ICM 2026 Public Lectures - Terence Tao

Summary

Terence Tao delivers the ICM 2026 Public Lecture titled “Mathematics in the age of AI,” introduced by Alex Kontorovich at the International Congress of Mathematicians in Philadelphia on July 24, 2026. Tao argues that mathematics is entering a crisis in its foundations of values and practices rather than formal logic, examining how the advent of capable AI tools challenges community priorities around problem-solving, verification, exposition, and canonicalization.

What is shown

• [00:00 - 00:59] Introduction of Terence Tao by the session chair at the podium.
• [01:46] Historical prologue slide outlining the foundational crisis of the early 20th century (Russell’s paradox, Gödel’s incompleteness theorems, ZFC).
• [05:19] Framing of the “Community Response Question” regarding how mathematics should respond to modern AI tools.
• [06:50] The “AI Capability Conjecture” template slide and its role in current mathematical discourse.
• [12:09] Overview slide of the “First Proof” benchmark evaluation initiative (1stproof.org).
• [14:43] Discussion of community goals, values, and Goodhart’s law applied to AI-driven mathematical optimization.
• [23:35 - 35:36] Step-by-step deconstruction of the problem-solving workflow: from proof generation and verification to exposition, publication, and canonicalization.
• [30:24] Slide displaying Tao’s personal heavily annotated copy of Jean Bourgain’s 1991 paper on the Kakeya and restriction conjectures.
• [43:31] Discussion of the Leiden Declaration on Artificial Intelligence and Mathematics (leidendeclaration.ai).
• [45:13 - 48:26] Concrete community recommendations: tool disclosure, supporting reviewing needs, retaining human responsibility, and proper attribution.

Claims & numbers

• First Proof Challenge results: The presenter states that for the second batch tested on May 28, 2026, against four AI harnesses, 7 of the 10 unpublished research-level problems were solved at a publication-level quality by at least one team, with compute costs ranging from $10 to $1,000 USD per problem [12:09].
• Historical timeline: The crisis in mathematical foundations occurred roughly from 1900 to 1930 [03:17].
• Goodhart’s Law (1975): Quoted directly: “When a measure becomes a target, it ceases to be a good measure” [19:40].
• Shift from proof scarcity to abundance: Tao asserts the mathematical community is shifting from an era of “proof scarcity” to an era of “proof abundance,” creating “proof indigestion” due to impedance mismatches across proof generation, verification, and human digestion [24:46].

Notable quotes

• [04:23] “I believe we are entering a similarly turbulent period — a crisis in the foundations of mathematical values and practices.”
• [24:46] “In short, we will transition from an era of proof scarcity to an era of proof abundance.”
• [48:08] “If the authors cannot convincingly demonstrate that they can give a clear, expert-level talk on their result, that is correct and properly attributed, then the result should not be published.”

Assessment

This is an official recording of an academic plenary lecture given at ICM 2026. The lecture presents conceptual arguments, historical parallels, and policy frameworks rather than live software demonstrations, citing empirical results from community benchmarks (First Proof) and community manifestos (Leiden Declaration).

Described by gemini-3.8-flash on 2026-10-12 from the video’s audio and frames.