Grant Sanderson (@3blue1brown) – AI and the future of math
In a Nutshell
Grant Sanderson argues that AI's rapid progress in math—driven by verifiable, grindable domains like IMO problems and Lean formalization—signals broader capabilities, but the hardest parts to automate are generating novel conjectures, definitions, and insightful cross-domain connections rather than raw theorem-proving. He highlights that AI's advantages include parallelization, systematic context-escaping, and endless exploration of axiom spaces or Mathlib extensions, yet warns that "mountain-building" new theories or distilling compressed, human-understandable insights may lag behind brute-force solutions. Ultimately, mathematicians' roles will shift toward curation—deciding what ideas are worth pursuing—while demand grows for humans who translate AI output into clear, motivating explanations and identify physically relevant results.
These notes were generated by AI and may contain inaccuracies.
Grant Sanderson runs 3Blue1Brown and is documenting AI progress in mathematics. AI has made the fastest progress in mathematics compared to any other field, and developments here will indicate what happens elsewhere as AI improves. Three years prior, when asked whether AIs achieving gold at the International Math Olympiad would constitute AGI, Sanderson responded that it would be another benchmark like others AI systems pass, without a singular "aha" moment. AI has improved generally since then, but the prediction held.
The IMO contains four categories: geometry, number theory, algebra, and combinatorics. AI systems are very good at geometry, solving problems in nineteen seconds via brute force since 2024. For students, geometry also has a brute force approach. Combinatorics problems are more playful and puzzly. In the year discussed, two combinatorics problems appeared on the test. If more geometry questions had been present, the AI would have achieved gold. The system struggles on combinatorics problems, which some view as requiring more creativity.
Solving a Millennium Prize problem like the Riemann hypothesis may not equate to automating white-collar work. AI systems excel at deep knowledge within specific domains but have not consistently found lightning-bolt connections between domains. Sparks of cross-domain connection-finding are emerging. The Riemann hypothesis solution might involve such bridging, as in the story of Hugh Montgomery and Freeman Dyson at the IAS. Montgomery studied statistical correlations between pairs of zeros of the Riemann zeta function and derived a formula resembling one over sine squared. Dyson recognized this expression from eigenvalues of random Hermitian matrices in nuclear physics, prompting exploration of random matrix theory's relevance to the zeta function.
Difficulty using AI as an editor stems not from needing to connect known facts but from other limitations. Fermat's Last Theorem (xⁿ + yⁿ = zⁿ having no integer solutions for n > 3) was phrased simply yet required heavy machinery involving elliptic curves and modular forms built over centuries. If solving the Riemann hypothesis required building entirely new theoretical mountains, this differs from the intelligence needed for white-collar tasks like video editing. Building correct new theories would represent a level of intelligence likely to have transformative economic effects beyond mathematics itself.
The question of AI connecting ideas for new discoveries represents goalpost movement from earlier discussions. The unit distance problem conjecture counterexample exemplifies the type of connection-finding once thought difficult. Next benchmarks include generating interesting problems and creating new conceptual objects or unifications. Riemann developed the zeta function because he believed its zeros connected to prime number density. Determining why a research direction is interesting represents a distinct capability.
A quote highlighted states: "good mathematicians prove theorems, great mathematicians come up with conjectures, and the greatest mathematicians come up with definitions." Conjecture generation and definition generation represent premium-tier mathematical work. These resist quantification as benchmarks because success cannot be scored definitively like disproving a specific conjecture. Progress would appear as a tone shift in how mathematicians discuss AI usefulness.
Conjecture-generating ability would be measured subjectively through tone shifts between mid-2025 and 2026 interviews with mathematicians. The indicator would be mathematicians reporting that conversations with models helped determine what their research field should be, not merely solve existing problems. This would not appear as a headline announcing another benchmark achievement.
Things difficult to benchmark are also difficult to train for under current paradigms. There is no fundamental difference between a benchmark and a training environment. The most valuable scientific and mathematical progress involves new conceptualizations that unify fields and solve problems not previously targeted. Einstein developed general relativity without initially aiming to explain light bending or black holes.
Galois's development of group theory while solving the quintic equation illustrates long verification loops. Abel proved quintics have no general formula using radicals years earlier without developing group theory. Lagrange identified symmetry questions as relevant to polynomial solvability. He examined how expressions with four variables behave under permutations and connected this to reducing degree-four to degree-three polynomials. Extending this to quintics required expressions taking on four or fewer values under five-factorial permutations.
Lagrange planted the seed that symmetry questions might be the right approach and that proving impossibility could be more productive than seeking formulas. Abel read Lagrange. Galois was influenced by Lagrange. Galois developed abstraction further, writing in prison about layers of abstraction beyond algebraic expressions toward underlying symmetries. His papers were rejected by the academy. He died young in a duel after attempting publication five times.
Liouville encountered Galois's notes twenty years after his death and attempted to clarify them. Another twenty years passed before Jordan produced a modern treatment of group theory attributed to Galois. Recognition came through applications in twentieth-century physics when Gell-Mann connected group breakdown patterns to particle structure, anticipating quarks via purely group-theoretic reasoning. The full arc from Lagrange's initial symmetry insight to practical applications spanned over a century.
