A New Foundation of Physical Theories (eBook)

Foreword 5
Vorwort 6
Preface 7
Acknowledgments 8
Contents 9
Intention of the Book 12
Motivation and Problem Setting 12
Current State of Art 14
Main Ideas of Our Approach 14
Outline of the Book 16
Part I A New Form of Physical Theory 19
1 Reality 20
1.1 The Structure of Reality 20
1.2 The Physical Reality 21
1.3 Fairy Tales 25
2 Building of a Mathematical Theory 26
2.1 Formal Language 26
2.2 Axioms and Proofs 28
2.3 Logics 30
2.4 Set Theory 36
3 From Reality to Mathematics 41
3.1 Recording Process 42
3.2 Mathematization Process 54
3.3 Idealization Process 61
4 Species of Structures and Axiomatic Basis of a PT 70
4.1 Mathematical Structures 71
4.2 Deduction of Structures 74
4.3 Axiomatic Basis and Fairy Tales 80
4.4 Pure Laws of Nature 83
4.5 Change of the Mathematical Form of an Axiomatic Basis 85
4.6 Inaccuracy Sets and Uniform Structures 92
4.7 Do the Laws of Nature” Describe Realities? 99
4.8 Classi.cation of Laws of Nature 103
4.9 Skeleton and Uninterpreted Theories 108
5 Relations Between Various PTs 112
5.1 Relations Between Two PTs with the Same Application Domain 113
5.2 Relations Between Two PTs with a Common Part of an Application Domain 118
5.3 Pre-theories 119
5.4 Relations Between PTs with Di.erent Application Domains 123
5.5 Approximation Theories 124
5.6 The Network of PTs 125
6 Real and Possible as Physical Concepts 128
6.1 Closed Theories 131
6.2 Physical Systems 135
6.3 New Concepts in a PT 138
6.4 Indirect Measurements 140
6.5 Classi.cations and Interpretations 145
6.6 The Reality Domain of a PT 150
Part II Examples of Simple Theories 152
A A Description of the Surface of the Earth, or of a Round Table 153
The Physical Reality 153
The Basic Language Bl 153
The Application Domain Ap 153
The Text A 154
The Basic Mathematical Theory MT 154
The Standard Mathematical Theory MTT 154
The Enrichment of MTT by A 155
The Idealized Mathematical Theory MT. 155
The Enrichment of MT. by A 156
The Fundamental Domain G 157
The Axiomatic Basis MT 158
The Inaccuracy Sets and Uniform Structures 159
The Description of the Reality 160
A Closed Theory 161
New Concepts 162
Classifications and Interpretations of the New Concepts 162
B A Simpli.ed Example of Newton’s Mechanics 164
The Physical Reality 164
The Basic Language Bl 164
The Basic Mathematical Theory MT 166
The Standard Mathematical Theory MTT 166
The Idealized Mathematical Theory MT. 166
C The Structure of the Human Species 171
The Standard Mathematical Theory MTT 172
The Idealized Mathematical Theory MT. 173
References 176
List of Symbols 177
Index 179