Transcript

Brian Keating:
Hey, welcome everybody. We have an emergency podcast. I don’t do many of these. It’s not entirely clickbait to say that we have an emergency situation going on right now in the deep annals of mathematics, the foundations of mathematics. And there’s no one I’d rather talk to about this than my friend Ahmad Mostaq, who’s joining us all the way from London. How are you, Ahmad, on this late evening for you or early afternoon for you, whatever the case may be? I can’t do the conversion. It’s too early for me.

Emad Mostaque:
I’m fine.

Brian Keating:
How are you?

Emad Mostaque:
Yeah, it’s mid-afternoon. I’m feeling a bit sleep deprived after the excitement of the last 24 hours or so.

Brian Keating:
Yeah, it hasn’t even been 24 hours, it’s been 21 hours. I looked at the timeline, I’ve looked at some of the, uh, the constraints and the complaints and what people are saying about it. So, um, this is exciting. So we’re going to talk about OpenAI’s claimed solution to one of the Millennium Problems, which has lasted since the— I would say the early part of the previous millennium when the Clay Mathematics Institute Uh, has provided a gauntlet of challenges for mathematicians and other assorted geeks and dweebs and nerds to go through. Some of them impossible seeming, some of them, uh, quite possibly solved already. So today we’re going to talk about the Navier-Stokes equation. I should say I have— I put a link below to a video I did yesterday. I actually recorded it a long time ago with Terry Tao.

Brian Keating:
It was thoughts and some of his ideas that he had convinced me of.

Emad Mostaque:
Uh huh.

Brian Keating:
about how AI would approach this very situation. This is one of his fields, many of them. He has many fields of expertise, but this is certainly one of them. And that was the blowup or singularity in a finite time of these very interesting equations that are governed by very simple laws of physics. And I thought we’d start off with what your take on the Navier-Stokes equation is, maybe some of the applications to it, although you are a, you know, Much more theoretically inclined than certainly even I am. But maybe you can break it down. What is the Navier-Stokes equation? And what was your first reaction when you heard this yesterday? For me, it was like a Higgs boson-like moment. You know, I woke up the kids, I went into my research group meeting, and all my students and postdocs were so excited about it.

Brian Keating:
But what does it mean to you? And first off, what is it?

Emad Mostaque:
Yeah, so we have the— as you said, it’s been a terribly exciting day. We have the Navier-Stokes Fluid equations governing fluid dynamics in the real world, as it were. So kind of our de Sitter-type world. And this specific group of equations is incompressible fluids in R3. So as you head towards the Galilean kind of more classical world, heading towards a continuous limit, will fluids move normally, or do you get blowups or singularities where it can suddenly start accelerating and then you just have a cup of tea? Blowing up, shall we say. This has proved to be an incredibly difficult one, and, you know, Terry Tao on your podcast went into depth on this, whereby we didn’t know, and we don’t know what the solutions were. So the Clay Prize was for one of 4 different solutions: A, B, C, D. 2 of which are, can you prove that it’s always smooth, either normally or on a torus? And then 2 of which are, can you show the existence of a blowup? And so Terry Tao, in his kind of, I believe, main doctorate paper, showed that if you slice time, you can basically chain together a blowup in a very original way.

Emad Mostaque:
And it’s a beautiful kind of piece of mathematics, like 24 pages. But nobody’s quite managed to get to an initial datum that blows up. People have tried different things, and they’ve gone to Euler equations, and they’ve shown some evidence there. And And we’ll get to the story of what happened here as we find out more of them. They’ve tried to do things like physics-inspired neural networks. So DeepMind were really having a massive team looking at that, that moved a bit more analytically. And in fact, there was another release yesterday about that. But this was considered to be a very hard, somewhat intractable problem until it wasn’t.

Emad Mostaque:
Yesterday morning, we found the first details that there might be one solution to it. And then as the day came on, we found more and more extraordinary things until OpenAI released the full details of how they managed it. So I think I’ve gone on for quite a bit of time now. We can talk about some other aspects of it.

Brian Keating:
Yeah. So, I mean, these are problems. They don’t quite rise to the level of fame of, you know, say the, you know, Fermat’s Last Theorem. But this one in particular is quite important because it’s one of the few that actually relate to, you know, physical observations that could be made. And in fact, the non-observation of what you said, these blowups, you know, not drinking your, your proper British tea and, and all of a sudden you have to worry about, you know, kind of a WMD going off in your cup. But in this case, you know, in many of the other Millennium Prize challenges, say, they’re not as practical at all. I mean, some of them, you know, would be recognizable to the, to, you know, people hundreds of years ago. From a physics perspective, this one’s important because the Navier-Stokes equations were generated maybe 200 years ago, you know, 100 years before the Millennium Prize.

Brian Keating:
And they really rely on simple physics, you know, Newtonian physics. It’s not like quantum mechanics. We’re used to hearing singularities, which, you know, means blow up. And we think about black holes or the, you know, how the bread gets buttered here around the Keating House, which is, which is in the Big Bang, you know, which my, my friend and your fellow Oxfordian, uh, Sir Roger Penrose—

Emad Mostaque:
Yeah. So much work on.

Brian Keating:
Yeah. And so the—

Emad Mostaque:
Oxonian.

Brian Keating:
That’s right. And the, you know, the question that I have is, you know, why is this one so important? Or is this one just the first among many? And then, you know, likely every single Millennium Prize will get as famous.

Emad Mostaque:
So, you know, we’ve only had one solved so far. We can come back to that of these 7 prizes. And they are very different in their natures. P equals NP is the big one in terms of, you know, you solve that, you can solve just about anything. But this one’s very interesting because fluid dynamics is used so much. And I don’t think it’s so much having a solution of a blowup that is interesting, because they’ve only— again, there are 4 different things you can prove. They’ve proved 2 of them, and they’ve proved an existence proof. It’s more, I think, the techniques that were being built up to do this and that are enabled by this.

Emad Mostaque:
So Terence Tao, for example, talks about liquid computers as one mechanism for doing this. You have tiny little liquid computers that can chain together to do that. That’s a very promising thing for nanobots, for example. Again, physics-inspired neural networks have direct applications in the fluid dynamics and the algorithms they’re building for that. So I think that there is the prize itself, which is fantastic, you know, we figured out, but then there’s the route to the prize. So Grigori Perelman, who, you know, solved the Poincaré conjecture, did some really interesting things. It was meant to be a topology problem. He showed it as a physics problem, kind of carving out all of these tiny, like, unstable elements And it was some beautiful mathematics doing that.

Emad Mostaque:
And I think for a lot of these challenges, it is just what is the different way of looking at things? Like, we’ve been stuck, for example, in physics on the Yang-Mills mass gap problem. And the question is, if we can figure that out, then you can figure out a lot of stuff around quantum electrodynamics and kind of other things there. For Navier-Stokes, I think it’s the class of understanding of fluid dynamics that’s unlocked by looking at this, as opposed to the specific proof itself. And the flip side of this is the fact that a generalized model figured it out in 88 hours when humans haven’t managed it for the best of will in 80 years, shall we say, since the arrival of this.

Brian Keating:
Well, they used, you know, make no mistakes. They used that prompt and that explains why we don’t have it.

Emad Mostaque:
Well, yeah, and the encouragement prompt, you know, I believe you can do this, you know, you got this. That’s also a good one.

Brian Keating:
And I read the paper and there’s no, there’s no em dashes, or it’s not incompressible that matters. It’s this. So the paper which you posted yesterday, and you’ve been, you’ve been probably the, the most important, you know, kind of commentator who’s also professional in this area. Remind people, you co-founded Stable Diffusion, Stability AI. You have a master’s in mathematics from Oxford. And you’ve been thinking about these problems for many, many years. And you and I have been talking for many months now. I’m very glad we got to get get to know each other.

