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Tao: Open math problems being non-renewably mined by AI

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Re: Tao: Open math problems being non-renewably mined by AI

#211

Post by bityard »

rzerowan wrote: post #201
dvt wrote: post #52

I'm with @nilesh on this one, and not exactly sure how merely the existence of a solution precludes the advancement of human knowledge. If a problem is "solved" (say, symbolically verified) without any insights gained, it doesn't seem very interesting to the profession. Navier-Stokes is a bit different (because there's a prize attached, so "scooping" matters), but almost all interesting problems don't have any prizes…

More of a 'its the journey' rather than the destination type of thing.Since the insights , quirks, tricks and procedures gained along the way allows insights intoother at that moment unknown problem/domains in the future. As far as researchers sharing their data/notes with the AI hyperscalars looks like that would be coming to an end wihth a mor guild-like structure going forward to prevent their progress being fron-…

I wonder if it would be possible for researchers and scientists to submit their papers to an organization which would then collect them, submit them for peer review by other experts in the field, and then release them in periodical form ONLY to individuals and organizations who pay a subscription fee in order to read them while suing those who try to redistribute them without permission?

Re: Tao: Open math problems being non-renewably mined by AI

#212

Post by mizzao »

colinhb wrote: post #50

Seems like in current cultural and economic context, short term extraction is what we’re going to do > In short, the indiscriminate use of powerful solution-extraction tools can achieve the immediate short-term goal of solving problems at hand, but at the cost of sustaining the ecosystem for the next wave of progress, or in understanding the progress already obtained.

Is there an analogy here to the phenomenon that senior {engineers, designers, PMs} are now able to be insanely productive with AI, but it's also very hard to train junior folks to develop the sense of judgment that senior folks have?

Re: Tao: Open math problems being non-renewably mined by AI

#213

Post by meken »

I don’t see why it makes a meaningful difference if a human solves a math problem versus AI - it seems like the same amount of understanding will come out in the end. Either the understanding will come from humans arriving at the proof in the former case, or the understanding will come from humans understanding the proof that the AI came up with in the latter.

Re: Tao: Open math problems being non-renewably mined by AI

#214

Post by magicalist »

modemNoises wrote: post #150
magicalist wrote: post #130

Earlier quoted context omitted.

Surprised to see someone on HN arguing against open science. Seems like the opposite of the lessons we should learn from Newton and Gauss, actually, hoarding results for decades at the expense of progress. (the Pythagorean thing isn't really competition either, is ahistorical, and from what we actually do know it's again people hoarding results instead of sharing them). FWIW, your post comes off as a middlebrow dismi…

Nothing in their comment reads to me as "arguing against" Reads like nothing but historical context

> Nothing in their comment reads to me as "arguing against"

If competition is somehow the opposite of "centuries of traditions of open science", and "mathematics has always been highly competitive", then open science is neither sufficient or necessary for the future of mathematics. Their clear implication is that we don't need to worry about it, though, because it's always been that way.

> Reads like nothing but historical context

They literally accuse Tao of "a misstatement of mathematical history".

Re: Tao: Open math problems being non-renewably mined by AI

#215

Post by twotwotwo »

kccqzy wrote: post #191
twotwotwo wrote: post #167

This is worsened by OAI/Ant's strategy of grabbing the glory and running rather than spending effort trying to to advance understanding of math. Tao, who is quite sophisticated in use of AI, has said a lot about this, including in meme form: https://mathstodon.xyz/@tao/117068266026618494 The AI labs' approach to math is immature in a way they can't get away with in coding. In coding, they realize that a pile of code…

Besides Sendov’s Conjecture, Terence Tao has also shared his AI-assisted digestion of the counterexample to the Jacobian conjecture. In simple words, digestion just means full human understanding of the results. It could come from understanding the result from a different perspective, or perturbing the result and seeing what breaks.

The Jacobian one is great, and fits here -- 1) the counterexample came out as a tweet of a single expression which seems exceptionally hard to turn into something sensible, which crystallizes the lab's approach; 2) it links (like the Sendov post) to a chat transcript showing a little of what was involved in untangling it (though of course all that went into asking the right questions is invisible!).

For other folks, the post: https://terrytao.wordpress.com/2026/07/21/a-digestion-of-the...

The transcript: https://chatgpt.com/share/6a5fdc7a-d6f8-83e8-bbea-8deb42cfed...

Re: Tao: Open math problems being non-renewably mined by AI

#216

Post by unified101 »

gowld wrote: post #114
monktastic1 wrote: post #93

Earlier quoted context omitted.

