CONTENTS
PAGE | |
Henri Poincaré | xi |
Author’s Preface to the Translation | 3 |
SCIENCE AND HYPOTHESIS | |
Introduction by Royce | 9 |
Introduction | 27 |
Part I. Number and Magnitude | |
CHAPTER I.—On the Nature of Mathematical Reasoning | 31 |
Syllogistie Deduction | 31 |
Verification and Proof | 32 |
Elements of Arithmetic | 33 |
Reasoning by Recurrence | 37 |
Induction | 40 |
Mathematical Construction | 41 |
CHAPTER II.—Mathematical Magnitude and Experience | 43 |
Definition of Incommensurables | 44 |
The Physical Continuum | 46 |
Creation of the Mathematical Continuum | 46 |
Measurable Magnitude | 49 |
Various Remarks (Curves without Tangents) | 50 |
The Physical Continuum of Several Dimensions | 52 |
The Mathematical Continuum of Several Dimensions | 58 |
Part II. Space | |
CHAPTER III.—The Non-Euclidean Geometries | 55 |
The Bolyai-Lobachevski Geometry | 56 |
Riemann’s Geometry | 57 |
The Surfaces of Constant Curvature | 58 |
Interpretation of Non-Euclidean Geometries | 59 |
The Implicit Axioms | 60 |
The Fourth Geometry | 62 |
Lie’s Theorem | 62 |
Riemann’s Geometries | 63 |
On the Nature-of Axioms | 63 |
CHAPTER IV.—Space and Geometry | 66 |
Geometric Space and Perceptual Space | 66 |
Visual Space | 67 |
Tactile Space and Motor Space | 68 |
Characteristics of Perceptual Space | 69 |
Change of State and Change of Position | 70 |
Conditions of Compensation | 72 |