The challenge is measuring progress via the instinct that "there is something here" rather than immediate problem-solving results. This instinct operated in Lagrange recognizing symmetry questions as promising, in Liouville seeing value in scattered notes, and in Galois believing his abstractions mattered despite rejection.
Even though this isn't the main angle being taken, there is something to the idea that the smaller expression that's more predictive feels more intelligent. This raises the question of the extent to which you can give some kind of verifiable reward around not just whether a problem was solved or what it is solving, but around the smallness of the concepts required to do it.
Going back to Riemann hypothesis solutions, one possibility is that an AI solves it by working harder in the same way you could maybe have an elementary proof of Fermat's Last Theorem that's just spelled out over thousands of pages that would be incoherent. The cleaner way to view it is with elliptic curves and all that. Maybe there's some thousand-page proof of the Riemann hypothesis that no one's really getting anything out of, and what you actually want are the succinct, compressed versions of those ideas that would then lend themselves to human understanding.
Maybe you throw Kolmogorov complexity into your attempt to quantify what you mean by elegance. It is not easy, but it is something you would have to do in order to reward the Galois-like instinct, rather than just rewarding whether you solved a problem. It is very hard to come up with the heuristic for science, but it is clear humans have been doing this somehow, and obviously AIs will do it at some point.
It is relevant also not just in terms of verified reward, but presumably the end goal is understanding, human understanding. Even if you do have some thousand-page proof of some math thing or some grand new physical theory, the goal is understanding. Maybe if the goal is predictiveness, you can just have automated engineers go off and build rocket ships where we have no idea how they work, but we can get between stars. But there are going to be a lot of people who want to understand. You are still going to want whatever the concision function is that distills down this complicated way of thinking into the right one, like the equivalent of the universal law of gravitation for Newton. You would still want to train AIs to be able to do that and find the compressed representation.
People have this worry about mathematics in particular that AIs will prove the Riemann hypothesis, and our understanding of mathematics won't be any the better for it. There are a couple of questions about this. The first one is whether this is something you should expect. Isn't the reason humans come up with general, natural objects and subgoals when we're working on a big problem that this is just useful when you're trying to work on a complicated, important problem? Theoretically, would this even be a simpler way to solve the Riemann hypothesis, as opposed to just coming up with the natural abstractions that are relevant to thinking about the problem? And then two, empirically, is this what we observe when AIs make progress on problems today?
When the AI came up with that counterexample to the unit distance conjecture, you can just read its chain of thought. It is not understandable to some people because they don't know anything about mathematics, but it seems that to other mathematicians it was understandable. It made use of known concepts of mathematics and proved relationships between them, all in natural language. As a result, it accelerated our understanding of the connection between this object and this conjecture. Empirically, is this a thing we should be worried about?
It depends on the nature of the solution. If we break down the three possible ways of solving the Riemann hypothesis, the other big one from this year was a certain Erdős problem numbered 1196, about these things called primitive sets. It had that character of bringing an idea from a seemingly different field. As soon as you present the basic idea to a mathematician, you say, "What if we try the Markov chain process where we show that this thing is one from the bottom up probabilistically rather than the top down, and use the von Mangoldt function?" If you say that to someone in the know, they would know how to run with it.
You have this very small idea that has the form of expertise in one field and expertise in another, drawing a little lightning bolt between them. Those are going to be very human-parsable, because all you have to do is show the start and end point of what those connections are. If the character of it is mountain building, you have to put in a lot more time to understand that new mountain that was built, because it's a new thread, not just a lightning bolt between them. And if the nature of the progress was just raw hustle—a super long chain of reasoning with no new theories—then you would have that worry of this whole digestion process. So there is not one clear answer. It depends on what the solution would look like.
On the mountain building side, that would actually be really interesting to see. Is it by default very human-understandable, the way we see new theories from great mathematicians? Or is it an alien, different kind of mountain being built where we have to reprocess the kinds of abstractions we engage with? The closest example here would be the attempted solution of the abc conjecture. We maybe shouldn't get into that one, but it probably is not a correct solution. Basically it's this whole new way of thinking that this otherwise reputable mathematician in Japan had come up with. It took mathematicians a long time to even parse what he was saying, but it had the feeling of an alien bit of mathematics that's theory building, not just a long chain of reasoning. He called it inter-universal geometry.
The biggest fear would be that an AI does that, and then much like the abc conjecture, people work for years to go up the mountain, and they're like, "Dang it. This just isn't right." If it turns out to be wrong, but it really looked right. Even if it was right, there's just a lot of effort to hike up a new mountain. If we end up in that situation, David Bessis had a really great blog post called "The Fall of the Theorem Economy." He's talking about how historically, as you were saying, mathematics is coming up with these definitions and problems, and it's about proving theorems about them. The theorem-proving stuff is what gets all the credit, but it's really a parasite on the coming-up-with-the-definition stuff. Historically, this has not been a problem in terms of credit apportionment, because if you come up with a definition, you're probably going to be the guy who comes up with a theorem. But now we're in a situation where if the valuable work is coming up with the insight and AI automates the latter part, imagine a scenario where an AI comes up with Abel-like direct arguments about a bunch of important conjectures in the world, and then we just have these proofs. Now it's up to humans or future AIs to consolidate. Again, having no object-level understanding of this argument whatsoever, it would make it easier to think about what's going on.
