A Defense of Arithmetical Platonism (Variations on a Theme by Kitcher)

Abstract

In The Nature of Mathematical Knowledge (1984) Philip Kitcher develops an exciting and insightful picture of arithmetical reality by considering mathematical activities--not objects--as ontologically primitive. Yet, his account goes astray because of his nominalistic interpretation of the properties of human activity, resulting in an implausible modal view of the truth of arithmetical statements.

After critiquing Kitcher's conceptions of arithmetical truth and knowledge--and finding them deficient--I develop an interpretation of standard first-order number theory which finds the referents of numerical expressions to be platonistically construed properties of human collecting activity. I proceed by specifying three "meta-arithmetical" criteria which, when conjoined with Kitcher's assumption of the ontological primacy of human operations, serve to delimit the range of possible candidates for a plausible interpretation of number theory.

I require the referents of arithmetical expressions to be (1) abstract objects, (2) epistemically accessible in an appropriately causal way, and (3) fixed in part by the way that numerical expressions are used. Clearly, reconciling the first two criteria is tantamount to answering Benacerraf's (1973) challenge to platonism: provide a naturalistically acceptable account of mathematical knowledge. Further, I argue that a causal account of reference which parallels the Kripke/Putnam view for natural kinds arises naturally from my account, buttressing its plausibility.

That we have arithmetical knowledge is obvious. The more interesting question is whether or not it is posssible that any of our mathematical knowledge can be a priori. Providing an affirmative answer to this query closes this defense of arithmetical platonism.