mathematicians have been using computers for well over half a century, but this was after "bounding" the problem first and then running through the cases with a computer. Now AI is doing the first part. However, mathematicians are still needed at crafting prompts, and knowing where to look, still. The prompt for the Jacobian conjecture was obviously not random. the search space is too big to just try all the combinations of 3 variable polynomials.

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skinner_ 17 hours ago | parent | next [-]

Okay, I'm not sure about the original one, but here is the prompt of a successful reproduction:

https://aaronlou.com/jacobian_counterexample_prompt.pdf

Obviously it is not random, but it's very generic. No mention of search space or how to reduce it.

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hobonation 20 hours ago | parent | prev | next [-]

This is the best take. Computers don't care about this stuff. A computer could make a movie, but only a human can appreciate it.

We're a good team, and that's ok.

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criddell 19 hours ago | parent | next [-]

For now. I wonder if we will ever get to the point where the computer starts doing mathematics that we just can't understand. Surely there must be some limit to what we can understand (like how a gorilla will never understand prime numbers, there are probably limits to our intelligence as well).

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inigyou 4 hours ago | parent | next [-]

https://mathstodon.xyz/@logicbot

This bot posts a tautology (always true statement) every day. They aren't very useful. Sampling from all true mathematics is like sampling from all instruction sequences that don't crash. The important part of computer programming isn't finding valid instruction sequences, it's finding the useful ones that do what the customer wants.

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zeroonetwothree 19 hours ago | parent | prev [-]

Mathematics only really matters insofar as humans can understand it.

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hgoel 18 hours ago | parent | next [-]

Aren't deep learning models themselves a case where we have hints of some deeper underlying logic to why some things are more effective than others, but we lack the mathematical tools to properly work it out for anything of practical size?

All we're able to do is apply flawed analogies, generic information theoretical models, trial and error, post-hoc rationalizations and benchmarks without really understanding why.

	▲	AlotOfReading 14 hours ago \| parent [-]
		You don't need anything as recent as deep learning for that. Look at economics or social sciences, which have been influencing national politics for well over a century now.

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drivebyhooting 18 hours ago | parent | prev | next [-]

Doesn’t this generalize? Mathematics matters less than less as fewer people are capable of understanding it. So whatever cutting edge, deep insight about the nature of groups matters less than different equations which matters less than solving linear equations, etc.

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stabbles 19 hours ago | parent | prev [-]

Not really, a theorem with a hard proof can have simple but important corollaries. It's also not unthinkable that theorems exist with proofs that cannot reduce to something simple/short.

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nullsanity 19 hours ago | parent | prev [-]

[dead]

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skinner_ 19 hours ago | parent | prev | next [-]

> The prompt for the Jacobian conjecture was obviously not random. the search space is too big to just try all the combinations of 3 variable polynomials.

Maybe the prompt contained a part like this: "the search space is too big to just try all the combinations of 3 variable polynomials, so be clever about it". Or maybe this part was omitted from the prompt, because modern LLMs are smart enough to figure this out without us having to mention it.

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jknoepfler 19 hours ago | parent [-]

If someone has written that in a paper or article they've ingested, as they no doubt have, then sure.

	▲	jameshart 17 hours ago \| parent [-]
		The ability of LLMs to solve problems is not confined to the training data they ingested. Claude knows how to read mathematical papers because of its training data, but it can and will pull in literature relevant to a specific problem into its context. We really need to stop thinking about LLM training data as the knowledgebase they build from and instead consider it more the skillset they start with.

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sesm 19 hours ago | parent | prev [-]

I don't think there was a 'prompt' for it, rather a long and dedicated work of a professional mathematician which involved LLM in some capacity. I'm sure the search step wasn't an ad-hoc script running in a Claude Code session (as somebody would naively assume), it was an optimized numerical code running in Anthropic's compute cluster. Note that details are not published yet and `__alpoge__` is officially working at Anthropic.