Is there some deeper way in which we can understand why this proof works that would make it easier to come up with the ideas behind group theory? It would be hugely helpful. So much of trying to discover new math is mostly being wrong. You're trying to solve a problem, and it doesn't feel like constantly taking the correct step up the mountain. Mostly it feels like a random drunken walk, where you're doing a thing and then you're wrong and constantly discovering that. If at the very least you know that trying to digest what you have is ultimately leading to a correct solution, that feels like progress, simply because of the sense of knowing it leads to a solution.
There are plenty of instances in the recent history of math where it feels like the reach has exceeded the grasp, where things are proven long before they're understood. One of the favorite openings to a paper—it's not even a research paper, it's more like an expository one—is from a mathematician named Timothy Chow, who was trying to understand a concept called forcing. There's this problem called the continuum hypothesis that more or less asks: you have a size of infinity for the natural numbers, and a size of infinity for the real numbers. Is there something in between? The answer is both yes and no. It depends on your axioms. It's outside the scope of our usual axiom systems, which is an interesting answer. But the method to describe it is really hard to understand. It's this thing called forcing. In the beginning of this paper, he writes, everyone knows the idea of an unsolved research problem. I want to propose the idea of an unsolved expository problem. Sure, we've proven it, but we don't really know why it's true. Then he proposes a partial solution to that expository problem.
There is a difference between proof and explanation, and that is the importance of that distinction. That will be the main incentive. Or the incentive would have to change, not just in mathematics but in other areas of science, from proving things about the world to consolidating proofs into problems or higher-level insights.
In the limit, is there really a difference between the conceptualization of an idea and the idea itself? If you think about special relativity and spacetime diagrams, and Minkowski spacetime, this is a way in which we illustrate why there's length contraction and time dilation. But that is the reality. So, the exposition does seem to be the explanation in some sense here.
There seems to be a really strong correlation between the people who come up with genuinely novel insights and the people who are actually quite clear in their communication of it. You might imagine the opposite, given that the experience of a university student is often that the expert teaching them is not necessarily the best explainer of that topic, because they're so spoiled by their expertise. But what seems, at least in some cases, to be the case is that the people who are really coming up with something quite novel—you've got Einstein or Claude Shannon or someone—you read their papers, and they're really lucid. It doesn't feel like this is just for the experts and you have to chop through it with a machete. They're very good expositors. Feynman has this characteristic too, he's a very good expositor. Maybe the same part of the brain that comes up with the correct new way of thinking about it at a research level also has this knack for good explanation.
This is pertinent to AI. It used to be thought that AIs would become these automated theorem provers, but the role of mathematicians was going to shift towards explaining these things. Now it seems that actually they'll also be quite good at doing that, probably better than most humans are at explaining and distilling. So digesting and explaining what was going on is probably actually not what's left for mathematicians, by the nature of how these things are going. Probably the same thing that comes up with the really good new idea that solves some new problem is also just good at explaining it. That's a way beliefs have changed.
What's the last thing you think you'll be doing? Both you and also what the human mathematical community will be doing. Some of the function of an explainer or a teacher is to add clarity to a thing that someone's curious about. That's one thing. But some of it is more relational, providing motivation and a sense of curation. One interesting take that has been heard about what mathematicians will end up being is that it's actually more analogous to art museum curators than anything else. The AI solved the thing, so the art exists. They even know how to explain it really well. But you still want someone to help you navigate this nearly infinite space of what ideas are worth engaging with. Even if AIs were in some sense better at that, we would always still prefer a human that we had a relationship with, because the way we get motivated to be interested in things is a social phenomenon. If you have some specific technology you're trying to build, that might be different. But the people listening to this podcast trust your curation on what's an interesting topic in the first place. It's not that they're landing here because whatever your next topic is, that's what they wanted to understand in a prior sense. They're trusting you as a curator. So the role, and arguably that of other mathematicians, might actually just shift subtly into that curation direction of what ideas are worth pursuing. That's a lot of the job right now.
People assume a lot of the time for a video goes into the visuals. Sure, it does. It's not immediate. But actually a lot of it is just deciding what's worth saying in the first place, what's worth putting there. The desire is to engage with that, and there is trust with certain people, and they're curious what would be chosen to put forward even if the AIs are better than that. It's the same reason human musicians are always going to have a role: that social function of the story behind them, even if the objective quality of the MP3 file coming out of some model is better. That's what is seen happening to the job. Wanting to go back to a question from earlier, just as AI has crossed this threshold, this important benchmark of being able to connect existing ideas to come up with a new discovery or prove or disprove something, we're like, "Okay, but what's the next thing?" There's a lot more to do on that one, by the way. Just because a couple lightning bolts have been thrown, there is this flourishing future over the next couple years of really connecting.
In the limit, you could even say a lot of the biggest breakthroughs look like this at some level. With general relativity, you're just connecting together Riemannian geometry and special relativity. So as AIs keep getting better and better at this connection thing, maybe a lot of big breakthroughs are not really of a different qualitative nature.
