Terence Tao: The Era of Mathematics 2.0 Has Arrived!

Terence Tao has formally published an article proposing that "the Mathematics 2.0 era has arrived." The core judgment is that AI is making mathematical proofs abundant, and that the Mathematics 1.0 era, built on the scarcity of proofs, is coming to an end. Within the Mathematics 1.0 framework, proving an open problem requires deep expertise and a great deal of time and energy, so "who first solved what problem" naturally became an important way to measure contribution. When the threshold for obtaining solutions drops, the premise of this evaluation system is shaken.

He states plainly that with sufficient compute and access to frontier models, people can already obtain, in a relatively short time, solutions to many open problems previously considered very difficult. But judging whether a solution is correct, how great the contribution actually is, and how to connect it with existing research still requires serious work. Otherwise, a worrying misalignment arises—problems get solved, but human understanding does not increase correspondingly.

Tao summarizes three mutually overlapping conditions that make mathematics vulnerable to AI: first, proofs can be objectively checked; second, mathematical objects and reasoning processes are highly digitizable; third, a large body of high-quality research literature already exists in digital form. These three together, plus heavy investment from frontier AI labs, quickly pushed mathematics to become a core battleground for rapid model capability advances.

This summary also explains why mathematics feels the impact earlier than other fields: objective checkability means results are easy to judge, high digitizability means training and inference material is abundant, and already-digitized literature means the knowledge base is ready-made. If any one of the three were missing, AI's pace of progress would be different.

Beyond the optimistic narrative, Tao holds up the classic aircraft bullet-hole diagram to warn against "survivorship bias." He points out that most problems at the current frontier of mathematics remain unsolved, and AI's capabilities are uneven—it performs astonishingly in some directions and remains insufficient in others. What AI vendors want us to see is often only carefully selected, published success cases, while a large number of unsuccessful AI attempts may never have a chance to enter public view.

This practice of counting only successes cannot answer key questions such as success rates, resource consumption and the degree of human involvement. He therefore calls for an urgent need to establish a more transparent and scientific evaluation mechanism that includes failed results and AI compute costs in reporting.

As for what the goal of mathematical research actually is, Tao offers a metaphor: open problems are like lighthouses—they guide people to explore the surrounding mathematical world, and reaching a lighthouse does not equal completing the exploration. In the process of trying to solve a problem, whether one succeeds, fails, or achieves only partial progress, insights may emerge that are more valuable than the final answer.

He warns that if automated tools hand over answers too early, while the evaluation system rewards only "solved," paths not yet traveled may lose the motivation for further exploration. The Millennium Prize open mathematical problems are finite, but the paths, understanding, connections and new problems that grow out of a single problem are almost infinite. He therefore explicitly warns: continuing to blindly optimize the "problem-solving" metric will harm the long-term health of mathematics.

In "Mathematics in the Age of AI," Tao breaks problem-solving into five stages: generating a proof, verifying correctness, explaining it clearly, being understood and accepted by the community, and finally integrating into the field's standard theory. The last step is especially slow—researchers need to find more natural formulations and more suitable proofs, and connect the result with other knowledge, before it can enter textbooks and the toolbox of the next generation of mathematicians.

And this is precisely what AI finds hard to accelerate. If AI mainly accelerates the earlier stages, "proof indigestion" may follow. He therefore advocates reducing the emphasis on being first to generate a proof and increasing attention to explanation, review and knowledge organization. In his view, formal verification is far from enough; authors should also be able to explain their results clearly. Results lacking this explanatory ability remain incomplete.

To illustrate why human understanding still matters, Tao proposes a thought experiment: suppose an advanced AI is asked to find a cancer therapy that can pass Phase III clinical trials. After burning through compute, it gives a mixture of unknown chemical substances; the model predicts it can kill cancer cells, Lean verification passes, and Phase III trials also pass. But no one knows how it found it, and no one can be sure whether the AI truly found a therapy or exploited a loophole in the trial process.

Before the needle goes into the vein, would you want at least one cancer specialist to understand its mechanism of action, one mathematician to understand the mathematical reasoning behind the model, or at least one scientist to understand the model that found the therapy? This thought experiment pulls the abstract mathematical discussion into a concrete scenario: a conclusion inside the model has passed verification, yet someone still needs to judge whether it aligns with the goal in the real world. Models can generate answers, verify reasoning and accelerate exploration, but they cannot gatekeep for humans.

Tao's concerns are also reflected in his criticism of AI companies' acceleration. In his latest SAIR talk, he said AI companies accelerate endlessly yet know nothing about what will happen afterward: "We have to slow down; there is absolutely no reason to go this fast." Dramatically, he revealed that the original theme of this speech draft was "Machine-Assisted Proving," and it was ultimately changed to "Mathematics 2.0" because AI has recently been "bombarding" the mathematics community.

After OpenAI released 722 mathematics papers on October 7, Tao led the establishment of the Association for Human Mathematics (AHM) and issued a joint statement, saying that releasing more than 700 documents at once is not research but a show of compute hegemony, and urging mathematicians to stop collaborating with OpenAI. On the question that concerns young researchers, he believes whether a PhD is necessary in the AI era depends not on whether one can produce papers, but on cultivating the systematic ability to absorb difficult material, synthesize knowledge and ask good questions.

The SAIR Foundation, which he co-founded, has launched an open mathematics model program. Its first phase targets everyday research needs such as understanding difficult arguments, checking citations, exploring examples, writing code and formalizing proofs, and it is announced to be released soon. At the end of the talk, he identifies imagination as the main bottleneck to realizing Mathematics 2.0: the supply of proofs is changing, but how to use this change to advance understanding, training and exploration together still requires the mathematics community to make a collective choice.