Sanford Shieh

This book is a study of the philosophy of arithmetic in one of the most significant periods of its history—from Frege to Carnap—prefaced by an account of Kant. Potter aims at a philosophical history, a story told from an explicit interpretative perspective. These theories of arithmetic are seen as attempts to account for its “source of content” and “source of concepts.” Potter never explains these terms; I take the former to be the thing that, when we have knowledge of it or insight into it, provides the ultimate justification for pure and applied arithmetical discourse. That is, Potter’s principal concern is both ontological and epistemological: what determines, and how do we know, the truth of the statements that express our putative arithmetical knowledge? The criterion of success of a theory of arithmetic is how well it explains arithmetic’s simultaneous necessity or apriority and empirical applicability. Potter divides the theories he discusses into four main types, determined by their accounts of the “source of content” and “concepts”: sensibility, thought, language, and the world. At the end, Potter finds that none succeeds in accounting for all of arithmetic: at best some capture the infinity of the natural numbers, but none gives the “source” of arithmetical concepts. But the “similarities … between the scopes of the various approaches” suggest a Gödelian moral for Potter’s tale: the philosophy of arithmetic must treat concepts of “thought structures or … contents,” where “reflection upon the meanings” of “propositions about these mental objects” is needed for their proofs.

This article treats three aspects of Frege's discussions of definitions. First, I survey Frege's main criticisms of definitions in mathematics. Second, I consider Frege's apparent change of mind on the legitimacy of contextual definitions and its significance for recent neo-Fregean logicism. In the remainder of the article I discuss a critical question about the definitions on which Frege's proofs of the laws of arithmetic depend: do the logical structures of the definientia reflect the understanding of arithmetical terms prevailing prior to Frege's analyses? Unless they do, it is unclear how Frege's proofs demonstrate the analyticity of the arithmetic in use before logicism. Yet, especially in late writings, Frege characterizes definitions as arbitrary stipulations of the senses or references of expressions unrelated to pre-definitional understanding. I conclude by examining some options for conceiving of the status of Frege's logicism in light of this apparent tension, and outline a suggestion for a philosophically fruitful way of resolving this tension.