which those groups naturally represented, not necessarily linear vector spaces with structures, but some sort of uniform set of objects that, uh, chose those as the atoms of the symmetry, that would be the sort of a, an answer that you would be looking for?
01:38:10
Richard Borcherds: Yeah, I, I’d, I’d like some sort of explanation of all the sporadic groups that fits on half a page, in the same way that we’ve got an explanation of the exceptional Lie groups that, uh, fits on half a page.
01:38:26
Eric Weinstein: Just thematically, like in other words, um, the way I would think about the Lie groups, uh, my guess is that you would reject it as an explanation. I think it has to do with collisions between division algebras. Is that how you see it, or do you see it more as just you take the structure requirements for a Lie algebra, you know that everything is re-realized in one compact, simply connected form, and then you say, “Well, here are the possible lattices that can, that can be associated with the Lie group, and that that’s an explanation”?
01:38:55
Richard Borcherds: Well, yeah, I’ve, I-I’ve got a different way of viewing Lie groups from you. I, I, I tend to approach it from the infinite dimensional side rather than from the exceptional-
01:39:06
Eric Weinstein: You mean Kac-Moody type?
01:39:07
Richard Borcherds: … Kac-Moody. Yeah, yeah, yeah.
01:39:09
Eric Weinstein: I see.
01:39:10
Richard Borcherds: That, that, that doesn’t, that doesn’t tend to see the exceptional finite dimensional algebra so much, so it’s– I don’t have a good feel for them.
01:39:19
Curt Jaimungal: What’s at stake?
01:39:21
Eric Weinstein: To me, everything’s at stake. Uh, this is the thing… I mean, I never wanted to be a mathematician. I wanted to go into physics, and I saw that physics was gonna go into the toilet over, uh, quantum gravity, and so you could sneak up on physics through pure geometry. Uh, to me, mathematics is the fabric of existence. I see the world through mathematics. I see, uh, all of, you know, just the way as I see the world through, through markets or see the world through evolutionary fitness. I don’t understand how we can casually take this lens into the world that is so far beyond our normal experience. With everything we’ve learned, we realize that everything is on the line because of it, even our own survival, uh, the Teller-Ulam design, cryptography, all these sorts of issues. And right now, we are right at the finish line, as far as I can tell, for writing down a relatively complete theory, um, of physics, which is going to immediately trigger the usual, uh, comparisons to Lord Kelvin. But le-leave those aside because that’s an automated reaction. Um, this is the moment of greatest urgency, greatest excitement. I think we’re going to be able to get off of this planet. And by off of this planet, I don’t mean just to go to Mars with chemical rockets. I mean, if we don’t break the Einsteinian paradigm, because we don’t live in space-time, we live in a successor that we don’t yet accept, uh, we don’t know whether that has the same kinds of speed limits and restrictions and limitations. I think we’re right now going to end a stagnation where things have changed very little over the course of a life. I mean, I always expected that the Riemann hypothesis wouldn’t fall. When Poincaré and Fermat fell, those were incredible events, and in general, most lives have very little tumult. And if you look at the first half of the twentieth century, it was just w- it was, you know, thrills, chills, wall-to-wall horror, wonder, whatever. We’re about to go right back into that frame of human existence, and mathematicians are absolutely at the forefront of this. This could be viewed as the deep blue moment for the human mind, where this is the hardest, important thing that we know how to do. And so the fact is, is that so many mathematicians are throwing in the towel basically tells you that, um, they’ve passed us and this is super intelligence. I don’t see that. I’m so curious about good cycles for elliptic cohomology a-and feel it– figuring out what the real meaning of the Kervaire invariant one problem is, and, you know, talking about it with Dan Freed and Mike Hopkins, and all of these sorts of wonderful new vistas that we’re about to see that I’m looking at the, at the people who are, who are sort of pessimistic and not figuring how much this is on the line. And I realize when you say mathematician and you say the community, we’re not really a community. I just think so differently about this. This is the most exciting possible moment, and the thing I keep trying to tell Sam is please pay a little bit of attention to the community. But some of us are absolutely excited about getting rid of peer review, getting rid of the leadership of the community, and allowing more v-views on what is out there to explore this. And in particular, if people are gonna take themselves on the chest– off the chessboard, let’s hope that we’re bringing up a group of people who’s excited to use this tool to do things never thought possible, and we’ll look back on the Riemann hypothesis and say, “Man, were we stuck on that thing for way too long.”
