one of the reasons it's interesting at all is that there's a spiky frontier to AI, and math is just right there in one of the spikes. But there's a fractal nature to that spikiness, because when you zoom into the specific progress within math, you have some things that are a lot easier than others. If we just think about IMO, which is old news at this point. It's been two years since they're really doing quite well. They would have gotten a gold in 2024 if not for the following reason. They're very good. They just cold-solved geometry basically.

The 'fractal spikiness' framing reveals that AI's uneven progress within math mirrors its uneven progress across fields — a key insight for predicting where AI will and won't excel.

If we focus on the Riemann hypothesis, what would it look like to solve that? These things are extremely good at a specific domain of knowledge, knowing it very deeply, and then knowing another domain, and another. You've pointed this out. It's bizarre to have something with this superhuman breadth that knows all the fields so well, and yet isn't finding those lightning bolts that connect them. I think we're starting to see sparks of it actually finding connections between the things it's an expert at.

This pinpoints the precise gap between current LLM capability and frontier mathematical discovery: possessing knowledge across fields versus synthesizing it into genuinely novel connections.

This is a side tangent, but it's a fun story. I don't know if it was over lunch or something like that, but you have this number theorist who is just trying to understand the statistical correlation between pairs of zeros of the Riemann zeta function. Freeman Dyson, a physicist, is like, 'I know that expression. That expression comes up in studying the eigenvalues for random Hermitian matrices,' which was something that comes up in studying the energy levels of a nucleus. The idea that the statistics of those two seemingly different things were the same prompted an exploration of whether there are aspects of random matrix theory that might be relevant to the Riemann zeta function.

The Montgomery–Dyson story is a perfect concrete illustration of how serendipitous cross-domain connection — the very thing AI might systematize — has driven some of the deepest mathematical progress.

good mathematicians prove theorems, great mathematicians come up with conjectures, and the greatest mathematicians come up with definitions. That's more or less exactly your framing here. We need the conjecture generator and then the definition generator. That's the premium-tier mathematician. I don't understand how exactly you'd make that a benchmark. Usually, when I think of the word benchmark, I'm thinking of something that is a goalpost. The ball is through the goal or it's not. You can clearly say, 'Yes, this is done.'

This hierarchy — theorem-prover, conjecture-generator, definition-creator — clarifies exactly which cognitive levels AI has reached and which remain unmeasured because they resist quantification.

The kinds of things you can't make benchmarks for are also the kinds of things, at least in the current paradigm, you can't easily train for. There's really no fundamental difference between a benchmark and a training environment. It's very easy to come up with some dichotomy of, 'here's a deep reason why AI can't do a certain thing', and then it turns out you're just thinking about it the wrong way, and actually it can do it pretty soon thereafter.

The equivalence between benchmarks and training environments is a crucial epistemological point: our inability to measure something may itself be the barrier to AI achieving it.

It's a perfect example for your case, because describing why it was a valuable insight does not come from immediate utility. Certainly, if you're thinking about RLVR environments, this is going to be really hard to do. But it's interesting to note that even with human verifiers at the time, it took a really long time to recognize it as being useful. With Einstein and GR, people could feel this was a good theory right away. What makes Galois theory such an interesting example is that you literally have this hundred-year segment of an idea that flows through many different people's heads before it settles into something the math community agrees is good.

Galois theory's century-long verification arc shows that the most transformative mathematical ideas may be fundamentally untrainable with any reward signal humans could construct today.

what is the way of measuring progress that's not based on solving a problem, but that is somehow capturing the instinct inside Galois's mind when he says, 'I think there's something here'? What's the instinct inside Lagrange's mind when he says, 'I think this is the right way to think about it'? What's the instinct inside Liouville's mind when he says, 'These scattered notes from this long-dead youngster might have something to them'? It's so hard to put a finger on that. A different series of videos I'm making right now is about the whole 'compression is intelligence' idea. Even though this isn't really the angle I'm taking, there is something to the idea that the smaller expression that's more predictive feels more intelligent. So I wonder the extent to which you can give some kind of verifiable reward around not just whether you solved it or what it is solving, but around the smallness of the concepts required to do it.

Sanderson proposes compression — conceptual minimality — as a potential proxy for mathematical insight quality, offering a rare concrete handle on how to reward Galois-like intuition.

There is a difference between proof and explanation, and I think you're getting at the importance of that distinction. Yeah. 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.

The shift from proof-generation to explanation and consolidation as the core human contribution may be the defining reorientation of mathematical practice in the AI era.

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. 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. 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. I think this is pertinent to AI. I used to think that AIs would become these automated theorem provers, but the role of mathematicians was going to shift towards my job, explaining these things. Now I suspect that actually they'll also be quite good at doing that, probably better than most humans are at explaining and distilling.

Sanderson's reversal — from 'AI proves, humans explain' to 'AI does both' — forces a harder question about what genuinely remains for human mathematicians.

