Terence Tao and Deng Yu in conversation: AI math results are "toxic waste," the problem lies in how they are released

In September of this year, OpenAI claimed that its model had cracked the Navier-Stokes equations, and 25 Fields Medalists promptly issued a joint open letter, pushing the AI-versus-mathematics debate into the public eye; on October 6, OpenAI released 722 AI-generated mathematics preprints all at once, and the next day withdrew three of them due to notational errors; the results update log of October 8 further showed that another 14 had been revised and 13 had updated citations, with the total number of manuscripts dropping from 722 to 719. At the time, the core question in public discourse was: will AI replace mathematicians? (See the article: AI Drops 722 Mathematics Papers at Once, 25 Fields Medalists Can't Sit Still)

Now, the matter has seen new developments. Recently, on the podcast of the SAIR Foundation, a nonprofit founded by Terence Tao, two mathematicians at the center of the controversy—SAIR chair and Fields Medalist Terence Tao, and this year's newly minted Fields Medalist, University of Chicago professor Deng Yu—held a conversation about the future of AI and mathematics. The sequence of events can be seen in the timeline in the figure below.

Solving a problem is not the same as understanding it: the two boundaries begin to separate

First, why are these two interlocutors qualified to respond to this controversy? Terence Tao is a professor at UCLA, won the Fields Medal at 31, and is one of the founders of the SAIR Foundation; in recent years he has continuously focused on formal verification and AI-assisted research; Deng Yu grew up in Shenzhen, won an International Mathematical Olympiad gold medal in 2006, and in July of this year, together with Peking University alumnus Wang Hong, received the Fields Medal, attracting wide attention for key progress on Hilbert's sixth problem. Both are among the signatories of the joint open letter. That "the people best at solving problems" discuss "the meaning of solving problems" itself constitutes the tension of this conversation.

The most illuminating judgment in the conversation comes from Deng Yu's recharacterization of the "boundaries" of mathematics. He said that doing mathematics is for understanding the world, and there are two boundaries: one is the boundary of problems that already have proofs, and the other is the boundary of proofs that humans can understand.

Over the past several centuries, these two boundaries have largely coincided. When a person solves a difficult problem, they usually also truly understand it: they can explain the proof clearly, can raise new questions from it, and can train students. Terence Tao further pointed out that the mathematics community rewards "solving problems," but what it actually rewards is this entire set of activities; solving problems is merely the easiest indicator to measure among them.

What AI changes is precisely this premise. When machines can generate large numbers of proofs in a short time, "obtaining an answer" and "grasping the logic behind it" begin to become two different things. Deng Yu summarized it as: AI has made the second boundary visible. His view is: AI's proofs are astonishing, and they contain very good new ideas; AI is learning human ideas and also creating new ideas of its own that can be understood—this is the co-evolution of humans and AI. But the premise is that these proofs must enter the range that humans can understand.

"Already proven" and "can be understood" are becoming two different boundaries

This framework also explains why top mathematicians have not fallen into the panic of "AI replacing humans." The question truly worth attention is not how many new problems AI has solved, but whether these proofs can be understood, whether they can be verified, and whether exploration can continue on their basis. Terence Tao said bluntly that among the results released by OpenAI, there are about a dozen problems he cares about, and he will look at them later, but for now they are not among his priorities; under the current way of explanation, they have not yet convinced him.

However, these results are not all empty. Among them, the result related to the quasi-Riemann hypothesis numbered 003 pushes the zero-free region of the zeta function's zeros to 7/8 (previously, humans could only establish a zero-free region that narrows continuously as the imaginary part increases and approaches real part 1 infinitely, and could not even rule out the possibility of anomalous zeros in a vertical strip region with real part greater than 0.99); Terence Tao called it "the greatest progress on this problem in fifty years," and believed that this arrangement, pieced together from three or four existing ideas, "probably could not have been assembled by any living mathematician alone." This precisely shows that the problem is not that the results are no good, but in the way they were released.

The severed chain of transmission: the provenance crisis matters more than solving problems

The understanding gap is not groundless worry; quantitative evidence has already appeared in this controversy. According to statistics on the repository's formalized directory at the initial stage of release, among the 722 preprints, only about 22% had undergone formal verification such as Lean, and among the 143 newly added from October 3 to 6, not a single one had undergone Lean formal verification. A deeper problem lies in the fact that Lean only verifies the formalized statement, and does not guarantee that it is consistent with the original problem, just as The Paper's paraphrase put it: "Lean saying your code has no bugs does not mean your code implements the function you want." Terence Tao called this phenomenon "proof indigestion": the speed of output has run far ahead of the capacity to digest.

If the "understanding gap" is an internal worry of the discipline, the second problem revealed by the conversation concerns the operation of the entire academic system: the chain of provenance for ideas is breaking.

Terence Tao pointed out that tech companies using AI to research mathematics are essentially extracting others' research results and obtaining data from others' papers and research. Academia has followed the principle of open science for hundreds of years: papers are public wealth, citations make ideas traceable, and this is how science progresses. But AI "opaquely transforms" the literature accumulated over thousands of years into answers; sometimes it can indicate which literature was referenced, and sometimes it cannot be traced. Tech companies have an incentive to play this down and make people think the answers appeared out of thin air.

Deng Yu gave a more specific example: there have already been cases where people directly used ideas that others entered into ChatGPT to generate proofs. He argued that AI should at least disclose its chain of thought so that people know what materials it referenced; not even being able to see the chain of thought is "too absurd."

