The future of mathematics
3 Aug 2026
AI cannot walk into a university, enter a classroom, pick up chalk, and write on the blackboard. It can't even understand the physical world as an ordinary child does. Yet it can solve Olympiad problems, produce research-level proofs, and resolve conjectures that have resisted mathematicians for decades.
Human children first acquire mobility, perception, and a practical understanding of the physical world before they learn mathematics, scientific reasoning, and other intellectually demanding subjects. It might therefore seem natural to expect artificial intelligence to follow a similar path. Yet it has not. Modern AI's path of development exemplifies Moravec's paradox, according to which capacities that humans acquire effortlessly (perception, movement, interaction, and understanding of physical environments) may be computationally more difficult to reproduce than certain forms of complex reasoning.1
The speed with which this reversal has occurred is equally striking. Initially, large language models (LLMs) could do little more than write fluent and grammatically coherent English. Although LLMs are still, fundamentally, systems trained to predict linguistic sequences, they've improved considerably and can now reason, plan, and act.
AI has already transformed coding from a manual activity, in which human programmers write, test, and debug individual lines of code, into an increasingly collaborative one between humans and AI. Agentic AI has pushed software developers into a role resembling that of a project manager. They no longer need to manually write code; instead, they can describe (in natural language) what they want to achieve, and AI proceeds to generate the code, run tests, and debug it. Thus, the human's role becomes that of an evaluator who judges the output and asks the AI for further changes as needed.
Recently, AI has started to do to mathematics what it has done to software development over the last few years. Just two years ago, in 2024, Google's specialised models achieved the equivalent of a silver-medal score at the International Mathematical Olympiad, and a year later, Google's Gemini reached gold-medal standard.2 Soon enough, models moved from standarised, self-contained competition problems into original mathematical research.
In the last year, AI models have resolved several Erdős problems, including the unit-distance problem in May 2026.3 In July, Anthropic researcher Levent Alpöge shared an explicit counterexample produced by Claude Fable 5 to the Jacobian conjecture in three dimensions, thus refuting a conjecture that had remained open since 1939.4 Since then, myriad results have followed, including a proof of the cycle double cover conjecture,5 a refutation of the Dinitz-Garg-Goemans conjecture,6, and advances on ten long-standing problems in various areas including sphere packinig, Ramsey theory, and extremal graph theory.7
OpenAI, Anthropic, and Google have recently been very active in this area. I suspect the reason is threefold. First, we view advanced mathematics as a “hard” subject and a true test of intelligence. Therefore, any significant research progress is taken as evidence of a model's capabilities. It's also good marketing. Second, unlike other more subjective areas of knowledge, mathematical claims are objective and comparatively easy to verify. And third, for the first time in history, long-standing open problems that have eluded mathematicians are within the reach of AI models.
Given the current rate of improvement of AI models' mathematical capabilities, together with the growing interest of frontier labs in applying AI to advanced mathematics, I predict that AI will solve (or make a decisive contribution to the solution of) at least one Millennium Prize Problem within the next year.
In response to the sudden improvement of AI's mathematical capabilities, some people have wondered what the role of mathematicians will be in the future. If AI models become so advanced that they can prove or refute virtually any theorem, open problem, or conjecture, what will mathematicians do?
This view stems from a misconception of what a mathematician is. Mathematicians do prove a lot of theorems, but theorem-proving is not done for its own sake. It is a means to an end: the formulation of new questions, the development of new theories and tools, the discovery of unexpected connections, the creation of powerful abstractions, and the deepening of mathematical understanding.
If we take this view of what a mathematician is, it becomes clear that, far from ending mathematicians' jobs, AI will empower them to an extent never seen before. Technical lemmas that might have taken weeks of arduous work to prove by hand, may now take minutes; ideas can quickly be validated or discarded, saving mathematicians from wasting time on paths that lead nowhere; alternative proofs can be generated and compared, sometimes revealing simpler arguments or a deeper understanding of why a result is true.8
Therefore, I believe that we stand at the advent of a golden age of mathematics. There will be a Cambrian explosion of mathematical results, powered by AI. Problems that once demanded years of specialised labour may be solved in days. If proving results is no longer the bottleneck, mathematicians will be able to devote their attention to generating ideas and building theories, free to pursue questions and speculative lines of thought for which they previously lacked the time.
