[optimalsolver]

If the shortest proof for some theorem is several thousand pages long and beyond the ability of any biological mind to comprehend, would mathematicians not care about it?

Which is to say, if you only concern yourself with theorems which have short, understandable proofs, aren't you cutting yourself off from vast swathes of math space?

[nicf]

Hm, good question. It depends on what you mean. If you're asking about restricting which theorems we try to prove, then we definitely are cutting ourselves off from vast swathes of math space, and we're doing it on purpose! The article we're responding to talks about mathematicians developing "taste" and "intuition", and this is what I think the author meant --- different people have different tastes, of course, but most conceivable true mathematical statements are ones that everyone would agree are completely uninteresting; they're things like "if you construct these 55 totally unmotivated mathematical objects that no one has ever cared about according to these 18 random made-up rules, then none of the following 301 arrangements are possible."

If you're talking about questions that are well-motivated but whose answers are ugly and incomprehensible, then a milder version of this actually happens fairly often --- some major conjecture gets solved by a proof that everyone agrees is right but which also doesn't shed much light on why the thing is true. In this situation, I think it's fair to describe the usual reaction as, like, I'm definitely happy to have the confirmation that the thing is true, but I would much rather have a nicer argument. Whoever proved the thing in the ugly way definitely earns themselves lots of math points, but if someone else comes along later and proves it in a clearer way then they've done something worth celebrating too.

Does that answer your question?

[diamondage]

So Godel proved that there are true theorems that are unprovable. My hunch is that there is a fine grained version of this result <-- that there is a some distribution on the length of the proof for any given conjecture. If true that would mean that we better get used to dealing with long nasty proofs because they are a necessary part of mathematics...perhaps even, in some kind of Kolmogorov complexity-esque fashion, the almost-always bulk of it

[photonthug]

Agree that something like this does seem likely. And this line of thought also highlights the work of Chaitin, and the fact that the current discussion around AI is just the latest version of early-2000s quasi-empiricism[1] stuff that never really got resolved. Things like the 4-color theorem would seem to be just the puny top of really big iceberg, and it's probably not going away.

The new spin on these older unresolved issues IHMO is really the black-box aspect of our statistical approaches. Lots of mathematicians that are fine with proof systems like Lean and some million-step process that can in principle be followed are also happy with more open-ended automated search and exploration of model spaces, proof spaces, etc. But can they ever be really be happy with a million gigabyte network of weighted nodes masquerading as some kind of "explanation" though? Not a mathematician but I sympathize. Given the difficulty of building/writing/running it, that looks more like a product than like "knowledge" to me (compare this to how Lean can prove Godel on your laptop).

Maybe it's easier to swallow the bitter pill of poor quality explanations though after the technology itself is a little easier to actually handle. People hate ugly things less if they are practical, and actually something you can build pretty stuff on top on.

https://en.wikipedia.org/wiki/Quasi-empiricism_in_mathematic...

[dullcrisp]

I’m not sure that’s quite true. Say the proof of proposition P requires a minimum of N symbols. You could prove it in one paper that’s N symbols long and nobody can read, or you can publish k readable papers, with an average length on the order of N/k symbols, and develop a theory that people can use.

I think even if N is quite large, that just means it may take decades or millennia to publish and understand all k necessary papers, but maybe it’s still worth the effort even if we can get the length-N paper right away. What are you going to do with a mathematical proof that no one can understand anyway?

[mb7733]

> If the shortest proof for some theorem is several thousand pages long and beyond the ability of any biological mind to comprehend, would mathematicians not care about it?

Care or not, what are they supposed to do with it?

Sure, they can now assume the theorem to be true, but nothing stopped them from doing that before.