Terence Tao's ICM 2026 public lecture 'Mathematics in the age of AI' calls a crisis in the foundations of mathematical values

Description by Gemini, which watched the video:

Summary
Terence Tao delivers a public lecture titled "Mathematics in the age of AI" at the International Congress of Mathematicians 2026 (ICM 2026) on July 24, 2026. He evaluates the impact of advancing AI systems on mathematical research, comparing current shifts to historical foundational crises and warning that optimizing purely for automated problem-solving risks breaking the consensus-building, human understanding, and exposition that underpin mathematics.

What is shown

• [00:00] Title slide introducing Terence Tao's ICM 2026 public lecture on July 24, 2026.
• [00:46] Historical overview slide tracing the crisis in mathematical foundations (c. 1900–1930) and the formalization of naive concepts (sets, numbers, limits).
• [03:01] Formalization of the "AI Capability Conjecture (template)" framing AI capabilities in terms of expense, supervision, domain, and success rates.
• [04:40] Presentation of the "First Proof" benchmark evaluation slide assessing four frontier AI harnesses against novel research-level problems.
• [05:25] Analysis slides outlining the "Goals and Values Question" and examining Goodhart's law applied to mathematical goals.
• [08:56] Diagram showing how AI optimization causes divergent pressures on core mathematical goals (theory building, Erdős problems, Olympiads, teaching, community).
• [11:11] Workflow diagram illustrating the pipeline of mathematics: open problems $\to$ proof generation $\to$ unverified solutions $\to$ proof verification $\to$ verified solutions $\to$ proof exposition $\to$ well-written solutions.
• [12:41] Personal artifact: Tao shows heavily annotated scanned pages of a 1991 paper by Jean Bourgain from his graduate student days, explaining how struggling through dense proofs is essential to learning.
• [14:16] Slide citing William Thurston's 1994 paper "On proof and progress in mathematics".
• [17:50] Slide detailing the concept of "proof indigestion" and the shift from an era of "proof scarcity" to "proof abundance," drawing an analogy to dietary health and food abundance.
• [19:43] Recommendations slide urging the math community to tightly restrict AI in foundational education/training while developing new workflows for research.

Claims & numbers

• The presenter notes that for the "First Proof" benchmark, the second batch was tested under controlled scientific conditions against four AI harnesses on May 28, 2026, using ten novel research problems; seven of the ten 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.
• Tao notes that problem repositories such as erdosproblems.com already receive dozens of AI-generated proof submissions where submitters often cannot personally verify or explain the arguments.
• Tao argues that mathematical infrastructure faces "proof indigestion" under proof abundance, where generation and verification outpace human refereeing, exposition, and canonicalization.

Notable quotes

• [14:30] "We are not trying to meet some abstract production quota of definitions, theorems, and proofs. The measure of our success is whether what we do enables people to understand and think more clearly and effectively about math." (quoting William Thurston)
• [15:28] "Community acceptance of a result, by its nature, is slow and human. It can be encouraged with good exposition and careful writing. But it is ultimately an external process that cannot be optimized purely by the authors and their AI tools."
• [18:07] "In short, we will transition from an era of proof scarcity to an era of proof abundance."

Assessment
This is authentic footage of Terence Tao's live public lecture delivered at ICM 2026, captured from the audience. The talk contains no fabricated claims or product hype, focusing on meta-mathematical methodology, community governance, and philosophical reflections on AI integration into mathematical research.

Described by gemini-3.8-flash on 2026-09-29 from the video's audio and frames.