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A misalignment of AI in mathematics

mathandai.org

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Re: A misalignment of AI in mathematics

#951
I know it's incredibly presumptuous for me, a nobody, to say this to 25 Fields Medalists, but:

Perhaps you are misaligned.

Who decided the goal of math must be human insight?

First off, some mathematical truths might simply be far beyond our biological comprehension.

Second, for us non-mathematicians, the value of math isn't in understanding exactly why a result is true, it’s in how those results can be applied to actually improve our lives.

Isn't this why we have math in the first place? To solve our real problems? Over time it morphed into this pursuit of pure theoretical insight, probably out of necessity at the time, but is it still necessary?

Re: A misalignment of AI in mathematics

#953

Im surprised about the sentiment in this discussion. I totally see the problem Terence is describing. We are loosing a lot in understanding and focus if it continues like that. The solution found for Navier Stokes doesn’t have much „real value“ - but what almost always happened in the past when people worked on the difficult problems, these sparked new ideas / new theorems that broadened our knowledge. Think back at…

His complaint is a very weak argument. If the complaint is about AI just giving answers and not good understanding around them, or frameworks that lead to more solutions from them, then that means there is a gap that can be filled by human mathematicians. So all the AI has done has actually made mathematicians lives easier and provided them with an opportunity to fill that gap. They should be using it to do that, ins…

You have a point, but I think a concern is that the environment in which to fill this gap is threatened since the unsolved problems have functioned as a guiding lighthouse and measure of progress. Frankly: will someone pay for mathematical research to understand things we already have proofs for?

Re: A misalignment of AI in mathematics

#954

As a consumer of mathematics, I want to attack the author’s idea of “what that work was meant to achieve in the first place”. I do a lot of programming that uses mathematics. I designed the error recovery file format “par2”. I worked as a quantitative trader. What do I want from math? To be able to search math results, understand them, and apply them. But academic mathematicians are so inwardly focused - as these aut…

different fields have different communication cultures, that’s not inherently bad. sacrificing inward focus is expensive from the pov of the insider; i see your points but not everything can be written for a general/wide audience. after all, the audience of the author are the specialists, since only specialists spent the effort to be able to engage with the work in depth. perhaps we disagree that mathematics is not a product that mathematicians are making?

what the work was meant to achieve in the first place is what the work was achieving in the first place. it was developing mathematics. if someone wants to use it, they are free to do so, or to write a blog article how it can be used by non experts. mathematicians do that themselves sometimes, but it’s not their job. myb there should be more incentives for them to do more of such work

i keep seeing this myth that mathematics has some purpose to do this or that, but somehow its purpose is least known to mathematicians, everyone else seems to know better

Re: A misalignment of AI in mathematics

#955
post #951

I know it's incredibly presumptuous for me, a nobody, to say this to 25 Fields Medalists, but: Perhaps you are misaligned. Who decided the goal of math must be human insight? First off, some mathematical truths might simply be far beyond our biological comprehension. Second, for us non-mathematicians, the value of math isn't in understanding exactly why a result is true, it’s in how those results can be applied to ac…

> Second, for us non-mathematicians, the value of math isn't in understanding exactly why a result is true, it’s in how those results can be applied to actually improve our lives.

This is just the problem with the Navier-Stokes solution. We now (supposedly) know the truthiness of the statement, but we have no more understanding of why or how it applies to our quality of life. This is exactly the issue Tao is pointing at.

Re: A misalignment of AI in mathematics

#956

Earlier quoted context omitted.

It’s frustrating that this comment is at the top because it, along with lots of the replies it inspired, absolutely misrepresents the actual declaration. The declaration is not making any statements about not using any AI in mathematics. The entire point is to push the use of the technology in a direction which is compatible with positive pre-existing features of the math community, and to make it better known what s…

The declaration literally opens by writing that AI companies (or really anybody) saying, “Hey, let’s see if this powerful reasoning engine can solve an open problem in mathematics” is “detrimental” to the “science” of mathematics. Full stop. So I don’t think it’s being misrepresented. If it’s a science, why isn’t progress good in itself? You can’t have it both ways. The online comments I’ve read that side with the le…

> The declaration literally opens by writing that AI companies (or really anybody) saying, “Hey, let’s see if this powerful reasoning engine can solve an open problem in mathematics” is “detrimental” to the “science” of mathematics. Full stop.

