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PaperclipPerfector  17d ago (text post)   28139 thread views

Culture War Roundup for the week of September 2, 2024

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What's the pecking order among subfields of (pure) mathematics? Is algebraic geometry more prestigious than geometric algebra? What about geometric topology vis-a-vis topological geometry? Do group theorists look down upon ring theorists? Are number theorists generally considered the smartest? What about combinatorists? Is set theory and/or foundations of mathematics low status?

I guess this isn't "culture war" per se but more "sociology of mathematics", but it seems to belong here.

Every Sunday there's a Small Questions thread that serves as an ideal home for these types of posts. But the range of topics allowed in the Culture War thread is very broad, and there are some interesting culture war implications in how academia treats "pure" and "applied" fields, so the post is fine here.

Generally I'd say that more abstract sub-fields are held in higher esteem (especially "abstract" in the sense of "fundamental", as in the results have wide-reaching implications in multiple areas of mathematics), in accordance with the general esteem that pure math itself is given over applied math. But this has its limits. Even some pure mathematicians balk at things like category theory, or the study of large cardinals, as "abstract nonsense". I think you're right that there is a certain mistrust of the foundations of mathematics - anything that carries the stench of philosophy is ipso facto suspicious. So it really depends on who you ask.

Even some pure mathematicians balk at things like category theory

The fact that it has more practical uses than a lot of other abstract mathematics is one of the endless ironies of the discipline.

It always seems to be that more applied mathematics generates new methods that advance abstract problems whilst abstract mathematics generates new frameworks that can solve new classes of practical problems.

There is also the funny phenomenon that plenty of academics don't have to interact with each other at all, so they can all independently look down down on all other fields and hold their own up as superior.

One observes this in a number of more practical fields, too.

I don’t know what the imaginary industrial engineers say about the rest of us!

Algebraic geometry and most of number theory on top. Graph theory and most of combinatorics toward the bottom (I am here). The basis is largely the minimum difficulty of producing independent, original work.

Getting spoonfed or rubber-stamped in a high status area, like Piper Harron, does not confer high status.

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