Intuiting the infinite
This paper offers a defense of Charles Parsons’ appeal to mathematical intuition as a fundamental factor in solving Benacerraf’s problem for a non-eliminative structuralist version of Platonism. The literature is replete with challenges to his well-known argument that mathematical intuition justifies our knowledge of the infinitude of the natural numbers, in particular his demonstration that any member of a Hilbertian stroke string ω-sequence has a successor. On Parsons’ Kantian approach, this amounts to demonstrating that for an “arbitrary” or “vaguely represented” string of strokes, we can always “add” one more stroke. Critics have contested the cogency of a notion of an arbitrary object, our capacity to vaguely represent a definite object, and the role of spatial and temporal representation in the demonstration that we can “add” one more. The bulk of this paper is devoted to demonstrating how to meet all extant criticisms of his key argument. Critics have also suggested that Parsons’ whole approach is misbegotten because the appeal to mathematical intuition inevitably falls short of providing a complete solution to Benacerraf’s problem. Since the natural numbers are essentially and exclusively characterized by their structural properties, they cannot be identified with any particular model of arithmetic, and thus a notion of intuition will fail to capture the universality of arithmetic, its applicability to all entities. This paper also explains why we should not reject appeal to mathematical intuition even though it is not itself sufficient to fully “close the gap” on Benacerraf’s challenge.
KeywordsCharles Parsons Benacerraf Platonism Structuralism Intuition Axiom of infinity
I presented an ancestor of this paper at the 2013 NYU La Pietra workshop on the a priori. Portions of the dialectic were discussed in a colloquium at the University of Texas, Austin in 2002 and in my philosophy of mathematics seminar at Yale University in 2004. Warm thanks to Crispin Wright, Christopher Peacocke, Tim Williamson, and Jane Friedman. I am especially grateful to Bob Hale for excellent criticisms of an ancestor of this paper.
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