Frege on the Foundation of Geometry in Intuition

Abstract

I investigate the role of geometric intuition in Frege’s early mathemat- ical works and the significance of his view of the role of intuition in geometry to properly understanding the aims of his logicist project. I critically evaluate the interpretations of Mark Wilson, Jamie Tappen- den, and Michael Dummett. The final analysis that I provide clarifies the relationship of Frege’s restricted logicist project to dominant trends in German mathematical research, in particular to Weierstrassian arith- metization and to the Riemannian conceptual/geometrical tradition at Göttingen. Concurring with Tappenden, I hold that Frege’s logicism should not be understood as a continuing a project of reductionist arith- metization. However, Frege does not quite take up the Riemannian ban- ner either. His logicism supports a hierarchical understanding of the structure of mathematical knowledge, according to which arithmetic is applicable to geometry but not vice versa because the former is more general, as revealed by the strictly logical nature of its objects in com- parison to the intuitional nature of geometric objects. I suggest, in par- ticular, that Frege intended that foundational work would show the use of geometric intuition in complex analysis, a source of error for Rie- mann that Weierstrass was proud to have uncovered, to be inessential.