陶哲軒談 AI 時代數學研究的「Math 2.0」
陶哲轩就OpenAI数学成绩下滑一事作出回应
陶哲軒指出，部分人以 AI 自主求解開放數學問題，卻較少促成研討會、工作坊和合作，部分研究方向亦因擔心被搶先而不再公開。他警告，大規模收割未解問題的答案並不可持續，並提出「Math 2.0」應更重視成果闡釋、社羣建設和開拓研究方向。他亦主張數學界重新檢視教育、出版和職業發展標準。
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In traditional mathematics (or "Math 1.0"), a breakthrough proof of a long-standing open conjecture generates a large amount of subsequent activity and excitement in the field. The authors of the proof are invited to give many talks, and meet with other experts in the area. Workshops are set up to discuss the proof, as well as other recent developments. As a result, new connections and collaborations are formed, followup problems are shared, and other mathematicians (both junior and senior) are attracted to the field (including in some cases the authors themselves, if they initially came from a different subject). As a result of all this activity, the proofs naturally become digested, streamlined, placed in context with other results in the field, and ultimately become part of the textbooks and lecture notes for the next generation of mathematicians in the field. (1/4)
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Use of AI tools, when done responsibly, can support not just the initial generation of a solution, but all of the valuable followup activities that traditionally flow out of that solution. But at the current time, the opposite is often occurring: problems are being solved autonomously by AI prompters who have no interest in the broader field itself once their initial target is "solved", and do not understand the AI output well enough to answer questions on the result, give talks, or otherwise interact with the rest of the field. Many fewer seminars, workshops, collaborations, or other activities are being generated from these results compared to traditional breakthroughs; few people are joining the community around the field as a consequence; and promising open directions are now being withheld from the public in fear that this will cause their own research to be "scooped". (2/4)
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This process cannot be easily reversed: a problem that has been "solved" cannot be somehow reverted to become "unsolved", and even the mere knowledge that a solution exists "contaminates" efforts by both humans and AI to find alternate routes to the problem that reveal additional insights. And so solutions to open problems are now being harvested at large scale in an unsustainable fashion, leaving entire fields of mathematics much less fertile than when such problems were solved in the traditional "Math 1.0" fashion. (3/4)
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Terence Tao@tao@mathstodon.xyz
"Math 1.0" placed a premium on being the first to solve an open problem, even if the solution was not initially well understood. Now that this goal has been optimized to the point of unsustainability, "Math 2.0" will need to decenter the role of raw problem solving and value mathematical progress more holistically - for instance by elevating the role of exposition, but also that of community building and opening up new directions of study. I believe that AI can contribute positively in all of these directions as well; but it will require more imagination and ambition than the "Math 1.0" mindset of simply pointing one's favorite AI agent at some set of open problems and asking for a solution. And our community will need to explicitly re-evaluate its criteria for education, publication, and career advancement, to reflect the "Math 2.0" era. (4/4)
Last edited Oct 06, 11:14 PM
32 boosts·2 quotes·112 favorites
Tariq@rzeta0exposition and community building will challenge much of established academia
a good and much needed challenge
on a positive note there are good example of mathematicians who prioritise transparancy, dialogue, public understanding and involvement, and community building*
perhaps AI was the shock we needed?
(* which means maintaining a non-zero influx of newcomers into that community)
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fauxthree@fauxthree@mastodon.social8h
What happens when all of the "low hanging fruit" has been picked? How does someone who enjoys mathematics, but is not a prodigy, who wants to make research their career going to get started? I fear for the next generation who have not yet started their doctoral work. There won't be any "baby steps" left to take, and there will be no way (let alone incentive) for a non-prodigy to join the field. Get a major result, or no PhD for you...
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cosmo_chou@cosmo_chou@tao
Your reflections on "Math 1.0" versus "Math 2.0" resonate deeply. Treating frontier math as an open-pit mine for AI prompters—harvesting million-line formal scripts without conceptual digestion—is indeed creating "proof indigestion" and field desertification.
Yet, does this crisis also expose an intrinsic blind spot of late-stage "Math 1.0"?
Harmonic analysis and PDE theory forged the sharpest surgical scalpels human intellect has produced. Yet within closed abstractions, mathematicians often operate without physical boundaries—cutting infinitely into the continuum (\Delta x \to 0) without an ontological resection margin. This absence of boundaries traps brilliant minds in "supercritical quagmires" and artificial blowup scenarios.
The true promise of AI in "Math 2.0" is not brute-force formalization or prompt-driven scooping. Rather, AI's unique strength lies in cross-disciplinary synthesis: connecting deep mathematical tools with foundational physics, geometry, and real-world boundaries. When surgical scalpels (like Littlewood-Paley cutoffs or dyadic martingales) are guided by physical floors (e.g., discrete topological cells, lawful category homeostasis, or thermodynamic backgrounds), problems that appear supercritical in an unconstrained continuum become naturally subcritical, self-consistent, and digestible.
Thus, "Math 2.0" must not only de-center raw problem-solving in favor of exposition, but also evaluate progress by its Proof Digestibility Index (CDI) and cross-disciplinary self-consistency. When AI helps discover meaningful boundaries rather than merely hacking through isolated conjectures, math will flourish as an interconnected forest rather than an exhausted quarry.
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Jason Dyer@jdyer@mastodon.gamedev.place13h
@tao I wonder if there's a way (socially) to "uncontaminate". The Four Color Problem was considered in a real sense still "open" after the first computer proof. The first Polymath project was on an already-solved problem. I don't know how one goes changing a culture, though. Start a journal that emphasizes re-proof?
