Terence Tao on AI at the ICM: AI proofs will multiply, but math won't get faster
- AI can now genuinely solve research-level math problems: First Proof gave four AI systems ten brand-new problems; seven of them received at least one solution deemed publishable. Meanwhile, the Erdős Problems website has accumulated nearly twenty AI-generated solutions, and no human expert is willing to verify them. The problems get solved; everything after that lags behind.
- On July 24, at a public lecture during the International Congress of Mathematicians, Terence Tao addressed exactly this. He didn't argue about whether AI is up to the task. Instead, he asked the audience to assume it is—then asked: what does that mean for the field?
- He framed mathematical research as a pipeline: produce a proof → verify it → write it readably → get peer acceptance → get it into textbooks. AI massively accelerates only the first step. The further along you go, the slower the work gets—and the more human, and the more valuable.
- The most counterintuitive point: a proof that reads too smoothly may actually be harmful. He showed a photo of his youthful annotations on a 1991 Bourgain paper, scrawled with "AARGH!" and "I hate Jean Bourgain." The places that trip readers up are exactly what force them to slow down and actually learn.
- His prescription for colleagues has three parts: normalize disclosing AI use; shift effort from racing to solve problems toward proof digestion; and the hardest one—if you can't explain your result, you shouldn't publish it.
AI can already solve research-grade problems. That's where the trouble begins.
The Erdős Problems website now holds dozens of AI-generated proofs.
Many of them are probably correct. The problem is that nobody is checking.
The people qualified to check definitely exist. The hard part: no mathematician of sufficient standing is willing to spend days reading an anonymous, potentially dozens-of-pages-long document, then sign their name to "I confirm this is correct." Some submissions even come with a disclaimer from the submitter: I'm not qualified to judge whether this is right.
From the stage, Tao posed the question: might we one day have a verified proof of a major result that no human understands well enough to explain?
And the fact that AI can solve research-level math is no longer hypothetical.
First Proof is an independent evaluation project that tests AI on research-level math. The approach: a group of mathematicians contributed problems from their own research, already solved but never published—nowhere online or in the literature. The second batch had ten problems spanning computability theory, discrete geometry, stochastic PDEs, von Neumann algebras, and more. In late May, under controlled conditions, four AI systems each attempted all ten, with one shot per problem and no human intervention allowed. The solutions went to blind review: roughly thirty experts in relevant fields, at least two reviewers per submission.
For one stochastic PDE problem, a system's approach diverged entirely from the human author's—the reviewers were impressed.
July 24, Philadelphia. The quadrennial International Congress of Mathematicians—the field's highest-profile gathering—is underway. Tao delivers a public lecture titled "Mathematics in the Age of AI." Fifty-two slides.
His topic: AI can already do some mathematics. What should the field do about it?
On whether AI is "really" capable, he wrote a whole slide—then said he wouldn't discuss it
He phrased the question as a mathematical conjecture, deliberately leaving blanks:
At some point in the near future, some AI tool will, at some cost, under some degree of human supervision, correctly complete some research-level mathematical tasks in some domains, with some non-trivial success rate, and some degree of correctness and quality.
Roughly speaking, he said, a weak form and a strong form are enough. The two versions imply completely different responses from the field:
Then everyone can treat AI as a long-term irrelevance and go about business as usual.
Then sustaining the current culture and practice becomes very difficult — especially when "solve as many open problems as possible" is our top goal.
Then he said: this lecture is not about that conjecture.
His reasoning: the debate has stalled. There's a pile of evidence on both sides, but most of it wasn't collected under controlled scientific conditions. What's publicly visible is heavily distorted by reporting bias and assorted non-scientific motivations, and key costs and variables simply aren't disclosed. He also flagged a distinction worth not conflating: whether a claim is true is a completely different question from whether we want it to be true.
So he took another approach: he asked the audience to assume AI can do it, to accept it as a premise. He called it a "working hypothesis," and made it explicit: I'm not asking you to want it to be true, believe it's true, or accept it as true. This is a conditional analysis. Evidence for or against the hypothesis is irrelevant to what I'm about to say.
