There has been a dearth of philosophical threads recently in GD, so I am going to try to correct that
Summary of the debate inside spoiler tag for those not familiar with it:
Basically the debate is largely based on refuting the other side’s position.
If mathematics is invented then the main retort is why does it work so well? Why is it we can use our man-made mathematics to build bridges and launch spaceships? In fact, we have often predicted phenomena from abstract mathematics.
OTOH if mathematics is discovered, how is it discovered? What did we actually see to make us think of quaternions, or complex numbers, say?
Quite a lot of books have been written about this topic but most just compile many examples, and the arguments are not much more sophisticated than I’ve presented here.
Many mathematicians and physicists quite like the idea of mathematics being “out there” and so go for option 2. In terms of the counter-argument, in as far as any engage with it, they suggest we have some sort of “mathematical intuition” that happened to fall out of evolution because our brains were made in a mathematical universe.[/spoiler]
Here’s my view: it’s invented. It’s better to think of mathematics as a verb than a noun: it’s a toolkit for manipulation information and deriving non-obvious facts from obvious facts. Essentially formalizing and building on our reasoning power, which is obviously very useful.
But it would be trivial to define a branch of mathematics that doesn’t “work” – that gives incorrect predictions about the external universe. The reason our mathematics does not do this is that new mathematics must follow certain rules of basic logic e.g. self-consistency.
So indeed it is a fact about the external universe that it follows fundamental logic (e.g. it’s self-consistent) but that’s a much smaller claim than saying every branch of mathematics actually exists “out there”.
I am aware of the standard retort to this position: that I’m alluding to Logicism and that was soundly defeated in the early-mid 20th century. I have a response, but I’ll put it in a separate post so that the OP is not too long.