Andrew Arana

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. 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

The architecture of modern mathematics

edited by José Ferreirós and Jeremy Gray Oxford University Press, 2006, 442 pp.

Plane and Solid Geometry: A Note on Purity of Methods

69.50 US, ISBN 0198567936 REVIEWED BY ANDREW ARANA This collection of essays explores what makes modern mathematics ‘modern’, where ‘modern mathematics’ is understood as the mathematics done in the West from roughly 1800 to 1970. This is not the trivial matter of exploring what makes recent mathematics recent. The term ‘modern’ (or ‘modernism’) is used widely in the humanities to describe the era since about 1900, exemplified by Picasso or Kandinsky in the visual arts, Rilke or Pound in poetry, or Le Corbusier or Loos in architecture (a building by the latter graces the cover of this book’s dust jacket). Though it is hard to say precisely what modernism is, or what distinguishes it from other eras, Gray attempts a definition in his closing essay in this collection:

In the light of logic

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.

Purity of Methods

I: FOUNDATIONAL PROBLEMS 1. Declining the undecidable: Wrestling with Hilberts Problems 2. Infinity in Mathematics: Is Cantor necessary? 3. The logic of mathematical discovery vs. the logical structure of mathematics II: FOUNDATIONAL WAYS 4. Foundational Ways 5. Working Foundations III: GODEL 6. Godels life and work 7. Kurt Godel: conviction and caution 8. Introductory note to Godels 1993 lecture IV: PROOF THEORY 9. What does logic have to tell us about mathematical proofs? 10. What rests on what? The proof-theoretic analysis of mathematics 11. Godels Dialectica interpretation and its two-way stretch V: COUNTABLY REDUCIBLE MATHEMATICS 12. Infinity in mathematics: Is Cantor necessary? (Conclusion) 13. Weyl vindicated: Das Kontinuum 70 years later 14. Why a little bit goes a long way: Logical Foundations of scientifically applicable mathematics