Comment by laichzeit0
2 months ago
The great thing about these mathematical mopping up type operations is that no person will waste their time trying to prove it to be true anymore. If anything that’s a win.
It would be great if an LLM could settle the Collatz conjecture next, god knows how many man-years have been burned on that by unsuspecting victims.
The reason this was "easy" is because the conjecture turned out to be false. If the collatz conjecture holds true (and most mathematicians seem to think it will), it will be much harder to prove than your average Erdos problem.
I think parent’s point is that every false conjecture can cost a lot of time to be spent on futile affirmative proofs. So if we “clean up” a bunch of false conjectures, then more effort can be spent on interesting proofs of the others. (Probably a rather naive view of the value of conjectures but I’m just offering an alternative interpretation of the comment.)
The opposing argument there is that the hope is that solving these problems reveals other interesting maths knowledge along the way. Finding a counter example all but ensures that won't ever happen.
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The Collatz conjecture is a question about positive integers, so enumerating and checking all the possible counterexamples is trivial, albeit requiring infinite time. It has been verified up to 2.36×10^21. It could turn out to be false, but nobody's going to find a counterexample as surprisingly simple as the one Claude found for the Jacobian conjecture, which would be like finding a Collatz counterexample in the first few billion integers or so.
... Or would it? The Jacobian counterexample seems like an especially simple, near-trivial integer-coefficient polynomial, but I haven't seen any thorough analysis of how "hard" it would have been to find by brute force, and I haven't seen Claude's reasoning.
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Did you read what you responded to? The Collatz conjecture is almost certainly not false, so no "clean up" is possible.
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True but Noam brown (openai researcher) said that in 2 years AI will start creating new math.
As long as they are not poisoning pigeons in the park.
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well if he said it then it must be true
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Give the llms a few years, they'll be smart enough to make progress on that.
I used to be a young mathematician sometime at the turn of the century...
"waste their time trying to prove it" is the MBA approach, where you should spit out results and articles.
Outside the MBA-thinking box, attacking hard problems, even unsuccessfully, is the way to gain deeper insight into various results and tools that you can later apply to other problems, i.e. no waste of time at all, unless you go to the extremes (like spending years on a single problem and nothing else).
Yeah but the Collatz probably has one of the highest man-hours of actual waste.
Ergo: > “This is a really dangerous problem. People become obsessed with it and it really is impossible,” said Jeffrey Lagarias, a mathematician at the University of Michigan and an expert on the Collatz conjecture.
and
> “Collatz is a notoriously difficult problem — so much so that mathematicians tend to preface every discussion of it with a warning not to waste time working on it,” said Joshua Cooper of the University of South Carolina in an email.
(from https://www.quantamagazine.org/mathematician-proves-huge-res... )
We won't be running out of hard problems to attack anytime soon, even if we produce a bunch of counterexamples to some of them.
I don't think this is how this works. The next step is to determine for what kinds of polynomials the jacobian conjecture is true and for what kinds of polynomials it's false.