Pollard Essays on the Foundations of Mathematics by Moritz Pasch

1. Auflage 2010
ISBN: 978-90-481-9416-2
Verlag: Springer-Verlag
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 248 Seiten

Moritz Pasch (1843-1930) is justly celebrated as a key figure in the history of axiomatic geometry. Less well known are his contributions to other areas of foundational research. This volume features English translations of 14 papers Pasch published in the decade 1917-1926. In them, Pasch argues that geometry and, more surprisingly, number theory are branches of empirical science; he provides axioms for the combinatorial reasoning essential to Hilbert's program of consistency proofs; he explores 'implicit definition' (a generalization of definition by abstraction) and indicates how this technique yields an 'empiricist' reconstruction of set theory; he argues that we cannot fully understand the logical structure of mathematics without clearly distinguishing between decidable and undecidable properties; he offers a rare glimpse into the mind of a master of axiomatics, surveying in detail the thought experiments he employed as he struggled to identify fundamental mathematical principles; and much more. This volume will: Give English speakers access to an important body of work from a turbulent and pivotal period in the history of mathematics, help us look beyond the familiar triad of formalism, intuitionism, and logicism, show how deeply we can see with the help of a guide determined to present fundamental mathematical ideas in ways that match our human capacities, will be of interest to graduate students and researchers in logic and the foundations of mathematics.

Autoren/Hrsg.

