Fwiw, I believe the romanticization of advanced mathematics needs to end. I'm 15-20 years out of university, and I had thought I might delve into some specific area of math after graduating. I studied EECS and only took courses in abstract algebra and real analysis. But pursuing further math never materialized for me. Moreover, all of my friends who majored in math transitioned away from it within five years of college. Most became developers, a few became product managers, but I haven't seen anyone maintain or even express an interest in advanced math outside of academia, especially in their 20s and 30s.
I have a PhD in pure math. Nothing I learnt specifically in that PhD is needed for my job, and in fact, I do a job where most people who do the same thing as me don't have a PhD. So its value even for signalling is limited.
However, I'm glad I did the PhD.
The 4 years I spent on it were time in my life well spent. I enjoyed the work and it didn't bother me to be earning less than I otherwise would have.
The main skill I gained from the PhD was being able to read technical papers - typically involving stats or financial math. I have needed to do this on and off for my work, and I've found that lots of people who have good high school or college math find this much much harder than I do.
I study math for fun. I find the ability to do this precious and life affirming. When I read an interesting article in Quanta, I can look at the papers cited, if it's a field I have some background in, and make more sense of them than the average reader. In ten years of doing this, I have once published a short paper which added a small improvement to a recreational problem. So studying math isn't really about external achievement - I am the equivalent of someone who plays the guitar at home but will never have a gig or record a song. But I feel very lucky to exist in a time period when I had access to this educational opportunity, and when so much interesting math is available to read and study essentially for free.
You see that here on HN with all the articles on Category Theory. CT appeals to people who majored in Comp Sci but long for some advanced and abstract mathematics to get away from the daily dose of Node. It is simplistic enough that you can read the first few chapters of a CT book and feel like you're really getting some deep math topics.
I'm a huge fan of Haskell and the language uses many concepts from category theory, such as monads, functors, and applicatives. However, these are all abstractions which are simple enough to not need any basis in mathematics. I tried to read resources on category theory to understand its motivation, but unlike real math where theorems actually seem to contain new insight, I never found such a thing in category theory.
The entire body of CT has just a small handful of what anyone would consider theorems or lemmas. It’s mostly just stacked definitions.
This is a set. This is a mapping. This is a set of sets. This is a mapping from elements to elements. This is a mapping from elements to sets. This is a mapping from sets to sets. This is the inverse mapping…. Chapter after chapter.
But then there’s “insight” that all of mathematics can be cast as CT, because… all math is just things that map to things! Whoa, far out, dude. Mind is blown.
I think what's happening is a lot of people who studied very little or no math are realizing a few 100 level courses in things like linear algebra, probability, stats, calculus etc would be very helpful. They might not realize that the cost/benefit beyond that is minimal. It's similar for many subject areas. A couple 100 level courses in accounting is super helpful, same for finance, law, mechanical engineering, electrical engineering, computer science, biochemistry and on and on.
Nothing you said supported your first sentence. If people enjoy pure math why shouldn't they study it? Now people should not feel inadequate if they decide not to study it.
What math helps with is signaling. Employers care about your problem solving skills (as opposed to actual mathematical knowledge) and there’s virtually no better signal than IMO/Putnam or their easier variants. On the other hand a mediocre math phd is not all that useful at all outside of publishing mediocre papers in mediocre journals, since the student probably just has a lot of knowledge but not many problem solving skill.s
