Essays on the Foundations of Mathematics by Moritz Pasch (eBook)

Preface 6
Contents 8
Translator's Introduction 13
0.1 Pasch of Giessen 13
0.2 Chains and Lines 15
0.3 Existence of Lines 16
0.4 Extracting Lines from Lines 18
0.5 Justifying Induction 20
0.6 Initial Segments 21
0.7 Conformity and Abstraction 23
0.8 Finite Ordinals 24
0.9 Addition and Multiplication 27
0.10 How Much Arithmetic? 30
0.11 To Justify the Ways of Peano to Men 35
0.12 Empiricist Arithmetic? 37
0.13 Ideal Divisors 42
0.14 Implicit Definition 48
0.15 Sets 51
References 54
1 Fundamental Questions of Geometry 56
1.1 Deductive Presentation of Geometry 56
1.2 Applicability of Geometry 57
1.3 Empiricist Geometry 57
1.4 The Levels of Concept Formation 58
1.5 Proof Procedure 59
1.6 Core Propositions for Straight Lines and Planes 60
References 60
2 The Decidability Requirement 61
2.1 Rigid Mathematics 61
2.2 Kronecker's Requirement 62
2.3 Core Concepts and Propositions 63
References 64
3 The Origin of the Concept of Number 65
Introduction 65
I Preliminary Facts 68
3.1 Things and Proper Names 68
3.2 Specifications and Collective Names 69
3.3 Earlier and Later 69
3.4 First and Last 70
3.5 Inferences 71
3.6 Between 72
3.7 Immediate Succession 73
3.8 Immediate Precedence 74
3.9 The Possibility of Specifications 75
3.10 Chains of Events 77
3.11 Lines of Things 78
3.12 Neighbor-Lines 79
3.13 Pacing Off a Line 81
3.14 Application to Collective Names 82
3.15 Proof by Pacing Off 83
3.16 Collections of Things 85
3.17 Implicit Definition 86
3.18 Consequences of Implicit Definition 87
3.19 Applications of Proof by Pacing Off 88
3.20 Backwards Pacing 89
II Summary of the Preceding Results 89
3.21 Summary of 3.1 89
3.22 Summary of 3.2 90
3.23 Summary of 3.3 90
3.24 Summary of 3.4 91
3.25 Summary of 3.5 91
3.26 Summary of 3.6 91
3.27 Summary of 3.7 91
3.28 Summary of 3.8 92
3.29 Summary of 3.9 92
3.30 Summary of 3.10 92
3.31 Summary of 3.11 93
3.32 Summary of 3.12 94
3.33 Summary of 3.13 94
3.34 Summary of 3.14 95
3.35 Summary of 3.15 95
3.36 Summary of 3.16 95
3.37 Summary of 3.17 95
3.38 Summary of 3.18 96
3.39 Summary of 3.19 96
3.40 Summary of 3.20 97
III Pairings Between Collections 97
IV The Natural Numbers 98
Conclusion 101
References 103
4 Implicit Definition and the Proper Grounding of Mathematics 104
4.1 Introduction 104
4.2 The Rise of Projective Geometry 105
4.3 Core Concepts and Core Propositions 106
4.4 The Fundamental Principle 107
4.5 Euclidean Definitions 108
4.6 Some Core Propositions 109
4.7 Notation for Segments 110
4.8 Straight Lines 111
4.9 Implicit Definition 112
4.10 Justifying Implicit Definitions 113
4.11 Employing Implicitly Defined Terms 114
References 116
5 Rigid Bodies in Geometry 117
5.1 Background 117
5.2 Introduction 119
5.3 Bodies and Their Shapes 120
References 124
6 Prelude to Geometry: The Essential Ideas 125
6.1 Introduction 125
6.2 Composition and Decomposition 126
6.3 Thickness 127
6.4 Width 129
6.5 Constitution of Bodies 130
6.6 Lines 131
6.7 Congruent Lines 133
6.8 Straight Segments 136
6.9 Length 138
6.10 Surfaces 140
6.11 Planar Surfaces 141
6.12 Exterior Surfaces 143
6.13 Motion 144
References 146
7 Physical and Mathematical Geometry 147
7.1 Introduction 147
7.2 From Physical to Mathematical Points 148
7.3 Summary 154
References 155
8 Natural Geometry 156
8.1 Hjelmslev's Complaint 156
8.2 Empiricism in Geometry 157
References 157
9 The Concept of the Differential 159
9.1 Introduction 159
9.2 Preliminaries 160
9.3 Differences and Difference Quotients 161
9.4 Limits: Some Background 165
9.5 Limit Taking 166
9.6 Infinitely Small and Infinitely Large 168
9.7 Differentials 170
9.8 The Inverse of a Function 174
9.9 Vaihinger's Interpretation of Fermat 176
References 179
10 Reflections on the Proper Grounding of Mathematics I 180
10.1 General Remarks 180
10.2 Some Details 183
References 183
11 Concepts and Proofs in Mathematics 188
11.1 Proof and Definition in Mathematics 188
11.2 Equality in Mathematics 195
11.3 The Decidability Requirement in Mathematics 198
Conclusion 201
11.4 Approximations of Arbitrary Numbers: The Indefinite Infinite 202
11.5 The Imaginary in Mathematics 205
References 208
12 Dimension and Space in Mathematics 209
12.1 Introduction 209
12.2 Dimensions in Elementary Geometry 210
12.3 Dimensions in Algebra 212
12.4 Dimensions in Analytic Geometry 213
References 217
13 Reflections on the Proper Grounding of Mathematics II 218
13.1 Introduction 218
13.2 Collections Implicitly Defined 219
13.3 Unrestricted Sets 220
13.4 Conclusion 222
References 222
14 The Axiomatic Method in Modern Mathematics 224
14.1 Introduction 224
14.2 Statements and Sentences 225
14.3 A Sequence of Statements 227
14.4 Names and Formulas 230
14.5 A Sequence of Statements: Discussion 233
14.6 Formalization 237
14.7 Inferences from a Stem 241
14.8 Conclusion 245
References 245
Index 246