It's a shame that match curriculum for non-math-majors is an all-roads-lead-to-calculus affair. I think we scare a lot of potentially talented people away from math with that approach.
Discrete math is very important to computer science. But in the rest of the world, calculus (and differential equations) dominate.
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.
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.
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.
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'll start with sets and you can start with the real numbers, and we'll see whose list is longer.
To a certain degree the whole game is nonsense, there's a countable infinity of axioms and a countable infinity of theorems that follow therefrom, so we're never going to get anywhere rigorous with this, but just like it's not unreasonable to say that there are more multiples of 2 than there are of 2000, I think it's fair to say that continuity games on the reals represents a relatively small share when compared to mathematics in general.