Terence Tao has turned his public lecture at the 2026 International Congress of Mathematicians into a twelve-page paper posted to arXiv, titled "Mathematics in the age of AI." He spends almost no space assessing how capable AI actually is. Instead he asks a different question: what does mathematics, as a discipline, actually want?
His method is to start from an assumption — treat AI as if it can already do research-level mathematical work — and then trace where that premise pushes the field.
The last crisis shook the foundations
The paper opens by comparing the present moment to the period from 1900 to 1930, when Russell's paradox and Gödel's incompleteness theorems arrived in quick succession, forcing mathematicians to go back and ask what a proof even is. That earlier crisis shook the field's formal foundations, and was eventually settled by a rigorous framework that has held for a century.
Tao says this time is different: what's being shaken are the unwritten norms mathematics never had to spell out — who counts as an author, what counts as understanding, what earns a paper the right to be published. None of this needed stating before, because doing a proof was hard enough that the difficulty itself acted as a filter.
The price tag on seven problems
The hardest number in the paper comes from the second round of the First Proof project. Ten unpublished, research-level problems were tested against four AI systems, and seven of the ten received a passing grade from at least one system — judged flawless or needing only minor fixes. The models evaluated were versions publicly available before May 28, 2026.
The compute bill for each problem ran to tens to hundreds of dollars.
That price is worth holding up against the alternative. Handing an unpublished, research-level problem to a PhD student typically costs months of work, translating to a labor cost in the low four figures of dollars at minimum. Once the holes in the filter widen, what comes through stops being the same order of magnitude.
From this, Tao draws the shift he's more worried about: mathematics is moving from a scarcity of proofs to a glut of them. He invokes Goodhart's law to explain the consequence.
"When a measure becomes a target, it ceases to be a good measure."
Paper counts, theorem counts, who-solved-it-first — these have historically tracked mathematics' real goals precisely because they were hard to game. Once they can be gamed, that link loosens. He also cites Thurston's old line: success should be measured by whether what we do helps people understand mathematics more clearly, not by hitting some abstract production quota.
The prescription he writes
Tao breaks problem-solving down into a full chain: generating a proof, verifying it's correct, explaining it clearly, winning peer acceptance, and getting it into textbooks. In the past, only the first link was visible — everything after it was quietly absorbed by the academic community. Now that AI has pushed the cost of that first link close to zero, the remaining links have to be made explicit.
His recommendations largely track June's Leiden Manifesto: add a "tools and compute disclosure" section to papers; downweight the value of being first to solve a problem and redirect resources toward writing, review, publication, and standardization; credit and responsibility still belong to human authors. The toughest line concerns the bar for publication:
If an author can't produce a clear, expert-level explanation of the result, it shouldn't be published. The companion standard, quoted repeatedly, is this: a proof nobody can explain should be treated as incomplete, even if it passes formal verification.
He flags another trap: AI-written proofs tend to be over-polished — smooth to read, but they teach nothing. Removing the friction of learning looks like a good thing, but it can cut off the on-ramp for the people coming up behind.
A personal opinion, not an institutional ruling
It should be said plainly: this is a paper Tao wrote in a personal capacity, not a policy position of any institution. Over the past year, several labs have taken turns announcing that AI had cracked an open problem, and most of the discussion in mathematics has circled around whether it can actually do the work. This paper tries to move the conversation elsewhere: treat "it can" as settled for now, and ask instead how much the field is prepared to absorb.
Infrastructure like Mathlib and the Erdős Problems database is named in the paper as part of the response already underway. The tools are already there — what's missing is a consensus on how to use them.
Sources: Terence Tao's arXiv paper "Mathematics in the age of AI", CocoLoop, reporting by The Decoder; the problem count, pass rate, and per-problem cost range for First Proof's second round are verified against the original paper, and quotes are kept in their original English.