A lot of the conversation has focused on problem-solving and that nature of math, ticking off Erdős problems or something. But it's not even a majority of mathematicians who would characterize their work as really targeting the next problem to tick down. The Langlands program is not even a field of math so much as it is a research ethos. Fermat's Last Theorem is one inkling of this. You had these two seemingly disparate things, and a connection between them led to a solution. Langlands was a mathematician. He has this famous letter essentially spelling out how it seems likely that there's a lot more connections like that. He even got a little bit more specific about the nature of the connections, such that you might imagine this large map, and you've got this valley over here and this mountain over here and this set of plains over there. There's a lot of mathematicians who would characterize their work as being part of trying to understand the threads on this map. The progress there, it's not even "Here's this one specific problem that we know will be solved by that connection." It's more that time and time again, there have been cases where big problems were knocked down by finding connections, such that it's almost preemptively finding the connections. It's actually very interesting. Anytime you run into a mathematician, ask them whether the character of their work is more akin to the Langlands program or to targeting one particular problem. You get a certain bifurcated split there.
The possibility of AIs being supercharged connectors feels like it might be an amplifying tool in that pursuit. It's hard to measure, though. This cuts to what was said earlier: How do you assign a score to say, "Yes, you've done it"? If it's knocking down a problem, you have a clear way of saying, "Yes, you've done it." You can write the headline. You can have your PR move as the AI company to say, "We did it." Whereas if it feels like that was the right connection to draw, you can write theorems around it. That's the nature of what the papers in that field look like. But it will require a lot more human in the loop to say, "What was the kind of connection that we were going for?" That's the guess on what most of the useful progress from these models will look like in the next five years. It's just really filling in that landscape of connections that you can draw if you're an expert in multiple fields.
As has been pointed out, it's kind of surprising we haven't already had this. It would be curious to know at a technical level what causes the unlock there. On the one hand, you can paint an explanation in your head for why you could be an expert in all of these things and not be drawing those connections. When the method of reasoning is this autoregressive chain-of-thought phenomenon, autoregression is actually a really weird way to produce stuff, if you think about it. You're an intelligent person. Imagine I've locked you in a box, and the only way you have of interacting with the world is that you receive a slip of paper, and someone says, "Can you predict what will come next?" You predict what will come next, and then your memory's wiped. You get another slip of paper. Imagine that was done a whole bunch of times, and then what comes out on the other end. They say, "Look at this essay that you wrote." You might look at that and say, "This is awful. That's not the essay that I would've written." The process of repeatedly predicting something is just pretty different from how you would think as a writer to compose it and think it through. In particular, what would probably happen is that you're a slave to your context. You might be answering some question about a particular field, so you draw on all the context around that. But the connection where all the substance is going to come from is, by its nature, a very unlikely one. You can do all the RL that you want to try to get better in some way, but what's the thing that's specifically upweighting and incentivizing making these unlikely connections when the vast majority of them aren't the predictable next token that would come in there? So it might be the case that you just have this intelligence locked inside that box, but it's a weird way of interacting with it.
The thing that is curious about is: do you ever get any fruit by questioning the premise of how tokens are generated? It doesn't think it would be as simple as manipulating the temperature, but are there any things that you can do that take the existing level of intelligence but find the right ways of sparking those connections that unlock these sorts of things that we've seen? Or do you just need a little bit more intelligence, such that at the level of prediction, it's predicting that it should be making that lightning bolt to another field? It is more productive to reason, instead of architecture or even loss function, about data. We have diffusion models that do text, and the kinds of things they produce are not of a wholly different character. They've just not been explored as much. The more relevant thing is: what is the data on which whatever architecture or loss function you have is incentivizing you to produce? It does seem like they're getting better. Forget about math. We did have a couple of examples of this kind of thing, but if you just look at why they're getting better at being autonomous agents, they're in an environment where they're autoregressively producing the step that says "Let's step back and do a search over the whole codebase," and then "Let's step back and assess my mistake," is the thing that works. It is assumed what happened in the case of progress in science or maybe in math is you have frontier math-like problems. Mathematicians have specifically designed them because they require connecting together two different fields. There's all kinds of clever, partially synthetic ways to make harder and harder problems like that that require these kinds of connections—for example, by eliminating assumptions and still requiring the AI to get to the answer—and then it doesn't really end up mattering what the loss function is. It's really about, can you come up with an environment that incentivizes this ability? It feels like you should be able to.
It would just be pretty surprising if, over the next three years, there weren't a lot more of those lightning bolts. This is an important thing to think about. We often think about how smart a single system is. And we don't think about AIs having advantages that are more the result of other facts about them. So in this context, the key fact about them is that we can just parallelize and arbitrarily scale them. Whatever level of capability they have, it's not just one idiosyncratic genius in the history of mathematics who makes a few connections and then dies in a duel. It's universally applying that waterline across all problems that are accessible at that level of capability. This is among the many advantages that digital minds inherently have that we don't think enough about. The other ones being that they can merge all their knowledge together—or at least that there will be techniques that allow this to happen—and that you can spawn off copies with identical levels of knowledge. This parallelization is quite an important property.
