Calegari: AI's math breakthroughs demand new norms, not denial
Frank Calegari argues mathematicians must adapt to AI's capabilities while fixing its poor standards of rigor and attribution.
What to know
- OpenAI's October 6 release of 700+ AI-generated math papers represents a genuine capability milestone but triggered mathematician unease over presentation quality, attribution standards, and correctness.
- Calegari argues the real problem is not AI itself but AI companies' disregard for mathematical practice norms—and that mathematicians have more control over their own hiring, publishing, and education than over trillion-dollar tech firms.
- The psychological impact mirrors the 1983 classification of finite simple groups: seeing all major open problems claimed solved at once creates false closure, even though real mathematics lies beyond the solved problems.
- The debate hinges on whether AI is a tool mathematics must adapt to or a threat to the discipline's collaborative and exploratory nature.
The dispute Whether Calegari's periodic table analogy holds up: optimists see it as reassurance that solved problems create new frontiers; skeptics argue it dismisses AI's capacity to solve not just specific problems but entire domains of mathematics at once. · positions read across 85 posts and comments
AI represents an inevitable shift from mathematical craft to technology; mathematicians must adapt their role to curation and problem selection rather than resist.
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“It's amazing for me to see math research going from craft to technology. CS had this moment decades ago, sans some theoretical parts that are joining the bandwagon now.”
fooker · Hacker News ↗
Calegari's reassurances underestimate AI's scope; AI will solve problems across many fields at once, not leave a manageable frontier of new questions.
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“This analogy is pants on head stupid. They're talking as though the only thing AI will ever solve is the Langlands program, not as though we have discovered a generic tool.”
0c3ca83 · Hacker News ↗
Questions remain about whether AI can generate genuinely novel mathematical concepts or only combine and extend known frameworks.
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“What is the current thought on AI creating new emergent forms of mathematics? Even the most stunning results like the proof of the quasi-RH are a clever combination of known mathematics. Humans have created (discovered?) stunning new ideas…”
waynecochran · Hacker News ↗
Frank Calegari Mathematician, guest authorTerence Tao Mathematician, blog hostHugo Duminil-Copin MathematicianOpenAI AI research company
How it unfolded 1 development · click the chart to see its coverage articlespostscomments
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Calegari reframes solved problems as doorways, not endings
Calegari argues that solving a major open problem is not the end of a field but the beginning of exploration into new territory. He uses the periodic table's relationship to chemistry—its construction did not solve chemistry but opened new directions. He calls for mathematicians to better communicate this reality to counteract the feeling that AI has "solved" their fields.
“But even if it had solved every problem in the Langlands program, to imagine that would have 'solved number theory' would be as naive as saying that the construction of the periodic table solved chemistry.”
— Frank Calegari -
The quote reads to me like “as long as you can analyze the solution after the machine solves it and make it suggest new ideas based on that, you’re almost the same as a discoverer yourself.” I’m deliberately avoiding “mathematician” here because I think Tao probably didn’t intend to take it that far. But, for many people that difference will be…
2 more of the top 3 · 85 posts in this stretch
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There's a long collection of comments by mathematicians at [1]. A large number of workers in the field are rather discouraged. The main quality complaint is that the proofs are ugly, long, and incomprehensible. One mathematician comments that the first proofs of new major results are often like that. Elegance comes later. The main social complaint…
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That’s a great paper, I was tripping on these same thoughts last night. The thing that rang true was kind of getting back to these roots, the Philo-Sophia as love of knowledge, and a community that loves knowledge. It doesn’t matter if one guy has a 70 IQ and loves fables, or another is 170 and loves something I can’t comprehend, they are one…
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background
Calegari compares AI math release to finite groups classification precedent — Calegari draws a parallel to Dan Gorenstein's 1972 announcement of a program to classify finite simple groups, completed in 1983, which he argues cooled interest in the field despite leaving many open questions. He warns mathematicians that seeing all their major open problems claimed solved at once creates a psychological sense that the field has been "hollowed out," but this parallels how group theorists felt when the classification appeared to close their subject.
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background
Calegari argues mathematicians must adapt to AI, not resist it — In a guest post on Terence Tao's blog, Frank Calegari acknowledges AI has made real mathematical contributions but identifies three core problems: the papers are poorly written, violate attribution norms, and some contain errors. He argues the mathematics community should redirect energy toward maintaining its own standards rather than opposing AI companies.
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background
OpenAI releases over 700 AI-generated research papers — OpenAI publicly released a large collection of papers produced by AI systems, marking what Calegari describes as a watershed moment in mathematics.
Also covered reported alongside — the timeline has no entry for these yet
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first by What's new, 6d ago · also HN Best, HN Frontpage
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2 outlets Egoless Mathematics in the AI Era
first by HN Frontpage, 5d ago · also What's new
What people are saying 21 voices from 1 site · best of 85 · verbatim
- Can AI generate genuinely new mathematical concepts and frameworks, or only prove and extend within existing ones?
- Will AI's speed at solving problems across many domains simultaneously create a different kind of field-cooling effect than the finite groups classification did?
- What institutional changes does mathematics need to maintain rigor and attribution standards as AI-generated papers become routine?
