View metadata, citation and similar papers at core.ac.uk brought to you by CORE
provided by PhilPapers
THE REVIEW OF SYMBOLIC LOGIC Volume 5, Number 2, June 2012
ON THE RELATIONSHIP BETWEEN PLANE AND SOLID GEOMETRY ANDREW ARANA AND PAOLO MANCOSU Department of Philosophy of Kansas State University and University of California-Berkeley
Abstract. 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 mathe- matical practice, epistemology, semantics, ontology, methodology, and logic. In addition, issues of psychology and pedagogy are also important here. To our knowledge there is no single contribution that studies in detail even one of the aforementioned areas. In this paper our major concern is with methodological issues of purity and thus we treat the connection to other areas of the planimetry/stereometry relation only to the extent necessary to articulate the problem area we are after. Our strategy will be as follows. In the first part of the paper we will give a rough sketch of some key episodes in mathematical practice that relate to the interaction between plane and solid geometry. The sketch is given in broad strokes and only with the intent of acquainting the reader with some of the mathematical context against which the problem emerges. In the second part, we will look at a 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. We conclude this part of the paper by remarking that only through a foundational and philosophical effort could the issues raised by the debate on “fusionism” be made precise. The third part of the paper focuses on a specific case study which has been the subject of such an effort, namely the 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, we begin in the fourth section the analytic work necessary for exploring various important claims about “purity,” “content,” and other relevant notions.
§1. Mathematical practice. 1.1. The Greeks. If we look at Euclid’s Elements we find a separation between plane (Books I–IV, and arguably VI and X) and solid geometry (books XI–XIII). The separation is not that neat, for in the stereometric books Euclid establishes many results that pertain to plane geometry: for instance 12 out of the 18 theorems of book XIII are theorems of plane geometry. While elements of plane geometry are obviously needed in the development of solid geometry, Euclid has no applications of solid geometry in establishing results of plane geometrical statements. Nevertheless, the application of stereometric considerations for solving problems of plane geometry was not unknown to the Greeks. Such constructions or demonstrations using solid geometry to establish results of plane geometry remained confined to the advanced branches of the subject, however, as opposed to the elementary
Received: January 10, 2011.