From a physicist's perspective there are several situations in which somehow arbitrary choices of mathematical structures can be made. One can describe a system from different perspectives, etc. without changing anything in the physical properties. (see gauge invariance, picture changing etc.) In the situation when these choices impose a mathematical structure one has limitations in the subsequent choices (are these limitations physical???). However, there are situations when this prescription does not hold. The best example is in this case the Black-Hole complementarity principle (if it is well defined in the first place). So, the question about how mathematical structures "interact" in a logical way is not a bad one. Again: what is the interaction between, say, locality and Hausdorff-ity in the case of a black hole? Or what is the true relation between metrizability and hausdorff-ity of a space in the context of a black hole? The question about how mathematical structures assigned (innocently) at some point interact in different situations is in my opinion a quite relevant one...
closed as unclear what you're asking by Brandon Enright, Emilio Pisanty, user10851, John Rennie, Alexander Feb 3 '14 at 11:23
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I guess you mean the sort of 'interaction' between structures (though I would call it 'interlocking' or something similar) that Gowers describes in this blog post. It is of course crucial, when one postulates two different structures on an object, to have some interplay between the two, or you will not get any result that cannot be derived from one of the structures alone. A topological group, for example, is not simply a group that has a topology on it: the group operations must play well with it (i.e. be continuous) for the theory to be more rich than just the results of group theory and topology.
However, it is ridiculous to assert that such interlocking of structures would have "unexpected physical effects". On what? Mathematics isn't physically real by itself. You can't "add" mathematical structure to an object, it either has it or not. A physical theory either has one or more mathematical structures, along with their 'interplay' properties, or not. Those structures determine what comes out. If you impose additional structure on the mathematical problem, of course it will predict 'different results'. But then you're talking about a different situation.