- Nov 3
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Yes, we should apply this same reasoning to AI and the companies that produce it, since they claim its the future. All that matters is truth, no valuations, no profits, not IPOs. All that matters is uncovering the truth and having the best models -- so the AI companies should just release everything for free, and not hope for notoriety as a…
- Nov 2
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Sure, mathematics will survive and even prosper with AI, but what about the mathematiciansIt's not a given that mathematics will survive without mathematicians. It's not even likely. The proofs OpenAI threw over the wall are unpublishable garbage, top to bottom. A big reason mathematicians are so upset is that they feel like they've been drafted…
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I know it's against the rules to be a curmudgeon, but this just reads like a lullaby for people preparing to drink the SF kool-aid. The opportunity cost of becoming an expert will remain, but the reward of pushing the frontier is mostly gone. Now, if you want to save universities as institutions(*), you'll have to offer people some other positive…
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The lens to use is the Venn diagram of: 1. What you like doing 2. What you are good at doing 3. What people will pay you to do For math, if your goal is to prove something that no one else has proven, then all three are under attack.I think the "egoless" path is taking a 50,000-foot view of mathematics that ignores the above lens. Sure…
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> Human pursuits cannot be separated from the humans who pursuit it.But it's pretty clear you can chuck "human pursuits" in the garbage, in favor of what really matters: intensification of the market system and wealth accumulation for its own sake.Your technology leader thinks: Humanity? People? Who the fuck cares about that BS? Fuck 'em (or "I'll…
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Watching Star Trek TNG as a kid, I vividly recall hearing Geordi or Data often say things like "run a level 3 diagnostic on the Worf Barrels" or whatever to the computer and wonder what the heck that would even be. I wondered where are all ship devops and programmers were hidden. Though I recall the one accurate part was Barclay in the Holodeck…
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This is one of the best posts I've seen on this topic in a while.I finished a math PhD before I decided to get out of academia and into startups. Since then, the older I get the more of a Platonist I become, especially now that I've taught my daughters a bunch of math (I'm their homeschool math+science teacher). And the more of a Platonist I…
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I think that math can still be a creative endeavor, ultimately. In many cases is not so creative, and really is an exercise in searching the potentially solution space. However, math can also be about the creation (discovery?) of new language itself, which in turn enables us to ask questions that we couldn't previously ask in the first place.As a…
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Mathematicians both build/formulate abstract conceptual frameworks and formulate problems within those frameworks. Solving the problems has historically been one of advanced the conceptual frameworks.Now, what has been useful for advancing physics and biology has been the abstract conceptual frameworks - using Hilbert Space for model quantum…
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> Mathematics isn't a way to keep a specific kind of intellectual happy by providing fun, meaningless puzzles to solve in isolation.Yes it is. At least in part.> It's the bedrock of all other science.Yes it is. At least in part.> The goal is to find solutions and to advance the fieldThat is only one of the goals - but it is not the goal.> we seem…
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My perception of mathematics when I was young was that it is a field of the absolute. It doesn't matter what anybody thinks, it matters what's true.To be fair math does have a great deal of truth, but it also has, surprising to me, a great deal of social cohesion. Some things are sufficiently true because the important people are pretty sure it's…
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Math envisions and communicates logic-based structures and objects that, as a side effect, may be relevant to certain real world problems decades to centuries after the initial research.I'd argue your statement is treating this discovery process as an engineering exercise: there exists an unexplored bounded space of logical axioms within which we…
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The field before AI was build up so that humans can learn and understand and teach other humans how to advance these fields.AI is here 'suddenly'.Let them discuss this out, let them go through the emotions.And its unclear today IF we still need anyone doing anything in 5 years or not. In theory we could also just freeze 90% of things we do and put…
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As i understand it. The difference is the journey.By short circuiting the exploratory process via AI to find a solution. Some of the questions that would normally be asked during the struggle of said process will also be circumvented.Questions that would lead to other discoveries, ideas, and perhaps entire fields of study.At the end of the day AI…
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I guess the issue is -To advance math you need people who could be really high earners in any other field and stuck to math mainly because they find it intrinsically motivating.You change the calculus for motivation you potentially devastate a generation of talent who just decide to go to wallstreet or something since it seems there is no longer…
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> It's the bedrock of all other science. The goal is to find solutions and to advance the field…That may be true but that may not be the feature that attracts people to the field.> As we solve lower level problems, we can unlock real world math solutions.That is an argument often made for funding mathematics research. But again it may not be the…
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I really struggle to understand this discussion. Mathematics isn't a way to keep a specific kind of intellectual happy by providing fun, meaningless puzzles to solve in isolation. It's the bedrock of all other science. The goal is to find solutions and to advance the field - but we seem to be treating it like some sort of occupational therapy for…
- Nov 1
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> understands it thoroughly, and uses it to develop interesting further questions or directionsHe phrases it like these criteria are requirements, but they really aren't for every problem. They make sense for those theoretical problems whose solutions' main values lie in their understandability and further development, but not every problem is…
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Given the capabilities of OpenAI's unreleased model, I think the model helps mathematicians more than it hurts them. Some of the open problems that were solved were "It would be nice to know that", but many of the others represented ways in which we had been stuck for many years. There had been very little progress on analytic number theory…
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Why do nearly all the articles exploring the development of AI applied to math assume that capabilities won’t extend beyond theorem proving?> other skills will matter more: choosing which terrain is worth exploring, steering fleets of airplanes, and explaining what they find to the rest of the world.These all seem totally achievable by LLM agents…
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> To me, someone who chooses a worthwhile problem, obtains a proof completely using AI, understands it thoroughly, and uses it to develop interesting further questions or directions has made a substantial mathematical contribution. I would give that person substantial credit even if they supplied no new ingredient in the original proof.Yup. This…