Why Vlad Tenev and Tudor Achim of Harmonic Think AI Is About to Change Math—and Why It Matters

0:00Last I checked, there's a 43 % chance that the next Millennium Prize will be solved by AI, or like human with significant AI assist. I think that's an underestimate. I mean, we could be lucky and Larry Gooth might be on the path to the Riemann hypothesis, which would be amazing. But I think that, you know, if the next one is solved by a human, it would probably have to be in the very near future, and for sure the next next one will probably be significantly sold by AI.

0:51We're excited to welcome Vlad and Tudor to the show. We've had the pleasure of knowing Vlad for many years at Sequoia. But what many of you may not know about Vlad is that in addition to being founder and CEO of Robinhood, he's also an enormously talented mathematician. Vlad and Tudor have teamed up to create Harmonic, an AI research lab with the goal of pushing the frontiers of human knowledge. Specifically, they hope to create mathematical superintelligence, with the thesis that understanding math allows you to better understand and reason about much of the broader world. We're excited to talk to Tudor and Vlad about Harmonic, about the ingredients that go into creating mathematical superintelligence, including synthetic data, reinforcement learning, and self -play.

1:33And when AI will win the IMO or a Millennium prize, or even solve the Riemann hypothesis. All right, Vlad, Dutor, welcome to the show. Oh, thanks for having us. All right, so you guys, you have this core belief that math is reasoning. And you have what might be a contrarian belief, that the best way to train a model to perform well in math is to directly teach it math versus allowing math to emerge as a property of scale, which is what a lot of the other foundation model companies are doing. Can you talk a bit about that core belief? Why do you need to teach the model directly math? And also maybe what does it mean that math is reasoning?

2:14When we started the company we you know had a really big focus on math and maybe we can get to it later but you know if you look around at all fields of science and engineering will almost all fields. Math is really at their foundation. And math has essentially become the way that people understand the universe. It's the way you model phenomena from black holes to atoms. It's the way you design things and engineering. And there's a couple of reasons for that. First of all, it just happens to be the case that the universe is explained all by math. So you can write down fairly complex set of symbols that explain things.

2:47But the other really important thing is that it's a way to build these shared cognitive theories that are very objective and clear and transparent. And if you and I are discussing something that's rigorous, we can write down a set of deductive rules, we can agree on what the ground rules are of whatever we're modeling, and then from there we can derive conclusions. And so with humans, what you see is that when people become very good at math, they tend to be good at other quantitative areas in science and engineering. And so our bet is that if you make assistance that's really good at math, you're probably going to see the same phenomenon where it's true.

3:22It might not immediately write the world's best history essays, but when you ask it to do something scientific or something in engineering, it's just going to be really, really good at that. That's why we started with math. And where is that boundary between, you know, help me with my math homework and write a history essay. There is some boundary that it's hard for math to cross. What do you think the outer edges of what's possible with a model with sort of math at its core? Where are those outer edges? I'll give you a sort of the non AI perspective. So I studied math and you know I was really good at math from when I was a little kid and I remember there were there were there were always like the Seventh graders in math class that would raise raise their hands whenever the teacher would come up with something and it was always like a abstract thing like you know the Side -angle side of triangles.

4:13There'd be the annoying kid that was like When are we ever going to use this? And you know, the teacher would kind of mumble a little bit and they'd be like, well, you know, like math just like, you're probably not going to use it soon, but it'll make you really good at other things. And you know, the other kids were always skeptical of that. And I bought into it. And so I just kept taking more and more advanced math. You know, I went to Stanford and I majored in it. Then I went to grad school to do a math PhD. And my belief was that, okay, if I just focus on math, then I'm gonna get really good at problem solving.

4:51And business problems and other problems will be easy compared to solving these really tough abstract algebra problem sets that I was banging my head against the wall for 10 hours every week trying to do. And I think it basically ended up being correct, right? It's like I didn't really pay attention to anything else. I took maybe like one computer science class intro to computer programming at Stanford. And five years later, when I became an entrepreneur, I found it really easy to pick up code. I found it really easy to pick up contracts. And of course, I'm no lawyer, but you could understand that stuff.

