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Breaking public key cryptosystems would be a better counterexample. Most NP-hard optimization problems can be solved in practice. While there are always some degenerate instances nobody knows how to handle, realistic instances tend to be easy enough to solve or at least approximate accurately.

by dataflow3 hours ago|

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> Breaking public key cryptosystems would be a better counterexample.

I deliberately avoided giving that example. Calling that a positive contribution to humanity and expecting credit for it feels a bit like calling the atomic bomb a positive contribution to humanity and expecting credit for it. Much of the world would rather stay away from that timeline.

> Most NP-hard optimization problems can be solved in practice.

Hence "a solution that is empirically sufficiently faster than the state of the art" and not merely "a solution".

by jltsiren2 hours ago|

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Once you start solving problems in practice, you are probably no longer interested in the simple clean versions of the problem theoreticians think about. For example, instead of trying to find the shortest common superstring, you may be doing phased genome assembly. The problem is no longer unambiguously defined. The constraints and assumptions evolve over time, as the problem space becomes better understood and the nature of the data changes. While algorithmic ideas often transfer from one problem subtype to another, everything needs to be tailored for the specific problem.

by bbeonx1 hours ago|

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I mean, of course, if you make the benefit of the result itself large enough than yes, by all means; I'm sure Terry would agree: if there was some algebraic geometry proof that could be used to build a life-saving drug, then I don't think folks would have a problem with using AI to prove it. But most math isn't like this. Most math has very little immediate effect on the real world.

There is a tradeoff that is implicit in this entire discussion: if I can go out and use a result to do a thing much much better/faster/easier than I could yesterday, like a real and useful thing in the real world, then the result itself has value. The cost incurred by swarm solving that individual problem is relatively low and isolated, and the value added is clear, so great, we should do it! But if proving a result only allows us to apply it to more mathematical questions, then there isn't really value added aside from our own value we place in the results, and that value comes from our understanding. So now, swarm solving random math problems is taking away our opportunity to solve these things ourselves.

And people who don't have math backgrounds will poo poo this whole "opportunity to solve problems ourselves" stance, but it is 100% the way mathers think about this stuff. When I was a math major in undergrad I actually had a friend who told me that he didn't want to give me hints on a difficult homework problem because he wanted me to have the opportunity to solve it myself. Like, his _present_ to me was to give me an opportunity to solve a hard but rewarding problem.