Reddit reviews Mathematics and Reality

There's a sense in which "mathematics exists" is almost trivially true. People tend to forget that math is a human endeavor -- it's a field of study, an intellectual enterprise. So, obviously, math, as a human endeavor, exists. (And it's not as monolithic as we tend to think of it either. There's good evidence that revolutions occur in mathematics -- i.e., that math changes and grows something like science. Cf. D. Gillies, Revolutions in Mathematics.)

Since I take it this isn't your concern, you could be wondering one of two things: Do mathematical objects really exist? Is mathematics true (in some sense) of the physical world?

The first metaphysical question has been addressed ad nauseum, and in many respects could be seen as boiling down to the debate between realism vs nominalism. Realism in math almost always relies on the existence of abstract objects -- things that don't exist in spacetime; ideal, Platonic numbers, lines, triangles, etc. In your post, you seem to be touching on the idea of fictionalism ("our collective imaginations"), which is another way of denying realism. I haven't seen any statistics on this, but I'd wager that realism is the de facto position amongst most philosophers. But it does suffer from some pretty large epistemological hurdles. How the hell are we supposed to have knowledge of things that don't exist spatiotemporally?

Penelope Maddy has a great book on realism in math. Mark Balaguer has a great book on the epistemological problems of realism in math. The seminal work on said epistemological problems comes from Benacerraf: "Mathematical Truth" (1973, I think). A great recent book on fictionalism in math is from Mary Leng.

The second question relies on the answer to the first. If math is just about the nominal world, then there's no mystery as to why it applies so well to it. But if math is about abstracta, or is some sort of collective fiction, then we have a real mystery on our hands. How the hell could something about abstracta apply to the real world in such a meaningful way. /r/Crushnaut has touched on that topic in these comments, so I'll leave it at that.