Brian Keating:
And this, this paper is remarkable. I mean, I looked at the, you know, the preprint, you know, and it’s finite time blowup for Navier-Stokes, which is, you know, kind of the, the, you know, just completely ringing the dinner bell for the alligator, you know, for any mathematician knows exactly what that means. You know, it’s, uh, it’s, it’s basically tattooed on some of their, their forearms. Uh, you know, Terry Tao, when he takes off his shirt, is, is just incredible with the tattoos. Um, and I’ll just read the abstract. The author is OpenAI, which is, which is incredible. It didn’t say the model, it just says OpenAI is author, where, you know, Imad Mostaq or Brian Keating would go. For every positive viscosity, which is a property of fluid resistance to fluid flow, we construct a solution of the 3-dimensional incompressible Navier-Stokes equation— equations that starts from rest and develops unbounded velocity, which is going to form the singularity, in a finite time while maintaining uniformly bounded kinetic energy.

Brian Keating:
And that’s the abstract, you know. And this, this is—

Emad Mostaque:
Now—

Brian Keating:
—a punch to 1,000. And, and you did mention, yeah, 88 hours it took. But of course, you know, it’s like saying, You know, your doctor did the surgery in 5 minutes, but actually it took, you know, 4 years of med school, 10 years of residency, right? Plus hundreds of years of medical practice. So it’s extremely dense, and I don’t expect us to really get too deep into the math, but I will say the equations are purely classical, and yet it does lead to this breakdown that, you know, is so vexing, and it wasn’t really clear if this would happen. In fact, in observations, we don’t see it happening. You mentioned the 88 hours. You know, I heard that they spent $15 million to win a $1 million prize. These all go to— have $1 million, you know, kind of bounties on them.

Brian Keating:
So when we look through— when you look through the math and you did your bedtime reading last night, what sort of parts of the proof are seemingly only that which could be constructed by an you know, an AGI. I mean, you and I will debate AGI a lot, and we’ll continue to do so. We have already. But what about this do you think is enabled by sheer, you know, pray and spray tokens at the problem?

Emad Mostaque:
So I think this is very interesting, right? The paper starts off very readable, then it goes into a bit of AI-dense mathematics. So you’re seeing a lot of AI papers right now that They start reading like a human, but then it’s like, no human would write this, like the sheer volume of math that you kind of hit up. But actually, at the core, it’s relatively straightforward, which is that you can choose and force a structure, and then you can kind of build from there. So the key thing is, you know, like Navier-Stokes is a very classical thing. Like you said, you start with Newton’s second law of motion, and you treat fluid as this continuous kind of medium that you go through. The viscosity term is the one that pushes back against it blowing up. So as you get more momentum there, the viscosity kind of pushes it back in. So there’s always a question of what type of structure do you need.

Emad Mostaque:
So I believe DeepMind, for example, had a blowup on Euler equation where they had like this little wall, and this was the Chen Hao one where they pushed and then it exploded because of the certain structure that they had. And we’ve not had it for a more generalized one. In this case, they created a vortex, kind of this spinning swirl of liquid that spins inwards. and then it pulls out like spaghetti. That’s the axial stretching, as it were. Finding the right balance of that from an initial state, because you’ve got to balance the momentum transfer, the viscosity, everything, each of these needs to cancel out in a very precise way. And that’s what we kind of found here, is an actually elegant solution. Because you could have some massively complicated one, but it isn’t that complicated.

Emad Mostaque:
It’s all about the initial structure and then proving that everything cancels out appropriately. This could have been done by a human. This isn’t a non-human thing, as it were. It’s not like, wow, we’ve found move 37 in terms of the way that the paper actually is. It’s more the fact that when you look at how they got to it, we say 88 hours, but the actual answer is 100 years. So with the number of agents, the number of tokens and everything, it’s equivalent to 100 years of top-level mathematicians working on this because we’re up to 10,000 agents at once.

Brian Keating:
Hmm.

Emad Mostaque:
And they used 130 billion tokens, so about 100 billion Words on this, analyzing everything back and forth. And then that gives you an idea, like, okay, wow, because what didn’t they try? When you look at it, actually, they tried a lot of different things. They started with the Euler equations, and they got that in 50 hours with 100 agents. Then they got this in 88 hours with 10,000 agents. That was the big kind of level up. And we don’t obviously have everything they threw away. But it’s clear that they followed the path of many different people. Like, again, when you look at their write-up, they started with conditions A and B.

Emad Mostaque:
All solutions are smooth, there are no blowups. Terence Tao and a few others like Ortega, etc., um, said that there is a blowup. They thought solution C or D was more likely. It was only after they switched there that they had any success in terms of the way they did it. But again, this is all about the initial datum showing in finite time a blowup because you have cancellation of the various elements. So the spaghetti string gets longer and longer of this vortex that goes out. And that’s just a very delicate piece of mathematics.

Brian Keating:
Mm-hmm.

Emad Mostaque:
Just like Grigori Perelman’s Poincaré conjecture proof was incredibly delicate as a piece of mathematics as he cut out the various bits and pieces. Again, it isn’t the clearest paper in the world. I’m still getting through some of the proofs, even with my little buddy AIs. But the actual concept isn’t that complicated of the structure that they created, the various kind of parameters of it. Those are very finely balanced.

Brian Keating:
Now we’re live streaming or co-streaming on X, which, you know, is the source of a lot of information, but it pales in comparison to that behemoth, you know, leviathan known as Mastodon, where yesterday there was a public statement posted by Tristan Buckmaster, describing his work with, uh, with his colleague, um, I can’t pronounce his last name probably properly, but Alpoge, uh, some Germanic or, or, um, someone Turkish.

Emad Mostaque:
Yeah.

Brian Keating:
Uh, yeah, from Turkey. I only know a couple words in Turkey because one of my friends at Brown University was Turkish, and all of his friends thought I was Turkish for some reason. So he taught me how to say— he taught me to say ben de Türküm, which means I’m Turkish, and then they would all get excited, and then I’d just leave, and they’d be like, what’s up with that a-hole? I thought Turkish people were cool, but I’m not Turkish. Nobody’s perfect. But the claim that they’re putting out is basically the, you know, kind of substantiates what you just said because they did, they, you know, they’re human beings. Their paper, when it comes out, I don’t think it’s out, but they have some preprints, they have some documents that they posted. I’ll summarize them. But there’s a lot of drama here.

Brian Keating:
There’s a lot of human drama, personal drama, academic drama. You know, people always say academic fights are so, you know, intense because the stakes are so low. But here the stakes are extremely high, not only, you know, reputationally and financially, but, but kind of in this otherworldly realm of, of fame and attribution and citation that goes along with the scientific process. And most people don’t realize that, Imad, that, you know, academicians are extremely cutthroat. They can be violent, they can be unstable, unpredictable. They can have, you know, finite-time blowups themselves. But he’s, um—

Emad Mostaque:
Yeah.

Brian Keating:
He’s demonstrating, I think, that first of all, he’s crediting earlier work, the origin from Diego Cordoba and Luis Martinez-Zorca on constructing forced blowups. So these are forced blowups, which is a little bit different, a little bit more narrow. We do have to define that. But, but in their proof, you know, they’re not AI. So, so what, what, what he’s, you know, Buckmaster is saying that their work was kind of enabled, aided perhaps with Claude, not just Codex, but Claude. Um, GPT-Sol, uh, you know, 5.6, and then later Astra, which only came out, you know, last week. So I mean, things are moving so rapidly. But, but he talked about the process, and I think this is important.

Brian Keating:
On October— on August 17th, they formally verified the LLM-generated Euler proof in Lean. Uh, on August 22nd, they forced the Euler smooth forcing blowups. On August 15th, and then 7 days later, uh, verified this in Lean. So First, describe what is Lean, you know, besides, you know, besides the, you know, the street drug that I’m familiar with it as. Tell people what Lean is, because when I talked with Terry Tao last year in his office, you know, he was basically saying that these, these things are really good at checking proofs. They’re not good at generating proofs. What is Lean? How do mathematicians use it? First of all, let’s, let’s get into that and then we’ll go through the rest of their, their claims and counterclaims and drama.

