Bridging the Gap: Philosophy, Mathematics, and Physics: Lectures on the Foundations of Science (Boston Studies in the Philosophy of Science)

"Foundational questions in logic, mathematics, computer science and physics are constant sources of epistemological debate in contemporary philosophy. To what extent is the transfinite part of mathematics completely trustworthy? Why is there a general 'malaise' concerning the logical approach to the foundations of mathematics? What is the role of symmetry in physics? Is it possible to build a coherent worldview compatible with a macroobjectivistic position and based on the quantum picture of the world? What account can be given of opinion change in the light of new evidence?" "These are some of the questions discussed in this volume, which collects 14 lectures on the foundations of science given at the School of Philosophy of Science, Trieste, October 1989. The volume will be of particular interest to any student or scholar engaged in interdisciplinary research into the foundations of science in the context of contemporary debates."--BOOK JACKET. 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 Why is there a general 'malaise' concerning the logical approach to the foundations of mathematics? What is the role of symmetry in physics? To what extent is the transfinite part of mathematics completely trustworthy? This volume discusses some of these questions which collects 14 lectures on the foundation of science.