Grant Sanderson on AI and the Future of Math, Conceptual Breakthroughs, and Human Curation

Grant Sanderson, creator of 3Blue1Brown and now documenting AI’s progress in mathematics, argues that because AI is advancing fastest in maths, the field is a live preview of what AI progress everywhere will look like. He and Dwarkesh Patel work through the anatomy of a mathematical breakthrough — the difference between throwing a ‘lightning bolt’ between two known fields and ‘building a mountain’ of genuinely new theory — why the deepest advances resist benchmarking, why maths is uniquely ‘grindable’, and what is left for humans when the machines can both prove and explain: curation, taste, and teaching.

Key ideas

1. Gold at the Math Olympiad was ‘just another benchmark’. Asked three years ago whether an AI winning IMO gold would amount to AGI, Sanderson said it would be one more benchmark passed with no ‘aha’ moment — and was right. AI now cold-solves IMO geometry in nineteen seconds by brute force but stumbles on playful combinatorics. The capability frontier is not just spiky but fractal: zoom into maths and some tasks are far easier than their neighbours.

2. A great proof can take three shapes, with very different consequences. A solution to something like the Riemann hypothesis might be (a) a lightning bolt connecting two fields the AI already knows deeply — like the chance Montgomery–Dyson lunch that linked the zeta zeros to random-matrix theory — which is human-parsable and exactly what LLMs’ ‘superhuman breadth’ should be good at; (b) mountain building, a genuinely new theory demanding a different kind of intelligence and much harder to digest; or (c) raw hustle, a correct but incoherent thousand-page slog. The nightmare is an AI building an alien mountain that looks right and isn’t — the abc-conjecture / inter-universal-geometry scenario at machine scale.

3. Theorems, conjectures, definitions. ‘Good mathematicians prove theorems, great mathematicians come up with conjectures, and the greatest mathematicians come up with definitions.’ The premium tier — generating good conjectures and definitions — resists benchmarking: you cannot headline ‘we promise everyone thinks it’s a good conjecture’. Progress there will register not as a toppled benchmark but as a subjective tone shift in how working mathematicians describe collaborating with the models.

4. The verification loop on a real breakthrough can run a century. Group theory took roughly a hundred years to be recognised as valuable: Lagrange planted the idea that symmetry was the right lens, Abel and Galois pushed it (Galois’s notes rejected by the academy, then rescued by Liouville twenty years on, and formalised by Jordan twenty years after that), and only in the twentieth century did Gell-Mann use it to predict quarks. No ‘verifiable reward’ fired for a century — so any system that only rewards solved problems misses the Galois instinct of ‘I think there’s something here’. Sanderson’s suggestion: reward compression and elegance (a Kolmogorov-complexity flavour), not just correctness.

5. Proof is not explanation — and curation is what survives. Citing Timothy Chow’s notion of an ‘unsolved expository problem’ (proven, but not yet understood), Sanderson notes maths often has ‘the reach exceeding the grasp’. He used to think AI would prove theorems while humans explained; now he expects the models to out-explain us too. What remains is curation — the mathematician as an art-museum curator, or the trusted human whose taste you follow — because motivation is a social phenomenon. Tellingly, the great originators (Einstein, Shannon, Feynman) also tend to be lucid expositors.

6. Maths races ahead because it is verifiable and grindable. Verifiability alone is not enough — computer use is verifiable yet slow, because you cannot run a thousand parallel checkout rollouts without tripping bot detectors, and sample efficiency isn’t solved (‘sucking supervision through a straw’). Maths and code can be containerised and run in massively parallel deterministic rollouts, which cracks credit assignment; most real-world tasks cannot. Formalisation (Lean) is probably overrated for current progress — DeepMind’s IMO work went from Lean to natural language — but has one unique superpower: a Mathlib-extender that, like AlphaZero at Go, could run for a decade without a human ever checking in, plus the green checkmark ‘every other field would kill for’.

7. Digital minds’ underrated edge is parallelism and context control, not raw IQ. Because autoregression makes a model ‘a slave to its context’, the winning move may be to spawn many agents with deliberately different contexts and biases — one trying to prove, one to disprove — systematically raising entropy at the prompt level to escape the trap that kept the unit-distance conjecture open (everyone assumed it was true). The caution: don’t accidentally make every model an Einstein, or you’d halt progress on quantum mechanics.

8. Why AI is bad at writing — and advice for students. Writing is not modular like code: each word is the substance, and good writing needs a deliberate unpredictable move plus theory of mind — projecting the reader’s future state (the Botox-and-empathy study as an analogy for why a model that can’t ‘mimic your face muscles’ struggles to mentalise). For students watching AI transform their field, Sanderson’s counsel is ‘who matters more than what’ (choose teachers and authors, not topics), use LLMs as a ‘souped-up Google’ to find the right human-made resource, and above all ‘understand where the money is coming from, what value you’re actually adding, and the connection between those two’. Teaching, being deeply relational, is among the most stable post-AGI jobs.