5:40I mean, the logical underpinnings are relatively simple compared to abstract algebra and analysis. So I think for humans, at least I consider myself an example of like math transferring to other very monetizable things. And I think for AI, my intuition seems to suggest that it should be the same. Yeah. And you already see a little bit of evidence of this. You know, at this point, and it's an open secret in the industry that, you know, code training on a lot of code data leads so much better performance on reasoning benchmarks. So you can imagine what that'll look like when you have incredible math data sets that encompass a lot more general types of reasoning.

6:20Yeah, yeah, that resonates. The idea that, you know, math kind of, math teaches a human how to think critically, how to think logically, and that skill can be ported to a bunch of other domains. It stands to reason that that would work in AI also. Glad you casually mentioned that you studied a little bit of math and just for anybody who's not quite familiar with your background in math, I believe you studied briefly under Terry Tao, who is perhaps the world's greatest living mathematician. Yeah. And then one of the things you mentioned to us was that you still catch up with them every now and then you have lunch when you're in LA and that sort of thing.

6:58So I'm curious, when you have lunch with Terry Tao, what do you guys talk about? Do you get many stock tips? No, no, I'm not allowed to do that. Yeah, one of the unfortunate things of being a public company CEO in the financial spaces, I have lots of stock tips, but I can't share any of them. I have to keep them in turn, and I can't even use them. I basically can't trade anymore, which is unfortunate because I love trading. But yeah, I, so to backtrack how I got to UCLA, I, so Terry Tao is a professor at UCLA. And I think what's really amazing about him is the breadth of the work. So most mathematicians get like very deep into a pretty narrow domain.

7:53and Terry can get very deep across like dozens of domains. You know, he's made contributions to number theory, combinatorics, harmonic analysis, applied math. He's one of the leading lean contributors at this point. I'm sure he's like formalizing his papers in lean and actually hopping on the community's zoolup and like engaging with students. and then he has a very popular blog. And I think the way that he's been able to do this is he's just smarter than 99 .99 % of people, probably even more than that. So from a very early age, it was very clear that he was like on another plane. And I studied, I did my math honors thesis in undergrad under this professor Larry Gooth, who's also really amazing.

8:49I mean, he actually had a recent result that came out that was a groundbreaking result. And I want to say, number theory or something about the Riemann hypothesis. But yeah, this result in non -AI math really was quite something. And, you know, he kind of suggested I look at UCLA. I was like really interested in his field and I ended up going there. And being fortunate to study under Professor Tao. But I should be clear, I am a dropout. And it's amazing that I can claim that after grad school. But I will claim dropout status. So I only did one year of UCLA. So it was an intro to graduate level analysis.

9:48Terry Tao taught my first year, which was pretty amazing. And the one thing I remember was I was doing some reading with Professor Tao. And he gave me this book, and he signed it. And I think he signed it because he wanted to make sure I would give it back to him when I was done reading it. And little did he know that by signing it, he guaranteed he would never get that book back. I bring it up every time I see him. I'm like, Hey, you're not getting that book back. On my shelf next to all my other autographed first editions. What does the math community think of AI math? Are people split or do people think it's, you know, the path to the promised land and the way that we're going to solve Riemann and everything else?

10:35I think it's split. I think it seems to be split and there's sort of like the younger mathematicians. I think are very like pro AI and pro verification and tools like Lean. And I think the older folks are a little bit more skeptical. So not surprising. I think you see that in pretty much every field. I think that my guess would be that this will evolve. My mental model is something like chess where, you know, at first, there will be perhaps a lengthy period of, you know, humans plus AI assist. And that will lead to a lot of really good results. But over time, I think the AI will get better and better.

11:28And you know, you look at chess right now, and it's sort of like, you know, if there's a human assisting the AI, the AI would be like a noise of it. They would just want to just delete all the input because it would only make the results worse. So, I'm not sure if we're going to get to that point. I think humans will, at some point, I mean, they'll need to guide the algorithms. But I think the kind of definition of what a mathematician will do will fundamentally change. I was talking to one of my friends who's a mathematician at MIT. And I asked him, you know, when we were kind of first starting this, what do you think?

