The Incredible Shrinking Manifold
Graduate studies at Western
|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..|
|Keywords||No keywords specified (fix it)|
No categories specified
(categorize this paper)
|External links||This entry has no external links. Add one.|
|Through your library||Only published papers are available at libraries|
Similar books and articles
John L. Bell, Two Approaches to Modelling the Universe: Synthetic Differential Geometry and Frame-Valued Sets.
Roberto Torretti (1983/1996). Relativity and Geometry. Dover Publications.
Sébastien Gandon (2009). Toward a Topic-Specific Logicism? Russell's Theory of Geometry in the Principles of Mathematics. Philosophia Mathematica 17 (1):35-72.
Frank Arntenius & Cian Dorr (2012). Calculus as Geometry. In Frank Arntzenius (ed.), Space, Time and Stuff. Oxford University Press.
D. Farnsworth (ed.) (1972). Methods of Local and Global Differential Geometry in General Relativity. New York,Springer-Verlag.
Gordon Belot (2003). Remarks on the Geometry of Visibles. Philosophical Quarterly 53 (213):581–586.
Lisa Shabel (2004). Kant's "Argument From Geometry". Journal of the History of Philosophy 42 (2):195-215.
Emil Badici (2010). On the Compatibility Between Euclidean Geometry and Hume's Denial of Infinite Divisibility. Hume Studies 34 (2):231-244.
Ladislav Kvasz (1998). History of Geometry and the Development of the Form of its Language. Synthese 116 (2):141–186.
Phillip John Meadows (2011). Contemporary Arguments for a Geometry of Visual Experience. European Journal of Philosophy 19 (3):408-430.
Jan Platvono (1997). Formalization of Hilbert's Geometry of Incidence and Parallelism. Synthese 110 (1):127-141.
Jan von Plato (1997). Formalization of Hilbert's Geometry of Incidence and Parallelism. Synthese 110 (1):127-141.
Added to index2010-12-22
Total downloads6 ( #154,923 of 739,575 )
Recent downloads (6 months)1 ( #61,680 of 739,575 )
How can I increase my downloads?