Comment by dataflow | Hacker News Reader

Comment by dataflow | Hacker News Reader

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> understands it thoroughly, and uses it to develop interesting further questions or directions

He phrases it like these criteria are requirements, but they really aren't for every problem. They make sense for those theoretical problems whose solutions' main values lie in their understandability and further development, but not every problem is like that. Some problems' solutions are just revolutionary on their own even if you contribute or understand absolutely nothing else. AI technology itself is an example of this!

To give a trivial hypothetical counterexample: if you came up with a solution for the traveling salesman problem that is both empirically sufficiently faster than the state of the art, that is an immense contribution and you absolutely deserve the credit for it regardless of whether you or anyone else understands how/why it works. It could literally even produce wrong results 50% of the time and still revolutionize humanity if it's efficient enough when it's correct. Denying them credit because they didn't understand the algorithm would be ludicrous - the results stand & speak for themselves.

Statisticians, traditional-AI researchers, etc. had to learn this the hard way when (basically black-box) neural nets became so good that their otherwise-elegant & explainable classical algorithms just stopped being able to compete in more and more applications, and got left behind. Are mathematicians looking to repeat that history?

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.

> 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".

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.

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.

> But most math isn't like this. Most math has very little immediate effect on the real world.

Then isn't it fair to ask why that math deserves to be done right now? Just because a problem can be formulated and piques your interest while leading to more problems that pique your interest, that doesn't mean someone on this planet has to work on it right now.

If you're one of the brightest minds in the world who wants society to credit you for your work (and fund your career!), and are finding that the rest of the world is replacing you with automation, and you're agreeing you don't anticipate your work to even have an effect on the real world in the foreseeable future... why is it unreasonable to expect you to start working on problems for which you can make a clearer case of usefulness in the horizon, at least for the time being?

> 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.

That's a wonderful anecdote, and it's what I feel is missing from the discussion of this so-called "AI era" of mathematics.

When the AI companies harvest all the low-hanging fruits of unsolved problems, they're robbing the mathematical community of the opportunity to learn and struggle for the solution themselves. That friction is the process of learning and understanding.

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