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Grant Sanderson – AI and the future of math
Overview
Dwarkesh Patel talks with Grant Sanderson (3Blue1Brown), who is producing a documentary series on AI progress in mathematics. The premise: math is the field where AI is advancing fastest, so it is a concrete preview of what AI progress will look like everywhere else. Three years ago Grant predicted that IMO gold would be 'just another benchmark' rather than an AGI moment — he was right, and the conversation digs into why the frontier stays jagged even inside math itself.
The spine of the episode is a taxonomy of mathematical breakthroughs and which kinds AI can currently produce. AI is now genuinely good at 'lightning bolts' — connecting expertise across two fields, as in the disproof of the unit distance conjecture and an Erdős problem solved by importing a Markov chain idea into number theory. What it hasn't shown is 'mountain building': inventing new conceptual frameworks like Galois inventing group theory, where the verification loop on whether the concept was even a good idea ran roughly a century. They also explore why math is uniquely 'grindable' for RL (unlike computer use or writing), whether we'll understand an AI proof of the Riemann hypothesis, why LLMs remain bad writers (theory of mind, autoregression), and what careers in math and explanation look like after automation — Grant's bet: curation, teaching, and pointing the machine in useful directions.
Topics
The jagged frontier, even inside math
Grant's 2023 prediction held up: IMO gold arrived and was absorbed as just another benchmark. His explanation is that spikiness is fractal — math is a spike of AI capability, but within math, geometry problems have been brute-force solvable in nineteen seconds since 2024, while combinatorics (the playful, puzzle-like category) still resists. Models would have taken IMO gold in 2024 had the problem mix leaned more toward geometry.
The deeper question is what a solution to a Millennium Prize problem would even imply about wider automation. Grant sketches three possible shapes for a Riemann hypothesis proof: (1) a cross-field connection, like Montgomery and Dyson's famous IAS lunch where a number theorist's formula for zeta-zero correlations turned out to be the eigenvalue statistics of random Hermitian matrices — exactly the kind of bridge LLMs with superhuman breadth should excel at; (2) 'mountain building,' as in Fermat's Last Theorem, where whole edifices (elliptic curves, modular forms) had to be constructed before the connecting question could be asked; or (3) raw hustle — a thousand-page, incoherent grind. Only the mountain-building version, he argues, implies an intelligence that would clearly spill over into the rest of the economy.
Galois and the century-long verification loop
The centerpiece story: why conceptual breakthroughs can't be captured by verifiable-reward training. Lagrange noticed that solving polynomials related to symmetries of algebraic expressions — a hunch with no solved problem attached. Fifty years later Abel proved the quintic unsolvable, but it was Galois who saw the right abstraction layer: study the symmetries underlying formulas rather than the formulas themselves. And the 'verifier' of his day rejected him — his papers were refused five times, he wrote his manifesto on abstraction from prison, and he died in a duel at twenty. It took Liouville twenty years to recognize something in his notes, Jordan another twenty to build modern group theory from them, and the twentieth century (Gell-Mann predicting quarks from group-theoretic structure) to prove the concept's fertility.
The upshot for AI: the reward signal for the most important kind of mathematical work — good definitions, good conjectures — can take a hundred years to arrive and was negative in the short run. 'Good mathematicians prove theorems, great mathematicians come up with conjectures, the greatest come up with definitions.' Grant's tentative idea for a trainable proxy is compression: reward the smallness and predictiveness of the concepts required, something like Kolmogorov-complexity-flavored elegance, because the end goal is human understanding, not just a checkmark.
Will we understand an AI proof? Proof vs. explanation
Whether an AI Riemann proof helps or hurts human understanding depends on its shape. Lightning-bolt proofs are inherently parsable — you state the start and end of the connection and experts can run with it (as happened with the unit distance counterexample, whose chain of thought mathematicians could actually read). A new mountain might resemble Mochizuki's attempted abc-conjecture proof: an alien theory that took years to parse and probably isn't right — the nightmare case being an AI version that looks right, absorbs years of climbing, and fails.
Grant invokes Timothy Chow's idea of the 'unsolved expository problem' (we proved forcing works for the continuum hypothesis, but we don't really know why it's true) to distinguish proof from explanation. He has also updated a personal belief: he used to think automated theorem provers would leave mathematicians the distillation job — his job. Now he expects models to be better than most humans at explanation too, so the durable human role shifts to curation: like art-museum curators guiding people through an infinite space of ideas, a relational and social function people will still want from humans they trust.
Why math is grindable and computer use isn't
Dwarkesh's theory of RL progress: verifiability is only half the story; the neglected half is grindability. Computer use is perfectly verifiable (did the package ship?) but you can't run a thousand parallel rollouts of an Amazon checkout without bot detection shutting you down. Code and math can be containerized and replayed deterministically, which makes credit assignment tractable. Most of the real world — building a business, trading a market that changes daily — can't be farmed this way.
On formalization, Dwarkesh argues Lean matters less than people think for current progress (the unit-distance chain of thought contained no Lean; DeepMind went from Lean-based to natural-language IMO systems), and DeepSeek's math model showed natural-language verification with meta-verification works. Grant pushes back on writing Lean out of the story for two reasons: it enables the AlphaZero move — an endlessly self-running system extending a fork of Mathlib for ten years with no human check-ins, possibly discovering island axiom systems that turn out to be motivated (Terry Tao's exhaustive-algebra-search flavor of project) — and it solves the incoming slop problem Alex Kontorovich warns about: if AI mathematicians generate ten papers a day with even a 1% error rate, human review becomes insufferable without a green checkmark that at least guarantees correctness.