12:13And this is a young professor, very excited about the field. I was like, are you worried that you're kind of in a field that is going to fundamentally change? He's like, the field of math has always changed. Back in the 1800s, mathematicians used to be kind of like in the royal court and they would be glorified calculators. They would solve quadratic equations by hand. And of course, they were worried that when computers and calculators came out, the job would no longer exist. But mathematicians get to define what math is. So I'm sure at some point it'll be prompting and guiding these AIs to solve problems.

12:57And I think that's going to be very huge. Even if an AIs solves the Riemann hypothesis, a human will always be in the loop, because the humans pose the Riemann hypothesis to begin with. Yeah, just to hop on that. I think there's like, like in the future, you're going to have a lot of compute resources dedicated to math. And the question will be like a very human one, which is like by which procedure do humans decide where to like direct all that like reasoning firepower. And I think that's going to be like the job of mathematicians. They have to choose what do we work on? How do we interpret the results?

13:33How do we interpret failures to find answers is that kind of thing? Do you think an AI, an AI math system can solve Fremont? Or where is the ceiling, do you think? I think that it should be able to solve it, or prove that it's undecidable. That would also be an interesting result. Yeah, I think if we think about what a great mathematician, like a Terry Tau, for instance, is capable of doing, you know, they're able to like synthesize lots of papers, lots of frontier results, and learn from them in a way that the other top human mathematicians can't, and kind of find connections between these things and sort of use them to create new and more complex theories.

14:25I mean, that's exactly kind of how the system where engineering works and that's what computers are great at and AI models are great at synthesizing large amounts of information, finding patterns, recursively self -improving. I think now on Metaculous last I checked there's a 43 % chance that the next millennium prize will be solved by AI or like human with significant AI assist. And I think that's an underestimate. I mean, we could be lucky and Larry Gooth might be on the path to the Riemann hypothesis, which would be amazing. But I think that, you know, if the next one is solved by a human, it would probably have to be in the very near future.

15:14And for sure, the next next one will probably be significantly solved by AI. One of the things that you said I want to hit on, which is this idea of recursive self -improvement, because it seems like in the world of AI there are, if you were to draw a spectrum of human only to AI only, and then human in the loop is sort of the spectrum in the middle from lots of human, a little bit of AI and a lot of AI, a little bit of human. One of the things that is interesting about harmonic, at least the way I understand it is, because of lean, you can encapsulate math in code because of formal verification, you can objectively determine whether things are right or wrong, which means that you have an objective reward function that you can use with self -play to have very fast cycle times with reinforcement learning, which means that the progress of your model has a chance to be extremely fast, because there are no humans in the loop with that recursive self -improvement.

16:20like objective function is clearly defined. You can do self play to just make the model better and better and better and better and better. Which is not something that we see in a lot of domains of AI. Most domains of AI, it's a lot messier to kind of get the cycle time on improvement. Can you just talk through the system a bit, a little bit of what I described, like what feeds into your model, what governs the rate at which it can get better, because it seems like something that we'll be able to get better at a pretty quick rate. Yeah, happy to cover that. One point before going to that is just that I think the most interesting part about this is that, you know, there are other areas where recursive self improvement can work.

16:59For example, again, in those board games like alpha go, but I think what people don't realize. Well, what a lot of people don't realize is that in these let's say, you know, perfectly observed zero some games, you improve recursively just by playing against yourself, but you hit an optimal strategy. Yes, so at that point you know it doesn't matter what system you have it won't do better The most exciting part about math is that there is no offer bound So you're just gonna keep putting compute in and it's gonna keep getting better and there's no end to it And so when we talk about do we think I can solve a reman hypothesis or get a millennium prize like those are very human milestones And I think the real question is like will it ever stop?

17:39I mean yeah, cuz it clearly will get there And I think we're gonna end up solving problems that are much much harder than three -month hypothesis which we haven't even conceived of yet, because it's almost like beyond us to write down such a hard problem. But coming guys, I've ever seen that Minecraft video of like the AI beating Minecraft in 20 seconds. No, that sounds so good. I think it's a good analogy. It's like, you know what Minecraft is? How a human would play it? And then the AI beating it in 20 seconds is just like incomprehensible. Like you can't even kind of grok what's going on in the video feed.

