From Proof Scarcity to Proof Abundance: Terence Tao on Mathematics and AI

Introduction

Mathematics has become the first field where AI can do real research work, and according to Terence Tao that success is now the problem. In autumn 2026 the question is no longer whether machines can prove theorems. It is whether the profession can keep what proving theorems was for.

A note on sources. Everything below is compiled from Terence Tao's own public positions, mainly his living summary Terence Tao on AI in mathematics (and beyond), last updated 11 October 2026, and the primary sources it links. That summary was itself drafted by an AI assistant from his posts, talks and interviews, then reviewed and corrected by Tao. The views and facts are his; the arrangement and any errors of compression are mine. I make no claims of my own here. Tao also discloses his ties to the industry on that page: he is not paid by any AI company, but has received access to premium models, collaborates with Google DeepMind staff, helped organize an OpenAI-sponsored IPAM workshop, and co-founded the AI-for-math non-profit SAIR. Readers should weigh what follows with that in mind, as he invites them to.

The latest automation

Tao frames AI as the newest chapter in a very old story of offloading mathematical labour. Human "computers" once built tables of logarithms and sines; Hendrik Lorentz ran teams modelling a Dutch dam; later came computer algebra and SAT solvers. Each wave took over a routine layer while humans kept directing the work. Tao likes to point out that solving differential equations, once a mathematician's main use, is now done by machine in seconds, and the field survived.

What is new is the shape of the tool. Tao doubts that current systems are on the way to general intelligence. He argues instead for a weaker but real "artificial general cleverness": solving broad classes of problems by ad hoc, often brute-force methods that are fallible and hard to interpret, but succeed at useful rates when paired with strong verification. In humans cleverness and intelligence tend to go together. In machines, he says, they come apart.

He describes the tools in three recurring images:

• A probability kernel, not a function. Classic software gives the same answer every time. An AI tool gives a random answer clustered near the right one, with plausible-looking errors mixed in, so it must be used interactively.
• Spiky competence. Superhuman in places, and capable elsewhere of absurd mistakes, such as claiming all odd numbers are prime.
• Fluency ahead of substance. He tells students to think of LLMs as children who can pass as competent adults for longer and longer stretches.

He also rejects the one-dimensional scale from "sub-human" to "super-human". The space of cognitive tasks has very many dimensions. The frontier that still resists machines is reasoning from scarce data: extrapolating from five or six facts and a vague analogy, which is exactly what good mathematicians do.

Why mathematics went first

In Tao's reading, mathematics is where AI's successes are clearest for one reason: its output can be checked. Elsewhere, the weak point of AI is mistakes nobody can catch. In mathematics a proof can be verified, in the limit mechanically by a proof assistant, so hallucination stops mattering wherever checking is cheap.

In his October 2026 Caltech lecture "Math 2.0" he names three properties that together made mathematics the ideal target for machine learning:

1. Objective verifiability. Mathematicians argue about what is interesting, but broadly agree on what counts as a correct, complete proof.
2. Full digitizability. Proofs need no experiment, no physical object, no input from the real world.
3. A large, high-quality, digital literature to learn from.

Frontier labs therefore prioritized mathematics over applications lacking one of these features. Tao's consequence is that capability is "extremely jagged": astonishing where tasks are verifiable and data-rich, weak where they are subjective, data-poor or tied to the physical world. It is also why he treats mathematics as a sandbox for every other field: it sets the best-case upper bound for safe AI use, which law, economics or medicine must then discount. (I explored that argument in an earlier post, Other Planets of Intelligence.)

Where capability stands in autumn 2026

Tao's 2023 forecast, that AI combined with formal verifiers and search would be a trustworthy research co-author by 2026, arrived roughly on schedule by his own judgement. The trajectory, as he has recorded it:

Two cautions accompany the numbers. First, Tao insists capability is not binary: results swing by orders of magnitude with compute, human help and reporting format, and negative results go unreported. Second, he has given up long-range forecasting. The confidence he had in 2023 about the next three years is gone; he doubts anyone can forecast reliably beyond a year.

He is equally clear that the frontier has not collapsed. Despite the impression from press releases, most problems at the current frontier remain unsolved even with massive compute, and since difficulty is unbounded, the frontier will keep expanding.

Working with an unreliable but powerful tool

The load-bearing idea across Tao's writing is that verification turns an unreliable tool into a useful one. His image for students: traditional mathematics is a tap delivering a trickle of drinkable water; AI is a firehose of undrinkable water; formal verification is the filter. The filter, however, is only as good as the person holding it. Students who do not understand a subject cannot check an answer.

