A new problem for naturalizing logico-mathematical knowledge?

Abstract

This article advances a categorial challenge to physicalistic versions of naturalized epistemology as an account of logico-mathematical knowledge and practice. It argues that, qua physically constituted, any putative rule-governed system is logico-mathematically indeterminate: physical facts are systematically insufficient to fix a unique logico-mathematical (L–M) profile to the exclusion of incompossible alternatives.  Building on a Kripkean line of thought concerning the truthmakers of rule attributions and rule applications, the paper develops the point through a Stabler-style mechanism allegedly computing a function f over the natural numbers, and argues that the same finite physical profile equally supports an incompossible alternative function g that coincides with f on all physically realizable cases. The result is that the physical facts fail to determine whether the mechanism determinately and objectively implements any unique or definite L–M rule at all. Crucially, the indeterminacy-generating features exploited by the Stabler-style case are structural: they are characteristic of any finite physical mechanism. Attempts to restore determinacy via counterfactual robustness, normal-operation conditions, or supervenience are argued to be question-begging, since they covertly presuppose precisely the rule whose objective implementation is at issue. The upshot is a dilemma: either determinate L–M rule-following is possible, in which case physicalistic NE is false; or physicalistic NE is true, in which case determinate L–M rule-following (and knowledge) is impossible.