Fri Sep 2, 2011
60, Evans Hall, 4:10–6 PM
Paolo Mancosu (Department of Philosophy, U.C. Berkeley)
Axiomatics and purity of methods: On the relationship between plane and solid geometry
Traditional geometry concerns itself with planimetric and stereometric considerations, which are at the root of the division between plane and solid geometry. To raise the issue of the relation between these two areas brings with it a host of different problems that pertain to mathematical practice, epistemology, semantics, ontology, methodology, and logic. In addition, issues of psychology and pedagogy are also important here.In this talk (which is based on joint work with Andy Arana), my major concern is with methodological issues of purity. In the first part I will 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, I will look at a late nineteenth century debate (on “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. I conclude this part of the talk 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 of the talk 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 section I point the way to the analytic work necessary for exploring various important claims on “purity”, “content” and other relevant notions.