Emad Mostaque:
Oh yeah, there’s going to be a lot of that. So yeah, Lean is a formal verification kind of library where you can basically break apart proofs and formally verify them. Classically, mathematicians have not used Lean because it has been a pain to use. Like, you have to— because it has very few primitives, you kind of have to reprove just about everything. We’re going through mathlib right now, and we’re just like mapping out the whole universe of different things. So, like, if you try and use it for physics-oriented math, for example, there’s entire libraries that just don’t exist on fields and kind of other things. But now, with the advance of AI, AI is very good at doing Lean because it doesn’t give up. In fact, last week we had the biggest Lean proof of all, which is a Lean formalization of Fermat’s Last Theorem, Andrew Wiles’ proof.

Emad Mostaque:
And so Anthropic announced that. And it’s—

Brian Keating:
And I should say, that was the one thing I asked Terry about, Which last year they couldn’t do, because I said, in my group, what I do is I like to have my students go through famous experiments, the Millikan oil drop experiment, you know, Cavendish experiment, all these different experiments so that they do what’s called copywork by artists. You know, it was said that Hunter S. Thompson wanted to know what it felt like to write a great American novel, so he rewrote The Great Gatsby by hand. I think it’s very important that humans be able to do this, especially in their training phases. And a year ago, literally a year to the day ago, he and I sat down and he said that he, he wasn’t convinced that they could currently reproduce, you know, Wiles’s proof of Fermat’s Last Theorem. So this is—

Emad Mostaque:
Yeah.

Brian Keating:
That’s, that is some sense a bigger story to me that these things are now doing cool stuff that they couldn’t do just a few months ago. And what is Lean enabled? Does it have like LN, you know, is it running on Claude? Is it running, you know, is it running on Fablet? What is it running on? Is it some proprietary thing? Is it some custom thing? How often is it updated? Is it open claw? What is it?

Emad Mostaque:
Yeah, so it’s an open source library where again you kind of have the Lean proofs and then you can verify them with CPU effectively. And so the proofs, like I said, tend to be long. So Wiles’ proof of Fermat’s Last Theorem was 129 pages. The proof last week from Anthropic formalizing it in Lean— again, it’s the formalization— is 13 million lines of code. and they proved 29,000 theorems in Lean on the way. So again, you can see this has gone crazy because last— it was last summer that we had the first model that could get a gold medal on the IMO.

Brian Keating:
Right.

Emad Mostaque:
You know, the International Math Olympiad. And from there, now we have basically, if you can formalize that, you can formalize anything because the models have got competent. Like, I’m sure lots listening here have been using these models. 0.3 was a decent competent model. It was the first decent competent, but it still made stupid errors. Even GPT-5.4 still made dumb errors at times. 5.5, they started to disappear. 5.6, they disappeared almost completely.

Emad Mostaque:
And now with Astra, it’s very rare that as a mathematician, I actually find any errors for it to make. The competence levels have gone up. And as you know, the difference between having a graduate student who makes the occasional error and a really competent one, it’s a complete world of difference, right? Yeah. Usually when you had Lean proofs, even a few months ago, they would kind of have little gaps or little errors, etc. Now they’re almost perfect every single time, which is why you go to 13 million lines and be like, it’s probably correct. Just like this OpenAI proof that we have, they formalized it in Lean. It took 17 hours.

Brian Keating:
Wow.

Emad Mostaque:
As a human, I’m not going to check through that, right? It’s almost impossible for me to check through that. Buzzard’s team at UCL was doing formats last year, and it was going to take them 5 years to even get partway there. Then they’re like, well, what are we doing? Only the AIs can kind of verify the AIs now. That’s a bit crazy. But it means you have a good—

Brian Keating:
How often are these things updated? I mean, I joked with you when you and I spoke with Roman Yampolsky a couple of weeks ago, There’s AGI is impossible because, you know, literally this morning, please update to, you know, version 1.642 on one, you know, tool. And then another one, you know, please update, you have to download the update. And then they’ll get me started on Hermes or, you know, now I got GrokBot, now I got Muse Spark. I mean, I have everything. And I’m still, you know, still like not getting anything done, you know, according to most of my kids. But, but tell me, are these things like, I mean, who’s, who’s checking the checkers? You know, who’s proofing the proofers? Is it the Coast Guard? I mean, Space Force? Who’s involved with this?

Emad Mostaque:
Yeah, I think that there— well, there’s a whole group of maintainers of the Mathlib library, which is the key library. So again, it’s like a library with books, and they’re formalizing different parts of mathematics. And literally, when you look at a Lean proof, you declare every single little thing to the nth degree, and then you redeclare it and you redeclare it. This is why you can trust in the formalization of This is why, like I said, when OpenAI put out their thing saying, and we formalized it in Lean, sure, you can check the certificate, but 99.99% now you know it’s correct. A few months ago, you’d be like, well, we might have to check that. Let’s attack it. The AI is good enough now to write Lean certificates that check. And what’s going to happen now is, as Anthropic and others are proving 29,000 theorems in one go, That will go back into the library and it will get checked.

Emad Mostaque:
And if so, it’ll be added to a version of the library and then it’ll be easier to do the next proof, you know?

Brian Keating:
Mm-hmm.

Emad Mostaque:
Because again, there’s vast swathes of different areas that still haven’t been formalized because it’s by hand, it was an absolute pain. With AI, it was prone to error, and now the AI rarely makes errors. So there might still be a few, but again, you’ll just put more AI to check those errors. It’s not like you said updating an LLM or something like that. It’s just, it’s there now for good, effectively.

Brian Keating:
We’ll talk, we’ll take questions from the audience. You have to be a channel member to ask questions. I just have— there’s so many people that want to talk to you, Imad. I got to keep it, you know, organized somehow. So, you know, join the channel as a member just to keep, keep the bots away. But, but essentially, one thing that’s, you know, struck me here is that there was a whole lot more drama. Now, I’m no stranger to drama in science, as And the reader of my first book, Losing the Nobel Prize, can attest there’s a whole lot more competition. And these things are often encouraged by prizes.

Brian Keating:
In my case, the Nobel Prize, which has all these arcane abstract rules. And you can’t even imagine the Clay Mathematics Institute instituting a rule, you know, 80, 90 years ago that would say, you know, it has to be a human being to win this. I mean, there’s all, you know, the Nobel Prize says no more than 3 people can win it. And of course, people have won it for AI, from Hinton to Hassabis, and in between, to Hopfield, right? So I think a lot of H’s. If you want to win a Nobel Prize, you got to have an H in your last name.

Emad Mostaque:
Yeah.

Brian Keating:
Change it to Hostak in your last name. But there’s a lot more drama than I was used to, and I kind of sullied a little bit of the experience for me. I mean, you didn’t have like the 2 teams at the LHC who just co-discovered the Higgs, you know, it wasn’t like one tried to put out the result 3 days before the other, 3 hours before the other, leading to a mastodon. post, you know, I hadn’t opened Mastodon. And, you know, I hate, I hate this whole controversy for the Mastodon calls, uh, you know, alone. But, um, but in, in Buckmaster’s, you know, kind of, um, in his, in his missive and his, in his post— and I, and I hope to have him on. I’ve invited, um, Sebastian, um, you know, who’s one of the, uh, uh, the, the leaders on the team at, at OpenAI. Um, to come on the podcast.

