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Mathematics Has Changed: A Number Theorist's Action List
On November 1, 2026, Terence Tao’s blog What’s new published a guest article titled “Mathematics has changed. Let’s act like it” (Mathematics has changed; let’s act according to its new state). Let’s clarify the author’s identity: the author of this article is Frank Calegari, a number theorist from the University of Chicago. Tao was merely the publisher, and his opinions belong to Calegari. In discussions on social media, many people mistook this long article for Tao’s personal statement, until someone pointed out that it wasn’t written by him. This misunderstanding itself highlights an issue: nearly a month after OpenAI uploaded over 700 mathematics papers, the mathematical community was so sensitive to “who is speaking” that even the attribution became a point of contention.
Underlying Event: The Paper Upload on October 6
The source of the controversy was OpenAI’s announcement page “Sharing AI progress in mathematics” on October 6. According to the official statement, these results were generated by an unpublished internal advanced model, stored in a GitHub repository along with protocols for revising and citing papers. Many of these proofs used Lean formalizations (proofs that can be checked by computers), and the repository also disclosed 10 model reasoning summaries and computational resources: on average, each result required approximately 3 hours of computing power equivalent to what ChatGPT Pro would use. OpenAI claimed that they consulted the Mathematics and Artificial Intelligence Advisory Group at the Institute for Advanced Study (IAS) before publishing these guidelines. On October 15, they updated their announcement to donate $5 million to the Simons Foundation’s new program to support the mathematical community in testing, understanding, and continuing to develop these AI results.
The number “over 700” comes from the first paragraph of Calegari’s article. He described this release as “a watershed in the history of mathematics.” But a watershed does not mean endorsement.
Further Controversy: The Long Article on November 1, First Criticism of OpenAI Papers
The most widely circulated criticism in the article is its harsh assessment of OpenAI’s papers. The text says: “OpenAI’s papers—despite some excellent ideas—are horribly written and violate our community’s expectations regarding attribution. Some of them are even wrong.” In plain English: OpenAI’s papers, despite some brilliant ideas, are poorly written and do not meet our community’s standards for attribution; some of them are even incorrect.
The person who made this statement is not an old-fashioned anti-AI supporter. Calegari is optimistic in his article: he acknowledges that AI can make groundbreaking contributions in many areas of mathematics. Using someone else’s words, he said, “I’m ready to board and see what’s on board.” Because the criticism comes from someone who is already on the optimistic side, its weight is greater: the problem lies not in the model’s capabilities but in the company’s approach to mathematics, ignoring the rules of the industry.
This serves as the pivot of the entire text. The original text says: “We have much more control over how we hire mathematicians, publish papers, and train students than over the behavior of trillion-dollar companies, and that is where our efforts would be better spent.” Translated: “Compared to the behavior of trillion-dollar companies, we have far more control over how we recruit mathematicians, publish research, and train students. Our efforts should be directed at areas where we truly have control. Criticizing OpenAI won’t change it; only by changing our own rules can we make a difference.”
He also believes that the panic is unfounded. He said that this announcement “almost didn’t touch” the general issues of functorality and reciprocal laws in Langlands’ program. Even if it solved all the problems in Langlands’ program, imagining that it “solved number theory” is as naive as claiming that the periodic table’s creation solved chemistry.
But he didn’t avoid the most damaged areas. He quoted Hugo Duminil-Copin, a Fields Medalist, who lamented: “None of the major unsolved problems I’ve openly discussed in my career—whether in reports, lectures, articles, or grant applications—escaped this announcement. They were all claimed to have been solved. I expected to see some of them on the list, but not all at once, and not with such indifference.” It should be noted that Calegari added the second half of this quote in his notes: “We are a resilient community, and I have no doubt we will adapt.” He quoted Duminil-Copin’s lament, but not to portray him as a pure pessimist.
Historical analogy: After 1983, famous problems disappeared
Calegari’s response is a piece of mathematical history. In 1972, Dan Gorenstein announced the classification program for finite simple groups. About ten years of intensive work later, he announced in 1983 that the classification was complete, but somewhat early: it took another twenty years for Aschbacher and Smith to complete the case of quasithin groups.