From this follow a handful of practical rules:

• Comparative advantage and acceptable failure rate. AI helps least where you are most practised, and most on tasks you can do but rarely do. The deciding factor is often not difficulty but how much failure the task tolerates: a weeknight dinner, not a state banquet.
• Only use AI where you could red-team its output yourself. His test: if you could not present the AI's output in class and answer questions on it without further AI help, it should not be in your workflow. Stuck? Propose your own strategy and ask the AI to critique it.
• Red team over blue team. Unreliable contributors are safer finding flaws in human work than building load-bearing structure.
• Keep some friction. Offload artificial friction, like tedious computation you can spot-check. Keep natural friction, the conceptual difficulty that produces understanding.
• Ask for honesty. Tools rarely rate their own confidence accurately, and push-a-button autonomy is the wrong interface for hard problems, which need a conversation.

Tao's own practice illustrates the style. When a counterexample to the Jacobian conjecture was announced in July 2026, he digested it over two days with a chatbot, driving the algebra himself and using the tool to check steps and surface literature.

Formalization and "big mathematics"

Proof assistants such as Lean change how many people can work together. Ordinary collaboration rests on personal trust and line-by-line checking, which caps a project at around five people. A compiler that guarantees correctness removes the need for trust. Tao's worked example is the Polynomial Freiman–Ruzsa conjecture: a 33-page paper split into independent pieces and formalized in three weeks by about twenty people, most of whom had never met. Because trust flows from verification rather than reputation, an amateur's idea can be taken seriously if it comes with a formal proof.

The cost picture has changed fast. In 2023–24 Tao put the overhead of formalizing at roughly twenty times the effort of writing. By 2026 autoformalization completes many tasks within hours and had emptied the queue of unclaimed tasks on at least one of his projects. He keeps two boundaries in view. Verification certifies the formal statement, not that it matches intent, so humans should author or carefully review theorem statements while automation handles proofs. And an eager model will happily prove spectacular results from a subtly wrong hypothesis, so a sudden run of easy successes is a warning sign.

His positive vision is what he calls "big mathematics": large, decentralized human–machine collaborations where humans take the creative parts and AI does most of the technical grunt work. He compares the shift to moving from a craftsperson making one toy at a time to a factory, or from the lone hacker to software engineering's specialized roles. He expects new roles to appear, including people who turn ugly machine proofs into readable ones, and a "citizen mathematics" where students and amateurs contribute modular pieces. A UCLA team including Tao now holds a three-year DARPA grant, ALPHA, to scale up libraries of formalized mathematics.

From proof scarcity to proof abundance

Tao splits problem-solving into generation, verification and digestion: understanding a result, placing it in context, explaining it. Historically all three were hard, and digestion happened as a by-product, so the profession rewarded the first two. AI now accelerates generation, and increasingly verification, far ahead of digestion. Mathematics is moving from proof scarcity to proof abundance faster than its culture can adapt. Two of his sharpest observations: it is now easier to generate long correct proofs than short ones, and faster generation has not produced faster mathematical progress.

In August 2026 he refined this into five stages a proof must pass through: generation, verification, exposition, publication and canonicalization into textbook knowledge. Each stage is accelerated less by AI than the one before. He compares a proof to a child its authors are expected to raise to adulthood, and finds it disturbing when individuals or companies generate and verify a proof with AI, then show no interest in talks or peer review. His proposals:

• Anyone claiming credit for a proof should commit to taking it at least to publication.
• Where that is impossible, mathematics may need a formal way to put a proof "up for adoption" by other authors.
• Priority should go to the first to give a public, clear exposition of a result, not the first to prompt an AI to it.

He also describes a variant of Simpson's paradox. A toy model: if generating a solution falls from six months to a day, while careful write-up falls only from a month to three weeks, every individual produces more and better work, yet the share of well-written literature collapses from about nine-tenths to about one-tenth. Better AI exposition does not fix this, because generation improves at the same time. The curated core shrinks relative to an ocean of technically correct but poorly digested results, and becomes, by the same token, the most valuable data the field has.

Answers versus insight

Faced with a problem, the obvious question is "what is the answer?" In basic research the better question is usually "what can be learned by studying it?": what makes it hard, which techniques must be invented, how it connects to the rest of the field. Until recently the two questions were aligned, because the only road to an answer ran through the insight. Point a powerful AI at a problem without expert guidance, Tao argues, and they can become negatively correlated.