Obviously, AI in math is making much faster progress than everything else, and people point to the verifiability of the domain as the key reason this is happening. I think that's 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: why has it been so slow at computer use? A computer is very verifiable. Is my Etsy package coming? Is my event booked? These 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's very hard to run a thousand parallel rollouts of the same checkout flow on Amazon. It's not just verifiability; it has to be grindable.

The 'grindability' concept — the ability to run massive parallel rollouts in a deterministic, containerizable environment — explains why math and coding outpace all other AI domains and predicts which future domains will see similar breakthroughs.

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's 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.

This frames automated mathematics not as proving known conjectures but as discovering entirely new algebraic worlds — a much more radical and underappreciated use case.

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 don't know if it's 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're training this model to solve in the art of problem-solving, the verifier is giving good feedback. It works. It's interesting that natural language verification with some sort of meta-verification seems to work so far in the published literature.

This reveals that formal proof languages like Lean may not be strictly necessary — natural language math verification with meta-verifiers is already showing real results.

If there's 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 don't know if it's worth your time. Even if 99 out of 100 are right, I don't know if it's worth my time because it's really labor-intensive to find what that error would be. It's really frustrating to spend all your time on a paper that was trash. Having something that's able to give you that green checkmark that says, 'Even if this is going to be complicated to understand, even if it's going to be a pain, you at the very least know it is correct,' every other field would kill for that.

This pinpoints the killer app of formal verification in an AI-flooded research landscape: not proving theorems, but certifying them so human attention isn't wasted.

I also love this extension of Mathlib as a metaphor for what's going to happen to our 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.

Grant reframes AI progress not as tool use but as a civilizational inflection — the moment humanity's knowledge library begins writing its own next chapters.

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's different from the way it's 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.

This offers a structural explanation — not just a qualitative one — for why AI excels at code and math but lags in writing: the output IS the process, leaving no room for slop.

This is where autoregression is a very weird way to generate things. When you're writing, you sort of know that in order for it to be good, you have to have an element of the unpredictable. It's not just increasing the temperature in your mind. It's knowing exactly the correct point when you want to make an unpredictable move, and that that's going to be what's more insightful. Even if it's better at explaining a preexisting thing, what generated that book that you wanted distilled in the first place? It wasn't 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.

Grant identifies a deep architectural tension: good writing requires strategically placed unpredictability, while autoregression is structurally biased toward the predictable.

The key constraint, it seemed to me, was that writing a good card is about projecting somebody's mind in three months. What is the way in which they'll 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're trying to make cards about? I think writing is similar to this. If you're writing something, the reason it's 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?

Grant reframes good writing as an act of mentalizing — continuously modeling the reader's evolving cognitive state — which explains why AI, weak at theory of mind, struggles with it.

The thought is that part of understanding the emotion you're looking at is doing it yourself. At a facial level, you're moving your face muscles. You see that, you mimic that, and you're like, 'Oh yeah, that's anxiety,' at some very subconscious level. So in that sense, if it is the case that models have bad theory of mind, sure, they know everything because they've read what everyone wrote. But at the level of actually being able to put themselves in your shoes in the same way that my face muscles are mimicking your face muscles—that's what helps me understand how you feel—it's not surprising at all. They don't have face muscles. Their brain works completely differently. It's like an alien trying to empathize.

The Botox study becomes a striking analogy for why LLMs' theory-of-mind gap is structural, not just a training data problem — embodied simulation may be fundamental to understanding others.

Even pre-LLM, I feel like a relevant insight in learning was recognizing who matters more than what. My advice to any college student when they're choosing what courses to take: care a little bit less about your preexisting interests, because they're kind of arbitrary right now, and care a little bit more about whether the person teaching it is a good educator and someone you resonate with. In choosing what books to read, who the author is maybe matters more than whether it's a prior interest. LLM explanations feel to me at the moment a lot like Wikipedia, which is to say, amazing. Imagine a world before Wikipedia, how long it would take to find and suss everything. But nevertheless, what's the most useful part of a Wikipedia page? It's often just the references at the bottom.

Grant's 'who over what' heuristic for learning recontextualizes LLMs as sophisticated reference-finders rather than primary teachers — with practical implications for how to use them.

If we have almost-automated theorem proving, and let's say they're also really good explainers so you even get the human understanding—I think a lot of the social role that mathematicians serve actually doesn't change that much. As a public, we still feel there's value to basic science, and we trust the judgment of mathematicians to determine where their time is best spent. The prestige comes from within that community. Maybe it shifts away from theorem proving and towards good definition writing. Maybe it's that museum curator idea. I actually think teaching is one of the most stable post-AGI jobs that there is, because it's so relational. This is where parents want to spend their money if they have an abundance of wealth: on good teaching and good educating.

Grant argues that the social and relational functions of mathematicians and teachers are largely AGI-proof, offering a concrete and non-obvious view of which academic roles survive automation.