【AI×数学】フィールズ賞数学者ヴィラニさんが語る、OpenAIの衝撃発表と数学者たちの批判声明 / AIで数学はどう変わる？【フランス潜入】

AI Is Changing Mathematics Unusually Quickly

Cédric Villani says that a year earlier, mathematics seemed less affected by AI than empirical sciences such as medicine. That has changed: AI systems can now work with mathematical rules, generate candidate arguments, and check proofs, making mathematics one of the fields changing most rapidly.

He describes mathematical proof generation as combining two activities:

• Exploration: trying possible lines of reasoning and assembling ideas from existing work.
• Verification: checking whether the resulting argument is logically valid, including with formal proof systems such as Lean.

Villani says that translating mathematics into Lean has become dramatically faster. For example, he contrasts an earlier estimate that formalizing a major proof would take years with a later AI-assisted effort that reportedly completed it in under two weeks.

AI-Related Mathematical Announcements and Disputes

Villani recounts a rapid sequence of developments, which he places in 2026:

• AI work on the unit distance problem reportedly found configurations that challenged an existing conjecture.
• A result concerning the Jacobian conjecture was also described as a disproof.
• OpenAI publicized a set of mathematical contributions, accompanied by explanations and checks intended to help mathematicians assess them.
• Anthropic announced a formalization related to Fermat’s Last Theorem.
• On September 8, researchers associated with Anthropic announced progress on a case related to the Navier–Stokes regularity problem. OpenAI followed with a larger announcement, claiming a full solution to the Clay Millennium problem and presenting additional results.

Villani stresses that the mathematical community would need time to examine these claims. He considers an announced result concerning the Euler equation especially striking, while noting that the Navier–Stokes claim and other results still require scrutiny.

The announcements also raised priority and credit disputes. Villani says researchers feared that one group’s work might have been overtaken or used by another, and that similar disputes had occurred several times. He describes more than half of living Fields Medalists as supporting a statement warning of serious misalignment between AI development and scientific practice, including problems around attribution.

Commercial Incentives and Academic Resources

Villani argues that major AI companies pursue scientific demonstrations not only for scientific reasons, but also for publicity and to improve their broader capabilities. He says that high-profile projects within companies may receive computing resources ahead of other research, and that the companies’ primary goals remain money and power rather than a culture of science for its own sake.

He also points to the imbalance between corporate and academic resources: companies can spend vastly more on a problem than universities can. That raises concerns about academic independence, researchers’ dependence on commercial tools, and whether academic institutions can compete.

Uncertainty for Researchers and Students

Villani describes a sharp change in mood: discussions that seemed exciting only a month or two earlier began to feel threatening. Some young mathematicians, he says, worry that their careers could disappear; he recounts hearing of students crying over that prospect. In his telling, younger researchers can be more alarmed than senior mathematicians, who may be more inclined to experiment with the tools.

A particular concern is doctoral training. Villani reports that a Cambridge professor told him a public AI model could solve many problems at the level of the first years of a PhD within hours. If students can no longer work through problems that previously served as training, professors may struggle to explain what a PhD is for or how to assess students’ individual contributions.

Villani is still taking on two doctoral students himself—one with experience using AI and one focused on mathematics—but expects the number of students taken on may decline and selection may become more intensive. He also notes that mathematics already has relatively small master’s cohorts.

Education and the Development of Basic Skills

Villani is concerned that routine AI use can weaken the development of reading, concentration, calculation, and logical reasoning. He describes a French school exercise in which many students used AI-generated material in their essays, including following a hidden instruction. He argues that AI assistance can act like an exoskeleton: it may help someone complete a task without building the underlying ability.

His proposed distinction is between training and later use:

• During formative education, students should develop core abilities themselves.
• Some researchers and educators may choose to prohibit AI assistance during a PhD, at least while students are learning to reason independently.
• After that training period, students can decide how to use AI.

He compares this to restrictions on certain activities before adulthood. The point is not necessarily to forbid them forever, but to preserve a period in which people build the relevant capacities.

Possible Future Roles for Mathematicians and Universities

Villani says no one knows whether AI progress will continue exponentially, slow down, or reach a limit. He sketches two possibilities:

• If AI can solve most problems mathematicians pose, the profession may shift toward identifying which questions are important.
• AI may first exhaust or recombine ideas already available, leaving room for new developments beyond that stage.

He suggests that universities may increasingly teach people how to use AI responsibly, including prompting and ethics, while also preserving mathematical reasoning as a human practice—“the ancient art” of reasoning without relying on an algorithm. Universities might also cultivate mathematical taste and judgment in areas where humans remain valuable.

Villani identifies possible human strengths not captured by straightforward evaluation: emotion, non-logical or physical intuition, personal history, and experience. These may matter especially when defining new concepts, choosing problems, or recognizing what is mathematically significant—not just when producing a proof.

Broader Economic and Cultural Concerns

Villani highlights the irony that many researchers have contributed ideas, labor, or money to improve commercial AI tools without being paid by the companies that benefit. Referring to the economist and philosopher André Gorz, he frames this as a broader feature of capitalism: companies find ways to turn shared culture, creativity, and other forms of value into concentrated economic power.

Speakers and Sources

Speakers in the Interview

• Cédric Villani — mathematician and interviewee.
• Takahiro Anno — interviewer.

People, Works, Organizations, and Examples Mentioned

• OpenAI and Anthropic — AI companies whose mathematical announcements are discussed.
• Lean and its mathematical community — the formal proof system and community discussed.
• Terence Tao — mathematician whose blog coverage of the Navier–Stokes-related announcement is mentioned.
• Tristan Buckmaster — apparently the Australian mathematician involved in the announcement; the name is imperfect in the auto-generated subtitles.
• Charles Fefferman and Stein — mentioned in connection with the Navier–Stokes problem and Tao’s mathematical background.
• Córdoba and Martínez — researchers Villani says provided early ideas that later work built on; the names may be imperfectly transcribed.
• Karen Hao, author of Empire of AI — cited in Villani’s discussion of AI-company leadership and power.
• André Gorz — philosopher and economist whose book The Immaterial is discussed.
• The Clay Mathematics Institute / Millennium Prize Problems — referenced in connection with the Navier–Stokes claim.
• PISA — cited in Villani’s remarks about educational performance.
• An unnamed Cambridge professor, French mathematician, and French secondary-school teacher — cited through Villani’s anecdotes.