His framing: this is a foundational crisis — the second one
The first happened a century ago. For centuries before that, mathematics ran on a "naive" foundation: what sets, numbers, and infinity are, which axioms math rests on — these questions were mostly left to philosophers. The Russell paradox in 1901 and Gödel's incompleteness theorems in 1931 forced working mathematicians to re-examine the assumptions they'd been taking for granted.
Those three decades were turbulent but hugely productive: a clear, rigorous, standardized foundational framework emerged. It survived intense scrutiny and remains a trustworthy working environment for mathematics today.
Tao says we're entering a similar turbulent period. Last time, it was the logical foundations that needed re-examination. This time, it's the field's values and working practices. He also offered the same conclusion: once we've thoroughly examined and explicitly written down these things, the community will emerge stronger and more resilient.
AI can help you solve more problems. But is solving more problems really the point?
AI writing fast is just the surface. The real reason proofs pile up: "solving a problem" was never the only thing this field is after.
In the lecture, Tao listed reasons for doing mathematical research. There were so many they barely fit on one slide:
In the past, he said, these goals were roughly aligned: progress on one usually carried the others along. So you could use one or two as proxies for the rest; many goals didn't even need to be stated explicitly.
But when any metric gets optimized to death, it runs into Goodhart's law: once a measure becomes a target, it ceases to be a good measure.
Tao added: generative AI is inherently "ungrounded" — its outputs aren't anchored in any fact, and it has no internal mechanism guaranteeing its statements are true. Combine that with the financial incentives of AI companies, and you have a recipe for triggering this law particularly easily.
So excessive AI optimization makes these once-aligned goals diverge from each other. He showed this on three consecutive slides:
Goals roughly aligned. Push one, the others follow. So you can use one or two as proxies.
The same goals, now diverging. Proxies break down: the problem-solving number goes up, but the others don't follow.
This diagram is worth holding up to your own industry. Every field can list a set of goals like this, and every field got used to using one or two as proxies for the rest because they used to move together. AI excels precisely at driving a single metric to extreme heights on its own.
So which things should we actually want? That's the main body of the lecture. On stage, Tao revised the goal five times.
From solving problems to writing textbooks, AI only speeds up step one
He picked "problem solving." He was careful to note this is only one facet of the field — theory building is an equally important other side that needs its own analysis — but problem solving is the part most exposed to AI.
The pipeline below is what his five revisions produced. Click the tabs to watch it grow:
v1: Solve as many open problems as possible
On its face this goal sounds fine, and the network is simple: an open problem on one end, a solution on the other, a single arrow called "proof generation" in between.
The risk was known long before AI: you'll receive a flood of incorrect solutions to major problems. Anyone who works in number theory has gotten emails claiming a proof of the Riemann hypothesis.
v2: Add "and verify that they're correct"
The pipeline grows a second stage: what's generated is first called an "unverified solution"; it only counts once verified.
Here AI genuinely helps. Proof assistants like Lean, Rocq, and HOL can encode mathematical proofs as code a computer checks line by line. Having AI do that translation is called automatic formalization.
Read the whole argument, judge whether each step holds. Requires an expert in the subfield, days to weeks of time, and willingness to put their name on the conclusion.
It's tiring, it misses things, and often nobody picks it up because "it's not my research contribution."
Translate the proof into Lean or similar, and the compiler checks it line by line. If it passes, there's no logical gap.
The catch: someone (human or AI) has to do that translation first, and the translation is itself a heavy piece of work.
Tao said that on both the generation and verification ends, AI has already dramatically accelerated things in many cases, and under the working hypothesis, that acceleration will continue.
Then he posed the question: what if AI produces a long proof that nobody understands — not even the person who fed it the problem?
That's exactly the pile on the Erdős Problems website from the opening. A machine might verify it, but no human can step up and say "I've checked this, I vouch for it."
v3: Add "and ensure it can be clearly communicated and understood"
The pipeline grows another stage: a verified solution must become a "well-written solution." This step is called proof exposition.
And this is where AI behaves the strangest.
When the writing is too smooth, readers don't learn
Tao's assessment of AI's current mathematical writing is split.
