Background and motivation

Question

Main questions

What are the different notions of infinity as used by mathematicians? To what extent does the use of infinity inform our understanding of the relationships between mathematics and the real world? Specifically: Does this mean

The following is copied from my blog since someone thought it was the clearest statement I had made regarding a problem I recently posed. On their advice, it is a community wiki. I would encourage, however, I discussion somewhere else (meta.MO perhaps?) on what seems to be a serious disconnect between pure mathematicians and physicists (note: I straddle the line but my PhD is in Math). I should add that nowhere on this site does it say that questions are to be restricted to pure mathematics (as opposed to applied mathematics). Some of the greatest discoveries in the history of mathematics came out of the consideration of physical problems. And I can't emphasis enough how important it is that the communities not drift apart (and I see it happening even within my own sub-discipline). On to the question, then.

Background: 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.

Example: 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.

Question: 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?