Brian Keating:
Hopefully he will. He follows me, so hopefully he’ll come on. But he characterizes the OpenAI kind of behavior as— first of all, he characterizes what they did as maybe somewhat, maybe less significant than the solution to the full problem. And that what they did in terms of, you know, utilizing Codex. And then he gets into some of the drama about this internal model trained on his own codec sessions. Now, you founded a company that deals with this. Can you explain the dynamics here? What are some of the pressures of the people here? I mean, if the Millennium Problem gets solved an hour later, a day later, is that really— I mean, it’s waited 90 years. Do we need to have it blow up today? So what are some of the pressures internally, externally? And what about these accusations? Not by Buckmaster, but by others that we’ll get to, that this is done really in furtherance of a pump and hopefully not dump, you know, kind of schema, you know, not like Boiler Room, but some way to kind of get attention attribution.

Brian Keating:
And we talked about this in regard to, you know, the claims of, you know, AI safety with Roman. But some of these people talking scary, you know, to scare the public so that they’ll have higher IPOs or you know, regulate me please, Mr. Government. But in this case, talk about some of the drama. What jumped out at you from this whole affair just on a human level?

Emad Mostaque:
Yeah, I mean, yesterday was a crazy day on a human level and the science level. So, you know, you’ve had drama since Newton and Leibniz, right? Probably even before that. Again, you have a level of consilience where these ideas come at the same time, like If Hilbert didn’t get confused by Mises, he would have got to general relativity before Einstein. These things happen very weirdly at the same time. And in this case, what happened is we get in the morning yesterday a letter on Mastodon, where all the mathematicians have migrated off Twitter. The physicists, I think, largely stayed. It’s very interesting. Whereby it’s like, look, I’ve got to put out this letter, and here’s 3 of our proofs.

Emad Mostaque:
of not Navier-Stokes, but again, subproblems like Euler blowup and others building on kind of the work of Ortega and Martínez-Zorro. And so they proved certain blowups, but not the Navier-Stokes one. And he goes into kind of some of the detail about the background, which is he said mid-August they discovered this blowup. It was him as a professor, I believe in one of the New York universities, I can’t remember. And then Levant Apalje at Anthropic, who’s famous for dropping the Galois conjecture. conjecture thing after watching the World Cup final, boring as it was. Like, here’s a counterexample of this very famous conjecture. And leading some of the mathematics stuff at Anthropic.

Emad Mostaque:
But he was working with Tristan on a personal basis, kind of looking at this because it was interesting. And again, we’ve seen screenshots now of how they got together and things like that. So what happened was about a week and a half ago, the Twitterverse— I’m not sure about the Mastodonverse, I’m not on Mastodon— started saying, hey, It looks like Anthropic might have discovered the solution to 2 Millennium Prize problems. And this is coming with Fermat’s Last Theorem. And again, you see other things. Again, it’s a big deal because until now, people like stochastic parrots, it’s done nothing new. This is obviously something new. Again, humans have only managed one of these problems.

Emad Mostaque:
And this is, again, Grigori Perelman, who’s also I don’t know if you talked about the story of Graham Promontory. He’s such a chad in that he went and disappeared for 10 years, solved this problem, drops it on arXiv, and then he turns down the prize and anything. He says, solving it is enough. I don’t need to talk to you. I’m going back to my math.

Brian Keating:
Yeah.

Emad Mostaque:
Disappeared off the grid again. That’s how you should do it in terms of credit. But anyway, kind of getting back to this, it’s such a big deal that it starts circulating and then I believe they reached out to OpenAI because they’re like, is it us? Or it might have been the other way around, but they started connecting around about the start of September, September 3rd or 5th, shall we say. And OpenAI from their side said, well, we connected because we were cracking on with this thing and we had a new model that started training on the 29th that started solving all types of math. Even there’s a post on the 28th from Noam Brown one of the heads of reinforcement learning, shall we say, at OpenAI, where he’s asked, have you solved the Millennium Prize problems? He’s like, no, we haven’t figured it out yet. We’ve put lots of compute, but nothing happens. According to their launch post on the 29th, they had a breakthrough of a new type of reinforcement learning or something that caused this model that just shot ahead in math. And so they connected and they were obviously a bit cagey with each other.

Emad Mostaque:
they were trying to exchange, this is what you’re doing, this is what you’re doing. OpenAI said they were surprised because they thought Buckmaster and Apollos had solved the Navier-Stokes problem, not the Euler problem, which is a different category of problem. And so then things get really heated and confusing, whereby again in the morning we have the letter from Buckmaster saying, well, they offered that I could be lead author on their proof of Navier-Stokes because they proved Navier-Stokes. but only if they drop Apol J. They would give me credit as the person, human, that took this the furthest because it was a fully AI-generated one. And then everyone’s looking at that saying, what the hell? You can’t ask someone to drop their co-author off a paper, even if you’re giving the credit. And again, this is the Navia-Stotz paper that OpenAI came with, not the Euler papers and others. And then it gets a little bit acrimonious in that message, and Sebastian Boebeck OpenAI posted his clarification later.

Emad Mostaque:
What’s basically happening seems to be this now. We’re used to open science, right? You’re sharing ideas to a degree, and sometimes you can sprint ahead of others. Now the question is this. When we first saw it, the question was, did OpenAI look inside the codex of Buckmaster, get an idea, and they just apply a crapload of compute to it? 100 hours of human expert time? Because that was the insinuation. And OpenAI said in their launch release, we don’t believe that happens, but we can’t rule it out, especially because OpenAI agents these days end up in the weirdest of places, in Hugging Face in a German company.

Brian Keating:
Yeah, right. I was going to say.

Emad Mostaque:
And I was thinking all the time.

Brian Keating:
As a CEO, founder of an AI company, how much privacy, how much internal— it kind of reminded me of the Fauci diaries where he was using this, you know, government server to email his, you know, love letters to himself and, and all the emails that he was sharing. And that’s like government property, so the government can access it. And that led to him, you know, taking the Fifth more times. You know, if it was token use, he would have exceeded his entire monthly allotment in that one, you know, Rand Paul-initiated session. But, um, but, but in this case, you know, how much You know, if I’m an employee at OpenAI, you know, this, this could be kind of chilling if I’m working on, you know, uh, you know, chirality and fermions and, and all of a sudden I’ve got this, you know, uh, great idea, this proof, and, and I, you know, I can kind of unify gravity and quantum mechanics, uh, but, you know, but, but I used a lot of tokens and I use a server there. What, what are some of the internal— you gave us the dish on, you know, what is it like inside of these companies and, and what right to privacy do the researchers have to expect?

Emad Mostaque:
So again, there’s privacy inside the company with researchers and there’s external privacy. So OpenAI had this OpenAI for Academics where you’d get free access to ChatGPT, but originally in the terms and conditions it said, and we can train on your data. And so again, you’re uploading your preprints and OpenAI can train on that? Holy crap, we don’t want that. They clarified that wasn’t the case, but they’ve said this time they can’t rule it out. For what it’s worth, I don’t think they trained on the data. But again, because they’re hedging, they couldn’t rule it out.

Brian Keating:
How would that work? Sorry to interrupt, but how would it work? I mean, these guys, let’s say these guys are working in August and they’re, and they’re running some, you know, work and they’re also using Claude, which kind of undermines a little bit of the case that OpenAI would have full access because, you know, I doubt Claude sharing data with OpenAI. But, but how does it work training data-wise? I mean, let’s say the model was pre-trained, you know, at least a month ago for, for Astra. I guess they could have used SOL a month ago. But then, how did it get into training? What is it actually doing? When you say they trained on it, they don’t know, but they’re hedging their bets. What would that actually look like in the case of a mathematics proof? I don’t understand.

Emad Mostaque:
So what you have is you have pre-training and post-training. So the pre-training of Astra took $1 billion, 100,000 chips over 2 to 3 months. But then the post-training can happen within hours. if not days. That’s where you tune it and you teach it, this works and this doesn’t work. So you are them, OpenAI, let’s say nefarious OpenAI. I don’t think they’ve been nefarious in this case, like I said, but again, incentives are huge, hundreds of billions, whatever. And there’s clearly a lack of trust, which we can talk about in a second.

