Philosophy of Geometry from Riemann to Poincaré / Edition 1

Philosophy of Geometry from Riemann to Poincaré / Edition 1

by R. Torretti
     
 

Product Details

ISBN-13:
9789027709202
Publisher:
Springer Netherlands
Publication date:
11/30/1978
Series:
Episteme Series, #7
Edition description:
1978
Pages:
461
Product dimensions:
6.14(w) x 9.21(h) x 1.06(d)

Table of Contents

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.

Read More

Customer Reviews

Average Review:

Write a Review

and post it to your social network

     

Most Helpful Customer Reviews

See all customer reviews >