Autoregression, entropy, and engineering serendipity
Grant offers a vivid model of why cross-field connections might be structurally hard for LLMs: autoregression is like being locked in a box, handed slips of paper to predict the next token with your memory wiped between slips — you are 'a slave to your context,' and the valuable connection is by definition the unlikely next token. The open question is whether unlocking connections needs more raw intelligence or different generation-time machinery.
They land on an intriguing inversion of the usual entropy-collapse worry: the real advantage of digital minds may be systematic context manipulation. Spin off agents that deliberately try to prove and disprove the same statement (the unit distance conjecture survived so long partly because everyone assumed it was true); assign agents different biases the way Einstein's reference-frame bias was formative (while remembering his 'God does not play dice' bias stalled him on quantum mechanics — don't make every agent Einstein); use old-school software to enumerate an ontology of approaches and explore all of them. Institutes are smarter than individuals because of serendipitous lunch conversations — Montgomery-Dyson again — and you can engineer those between agents at scale, applying a uniform capability waterline across every accessible problem instead of waiting for one genius who dies in a duel.
Why AI writing still fails: theory of mind
Writing progress lags because judges get fooled: models can rank a B essay against an A essay but prefer B* — the essay that hits all the surface bells and whistles — so reward hacking derails training. Dwarkesh adds that writing isn't modular the way code is: the paragraph IS the product, so it can't be slop internally yet still work. Grant counters that both admit LLMs are excellent explainers of existing ideas (Dwarkesh's revealed preference is pasting human writing into an LLM and asking for the explanation) — what's missing is the generative act: knowing exactly when to make the unpredictable move, which contradicts next-token prediction.
The deeper deficit is mentalizing. Andy Matuschak's spaced-repetition experiments found models can't write good flashcards because that requires projecting a specific person's mind three months into the future. Grant's Botox anecdote (people fresh from Botox got worse at reading facial emotions, suggesting we understand feelings by physically mimicking them) frames LLM theory-of-mind failure as unsurprising: an alien without face muscles trying to empathize. Great teachers 'jujitsu' a student's misframed question into their creative frame; LLMs placate and run off instead.
Advice for students in AI-transformed fields
Grant's core advice predates AI and most students skip it: understand where the money comes from and what value you add — whether that's university brand value, NSF public-good funding with its grant-writing song and dance, or straight teaching. Students who've been rewarded for jumping through hoops choose math to keep engaging with hoops, not because they've mapped the value chain. He stumbled into 'math exploration monetized as entertainment' by accident and wishes he'd designed it deliberately.
Post-automation, he expects the social structure of mathematics to survive: prestige comes from within the community, and the valued contribution may shift from theorem proving toward definition writing and curation. Teaching, he argues, is one of the most stable post-AGI careers because it is relational — coaching and mentorship, not explanation — and it's where wealthy parents spend abundance. Dwarkesh adds that if AIs see furthest beyond human horizons in math, distilling what the AIs have seen becomes one of the last jobs standing; Grant notes the curator who can point the math behemoth in economically useful directions (his Boeing PDE-simulation example — one research group's insights saved billions and got funded for it) is suddenly a highly levered position. The honest coda: 10x math progress might also expose that some pure math was never going to touch the physical world — 'trust us, elliptic curves help cryptography' may stop being a usable grant line.
Visuals
| Lightning bolt | Mountain building | Raw hustle | |
|---|---|---|---|
| What it is | Connect two fields the model already knows (Montgomery–Dyson style) | Invent new conceptual edifices (elliptic curves / modular forms style) | Thousand-page grind, no new theory |
| Can current AI do it? | Yes — sparks already visible (unit distance, Erdős 1196) | Not demonstrated | Plausible with enough compute |
| Human understanding | High — state the two endpoints and experts run with it | Slow, alien — abc-conjecture risk of a wrong mountain | Low — proof without explanation |
| Implication for wider economy | Modest — distinct from white-collar skills | Huge — would spill over everywhere | Modest — compute story, not intelligence story |
Takeaways
- Watch math as the leading indicator of AI progress everywhere: capability is fractal-spiky, so expect 'solved' headlines and stubborn holdouts to coexist within the same field for years.
- The current unlock is cross-field connection ('lightning bolts'); the unsolved frontier is inventing new conceptual frameworks whose value can take a century to verify — no RL environment captures that reward signal yet.
- Verifiability alone doesn't drive RL progress — grindability (cheap, parallel, deterministic replay) does. That's why math and code race ahead while computer use crawls.
- The durable human roles Grant bets on: curation of what's worth understanding, relational teaching, and directing automated research toward useful ends — not proving, and probably not even explaining.
- A practical LLM learning pattern: use a human-curated artifact (book, lecture) as the spine and the LLM for local pruning; ask 'who should I read?' rather than treating the model as the teacher.
- For AI systems design: deliberately splitting agent contexts (prove vs. disprove, different heuristic biases) may matter more than merging knowledge — engineered serendipity as a substitute for the Montgomery-Dyson lunch.
Notable quotes
Good mathematicians prove theorems, great mathematicians come up with conjectures, and the greatest mathematicians come up with definitions.— Grant Sanderson (quoting a line highlighted in Polylog's video)
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 might look at that and say, 'This is awful. That's not the essay that I would've written.'— Grant Sanderson
Sure, we've proven it, but we don't really know why it's true.— Grant Sanderson (paraphrasing Timothy Chow's 'unsolved expository problem')
It's like an alien trying to empathize. How could it have theory of mind? ... They don't have face muscles.— Grant Sanderson
Quantity has a quality all of its own.— Dwarkesh Patel
If you're a young genius, don't work on the quintic.— Grant Sanderson