Maybe we're having a definition difference. My understanding of the term "discrete math" is that it refers to any math whose characters aren't part of a continuum.

Anything to do with numbers besides the reals. Anything to to with finite sets. Anything to do with groups, rings, polynomials, trees, graphs, ordinals, lattices, compass-straightedge-constructions, polygons, knots, categories, sheaves, topological spaces, vector spaces, and anything to do with oddities like map coloring or plane tiling, and a lot else too.

It's a course in "everything else" for students that are being pigeonholed into a mathematical specialization by the fact that it's been fashionable to use real numbers to describe the world since Newton.

Have you studied math? No mathematician would call algebraic topology discrete math, nor for that matter almost anything else on that list.

Yes, I have a degree in it.

I'd have to ask around but I know a few algebraic topologists and I'd bet that if pressed to describe their work as more-continuous or more-discrete they'd first tell you that this is a silly way to classify subfields in math and then they'd says it's probably more on the discrete side since it's all about categorizing topological spaces based on whether they have certain discrete properties:

- separable - countable - metric - compact

Sure the spaces themselves might be continuous, but topology is for telling those spaces apart based on where and how continuity fails. It's a discretization of things formerly suspected to be the the same.

I mean the notion of compactness is inspired pretty clearly from the continuum. I also have a degree in pure math and disagree strongly with your characterization of topology, which is literally defined up to homeomorphism which is itself defined by it's continuity. Arguably the whole point of topology is characterized by that which is preserved by continuous deformation, which clearly is inspired by more continuous math. I think you are using incredibly strong language and in a misguided way.

Yeah, I think you're right.

There's something about real analysis that makes me uncomfortable. I've been struggling for years to put my finger on it. Whatever it is, topology doesn't have it.

I had mistakenly decided that it was an overappreciation of continuity, but I think it must be something else.

Maybe it's the differential structure itself? Or an (over?)emphasis on the study of functions from space to space instead of the study of the space itself? Or the specificity of calculus (the study of one specific space) instead of the generality of topology (thinking about a lot of spaces and comparing them to each other)

Hmm, I'll have to ponder those. Specificity seems closest. Whatever it is, it's not a rational critique. Despite the discomfort, I'm also fascinated by it because one should not have an emotional response to specific types of math, but I very much do.

Something about the homework in Real Analysis left me feeling angry. Not because it was difficult or presented poorly, but because it was somehow... untrustworthy? As if my betters had decided which ideas were the good ones and the only thing left for me to do was optimize along the one dimension that they had assigned me. I realize that this is nonsense, but I can't seem to shake it.

I think it's totally good and human to have preferences for one sort of math over another :)

Double integration - brings not very nice memories back of my hand aching from writing upto 5 pages of workings out

It was especially bad in quantum mechanics where they made us do it the hard way (using calculus) before showing us the easy way (using noncommutative algebra).