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:

[spoiler]

Basically the debate is largely based on refuting the other sideâ€™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 Iâ€™ve presented here.

Many mathematicians and physicists quite like the idea of mathematics being â€śout thereâ€ť 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 â€śmathematical intuitionâ€ť that happened to fall out of evolution because our brains were made in a mathematical universe.[/spoiler]

Hereâ€™s my view: itâ€™s invented. Itâ€™s better to think of mathematics as a verb than a noun: itâ€™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 doesnâ€™t â€śworkâ€ť â€“ 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. itâ€™s self-consistent) but thatâ€™s a much smaller claim than saying every branch of mathematics actually exists â€śout thereâ€ť.

I am aware of the standard retort to this position: that Iâ€™m alluding to Logicism and that was soundly defeated in the early-mid 20th century. I have a response, but Iâ€™ll put it in a separate post so that the OP is not too long.