Why does mathematics "work"? How does it manage to describe the physical world?

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Which mathematics manages to describe the physical world? Mathematicians offer us, e.g., Euclidean and non-Euclidean geometries of spaces of various dimensions (and the non-Euclidean geometries come in different brands). They can't all correctly describe the world, since they say different things even about such simple matters as the sum of the angles of a three-dimensional triangle. But we hope that one such geometry does indeed describe the sort of structure exemplified by physical space (or better, physical space-time).

That's pretty typical. Mathematicians explore all kinds of different possible structures. Only some of them are physically exemplified. For example, group theories explore patterns of symmetries; some of the patterns are to be found in the world -- but I guess that no one thinks e.g. that the Monster Group is physically instantiated. Mathematical physicists tell us which kinds of structures are to be found in the physical world and then use the appropriate mathematics to deduce more facts about the relevant structures.

Perhaps though it is different with elementary arithmetic. Perhaps, whenever we talk about objects at all, we talk about things that in principle can be counted and to which arithmetic can be applied. 'Logicists' like Frege hold that arithmetic thus has the same kind of necessary applicability as logic. But if so, it is a rather special case.

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