On the Relationship Between Plane and Solid Geometry

Abstract

<jats:p>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. To our knowledge there is no single contribution that studies in detail even one of the aforementioned areas.</jats:p><jats:p>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.</jats:p><jats:p>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.</jats:p>

Are There Genuine Physical Explanations of Mathematical Phenomena?

“…12 Arana and Mancosu ([2012]) discuss another potential counterexample. Desargues' theorem says that if the lines through corresponding vertices of two triangles (lying in the plane) meet in a point then the intersection points of lines along corresponding sides lie on a single line (that is, if two triangles are in perspective from a point then they are in perspective from a line).…”

Purity in Arithmetic: some Formal and Informal Issues

“…These together are constitutive of α's understanding of the problem, and hence of the identity of the problem (to α). 4 A solution to a problem is "topically pure" (for α) if it draws only on what belongs to that problem's topic. In other words topically pure solutions to problems draw only on what is constitutive of the identity of that problem.…”

“…There is a discussion of a related distinction in[4] between "informal" or "intuitive" content of a statement, by which is meant what someone with a casual understanding of geometry would (be able to) grasp, and "formal" or "axiomatic" content, by which is meant the inferential role of that statement in an axiomatic system.UnauthenticatedDownload Date | 6/14/16 12:59 PM…”

Explanation, Existence and Natural Properties in Mathematics - A Case Study: Desargues’ Theorem

“…There are non-Euclidean geometries where Desargues' theorem in two dimensions fails, but the axioms of incidence regarding two dimensions hold. See Baker (1922, 120) and Arana and Mancosu (2012). The latter presents a passage where "the eminent algebraist" Marshall Hall (1943) says that the explanatory contribution made by exiting to three dimensions is to supply another way of picking out line LMNnamely, as the intersection of two planes.…”