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Philosophical aspects of logic and set theory; truth status of mathematical axioms; Philosophy of Mathematics; philosophical aspects of mathematics in general; relation of mathematics to philosophy; etc. Consider also posting at http://philosophy.stackexchange.com/, where philosophy-of-mathematics is one of the most popular tags.
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Taking a theorem as a definition and proving the original definition as a theorem
An example which is not quite what is being asked, and which is certainly much less sophisticated than the other answers, but which comes up in my calculus teaching around this week each year: continu …