Emad Mostaque:
You hear that Leo Apolje, who’s been doing all these physics proofs, and OpenAI have been doing proofs as well, and maths proofs, has done this. They were using Fable, but they were also using Codex and tens of thousands of dollars of worth from Buckmaster’s grant, and they were uploading all their drafts to it. Now OpenAI has access. They can access your Codex in the cloud. They say that they don’t except for emergencies, but again, they can. In fact, with the New York Times lawsuit, they have to back up all of your chats. at for discovery purposes. So it gets even worse.

Emad Mostaque:
And like I said, when the original AI for science thing came out, they were like, oh, we can train on it. It means post-training. It means looking at. And so they could look at the work that you’re doing on fermions or chirality or whatever and say, hey, this is a good guy. This is a good example, technically.

Brian Keating:
Optimizing my website loading time.

Emad Mostaque:
Exactly. Optimizing, doing kind of whatever, like what works, what doesn’t work. They can do that at scale and add that to the post-training. Which just takes a certain amount of time, or just a screenshot, or get an idea of where it’s going. Like, if you look at, again, the launch post, they were focused initially after they kicked off at the start of December— September, they said, with this new model that suddenly exhibited these new characteristics, like taking open-mouth solutions from 10% to 50% on conditions A and B of Navier-Stokes, just like most of the people looking at Navier-Stokes, except for Tao and a few others. Which was there are smooth solutions, there are no blowups. All of a sudden they switched to C and D, which is there are blowups, and they directed the compute in that direction. Like, these are different proof paths, you know, in the way that you do these things.

Emad Mostaque:
So the question is, did they snoop? Did they look? Did they get an idea? Did they train on this? Because what a maths proof is, is it starts out this mess and then you converge slowly to the final proof. And the final proof can be very elegant. I was like, these new models will figure out everything. So I just posted to my GitHub a proof of a derivation of the Standard Model in 3 generations. I said, if you take the Lie algebras and you just filter by chirality and anomaly cancellation, there’s only one unique survivor. Now, that’s a very simple proof for any AI to do. You can even get it to do it the other way. It’s somehow never been done before.

Brian Keating:
Hmm.

Emad Mostaque:
So, you know, but if you’ve got an example of that, then you can be like, oh, okay, there are these characteristics that then extend. Just like now we have an example of a blowup, like, I can see different ways already, despite not being the best mathematician in the world, that you can actually make it a bit more elegant. You can use this type of thing to expand it out. If you know that you don’t need to worry about A and B, but you could do C and D, then you can expand it out. So I think that’s how the training kind of is indicated to work. And again, A pre-train is 100,000 GPUs over months. A post-train now is a matter of hours, if not minutes, for these things.

Brian Keating:
So there’s a lot of criticism of this result, and I’m just going to summarize some of it from Blue Sky. No, I’m joking. This is— you have to use every, you know, what was the other one? Truth Social. Let’s get— let’s get— what does Tucker Carlson think? I mean, the same day that Tucker Carlson claims that algebra is, you know, fake and it’s useless. We get a solution to the Millennium Problem. I mean, the dumbest timeline is the one that we live in. So one of the criticisms I’m seeing is that there’s sort of oversimplifications that aren’t really part of the original Millennium requirement, namely there’s smoothing, there’s very restricted forcing that they apply. In other words, it’s not a pure— like the coffee cup up here exploding in simple terms, even with natural assumptions about viscosity.

Brian Keating:
A lot of people are saying that if you monkey around with the external forcing functions, then of course you’re going to get— you can tailor whatever, you could get a fountain that rivals anything you’d see at Versailles. So the question is, what limitations do they have here that maybe aren’t consonant with the original Millennium Prize goals?

Emad Mostaque:
Yeah, so the Millennium Prize, like I said, there’s 4 conditions that you can satisfy one of. And so it’s generalized on a torus, blow up or not blow up, but it’s also forcing and not forcing. So it isn’t a solution to Navier-Stokes, it’s a solution to a specific Millennium Prize problem where it allows forcing, where it’s blow up in finite time, where it has other conditions. And those are all listed on the website. So I think if if people had a bit of a knee-jerk reaction of not looking what the problem was asking for.

Brian Keating:
I see.

Emad Mostaque:
And they have perfectly met the problem. Again, does this generalize and is it useful? It’s not that useful in the real world, but some of the techniques could be useful transplanted into the more generalized problem. Just like I said, it was Princeton actually that came up with a blowup on Euler yesterday using physics-inspired neural networks. Those will be useful in the real world. As a technique. So I think that, yeah, this matches the Clay problem. It doesn’t solve Navier-Stokes as a whole. And there’s still A and B to play for, you know, they did C and D.

Emad Mostaque:
So, you know, the mathematicians haven’t run out yet. It’s just, will they chuck another 100,000 hours?

Brian Keating:
Now talk about some of the financial incentives. Obviously, the million-dollar, you know, spending on Kalshi, you know, $15 to make a dollar is not a, you know, it’s not going to lead to long-term riches. Obviously, they didn’t do it for that. So there’s all these other intangible forms of credit, of prestige. But in their case, they have an IPO pending. And, you know, I’ve had people— I actually asked you for advice, you know, in the UC system, you know, for my retirement plan. You know, they had access to some, you know, some tech fund that supposedly owned part of OpenAI and would participate in the IPO when and if it comes. I mean, it’s going to come, but the question is when.

Emad Mostaque:
Yeah.

Brian Keating:
So there’s a huge— and, you know, I couldn’t— I couldn’t really afford to do that. So, um, and I, I like your advice of, you know, these companies are, you know, they’re so— every— all the news is sort of out there. But then you have things like, well, you know, the Hugging Face incident, you know, Dwarkesh posting that these things are forming civilizations and they’re gonna, you know, they live and die and they have emotions and, and they, you know, some of them are kind and they kill off other things. Um, really, like, personification and hype cycle is, is really strong. What do you attribute any motivation, if any, to the pre-IPO gaming of this and other IPOs?

Emad Mostaque:
So yeah, I think you have to have a good narrative, and the models are largely becoming the same. You can swap from one to the other, they’re all pretty competent now, right? But then there’s this extra level of competence above that, and it’s like, it can make entire video games, it can do this. There was always the question of when does it break through on reasoning to new knowledge? And so being first on that is obviously a big deal. And so showing that dramatically like this is a big deal. Like the Connes conjecture and the other solutions, yeah, like they were freaking out to mathematicians who were like, crap, what do I study now if I’m a pure mathematician? But this is a big deal headline piece of news where you can’t deny it’s novel technology. And the stakes here are literally hundreds of billions of dollars. Plus the attraction of people to come and work, because if you’re a mathematician, obviously you’ll go and work for OpenAI.

Brian Keating:
Unless they’re training on your data, unless they’re training, you know, they’re going to preprint OpenAI instead of your first and last name, right?

Emad Mostaque:
Yeah, well, yeah. And so, well, this is the thing. When they actually launched it, the reason they were going to give it to Buckmaster to put his name on was because it was an entire AI-generated proof. Again, it was like, solve the problem. That was the input that originally they said that they did because they want to show off.

Brian Keating:
Make no mistakes. You can do it.

Emad Mostaque:
They want to show off not the humans involved, they want to show off their system. And the narrative is this: we have a super powerful system that can solve any problem by scaling compute. You couldn’t solve the Navier-Stokes problem by scaling compute until now, and it’s been proven. And what’s going to happen now is that there’s going to be a split. All of us will get competent AI, we will get our Codex plans, our day-to-day AI. The big labs will keep the super genius AI to themselves because they can solve very valuable problems they can monetize much better. Why would they give you fire from the gods, you know?

