# The Problem of Existence in Mathematics

# The Problem of Existence in Mathematics

Concerns the ‘problem of existence’ in mathematics: the problem of how to understand existence assertions in mathematics. The problem can best be understood by considering how Mathematical Platonists have understood such existence assertions. These philosophers (‘Literalists’) have taken the existential theorems of mathematics as literally asserting the existence of mathematical objects. They have then attempted to account for the epistemological and metaphysical implications of such a position by putting forward arguments that supposedly show how humans can come to know of the existence of mathematical objects. The reasoning of the most prominent philosophers in this Literalist tradition, Quine and Gödel, are critically discussed. By way of contrast, an alternative position, Heyting's Intuitionism, is examined.

*Keywords:*
Gödel, Heyting, Intuitionism, Literalism, mathematical existence, mathematical objects, Mathematical Platonism, Quine

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