Spelling, grammar, and formatting are nearly flawless.
He added a footnote here: arguably too flawless.
It often goes on at length about trivial details, then rushes past — or actively obscures — the most interesting and novel parts of the argument.
It also frequently fails to relate the result to existing literature or give a high-level overview.
The same observation shows up elsewhere. In the blind review reports from First Proof's second batch, the same complaint recurs: AI solutions tend to be extremely thorough on routine parts, then hand-wave through the hardest steps — sometimes asserting a key conclusion "follows by standard arguments" without a reason, sometimes citing a paper that doesn't actually contain the stated result.
The reports also recorded something more damning. Several AI solutions to one problem borrowed line-by-line phrasing from the proposer's own earlier paper, including made-up terminology (T-patterns, bends) and equation labels (B, T, D, H), without citing that paper even once. The reviewer's exact words: if a human had submitted this, it would be plagiarism.
One level deeper: exposition can also be over-optimized
Exposition is a much fuzzier optimization target than verification. Under the working hypothesis, AI's exposition will improve eventually. But even once it improves, there's another problem: exposition itself can be over-optimized.
A proof can become too smooth: the routine parts and the genuinely hard parts are presented as equally easy to digest.
In human-written proofs, the places the author found difficult typically leave some kind of natural friction: sentences get rougher, gaps get wider, the tone tightens. These traces tell the reader: slow down here, look closer.
Excessive AI polishing sandpaper away both kinds of friction at once: the artificial (author was lazy, didn't write clearly) and the natural (this was genuinely hard). Once smoothed over, the reader glides through without being pushed to actually understand the key ideas. He used the phrase "paradoxically": the "flaws" in human exposition may end up helping readers.
After this section, he showed a photo.
This photo is the physical evidence for the argument.
Bourgain wrote "we skip the details." The young Tao got stuck right there. He underlined it, put question marks, wrote "Sobolev norms!" to remind himself which direction to think, and finally scrawled in the margin: "I hate Jean Bourgain."
More than thirty years later, he photographed the page and put it up on the big screen at the ICM. These very sticking points forced him to slow down and genuinely learn the material. A proof where every step is equally smooth leaves no such place for the reader.
Immediately after, he quoted William Thurston's 1994 essay "On Proof and Progress in Mathematics":
We are not trying to meet some abstract production quota of definitions, theorems, and proofs. The measure of our success is whether what we do enables people to understand and think more clearly and effectively about mathematics.
William Thurston, "On proof and progress in mathematics," 1994
More proofs, but math isn't getting faster
Making it readable for humans is only step three.
v4: Add "and have it digested and accepted by the community"
For a proof to actually contribute to a field, being correct and readable isn't enough. Other mathematicians need to digest it, absorb it into their own work.
The author can help: explaining where they got stuck, how they figured it out, why they took this path — these things let others absorb it faster. But current AI tools are quite opaque about their own solution process — especially proprietary models whose inner workings are trade secrets.
And community acceptance is, by nature, slow and human. Good exposition and careful writing can facilitate it, but ultimately it's an external process that can't be optimized unilaterally by the author and their AI.
What keeps this whole system running right now? Volunteer labor from human editors and reviewers. Tao pointed out: this work is often seen as less prestigious than generating proofs, but it's indispensable — it's how an individual mathematician's achievement becomes collective progress and understanding.
AI can act as a filter here — journals could automatically reject papers flagged as insufficiently verified or poorly exposited. But a filter can't substitute for acceptance itself.
v5: Add "and have it incorporated into the canonical theory of the field"
Even being published isn't always the end point. Key results eventually make their way into the field's authoritative textbooks and reference works, becoming the standard account taught to the next generation of students. Tao calls this step canonicalization.
It's the slowest stage of all, requiring broad, deliberative consensus across the community — and the least suited to AI optimization.
But, he said, it's the most valuable part of the entire pipeline. Two reasons. First, many applications only become feasible once the underlying mathematics has been thoroughly digested. Second, and more importantly: AI tools' success in mathematics today depends critically on the canonical theories human mathematicians built up over centuries. Models can solve problems because previous generations organized the field into a form that was learnable.
