Philosophy of Geometry from Riemann to Poincaré

Geometry has fascinated philosophers since the days of Thales and Pythagoras. In the 17th and 18th centuries it provided a paradigm of knowledge after which some thinkers tried to pattern their own metaphysical systems. But after the discovery of non-Euclidean geometries in the 19th century, the nature and scope of geometry became a bone of contention. Philosophical concern with geometry increased in the 1920's after Einstein used Riemannian geometry in his theory of gravitation. During the last fifteen or twenty years, renewed interest in the latter theory -prompted by advances in cosmology -has brought geometry once again to the forefront of philosophical discussion. The issues at stake in the current epistemological debate about geometry can only be understood in the light of history, and, in fact, most recent works on the subject include historical material. In this book, I try to give a selective critical survey of modern philosophy of geometry during its seminal period, which can be said to have begun shortly after 1850 with Riemann's generalized conception of space and to achieve some sort of completion at the turn of the century with Hilbert's axiomatics and Poincare's conventionalism. The philosophy of geometry of Einstein and his contemporaries will be the subject of another book. The book is divided into four chapters. Chapter 1 provides back­ ground information about the history of science and philosophy.

Contenu

1 / Background.- 1.0.1 Greek Geometry and Philosophy.- 1.0.2 Geometry in Greek Natural Science.- 1.0.3 Modern Science and the Metaphysical Idea of Space.- 1.0.4 Descartes' Method of Coordinates.- 2 / Non-Euclidean Geometries.- 2.1 Parallels.- 2.1.1 Euclid's Fifth Postulate.- 2.1.2 Greek Commentators.- 2.1.3 Wallis and Saccheri.- 2.1.4 Johann Heinrich Lambert.- 2.1.5 The Discovery of Non-Euclidean Geometry.- 2.1.6 Some Results of Bolyai-Lobachevsky Geometry.- 2.1.7 The Philosophical Outlook of the Founders of Non-Euclidean Geometry.- 2.2 Manifolds.- 2.2.1 Introduction.- 2.2.2 Curves and their Curvature.- 2.2.3 Gaussian Curvature of Surfaces.- 2.2.4 Gauss' Theorema Egregium and the Intrinsic Geometry of Surfaces.- 2.2.5 Riemann's Problem of Space and Geometry.- 2.2.6 The Concept of a Manifold.- 2.2.7 The Tangent Space.- 2.2.8 Riemannian Manifolds, Metrics and Curvature.- 2.2.9 Riemann's Speculations about Physical Space.- 2.2.10 Riemann and Herbart. Grassmann.- 2.3 Projective Geometry and Projective Metrics.- 2.3.1 Introduction.- 2.3.2 Projective Geometry: An Intuitive Approach.- 2.3.3 Projective Geometry: A Numerical Interpretation.- 2.3.4 Projective Transformations.- 2.3.5 Cross-ratio.- 2.3.6 Projective Metrics.- 2.3.7 Models.- 2.3.8 Transformation Groups and Klein's Erlangen Programme.- 2.3.9 Projective Coordinates for Intuitive Space.- 2.3.10 Klein's View of Intuition and the Problem of Space-Forms.- 3 / Foundations.- 3.1 Helmholtz's Problem of Space.- 3.1.1 Helmholtz and Riemann.- 3.1.2 The Facts which Lie at the Foundation of Geometry.- 3.1.3 Helmholtz's Philosophy of Geometry.- 3.1.4 Lie Groups.- 3.1.5 Lie's Solution of Helmholtz's Problem.- 3.1.6 Poincaré and Killing on the Foundations of Geometry.- 3.1.7 Hilbert's Group-Theoretical Characterization of the Euclidean Plane.- 3.2 Axiomatics.- 3.2.1 The Beginnings of Modern Geometrical Axiomatics.- 3.2.2 Why are Axiomatic Theories Naturally Abstract?.- 3.2.3 Stewart, Grassmann, Plücker.- 3.2.4 Geometrical Axiomatics before Pasch.- 3.2.5 Moritz Pasch.- 3.2.6 Giuseppe Peano.- 3.2.7 The Italian School. Pieri. Padoa.- 3.2.8 Hilbert's Grundlagen.- 3.2.9 Geometrical Axiomatics after Hilbert.- 3.2.10 Axioms and Definitions. Frege's Criticism of Hilbert.- 4 / Empiricism, Apriorism, Conventionalism.- 4.1 Empiricism in Geometry.- 4.1.1 John Stuart Mill.- 4.1.2 Friedrich Ueberweg.- 4.1.3 Benno Erdmann.- 4.1.4 Auguste Calinon.- 4.1.5 Ernst Mach.- 4.2 The Uproar of Boeotians.- 4.2.1 Hermann Lotze.- 4.2.2 Wilhelm Wundt.- 4.2.3 Charles Renouvier.- 4.2.4 Joseph Delboeuf.- 4.3 Russell's Apriorism of 1897.- 4.3.1 The Transcendental Approach.- 4.3.2 The 'Axioms of Projective Geometry'.- 4.3.3 Metrics and Quantity.- 4.3.4 The Axiom of Distance.- 4.3.5 The Axiom of Free Mobility.- 4.3.6 A Geometrical Experiment.- 4.3.7 Multidimensional Series.- 4.4 Henri Poincaré.- 4.4.1 Poincaré's Conventionalism.- 4.4.2 Max Black's Interpretation of Poincaré's Philosophy of Geometry.- 4.4.3 Poincaré's Criticism of Apriorism and Empiricism.- 4.4.4 The Conventionality of Metrics.- 4.4.5 The Genesis of Geometry.- 4.5.6 The Definition of Dimension Number.- 1. Mappings.- 2. Algebraic Structures. Groups.- 3. Topologies.- 4. Differentiable Manifolds.- Notes.- To Chapter 1.- To Chapter 2.- 2.1.- 2.2.- 2.3.- To Chapter 3.- 3.1.- 3.2.- To Chapter 4.- 4.1.- 4.2.- 4.3.- 4.4.- References.