So let's just quickly agree that the actual quote is “However, the push by AI companies to solve mathematical problems as a benchmark is detrimental to the science of mathematics, and to the mathematical community.” And that in this, "as a benchmark" is load-bearing.

We wouldn't see nearly the same amount of contempt from researchers had OpenAI picked a research-friendly approach.

What they did: Hear a rumour about the problem being solved by other researchers, then rush to scoop them (unethical), then, when they actually go talk to them, they try to oust an author (also unethical), and when they finally decide to share their own work, do so in the least useful way possible.

What they could have done: Upon hearing the rumours, connect with the researcher in question and propose that they join efforts instead; set up a joint project to test if the machines are useful in any way, and if that's not appreciated, back down again. And instead of dumping only an undigested paper* and a Lean proof, do the digestion prior to publishing anything (as Buckmaster was in the process of doing). If their own lack of competences was keeping them from digesting it, then again, reach out to the researchers to understand if anyone would be willing to do so.

In the second of those two worlds, we wouldn't be seeing nearly the amount of outrage that we are seeing right now.

*: Here, “digestion” is the process of turning an AI slop paper into something humans can read. LLMs can indeed sometimes (if much more rarely than marketing material from the large LLM companies will suggest) produce correct proofs, but they are often written in bizarre ways – they'll use lingo that doesn't exist, seem overly pretentious, dwell on extremely easy steps while glossing over the hard ones. Currently, a real researcher will take that output and transform it into something that others can understand, use, and build upon. This is not so different from what happens when using it to write software, although as someone who does both, I will say that the amount of digestion needed for proofs tends to be orders of magnitudes larger than for code. This meme is quite accurate: https://mathstodon.xyz/@tao/117068266071803252

Re: A misalignment of AI in mathematics

#958

Earlier quoted context omitted.

It’s frustrating that this comment is at the top because it, along with lots of the replies it inspired, absolutely misrepresents the actual declaration. The declaration is not making any statements about not using any AI in mathematics. The entire point is to push the use of the technology in a direction which is compatible with positive pre-existing features of the math community, and to make it better known what s…

The declaration literally opens by writing that AI companies (or really anybody) saying, “Hey, let’s see if this powerful reasoning engine can solve an open problem in mathematics” is “detrimental” to the “science” of mathematics. Full stop. So I don’t think it’s being misrepresented. If it’s a science, why isn’t progress good in itself? You can’t have it both ways. The online comments I’ve read that side with the le…

Exactly. They can continue doing the sport version, just like sprinting is still there at the Olympics and people admire Usain Bolt or whoever is more recent. But we use cars and trains and planes when we want to get somewhere and don't confuse Bolt's run with transportation. So you can do this kind of recreational intellectually satisfying human math in a similar way, while productive freight-train-like math will be done with AI.

Re: A misalignment of AI in mathematics

#959
post #739

Earlier quoted context omitted.

I really hate this overly condescending takes. First of all, what do you know about the internals of math research that allows you to speak with so much confidence. Second, you're not even addressing the issues raised by the letter! This is not about "oh they made a bunch of problems easier". There are huge economical interest behind: who owns and has access to models? are these companies interested in developing res…

>> "are these companies interested in developing research" judging from the money, resources spent and the value they derive from this the answer is very definitively yes. What makes you think these companies (and I'm not a fan of all their motives) are not interested in developing research? The motives may be self-serving, but it is undoubtedly and objectively accelerating research.

> but it is undoubtedly and objectively accelerating research.

Part of the point of the letter is that it is quite possible to act in a way that is a net negative to research. The most obvious case is when the companies violate ethical standards in research.

The subtler case, the one for maths in particular, is what happens when you fail to follow well-established patterns for making maths research productive. Tao himself spelled out how that can look in https://mathstodon.xyz/@tao/117207856734787448 (which notably came before any of the news on Navier–Stokes).

Re: A misalignment of AI in mathematics

#960
post #453

Earlier quoted context omitted.

Dont confuse mathematics with the formal system. If you beleive mathematics = formal system then AI is obviously better at it, and we dont need humans. But then who decides why a statement is mor important than another? In the eyes of a formal systems all statements are born equal.

I didn't know some statements in math are more important than others?! Some are more useful, and that is something you can quantitatively ask.

How can you quantitatively evaluate that? By how many times it is used in the literature? That can be gamed and there is an inherent bias on that
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