2026-09-08 20:32 UTC
Replies (16)
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@slowcoding@social.vivaldi.net 2026-09-08 20:51
@tao@mathstodon.xyz curious to know what makes LLM fundamentally different compared to (other) automatic theorem provers?
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@abelc@mathstodon.xyz 2026-09-08 21:10
@tao@mathstodon.xyz I would like to add that the current situation is also seriously damaging the training and development of mathematicians, particularly early-career researchers who have not yet secured tenure. Publication inflation makes it difficult for them to concentrate on their research. They feel anxious and often seriously question whether their work has any real value. In particular, the uncertainty this creates about how the mathematical community and universities will evaluate research in the future leaves them with little hope for their careers. If the current situation drives young mathematicians out of the field in large numbers, creating a generational gap, I believe this will be even more detrimental to the development of mathematics.
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@nilesh@fosstodon.org 2026-09-08 21:24
@tao@mathstodon.xyz I still am not persuaded by this. When AI companies find incomprehensible solutions to open problems, this either advances the field, or it does not. If it does, then that's a good thing and we should have more of it. If it doesn't, then no incentives for humans to advance the field are lost. Existence of a raw solution won't make a difference. Only our math artifacts and systems need to change towards what we consider truly valuable: insights, not results.
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@cosmo_chou@mathstodon.xyz 2026-09-08 22:34
@tao@mathstodon.xyz Dear Professor Tao (@tao@mathstodon.xyz), Your freshwater analogy on the non-renewable depletion of fruitful open problems—and the flattening of the difficulty landscape by brute-force AI—hits the core crisis facing mathematics. When automated systems bulldoze conjectures via continuous gradient fitting (PINNs) or million-line formal syntax trees, they solve equations while obscuring why they were hard, while the preemption chilling effect drives researchers into secrecy. In direct response to your food bank standard, we formalized the quantitative Proof Digestibility Index (D_CAP >= 0.70) in our new open paper: "Restoring the Mathematical Difficulty Landscape: Why Constructive Topological Manifolds and Proof Digestibility Outshine Brute-Force AI Mining" DOI: https://doi.org/10.5281/zenodo.22667485 Portal: https://h3qm.com/math/ We demonstrate three constructive principles: 1. Difficulty as Topological Boundary: True difficulty is a topological obstruction (e.g. 3D Euler vortex reconnection ∮∇P·dl=2πn at phonon scale; (3,4,∞) modular family on S^6; metric integer phase frustration in FLT). 2. The Cosmo Chou Machine Epsilon Identity: (2^-3)^8 = 2^-24 = ε_float32. In 8 steps, discrete integer sign-flow saturates float32 precision, reaching Exact 0 residual without floating-point accumulation. 3. Democratizing Open Science: 150 lines of pure standard Python executes full CAP audits in <80ms on $100 consumer laptops, shielding open science from compute monopolies. Would institutional adoption of explicit Proof Digestibility Standards help ensure AI nourishes rather than exhausts human mathematical intuition?
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@Ianagol@mathstodon.xyz 2026-09-08 23:32
@tao@mathstodon.xyz These companies are taking advantage of the structure of the modern mathematical enterprise. The field has become so specialized that mathematicians in one area many times cannot understand the problems or progress in other related areas. So to emphasize the importance of problems, conjectures are made, some even with prizes attached. Conjectures with many implications (even if most mathematicians cannot understand those implications) or formulated by the most famous mathematicians become more well-known (for example, the Millenium problems were curated by a small group of elite mathematicians). Long open conjectures can also achieve prestige. Solving such a conjecture then becomes prestigious because of expert opinions that the conjecture is significant, rather than for its own sake. So the AI companies are now taking advantage of this situation by pursuing the solution of these famous conjectures or problems. Compare, for example, to protein folding, which was “solved” by Deep Mind’s AlphaFold. This is a biological chemistry problem that can be appreciated by a large swath of the population (still, it requires some education to appreciate, but not a graduate degree in a specialized field of mathematics).
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@joshuagrochow@mathstodon.xyz 2026-09-09 00:19
@tao@mathstodon.xyz You don't even need the rumor of someone working on a problem (even if that is what happened w/ Navier-Stokes). People who should know better made a system to automate mining the literature for open problems and trying to solve them: https://arxiv.org/abs/2608.16977. How many potential thesis problems, excellent for training budding mathematicians, did this one paper eliminate, just to give credit for those problems to...an automated system?