The mathematical community, however, is not quite prepared for this transformation. The present system of journal submission and peer review is too slow and, if AI enables mathematicians to produce groundbreaking results at a scale never seen before, journals will have to adapt to a potentially overwhelming volume of (high quality) submissions. Two obvious ways to address this problem are to use AI to assess papers (if not fully independent of human peer reviewers, at least to a high degree), and to fast-track submissions whose main results have been formally verified with proof assistants such as Lean.9
In the short term, therefore, the rapid emergence of AI is likely to benefit both mathematicians, and mathematics itself. In the medium to long term, however, things may look rather different. So far, AI's mathematical achievements have largely involved proving (or refuting) theorems, open problems, and conjectures. In other words, models have taken well-defined problems, and solved them. It is yet to be seen if AI can create genuinely new, original, and revolutionary insights. Will models be capable of conceptual innovations comparable to the development of calculus by Newton and Leibniz? It is far from clear if LLMs can go beyond deductions and formal proofs to formulating fundamentally new frameworks, and there is evidence that current models can't produce such conceptual “jumps.”10 For the time being, then, even if AI becomes superhuman in theorem proving, we still need human mathematicians to come up with out-of-the-box, original, creative ideas that further develop the field. In this sense, mathematically advanced AI would not displace human researchers, but would allow them to concentrate more fully on the creative aspects of the field.
This equilibrium may not last long, however. Until only a few months ago, frontier models were incapable of producing the kind of groundbreaking mathematical research that they have generated in recent weeks. Given the current rate of improvement and the growing possibility of reaching recursive self-improvement (AI models that increasingly contribute to the research and development of their successors, in a recursive and self-sustaining way), I would not be surprised if original mathematical ideas of this kind soon became possible. In this case, the bottleneck may shift from mathematical discovery to interpretation and understanding. AI might advance mathematics to such an extent that we have an abundance of correct, formally verified results, but lack the time and conceptual resources needed to understand fully their significance or determine how they fit into the existing body of mathematics. AI might eventually produce new mathematics faster than humans can comprehend it, and sufficiently advanced models might even end up producing theories and proofs that escape the limits of human intuition.
Whatever happens, mathematics is unlikely to remain unchanged. AI is here to stay, and mathematicians will be forced to adopt it lest they fall behind. The more interesting question is whether AI will merely extend human mathematical thought, or eventually lead it somewhere humans can no longer follow.
Notes
1 H. Moravec, “Mind Children” (Harvard University Press, 1988).
2 Google DeepMind, “AI Achieves Silver-Medal Standard Solving International Mathematical Olympiad Problems” (2024); Google DeepMind, “Advanced Version of Gemini with Deep Think Officially Achieves Gold-Medal Standard” (2025).
3 OpenAI, “Planar Point Sets with Many Unit Distances” (2026).
4 See T. Tao, “A digestion of the Jacobian conjecture counterexample” (What's new, 2026) for an overview of the counterexample.
5 OpenAI, “A proof of the cycle double cover conjecture” (2026).
6 Dmitry Rybin, https://x.com/DmitryRybin1/status/2079904005652893709.
7 OpenAI, “Ten Advances in Mathematics and Theoretical Computer Science” (2026).
8 Cf. T. Tao's overview of the counterexample to the Jacobian conjecture found by Fable 5.
9 However, formal verification should not be treated as infallible. It guarantees that a proof is accepted relative to the theorem's formal statement, its assumptions, and Lean’s trusted kernel. Errors in any of these can undermine the result, allowing the proof of a false proposition to pass Lean's checks.
10 T. Zahavy, “Position: LLMs Can't Jump” (Proceedings of the 43rd International Conference on Machine Learning, 2026).