The fall of the theorem economy

Not that it is wrong for them to be doing this---we do want a society where people get to devote their life to what interests them---but it is bizarre because of the framing. For some reason it is ambiently understood in our society that this work is of incontrovertible value, when in fact it is largely not. And the value-producing parts of the work, the parts that end up having applications to other fields, largely run contrary to the actual daily goals of the cloistered devotees: it is mostly the intuition and pedagogy and the compactification and refactoring of knowledge that have value at this point, not the production of esoteric theorems, yet that is expressly not rewarded in the incentive structures.

That latter point is more due to the sorry state of academic incentives in general than to a particular failing of mathematics, though. Were I somehow given the ability to restructure things by fiat I would immediately create journals which publish only useful articles that refactor knowledge, communicate intuition, better explain things, argue for structural improvements to notation and terminology, etc, and this would immediately create an incentive to do that kind of work for working researchers to do work which aligns with the actually-useful output of their fields. I suspect most fields could use something like this. New knowledge is just not that valuable if it is all dumped into a giant pile and unprocessed, and I have seen firsthand a bunch examples where entire subdisciplines are hamstrung in their actual application-heavy work because they don't have easy access to basic tools that are hidden behind hard-to-learn theory.

How is it not already this? Jon von Neumann was already calling most math this many decades ago. Pull up any random arxiv math paper and it’s abstract nonsense with no applications to the real world.

First, math, generally, is useless. I mean, yes there are of course practical uses of basic thru undergrad-level math, and some beyond that. But for many mathematicians, the sum result of their entire career may lead to exactly zero results that have any real-world value. The entire field they work in may have meaning only to the handful of other individuals on the planet that also work in that field. But to those handful of people, the meaning defines their lives. From a socio-economic perspective, those departments should have been defunded a century ago. Yet they continue. Why? Because it scratches an itch. Not just for those individuals in the field, but also for us as a species. To stop exploring, to eliminate the search for pots of gold that may be buried in some odd corner of sphere packing, or coloring theorems, or Garside categories, and to put a boundary on the limits of our understanding, just because they aren't immediately applicable, is an idea that most humans would not be willing to sacrifice, even if it reduced their tax burden a couple cents. If it was going to happen, it'd have happened already.

The second is, even with AI, it's not free. As the software industry is discovering, far from it. So, given that, who is going to decide what theorems to research and how much it's worth? Congress? Of course not. AI itself? In theory that sounds plausible, but that falls victim to thing 1 above: most math is useless, so AI itself has no value metric it can assign to things, and besides which, without the human element, once the initial curiosity has subsided, there'd be no reason to continue any funding for AI to do it. So no, the only possible owners of this is going to be mathematicians themselves, the ones who care about the field and deeply understand the kwah of their vision.

Combining these, there's a future where, humanistically, "nothing changes". The method changes, the efficiency changes, the scope changes, but the work itself: publishing proofs, remains the domain of professional mathematicians. AI will enable them to be dramatically more daring and broad in their investigations and scope, and will likely write the entirety of the proof. However it will remain the work of the mathematicians to determine, what areas are worth spending limited AI resources on to investigate further, how far to go down rabbit holes, how to prioritize potential connections, and what the ultimate meaning of the findings is. So rather than being an end of mathematics, it could be a dawn of something far greater than anything we've ever seen before.