Comment by TrackerFF
7 hours ago
If we're going to be frank about it, higher level math is:
1) Intellectually challenging, to such a degree that those wishing to enter the field need to have a certain level of intellectual prowess to do so. This creates some levels of mystique, with a sprinkle of elitism and gatekeeping.
2) Driven (among other things) by prestige. And the more pure the math is, the more prestigious it is.
3) So complex that people can spend their entire working careers chasing a handful of problems. The amount of time researchers spend on very specific problems is mind-boggling, if we think about the results.
4) Intensely captivating for the people deep in the weeds.
And the deeper you get, the longer you study, the more you start to value things like "mathematical beauty", and may start to view math as a form of art.
Like many similar fields, you end up with this ivory tower where people can dedicate their whole lives to thinking deeply about extremely niche and theoretical problems.
I'm not totally against "math as art" but I don't think that formulation goes deep enough towards what's really going on with mathematics. There is something deeper where I lean more towards the philosophers who have said that mathematics is essentially ontology.
Since the Greeks we've had the idea that "Being and thinking are one," or that Being (in the sense of all of existence as such) has some essential unity with thought, and therefore can be thought, and expressed or submitted to the logos or reason. Being is in some sense fundamentally intelligible, and mathematics is the most developed, exacting, and articulate expression of Being.
Logic was understood in this older sense up to roughly the the mid to late 19th century. This is why a work like Hegel's Science of Logic begins not with syllogisms or propositions but with Being and Nothing. But this was forgotten after logic was mathematized by the English around the time of Russell, and its connection to ontology was gradually overshadowed by a focus on epistemology (still, it should be remembered, originally as a means of getting back to ontology).
There may be truth in art, but it's always haunted by its own historicity or contingency, which is to say untruth. Mathematics seems on the contrary the only really timeless, absolute thing we have. Part of what makes it captivating is stumbling on a construction or concept or proposition or theorem that simply must be, independent of us.
The AIs are certainly now more than automatic theorem provers, mechanically traversing some space of true propositions. They are able to push things forward and connect seemingly disparate domains to get to a proof, but to my mind it remains to be seen how well they will be able to form new concepts and definitions.
Imagine the controversy surrounding Cantor, for example, but put an AI in the place of Cantor. If an AI proposed something like the (infinite) hierarchy of infinity, would we have accepted it? What would the intuitionism debates have looked like? Would they even have taken place? And aside from that, has it actually been shown conclusively that an AI could propose such a thing?
There are lots of attempts right now to recover a humanism for mathematics, or restore man's pride of place with respect to it, but maybe we don't need to worry about that. Tao's attempts to preserve the mathematical community, while allowing for practices to change through the crisis may look like a kind of rearguard action, but seems reasonable to me and not really dependent on any kind of humanism. It's a way to avoid the question for now while things play out (and not conservative/reactionary like Scholze and others), which may be exactly what we need, because after all, perhaps we still don't understand why we do mathematics, what it's really for, and what our relation is to it. Whether it's enough to preserve funding is another issue.
Similar things are happening in the competitive programming world.
Quite a lot of people are not happy they aren't elite anymore, and many have spent years to decades to arrive here.
Simply put you invest years of your life to establish a kind of distinction over others, and that goes away. That hurts.
But its not something surprising. Most of these competitive programming problems were actually English languages puzzles, because you couldn't dial up the mathematical difficulty anymore making it a fields medal problem. And in most cases in simple language weren't even that hard to begin with, and you could look up solutions to these problems in an hour of Google searching.