His assessment is straightforward: The classification was a huge success in many ways, but also a failure, as it cooled down the interest surrounding the entire field. “Famous problems” disappeared, and mathematicians shifted their attention elsewhere. Yet there are still many rich problems in finite group theory, such as one of John Conway’s favorites: why do monster groups exist?
Then there’s the most sharp metaphor in the text. The original text says that for some fields, OpenAI’s announcement might feel like a group theorist in 1970 seeing the classification results dumped into his own office: not in the form of formal papers, but printed out by an alcohol copier, with ink smudged, then scattered across the floor like colored paper scraps. It was received, but no one could digest it, and no one claimed it.
What to do? He pointed to the revised classification project led by Gorenstein, Lyons, and Solomon: What is needed now is work that transforms ruins into inheritable knowledge, and when giving positions, those who do this work should be rewarded. There is also a sentence in the original text: When mathematicians lament that “the galaxy highway” (a reference to The Hitchhiker’s Guide to the Galaxy) bypasses the winding old paths, they must realize that the old paths still exist and retrieve them. This is not consolation, but a recruitment signal: To digest the piles of AI papers scattered on the floor is not a dead end; it can become a reason for getting a position.
The Ugly Truth: The Attack Is On Our Own People
One section in the long article is titled “The ugly truth,” criticizing the mathematics community itself. Just look at arXiv: many people in this field are unreasonably competing to publish AI-assisted papers, striving to be the “first” to prove them. A rapid gap is widening between the ability to produce new proofs and the ability to demonstrate understanding or to push the discipline in new and interesting directions. He specifically mentioned that the Leiden Declaration (an initiative by the mathematics community on norms for using AI) is being “officially followed, but spiritually violated.” Everything remains the same, only the time scale has been compressed, with everyone facing the threat of premature publication.
He provides clear criteria for recognition: If someone selects a worthy problem, proves it entirely with AI, fully understands it, and uses it to develop interesting subsequent problems or directions, they have made substantial mathematical contributions. Even if their original proof does not contain any new elements contributed by them, they should still be recognized. Selecting worthy problems may ultimately be more important than producing proofs. For senior mathematicians, he has an uncompromising statement: If you are still anxious about being prematurely published by AI, it means your ambition is not big enough.
Behind this is a set of economic principles. He uses Nelson Cole’s example: Cole spent the famous “three years of Sundays” to find the prime factorization of 2^67−1, which was a mathematical contribution. Today, computer algebra systems can decompose the same number in less than a second; doing it manually is just an arithmetic feat, not a contribution anymore. AI is rewriting the economics of “difficult” and “easy”; the “theorem economy” is facing collapse. His conclusion is: In the future, there will be no reason to reward a paper just because a machine can produce the same result, regardless of whether it is machine-generated or not.
Action Plan: Six Points All Relate to Money and Positions
The long article is very straightforward about its own positioning: Comment articles that do not propose mechanisms to truly change people’s behavior are ultimately meaningless; the most effective tools are economic, not rhetorical. Mathematicians want recognition, positions, and funding, so the only real levers at their disposal are recruitment and journal acceptance. He outlines six points:
- Evaluate all work as if it has been assisted by AI. The author must explain: why this mathematics is interesting, what problems it raises, and how it connects to broader disciplines. He acknowledges that this is the most controversial and inevitable point. Currently, using AI still carries a stigma; many statements about disclosure read as an attempt to minimize the machine’s contribution, and this attitude won’t last long.
- Prioritize face-to-face interactions over paper resumes. Postdocs and graduate students should also undergo in-person interviews. He criticizes past lazy committees: they praised candidates based on a single paper from Annals, without considering their ability to narrate the direction of mathematics, their technical skills for explaining feasibility, their cross-disciplinary communication abilities, and their verifiable past records. There’s a poignant example: hiring a graduate student who claimed to work in “higher-dimensional homology theory” only to find out he couldn’t calculate the homology of circles.