Pollard, Stephen

Inhaltsverzeichnis

1;Preface;6
2;Contents;8
3;Translator's Introduction;13
3.1;0.1 Pasch of Giessen;13
3.2;0.2 Chains and Lines;15
3.3;0.3 Existence of Lines;16
3.4;0.4 Extracting Lines from Lines;18
3.5;0.5 Justifying Induction;20
3.6;0.6 Initial Segments;21
3.7;0.7 Conformity and Abstraction;23
3.8;0.8 Finite Ordinals;24
3.9;0.9 Addition and Multiplication;27
3.10;0.10 How Much Arithmetic?;30
3.11;0.11 To Justify the Ways of Peano to Men;35
3.12;0.12 Empiricist Arithmetic?;37
3.13;0.13 Ideal Divisors;42
3.14;0.14 Implicit Definition;48
3.15;0.15 Sets;51
3.16;References;54
4;1 Fundamental Questions of Geometry;56
4.1;1.1 Deductive Presentation of Geometry;56
4.2;1.2 Applicability of Geometry;57
4.3;1.3 Empiricist Geometry;57
4.4;1.4 The Levels of Concept Formation;58
4.5;1.5 Proof Procedure;59
4.6;1.6 Core Propositions for Straight Lines and Planes;60
4.7;References;60
5;2 The Decidability Requirement;61
5.1;2.1 Rigid Mathematics;61
5.2;2.2 Kronecker's Requirement;62
5.3;2.3 Core Concepts and Propositions;63
5.4;References;64
6;3 The Origin of the Concept of Number;65
6.1;Introduction;65
6.2;I Preliminary Facts;68
6.3;3.1 Things and Proper Names;68
6.4;3.2 Specifications and Collective Names;69
6.5;3.3 Earlier and Later;69
6.6;3.4 First and Last;70
6.7;3.5 Inferences;71
6.8;3.6 Between;72
6.9;3.7 Immediate Succession;73
6.10;3.8 Immediate Precedence;74
6.11;3.9 The Possibility of Specifications;75
6.12;3.10 Chains of Events;77
6.13;3.11 Lines of Things;78
6.14;3.12 Neighbor-Lines;79
6.15;3.13 Pacing Off a Line;81
6.16;3.14 Application to Collective Names;82
6.17;3.15 Proof by Pacing Off;83
6.18;3.16 Collections of Things;85
6.19;3.17 Implicit Definition;86
6.20;3.18 Consequences of Implicit Definition;87
6.21;3.19 Applications of Proof by Pacing Off;88
6.22;3.20 Backwards Pacing;89
6.23;II Summary of the Preceding Results;89
6.24;3.21 Summary of 3.1;89
6.25;3.22 Summary of 3.2;90
6.26;3.23 Summary of 3.3;90
6.27;3.24 Summary of 3.4;91
6.28;3.25 Summary of 3.5;91
6.29;3.26 Summary of 3.6;91
6.30;3.27 Summary of 3.7;91
6.31;3.28 Summary of 3.8;92
6.32;3.29 Summary of 3.9;92
6.33;3.30 Summary of 3.10;92
6.34;3.31 Summary of 3.11;93
6.35;3.32 Summary of 3.12;94
6.36;3.33 Summary of 3.13;94
6.37;3.34 Summary of 3.14;95
6.38;3.35 Summary of 3.15;95
6.39;3.36 Summary of 3.16;95
6.40;3.37 Summary of 3.17;95
6.41;3.38 Summary of 3.18;96
6.42;3.39 Summary of 3.19;96
6.43;3.40 Summary of 3.20;97
6.44;III Pairings Between Collections;97
6.45;IV The Natural Numbers;98
6.46;Conclusion;101
6.47;References;103
7;4 Implicit Definition and the Proper Grounding of Mathematics;104
7.1;4.1 Introduction;104
7.2;4.2 The Rise of Projective Geometry;105
7.3;4.3 Core Concepts and Core Propositions;106
7.4;4.4 The Fundamental Principle;107
7.5;4.5 Euclidean Definitions;108
7.6;4.6 Some Core Propositions;109
7.7;4.7 Notation for Segments;110
7.8;4.8 Straight Lines;111
7.9;4.9 Implicit Definition;112
7.10;4.10 Justifying Implicit Definitions;113
7.11;4.11 Employing Implicitly Defined Terms;114
7.12;References;116
8;5 Rigid Bodies in Geometry;117
8.1;5.1 Background;117
8.2;5.2 Introduction;119
8.3;5.3 Bodies and Their Shapes;120
8.4;References;124
9;6 Prelude to Geometry: The Essential Ideas;125
9.1;6.1 Introduction;125
9.2;6.2 Composition and Decomposition;126
9.3;6.3 Thickness;127
9.4;6.4 Width;129
9.5;6.5 Constitution of Bodies;130
9.6;6.6 Lines;131
9.7;6.7 Congruent Lines;133
9.8;6.8 Straight Segments;136
9.9;6.9 Length;138
9.10;6.10 Surfaces;140
9.11;6.11 Planar Surfaces;141
9.12;6.12 Exterior Surfaces;143
9.13;6.13 Motion;144
9.14;References;146
10;7 Physical and Mathematical Geometry;147
10.1;7.1 Introduction;147
10.2;7.2 From Physical to Mathematical Points;148
10.3;7.3 Summary;154
10.4;References;155
11;8 Natural Geometry;156
11.1;8.1 Hjelmslev's Complaint;156
11.2;8.2 Empiricism in Geometry;157
11.3;References;157
12;9 The Concept of the Differential;159
12.1;9.1 Introduction;159
12.2;9.2 Preliminaries;160
12.3;9.3 Differences and Difference Quotients;161
12.4;9.4 Limits: Some Background;165
12.5;9.5 Limit Taking;166
12.6;9.6 Infinitely Small and Infinitely Large;168
12.7;9.7 Differentials;170
12.8;9.8 The Inverse of a Function;174
12.9;9.9 Vaihinger's Interpretation of Fermat;176
12.10;References;179
13;10 Reflections on the Proper Grounding of Mathematics I;180
13.1;10.1 General Remarks;180
13.2;10.2 Some Details;183
13.3;References;183
14;11 Concepts and Proofs in Mathematics;188
14.1;11.1 Proof and Definition in Mathematics;188
14.2;11.2 Equality in Mathematics;195
14.3;11.3 The Decidability Requirement in Mathematics;198
14.4;Conclusion;201
14.5;11.4 Approximations of Arbitrary Numbers: The Indefinite Infinite;202
14.6;11.5 The Imaginary in Mathematics;205
14.7;References;208
15;12 Dimension and Space in Mathematics;209
15.1;12.1 Introduction;209
15.2;12.2 Dimensions in Elementary Geometry;210
15.3;12.3 Dimensions in Algebra;212
15.4;12.4 Dimensions in Analytic Geometry;213
15.5;References;217
16;13 Reflections on the Proper Grounding of Mathematics II;218
16.1;13.1 Introduction;218
16.2;13.2 Collections Implicitly Defined;219
16.3;13.3 Unrestricted Sets;220
16.4;13.4 Conclusion;222
16.5;References;222
17;14 The Axiomatic Method in Modern Mathematics;224
17.1;14.1 Introduction;224
17.2;14.2 Statements and Sentences;225
17.3;14.3 A Sequence of Statements;227
17.4;14.4 Names and Formulas;230
17.5;14.5 A Sequence of Statements: Discussion;233
17.6;14.6 Formalization;237
17.7;14.7 Inferences from a Stem;241
17.8;14.8 Conclusion;245
17.9;References;245
18;Index;246