18:14Yeah, but I think if we just talk about, you know, how harmonic works, you can just think of it as there's a collection of agents that are essentially trying to solve problems and it's true because we use lean we're able to check whether our answers are correct and thereby derive a variety of training signals that we used to improve the system. But just to be clear, you know, the use of lean just lets you verify things. Lean doesn't itself tell you whether you're getting closer to the answer or whether you're getting smarter or not. It's just telling you whether it's correct or not. So there's a lot of open scientific challenges to making it get better quickly.

18:49Can you just say word about what lean is just in case people aren't familiar? Yeah, totally. Lean is just another programming language, a really great one created by Leo Demora, the best programming language. We might all be writing lean or the A's might just be writing lean in the future. But the idea is that the mathematical statements are encoded in the type system of the language. So just very simply, you know, you have functions in lean and the input types correspond to the assumptions of the mathematical theorem and the output type is the conclusion. And the point of lean is that if you write a program that implements that function and it compiles, that means you can derive the output type from the input type, which in turn implies that you can conclude the conclusion from the assumptions.

19:31So that's really the the fundamental. That's that's how you use lean for math. I think one thing that's super interesting about Lean is if you look at Leo Demora the creator who's at Amazon AWS now He's not a mathematician and he wrote this as a software verification tool So he has you know the belief that you know in the future Software will be verified and the existing tools things like like Cawken is a bell, which are kind of multi -decade old software verification tools are just not good, you know, and they're frankly unusable. The experience for developers is poor. And so you wanted to create a better software verification tool in the hopes of if you build something better, more people will use it and will have better software, which is a super noble goal in its own right.

20:26But then what he didn't realize was software verification, all it is is just proving that software has certain properties. And this thing became very popular in the math community. And you had like thousands of mathematicians and math students building up an organic library called mathlib, which you can think of as kind of like the largest math textbook open source. It's on GitHub and it's just like growing at a pretty fast clip. And like the usage of lean for math, I think to some degree has surpassed anyone's expectations. It might be more than the usage for verified software at this point. And that might change as time goes forward.

21:18and with AI. One of the questions we always ask is why now? Is reinforcement learning as existed for a long time? Math has existed for even longer. And it seems like math is really hidden in inflection points. You guys have chosen to start harmonic at this point in time. Why now? Oh, I mean there's two really good reasons why now. The work's excited about. So first, you know, one is just that the AI systems have gotten better in an interesting way. So I was actually talking with with a friend, a close friend, about RL for theorem proving back in 2015 and 2016. And one issue back then was that there wasn't even a great notion of an AI system that could predict something in an infinite action space.

22:04So in Go, you can place a piece somewhere, right? And see either a black or white piece. But in math, you can really do anything. Like you can just generate any like next step. And so we didn't have great systems to do that. So auto -aggressive language models have gotten pretty good. So that's one thing that makes it possible. I am talking on the time scale of like a decade here, but that's really important. And the other thing that's kind of crazy is that lean has gotten really good. So if you had told a mathematician 20 years ago that a large fraction of the field would be excited about formal methods in math, they might have thought you were crazy because back then, formal methods were really isolated to formal logic or certain types of, you know, graph theory.

22:40Like if you guys have heard the four -color theorem, that was one big success for formal math. But what's changed is that lean is so flexible and so exciting for people that they've contributed this thing called math lips and there's a lot of body of knowledge that you can build on to prove things. And so the combination of AI starting to like even be a possible fit for this problem plus lean working really well and lean for was only released officially in September 2023. So those two things happening together really made it the right time to attack this. You guys say word about data and specifically synthetic data and what it is that fuels the model that you guys are building.

23:20Yeah, so synthetic data is the fuel for the model. There's a, there's an amazing resource called math libits that open source repository. So that's a lot of human written lean. And it's written a way that's very general and compact. So they're really proving advanced theorems, right. It's not necessarily the best fit for problem solving. And so as a result, almost the only data you can use for this is synthetic data to generate yourself because that original data is not very applicable. So it's kind of a data poor regime compared to most areas of AI. And so that process that I described where the agents themselves are trying to solve problems and thereby generate training signals, that's the primary way in which you can get data.

