alt Hacker News> I think the point is to prove the statement.
I couldn't disagree more.
A lot of mathematical "problems" are almost entirely pointless. Nobody genuinely cares about the moving sofa problem, or about square packing, or about the minimum number of colors needed to draw a map - it is the math that is developed during the solving process that is valuable!
An answer to a question like "what is the exact area of a unit circle" is a mere curiosity. Calculating a good-enough approximation is trivial, after all. But wanting an exact answer leads to developing calculus, which leads to most modern physics. Science was able to make a giant leap forwards due to the techniques developed, while the actual answer itself is mostly useless.
Replies
"Mathematics isn't about proving the statement! It's about the collection of substatements which lead to the proof of the statement."
"Okay, so what determines what is and is not allowed in the collection?"
"Whether the given substatement is true or not, of course."
Like this is obviously silly, right. In your view you could have two guys both trying to prove or disprove that the area of the unit circle is 3, and yet only the guy doing it with some vision of nobility where he's building up to this grand theory of approximations is the one actually doing "real" mathematics. The guy who's doing it just because he thinks it's neat and would like an answer to the problem itself doesn't count, and you suspect he couldn't even exist.
My instinct is to agree with you. I believe that the drive to a deeper understanding of the problems is what helps us unlock new areas of study, and find opportunities to transfer techniques or bridge otherwise unconnected domains.
But let’s consider a hypothetical: what if an intuitive understanding of the true “boundaries” of mathematics (if such things exist) is beyond the capabilities of a human mind? If there truly is no way to simplify some proofs down from 200,000 line incomprehensible gibberish to something you could teach to a high schooler or undergraduate or even a PhD. Is the proof still worthless? Sure, at the moment, it might be. Finding such a proof and understanding the implications of it are different skills, the latter of which AI almost certainly does not possess at the moment. But there may come a time where the AI can view the bigger picture and make the leaps you described (say, an eka-Calculus from an eka-unit-circle). These leaps may be as unintelligible to us as the proof in OP is.
I guess the question is: assuming that we can’t make the proof beautiful enough to spark deeper human understanding, do we still want it if it sparks deeper AI understanding?
Personally I would hate to live in a universe where the boundaries of science are beyond intuitive human understanding, but I think it’s almost certainly the case. The idea that the rules are all within our grasp reeks of anthropocentrism to me. I would love for the universe to prove me wrong though. It’d be a pleasant, hilarious coincidence if they do fit within the boundaries of our understanding.