Is mathematics mostly chaos or mostly order?

I’ve always considered math is something that is discovered, neither chaotic or orderly, it just… is. Really brilliant people make new discoveries, but they were there the whole time waiting to be found.

This article seems to kind of dance around yet agree with the discovery thing, but in an indirect way.

Math is just math. Music is just music. Even seemingly-random musical notes played in a “song” has a rational explanation relative to the instrument. It isn’t the fault of music that a song might sound chaotic, it’s just music. Bad music maybe. This analogy can break down quickly, but in my head it makes sense.

Disclaimer - the most advanced math classes I’ve taken: calc3/linear/diffeq.

Mathematics isn't monolithic—it depends heavily on the axioms you choose. Change the axioms, and the theorems change. ZFC, ZF¬C, intuitionistic logic, non-Euclidean geometry—each yields a different “math,” all internally consistent. So it’s not right to say math “just is” in some absolute sense. We’re not just discovering math; we’re exploring the consequences of chosen assumptions.

For instance:

Under Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC), every set can be well-ordered, but we do get the Hahn–Banach paradoxes.

Under ZF without Choice, analysis as we know it no longer holds.

In constructive mathematics, which avoids the law of the excluded middle, many classical theorems lose their usual formulations or proofs.

Non-Euclidean geometries arise from altering the parallel postulate. Within their own axioms, they are as internally consistent and "natural" as Euclidean geometry. Do non-intersecting lines exist in this universe? I've no idea.