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.
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.
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.
The Definition/Theorem/Proof (DTP) emphasis (to the exclusion of everything else) for axiomatization in mathematics arose in the late-19th century and solidified in the 20th century. Prior to this, while mathematicians always practiced rigour, they valued insight/intuition more for understanding.
An attempt was made to teach "Modern Math" in a "New Math" manner which though withdrawn, has left its detrimental mark on mathematics education to this day. See the "Reception" and "Legacy" sections at https://en.wikipedia.org/wiki/New_Math
Excerpts:
Physicist Richard Feynman argued, "first there must be freedom of thought; second, we do not want to teach just words; and third, subjects should not be introduced without explaining the purpose or reason, or without giving any way in which the material could be really used to discover something interesting. I don't think it is worthwhile teaching such material."
In a 1971 article, mathematician René Thom rejected the New Math as "a test of memory that poisons intelligence" because of its complete neglect of intuition.
Mathematician and historian of mathematics Morris Kline observed that it was "practically impossible" to learn new mathematical creations without first understanding the old ones, and that "abstraction is not the first stage, but the last stage, in a mathematical development."
Mathematician and author George F. Simmons wrote in the algebra section of his textbook Precalculus Mathematics in a Nutshell (1981) that by focusing on form rather than substance, the New Math produced students who had "heard of the commutative law, but did not know the multiplication table."
Mathematician Laurent Schwartz described the new reforms as "very poor" pedagogy. For him, "The goal of mathematics is not to prove rigorously things that everyone knows. Instead, the goal is to find rich results and then, in order to make sure they are true, to prove them."
See also;
Against mathematical proof - https://mathwithbaddrawings.com/2021/05/12/against-mathemati...
Proof and Understanding in Mathematical Practice by Danielle Macbeth - https://journals.openedition.org/philosophiascientiae/712?la...
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.
_We do not know in advance which efforts are going to pay off_. Abstract efforts in topology put us on the road to nuclear energy. Silly number puzzles enabled internet commerce. Non-euclidean geometry gave us synchronized universal GPS.
We should not let our inability to conceive of applications of weird abstract stuff prevent us from making these investments. If our ancestors had fallen in to that trap, we'd be far poorer as a society.
What we can do is ask that people trying new stuff attempt to fail quickly. And that's basically where we are with academia today. Most people who do mathematical work will not have a career in math. They try something new, work for a little while on it, and go do something else when the results turn out to be of modest interest. This leaves behind a messy undigested literature, which is unfortunate. But maybe AI can help us sift that for treasures we missed.