ABSTRACT
It is widely held that Frege’s opposition to the grounding of arithmetic in Hume’s Principle was due to the failure of that principle to provide truth-conditions for sufficiently many identity-sentences. This chapter argues that that view, and the corresponding understanding of the import of the famous “Caesar passages” of Grundlagen, are mistaken. It is argued (i) that Frege’s conception of numbers as objects does not imply that an analysis of number-theoretic discourse must settle truth-conditions on all identity-sentences involving numerical terms and indeed that such a requirement is inconsistent with Frege’s own treatment of number in Grundgesetze; (ii) that the Caesar passages demonstrate a quite different problem with the proposed accounts of the concept of number; and (iii) that the fundamental difficulty with a grounding of arithmetic in Hume’s Principle, after the paradox, is the one Frege notes in his letter to Russell, that is, that the attempt to introduce object-names via abstraction principles is flawed in the way revealed by the paradox. It is suggested that the account of Frege’s texts provided here attributes to him a more reasonable version of mathematical realism than is often associated with those texts.