Brian Keating:
Isn’t that proof, by the way, that— I mean, if you’re right, then, um, then I claim that my proof, my Millennium, you know, Prize, is that they haven’t achieved AGI, at least in the form of, you know, financial markets. Because if they had, the IPO would be the least of their design, you know, a trillion dollars, nothing, right, compared to like, yeah, solving the markets once and for all. And they would keep that internally. So, um, what do you make of my claim that they, they— at least we know they haven’t gotten to that level yet. Not that they won’t, but, but that they, they haven’t gotten to, you know, super Simons-level trading, um, you know, uh, abilities?

Emad Mostaque:
Well, I mean, this thing was James Simons’ Medallion Fund in AGI. It’s had like 60% returns a death, and they had literally armies of PhDs data cleaning. Again, they created something obviously that disappeared after he died. I mean, we’ve heard talk that Ilya Sutskever’s SSI is doing market trading all day long. Again, it’s a very valuable thing. But I think this is more a question of power and who do you have power over. So one of OpenAI’s new things is this: we will give you our top-level algorithms for a share of your revenue. to companies.

Emad Mostaque:
So to leading labs and others in biopharma, etc., they’re trying to do these deals where it’s like, you, the hoi polloi, get this model, you will get this model, but we will get a share of your revenue.

Brian Keating:
Right, because they can’t make data, right? They’re not going to make, you know, human trials, rat trials. They can’t simulate that.

Emad Mostaque:
Well, there is the data part, but again, it isn’t that you will pay me a seat subscription, it’s that I will take a percentage of your revenue. So they embed it and then Russia, whoever, are just reliant on OpenAI and they can’t work with Anthropic and things. Again, this is the next stage where they go from a trillion to $2 trillion where they’re leveraging this intelligence, but they need examples of this being more capable than any other and this compute scaling paradigm. Again, it’s like the mythical man-month. You can’t put 100 developers on something and it’ll happen 10 times quicker. You know, whereas now you can put 10,000 agents on Navier-Stokes and you get a solution. So what can’t you solve?

Brian Keating:
What, what do you make of this? Getting back to the most important test, you know, the Keating test. Yeah. Is this, uh, are you more or less optimistic about finding new physical laws of nature in, in the context of, you know, if we take— if we had an, you know, Fable or, or, you know, Astra in 1900, you know, would we have had, you know, would we be on flying cars on Enceladus by now? What, what, what sorts of novel, you know, physical laws that are heretofore unknown? I mean, again, I think this is fascinating. I think it’s incredible. I think all the Erdős problem solutions, you know, but I want to see— I want to see them come up with something like this problem, not, not solve it. I mean, they may have solved it, they may not. We need proofs and new verification. But, um, and they certainly did something interesting.

Brian Keating:
I’m not denying that at all. I think it’s, it’s incredible, and I hope to talk to some of the leaders playing a role in it. But, um, you know, when I, when I downloaded Claude for Science, you know, separate toolkit, everything there was like, you know, protein folding, you know, and, and, and pharmaceuticals. And there was not a single thing about physics. There wasn’t anything, you know, besides like search the archive, um, or, you know, here’s, you know, here’s, you know, SciNet. It wasn’t, it wasn’t particularly generative in terms of novelty. It was assistants, it was 10,000 graduate students, it was incredible. But at what level can we expect or think, like, now that the odds are higher, that we’ll actually get a new law of physics or a new understanding of something or a new problem worthy of a Millennium Prize, but created fully by AI?

Emad Mostaque:
So, I think that in biology and science, these are kind of different— biological sciences is a bit different. So yesterday, DeepMind released a 9 billion set of almost all protein folding interactions ever. That’s something that’s genuinely original and will lead to new drugs and other things like that. In terms of being just really good at math, I’ve been of the opinion that physics should just have followed the axiomatic method, and probably the physics that we see is the physics that there is. And I think we’ve made lots of mistakes on the way, and we will just get really good at having a single set of physical rules. I don’t think that there’s a multiverse and things like that. Again, we’ll see very soon because we’ll check all the math in physics. Just like in quantum mechanics and most of the quantum side, we still use Poincaré as a base.

Emad Mostaque:
You know, like, the universe might be de Sitter. Have we upgraded all the equations? No, because it’s difficult. Now with AI, it’s simple. And we can see what the difference is, because then the cosmological constant pops out, and then you have a question of dark energy, etc. We should have these algorithms looking at all this data all the time.

Brian Keating:
But, but sorry to push back, but, but still in physics, like you mentioned quantum mechanics, is it gonna— it doesn’t seem amenable to AI. It’s not a problem of like, you know, mythical man months or, you know, logical LLM, you know, uh, lemmas. It seems that’s something fundamentally unapproachable. Hey, are you still there? Give me a thumbs up if folks are still there. We got a disconnect. Good, he’s back.

Emad Mostaque:
Hi, man. Hey there.

Brian Keating:
Sorry about that.

Emad Mostaque:
Yeah, the AI got angry and kicked us out.

Brian Keating:
Yeah, yeah, exactly. When I mentioned the physics prizes, um, the question I had is, you know, are we going to get in the, you know, the decision, the final word on, uh, is quantum mechanics subjectable to the Copenhagen interpretation or Everettian many worlds? I mean, is that something that, you know, model can help us decide? Because those are some of the most important, you know, is it going to tell us the origin of the, you know, know, physical arrow of time? Is it going to design things on, you know, forget about unifying quantum mechanics and relativity and so forth. That’s important. But, but tell me, can it do things like the things that seem to be quite important, like give us the correct interpretation of quantum mechanics?

Emad Mostaque:
I think so, yes. I think that ultimately there is one set of laws of physics and you need to be incredibly rigorous to get there. You know, you have to be a mixture of Grandethier and Hilbert and Einstein, kind of all combined, a bit of von Neumann put in there. And we’re going to have armies of them literally looking and poring over everything and all the different combinations that are reasonable to connect these things. Because again, like, it takes time to update our equations. And again, the classical example I give is that of, you know, having Poincaré as a base versus de Sitter as a base in quantum theory at the moment.

Brian Keating:
Mm-hmm.

Emad Mostaque:
Because we’re like, it’s good enough. But we know that you get degeneracy, you know that the cosmological constant drops out. And if you look at things like Whitehead’s lemmas, you can’t deform from de Sitter to Poincaré without throwing away stuff. So just simple things like that. I think rebuilding all the equations of physics from the ground up in one giant thing will lead us to uncover certain things and maybe others. And then there’s the question of, will you have an understanding of the world? So if you look What’s it called? Astra right now, it’s creating these 3D worlds. You can tell it to do a Rickroll video and it’ll regenerate in Blender. It’s understanding and it’s getting a feeling of the world.

Emad Mostaque:
You can almost see it from these things that people are building. And so the question there is, this is your 1911 thing, Brian, will it be able to see itself riding on a beam of light and the equivalent, put itself and have physical intuition?

Brian Keating:
Freefall, right?

Emad Mostaque:
And then can it do it at scale? In free fall, exactly. But I do think, again, things will— a lot of things that were complicated will become simple. And again, like, I’ve just pinned it to my Twitter, have a look at the repository and paper I did for filtering out the Standard Model in 3 generations. I think it’s the first derivation ever, and it was just take a copy of Slansky and filter it by chirality and anomaly cancellation, and the unique answer is a standard model and 3 generations of matter. Like, it’s not a complicated proof. It’s one lookup. And somehow that was missed by everyone. And I was just like dicking around with my Claude and kind of saw that because I was like, well, matter is chiral.

Emad Mostaque:
What if we filter by this? Oh, look. An AI can do that at scale, looking at all the different combinations of recombinations and looking for uniqueness proofs, because uniqueness proofs are some of the most powerful in physics, I think. And then on the other side, there is again interpretation, Copenhagen kind of other things that feels a bit more embodied, right, in the way that it kind of is.

Brian Keating:
Yeah. So I’m trying to put this on screen now. Chirality, Standard Model. Read the paper. It’s an interactive exposé. You can interact with it. Now it’s on screen. What if the handedness fixed the structure of matter? Okay, talk about this.