Once the five steps are in place, the diagnosis lands
If AI can do this work, and policy and culture don't change accordingly, this pipeline will develop "impedance mismatches" everywhere. It's an electrical engineering term — two stages that don't interface properly, so energy gets stuck at the junction. He also offered a blunter phrase: proof indigestion.
Concretely, four bottlenecks:
In one sentence: we will move from an era of proof scarcity to an era of proof surplus.
The signs are already visible. In fact, he said, the pressure had been building well before modern AI.
He once used a more intuitive analogy himself
Three months earlier, on social media, he recast the pipeline in terms of food:
A society used to food scarcity is bottlenecked on getting food. The hard work of cleaning and cooking is appreciated, but the prestige goes to the hunter who brings down the prey. In that world, nearly any non-toxic meat or vegetable is welcomed to the communal table, and volunteers can always be found to turn it into a meal.
A potluck in a food-abundant society is different. Raw ingredients dropped off casually are no longer welcome: a stranger dumping the carcass of an unknown animal for others to clean and cook gets no thanks (unless the prey has an especially unique and interesting story behind it, and the meat is assured safe). Even a ready-to-eat meal that's been inspected, packaged, and certified is usually just one part of the table. What's truly valued is the home-cooked dish made with care by someone the community trusts — because the conversation that forms around those dishes is part of the gathering itself, and an opportunity to train the next generation of cooks.
In that same thread, he made an observation sharper than anything on the slides: the massive acceleration in proof generation has not actually produced a corresponding acceleration in mathematical progress itself.
He also noted a side effect: a problem "solved" by AI but understood by no one may actually kill others' interest in pursuing it. After all, it's been "solved" — even though no human understands the solution.
So what should mathematicians do?
Since the bottlenecks are in the later steps, that's where the effort should go.
He recommended the Leiden Declaration as a starting point. The initiative went live June 2 of this year, beginning with a workshop at Leiden University in September 2025, drafted by a working group of sixteen mathematicians from institutions including Cambridge, Columbia, Oxford, Leiden, and ETH Zurich. It has already received endorsement from the International Mathematical Union — the same organization hosting this congress.
Tao picked four clauses and added three comments of his own.
Worth noting: the Leiden Declaration is much broader than the four clauses Tao drew on. It has dedicated sections on AI in warfare, mass surveillance, undermining democracy, and environmental costs, calling for stronger public oversight of the AI industry — one clause aimed at policymakers is literally titled "Don't believe the hype." This lecture only used the operational parts. He was talking about how the mathematical community should do its own work.
He made two disclosures himself, in the slides
The footnote on the first page reads: all the em-dashes in these slides were typed by hand. (Em-dashes are now often treated as a telltale sign of AI writing.)
On the "normalize disclosure" slide, the footnote reads: this slide uses AI tools to autocomplete text and generate figures.
Neither was in the body of the slides. Both were just in the footer.
Finally
Problem solving, he said, is just one facet. The same analysis needs to be done for teaching, mentoring students, hiring, grant applications, and public communication. In some areas — especially education and training — the human side of the work needs to be emphasized, with AI tool use strictly limited. In others, the field should proactively define how to integrate these tools into workflows on our own terms. And new workflows and infrastructure are needed to supplement traditional ones — but that's another lecture.
His final line to the audience: our community needs to sit down together and have an open, honest discussion about AI capabilities and our goals and values.
The last page of the talk's main body, he added no comment — just Leiden Declaration clause 7:
AI can already solve research-level math. The bottleneck is the steps where no human is left holding the bag.
Terence Tao on AI at the ICM: a five-step pipeline, in one illustrated page, on why math hasn't gotten faster.
↓ One page · One animated figure
The Erdős Problems website is piled with dozens of AI-generated proofs. Many are probably correct. But no mathematician of sufficient standing will spend days reading something by an unknown author and then sign their name to "I confirm this." Some submissions literally come with the submitter's own disclaimer: I'm not qualified to judge.
AI solving research-grade problems is no longer hypothetical. On July 24 in Philadelphia, at the quadrennial ICM, Terence Tao gave a public lecture, "Mathematics in the Age of AI," about what to do.
