I realize this forum hasn't gotten much action lately, but I'm hoping enough people pop on here that I get some feedback on this (particularly some of the membership who may have worked on or thought about related problems).
What I am interested in from a non-speculative standpoint are those places that mathematics *diverges* from the physical world, i.e. those places in which mathematics describes something that is very clearly not physically possible. Now, this obviously does not mean any instance of infinity since, as an example, infinity is necessary to resolve such physical paradoxes as those conjectured by Zeno. But there *are* instances where mathematics in some infinite limit ends up describing something that is not physically possible. There are probably plenty of examples that don't involve infinity.
More succinctly, is this divergence necessarily always a discrete point or can it be continuous? For instance, take something simple like a set whose elements represent some physical object (perhaps we're simply numbering these objects and the numbers are the elements of the set). Call the number of elements N. As long as N is finite, it is presumably physically possible (regardless of how unlikely that may be) to realize N of these objects. But in the infinite limit it is *not* possible if both the universe and these objects possess a finite, non-zero volume.
Incidentally, could this be one way to talk about the many-worlds interpretation? In other words, maybe math diverges from a single physical universe and only perfectly matches reality if there are an infinite number of parallel universes (e.g. limits to infinity such as the one just mentioned could be physically realizable in this case). This would get at the heart of whether mathematics is discovered or invented. If it is always true that it diverges somewhere and that either there's only one universe or it can be shown that it still diverges even with an infinite number of them, then it seems as if it would be invented. Otherwise it could be discovered.
One motivation for this is the Quine-Putnam indispensability argument which basically says that if we believe in the concreteness of the physical theories described by mathematical objects then we also ought to believe in the concreteness of those mathematical objects themselves.
I am very curious to get some thoughts on this from people.