Terence Tao: Mathematics in the Age of AI [pdf]

Mathematics in the age of AI Public lecture, International Congress of Mathematicians 2026 Terence Tao University of California, Los Angeles July 24, 2026 Terence Tao Mathematics in the age of AI Historical prologue: the crisis in foundations For centuries, mathematics successfully operated using “naive” foundations, with some fundamental questions largely delegated to philosophers, such as: What is a set? What is a number? What is infinity? What are the axioms of mathematics? Terence Tao Mathematics in the age of AI But in the early twentieth century, discoveries such as Russell’s paradox (1901) or the Gödel incompleteness theorems (1931) forced practicing mathematicians to critically re-examine their implicit assumptions about the foundations of mathematics. Terence Tao Mathematics in the age of AI This crisis in foundations (∼ 1900–1930) was a turbulent period for mathematics. But the end product was extremely valuable: an explicit, rigorous, and standardized foundational framework. There is scope for further improvement. But our current foundations have survived strenuous testing and are a trusted environment for mathematics. Terence Tao Mathematics in the age of AI I believe we are entering a similarly turbulent period —1 a crisis in the foundations of mathematical values and practices. But once we thoroughly examine and codify these foundations, our community will emerge stronger and more resilient than before. 1All em-dashes in these slides were human-generated. Terence Tao Mathematics in the age of AI The motivating question This talk is focused on the following question: Community Response Question How should the mathematical2 community respond to the advent of modern AI technologies, and their real and/or claimed capabilities to perform mathematical tasks? This is a question for the entire community; I do not presume to have all the answers. Nevertheless, I have some things to say. 2The impacts of AI of course extend far beyond mathematics, but that is too vast a topic to cover in this talk. Terence Tao Mathematics in the age of AI Community Response Question How should the mathematical community respond to the advent of modern AI technologies, and their real and/or claimed capabilities to perform mathematical tasks? This question is not a mathematical question; it is a metamathematical (and political, ethical, and cultural) one.

However, in this talk, I will borrow from the precise, familiar language of mathematics in order to clarify the points I wish to make today. Terence Tao Mathematics in the age of AI The first subquestion: AI capability The answer to this question depends crucially on the following subquestion, which I will formulate pseudomathematically as a conjecture (or more precisely, a family of conjectures): AI Capability Conjecture (template) At some point in the near future, some AI tools will, at some expense, and with some level of human supervision, be able to accomplish some research-level mathematical tasks in some fields of mathematics, with some non-trivial success rate, and at some level of correctness and quality. The words “some” here should be viewed as placeholders (or “variables”). Terence Tao Mathematics in the age of AI AI Capability Conjecture At some point in the near future, some AI tools will, at some expense, and with some level of human supervision, be able to correctly accomplish some research-level mathematical tasks in some fields of mathematics, with some non-trivial success rate, and at some level of correctness and quality. One can create many formulations of this conjecture, depending on how one fills in the placeholders. The finer distinctions between these formulations is not the point of my talk today. I will only make only a coarse distinction between “weak” and “strong” forms of the conjecture. Terence Tao Mathematics in the age of AI If even weak forms of the AI Capability Conjecture are false, then we could safely choose to dismiss AI tools as being of no long term significance, and largely continue “business as usual”. But if the strongest forms of the conjecture are true, then it becomes challenging to maintain our current culture and practices, particularly if we prioritize such goals as obtaining as many solutions of unsolved problems as possible. Terence Tao Mathematics in the age of AI It is difficult to have a constructive discussion on the Community Response Question when the status of the AI Capability Conjecture remains under dispute. Thus, much of the debate about AI for mathematics has focused on which versions of the AI Capability Conjecture are true. I myself have devoted many lectures, writings, and posts on social media to such topics. Terence Tao Mathematics in the age of AI There are now many data points for or against various forms of the AI Capability Conjecture.

But most have not been gathered under controlled scientific conditions. In particular, much of the publicly available evidence is highly subject to reporting bias and non-scientific incentives, with some important costs and variables remaining undisclosed. Furthermore, it is important to distinguish the truth value of a given form of the conjecture from its desirability. Terence Tao Mathematics in the age of AI Despite the central relevance of the AI Capability Conjecture to the Community Response Question, my talk will not be about that conjecture. In this direction, I will only mention the recent results of the First Proof challenge 1stproof.org. Terence Tao Mathematics in the age of AI First Proof First Proof is an independent assessment of the state of frontier AI models and harnesses. Each “batch” consists of ten novel research-level problems in various fields. The second batch was tested under controlled scientific conditions against four AI harnesses on May 28, 2026. Results were refereed by experts for both correctness and exposition. Seven of the ten problems were solved at a publication-level quality by at least one team. Compute costs ranged from 10 to 1000 USD per problem. Further “batches” are planned in the near future. Terence Tao Mathematics in the age of AI The complement to the capability conjecture This talk will instead be about the “orthogonal complement” to the AI Capability Conjecture within the Community Response Question. The rest of this talk will therefore be conditional on the following (imprecisely stated) hypothesis: Working Hypothesis A reasonably strong version of the AI Capability Conjecture is true: 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. The precise definition of “reasonable” is not critical for my talk. Terence Tao Mathematics in the age of AI Working Hypothesis 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. For the rest of this talk, I will ask you to assume that the Working Hypothesis holds. I am not asking you to want, believe, or accept that this hypothesis is true — this will be a conditional analysis. Evidence for or against the Working Hypothesis is orthogonal to the rest of my talk.

Terence Tao Mathematics in the age of AI The orthogonal subquestion: our goals and values Once we condition on the Working Hypothesis, another fundamental subquestion comes into view. Goals and Values Question What are the precise goals, objectives, and values of our mathematical community, and the enterprise of mathematical research? Not just the explicit goals that we communicate to the public (or to funding agencies), but also the implicit goals that we actually seek in practice? Terence Tao Mathematics in the age of AI Goals and Values Question What are the precise goals, objectives, and values of our mathematical community, and the enterprise of mathematical research? In the past, we largely delegated this question to the humanities, and focused instead on the technical aspects of our profession. Assuming the Working Hypothesis, we will no longer have this luxury. However, I argue that a critical examination of this question will ultimately be highly valuable, regardless of the status of the Working Hypothesis. Terence Tao Mathematics in the age of AI What are our goals? There are many reasons to justify mathematical research. To list just a few: To solve unsolved problems (both pure and applied). To develop new theories and techniques. To understand the world around us. To build a community of mathematicians. To train the next generation of mathematicians to guide its future directions. To contribute to the shared network of mathematical knowledge. To create enduring works of aesthetic value. etc. Terence Tao Mathematics in the age of AI In the past, these goals have been largely positively correlated with each other: progress in one goal typically also led to progress on the others. As such, one could use one or two of these goals as proxies for the others. Many of the goals could be left implicitly stated only. Mathematical knowledge Goal 1 Goal 2 Terence Tao Mathematics in the age of AI However, all metrics, when excessively optimized for, are at risk of being subjected to Goodhart’s law: Goodhart’s law (1975) When a measure becomes a target, it ceases to be a good measure. The inherently ungrounded nature of generative AI, as well as the financial incentives of AI companies, make the use of AI tools particularly vulnerable to this law. Mathematical knowledge Goal 1 Goal 2 Terence Tao Mathematics in the age of AI Excessive AI optimization may in fact cause the many (previously largely aligned) goals of mathematics to diverge3 from each other.