Take Kant and Gödel, for example. Kant was neither just an intuitionist nor just a formalist, nor even absolutely a non-realist (the forms of space and time are, after all, empirically real and transcendentally ideal, or variously provide for such statuses). Gödel was a strong realist with abiding sympathies for constructivism, intuitionism, and perhaps even definabilism (where this is a distinct enough position). Hamkins has spaces in the logical flux of the set-theoretic multiverse to envision worlds based on different axiomatizations of various schools of philosophy-of-math: a world where the inscription axioms are all literally inscriptive (semiotic formalism) but so where we are still, ultimately, talking about abstract forms of symbols; and so on and on.
So it doesn't seem quite accurate to try to pigeonhole any eminent philosopher of mathematics into a strictly exclusionary standpoint, but we might redescribe these standpoints as ones such as if Kant, say, were to rank intuitionism highest, formalism next, and realism third or last, with some fourth or other if we had to keep going along, here.
Do philosophies of mathematics have to be advanced in a strongly exclusionary way as such, in the end, or instead is it epistemologically stable to imagine a system of epistemic weights and ranks? One can also think about a system where some methodologies are assigned nonzero weights while other methodologies do have no weight in such a system, although unless we were talking about inchoate numerology, I would not be minded to consider ruling out any specific methodology in advance.
