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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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Siegel zeros and other "illusory worlds": building theories around hypotheses believed to be...
This is more of a theoretical/mathematical physics example, but it can happen in mathematical physics that a lot of stuff is built around hypotheses which are essentially known to be false from the be …
0
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Modeling in pure math
Basically every branch of pure mathematics uses modelling of some kind, and the idea that pure mathematicians deal with 'exactness' and do not need to model or approximate the phenomena they are study …
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Small ideas that became big
Another possibility could be the problem of the brachistochrone, a famous but one might think relatively innocent problem, which then led to the development of the calculus of variations.