The Incredible Shrinking Manifold

Abstract

Traditionally, there have been two methods of deriving the theorems of geometry: the analytic and the synthetic. While the analytical method is based on the introduction of numerical coordinates, and so on the theory of real numbers, the idea behind the synthetic approach is to furnish the subject of geometry with a purely geometric foundation in which the theorems are then deduced by purely logical means from an initial body of postulates. The most familiar examples of the synthetic geometry are classical Euclidean geometry and the synthetic projective geometry introduced by Desargues in the 17th century and revived and developed by Carnot, Poncelet, Steiner and others during the 19th century. The power of analytic geometry derives very largely from the fact that it permits the methods of the calculus, and, more generally, of mathematical analysis, to be introduced into geometry, leading in particular to differential geometry (a term, by the way, introduced in 1894 by the Italian geometer Luigi Bianchi). That being the case, the idea of a “synthetic” differential geometry seems elusive: how can differential geometry be placed on a “purely geometric” or “axiomatic” foundation when the apparatus of the calculus seems inextricably involved? To my knowledge there have been two attempts to develop a synthetic differential geometry. The first was initiated by Herbert Busemann in the 1940s, building on earlier work of Paul Finsler. Here the idea was to build a differential geometry that, in its author’s words, “requires no derivatives”: the basic objects in Busemann’s approach are not differentiable manifolds, but metric spaces of a certain type in which the notion of a geodesic can be defined in an intrinsic manner. I shall not have anything more to say about this approach. The second approach, that with which I shall be concerned here, was originally proposed in the 1960s by F. W. Lawvere, who was in fact striving to fashion a decisive axiomatic framework for continuum mechanics..

Links

PhilArchive



    Upload a copy of this work     Papers currently archived: 90,221

External links

  • This entry has no external links. Add one.
Setup an account with your affiliations in order to access resources via your University's proxy server

Through your library

  • Only published works are available at libraries.

Similar books and articles

Relativity and geometry.Roberto Torretti - 1983 - New York: Dover Publications.
Essays on the foundations of mathematics.Moritz Pasch - 2010 - New York: Springer. Edited by Stephen Pollard.
Calculus as Geometry.Frank Arntenius & Cian Dorr - 2012 - In Frank Arntzenius (ed.), Space, Time and Stuff. Oxford University Press.
Remarks on the geometry of visibles.Gordon Belot - 2003 - Philosophical Quarterly 53 (213):581–586.
Geometry and Spatial Intuition: A Genetic Approach.Rene Jagnow - 2003 - Dissertation, Mcgill University (Canada)
Kant's "argument from geometry".Lisa Shabel - 2004 - Journal of the History of Philosophy 42 (2):195-215.
Contemporary Arguments for a Geometry of Visual Experience.Phillip John Meadows - 2009 - European Journal of Philosophy 19 (3):408-430.

Analytics

Added to PP
2010-12-22

Downloads
6 (#1,264,092)

6 months
1 (#1,027,696)

Historical graph of downloads
How can I increase my downloads?

Author's Profile

John L. Bell
University of Western Ontario

Citations of this work

No citations found.

Add more citations

References found in this work

No references found.

Add more references