One particularly contentious question in the foundations of both mathematics and physics is whether mathematics is discovered or invented. Related to this is whether mathematics is the way the world actually is or if it is simply a way in which we can model the world.
This is a particularly difficult question to answer since it is quite clear that there are physically impossible situations that can be spoken about in mathematics. The best example I can think of relates directly to some things I bitched about on this blog in recent weeks concerning Brikhoff’s theorem (which I won’t rehash in the interest of brevity): taking n copies of a quantum channel should start to approach something that can be approximated by unitaries as n approaches infinity, i.e. in the asymptotic limit, n copies of a channel have a unitary representation (roughly speaking and if the theorem is correct). But, from a physical point of view, this is ridiculous since it is literally impossible for an infinite number of channels (or anything else, for that matter) to physically exist. Yet whole branches of mathematics are devoted to working with infinity (infinite sets, for example).
Clearly some mathematical results have been originally thought to be entirely abstract only to much later find some application in the physical world. But it could be argued that concrete, mathematical analyses of infinite things could never, by their very definition, find an application in the physical world. Does this mean that mathematics is inherently discovered then? Or could it be taken as “evidence” for the Many-Worlds Interpretation (MWI), i.e. the only way to have a physically infinite result is to have a physically infinite number of universes?