01:43:06
Curt Jaimungal: Richard, there was a graph recently. It was showing that AIs can find more mistakes than humans can, than professional experts in a field of math, biology, and so forth. And that makes me wonder, for peer review, would you rather, professor, be peer-reviewed by an AI agent than your peers?
01:43:25
Richard Borcherds: Well, pe-peer review will very soon be dead. Um, I mean, it’s– Well, well, logically, it should be dead. I mean, I mean, publication in paper journals should have been dead twenty years ago, but it’s still going strong, so maybe peer review will keep going. But now peer review is rapidly becoming pointless. AI can Even if AI doesn’t prove a result, it can take the result you’ve proved, it can formalize it into something like Lean, and then you can check the Lean program and make sure it’s correct. Peer review to make sure that something is correct will very soon no longer be needed. Good riddance to it, too.
01:44:10
Eric Weinstein: Good riddance to it.
01:44:12
Curt Jaimungal: Why good riddance to it?
01:44:13
Richard Borcherds: Let’s just say I’ve had some experiences with, with referees doing peer review who, uh, I would rather have done without.
01:44:22
Eric Weinstein: Peer review is part of a political economy of mathematics. You know, there are things we don’t talk about like gripe and swipe, where you complain about a paper and then you take the idea and advance it yourself. You know, it’s in general an extremely fraught thing that doesn’t work. It’s very recent in origin. It was backfitted with an origin story to put it at the beginning of the Royal Society. Um, it’s a terrible institution, and, uh, my feeling is, is that it has to do with the control of science, in part to keep the barbarians, uh, in Washington, particularly conservatives of both fiscal and, and, uh, social natures, uh, out of the affairs of science. And that’s, that was its origin in nineteen seventy-five with a controversy over something called Man, A Course of Study. The, uh, the fact is, is that I really want to see more wild men, crazy women, people who think very differently, cowboys, and this oppressive move to uniformize the field, to not allow the truly eccentric who view things very, very differently to have at it. Um, I see a world in which, uh, I met many more people who were super exciting when I was young than I’m seeing now. There’s still some who are coming up who are quite young, but my feeling is, is that I’m really very interested in humans who are extremely discordant because they are the ones who tend to bring about the gi-giant mega breakthroughs. And it was, it’s very exciting to imagine that the, that the machines might be able to say, “Look, I know that you’re going to ruffle the feathers of your colleagues, but they’re wrong and you’re right, so let’s g- let’s go.” I mean, you know, just, you know, as you may know, Curt, at the top of my, uh, Twitter feed is, is an action which Sean Carroll claims doesn’t exist. And it’s very strange that I would have that o- for over a year and no colleagues would say, “How can he claim this doesn’t exist? You’ve had it at the top of your f- your Twitter feed.” Um, the cowardice of the community, in part, with its inability to call out bad deeds on behalf of those who hold the field back, either mostly in physics, but somewhat in mathematics, I’m very excited that the machine is going to be a form of liberation once we stop trying to make it human aligned. I, I really worry that by doing this human alignment thing, we’re walking right into the thing we’re doing with this group that’s supposedly, quote, shepherding the, uh, proof of Yang Mills and QCD as, as formal systems, where the idea is we call up David Gross. David Gross hand chooses, uh, a collection of people, uh, which of course includes Arthur Jaffe, and then they, uh, set about having custody, as, as the phrase that they’ve used, taking custody of the proof for the good of the community. And my feeling is, uh, I’d prefer to clear all that structure away with the back of my hand, and the problem that we have now is, is that the AI companies are being induced to think that they’ve done something terrible by letting loose all of this problem-solving on the community. My feeling is what they’re going to do is they’re going to clear out the low-hanging fruit, and once the low-hanging fruit is cleared out effectively, we’re going to have a new paradigm in which we have new problems, new things that the, uh, computers don’t find quite so easy, and we’re going to have an incredible renaissance that’s going to get outside of mathematics, and some of it’s going to be terrible and horrible, and some of it’s going to be wonderful, but that’s where we’re headed.
01:48:04
Curt Jaimungal: Professor, do you have anything to comment about that?
01:48:07
Richard Borcherds: This involves predicting the future. I’ve made several attempts to predict the future of mathematics and AI, and every single one of them has been wrong, so.