His images for this are vivid. Solving a hard problem is like a hike that lets you lay trail markers and draw maps; AI is a helicopter that drops you at the summit. Indiscriminate automated mining of open problems is like digging treasures out of an archaeological site with excavators, destroying the context that gave them meaning. And because a known solution contaminates later attempts, good open problems have become something like a non-renewable resource for the first time. To the objection that infinitely many problems exist, he answers that a region can run short of drinking water while surrounded by ocean.

He makes this concrete with two cases:

• Navier–Stokes. The value of the regularity problem lies in the theory generated by attempts on it. He fears an autonomous system with vast compute settling it behind closed doors, so that a famous problem is technically solved with almost nothing added to mathematics. In September, Alpöge and Buckmaster announced finite-time blowup for 3D Euler with a smooth forcing term, formalized in Lean and with significant AI input the authors spent weeks reworking. Tao called it remarkable, but said extending it by brute force does not particularly interest him; digesting the method does. (I wrote about the precise scope of these results in The Equation and the Problem.)
• Bounded gaps between primes. The real history ran for two decades, through Zhang, Polymath and Maynard's sieve, which became a textbook method. Rerun it with benchmark-driven AI in 2005, he suggests, and the numerical bound drops faster, but no write-up is produced, the field is abandoned, and the Maynard sieve stays buried in machine output.

None of this, he stresses, is intrinsic to the tools. The story of Erdős problem #126 shows AI used collaboratively to advance both a problem and the field's understanding. The damage comes from choices: not working with domain experts, not writing papers, not submitting to peer review. His suggested fixes include declaring some classes of problems off-limits to automated solvers while opening others, and having AI companies race to announce a new insight rather than a new solution.

Students and the "AI diet"

Tao's central analogy for education is nutrition. Food went from scarcity to abundance and traded famine for obesity; cognition is going from friction to abundance. The subtle risk is not that AI is unreliable, but that it becomes reliable enough to substitute for thinking. The harms he lists include cheating, deskilling (as calculators eroded number sense and GPS eroded spatial sense), loss of patience, sycophancy that makes real criticism harder to take, and a drift toward one bland default style. He is blunt that some students' grades rise while their understanding falls.

His prescription: discourage unsupervised access, emphasize process and verification over answers (students submit their prompts and how they checked the output), allow freedom to fail, and permit creative use in small doses. AI should be the vanilla extract of intellectual work, used sparingly. In-person, AI-free exams are back as a deliberate stopgap. He sees teaching a mindful cognitive diet as one of the great new purposes of education, and argues that the inequality it risks is not inevitable: open, locally run and distilled models can widen access.

The same logic reaches the PhD. A thesis and a paper or two can now be superficially imitated by AI. But a doctorate was never about the thesis; it was about training the harder-to-see skills of absorbing, synthesizing and asking good questions. Those now have to be made explicit and valued directly.

A crisis of values, not of capability

Tao's fullest synthesis is his ICM 2026 lecture and essay "Mathematics in the age of AI" (July 2026). He frames the moment as a second crisis in foundations. The crisis of roughly 1900–1930 concerned the foundations of reasoning and ended with a rigorous, standardized framework. Today's crisis concerns the foundations of mathematical values and practices.

His key move is to separate two questions. The first is how capable AI is, which he states as a deliberately open template he calls the AI Capability Conjecture. He sets that aside and asks listeners only to grant, conditionally, that a reasonably strong version holds. The second question then comes into view: what are mathematics' real goals? Historically, solving problems, building theory, understanding the world, training students and creating beauty were positively correlated, so any one could stand in for the rest. AI breaks that alignment through Goodhart's law: when a measure becomes a target, it stops being a good measure. Optimizing for solved problems alone can now pull loose from everything it used to carry. On the following day's panel he warned of an "AI monoculture" in which funding and prestige drift toward AI-amenable problem-solving and starve theory-building.

He draws three recommendations:

1. Normalize responsible disclosure of AI assistance, ideally including the chat logs and failed attempts, not just a checkbox.
2. Reduce the emphasis on proof generation and on being first; raise it on exposition, publication and canonicalization.
3. Do not publish a result whose authors cannot give a clear, correct, expert-level talk on it.

