Terence Tao Declares Mathematics 2.0 as AI Makes Proofs Abundant

In a recent essay and talk, Tao said Mathematics 1.0 operated in an environment where proofs were scarce. Solving an open problem required deep expertise, time and effort, so who solved what first became a key measure of contribution. AI is changing that environment. With sufficient compute and access to frontier models, people can obtain answers to many previously hard open problems in relatively short time. Tao said the harder tasks remain judging whether an answer is correct, how large the contribution is, and how it connects with existing research. Without that work, he warned, the field risks solving problems without increasing human understanding.

Tao listed three overlapping conditions that make mathematics vulnerable to AI: proofs can be objectively checked; mathematical objects and reasoning are highly digitizable; and a large body of high-quality research literature already exists in digital form. Combined with heavy investment from frontier AI labs, these conditions have pushed mathematics to the center of model capability races. Tao also invoked the classic aircraft bullet-hole image to warn of survivorship bias. Most current frontier problems remain unsolved, AI ability is uneven, and companies tend to show selected successes while unsuccessful attempts stay out of view. He called for a more transparent and scientific evaluation system that reports failures and AI compute costs.

Tao warned that blindly optimizing the metric of “solving problems” could damage mathematics’ long-term health. He compared open problems to lighthouses: they guide exploration of the surrounding mathematical world, but arriving at a lighthouse is not the same as completing the exploration. Success, failure and partial progress can all produce insights more valuable than the final answer. If automated tools deliver answers too early and the evaluation system rewards only “solved,” paths not yet taken may lose the incentive to be explored. Millennium Prize problems are finite, Tao noted, but the paths, understanding, connections and new questions that grow from a single problem are nearly infinite.

In a recent SAIR talk, Tao criticized model companies sharply: “It’s crazy. AI companies accelerate endlessly but know nothing about what happens after. We must slow down. There is no reason to go this fast—none at all.” According to the QbitAI report, the talk was originally titled “Machine-Assisted Proof” but Tao changed its theme to “Mathematics 2.0” because AI had repeatedly hit the mathematics world. In September, Tao had already written in a blog comment that AI can serve mathematical research, but chasing short-term goals may sacrifice the long-term sustainability of the mathematical community.

To show why human understanding still matters, Tao offered a thought experiment: ask an advanced AI to find a cancer therapy that can pass phase III trials. After burning compute, it returns a mixture of unknown chemicals; the model predicts it kills cancer cells, Lean verification passes, and the phase III trial passes. But no one knows how it found the therapy, and no one can be sure whether the AI truly found a cure or exploited a loophole in the trial process. Before the needle enters your vein, Tao asked, would you not want at least one cancer expert to understand the mechanism, one mathematician to understand the mathematical reasoning behind the model, or one scientist to understand the model that found the therapy? The experiment, he said, concerns goals, verification and understanding. A conclusion that passes validation inside a model still needs humans to judge whether it aligns with real-world goals. This is why human scientists cannot be replaced: models can generate answers, verify reasoning and accelerate exploration, but they cannot be the final gatekeepers.

Tao has divided problem-solving into five stages: generating a proof; verifying correctness; explaining it clearly; having it understood and accepted by the community; and integrating it into the standard theory of a field. The last step is especially slow. Researchers need to find more natural formulations and better proofs, and connect the result to other knowledge before it enters textbooks and the toolbox of the next generation of mathematicians. AI can hardly accelerate that step. If AI mainly accelerates the earlier stages, “proof indigestion” may follow. Tao therefore argues for less emphasis on being first to generate a proof and more emphasis on explanation, review and knowledge organization.

The dispute over “after the proof” intensified in early October. On Oct. 7, OpenAI released 722 mathematics papers, angering the slowdown camp led by Tao. Tao then helped establish the Association for Human Mathematics, or AHM, and issued a joint statement: releasing more than 700 documents at once is not research at all, but a pure display of compute hegemony. The statement strongly urged all mathematicians to stop cooperating with OpenAI and return to a scientific vision centered on human understanding. Tao said formal verification is not enough; authors should also be able to explain their results clearly. Work without that explanatory ability remains incomplete in his view.

These events also touch a question young researchers care about: if AI can rapidly generate papers, is graduate school still necessary? Tao said the key is not whether one can produce papers, but whether doctoral training builds a system of abilities to absorb difficult material, synthesize knowledge and ask good questions. Using food and exercise as an analogy, he said that in the AI era, a PhD still has value if it trains someone to chew through hard problems, synthesize knowledge and pose good questions; if the only goal is producing papers, its necessity is indeed declining. Just as abundant food and transportation require people to exercise deliberately, readily available answers require people to maintain cognitive ability and resist the convenience of bypassing thought. Tao also said he is advancing concrete efforts through the SAIR Foundation’s open mathematics model project. Its first phase targets daily research needs such as understanding difficult arguments, checking citations, exploring examples, writing code and formalizing proofs. The open math models will be released soon, according to the report.

At the end of his talk, Tao identified imagination as the main bottleneck for achieving Mathematics 2.0. The supply of proofs is changing, but how to use that change to advance understanding, training and exploration together remains a choice for the mathematical community. After AI proves a problem, the mathematician’s next questions may become: what can we learn from this answer, and what can we do next?