Gödel was a platonist. He was opposed to formalism because he didn't think reasoning could be reduced to symbol manipulation (and, of course, proved Hilbert's program for doing so wasn't adequate), but he was hardly any kind of post-modernist. The science wars and the like were irrelevant to mathematics because it doesn't say anything about the world; if you object to the axiomic method, that's cool, you're just not doing mathematics. It just doesn't matter for the purposes of doing math whether you think it's a game that happens to yield results useful to scientists and engineers sometimes, whether it's just a particular genre of literature or whether you think we're studying things that are somehow fundamental. We had a similar conflict much earlier, over non-euclidean geometries, but that was mostly settled by the time modernism was a thing. You could make the case for George Lakoff, but I don't think anyone cares what Lakoff thinks about math.in mathematics Modern is Bertram Russel - Post-Modern is Godel
I think Modernism is Aristotelian. I understand exactly where you are coming from and I don't think Gödel thought of himself as a post modern (I think he was a royalist politically) The effect of his proof was by my definition the definition of post-modernism.
The history of mathematics has a pre-Gödel Hilbert or modern period and a Post-Gödel or postmodern period perfect. :) [edit] I don't think that Gödel should be included in the nothing means anything crowd.
and I do agree that mathematics is or should be immune to the stupidities of the worst manifestations of post-modernism is other fields.