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This paper concerns the epistemic status of “Hume’s principle”— the assertion that for any concepts Fand G, the number of Fs is the same as the number of Gs just in case the Fs and the Gs are in one-one correspondence. I oppose the view that Hume’s principle is a stipulation governing the introduction of a new concept with the thesis that it represents the correct analysis of a concept in use. Frege’s derivation of the basic laws of arithmetic from Hume’s principle shows our pure arithmetical knowledge to arise out of the most common everyday applications we make of the numbers. The analysis of arithmetical knowledge in terms of Hume’s principle ties our conception of number to the interconnections of which our concepts of divided reference are capable; in so doing, it locates the origin of our conception of number in the structure of our conceptual framework. 1. The Neo-Fregean Background In Frege’s Conception of Numbers as Objects [18], Wright showed that it is possible to extract from Frege’s Grundlagen [11] a valid second-order proof of the Dedekind infinity of the natural numbers using only a suitable formalization of the following “partial contextual definition” for numerical identity: For any concepts F and G, the number of Fs is identical with the number of Gs if and only if the Fs and the Gs are in one-one correspondence.1 Wright argued that this proof can form the basis of a defensible, if modified, formulation of Frege’s philosophy of arithmetic, one that relies on Frege’s thesis that numbers are objects but makes no use of the problematic notion of the extension of a concept. Hale’s Abstract Objects [14], which appeared in 1987, is perhaps the next important landmark in the elaboration of a “neo-Fregean” philosophy of arithmetic along the lines of [18]. Received March 29, 2001; printed September 30, 2002 2001 Mathematics Subject Classification: Primary, 18A15

On the Origin and Status of our Conception of Number