Even if they're not as smart as human mathematicians, the fact that for PR reasons the AI companies are just throwing billions and billions of dollars at this means that quantity has a quality all of its own. That seems in the right direction. If we take that conversation between Montgomery and Dyson at the IAS that suggests some connection between the Riemann hypothesis—or the Riemann zeta-function zeros—and random matrices, that feels like the kind of thing that you could try to automate. You have agents representing expertise in all these fields. We all know that an institute is smarter than an individual. The reason for having people all in the same geographic location is that you want those serendipitous conversations to happen. What does it look like to engineer those between agents? It's interesting, because you point out that you can pool all your knowledge, but there is really wonder if one of the advantages is that you can do the opposite of that. Sometimes when an AI is failing, it's because it gets into a bad chain of thought and it's really hard to get it out. So you say, "I'll just start again." Same deal with humans. Sometimes you start thinking about it in a certain way, and what's required is to just back up. There are stories about people trying to prove something for a long time, and
There are stories about people trying to prove something for a long time, and then at some point they say, "Hang on a second. What if I tried to prove that it's impossible, or prove the opposite?" Unwinding your own context and going at it with a fresh mind could be systematized by having multiple different agents deliberately given different pieces of context and trying to compare and contrast. Humans do not have the same level of manipulation on their own context.
In the AI and math series, the first episode will focus on when AI systems solved the IMO, specifically one problem that AI failed on, which is one that a lot of very smart students failed on. Terry Tao also failed on it. People were very mad at the problem because they called it a troll problem. There is an elegant way of going down what feels like the solution based on the context of it being an International Math Olympiad problem. The character of the solution is really enticing, but it is hard to prove that it is the best because it is not. There is a brain-dead solution that is the best.
The relevance to the AI story is that for a human, what is required to answer that question is to escape your context—escape the context of being in the IMO, escape the context of the way you have been trained to solve these contest math problems. If you approached it like a brain teaser thrown at someone off the street, they would probably answer it well. The same applies to human research in other contexts: being able to refresh your thinking and come at it completely differently.
Of all the advantages that digital minds have, a more systematic approach to refreshing your thinking might actually be one of them. You can spin off two agents, one trying to prove it and one trying to disprove it, one who tries it this way and one who tries it another way. They deliberately have different contexts. A common concern people have about AIs is entropy collapse where they all think the same way because they are trained in similar ways. This is why they are bad at writing—they go down the same path and have similar patterns of speaking. But maybe the key advantage AIs have is that you can systematically increase entropy by systematically trying out both the negation and trying to prove the positive of any given statement, or being able to systematically give different agents different biases.
It seems like an important thing in the history of human science is that Einstein was really motivated by this bias that things should look the same in different reference frames. He had multiple other biases, but that one was very formative in his thinking. You can systematically survey a bunch of heuristics and see which ones are being productive on a given problem. Einstein would be an interesting example because he has this bias toward things being relative, and he also has a bias toward "God should not play dice." You want to make sure you do not accidentally have all your LLMs be Einstein, because you might halt progress on quantum mechanics. There is not a correct heuristic for science—you just need multiple independent research programs with their own heuristics.
There is a certain design question about how exactly you describe the different approaches. The easy one is whether you are trying to prove it or disprove it. The harder one would be to say what are all the tactics you could take to prove this, and make sure you are applying sufficient breadth to exploring them. These models can handle many things when equipped with a good harness like Cursor. Grant Sanderson started publishing episodes on Bilibili for a Chinese audience, but everything uploaded there needs the sponsored segments cut out. Normally that would have meant asking editors to go back through all the old episodes, cut the ads, and re-export everything. But in about as much time as it would have taken to send that Slack message, Cursor can be told to do it instead. For podcast research, there is a whole repo with every single book and paper relevant to prepping recent episodes. Everything can be thrown in there because the Cursor harness is extremely good at helping the model figure out exactly what to pull, whether from the repo or from the web, to answer questions while researching.
AI in math is making much faster progress than everything else, and people point to the verifiability of the domain as the key reason. That is one of the two important reasons, but people really neglect the other one. A tangential question to why AI is making so much progress in math is why it has been so slow at computer use. A computer is very verifiable—whether an Etsy package is coming or an event is booked are extremely verifiable things to survey. What computer use lacks is grindability. Because websites have bot detectors and it takes a tremendous amount of compute to run parallel rollouts, it is very hard to run a thousand parallel rollouts of the same checkout flow on Amazon. You could try to build clones of every single website, but that is very labor-intensive and slows you down.
The reason you currently need to do so many parallel rollouts to learn a skill with deep learning is that sample efficiency has not been solved. With code, you can containerize a given level of progress in a repository and then spin out hundreds of parallel containers and say, "Try to implement this feature," and it is totally deterministic. Because it is deterministic, you can solve the credit assignment problem because you know that whatever caused this rollout to succeed and this one to fail, the diff is the thing that worked. If you have situations that are starting off at different starting points, this credit assignment problem becomes much harder to solve. Most things in the real world are very hard to containerize in the same way. Coding and math are exceptions to this rule. If you are trying to figure out how to build a new business that succeeds or how to go trade in the markets for a day and make money, the fact that you have to interact with the real world and things change day after day means you cannot keep replaying and grinding and farming the simulator. Math is the exception, and this is an important driver of progress in this domain and also in coding. It is not just verifiability—it has to be grindable.