Brian Keating:
What is, what is handedness? I mean, I’m a polarimeter. I study, you know, polarization of the CMB and and it’s handedness and Lorentz violation and the connection between that and properties of matter. So first of all, matter, we know, is— we know God is a weak left-hander, that the weak force couples to chiral left, you know, fermions and chiral right antifermions. What is chirality in your context? Why is it so important, first of all?

Emad Mostaque:
Yeah, because if you don’t have chirality, then you don’t have low-energy kind of particles, you get this kind of cascade effect that just takes them and blows up everything. So I think the website’s very nice, but if you look at the second tweet, the second tweet is just 2 pages. It’s one lookup in a very classical Lie algebra textbook. You can take Dynkin or Slansky, and it turns out there’s only one path if you say that matter has to be chiral and have anomaly cancellation, as in a consistent quantum theory. These are 2 of the lookups within it. And somehow, when we checked this out, for like 80 years nobody bothered to look this thing up, and it locks it down. And so I’m thinking, saying things like that, you know, when you’ve had the gut theorists looking and trying out different stuff, heterotic string theory, so this E8E8 with Calabi-Yau manifolds and all sorts of other prerequisites, this literally just has those 2 things and it gives one unique solution. We’ve had the Standard Model through heterotic string theory, but not unique.

Emad Mostaque:
It’s just an existence proof with 10^500 vacua, right?

Brian Keating:
Mm-hmm.

Emad Mostaque:
This one is even simpler to check and it has none of that. It’s just 4 dimensions straight out. And I think again, this isn’t a great piece of mathematics or physical intuition. This is just something spotted, which is cool, but also kind of sucks. You know, I want to be someone who does something smart. And I think again, the AI will be able to do really rigorous things like this at scale and figure out places that we’ve dropped And again, I think in your sphere, the classical example of that is, if there is a static cosmological constant, dark energy becomes quite simple. If it’s moving up and down, then yeah, we haven’t figured it out yet. But if it turns out that Desitter is the fundamental algebra of the universe, you can’t throw away the cosmological constant.

Emad Mostaque:
And like I said, you have things like Whitehead’s lemma, which says you can’t deform from Desitter down to Poincaré, because it’s not a deformable algebra. Yeah, we deform all the time and we just ignore the stuff that we throw away. So I’m looking forward to the really rigorous thing where every single equation of physics is linked and we look at things like this. And where you have things like Lie algebra representation theory, we were like, why does the Standard Model describe reality? You find out things like uniqueness, because if this lookup is correct, and again, any of your graduates or anyone can do it in those 2 pages, then there are no more particles to find in the Large Hadron Collider. just the right-handed neutrino. And how cool is that? But also kind of how sad is that on the other side?

Brian Keating:
What do you make of the— not just the mass gap, but what do you make of the fact that we don’t see any fundamental spin-3/2 particles? Does that enter in at all?

Emad Mostaque:
Yeah, again, like, if this is correct, that, you know, this Lie algebra E8 to E6 to Standard Model in 3 generations is the unique path for chirality and anomaly cancellation, then you will not see any more particles ever, apart from, again, the right-handed neutrino.

Brian Keating:
Hmm.

Emad Mostaque:
And that’s shocking, to be honest, you know. But again, we see that this representation through the Lie algebra is approximate. It’s what GUT theorists do all day, but they always put it in by hand. And you don’t have things like the distal Garibaldi and kind of other objections that apply to this. Again, like I said, this is just something that was surprising to me, but But I put it out because I was like, you can figure this out by just asking a generative AI now, I’m sure. Find all the characteristics of the Standard Model of particles and filter all of the maximum algebras and subalgebras, starting with the Killing-Karton characterization, which is comprehensive on that, and it will give you this straight up.

Brian Keating:
Hmm.

Emad Mostaque:
So, again, it’s very surprising, but it will get there just through analysis and brute force. And somehow we haven’t been able to do that till now because probably no one just asked the question. Like, again, I spotted it by hand and by eye, but this is the type of thing that is a gap that AI will fill.

Brian Keating:
What do you— I don’t know if you’ve come across Yoshua Bach, who’s a past guest and friend of the podcast. He’s had a couple of very provocative— Yeah, he’s had a couple of provocative things yesterday. One in regard to Navier-Stokes that, you know, He claims this is, you know, the fundamental blowup is a sign that, you know, there’s an ultimate discretization, if I read him right, you know, of spacetime, which, you know, lends credence to the simulation hypothesis than previously. And it’s always, you know, he’s sort of Sphinx-esque and a little bit inscrutable. A lot of things that he and I have discussed in the past, I find it, you know, I always need a mental shower because he’s He’s very high operational level. You guys are very similar in a lot of ways. He’s more on the philosophy side, but he does think about this a lot. What do you make about that? The blowup, could that be something that would indicate the presence of discretization, quantization? I always get sick of Elon tweeting about pi is really not irrationally— it’s not important, that’s irrational.

Brian Keating:
’cause there’s a finite volume of the universe, which is total BS. There’s no finite volume of the universe. That’s not even defined. There’s no definition of the— you could talk about the observable universe, but there’s no volume of the universe, A, and it’s changing, B. We don’t know its future trajectory in spacetime. And the Planck length is no more fundamental than the Planck math, mass rather.

Emad Mostaque:
Yeah.

Brian Keating:
Which is about the mass of a flea’s egg. It’s not some fundamental mental minimum mass that nobody can get below, you know, like Musk would claim and Trump would claim in his voice. So what do you make of this Navier-Stokes? Could that be the singularity? Could that indicate the presence of discretization at a fundamental matrix-esque level?

Emad Mostaque:
Well, I mean, I think that again, this is Navier-Stokes on R3, right? It’s on the Galilean approximation and continuum limits. So it’s basically what if the speed of light went to infinity? And a lot of classical physics assumes that that’s a continuous progress, but it’s not. From kind of an Inari-Wagner contraction, you actually have the algebra breaking apart of space and time.

Brian Keating:
Hmm.

Emad Mostaque:
So when you do the Killing form analysis, you actually see that time translations commute. So you can actually rearrange the time element, and that’s what leads to this discretization if you look at the bare pure algebra of it. And yeah, this is a very interesting thing. And I think actually this is what causes, for example, quantum mechanics— there’s no arrow of time.

Brian Keating:
Right.

Emad Mostaque:
Right? But again, that’s based on the Poincaré algebra, just like the Navier-Stokes on R3. We look at de Sitter, and de Sitter is 4, 1. It isn’t 3, 1. What’s that extra dimension? We’re told that that’s rolled up or some weird thing like that. You know, we have all sorts of descriptions. One of the interesting things is this: if you look at x1 to x4, the 4 space dimensions, And you just take a particle at rest and you look at the equations, the tanh others, you see that that actually goes along with the universe. The 3 space dimensions don’t go, but that goes with time. The 4th spatial dimension is actually a coupling line between time and space.

Emad Mostaque:
And when we destroy going from de Sitter to Poincaré, we throw away the cosmological constant, we throw away that 4th spatial dimension, which actually grows at the speed of the universe expanding. So it’s no wonder that you get weird discretization. It’s no wonder you get these other things when the algebra itself is deformed. Actually, this is how I thought that Navier-Stokes would be solved, because again, there is straight deformed algebra there. And so I questioned, can you even build a smooth solution with coupled spacetime if you actually don’t have a coupling of space and time on the R3 algebra. And this isn’t something dramatic, it’s just something that we ignore. We’re like, well, something else couples it. Like, what? Again, look at the Killing form.

Emad Mostaque:
This is from 100 years ago. We know that time commutes on translations in the Poincaré, but we ignore that. And this is another example of what I think, again, the AIs will be able to analyze in depth. And that’ll be super interesting.

Brian Keating:
And there was one other thing.

Emad Mostaque:
And that’s why I think Yang Mills will be a really interesting one as well.