01:48:16
Eric Weinstein: Some people have been predicting the future pretty well. I’ve, I’ve noted that, for example, Jared Lichtman at, uh, Stanford has been right on all sorts of things that I thought he was somewhat wrong about. And I do think that, um, we all owe it to ourselves and each other to try to keep track of who has been, uh, adventurous in telling us what is going to be likely to be true. And I w- I do wish that we were… You know, I, I take Richard’s point that if he feels that his intuition pump isn’t good for this particular revolution, it’s been wonderful in other areas. Uh, I would say mine is a mixed bag. I’ve gotten a lot of stuff wrong. Uh, I’ve gotten some stuff right. But my feeling is, is there are people like Sam Altman, who’s been telling me the same story for a very long time, and Jared Lichtman, uh, who I think we sh- we need to listen to more. We do have people who seem to be calling this revolution pretty accurately. In particular, this, uh, mathematical autoformalization project, or MAP, uh, is something like a human genome project. Imagine if we autoformalized all of core mathematics, uh, and made it computer accessible. I think that’s extremely exciting, and I would like to see some younger people put at the head of it.
01:49:29
Richard Borcherds: Well, that seems to be happening much faster than I expected. I mean, I was, I thought the classification of finite simple groups was going to take at least a year, and somehow it took a few weeks.
01:49:43
Eric Weinstein: Well, you know, Ja- Jared is the one who’s been sort of telling me what he thought the time frames was. He says, “All of my colleagues are getting it wrong. You just watch.” And by virtue of the fact that he and I have been having this internal conversation, it’s incumbent upon me To say, here’s a gentleman sitting in mathematics who’s not getting the attention he deserves, who’s been calling it right when his colleagues have thought this was much farther away.
01:50:07
Curt Jaimungal: So suppose Tino from Toronto or Danny from Detroit proves the Riemann hypothesis, but this person doesn’t know what a zero is. They don’t know what ZFC is, even though they proved the Riemann hypothesis true by showing that it’s independent of ZFC, or maybe they showed it to be false. Who deserves the credit? Is it them for prompting it and just by happenstance getting the correct answer or what?
01:50:31
Richard Borcherds: This is a big problem that math departments now have. We used to award jobs based on people’s published results and the theorems they’ve proved. And if now some random person on the street can press a button and get the AI to give them a published correct proof of the Riemann hypothesis, what do we do about it? And I don’t think anyone has come up with a good solution. Maybe we have to put more emphasis on asking people for interviews and making sure they actually know what they’re talking about in their papers.
01:51:11
Curt Jaimungal: A few months ago, I was thinking that Christopher Hitchens had this razor that that which can be put forward without evidence can be dismissed without evidence. He was speaking about God, but it had me thinking that right now we’re living through that which can be put forward without effort can be dismissed without effort. But now you can’t even do that if it comes with a lean certificate. So Eric, what are your views?
01:51:35
Eric Weinstein: Look, this is the unbundling of some customs that I think were never very healthy to begin with. I’ll be honest, I really do feel like I can detect most of my colleagues inside the AI. I can tell whose voice is speaking. Oh, you know, that’s Andy Strominger, that’s Ed Witten. You know, I hear Grothendieck, you know, inside of the sensibilities. I think that what we have to recognize is that we’re going to have to reapportion credit. And in part, if Danny from Detroit is running somebody whose work is very original and getting the computer to use it effectively, I do think Danny from Detroit deserves something because there’s probably a reason that it wasn’t done by the person who actually did the work. But if, for example, you know, it’s the use of a spectral sequence, I think that I want to go back to, well, who developed the spectral sequences and who developed that spectral sequence in particular? You know, those are not easy things to think up. They’re very complicated to explain. So what I want to do in these situations is I increasingly want to stop with the mystery of how did the AI do it? So far as I know, most of these things are still recognizable human contributions being run inside the machine by companies. And so in effect, and I don’t know whether Richard would agree with this, but I very much, I can tell when my voice is present in the AI, in the corpus, when it’s a critic, when it’s a troll. I can tell by differential application of standards. So something is applied to Ed Witten differently than it’s applied to someone else. And I know the people who do that, and I know the people who don’t do that. So what I still see is I still see a heavy human hand when the AI outputs something. I know all these people. And it’s really uncomfortable because, you know, I can tell you the difference between Frank Wilczek and Ed Witten. I know those two men. I know their voices. I know the difference between Michael Atiyah, Raoul Bott, and Isadore Singer, even though they were working in almost exactly the same area. Those three voices couldn’t be more distinct. And they’re all people that I knew. My feeling is that we’re going to need to go back to the trace of what the AI is doing and start giving credit to those who are being automated within the machine.
