Terence Tao’s “Math 2.0” Talk: AI Is Making Proofs Abundant, and Blindly Pursuing Problem-Solving Is Harmful to Mathematics
Mathematician Terence Tao gave a public lecture at the California Institute of Technology on October 9. Originally titled “Machine-Assisted Proof,” it was ultimately retitled “Math 2.0.” In the lecture notes posted on his personal website, he said that mathematics is entering an era of “proof abundance.” The traditional way of evaluating mathematicians, centered on solving problems, is now out of step with mathematics’ long-term goals. “Continuing to blindly pursue problem-solving alone is now harmful to the long-term health of mathematics.”
The lecture was part of the public lecture series of the American Institute of Mathematics (AIM). Tao said on his website that he would use the title “Math 2.0” going forward, and that a video and transcript of the talk would be posted later.
From “Proof Scarcity” to “Proof Abundance”
Tao describes traditional mathematics as “Math 1.0,” arguing that it is built on the premise of “proof scarcity”: solving an open problem and proving the solution correct has always been difficult and rare, usually requiring extensive expertise. Many of the mathematical community’s institutions, traditions, and reward systems were designed around this premise.
He argues that mathematical proofs have three characteristics that make them particularly well suited to machine learning: proofs can be objectively verified, even checked by computers using proof-assistant languages such as Lean; proofs are highly digitized and do not require physical experiments or real-world data; and a large body of research literature has also been digitized. As a result, frontier AI labs have prioritized building mathematical problem-solving capabilities. “The result of this enormous investment is that we are entering an era of proof abundance in mathematics,” Tao wrote.
He notes that, with enough computing power and frontier models, many open problems once considered difficult can now be solved in a relatively short time—and increasingly, the people prompting the models do not need relevant expertise. By contrast, AI still performs poorly on tasks involving subjective judgment, real-world interaction, and scarce data. He describes AI’s capabilities as extremely “uneven.”
Using Survivorship Bias to Explain the AI Problem-Solving Boom
In his lecture notes, Tao included an image of bullet holes in a World War II bomber, alongside the caption, “Wow, every math problem I see on social media has been solved by AI!” The image uses survivorship bias to illustrate how the public sees only the success stories. He says that despite the impression created by press releases and social media, most frontier problems remain unsolved even after substantial computing resources are devoted to them, while failed AI attempts are rarely reported.
He called for more transparent and scientific assessments of AI capabilities, with full disclosure of key data such as resource consumption and negative results. As an example, he cited First Proof, an initiative that independently evaluates AI’s mathematical capabilities.
Open Problems Are Beacons, Not Destinations
Tao argues that although problem-solving has always been the most visible goal of mathematical research, in practice it is merely a “proxy” for more valuable long-term goals. In pure mathematics, open problems are more like “beacons,” guiding mathematicians as they explore the surrounding fields. What is learned along the way—whether through success, failure, or partial progress—is often more valuable than the final answer. If automated tools reach these beacons too soon, they may interrupt paths that have yet to be explored, “leaving the surrounding fields barren.”
He also offered a thought experiment: Suppose someone asks an AI to “find a cancer treatment that can pass Phase III clinical trials.” After using enormous computing resources, the AI finds a previously unknown combination of chemical drugs. A mathematical model predicts that it can kill cancer cells, the prediction is verified using Lean, and the treatment ultimately passes Phase III trials—but nobody knows how the combination was found. Tao asks whether this solution might have exploited a loophole in the trial process. “Before injecting this combination of drugs into your bloodstream, wouldn’t you want at least one human cancer expert to understand the mechanism behind the treatment?”
Open-Source Mathematical Models Promised “Soon”
Tao argues that the aim of “Math 2.0” should be to use AI to enhance every aspect of human understanding of mathematics, rather than using it only to produce proofs. The approaches he lists include finding more comprehensible explanations for theorems whose existing proofs are difficult to understand; integrating large numbers of papers into a continuously updated network; building large formalized mathematics databases such as Mathlib; and conducting “ablation studies”—removing a key condition to see whether a proven theorem still holds.
He also mentioned developing open-source mathematical models tailored to mathematical tasks, with high interpretability and safety. “Some open-source mathematical models will be released soon, so stay tuned,” he wrote, without specifying which team would release them or when. The final slide of his lecture notes says that the main bottleneck to realizing Math 2.0 is not technology, or even institutional culture, “but imagination.”
Tao was among the first signatories of a statement signed by 25 Fields Medalists on September 11, which likewise criticized the mismatch between AI companies’ problem-solving races and the goals of the mathematical community. The Caltech Mathathon mentioned in the lecture notes will be held November 13–15. Organized by students, it invites participants to use open-source AI tools to find new proofs or explanations for theorems whose existing proofs are unsatisfactory.
This article, Terence Tao’s “Math 2.0” Talk: AI Is Making Proofs Abundant, and Blindly Pursuing Problem-Solving Is Harmful to Mathematics, originally appeared on .
