I'm surprised no one has yet mentioned Cantor. His work leading to Transfinite Arithmetic, and from there to general Set Theory, overturned millenia of received opinion on the concept of the infinite by insisting on the mathematical existence of actual, completed infinities. He defended this work with a body of philosophical arguments arguing that the earlier conclusion - that infinity could of necessity only be potential - was in fact mistaken. [source: biography of Cantor by (I forget who, sorry)]
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