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This is also my stance. The fact that large numbers of people spend large amounts of publicly-funded time exploring what are essentially abstract puzzles is bizarre and not that different from, like, cloistered religious devotees who are supported in spending their time studying scripture and are considered to be the 'source' from which flows a certain kind of universal truth.

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.

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Right; this is my viewpoint too. All the "pure mathematicians" have a bleak future where AI can do all the puzzle solving better and faster. They existed in their own world elevating "theorem proving within a formal system" as the central aspect of "proper" mathematics and everything else as ancillary.

It always felt wrong to me that while the scientific method iterated starting with the "real world" viz. Observe, Measure, Hypothesize (includes modeling with mathematics), Test and Refine; pure mathematicians lost themselves in the formalization of hypothesizing/modeling and thus lost touch with mapping it to reality. The AI revolution is now showing them up.

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> pure mathematicians lost themselves in the formalization of hypothesizing/modeling and thus lost touch with mapping it to reality.

You’re describing a very small fragment of total current mathematical labor. Very few people work solely on “formalization” and even e.g. model theory or type theory have real consequences.

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Yes. Though even philosophy, which doesn't have the "real world" iteration that science does, arguably doesn't have the problems of pure mathematics.

Pure mathematicians create ever more abstractions and get lost in solving puzzles on how these abstractions logically relate to each other. But since these abstractions don't have any relevance outside of pure mathematics, it's an entirely self-referential game, like chess. Except that nobody confuses being a professional chess player with being a noble researcher.

Even in philosophy, at least analytic philosophy, that issue of getting lost in your own abstractions doesn't really exist. Because analytic philosophy doesn't analyze its own concepts, it analyzes the concepts that already exist in natural language. Like truth, knowledge, probability, causation, belief, desire, consciousness, rationality and so on. These concepts come from outside of philosophy, and they have independent relevance for non-philosophers.

In contrast, pure mathematics seems to be the part of mathematics that only has relevance to pure mathematicians. Similar to how a game like chess has only relevance to chess players, not to anything entirely unrelated to chess. But again, people who are into mastering some game or sport are fully aware that what they are trying to master is a self-contained game, or sport, not something that increases the amount of human knowledge beyond that.

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