Mathematician Terence Tao gave a public lecture at the California Institute of Technology on October 9. Originally titled “Machine-Assisted Proof,” it was ultimately retitled “Math 2.0.” In the lecture notes posted on his personal website, he said that mathematics is entering an era of “proof abundance.” The traditional way of evaluating mathematicians, centered on solving problems, is now out of step with mathematics’ long-term goals. “Continuing to blindly pursue problem-solving alone is now harmful to the long-term health of mathematics.”
The lecture was part of the public lecture series of the American Institute of Mathematics (AIM). Tao said on his website that he would use the title “Math 2.0” going forward, and that a video and transcript of the talk would be posted later.
From “Proof Scarcity” to “Proof Abundance”
Tao describes traditional mathematics as “Math 1.0,” arguing that it is built on the premise of “proof scarcity”: solving an open problem and proving the solution correct has always been difficult and rare, usually requiring extensive expertise. Many of the mathematical community’s institutions, traditions, and reward systems were designed around this premise.
He argues that mathematical proofs have three characteristics that make them particularly well suited to machine learning: proofs can be objectively verified, even checked by computers using proof-assistant languages such as Lean; proofs are highly digitized and do not require physical experiments or real-world data; and a large body of research literature has also been digitized. As a result, frontier AI labs have prioritized building mathematical problem-solving capabilities. “The result of this enormous investment is that we are entering an era of proof abundance in mathematics,” Tao wrote.
He notes that, with enough computing power and frontier models, many open problems once considered difficult can now be solved in a relatively short time—and increasingly, the people prompting the models do not need relevant expertise. By contrast, AI still performs poorly on tasks involving subjective judgment, real-world interaction, and scarce data. He describes AI’s capabilities as extremely “uneven.”
Using Survivorship Bias to Explain the AI Problem-Solving Boom
In his lecture notes, Tao included an image of bullet holes in a World War II bomber, alongside the caption, “Wow, every math problem I see on social media has been solved by AI!” The image uses survivorship bias to illustrate how the public sees only the success stories. He says that despite the impression created by press releases and social media, most frontier problems remain unsolved even after substantial computing resources are devoted to them, while failed AI attempts are rarely reported.
He called for more transparent and scientific assessments of AI capabilities, with full disclosure of key data such as resource consumption and negative results. As an example, he cited First Proof, an initiative that independently evaluates AI’s mathematical capabilities.
Open Problems Are Beacons, Not Destinations
Tao argues that although problem-solving has always been the most visible goal of mathematical research, in practice it is merely a “proxy” for more valuable long-term goals. In pure mathematics, open problems are more like “beacons,” guiding mathematicians as they explore the surrounding fields. What is learned along the way—whether through success, failure, or partial progress—is often more valuable than the final answer. If automated tools reach these beacons too soon, they may interrupt paths that have yet to be explored, “leaving the surrounding fields barren.”
He also offered a thought experiment: Suppose someone asks an AI to “find a cancer treatment that can pass Phase III clinical trials.” After using enormous computing resources, the AI finds a previously unknown combination of chemical drugs. A mathematical model predicts that it can kill cancer cells, the prediction is verified using Lean, and the treatment ultimately passes Phase III trials—but nobody knows how the combination was found. Tao asks whether this solution might have exploited a loophole in the trial process. “Before injecting this combination of drugs into your bloodstream, wouldn’t you want at least one human cancer expert to understand the mechanism behind the treatment?”
Open-Source Mathematical Models Promised “Soon”
Tao argues that the aim of “Math 2.0” should be to use AI to enhance every aspect of human understanding of mathematics, rather than using it only to produce proofs. The approaches he lists include finding more comprehensible explanations for theorems whose existing proofs are difficult to understand; integrating large numbers of papers into a continuously updated network; building large formalized mathematics databases such as Mathlib; and conducting “ablation studies”—removing a key condition to see whether a proven theorem still holds.
He also mentioned developing open-source mathematical models tailored to mathematical tasks, with high interpretability and safety. “Some open-source mathematical models will be released soon, so stay tuned,” he wrote, without specifying which team would release them or when. The final slide of his lecture notes says that the main bottleneck to realizing Math 2.0 is not technology, or even institutional culture, “but imagination.”
Tao was among the first signatories of a statement signed by 25 Fields Medalists on September 11, which likewise criticized the mismatch between AI companies’ problem-solving races and the goals of the mathematical community. The Caltech Mathathon mentioned in the lecture notes will be held November 13–15. Organized by students, it invites participants to use open-source AI tools to find new proofs or explanations for theorems whose existing proofs are unsatisfactory.
This article, Terence Tao’s “Math 2.0” Talk: AI Is Making Proofs Abundant, and Blindly Pursuing Problem-Solving Is Harmful to Mathematics, originally appeared on .