Plane and Solid Geometry: A Note on Purity of Methods

Abstract

Traditional geometry concerns itself with planimetric and stereometric considerations, which are at the root of the division between plane and solid geometry. In this note (which is based on Arana and Mancosu. The Review of Symbolic Logic 5(2): 294–353, 2012), our major concern is with methodological issues of purity. In the first part we give a rough sketch of some key episodes in mathematical practice that relate to the interaction between plane and solid geometry. In the second part, we look at a late nineteenth century debate (“fusionism”) in which for the first time methodological and foundational issues related to aspects of the mathematical practice covered in the first part of the paper came to the fore. We conclude this part by remarking that only through an axiomatic and analytical effort could the issues raised by the debate on “fusionism” be made precise. The third part focuses on Hilbert’s axiomatic and foundational analysis of the plane version of Desargues’ theorem on homological triangles and its implications for the relationship between plane and solid geometry. Finally, building on the foundational case study analyzed in the third section, in the fourth and last section, we point the way to the analytic work necessary for exploring various important claims on “purity,” “content,” and other relevant notions.