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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

#51

Post by kzz102 »

vessenes wrote: post #3

That’s not untrue. But it’s also a misstatement of mathematical history. Many leading mathematicians historically have been highly competitive — Gauss comes to mind. Woe betide the lesser intellect that sent Gauss some ideas. The Newton Leibniz controversy was very serious business at the time in the UK and the continent. It was considered at the least a sin to reveal that sqrt(2) was irrational to those outside Pyth…

The mathematical community was very competitive in its early years, but in the last 70 to 100 years, it has been generally less competitive and very collegial. The community was in a good place, and progress has been very good. In a few cases when competitiveness was ramped up, it lead to bad behaviour and destructive fights. Few would like to return to those competitive years.

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

#52

Post by dvt »

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 attached.

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

#53

Post by lalalanananana »

olalonde wrote: post #49

Can't mathematicians still gain novel insights by reverse-engineering AI-generated proofs? Just like chess players learn new concepts by studying what engines play.

It's possible but the approaches these tools take are usually verbose and strange. Think about it like anything else llms do. Even when the picture is right and there are only 5 digits on each hand all the textures are off and so is the lighting and postures. Or in code, the code is always way larger then it needs to be and tightened up strangely with weird loose ends. Or in writing weird idioms, words, structure, and a weaselly way to turn 3 sentences into 8 paragraphs.

People usually use these tools in math and science to find an answer. Then often they will work it back using more sane or human pathways. So it's shareable or even beautiful.

Knowing the answer has value. But, often in math the best thing was how someone got there.

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

#54

Post by dev_dan_2 »

contravariant wrote: post #39
dev_dan_2 wrote: post #12

Earlier quoted context omitted.

It is a strong argument in this case though, because Terence Taos expertise is directly linked to his ability to not misstate the history of mathematics. Also note how the quote by Tao is in all likelyhood not meant as an absolute; rather than a statement of a trend - a handfull of counterexamples do I no way change anything about the truth value of Tao's quote. On the other heand; consider how absurd it would be if…

I mean mathematics has enough history that something can both have been false for centuries of mathematics research and true for centuries. Sometimes in different places simultaneously. Terrence Tao can do his job perfectly well without being aware of any mathematical history, though I consider it unlikely that he is. I'm not seeing the direct link you're talking about, in fact history is frequently left out of mathe…

> in fact history is frequently left out of mathematical teaching even when the history would in fact help in the understanding of some concepts.

That is well-known I assumed and continue to assume.

> I'm not seeing the direct link you're talking about

You are stating that link yourself; indirectly: "though I consider it unlikely that he is [being unaware of any mathematical history]". Why is it unlikely, precisely?

- Maybe because it is unlikely that he recieved the mathematical teaching that frequently does not contain history of mathematics (wild! I wonder which university you have in mind in particular) that you seem to be refering to?

- Maybe because his writing is evidence that he is interested about, incorporates and refers to history of mathematics, refer for example to https://terrytao.wordpress.com/2008/01/04/pcm-article-genera... or https://terrytao.wordpress.com/career-advice/theres-more-to-...

- Or maybe because he is quite the opposite of a person that never ventures outside of their own area; being blind for other fields, or ones own history; as evidence by being famously collaborative across different fields, having a popular blog where he writes about non-mathematical topics too and last; him being one of the main proponents of foundational topics such as formalization of mathematics; or the use of LLMs for mathematical research.

Does all that really make it more likely to you that Tao is not aware of the existence of counterexamples like those the commenter above mentioned - more likely than the commenter simply having missed a nuance or taking something out of context?

If so; I would be genuinely curious why - people work differently, and I am always happy to learn, or close gaps in my own understanding.

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

#55

Post by _alternator_ »

olalonde wrote: post #49

Can't mathematicians still gain novel insights by reverse-engineering AI-generated proofs? Just like chess players learn new concepts by studying what engines play.

Yes, and they will. But what's happening here is that the system that cultivates mathematics (and mathematicians) is recieving likely the biggest shock of its history. How do you reward merit and identify talen when people can't absorb the number of proofs being generated, much less understand them? Perleman's proof of the Poincare conjecture took several years for the mathematical community to digest; the proof of Navier-Stokes will probably take a similarly long time. In the mean time, it looks like all open problems will be solved (or proved that they can't be solved).

It's not that the horizon is expanding because of this. It's more like a forest getting clear-cut.

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

#56

Post by ReflectedImage »

olalonde wrote: post #49

Can't mathematicians still gain novel insights by reverse-engineering AI-generated proofs? Just like chess players learn new concepts by studying what engines play.

Well no because it works by joining together existing novel insights.

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

#57

Post by CamperBob2 »

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.

The "ecosystem" is dead. Tao should be thinking about what will replace it. I don't understand why he's taking this tack.

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

#59

Post by johnsmith1840 »

calf wrote: post #35
johnsmith1840 wrote: post #28

I mean, hasn't it always been this way? If something valuable is within reach and you disclose it, someone else might reach for it? Just because the length of the arm is longer with an AI org doesn't mean it's somehow fundamentally a different system. The future is that if you don't use AI your work is a lot easer to reach against someone else who has it. That dude hand writing code with punchcards can be lapped by a…

The difference is no one writes punchcards so the comparison is inapt. It is more like supposing AI eats everything humans attempt to do. They will take your work as you are working on it, any interaction at all. It is meta-plagiarism, on another level.

AI didn't "take" anything. An openai researcher did?

The solution is also different to theirs? Literally no evidence of plagarism?

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

#60

Post by BeetleB »

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…

Mathematicians will be less likely to work on a problem if there is a solution - even an incomprehensible one.
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