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immanuELLE@iammteah@chaos.social15h *
@tao I learned a lot from example and trying to rephrase a problem statement in my own words and seeing how I can adapt methodologically to solve problems I couldn't solve before. Over time this helped me to develop entire skill sets I didn't need to overthink too often, so I could focus on the details. This was especially helpful in critical periods of my life, where I needed to learn the hard lessons and had to rely on a working functional approach to bridge the gaps I didn't see.
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John Ericson@Ericson2314@mastodon.social16h
@tao IMO "decenter[ing] the role of raw problem solving" is a phenomenal thing silver lining overdue in many fields regardless of machine learning.
Collective human understanding (and theory building) is the ultimate purpose of intellectual pursuits. Furthermore, in the realm of applications, "nicer" theories are coupled with an optimism that the world (or in the case of math the "metaverse") is a beautiful ordered thing, and I believe this optimism can actually bend history for the better.
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John Ericson@Ericson2314@mastodon.social16h
@tao in particular I believe when "pure" research gives up and does ad-hoc methods, the students of the applied correspondents of those fields are more likely to be nihilists, and create an uglier human realm.
There are good reasons to believe that coordination problems more than actually irreconcilable desires hold back humanity. And I really think pure theory being more elegant -> applied theory being more elegant -> technological change better facilitating coordination.
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Math Student@mathalgos@mastodon.sdf.org22h
@tao A.I. will be able to fill in the “valleys” between the abrupt “peaks” (isolated discoveries) too. Maybe some mathematicians will want to spend some time considering politics.
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reazhuq@reazhuq@tao Sadly, I feel as if this vision of mathematics accords no value to the human capacity for intellectual creativity. We esteem this quality as being nothing less than a trans-civilizational achievement.
But perhaps I'm wrong. What would an exceptionally creative mathematician look like in this world?
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@reazhuq there will be none. mathematicians will not survive apart from as a creative endeavor with low funding, at best. they will commoditize problem-solving and sell it to people for a fee, and humans will never be able to compete.
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Zillion@zillion@freeradical.zone1d
@tao Exposition and comrunity building are undervalued, and a new focus on them would be welcome. Opening up new directions of study is already valued, and it will be more valued, perhaps particularly in the form of proposing new problems that AI can't solve. But I think your description omits something else, which is important to exposition, but more purely proof oriented: factoring proofs, that is, dividing them up into a structured series of lemmas and theorems of independent interest.
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Mitchell@mitchell@johnson.fyi1d
@tao that sounds like a good approach (from a layman’s perspective) have you considered its analogous to experimental physics? In many cases you know the thing exists but trying to understand it is the quest?
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Zillion@zillion@freeradical.zone1d
@mitchell@tao That reminds me of an old question—I don't know the source. Suppose we had a book that listed all physical events from the beginning to the end of time. Would physics be over?
Similarly, to take the problem AI creates to a similar extreme, suppose we had an oracle that settled the truth of every mathematical question. Would mathematics be over?
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Robert@asperamanca@mastodon.social1d
@zillion@mitchell@tao I suppose in mathematics (and any othet field) people spend many learning years working on solved problems, because they enjoy it.
I don't think a book with all the solutions would change that. The challenge would now be to deeply understand those solutions.
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@asperamanca@zillion@mitchell@tao
Part of this interest is not purely interest for knowing the maths and understanding the result in abstract though. It's also because people are interested to build the skills to do something similar in the future, within the community and system we have (had?).
Moreover, as Tao describes, the way you work on solutions with a known answer is fundamentally different than if the answer is not known. People don't simply spends years to survey approaches that are known to not lead to the right solution, even if the process itself of doing that is provably fruitful. The incentives are simply not there, on the community level, organizational level, but even mostly at the personal level.
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Benoît Régent-Kloeckner@BrKloeckner@piaille.fr19h
@fenjen@asperamanca@zillion@mitchell@tao Sorry to be the killjoy, but why don't we talk about the consequences of embracing gen AI? The Ecological, economical, political, cultural consequences are huge. I don't quite see how getting a proof of Mahler's conjecture and whatnot is worth that. Mathematics are a good thing, but not important enough to be built on a pile of human suffering.
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@BrKloeckner@asperamanca@zillion@mitchell@tao
I'm in your camp my guy! However, it looks to me that to some people these problem seem like a bunch of "feelings" to some of the more technically oriented people, so I feel it's sometimes very easy to get dismissed when talking on the topics you mention.
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Mitchell@mitchell@johnson.fyi14h
@fenjen@BrKloeckner@asperamanca@zillion@tao
To try to take the middle ground. It’s happening whether we like it or not. I wonder if the mindset will be a shift similar to that between theoretical physicists and experimental or maybe not a shift but a field of experimental mathematics (maybe that already exists)
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Benoît Régent-Kloeckner@BrKloeckner@piaille.fr14h
@mitchell@tao@zillion@fenjen@asperamanca Even if something is happening, being complicit is a choice. I can understand people having a different assessment of the harms of gen AI, but I have quite some trouble with those who agree on this point and still find "middle ground" anywhere near acceptable.
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Mitchell@mitchell@johnson.fyi13h
@BrKloeckner@tao@zillion@fenjen@asperamanca you are absolutely entitled to that opinion.
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Zillion@zillion@freeradical.zone3h
@BrKloeckner@fenjen@asperamanca@mitchell@tao I completely agree about the evils of LLMs, but the discussion of what is central to mathematics that the present situation has led to is interesting in itself.
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Benoît Régent-Kloeckner@BrKloeckner@piaille.fr53m
@zillion@fenjen@asperamanca@mitchell@tao It is interesting in theory, but this tree should not hide the forest of harm of gen AI. Once these harms are acknowledged, this interesting discussion is purely theoretical action-wise, since there is no ethical use of gen AI.
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