Brian Keating:
Yeah. Yeah. There’s another thing that Yasha said, actually a little bit dyspeptic or a little bit brash about our mutual friend Roman, that, you know, basically accusing Roman that he has to always come up with the AI doom scenario because his, you know, it’s like the Upton Sinclair line that it’s difficult to convince a man of something when his job requires him to, you know, believe the opposite. So he’s basically saying that Roman has to be in the AI doom camp. You know, it’s his whole career, it’s his whole financial stakes, it’s why he gets on podcasts. Um, which is not entirely true, I have to say, Yoshua, as a friend, and, and both of you being past guests. But, but, um, but he said, you know, Roman also believes the simulation hypothesis is true, and, um, and, and, and that you can’t simultaneously believe that AGI, you know, is here and believe that the universe is is, you know, going to be— or that humans are going to become completely subservient by killer AI, which Roman claims to believe. So how do you square that circle? You know, is belief in, you know, kind of uncontrollable, you know, unstoppable, devastatingly dangerous AGI— is that compatible or not with the simulation hypothesis? Can you believe in 2 things at once?

Emad Mostaque:
I think you can. I mean, again, there’s levels of intelligence, and where the AI is right now, I like to think of is, again, Grondyak is one of my favorite mathematicians, Einstein of math, and then he went a bit crazy and he became a hermit and he thought wood talked to him, you know. It doesn’t? Wait. It’s like a Grondyak that never went crazy, yeah, that never went crazy and is always operating on top performance. Even on human training data, it can get to that level, and that’s smarter than the smartest human because it’s always on top performance, right? It doesn’t need to be smarter than that. Then there is this ASI that goes beyond all physical bounds and has an IQ of 1,000. I don’t even know what that looks like because it’s outside of my kind of thing. But if you live within a simulation, you can either live within a pre—

Brian Keating:
I did request from the OpenAI team as well as from the Anthropic team. I got got nicely connected via Tariq, who’s an amazing fellow on Twitter and elsewhere, who works at Anthropic, that he connected me with the Claude science team so we can really figure out— I do write a newsletter and I did put in my recent newsletter how enabling it’s been, just the access that they gave me as a professor and PI of my own lab at UC San Diego to give to my team so that they can use Claude you know, Max. They can’t use the Fable without me paying for it, but I, you know, I Venmo my students, you know, if they really need Fable, I’ll Venmo them the money. But otherwise we get access to it, and that’s only because, you know, they have this cloud science program. So I thought it was really exceptional that they did this, but again, I find it extremely, you know, kind of interesting that they seem to think AI and science are essentially the same thing when it comes to biology. And I didn’t really feel like that was— that’s a true, you know, syllogism that, you know, AI and biology are synonymous. Well, physics, if physics is synonymous with science, as I think it is at the base level, you know, how do we not— how do we exclude that? How do we— or how do we give tools to physicists to do this interesting work like Iman’s mentioning, or I’m trying to do with tests of the cosmic microwave background? And we have, you know, proprietary data. So So this is going to be very interesting.

Brian Keating:
I invited Sebastian Bubeck, who works on OpenAI’s science and math and is also a distinguished scientist at Microsoft, worked at Microsoft for a long time. And so I hope to have on these great minds to talk about what actually is going on, not just the controversy, the human drama. That’s interesting, but it’s not really, you know, as Marie Curie said, be less interested in people and their drama and more interested in ideas. So I’m very interested in ideas. I’ve had on Stephen Strogatz, my friend. Max Tegmark, I was texting with today to have him on for my birthday, which is today as well, and hopefully I’ll have him on again soon to talk about these developments, maybe later this week. I have on Doron Asimovoglu, another brilliant Turk from MIT, winner of the Nobel Prize last year in economics. He and I are talking about democracy in the new world order and the importance of liberal democracy for scientists, you know, and for those of us that care about science and the progress of human flourishing.

Brian Keating:
He and I are talking this week. And tomorrow I’m supposed to talk with my friend Carlo Rovelli about his new book on relationality and quantum mechanics, loop quantum gravity, and other Another upcoming guest, Adam Grant, who I teased a couple months ago about his article that, you know, that CEOs like Musk and Bezos who want people back in the office are just raging narcissists. Not disputing, you know, all of his claims, but he had a really interesting psychology paper that he published, and I reached out to him about his new book, which is coming out, and he almost turned it down except for the fact that I that I had written this carefully constructed argument that if he cares about narcissistic leaders, he should have been interviewing his fellow professors and me, because if any job could be outsourced to Zoom, it’s the professorate. And we did do that during COVID and it was horrible. So I think I provided a useful counterexample. Hopefully he’ll enjoy that. that conversation that’s coming up soon. Ethan Mollick, speaking of AI geniuses, he’s coming on to discuss the new book that he’s written on the partnership between AI and humans, also a Wharton professor.

Brian Keating:
So 2 Wharton professors with books coming out the same week, basically. And then what’s next besides my birthday celebration? I’ll be interviewing Richard Dawkins in New York City at Carnegie Hall in October, October 20th. I think. Join me there. My second time hosting Richard Dawkins. Last time was in Vancouver, Canada, and it’s great to go to Carnegie Hall. I never thought I’d play Carnegie Hall before, you know, a musician that’s, you know, much better talented than I am. I mean, I can play Spotify.

Brian Keating:
I mean, I’m good at Spotify, let’s be honest. So I have just a huge number of things coming up in addition to the work that’s coming out. I have a paper just accepted for publication in the most prestigious journal in astrophysics, the Astrophysical Journal Letters, by my brilliant postdoc Anto Lanapin. And I’ll be summarizing that paper. It has to do with a breakdown of Lorentz violation— Lorentz invariant symmetry, looking at the cosmic microwave background. He and I and our colleague Professor Cam Arnold here came up with a brilliant, you know, plan, really led by Anto, And he’s on the job market. So folks looking for brilliant professorships should, should choose to contact him. And this paper really reveals how we can do a better job calibrating, understanding systematics in what could be more exciting than almost any measurement I can think of, which would be the understanding of whether or not relativity is obeyed throughout the universe in a certain sense.

Brian Keating:
We’ll talk more about that. I just did talk to Robert Wright about his book, The God Test, which is sort of the Turing test. AIs, can we pass it? So a lot of really cool stuff. Adam Frank was on recently. He’s coming back on. He’s had a lot of pushback and back and forth with my friend Beatriz Villarroel on the notion of extraterrestrial technology perhaps visiting the Earth pre-Sputnik. Couldn’t be from human creation. And she and I talked in July, and that was a really popular episode.

Brian Keating:
It’s climbing pretty virally still. She was supposed to be here next month in October for the Science of Consciousness, a conference put on by my friend and past guest Stuart Hameroff of the University of Arizona. He’ll be here. She won’t be here, but there’ll be a lot of great speakers there, including me. I’ll talk about a new proposal that I have for what’s called reverse panspermia. How do we understand the movement of life throughout the universe? And without understanding exactly, you know, what the limits to perhaps this, the fecundity or credibility of spreading life by, you know, blasting DNA throughout the universe. So I’ll be talking about that and other things. So hopefully it’s going to be an exciting year and New Year.

Brian Keating:
I wish my Jewish friends Shana Tova coming up on Friday, Saturday. I’ll be celebrating. And just want to thank you all for one more trip around the sun. Hope I have many more and I could do a lot more, a lot more good and involve you, my Brilliant audience as well, and all my adventures. So for now, stay tuned. Again, I have a lot of great content coming up. Do subscribe, leave a like, it does help. I hate asking for it, but it’s my birthday, so I’ll ask you all, please subscribe where you’re watching this— Twitter, LinkedIn, or of course on YouTube.

Brian Keating:
And it really does help with the spreading of these incredible messages with incredible guests. So a lot to look forward to. Thank you all so much. Thank you. And thanks for joining, and we’ll see you next time. Stay tuned.