01:54:15
Curt Jaimungal: Atiyah had a question that he had a whole lecture, much like how Gromov had a lecture called What is a Manifold? And in it, he said four equals two plus two is an extremely deep equality. Atiyah had a lecture called What is a Spinor? Something like that. And it’s such an odd question on the face of it, because you’re saying, well, let’s just go to a textbook, open it up, and you’ll see a definition of a spinor. And in fact, Atiyah, of course, knows various definitions of spinors, such as there’s the exterior algebra and odd and even elements, or there’s group representations or square roots, et cetera. And then he said, but I still don’t understand what a spinor is. What I want to know from both of you, this is now almost a personal taste question. What is something seemingly simple that you keep coming back to asking yourself, but what is that? Professor.
01:55:10
Richard Borcherds: Well, all I can think of is the well-known quote that you never understand things, you just get used to them. And human brains were just not designed by evolution to understand or do mathematics. So it shouldn’t be surprising that we have great difficulty understanding them and have to keep coming back.
01:55:34
Eric Weinstein: I think, you know, in my case, it’s pretty close to actually what Atiyah, you pick a very interesting example, Curt. There’s a huge division culturally in the mathematics community over that question. Isidore Singer was of the opinion that we knew exactly what a spinor was by virtue of the fact that that was the question that motivated the Atiyah-Singer machinery, that the reason for the integrality of the A-roof genus on manifolds whose second Stiefel-Whitney class vanished was answered by the Atiyah-Singer theorem and the importing of the Dirac operator from flat Minkowski space onto curved manifolds meant to him we could define it, we knew exactly what it was. On the other hand, Dennis Sullivan would have taken exactly the opposite perspective. Nobody knows what a spinor is. Why is there an answer to the question of what is the square root of a tensor algebra? Why, for example, does the group spin 16, let’s say, have no irreducible faithful representations? You are always killing something. No irreducible representation sees it all. In such a situation, I don’t think we actually have the spiritual answer to what is a spinor. Now, many people will turn up their nose and say, I give no mind to such philosophical. That’s part of the mathematical aesthetic, which I find destructive. If you found out, for example, that spinors on, let’s say, I don’t know, an eight-dimensional manifold, which are dimension 16, actually formed meaningful four-by-four matrices and were the endomorphisms of something else, you might start to have an answer. If you found, for example, that you had an elliptic complex that had an obstruction but was elliptic at the symbol level, and then you said, oh my gosh, by virtue of spinors, I can take the square root of curvature, which is the obstruction, and save something from having a d squared equals thing obstruction, and I can get d squared equals zero, you’d have more of an answer as to what a spinor is. You’d probably understand what supersymmetry is, but my guess is that supersymmetry is really more about the question of what is the square root of a connection rather than what is the square root of momenta. So my feeling about this is that there are all sorts of everyday objects. What is the square root is probably my meta question. I’ve called the square root the psychedelic of mathematics. You put it under your tongue and suddenly you’re in a different world. You can ask the question of what is the square root of negative one in the real line and it pops you out of flatland into the complex numbers. You ask what’s the square root of a determinant, it gives you the Pfaffian. What is the square root of Pontryagin class? You suddenly discover the unstable Euler class. So over and over, what is the square root of the tensor algebra? It’s the spinorial algebras. My guess is that the most powerful question that I sit and ask is, one, what is keeping the exceptional algebra out of the mainstream? And what is the square root as a polymorphic operator that always allows you to ask a question in one space and land in a completely foreign world as if you were finding a panic room in a house that you’d lived in for years?
01:59:00
Curt Jaimungal: Thank you all for spending so much time with me. I appreciate you both, and it’s an honor to be on the same show as you both. Thank you.
01:59:07
Eric Weinstein: Kurt, thanks for having me. Great pleasure meeting you, Richard.
01:59:10
Richard Borcherds: Okay, well, thanks for asking me. As you say, it’s a great pleasure doing these things. It’s always interesting experience.
01:59:19
Curt Jaimungal: If you want access to the next conversation prior to anyone else, plus my actual opinions on these theories and concepts, which I keep out of the interviews, then visit my sub stack, kurtjaimungal.com. That’s C-U-R-T-J-A-I-M-U-N-G-A-L.com. And if today’s conversation grabbed you, there’s an episode on screen somewhere right now over here, which is the next one that I’d watch. Enjoy.
