1830--1930: a century of geometry. Epistemology, history and mathematics. Selection of articles based on lectures presented at the international conference
From the Foreword, ``While geometry holds a leading position in modern mathematics and physics, epistemology and the philosophy and history of science still seem to concentrate upon the problem of foundations, formal logic and classical analysis, or the epistemological status of quantum mechanics. But this is obviously insufficient. We therefore urgently need to revisit the philosophical problems concerning the status of geometry and to renew our present epistemological trends with some of the rich reflexions developed at the beginning of this century. The conference took risks by gathering together historians, philosophers, mathematicians and physicists to re-examine the recent history of geometry in the light of the contemporary developments of mathematics and physics, but we think that the talks and discussions showed that this mixture generated more light than heat, as we had hoped.'' Contents. Part 1. Pluralization of geometry: New foundations and continuity of problems: \textit{Christian Houzel}, The birth of non-Euclidean geometry (3--21); \textit{Erhard Scholz}, Riemann's vision of a new approach to geometry (22--34); \textit{J. J. Gray}, Poincaré and Klein -- groups and geometries (35--44); \textit{David E. Rowe}, Klein, Lie, and the ``Erlanger Programm'' (45--54); \textit{Bernard Teissier}, Apparent contours from Monge to Todd (55--62). Part 2. Historical and epistemological aspects of the connexion between physics and geometry: \textit{Luciano Boi}, L'espace: concept abstrait et/ou physique; la géométrie entre formalisation mathématique et étude de la nature [Space: abstract and/or physical concept; geometry in mathematical formalization and in the study of nature] (65--90); \textit{F. Balibar}, Geometrie und Erfahrung (English) [Geometry and experience] (91--97); \textit{Ruth Farwell} and \textit{Christopher Knee}, The geometric challenge of Riemann and Clifford (98--106); \textit{Giorgio Israel}, Poincaré et Enriques: deux points de vue différents sur les relations entre géométrie, mécanique et physique (107--126); \textit{Michel Paty}, Physical geometry and special relativity. Einstein et Poincaré (127--149); \textit{Jean-Pierre Bourguignon}, Transport parallèle et connexions en géométrie et en physique [Parallel transport and connections in geometry and physics] (150--164). Part 3. Formalism and intuition: \textit{Hourya Sinaceur}, De la géométrie formelle à l'algèbre abstraite [From formal geometry to abstract algebra] (167--174); \textit{Ludovico Geymonat}, Le principe de dulatié: sa signification historique et epistémologique [The duality principle: its historical and epistemological meaning] (175--177); \textit{Gilles Gaston Granger}, The formal and the transcendental in mathematics (178--183); \textit{René Thom}, Un panorama des mathématiques [A panorama of mathematics] (184--191); \textit{Klaus Volkert}, Mathematical progress as synthesis of intuition and calculus (192--198). Part 4. The philosophical problem of space: \textit{Hans Freudenthal}, What is space? (201--204); \textit{Dominique Flament}, La ``lineale Ausdehnungslehre'' (1844) de Hermann Günther Grassmann [The Lineale Ausdehnungslehre (1844) of Hermann Günther Grassmann] (205--221); \textit{Gilles Châtelet}, La capture de l'extension comme dialectique géométrique: dimension et puissance selon l'Ausdehnung de Grassmann (1844) [The apprehension of extension as geometrical dialectic: dimension and power according to Grassmann's Ausdehnung (1844)] (222--244); \textit{Gerhard Heinzmann}, Helmholtz and Poincaré's considerations on the genesis of geometry (245--249); \textit{Jean-Michel Salanskis}, Le continu contre l'espace [The continuum versus space] (250--264). Part 5. Some insights about modern physics: \textit{Gilles Cohen-Tannoudji}, Geometrical concepts in quantum physics (267--269); \textit{Tullio Regge}, Physics and differential geometry (270--272); \textit{Jean Petitot}, Actuality of transcendental aesthetics for modern physics (273--304). The articles will not be reviewed individually.
scientific article; zbMATH DE number 108065