alt Hacker NewsMy 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.
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> what if an intuitive understanding of the true “boundaries” of mathematics (if such things exist) is beyond the capabilities of a human mind?
By extension: why should we assume that a human would still understand the problems - or the answers? If all of it is complete gibberish to a human and can never be applied in any way, shape, or form, then what's the point?
The way I view it there are two options here: either you completely ignore it and end up burning a massive amount of electricity on what is essentially a bunch of LLMs jerking each other off, or you blindly follow it and end up with a Machine God who can justify a genocide with a "This is the correct thing to do. Trust me bro, I have irrefutable proof - you won't understand it". There's just no sensible way to do post-human math in an inherently human world.
Mathematics, and its empirical cousin science, are not about picking out individual things that are true in the universe. There is an infinite set of statements of things that are true about the universe, almost all of which are, necessarily, too complex (in a Kolmogorov sense) to be captured by a human mind, and each of which is so specific as to be largely useless. Once you have a statement, though, they're usually trivial to prove by observation.
Mathematics and science are both about building structure on large classes of facts about the (logical or physical, respectively) universe that allow us to generalize our knowledge and make predictions about facts we cannot yet observe. This also lets us make claims about things that are, individually, too complex for the human mind to grasp, by abstracting away the complex details into a simpler structure and then making the claim about anything that satisfies that structure.
To put it another way, mathematics is about finding beauty — specifically those things that exhibit structure that humans can grasp. Modern mathematics, for the most part, makes no claims about what lies outside that space: if it turns out that the universe consists only of things that humans can describe through mathematics that would be neat, but in the much more likely case that it doesn't mathematics continues as it is today.
It's an interesting and maybe even useful trivium if (according to some set of axioms, such as those implemented by your favourite proof assistant) a fact is true, but it's not (human) mathematics, and if there's no useful way for humans to generalize from it there's no point in including it in a library of mathematics. It wouldn't be that surprising (but would be very interesting!) if there were an entire parallel class (or, more likely, family of classes, one for each AI architecture) of AI mathematics comprising structures that AIs can usefully generalize from. Such a thing has no reason to bear any resemblance to human mathematics, which is based on mapping structures to innate human linguistic and spatial intuitions, and may not yield any insights to human mathematicians.