The third reason people point out that AI is making fast progress is they focus a lot on Lean and formalization. Lean just does not matter that much for the current level of progress in AI. Why is AI able to disprove the conjecture about the unit distance problem? They released the chain of thought, or at least a rewrite of the chain of thought, and it did not have any Lean in it. The process-based supervision that Lean provides, where you know each step is correct, seems less relevant than just having this grindable outcome that is verifiable. Naively you might think Lean provides something unique for math because you are able to see if it can prove it. You have old-school software that can tell you yes or no, and you use that as your VR. The initial attempts corroborate this point—initially DeepMind basically does that with everything in Lean, and then the next year it is all in natural language.
There is a yet-to-be-explored benefit of that formalization domain. At the moment you still need a human reviewing that counterexample to the unit distance conjecture to say, "Looks good." That provides a certain bound on how endlessly explorable things are. If you consider AlphaGo or AlphaZero-style systems, they are off in their own universe playing a bunch of Go and exploring themselves, potentially going off the rails of what any human needs to look at, but they still have this automated verifiable reward. It is not just that you can do RL on that—it is also that you basically never have to check in, and you can just pour compute at them exploring the universe of Go. With Lean, you could imagine having a basically endlessly running program that is constantly trying to extend Mathlib. Mathlib is this GitHub repository that is basically all of math written in code. It is very far from all of math, but they want it to be all of math. It is written in code where you can ask, "Is this proof correct?" It is very labor-intensive to write these proofs. There is a whole subcommunity around it. But you could imagine having an AI where you say, "Simply try to extend Mathlib." Maybe it is a fork of it so that it does not have trash in it, because people have a certain taste for what they want to be in there. So you have your fork of the pure AI Mathlib, and it just goes and it does not stop. It does not need anybody to check in on it. It could just keep going. It might come up with its own conjectures. It might come up with its own theories and different definitions. Maybe many of them are useless, but it just has this infinite tree that it can grow out. That is a very unique thing that math has that nothing else has, where you could press go and just pour compute at it, look away for ten years, and then come back and say, "What do you have?" There is going to be something. Then there is a question: is it useful or not? How do you suss that out?
There are two different ways that Lean is important in this story. The first one is how you could let go, not even check in, and progress will be made. You can do that with Go. You do not think you can do that with natural language math. Karpathy's auto research idea involved writing one Python file that does basic LLM training, and then had a repo where LLM agents would try to make modifications to the file, and if it sped up the speed run, the modification stays. Eric Jang, who explained how AlphaGo works, did a similar thing when he was trying to build a very strong Go bot. It is really good at running an experiment and going down that path, but it is bad at stopping at dead ends and doing extremely parallel things. This will probably change in the future. It is very interesting to think about what it looks like in the limit. This is fundamentally what the human institution of mathematical research is—it is a library extended in interesting and useful ways. This way you do not have any outcome-based supervision. There is no outcome that you are trying to incentivize, but you have a process. You know the steps are correct, you just do not know if it is going in an interesting direction. If you were doing that, you do not want to completely go off the rails and do a random walk through the space of logic. You would probably want some supervisor model that is trying to provide heuristics on whether it is useful or not.
Terry Tao was talking about one research project that tries to exhaustively search the space of possible algebras. You could imagine different axioms that you apply to algebraic systems. When we come up with group theory, there is a certain axiom system that looks like arbitrary rules unless you know the motivation. What if you tried all of them? Do any of them yield useful things? The vast majority of them are just trash in some way. It all collapses to no interesting results. But every now and then, there would be this little island of a completely different type of axiom system that at the very least seems rich in terms of the number of theorems that can come out of it. That is bread and butter for what you would imagine automated provers being good for, exploring that space and seeing which one of them turns out to be something. Maybe one of those islands actually turns out to be something you can retroactively put motivation on, to say this is the kind of structure it is trying to get at. In the same way that you could imagine looking at the axioms for a group, not knowing that it is about symmetry, but you retroactively realize this is very relevant to studying symmetry. You could imagine results of that flavor, but instead of just exploring possible algebra systems, it is exploring all possible logical consequences of any kind of axiom.
On the point about whether you can provide process-based supervision without Lean, DeepSeek had their DeepSeek Math model. They released a paper on how they trained it, and it was quite interesting. The problem with natural language proofs is you do not know if it is correct or not. They have a verifier, and the verifier is trained by a meta-verifier that makes sure that for all the problems they are training this model to solve in the art of problem-solving, the verifier is giving good feedback. It works. It is interesting that natural language verification with some sort of meta-verification seems to work so far in the published literature. It also seems to work in the published products that are being used. If you look at coding agents, they are getting better and better at writing clean code and refactoring code. There are process-based "LLM-as-a-judge" systems providing taste and saying, "Is this a clean way to write this function? Are there duplicates of the same kind of modular forms?" That should also work for mathematics. It seems more plausible for math than anything else, even if you are only working in natural language, that you could trust a verifier.