This problem is in the same vein as the September open letter. The letter concentrated its criticism on AI Companies are mass-producing "true/false" statements at an ever-faster pace, bypassing peer review, while the cost of verification and digestion is borne by the mathematical community. There are also tougher voices within academia: on October 7, the Association for Human Mathematics (AHM) issued a statement saying that "releasing more than 700 documents at once demonstrates not academic research but a display of power," and urged mathematicians to stop collaborating with OpenAI; Terence Tao reposted this statement on his personal blog. In this conversation, Tao put it more bluntly: the results produced by batch problem-solving have become a kind of byproduct, one might even say "toxic waste," which then has to be dealt with somehow, or by mathematicians.

Domestic academia has also responded. On October 11, the "Yau Mathematical and Artificial Intelligence Laboratory Alliance" was established at Tsinghua University. Responding to the controversy at Qiuzhen College, Shing-Tung Yau said: "It's like a tsunami coming—how should we respond to the tsunami?" He Xuhua, for his part, believes that this batch of content "does not all conform to academic norms, but there are many ideas and results in it worth absorbing."

From "cleaning up afterwards" to "participating beforehand": incentives need to be repriced

In response to these structural problems, the core of the two mathematicians' proposal is to change incentives rather than reject the technology.

First, change the standard by which results enter the academic literature. Tao argues that if an AI-produced result gives only an answer, without accompanying understanding, explanation, and traceable citations, it should not become part of the mathematical literature. Such output needs to undergo formal verification first and be given a human digestion period.

Second, have mathematicians participate beforehand rather than clean up the mess afterwards. Tao believes the ideal state is for AI companies to collaborate with mathematicians at project launch—helping to understand proofs and annotate sources—rather than releasing first and then asking academia to backstop it. On October 21, China Youth Online reported on this conversation under the headline "AI companies should invite mathematicians to participate beforehand rather than clean up the mess afterwards."

Third, redistribute the value of "verification labor." In the past, solving a problem meant the completion of a whole set of understanding activities, and metrics and purpose had not yet become disconnected; now, "solving problems" itself is becoming the pursued goal. Borrowing Goodhart's law, once a metric becomes a pursued goal, it is no longer a good metric. Academic evaluation needs to see broader understanding and insight, and labor such as formal verification and explaining results must be priced.

Open mathematical models: hand the tools to the community

Another important piece of information in the conversation is the Open Math OS project launched by SAIR. According to Tao's introduction, its vision is to provide the mathematical community with tools developed by the community itself and aligned with its own values and goals: in the initial stage, it will not release new self-developed models but will first integrate existing open-source models and research tools, while pooling data, training resources, and compute, hoping to build its own model on the basis of open-source models in a few months.

Deng Yu's motivation for joining this project is the same as Tao's: to build a tool that can truly trace the origin of ideas and does not depend on big companies. In his view, AI is a tool, and whether it is good or bad depends on the system formed around it; only by staying open can AI develop healthily and continue to benefit the community.

This choice has a realistic target. In recommendations released at the end of September, the "Mathematics and Artificial Intelligence Advisory Group" of the Institute for Advanced Study in Princeton warned that proprietary models could create a two-tier system in which "a laboratory outruns the entire field," making tool owners effectively set the research agenda for the whole discipline; the group's nine members include three Fields Medalists: Gowers, Hairer, and Witten. Open-source mathematical models are precisely a precaution against this possibility: when tools are jointly owned by the community, the power to set the research agenda will not be concentrated in the hands of a few companies.

It is worth noting that neither of them sets "slow" against "fast." Tao stresses that one should not simply divide into supporters and opponents; the ideal is to have both the benefits of "slow science" and of fast methods: preserving the deep understanding, strong community, and sustained human mastery of the discipline that slow-paced research brings, while using tools that can accelerate research. Speed is not the only goal that needs optimizing.

The question left for researchers: what still belongs to humans

The conversation repeatedly returns to one topic: in the AI era, what capabilities do mathematical researchers still need to cultivate? The two men's answers are highly consistent: posing valuable questions, designing the overall proof route, developing new ideas, and making value judgments—these remain key human capabilities.

Deng Yu particularly stresses that "understanding" is the precondition for using AI: AI can do things well only on the premise that humans already understand the problem sufficiently; understanding is used both to guide AI and to enable people to work independently when they leave AI. When humans already have a roadmap for solving a problem and only the implementation details are too cumbersome, AI can take over the derivation and leave researchers' time for more fundamental questions.

Tao's attitude toward the younger generation is restrained but optimistic. He acknowledges that many old rules in mathematics are changing, and that the younger generation currently has a lot of anxiety and even despair. But he believes that once the tools are truly in their hands, they will come up with highly creative new approaches. "I am full of hope for the younger generation; as long as they have the right tools, they can achieve amazing results."

The landing point of these capabilities is also entering institutional discussion. Previously, the Leiden Declaration on Artificial Intelligence and Mathematics (released in June 2026), supported by the International Mathematical Union and signed by 2,411 mathematicians, recommended that papers add a "tools and computational resources disclosure" section and, where feasible, provide formal proofs; for young scholars, rather than competing on the track of problem-solving, which machines have already broken through, it is better to shift energy to links that machines are not yet good at, such as posing new questions and judging the value of results. Although the conversation does not reiterate these recommendations one by one, the direction given by the two laureates is fully consistent with them.

From the shock caused by the 722 preprints to this conversation between two Fields Medalists, the focus of discussion has shifted from "whether AI can do mathematics" to "what kind of AI" the mathematical community needs. This may be the most valuable turn in this round of debate: the question is no longer about taking sides, but about how to rebuild, along the dimensions of openness, traceability, understanding, and incentives, an academic order compatible with the new tools.

Mathematics endures by lasting. It has digested the crisis of set theory and also gave birth to computer science. This time, what it faces is not a crisis of being replaced but a more modest demand: no matter who writes the proof, the responsibility for understanding it still lies with humans. (This article was first published on TMTPost APP; author | Silicon Valley Tech-news; editor | Jiao Yan)