FOM: Foundations of Geometry
dubucs
dubucs at ext.jussieu.fr
Fri Feb 12 00:02:14 EST 1999
From a logical viewpoint, the following book is still highly
informative: K. Borsuk & W. Szmielew, "Foundations of Geometry", Amsterdam,
North-Holland, 1960 (English translation from a former Polish version).
Here some questions and possible topics of discussion about FOG,
ranging from philosophical to technical:
(i) What is the status of the isomorphisms between R and the
geometrical lines ? between R and the "physical lines": testable
conjectures versus mere conventions ?
(ii) Geometry is often constrasted with logic and arithmetic as a
science specifically dealing with continuum. But at what extent are the
continuity assumptions really necessary to make geometry applicable to the
description of the physical world ? Could not the physical world be
suitably represented in a discretized way ?
(iii) Is there definite results concerning the geometrical
structure of the world (modulo some suitable physical interpretation of the
primitive predicates of the geometry), e.g. euclidean versus non-euclidean ?
(iv) After a long period of opprobrium, which goes back to
Russell's famous "Principles of Mathematics", there is an increasingly
fashionable tendancy to consider again geometric reasoning by means of
diagrams or figures as somehow irreducible to logical inference. Could we
get to the bottom of the matter ?
Jacques Dubucs
Institut d'Histoire et Philosophie des Sciences et des Techniques
Universite de Paris I Sorbonne Pantheon
13, rue du Four, 75006 Paris
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