Content

Is an AI that does research-level maths already AGI?

Sanderson’s core heuristic is that the very things you cannot build a benchmark for are the things you cannot easily train for — ‘there’s really no fundamental difference between a benchmark and a training environment’. That is why IMO gold came and went without transforming the world: it was benchmark-shaped. The open question is how long maths can stay a narrow spike. If solving the Riemann hypothesis turns out to be a lightning-bolt connection between analytic number theory and quantum physics, that is squarely what LLMs should do well and ‘totally different from white-collar work’; if it requires building a whole new mountain of theory, that is a level of intelligence that would be surprising not to see ‘permeate into other aspects of the economy’.

Conjectures, definitions, and the century-long loop

The next benchmarks — generating interesting problems, and inventing the objects and definitions that create fields — are exactly the ones that resist scoring. Sanderson’s extended Galois story is the case study: a hundred-year relay (Lagrange → Abel → Galois → Liouville → Jordan → Gell-Mann’s quarks) in which the idea repeatedly failed the human verifier of its day before proving foundational. The lesson for AI training is that rewarding only solved problems cannot capture the instinct ‘I think there’s something here’; you would need to reward the smallness and predictive power of the ideas themselves. And even if predictiveness were all you wanted — automated engineers building starships no one understands — many people will still want understanding, so you would still train models to find the compressed representation, ‘the equivalent of the universal law of gravitation for Newton’.

Will we understand an AI proof — and what’s left for mathematicians?

Whether an AI proof advances human understanding depends on its shape: lightning-bolt connections are human-parsable (you only need the start and end points), mountains take real effort to climb, and raw-hustle proofs are the ones to fear. Referencing David Bessis’s ‘The Fall of the Theorem Economy’, Dwarkesh notes theorem-proving has always been ‘a parasite’ on the harder work of coming up with definitions — a credit-apportionment problem that AI sharpens. Sanderson’s shift of view: he no longer expects explanation to be the human redoubt, because the models will explain well too. What is left is curation and the relational role of a teacher — a point he returns to repeatedly, since ‘the way we get motivated to be interested in things is a social phenomenon’.

Grindability, Lean, and the endless Mathlib

The heart of the technical discussion is that maths’ speed comes from being grindable as well as verifiable. Deterministic, containerisable rollouts solve credit assignment (the diff that made one rollout succeed is the signal); the real world, changing day to day and defended by bot detectors, does not containerise. Lean and formalisation, Sanderson argues, matter less than people think for current progress — but enable something no other field has: an open-ended prover that extends a fork of Mathlib forever, exploring its own conjectures with no human in the loop, and could be left for ten years before you ‘come back and say, what do you have?’. Terry Tao’s exhaustive search over possible algebras is the same idea — most axiom systems are trash, but the occasional ‘island’ is rich enough to retrofit meaning onto.

Entropy, context, and the writing problem

Because autoregression is ‘a weird way to produce stuff’ — a mind that predicts the next slip of paper and then has its memory wiped — models are prone to entropy collapse and being enslaved to their context. The counter-move is engineered diversity: agents with different assigned biases, systematically exploring both a conjecture and its negation. Writing exposes the limit most sharply. Unlike code, it is not modular — the artefact is the output, so it ‘can’t be slop and still produce the outcome you want’ — and it demands theory of mind, projecting the reader’s mind three months out (Dwarkesh’s example of the near-impossibility of getting LLMs to write good spaced-repetition cards). Sanderson’s Botox study — people worse at reading emotions once their own faces are frozen — dramatises why a system with ‘no face muscles’ might genuinely struggle to mentalise: ‘it’s like an alien trying to empathize’.

Learning, and advice for the AI era

On using LLMs to learn, both land on the same place: they are ‘a lot like Wikipedia’ — amazing, but most useful for pointing you to the right human-crafted resource (the Stanford Encyclopedia, the Princeton Companion, a Strogatz textbook). ‘Who matters more than what.’ For students fearing their field is being automated, Sanderson — couching it as advice from ‘a YouTuber’, an outsider to the academy — urges them to understand where their salary actually comes from and what value they add, and to weigh teaching seriously, since a relational, coaching role is among the most durable in a post-AGI world. He closes on the awkward possibility that a 10× acceleration of pure maths might mostly reveal how little of it touches the physical world — though he’d bet some ‘meaningful leakage’ (PDEs into engineering, say) does reach it.

See also

• Grant Sanderson — the guest; creator of 3Blue1Brown
• Dwarkesh Patel — the host
• Sergey Levine on Robot Foundation Models, Generalisation, and Embodied AI — a companion read on where AI progress is fast and slow, and why real-world grinding is hard
• Large Language Models — the underlying technology whose reasoning style (autoregression, RL, verifiability) this conversation dissects