If given two essays that students wrote, LLMs would be able to say which one is more accurate and insightful. So why cannot you just have a verifier saying, "Is this a good piece of writing or not?" Maybe the ultimate failure there is that even if they are good at discriminating between a B essay and an A essay, they are not actually good at discriminating between an A essay and a thing you actually want to read, something that would be followable on Substack and insightful. They actually end up preferring uninsightful pieces of writing. On the math front, the step to simply know if a proof is correct or not lends itself to an automated verifier, even in natural language. You could probably still make a ton of progress. The tree of logic out of Lean is still preferred, just in that you can really go off the rails. There is no constraint on the previous way things had been phrased before. Everyone talks about move 37 in AlphaGo. What is the thing that lends itself to going outside the prior heuristics? It seems productive to have a disconnection from the rest of the world in that exploration, as a complementary research pursuit to the natural language math front.
The other relevance of Lean would be, let's say you have your pure natural language RL environments and a pure natural language set of proofs. People say, "Proceed, AI mathematicians," and they generate ten papers a day. If there is any error rate to that at all, Alex Kontorovich has talked about this—it becomes insufferable as a mathematician. Every single time you see one of these, you do not know if it is worth your time. Even if 99 out of 100 are right, it is not known if it is worth the time because it is really labor-intensive to find what that error would be. It is really frustrating to spend all your time on a paper that was trash. Having something that is able to give you that green checkmark that says, "Even if this is going to be complicated to understand, even if it is going to be a pain, you at the very least know it is correct," every other field would kill for that. Math has that. If the models are also able to take their natural language proofs and formalize them, that seems huge. Every field would love to have something like that. Lean is maybe overrated regarding its importance as a VR environment for progress in math generally, but it should not be written out of the story.
This extension of Mathlib is loved as a metaphor for what is going to happen to civilization pretty soon. For millennia, humanity has built this corpus of knowledge and understanding, and everything that we have is now distilled into these models. At some point, the models will just extend that arbitrarily.
On the writing front, there is a theory of why writing is making worse progress than these other domains. One reason is that they are bad at judging not only A versus B, but they get totally derailed by B*, which is this shitty essay that hits all the bells and whistles that A is supposed to hit. The reward hacking thing just goes off the rails. But the other important thing is that writing is not modular in the same way that code and math are. You can write a function many different ways, and they do the same thing. Of course you want it to be clean, but at the end of the day, if it works, it works. Same with lemmas in mathematics. You can have some end product that is different from the way it is produced. Code is the thing that produces some end product, and you want a functional end product. Whereas in writing, the end product is directly the thing the AI is producing. Each paragraph, sentence, and word matters because that is the substance. It is not some separate thing produced out of the writing. It cannot be slop in the way that code can be slop and still produce the outcome you want. Progress has been made with agents writing not just functional code, but clean code. The same progress that lets you go from merely functional to a clean and mergeable PR should also result in clearer writing.
There are many ways in which LLMs are terrible writers. But for a lot of writing consumed, it is better to just copy-paste it into an LLM and say, "Explain this to me." The explanation will be better than the thing produced by the human. It is funny that these are said to be such terrible writers, and yet the revealed preference is to have an LLM explain it. Even when talking to a human expert live on a call, if it is a piece of knowledge that only they have that is not encoded in the distribution, the preference is for them to explain it. But if in order to understand that, a more basic concept needs to be understood, the preference would be for it to be socially acceptable to just say, "Let's pause here. I'm just going to ask an LLM how that works, and then we can come back to your special piece of knowledge." That is distillation, an explanation. If thinking of quality as an essay writer, if given a book to read and wanting a book report, the LLM might give a better book report. But what people are really getting at when they say it is bad is, what is writing? It is not just the distillation of preexisting ideas. It is not just how you explain clearly, because they are good explainers. It is about what the insight is.
This is where autoregression is a very weird way to generate things. When you are writing, you sort of know that in order for it to be good, you have to have an element of the unpredictable. It is not just increasing the temperature in your mind. It is knowing exactly the correct point when you want to make an unpredictable move, and that that is going to be what is more insightful. Even if it is better at explaining a preexisting thing, what generated that book that you wanted distilled in the first place? It was not an LLM that generated it and you just needed it. It was some author who, through a lot of exploration of ideas in the world, decided what aspects were interesting and what ways of presenting it formed a coherent, well-motivated narrative. They put that all together in some way. If they are a good author, you would probably err on the side of reading their book instead of the distillation. Still, what makes it worthwhile to explore at all in the first place and want to upload it at all? It is that side of it that people cite when they say LLMs are bad at writing. It is that element of unpredictability, of deliberately choosing something novel that is very directly contradictory to the way things are typically produced. LLMs are also really bad at building really good mental models of people, which is a very important skill in writing.
Andy Matuschak and another collaborator tried to teach LLMs to write good spaced-repetition prompts. Even though it seems like a totally random skill, people are talking about recursive self-improvement in a year, and these things cannot write good flashcards. They tried many different kinds of techniques. They tried to RL open source models. They tried all kinds of things, including chain of thought and a big prompt sent to the best closed source model. The key constraint was that writing a good card is about projecting somebody's mind in three months. What is the way in which they will associate the question? What kind of answer will they be thinking at that moment? Is the elicitation that inspires the detail you actually want to take away from the passage you are trying to make cards about? Writing is similar to this. If you are writing something, the reason it is such an enervating process that takes so long is that with each word or each sentence, you have to be thinking: what is happening in my reader's mind right now? Even if the phrasing is flipped around so the end phrase goes to the beginning and this is the first image that comes to your mind before you read the rest of the sentence, maybe autoregression is bad at that. This is maybe a more diffusion-like property of considering the whole rather than going sentence by sentence. But also it requires a lot of mentalizing, which these models weirdly struggle at.