He didn't debate whether AI is "really" capable — that debate is deadlocked, its public evidence heavily distorted by reporting bias. He asked the audience to assume it is, and then asked: what should we do? His framing: this is the second foundational crisis. A century ago, Russell's paradox and Gödel's theorem forced mathematicians to re-examine the logical foundations — thirty turbulent years that produced a rigorous foundation still trusted today. This time, the values and working practices are what need examining.
He listed reasons for doing math: solving open problems, building theory, applying knowledge, building community, training the next generation, creating work of lasting aesthetic value. These goals used to move together, so one or two could stand in as proxies for the rest.
The trouble starts when a metric gets optimized to death. Generative AI is ungrounded — no anchor in fact, no internal guarantee of truth — and that, plus the financial incentives of AI companies, makes it especially good at driving one metric to absurd heights while the others sit still.
Problem-solving went up, and theory, training, community roughly followed. So "how many solved" could proxy for the whole field's progress.
The problem-solving number gets pushed up on its own; the others don't follow. The proxy breaks down — the number looks good, but the field isn't moving.
Worth holding this side-by-side up to any industry: every field can list such goals, and every field got used to treating one or two as proxies.
He rewrote "solve as many open problems as possible" five times on stage. Each time he added a requirement, the pipeline grew a stage — and the part AI can handle shrank by one slot. This is the core thing to take away.
Tao's take on AI math writing is split: spelling, grammar, formatting nearly flawless — he added a footnote: "arguably too flawless." But it lingers on trivial details, rushes past or obscures the most interesting, novel parts, and skips relating results to existing work. First Proof's blind reviews independently logged the same complaint: key steps waved away with "by standard arguments"; several solutions borrowed line-for-line phrasing and made-up terminology from the proposer's earlier paper without a single citation. The reviewer's words: if a human had submitted this, it would be plagiarism.
Places the author found hard leave friction: rougher sentences, wider gaps, tighter tone. These traces tell readers to slow down.
We skip the details. → Margin: AARGH! ??? I hate Jean Bourgain.
Routine and genuinely hard parts are presented as equally easy. The reader glides through, never pushed to understand the key ideas.
Perfectly smooth → Margin: (blank)
The handwriting on the left is real. He put a photo on screen of his youthful annotations on a 1991 Bourgain paper: the printed "We skip the details" struck through, "AARGH" below, three question marks beside, and one line up, "I hate Jean Bourgain." Thirty-plus years later, he put that page up at the ICM. The places that stopped him are exactly what forced him to slow down and actually learn it.
The last two stages are the slowest: peer acceptance runs on volunteer human editors and reviewers — often seen as less prestigious than generating proofs, but it's how individual achievement becomes collective progress. Canonicalization is slowest of all, requiring broad deliberative consensus, and least suited to AI optimization — but it's the most valuable. AI can solve problems today precisely because past generations organized the field into a learnable form. All four junctions clog: waiting for verification, waiting for a readable write-up, review overwhelmed by volume, too many results to integrate into canonical form. The field will move from proof scarcity to proof surplus.
Where the numbers come from: 7/10 is from the First Proof second batch report (~30 expert blind reviewers); the Erdős backlog figure and all opinions are Tao's own, from the lecture and Mastodon. Note: he is a team member of one of the four systems tested (UCLA Moonshot Harness); the slides don't mention this.
sitting right here.
one or two
They might all be right!
sign your name,
"I confirm this."
"Mathematics in the Age of AI."
about whether AI is capable.
What should we do?
Isn't that good?
on the board, five times.
→ Get accepted → Enter textbooks
flawless!
then "by standard arguments"
on the hardest step.
learns nothing.
Margin: I hate Jean Bourgain.
Underlined, question marks, swearing.
forced him to actually learn it.
he puts this page on the big screen.
- × Awaiting verification
- × Awaiting a readable write-up
- × Reviews drowning in volume
- × Too many to integrate
sped up by AI.
at the seam.
but math isn't faster.
to proof surplus.
to digesting proofs.
Don't publish it.