- Set higher standards for journals and abandon those that no longer meet requirements. He is quite harsh here: a significant portion of papers posted on arXiv should not be published anywhere at all. If the editorial board cannot change this, a more radical idea is (at least temporarily) to completely abandon the journal system, removing the “publication” layer, and let papers stand on their own value: Is the story well told? Has it been verified by Lean? The goal should be fewer but better papers; fully automated AI-generated proofs, preferably checked by Lean, should also be included in the collective knowledge base.
- Graduate education should emphasize breadth, but not abandon technical training. Just as computers exist, it doesn’t mean we stop teaching children arithmetic.
- Organize better discussion sessions. Reports should not focus solely on minor improvements to latest results, but rather revolve around bigger questions: where we are stuck, where progress is possible, and where hope still exists, before finally discussing new ideas (if any).
- Have more ownership over AI’s results. The things machines produce are built upon thousands of years of human mathematical accumulation; instead of attributing these results to a Silicon Valley tycoon or a math Twitter prankster, it’s better to claim them on behalf of all humanity.
At the end, he shares an anecdote: when the dean went to New York to meet donors, regardless of whether they were in mathematics or not, they wanted to talk about mathematics; 3Blue1Brown has over six million subscribers. The world is watching us, he says, and this is a moment to explain new opportunities, not to mourn.
Outside Voice: Another Perspective on HN Discussions
This HN received 207 points and more than two hundred comments (based on the discussions I captured; this is my estimate, not statistics: a significant proportion was debating “whether this is Tao Zhexian’s opinion”). Here are a few noteworthy perspectives:
The most sharp comment came from user qasg17: Has the AI-friendly faction finally found a new face? Otherwise, it’s just the same ten people on Tao’s blog citing each other. He also posted a page showing Calegari hosting an AI seminar at the University of Chicago, implying that the author himself is a stakeholder. This reminder is worth taking: this long article is the opinion of the individual involved and does not represent the entire community, let alone Tao Zhexian.
Another category of doubts focuses on the program itself. Some ask: If significant results require a huge amount of computing power, will “approval” slide from an achievement of merit to a capital requirement? Others analyze the statement “average 3 hours of computing power”: This is the average after casting a wide net across all areas of mathematical frontiers, not equivalent to getting headline results for any given difficult problem in three hours. The most realistic concern comes from doctoral students who have just lost the opportunity to choose their research topics, with some comments comparing the mathematics community to Uber drivers standing in the way of Waymo.
There is also convergence of views. Several commentators note that mathematicians are converging towards the same narrative: Human value is shifting upward, toward topic selection, understanding, and presentation. Humans are more important than software engineering: Coding can be handed over to AI, but understanding requirements is actually more difficult and valuable.
Conclusion
The most easily cited sentence in this long article is: For pessimists, the game is over; for optimists, the game has just changed in style. Calegari acknowledges that part of our mathematical culture will disappear, and it’s natural to feel sad about this; but if it is said that “the current mathematics community” is more important than “mathematics itself,” that is a terrible mistake.
What I value more is the focus of criticism. The negative reviews of OpenAI papers, Duminil-Copin’s lament, the analogy from 1983, and the scattered dirty copies on the floor—all are ultimately tied to one set of specific questions: How do recruitment committees evaluate candidates? What criteria do journals use for submissions? To whom are positions assigned. This is not a lamentation, but an action plan, and one that明知 will be criticized yet still includes “assumption that you used AI during evaluation” as the first point. Whether these six suggestions will be adopted, and whether recruitment and publication will truly change, are the most值得 tracking aspects of this matter. Compared to the一次性 dumping of over 700 papers, how the mathematics community rewrites its reward structure may be the deeper legacy of this incident.
The last detail that is easily overlooked: Tao Zhexuan notes in the editor’s note that this long article was “translated into another file format using AI.” The article criticizing the chaos in AI papers itself is also a product of AI formatting. This little joke is placed at the first line of the entire text, as if the authors had already self-mockery done for the readers in advance.
Reference sources: Frank Calegari “Mathematics has changed. Let’s act like it,” Tao Zhexuan’s blog What’s new guest article (2026-11-01); OpenAI “Sharing AI progress in mathematics” (2026-10-06, updated from 10-15); Hacker News discussion thread id 50328251 (207 points); Tao Zhexuan’s blog sidebar Archives and Top Posts.
Author: Yongliang