Fields Medalist Terence Tao warns AI could produce more math proofs than humans can handle

Artificial intelligence is triggering a foundational crisis in mathematics by shifting the field from a shortage of proofs to an overwhelming abundance of them, according to 2006 Fields Medal winner Terence Tao. Tao, 51, an Australian and American mathematician recognized for his contributions to partial differential equations, combinatorics, harmonic analysis, and additive number theory, made the remarks in his lecture, "Mathematics in the Age of AI," delivered at the International Congress of Mathematicians (ICM) last weekend. His speech drew widespread attention from the global mathematics community.He began by reflecting on mathematics between 1900 and 1930, when breakthroughs such as Russell's paradox (1901) and the Gödel incompleteness theorems (1931) forced mathematicians to reexamine the discipline's underlying assumptions.He said this "was a turbulent period for mathematics, but the end product was extremely valuable: an explicit, rigorous, and standardized foundational framework."Noting that there is scope for further improvement, he stated: "I believe we are entering a similarly turbulent period, a crisis in the foundations of mathematical values and practices." Fields Medalist Terence Tao. Photo courtesy of the University of California, Los Angeles Tao cited his First Proof experiment, conducted in May, in which frontier AI models successfully solved seven out of 10 novel research-level mathematics problems under controlled scientific conditions.The results were refereed by experts for both correctness and exposition, with the resulting proofs considered to be of publishable quality.Based on those results, he developed a "Working Hypothesis" predicting that "AI tools will, reasonably soon, become capable of performing a reasonable fraction of research-level mathematical tasks, with reasonable levels of success, quality, supervision, and cost."Assuming the Working Hypothesis holds, he argued that the central question will shift from what AI is capable of to: "What are the precise goals, objectives, and values of our mathematical community, and the enterprise of mathematical research?"Traditionally, those goals have included solving unsolved pure and applied problems, developing new theories and techniques, and understanding the world. Additional goals involve building a community of mathematicians, training the next generation to guide future directions, contributing to a shared network of mathematical knowledge, and creating enduring works of aesthetic value.In the past, these goals have been largely positively correlated, with progress in one typically leading to progress in the others, Tao explained.However, he warned that "all metrics, when excessively optimized for, are at risk of being subjected to Goodhart’s law," which states that: "When a measure becomes a target, it ceases to be a good measure.""The inherently ungrounded nature of generative AI, combined with the financial incentives of AI companies, makes the use of these tools particularly vulnerable to this law. Excessive AI optimization may in fact cause the many previously aligned goals of mathematics to diverge from one another," he added.According to Tao, the process of bringing a mathematical proof into the broader community consists of five stages. These include "opening problems, solving unsolved problems, verifying their correctness, ensuring clear communication, and finally, having them digested and accepted by the mathematical community."However, he noted that the accelerating pace of AI is creating a severe mismatch across these stages, resulting in what he described as a "proof indigestion.""If the Working Hypothesis holds, then without suitable policy and cultural changes, significant impedance mismatches (or proof indigestion) will emerge all throughout this process, as AI-generated proofs will accumulate waiting to be verified andmany verified AI-generated proofs will await a readable write up.""AI-generated proofs, even when required to be correct and well-written, will overwhelm our traditional peer-review system. And even the published proofs will be too numerous for the community to work into a definitive form," said Tao."We will transition from an era of proof scarcity to an era of proof abundance."He urged mathematicians to disclose the AI tools and computational resources used in their research papers. He also called on authors to present their results clearly and accurately, complete with full citations, to make peer review more effective."One needs to avoid the worst case scenario in which authors use AI tools covertly to aid their work, but conceal that usage to avoid criticism from peers," he said, and urged mathematicians to "normalize the responsible disclosure of AI assistance."Tao is a mathematician of Chinese descent, widely known as the "Mozart of Mathematics" for his extraordinary talent. At age 13, he became the youngest gold medalist in the history of the International Mathematical Olympiad. He was appointed a professor at the University of California, Los Angeles (UCLA) at 24, a job he is still holding, and won the Fields Medal seven years later.The International Congress of Mathematicians (ICM), held every four years, is the world's most prestigious gathering of mathematicians, bringing together thousands of researchers to present groundbreaking work. This year's congress was held in Philadelphia, United States.