The call became more urgent through the summer. At the ICM he told mathematicians they had only months to reimagine their profession and should organize and become a little political, because the field had temporarily lost the narrative. In a statement for the record in September he said he is neither pro- nor anti-AI. He saw two paths from 2023: responsible incorporation, or indiscriminate use for short-sighted goals, and he now believes the field is rapidly approaching the second. The main harm in 2026 is no longer unverified output but benchmark-driven extraction, and many industry people who shared his view have left or been sidelined.

On 11 September 2026 he was one of 25 Fields Medallists who were initial signatories of "A Severe Misalignment of AI in Mathematics". The declaration concedes that LLMs can now solve major open problems, and argues that the goals of AI companies and of the mathematical community are severely misaligned: solving problems is a proxy for conceptual understanding, and rushed, uncited, unwritten-up announcements raise serious attribution questions. It locates the outcome not in the technology, but in the decisions of the people controlling it.

Math 1.0 and Math 2.0

By October 2026 the diagnosis had a name. "Math 1.0" was built on proof scarcity: solving an open problem was rare, hard and expert work, and the field's institutions, incentives and stated goals grew up around that fact. Under scarcity, problem-solving was a fairly accurate proxy for the field's deeper goals. Under abundance the proxy is misaligned, and Tao argues that further blind optimization of problem-solving is now actively harmful. "Math 2.0" must therefore both expand the research frontier and decenter problem-solving.

What has to be replaced is a cycle. A Math 1.0 breakthrough brought invitations, workshops, new collaborations, follow-up problems and newcomers, until the proof was digested and entered the textbooks. Problems solved autonomously generate little of this, because those prompting the AI often have no interest in the wider field and cannot give a talk on the result. A solved problem cannot be made unsolved, so fields harvested this way are left far less fertile.

His list of what to do instead is deliberately AI-positive:

• Open exposition problems: finding motivated explanations for results whose known proofs are opaque.
• Collating scattered papers into living networks, and building large formal libraries such as Mathlib.
• Using AI to surface new principles and methods, not just new proofs.
• Ablation studies: does a theorem survive removing a key input, or admit an elementary proof?
• Automatically exploring the neighbourhood of a theorem as hypotheses change.
• Open, interpretable, tailored mathematical models, and interactive texts that synthesize whole theories.

The binding constraint, he says in the Caltech slides, is neither technology nor even institutional culture: "It is imagination." What the community must supply is institutional reform: valuing exposition, community-building and new directions, and rewriting its criteria for education, publication and careers accordingly.

What mathematics is for

If machines eventually take over even the creative core, what is the profession for? Tao avoids the chess analogy and reaches for William Thurston: mathematics is not a quota of definitions, theorems and proofs, but understanding, to which he now adds the word human. His preferred comparison is music. We can reproduce and even generate music indistinguishable from human performance, yet we still go to concerts, because we value human connection. Mathematics, he argues, has always been human-centred at its core and should now be more honest about it. He also notes that the mathematical community already is a kind of superintelligence no individual can match, and that getting a new idea accepted and found exciting is a different game from solving a problem, closer to writing a hit song.

On risk he inverts the science-fiction ordering. Autonomous AI malfunction worries him least; malicious humans assisted by AI worry him most, with misuse, socioeconomic disruption and AI panic in between. His underlying message is about agency. AI forces questions about what a proof, a paper and the profession are, and if mathematicians do not answer them, technology companies and financial incentives will.

Asked what people most often get wrong, Tao asks for the opposite of a slogan. Neither "AI good" nor "AI bad" captures the subject; it is one of the most important issues of the era, and it rewards anyone willing to dig into its complexity and paradoxes. That seems the right note to end on.

Sources

All positions above are Terence Tao's, as compiled in his summary and the sources it cites.

• Terence Tao, Terence Tao on AI in mathematics (and beyond): a living summary, updated 11 October 2026, and its companion interview
• Terence Tao, "Math 2.0", AIM Public Lecture, Caltech, 9 October 2026 (slides)
• Terence Tao, "Mathematics in the age of AI", ICM 2026 public lecture and Proceedings essay, July 2026
• A Severe Misalignment of AI in Mathematics, declaration by 25 Fields Medallists, 11 September 2026
• Terence Tao, statement for the record on AI and industry, September 2026
• Klowden and Tao, Mathematical Methods and Human Thought in the Age of AI, 2026
• First Proof assessment
• Tao, The story of Erdős problem #126, December 2025

How this post was made: compiled and drafted with AI assistance from Tao's published material, then edited by me, in line with this blog's usual practice.

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Image Credit: Marcel Blattner | 2026 | Tulip Nebula

https://app.astrobin.com/u/blattnem?i=ieqz0m

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