Geometry and generality in Frege's philosophy of arithmetic

This paper develops some respects in which the philosophy of mathematics can fruitfully be informed by mathematical practice, through examining Frege'sGrundlagen in its historical setting. The first sections of the paper are devoted to elaborating some aspects of nineteenth century mathematics which informed Frege's early work. (These events are of considerable philosophical significance even apart from the connection with Frege.) In the middle sections, some minor themes ofGrundlagen are developed: the relationship Frege envisions between arithmetic and geometry and the way in which the study of reasoning is to illuminate this. In the final section, it is argued that the sorts of issues Frege attempted to address concerning the character of mathematical reasoning are still in need of a satisfying answer.I am indebted to many people for helpful conversations and comments on this paper, notably Stephen Glaister, Phil Kremer, Madeline Larson, John McDowell, Jim Conant, Charles Chihara, William Craig, Jan Alnes, Joan Weiner, Leon Henkin, Paul Benacerraf, Juliet Floyd, Bill Demopoulos, Jose Ferreiros, Tom Hawkins, Gideon Rosen. Two superb papers on Frege — Bill Demopoulos' (1994) and Mark Wilson (1992) played a significant role in the early stages of composition. Special thanks are due to Hans Sluga, Mark Wilson, Bob Brandom, and Ken Manders for comments, encouragement, information and advice.