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A misalignment of AI in mathematics

mathandai.org

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Re: A misalignment of AI in mathematics

#301

Post by globular-toast »

oytis wrote: post #184
sdeframond wrote: post #162

Earlier quoted context omitted.

Say AI becomes the best at everything. Best at chess/go, best at maths, philosophy, economics, romantic advices ... and so on. Then what's the point of thinking by oneself? Of talking to one another? What's the point of being human if we dont do human things but entirely rely on AI? I believe this is more or less these mathematicians' argument.

It has been best at chess for quite a while. Yet everyone knows who is Magnus Carlsen, even though at no point of his career he was stronger than the machine

Magnus Carlsen plays fellow humans at the game because people still care about human competitions. What motivation would a mathematician have for solving already solved problems by hand? Can you imagine someone spending years working on a proof for an already-proved theorem just in case it leads to new insight?

Re: A misalignment of AI in mathematics

#303

Post by fg137 »

david-gpu wrote: post #141

Tao's critique of AI in the field of mathematics reminds me of what French art critic Charles Baudelaire said in the 19th century about photography [0]. Baudelaire argued that photography became a haven for failed painters, the sorts of hacks that could not finish proper training. Photography, as a mechanical rendering of the world, could only record what already existed; it couldn't transform reality the way a paint…

We can start having a meaningful discussion when people use real reasoning instead of analogy.

Re: A misalignment of AI in mathematics

#304

Post by jscd »

jrflo wrote: post #276
9question1 wrote: post #221

Earlier quoted context omitted.

That's precisely the problem though. You cannot still read and understand an AI written proof at the current skill level of the AI being applied, because they're orders of magnitude longer than human written proofs even when they don't need to be, and spend most of that length on the parts that aren't important. This has been really thoroughly documented by expert mathematicians who are engaging with AI in public lik…

That doesn't seem to be true. The OpenAI NS paper was 166 pages. Wiles-Taylor proof of Fermat's last theorem is 129 pages. The length is not unprecedented for a difficult unsolved problem. To be honest, I feel like the difficulty of reading AI proofs is due to the fact that we are on the verge of being beyond human comprehension. This is a demonstrable fact as no human has figured this out despite the problem being o…

This is a very token-brained take. The length of a work has no bearing whatsoever on its comprehensibility.

Re: A misalignment of AI in mathematics

#305

Post by ceh123 »

yzydserd wrote: post #6

> solving problems is only a tool and proxy for achieving the primary goal of conceptual understanding and insight. This is the effect of AI on most intellectual disciplines, and it’s a real worry.

I published a substack about this just a few days ago [1], my core theory here is that we will absolutely have what I call a "highly productive dark age" in mathematics where knowledge vastly outpaces understanding driven by publish-or-perish incentives, but additionally this will lead to the loss of the skills necessary to understand.

The hopeful note is that I do think we are entering a golden age for the curious casual/semi-pro mathematician and for niche mathematics areas that won't get the attention of the top labs. Everyone is sprinting to solve the millennium problems, but this is a very exciting time to be in a sub-sub-field where you and 4 others are keeping things alive.

[1] https://substack.com/home/post/p-214740151

Re: A misalignment of AI in mathematics

#306

Post by adastra22 »

Mathematics is about discovering and understanding the logical implications of assumed axioms under various inference rules.

Alternatively, some claim that mathematics is about understanding these implications.

Under the first definition, AI is already, and forevermore will be faster and better at proving theorems. Just like it is better at checkers, chess, and now go.

The author asserts that AI proofs are incomprehensible to humans, and so under the second definition AI is merely a tool to overcome one hurdle on the way to understanding.

So which is it? The author seems to claim the second definition, but bemoan the end of mathematics under the first.

Re: A misalignment of AI in mathematics

#307

Post by BeetleB »

david-gpu wrote: post #141

Tao's critique of AI in the field of mathematics reminds me of what French art critic Charles Baudelaire said in the 19th century about photography [0]. Baudelaire argued that photography became a haven for failed painters, the sorts of hacks that could not finish proper training. Photography, as a mechanical rendering of the world, could only record what already existed; it couldn't transform reality the way a paint…

Keep in mind that his critique is very recent, and likely applying to a specific use of AI, as opposed to AI as a whole. If you've been following his Mastodon account, he's been happily using LLMs for math purposes for well over a year.

Re: A misalignment of AI in mathematics

#308

Post by pred_ »

jrflo wrote: post #276
9question1 wrote: post #221

Earlier quoted context omitted.

That's precisely the problem though. You cannot still read and understand an AI written proof at the current skill level of the AI being applied, because they're orders of magnitude longer than human written proofs even when they don't need to be, and spend most of that length on the parts that aren't important. This has been really thoroughly documented by expert mathematicians who are engaging with AI in public lik…

That doesn't seem to be true. The OpenAI NS paper was 166 pages. Wiles-Taylor proof of Fermat's last theorem is 129 pages. The length is not unprecedented for a difficult unsolved problem. To be honest, I feel like the difficulty of reading AI proofs is due to the fact that we are on the verge of being beyond human comprehension. This is a demonstrable fact as no human has figured this out despite the problem being o…

> To be honest, I feel like the difficulty of reading AI proofs is due to the fact that we are on the verge of being beyond human comprehension.

I can see where that's coming from, but I really don't think it's the case. Even with Astra, the proofs you get are just off in a way that doesn't signal superhuman comprehension. As 9question1 says, a common theme is that they dwell on insignificant steps. Another one is that they'll often be full of terminology that either doesn't exist, or has this weird quality where it looks like it is trying to make some minor insight seem much greater than it is. At first glance, that'll often make it look like it knows more than you, but when it's really just doing the same thing but in a more complicated and worse fashion, that to me isn't a signal of comprehension at all. The bizarre thing is that despite all the "stochastic parrot" style nonsense you'll get in individual proof steps, they still often combine to something valid.

In either case, what all of this means is that the working mathematician still needs to go through, and generally completely rewrite, any proof output by an LLM. Otherwise you are passing the burden of unreadability onto the reader.

Re: A misalignment of AI in mathematics

#309

Post by adastra22 »

omnicognate wrote: post #201
jeremysalwen wrote: post #20

To me it doesn't seem like what AI has destroyed is the ability for mathematicians to develop understanding and share it with each other, but rather it's destroyed the yardstick (solving open problems) that has traditionally been used to measure how much they have contributed to that understanding. I do see how this is a problem in terms of assigning credit, but I think the cat is already out of the bag in terms of t…

The statement is not about AI but about the behaviour of AI companies. OpenAI have put vast resource into solving open maths problems: many millions of dollars of compute just on the Navier-Stokes result, plus whatever they spent on the broader Millenium Prize problems initiative and the other results they have published. Anthropic are doing the same. The statement is asking them to stop doing this. AI companies are…

Humanity is better off for knowing these proofs. This strikes me as academic NIMBYism.

Re: A misalignment of AI in mathematics

#310

Post by bobzowak »

This is really only a short-term problem where the AI companies only have the internal models that can solve these. In the “long” term, which could honestly mean months, everyone will have access to Bel/C/D-level models capable of solving these anyway.
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