Scott asked if there is something mysterious about math and, naturally, in the comments "the unreasonable effectiveness of math" came up; i.e. "why the structures that mathematicians found to be important for their own internal reasons, so often turn out to be the same structures that are important for physics". I wrote the following in response.
One should keep in mind that “the unreasonable effectiveness of math” is possible because the weirdness (e.g. the Banach-Tarski paradox) can be contained and e.g. Goedel’s result does not show up (but it could have, e.g. sums over all possible manifolds in 4d quantum gravity).
But one should also keep in mind that large parts of e.g. the standard model are not even based on well-defined (axiomatic) math, rather a patchwork of “physicist’s math” and the real mystery is why this works.
In other words, the real mystery is "the effectiveness of unreasonable math", an idea which goes back to Vaihinger and his philosophy of "as if".
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