I thought one of the implicit points of the open letter was that unsolved problems are not something that falls out of the sky, they are a curated resource that people have spent time on and shared for the benefit of like-minded peers and humanity as a whole. And the AI companies treat them like they treat absolutely everything else: natural resources, literature, art, code etc. as something to be chucked into the ravening maw and pooped out the back as profit. They don't care if mathematics advances, they don't care if they strip-mine the available problems and damage the field. In fact, as with programming I think they see that as in their long-term interest - soon there will be no intelligence or creativity but the one that Sam Altman bills you for.
Ok, so unresolved math problems are often something people discover while trying to solve a different math problem.
However, math problems are really there to solve a real world problem. We have unlimited real world problems no matter how smart AI gets. Therefore, we will always have unresolved math problems.
> math problems are really there to solve a real world problem
I think this is totally wrong. Math problems are almost by definition problems with a particular theory. That theory might be inspired by the real world, but the problem itself is purely theoretical. I can't think of any theoretical problems like this that actually support a practical problem, as opposed to being an internal knot in the theory that indicates something is wrong with it. Not to say that cannot happen - certain optimization problems were historically actually hard to solve and solving them helped us to genuinely optimize a real thing (rather than just explain why the answer we already had was correct, which is much more common). In particular, none of the millennium problems have anything to do with a "real" problem, including the Navier Stokes one.
As an aside, it's amusing that this conversation is a re-statement of a main point in TFA:
> However, math problems are really there to solve a real world problem.
vs
> That theory might be inspired by the real world, but the problem itself is purely theoretical.
From TFA:
> In my essay The Two Cultures of Mathematics a quarter of a century ago, and which can be summarized by saying that there is a spectrum of attitudes in mathematics to the relationship between problem-solving and conceptual understanding.
The author thinks this letter was choosing only one of them as the "right" approach whereas the better stance is "porque no los dos?"
I only just skimmed the referenced essay, but a priori I don’t think “problem-solving” as Gowers uses it has anything to do with real-world practicality. The problems under consideration are entirely theoretical, regardless of which “culture” a mathematician belongs to.
You're right, but I also may have quoted poorly to give the impression that the first post was only about real-world problems. It goes on to point those out as an infinite source of theoretical problems, which sounded to me like an emphasis on the problem-solving culture.
The reply to that seems to say there are theoretical problems not necessarily connected to real-world problems, which I interpreted as an emphasis on the conceptual understanding aspect.
I may have misinterpreted either or both of them though!
fruitl00p · · focus · HN ↗
aurareturn · · focus · HN ↗
Ok, so unresolved math problems are often something people discover while trying to solve a different math problem.
However, math problems are really there to solve a real world problem. We have unlimited real world problems no matter how smart AI gets. Therefore, we will always have unresolved math problems.
tempfile · · focus · HN ↗
I think this is totally wrong. Math problems are almost by definition problems with a particular theory. That theory might be inspired by the real world, but the problem itself is purely theoretical. I can't think of any theoretical problems like this that actually support a practical problem, as opposed to being an internal knot in the theory that indicates something is wrong with it. Not to say that cannot happen - certain optimization problems were historically actually hard to solve and solving them helped us to genuinely optimize a real thing (rather than just explain why the answer we already had was correct, which is much more common). In particular, none of the millennium problems have anything to do with a "real" problem, including the Navier Stokes one.
keeda · · focus · HN ↗
> However, math problems are really there to solve a real world problem.
vs
> That theory might be inspired by the real world, but the problem itself is purely theoretical.
From TFA:
> In my essay The Two Cultures of Mathematics a quarter of a century ago, and which can be summarized by saying that there is a spectrum of attitudes in mathematics to the relationship between problem-solving and conceptual understanding.
The author thinks this letter was choosing only one of them as the "right" approach whereas the better stance is "porque no los dos?"
BalinKing · · focus · HN ↗
keeda · · focus · HN ↗
The reply to that seems to say there are theoretical problems not necessarily connected to real-world problems, which I interpreted as an emphasis on the conceptual understanding aspect.
I may have misinterpreted either or both of them though!