But this oracle doesn't just say true / false. It also gives a proof. That makes it much less exciting (not to mention beneficial for your career) to find another one (or even worse, the same one).

The "proof" is merely an appeal (unreadable program) submitted to a different oracle (Lean).

What do u think lean is? That's like saying a program that works, is inscrutable because it appeals to the oracle of "code test cases" to prove itself correct.

You're either being intentionally obtuse, or unintentionally ignorant.

Re: Tao: Open math problems being non-renewably mined by AI

#217

Post by morpheos137 »

gpm wrote: post #157
johnsmith1840 wrote: post #155

Earlier quoted context omitted.

So solving and discovering math is or is not the core value a mathematician provides? If AI can perfectly replicate their work but faster and better then what? SWE have nobody crying for them as they've been massively disrupted.

Mathematicians provide two complementary services bundled together. 1. Proving theorems - what AI can apparently replicate faster and better. 2. Creating definitions and new theorems from those definitions to prove, selecting which of the possible statements to work on. I.e. developing the "language" of mathematics. So far there is no evidence that LLM can do this at all well. And there's some reason to think that ma…

We don't understand why? Reality is mathematical. As evidenced by lawful physics.

Re: Tao: Open math problems being non-renewably mined by AI

#218

Post by pixl97 »

wrsh07 wrote: post #171
jfengel wrote: post #75

I didn't realize that open math problems were a finite resource. I recall a story about some famous mathematician (Gauss?) dismissing interest in Fermat's Last Theorem claiming that he could crank out problems of equivalent interest. Clearly Tao knows a hell of a lot more than I do about this, but I'm surprised that math that close to completion.

I'm surprised nobody has stated the obvious: a hard math problem that has been open for ten years (because many serious people have given it serious thought and been unable to make significant progress) is, in fact, nonrenewable. The only way to renew it is to make a new problem that is so hard systems and humans will be unable to solve it for the next ten years. And, in the spirit of trees, the best time to plant a…

Eh, if AI quickly solves most of our mathematics problems that are solvable then it might be time for us to hang up our hat as our little monkey brains aren't very good at this stuff.

Now, I think AI will solve some, but we'll find out that some are just either unsolvable or wildly huge that nothing is solving them any time soon.

And a whole lot of these problems have been around quite some time, when even knowing how to do advanced math meant you were a landed gentry or someone of high wealth. If those problems fall, they fall. They aren't pets we keep around forever. And new problems will crop up over time for both AI and men to scratch their brains over.

Re: Tao: Open math problems being non-renewably mined by AI

#219

Post by mrbungie »

meken wrote: post #213

I don’t see why it makes a meaningful difference if a human solves a math problem versus AI - it seems like the same amount of understanding will come out in the end. Either the understanding will come from humans arriving at the proof in the former case, or the understanding will come from humans understanding the proof that the AI came up with in the latter.

Probably an AI-written Lean proof is very different to how a human would write it, and some may say it's more like mathy neuralese. For sure it works but it is not human-friendly and needs to be transformed into something more readable and digestible to be able to extract insights from it.

Not that different from when trying to read an out-of-control vibe coded codebases, or an sloppy AI long email that someone may send you at 9 AM.

Re: Tao: Open math problems being non-renewably mined by AI

#220

Post by vikramkr »

evenhash wrote: post #197
dbmikus wrote: post #158

Earlier quoted context omitted.

An AI-generated solution always provides two pieces of info: 1. proof that there is a solution 2. a solution that you can work backwards from to build understanding Maybe the solution is pretty inscrutable, but it's almost always better than nothing. So, both of these pieces of info would be at least marginally useful for advancing human knowledge.

> An AI-generated solution always provides ... proof that there is a solution This is only true in the most trivial sense. A solution is a solution, sure... but how do you know it's a solution, and not an incoherent jumble of words? A human has to review and vouch for it. Just because the AI gives you an arxiv-worthy PDF, or a Lean proof which compiles, doesn't mean it proves what the AI says it does. The AI could gi…

Not an expert by any means but the assumption here as I understand it is that the arxiv worthy PDF would not be acceptable or meaningful for impossible to understand proofs. And the lean proof would be meaningless unless the specific expression being proven is human understandable as the direct translation of the question the human is asking in formal form. So proving the negation is not a thing but if you make a subtle mistake in translating the statement you want to prove then obviously the QI is going to be proving the wrong thing. And otherwise you're relying on the correctness of lean as a system and on identifying/preventing if the proof is adversarially exploiting bugs in lean to falsely prove things.
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