In a paper prepared for the 2026 International Congress of Mathematicians, mathematician Terence Tao argues that artificial intelligence will force a restructuring of mathematical research practices, publication criteria, and education. The essay, released on arXiv (2608.16753), outlines how the transition from proof scarcity to proof abundance creates operational and epistemological challenges distinct from earlier debates over automated theorem proving.
Tao frames the incoming disruption against the early 20th-century foundational crisis led by Russell's paradox and Gödel's incompleteness theorems. While that period forced mathematicians to formalize axiomatic systems, Tao argues the current shock tests implicit cultural values: how research credit is assigned, how comprehension is validated, and how educational pipelines maintain rigorous training standards.
The First-Proof Benchmark and Proof Abundance
As evidence for near-term research capability, Tao cites results from the second round of the First-Proof Project. In controlled evaluations against ten unpublished research-level mathematics problems, four evaluated AI systems produced passing solutions for seven problems at compute costs ranging from tens to hundreds of dollars per attempt.
Tao argues that as automated systems become capable of generating mathematically valid proofs at scale, the discipline faces an influx of results that outpace the human capacity to review and synthesize them. Online repositories, including the Erdős problem database, have already received batches of machine-generated proofs that lack human domain experts willing to conduct manual verification.
Goodhart's Law and the Loss of Pedagogical Friction
The essay highlights how commercial AI optimization targets quantifiable benchmarks rather than structural mathematical understanding. Applying Goodhart's law, Tao notes that optimizing generative models for formal acceptance criteria strips away the instructional friction inherent in human mathematical writing.
Human proofs frequently contain pedagogical signposts, deliberate lemmas, and transitional notation that reflect the author's path to discovery. In contrast, AI-generated or heavily polished machine proofs often eliminate intermediate conceptual cues, resulting in texts that Tao characterizes as mechanically valid but difficult for human researchers to learn from or build upon.
The Leiden Declaration and Publication Standards
To address ungrounded automated output, Tao endorses the Leiden Declaration on Artificial Intelligence and Mathematics, published in June 2026 by the International Mathematical Union. He proposes an explicit operational threshold for peer-reviewed publication:
- Explanatory Mastery Requirement: Research papers should not be accepted for publication unless the human authors can deliver a rigorous, expert-level technical presentation explaining the underlying mechanisms of the proof and accurately attributing all machine-derived steps.
- Formal Verification vs. Comprehension: A formally verified proof generated or assisted by automated tools should remain classified as incomplete if human practitioners cannot articulate why the proof holds.
Safeguarding Mathematical Training
Tao warns that unconstrained deployment of AI tooling in educational settings risks eroding deep mathematical intuition. While disclosing personal use of AI systems for literature retrieval, slide formatting, and text polishing, Tao contends that foundational coursework and doctoral training must restrict automated assistance so that emerging researchers develop independent problem-solving capabilities rather than prompt-tuning skills.
