Philosophy of Mathematics seminar

Patricia Blanchette, Notre Dame

Frege on Caesar and Hume's Principle

Frege's reaction to Russell's Paradox was, in the end, to abandon his logicist project. That he did so without seriously considering a now widely-discussed alternative, the alternative of grounding arithmetic in pure logic together with Hume's Principle, is significant. Clearly there is something in such a grounding that Frege took to be incompatible with either his view of the nature of logic or of arithmetic, or his view of acceptable ways to pursue a reduction of one to the other. The purpose of this talk is to investigate what, exactly, Frege took the difficulty to be, and what we can learn from this about his view of the nature of mathematics and of logic. I will be concerned to argue that there are some important difficulties with what might be called the ‘standard’ answer to these questions, an answer according to which the central difficulty with the Hume's-Principle strategy is to be found in the so-called “Caesar passages” of Grundlagen. The alternative answer offered here involves a different account of the importance of the Caesar passages, and a revisionary, though I think better-supported, account of Frege's view of the role of identity-statements in the reduction. Finally, I'll suggest that the alternative reading attributes to Frege a more plausible view of mathematical discourse than does the standard reading.