[andars]

In some sense, don't the modern formulations of real analysis, etc. already start from as close to discrete maths as you can get (set theory)?

Sets -> Naturals -> Rationals -> Reals

I don't understand how you could reformulate study of continuous structures into discrete math in any sense other than the above.

[ginnungagap]

Every mathematical object (ok, this is false but that's not the point here) can be constructed in ZFC (the standard axiomatic framework for set theory) so you can construct the real numbers in terms of sets (if you want more precise informations on this construction look up Dedekind cuts).

However this is irrelevant to, say, analysis, you could define the real numbers as the unique (up to isomorphism) complete, ordered, archimedean field and do analysis just as well, so I'd say that you are right in some sense and some formulation, but it's a bit of a stretch to consider analysis as starting from discrete maths.

I also don't see how set theory fits into discrete maths, apart from the basics it seems pretty far from the common structures studied in discrete maths.

[EtDybNuvCu]

Pick the Grothendieck-Tarski axiom instead, and use category theory to build ZFC via topos. This path is "big" enough to handle all the interesting sets; it can't deal with proper classes, but proper classes are kind of metaphysical anyway.

[0] https://en.wikipedia.org/wiki/Tarski–Grothendieck_set_theory

[ginnungagap]

Sure, but ZFC by itself also deals with every interesting set, I was just being nitpicky of my own assertion.

I'm not familiar with TG, what's the relation between it and ZFC+some large cardinal axiom?

[andars]

I understand that it's a stretch; thus the "in some sense" and "close".

See my last sentence. It's not clear to me how you could reformulate analysis, which in many ways is the study of the infinite, into discrete maths in any way other than the very loose sense of starting with ZFC.