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“Platonism” as a philosophy of mathematics refers back to Plato's dialogues on the Forms, which have been represented as existing in some eternal, unchanging, non-physical realm. Platonism in mathematics locates mathematical objects there. Many mathematicians believe that platonism as a philosophy of mathematics has been discredited, in part due to the contradictions of naive set theory, in part because of the question of how we physical beings could contact such a realm. However, in fact platonism remains the default philosophy of mathematics among philosophers, one that few are willing to defend in the strong form attributed to Gödel, but which will not be replaced until a satisfactory alternative has been found.
This chapter summarizes over forty years of such discussion, between those defending some version of platonism (called platonists, or realists) and those opposing platonism, usually called nominalists. Because it is summarizing discussion that has developed in several hundred articles and dozens of books, this chapter is not one to read casually. However, this chapter is very clearly and comprehensively structured, so that those who take the effort it to read it will be rewarded with a thorough survey of the many different schools of thought that have developed among philosophers during this period. Thus this chapter provides an excellent introduction for anyone who would like to be able to start reading original work by philosophers of mathematics.