24:04And the other issue is that, you know, you have to progress through levels of intelligence. So you're not necessarily proving the remunipotence as a front, you're proving really simple things, but then you kind of amplify yourself recursively throughout the process. Turns out there's not as much math data on the internet as cat videos. Unfortunately, no, unfortunately, no. Well, yeah, it's interesting though, because, you know, there's the data law that the foundation model, you know, the general purpose foundation model companies are running into. and at this point they've exhausted what's available on the internet and if you can generate most of the data that's required to train, that's kind of another advantage of having math at the core versus hoping for math as an emergent property of scale.

24:49Yeah, I think the data wall kind of manifests itself in two ways. One is just like you said we're out of internet data. Yeah, the The other is the actual internet data quality that's out there. You can think of that as providing kind of a ceiling to how smart these models can get, because if you train on the CAD videos and all the nice Wikipedia content and the internet content, it's an open problem how to get something that's significantly smarter than that. And so you do need to get into some kind of self -reinforcing, self -play regime, in our opinion, to get to a point where you can surpass the ability of human mathematicians and researchers at multiple tasks.

25:41And so I think in many ways, like the path, it's inevitable that it takes kind of the AlphaGo to Alpha0 approach, and we learn how to make these models create the vast majority of their data and have the data actually increase in complexity as these models continue to iterate. I think the great thing about math is there's a simple path to doing this. You can basically measure the complexity of a math problem and how difficult it is by how many lines of lean it takes to solve. So you can actually look at the complexity of a system. And a lot of problems are solved by breaking it down into smaller chunks and actually solving those chunks.

26:35And if you kind of think about how that works, the smaller chunks are then more manageable because they're sort of like few lines to solve them than the big one. So if you get really good at that and then you get good at solving the chunks, then you can kind of like Train your model to do better and as you kind of like keep Turning the gears on that the model gets better at solving incrementally harder and harder and more complex Problems I think that works very well at math because it mirrors how we solve math like on on pen and paper Yeah, and we've been able to start with like simple axioms and build up like just giant complex structures maybe the Riemann hypothesis would be hundreds of pages if not more to solve Fermat's last theorem was I think 200 pages very very complex so Yeah, I do think eventually you'll get to a level where you'll be able to solve these things and the math is to some extent like the original synthetic data.

27:46Yeah. What determines the rate at which your model can get better?

27:56The rate at which you can get better? Well, I think the highest level one is energy. So the more energy you can put into it, the more attempts can happen in parallel, which means you generate data faster. So, I mean, there's no rate limiting step. I mean, sorry, there's a bunch of rate limiting steps, but there's not like fundamental constraints on how fast it gets better. So it's really just about how much compute you put in. I think it's also, I mean, there's still a lot of unsolved problems in this field, right? Like, we benefit a lot from core theorems that have been proved in the past. And if you think about competition math context, there's theorems that every student would just learn and use AMGM inequality, things of that nature.

28:53And so to some degree, like math lib is incomplete. There's very little content about geometry, for instance. And it's very theoretical and abstract. And so a limiting step is like what's in math lib. And of course, at some point, the models have to solve the problem of creating new theories and new structures and expanding to other domains and getting really, really good at formalizing things that haven't been formalized by humans. I think that'll be a big unlock and that'll certainly happen within the next several several years. You'll be able to say, hey, here's just like this situation. It could be a simple as like a baseball team and they're like throwing balls back and forth to each other and you know, you systems would be able to kind of like auto -formalize that and turn that into lean code on the fly.

29:52And I don't think we're quite there yet to the point where that's reliable. But when it does get reliable, I think that'll be a really big unlock. If everything goes right, what do you think harmonic becomes? Well, our mission statement is to explore the frontiers of human knowledge. So it's very important to us that, you know, the things we produce are correct and useful to humans. So I think in the best case, you know, we're able to build a tool that a lot of mathematicians use to close all the millennium price problems and to go far beyond that. I think that would be a great service to humanity.