The foundations of mathematics : a contribution to the philosophy of geometry

in This Brief Treatise, Carus Traces The Roots Of His Belief In The Philosophical Basis For Mathematics And Analyzes That Basis After A Historical Overview Of Euclid And His Successors. He Then Examines His Base Argument And Proceeds To A Study Of Different Geometrical Systems, All Pulled Together In His Epilogue, Which Examines Matter, Mathematics, And, Ultimately, The Nature Of God. Title -- The Foundations of Mathematics: A Contribution to the Philosophy of Geometry......Page 1 Copyright......Page 2 Contents......Page 3 Axioms and the Axiom of Parallels......Page 5 Metageometry......Page 9 Precursors......Page 11 Gauss......Page 15 Riemann......Page 19 Lobatchevsky......Page 24 Bolyai......Page 26 Later Geometricians......Page 28 Grassman......Page 31 Euclid Still Unimpaired......Page 35 The Philosophical Problem......Page 39 Transcendentalism and Empiricism......Page 42 The A Priori and the Purely Formal......Page 44 Anyness and its Universality......Page 50 Apriority of Different Degrees......Page 53 Space as a Spread of Motion......Page 60 Uniqueness of Pure Space......Page 65 Mathematical Space and Physiological Space......Page 67 Homogeneity of Space due to Abstraction......Page 70 Even Boundaries as Standards of Measurement......Page 73 The Straight Line Indispensable......Page 76 The Superreal......Page 80 Discrete Units and the Continuum......Page 82 Different Geometrical Systems......Page 86 Tridimensionality......Page 88 Three a Concept of Boundary......Page 92 Space of Four Dimensions......Page 94 The Apparent Arbitrariness of the A Priori......Page 100 Definiteness of Construction......Page 103 One Space, but Various Systems of Space-Measurement......Page 108 Fictitious Spaces and the Apriority of all Systems of Space-Measurement......Page 113 Infinitude......Page 120 Geometry Remains A Priori......Page 123 Sense-Experience and Space......Page 126 The Teaching of Mathematics......Page 131 Epilogue......Page 136 Index......Page 143 Title -- The Foundations of Mathematics: A Contribution to the Philosophy of Geometry 1 Copyright 2 Contents 3 The Search for the Foundations of Geometry: Historical Sketch 5 Axioms and the Axiom of Parallels 5 Metageometry 9 Precursors 11 Gauss 15 Riemann 19 Lobatchevsky 24 Bolyai 26 Later Geometricians 28 Grassman 31 Euclid Still Unimpaired 35 The Philosophical Basis of Mathematics 39 The Philosophical Problem 39 Transcendentalism and Empiricism 42 The A Priori and the Purely Formal 44 Anyness and its Universality 50 Apriority of Different Degrees 53 Space as a Spread of Motion 60 Uniqueness of Pure Space 65 Mathematical Space and Physiological Space 67 Homogeneity of Space due to Abstraction 70 Even Boundaries as Standards of Measurement 73 The Straight Line Indispensable 76 The Superreal 80 Discrete Units and the Continuum 82 Mathematics and Metageometry 86 Different Geometrical Systems 86 Tridimensionality 88 Three a Concept of Boundary 92 Space of Four Dimensions 94 The Apparent Arbitrariness of the A Priori 100 Definiteness of Construction 103 One Space, but Various Systems of Space-Measurement 108 Fictitious Spaces and the Apriority of all Systems of Space-Measurement 113 Infinitude 120 Geometry Remains A Priori 123 Sense-Experience and Space 126 The Teaching of Mathematics 131 Epilogue 136 Index 143