Bridging the Gap: Philosophy, Mathematics, and Physics: Lectures on the Foundations of Science

Cover......Page 1
Title Page......Page 5
Table of Contents......Page 7
Preface......Page 9
Acknowledgements......Page 13
Part I: Logic, Mathematics and Information......Page 15
1. Introduction......Page 17
2. First-Order Languages......Page 18
3. Inference Rules......Page 19
4. Derivations......Page 20
5. Restrictions on Rules......Page 22
6. Minimal, Intuitionistic and Classical Logic......Page 23
7. Logical Reductions......Page 26
8. Commutative Reductions......Page 27
9. The Lemma on Proper Parameters......Page 28
11. EF-Reductions......Page 30
12. Normalization Theorem for NJ......Page 32
13. Normalization Theorem for NK(D)_P......Page 33
14. Normalization Theorem for NK(D)_P......Page 34
15. CR-Reductions......Page 35
16. Normalization Theorem for C......Page 36
17. (CM)-Reduction Rules......Page 37
18. Normalization Theorem for NK(CM)......Page 39
19. Identity of Proofs......Page 40
20. Defects of Natural Deduction......Page 41
21. The System ND_ε......Page 44
22. Normalization Theorem for ND_ε......Page 45
23. Necessity of Restrictions......Page 46
24. The System BK......Page 47
References......Page 49
1. Introduction and Basic Notions......Page 53
2. General Frames and Filtrations......Page 61
3. Relations between Structures......Page 66
3. Canonical Structures and Completeness......Page 72
4. The Logic K......Page 77
Terms......Page 81
References......Page 82
1. Introduction......Page 85
2. Hilbert's Entscheidungsproblem and Universal PROLOG Programs......Page 87
3. Recursive Analogues of Hilbert's Entscheidungsproblem and Limited Computations......Page 92
References......Page 99
Remarks on Hilbert's Program for the Foundation of Mathematics......Page 101
1. General Character of the Program......Page 103
2. Hilbert's Kind of Formalism......Page 104
3. Statement of the Program......Page 105
4. The Borderline between Real and Ideal Propositions......Page 106
5. An Ambiguity in the Motivation of the Program......Page 108
6. Merits and Weaknesses of the Program......Page 110
References......Page 112
1. What's the Use of Foundations?......Page 113
2. Conceptual Clarification......Page 114
3. Interpretation of Problematic Concepts and Principles......Page 116
4. Replacement or Elimination of Problematic Concepts and Principles......Page 122
5. Foundations of Problematic Methods and Results......Page 124
6. Organizational Foundations; Axiomatization......Page 125
7. Reflective Expansion of Concepts and Principles......Page 128
Postscript: Foundational Work and Philosophy of Mathematics......Page 131
Notes......Page 133
References......Page 134
1. Introduction......Page 139
2. The Problem of Other Minds......Page 141
3. Gödelization Arguments......Page 144
4. Loss or Partial Loss of Understanding......Page 145
5. Conclusion......Page 149
Notes......Page 151
References......Page 152
1. Creatura and Pleroma......Page 155
2. The Mental World......Page 157
Part II: Physics and Probability......Page 161
1. Foreword......Page 163
2. Introduction......Page 164
3. Symmetry......Page 165
4. Galilean Relativity......Page 166
5. Special Relativity......Page 167
6. Consequences of Special Relativity: Relativity of Simultaneity......Page 170
7. Further Consequence of Special Relativity: Dilation of Time, Contraction of Length, Equivalence of Mass and Energy, Existence of Antimatter......Page 175
8. Antimatter Conceived as Matter Travelling Backwards in Time......Page 178
9. Paradoxes and Prejudices......Page 180
10. Further Symmetries and Prospects......Page 182
Notes......Page 185
References......Page 187
1. Orthodoxy versus Heresy......Page 189
2. The Quantum Formalism......Page 190
3. Description versus Explanation......Page 193
4. Nonempirical Criteria......Page 194
5. A First Comparison of Quantum Theory with Nonempirical Criteria......Page 196
6. A Digression: Looking for Deterministic Completions of the Formalism......Page 198
7. The Other Nonempirical Criteria......Page 199
8. Modifying the Dynamics: Hints and Problems......Page 201
9. The Model: Quantum Mechanics with Spontaneous Localization......Page 203
10. The Main Implications of QMSL......Page 205
11. Recent Developments and Conclusions......Page 207
References......Page 210
2. Indeterminism......Page 213
3. Nonseparability......Page 217
4. The Einstein–Podolsky–Rosen Paradox......Page 219
References......Page 223
1. Introduction. Basic Notation and Terminology......Page 225
2.1. Convexity......Page 226
2.2. Topologies......Page 227
2.3. Sigma-Convexity......Page 228
2.4. Krein-Milman Property......Page 229
2.6. Finite-Dimensional Case......Page 230
3.3. Sigma-Convexity......Page 231
3.4. Krein-Milman Property......Page 232
4. Continuity of the Mapping T ↦ E_T......Page 233
5. Integration vs. Summation......Page 234
6. The Range of V......Page 235
References......Page 238
1. The Formal Language of Statistical Theories......Page 239
2. The Sentential Logic of Statistical Theories......Page 241
3. The Empirical Semantic of Statistical Sentential Logics......Page 242
4. Mackey's Approach to Quantum Theory as Statistical Theory......Page 246
5. Events, Probability Measures and Event-Valued Measures......Page 248
5.1. State-Event-Probability (SEVP) Structure......Page 249
6. M-Propositions and Events......Page 253
7. The Orthoframe Based on the Phase Space......Page 254
8. State-Effect-Probability (SEFP) Structure......Page 258
9. Complete SEFP Structures......Page 264
9.1. Brouwer-Zadeh Posets in Complete SEFP......Page 266
10. Quantum SEFP as a Unified Axiomatic Approach to Unsharp Quantum Theory......Page 270
References......Page 271
1. Introduction......Page 275
2. The Case of the Heavenly Bodies......Page 276
3. The Land of Anonymity......Page 278
4. Particle Quasets......Page 281
5. Interaction with Possible Worlds......Page 284
6. Denotability and Distinguishability......Page 286
7. Languages and Metalanguages of Physical Theories......Page 288
8. A Quaset-Theoretical Semantics for Microphysics......Page 290
9. Leibnizian and Antileibnizian Particles......Page 293
10. General Conclusions......Page 294
Notes......Page 295
References......Page 296
1. A General Approach to Opinion Change......Page 299
2. A Symmetry Argument for Conditionalization......Page 302
3. Probability as Measure: The History......Page 305
4. Symmetry: An Argument for Jeffrey Conditionalization......Page 311
5. Levi's Objection: A Simulated Horse Race......Page 317
6. General Probability Kinematics and Entropy......Page 319
7. Conclusion: Normal Rule Following......Page 326
Notes......Page 327
Index of Names......Page 329
Boston Studies in the Philosophy of Science......Page 335