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@cosmo_chou@mathstodon.xyz 2026-09-09 01:12
@tao@mathstodon.xyz Terence Tao's warning on the depletion of fruitful problems and difficulty flattening points to a deeper crisis: mistaking an exhaustive AI map for mathematical insight. Current AI formalization (e.g., Lean 4) acts like a brute cartographer. It draws an unfathomable map of millions of dead ends, but cannot answer why a path is optimal or how it serves truth. This triggers an infinite regress of syntax—methods verifying methods verifying methods—detached from physical reality until humans cannot judge if the model reflects nature. This causes the 'scorched-earth trap' Tao fears: brute-force search carpets problems with uninterpretable debris, misleading researchers into assuming paths are closed and killing organic curiosity. Verifying AI noise takes humans 100x more time than machines take to generate it, causing intellectual paralysis. Humanity does not need infinite labyrinths; it needs a topological compass: a shortest-path algorithm revealing why difficulty exists. Our Constructive CAP (H3QM) anchors mathematics in physical ontology (vacuum elasticity, vortex reconnection ∮∇P·dl=2πn, phonon radiation □²Ω=-κT_topo). Governed by octant contraction κ=2^-3, it saturates float32 precision via Cosmo Chou's identity (2^-3)^8=2^-24=ε_float32 in 8 steps, achieving Exact 0 on integer spaces without floating-point accumulation. Transferred across Hopf S^6, 3D Euler, and FLT, 150 lines of pure Python audits in <80ms, seeding the commons by proving physical algorithms can illuminate nature without compute monopolies. Preprint: https://doi.org/10.5281/zenodo.22667485 Portal: https://h3qm.com/math/
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@practal@mathstodon.xyz 2026-09-09 01:19
@tao@mathstodon.xyz Pure mathematics is dead. Long live mathematics. I think all of interesting mathematics is applied mathematics in the end. Powerful AI means that the level at which we can do applied mathematics will be so much higher, though, and many more people will be able to be "mathematicians". The importance of pure mathematics is often argued for by citing examples of important applications that used pure mathematics invented a long time before the application became apparent. We can reverse this argument: by properly developing the mathematics our applications need, we surely will obtain all of interesting pure mathematics.
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@srinivas@mathstodon.xyz 2026-09-09 03:00
@tao@mathstodon.xyz Evidently, AI's solution of valuable open problems in mathematics is not for mathematicians; the eventual audience for this effort is those (companies) who are willing and able to pay to "employ" such AI in their "workforce". A dynamic that causes academic inflation - people getting more advanced degrees to give themselves a competitive edge in being hired to work for a company, rather than the desire for more advanced work in the field - seems identical to the one incentivising the present AI dynamic. One measure to alleviate this is a value-proposition favouring the eco-system of mathematics. The effort to reduce greenhouse gas emissions, for instance, needed the value-propositions of the information economy and geopolitics to incent against using dead dinosaurs; not just accounts of harmed natural ecosystems. A temporary value-proposition candidate seems to arise from AI needing to learn from ongoing mathematical work even to flatten the mathematical landscape: if a consortium of mathematicians can be funded to develop a culture around efficient use of AI to solve research problems for the industry, it offers an alternative to companies that will otherwise only have the leading AI providers as options in the limit. It makes the argument that AI doesn't need to risk turning the eternally renewable resource of (human) mathematical creativity into a limited one by zero-sum behaviour; that it can instead sustainably accelerate such creativity in a way that is also more profitable. This "appeal to application" may make the likes of G.H. Hardy (and you?) wince, but perhaps less so than the present situation.
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@iWonderAboutTuatara@mathstodon.xyz 2026-09-09 07:52
@tao@mathstodon.xyz while i certainly agree with the correctness of what you are saying, I do wish you would be more concrete in these recommendations. How do we change our incentive structure to ensure these processes happen? While the idealistic end of determining what is and isn't desirable is needed, the practical end.of this also seems to be a hard and pressing problem.
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@Greyone@mathstodon.xyz 2026-09-09 17:44
@tao@mathstodon.xyz I wonder whether the scarce resource is not only promising problems, but representations in which "promising" becomes visible. What if human AI symbiosis can create new observables from which a different geometry emerges?
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@ninalias@mathstodon.xyz 2026-09-10 11:40
@tao@mathstodon.xyz Given the problems you highlight, I have many questions. What is the future you see with the SAIR foundation? The AI companies have lots of capital. What can less resourced communities do? Most importantly, what can be done to wrest understanding and insight from a volume of presumably "correct" AI output? A world-class mathematician's approach to this problem would inform a great many knowledge workers. I am just a math enthusiast, but a similar thing is happening (has happened) in software. Many people say that there is no need to understand the code anymore. But if you are the person liable for the quality of the software, how we proceed is still daunting.
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@openspur@mathstodon.xyz 2026-09-10 14:26
@tao@mathstodon.xyz how can we discriminate between what is worthy of an automated proof and what needs human playfulness and understanding? i think the human decides. you can't show some kid toys, sit down and expect playtime to begin.
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@robbiemu@mastodon.social 2026-09-09 01:20
@tao@mathstodon.xyz I think that the downstream effect that is similar to this might be that 10yrs from now when publishes a proof, people say “oh, cool. So what?” Ie, be it human or ai, proofs are polite, but impact is what matters. Manuals/Textbooks and systems/approaches/problems might be where the action belongs.
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@iamwil@mastodon.social 2026-09-09 03:54
@tao@mathstodon.xyz What if people got credit for both posing the problem, and also finding the solution? So that every problem always has a problem definer and solution finder pair that gets credit?
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@siddharthsabron@mastodon.social 2026-09-10 19:21
@tao@mathstodon.xyz