@MISC{_mathematicalconceptualism, author = {}, title = {MATHEMATICAL CONCEPTUALISM}, year = {} }

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Abstract

We must remember that in Hilbert’s time all mathematicians were excited about the foundations of mathematics. Intense controversy centered around the problem of the legitimacy of abstract objects... The contrast with today’s foggy atmosphere of intellectual exhaustion and compartmentalization could not be more striking. — Stephen Simpson ([15], p. 350) From the standpoint of mainstream mathematics, the great foundational debates of the early twentieth century were decisively settled in favor of Cantorian set theory, as formalized in the system ZFC (Zermelo-Fraenkel set theory including the axiom of choice). Although basic foundational questions have never entirely disappeared, it seems fair to say that they have retreated to the periphery of mathematical practice. Sporadic alternative proposals like topos theory or Errett Bishop’s constructivism have never attracted a substantial mainstream following, and Cantor’s universe is generally acknowledged as the arena in which modern mathematics takes place. Despite this history, I believe a strong case can be made for abandoning Cantorian