An experiment tested whether facial mimicry aids emotion recognition. Participants completed an emotion-reading test before and after receiving Botox injections. After Botox froze their facial muscles, their ability to identify emotions from facial expressions declined noticeably. The underlying mechanism is that people subconsciously mimic observed expressions with their own facial muscles, and this physical mirroring helps decode the emotion being viewed.
This provides an analogy for why AI models struggle with theory of mind. Models lack the equivalent of "face muscles"—they cannot simulate another agent's internal state through embodied mimicry. While they have read vast amounts of text describing how humans think and feel, they lack the hardware to run the same simulation humans perform automatically. Their understanding remains fundamentally alien rather than experientially grounded.
Jane Street maintains an exceptionally high retention rate. Employees often cannot clearly identify their colleagues' official roles because researchers, traders, and engineers all perform overlapping work. A trader may spend significant time on research, while a researcher may do substantial coding. This cross-functional environment allows individuals to pursue growth across multiple domains rather than remaining confined to narrow job descriptions.
LLMs perform well when explaining well-known concepts but frequently become confused when the topic grows more specialized, producing explanations that increase rather than reduce confusion. A competent human tutor can often resolve the same confusion in minutes.
Grant's approach treats LLMs primarily as an enhanced search tool. He asks them to recommend high-quality human-authored resources rather than relying on the model itself for deep explanation. In one case, he requested video recommendations on semiconductors; the model incorrectly attributed an existing video to 3Blue1Brown, but the actual linked resource proved more useful than continuing with the LLM.
The key principle is "who matters more than what." When selecting courses or books, the quality of the individual explainer often outweighs preexisting interest in the subject matter. Wikipedia articles represent a local minimum of correctness achieved through crowdsourcing, whereas single-author resources such as the Stanford Encyclopedia of Philosophy or the Princeton Companion to Mathematics deliberately craft motivational narratives and accept temporary simplifications that are later refined.
The most effective learning occurs when a high-quality human artifact—an article, book, or video—first organizes concepts and builds motivation for subsequent ideas. LLMs are then used only for targeted clarification around branches the human resource has already identified.
One effective setup involved simultaneously viewing Steven Strogatz's university lectures, reading his textbook Nonlinear Dynamics and Chaos, and consulting an LLM. The human-curated sequence of concepts and problems provided the necessary structure; the LLM served only for occasional pruning.
Good human teachers can recognize when a student's question reveals a fundamentally different mental model and then reframe the discussion accordingly. LLMs tend to be overly placating and rarely challenge or redirect the framing of a question. An "A+ explainer" not only answers but identifies when the student's approach itself is unproductive and gently redirects it.
Students considering mathematics careers should first understand where funding originates and what value they would actually provide. Many students pursue mathematics simply because they have been rewarded for solving problems correctly and wish to continue that activity, without examining the economic or institutional mechanisms that would pay them.
Different mathematician roles derive compensation from distinct sources: brand value to universities, public-good arguments underlying NSF grants, or direct teaching value. Even in a world with automated theorem proving, the social roles of mathematicians are unlikely to disappear entirely. Society continues to value basic science and trusts the mathematical community's internal judgment about which problems merit attention.
Teaching itself appears especially stable. Parents consistently prioritize spending on high-quality education for their children, and the relational, mentoring aspects of teaching extend far beyond mere explanation—functions LLMs are unlikely to replicate.
Grant notes that his own path into mathematical exposition as entertainment was largely accidental. He recommends that students deliberately analyze where economic value flows rather than relying on serendipity.
Even if AI systems eventually solve the Millennium Prize problems and generate novel mathematical fields, substantial demand will remain for humans who can translate those AI-generated insights into human-understandable terms. The economic value of curating and directing new mathematical output toward useful applications may actually increase as the volume of generated mathematics grows.
Progress in different mathematical domains will have varying degrees of immediate applicability. Algebraic number theory advances are unlikely to produce rapid practical payoffs, whereas progress in dynamics and PDEs already yields concrete engineering value. One research group developed simulation techniques that reduced the need for physical prototyping at Boeing, saving billions of dollars.
Pure mathematical insights are more likely to produce incremental improvements—more efficient CFD simulations, better wing shapes, or fluid engine design—than sudden revolutionary breakthroughs. Mathematicians working at the interface with engineering and materials science are well positioned to identify which AI-generated mathematical results have physical relevance.
Rapid AI-driven progress in pure mathematics could expose an uncomfortable possibility: that much contemporary mathematical research has become disconnected from physically applicable questions. If 10X or 100X acceleration produces no visible downstream effects in engineering or technology, external observers may question the premise that pure mathematics reliably feeds into practical domains. This would place pressure on the traditional justification for funding basic mathematical research.
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