of Epistemology III.- 1. The Chasm between S&T and the Humanities.- 2. Bridging the Chasm.- 3. Towards a Useful PS&T.- 4. Concluding Remarks.- 1. Formal Science: From Logic to Mathematics.- 1. Generalities.- 1.1 Two Main Types of Research Field.- 1.2 Some Peculiarities of Mathematics.- 2. Mathematics and Reality.- 2.1 Conceptual Existence.- 2.2 Mathematics and Reality.- 3. Logic.- 3.1 Logic Lato Sensu.- 3.2 Non-standard Logics.- 4. Pure and Applied Mathematics.- 4.1 Applications of Mathematics.- 4.2 An Example: Probability.- 5. Foundations and Philosophy.- 5.1 Foundations of Mathematics.- 5.2 Philosophies of Mathematics.- 6. Concluding Remarks.- 2. Physical Science: From Physics to Earth Science.- 1. Preliminaries.- 1.1 Physical Quantity, Convention, Measurement.- 1.2 Theory, Metatheory, Protophysics.- 2. Two Classics.- 2.1 Classical Mechanics.- 2.2 Statistical Mechanics.- 3. Two Relativities.- 3.1 Special Relativity.- 3.2 General Relativity.- 4. Quantons.- 4.1 Classons and Quantons.- 4.2 The State Function and its Referents.- 5. Chance.- 5.1 Probability.- 5.2 Double Slit and Double Logic.- 6. Realism and Classicism.- 6.1 Measurement and Projection.- 6.2 Hidden Variables, Separability, and Realism.- 7. Chemistry.- 7.1 Philosophy and Chemistry.- 7.2 Is Chemistry Reducible to Physics?.- 8. Megaphysics.- 8.1 Earth Sciences.- 8.2 Cosmology.- 9. Concluding Remarks.- Index of Names.- Index of Subjects.
of Epistemology III.- 1. The Chasm between S&T and the Humanities.- 2. Bridging the Chasm.- 3. Towards a Useful PS&T.- 4. Concluding Remarks.- 1. Formal Science: From Logic to Mathematics.- 1. Generalities.- 1.1 Two Main Types of Research Field.- 1.2 Some Peculiarities of Mathematics.- 2. Mathematics and Reality.- 2.1 Conceptual Existence.- 2.2 Mathematics and Reality.- 3. Logic.- 3.1 Logic Lato Sensu.- 3.2 Non-standard Logics.- 4. Pure and Applied Mathematics.- 4.1 Applications of Mathematics.- 4.2 An Example: Probability.- 5. Foundations and Philosophy.- 5.1 Foundations of Mathematics.- 5.2 Philosophies of Mathematics.- 6. Concluding Remarks.- 2. Physical Science: From Physics to Earth Science.- 1. Preliminaries.- 1.1 Physical Quantity, Convention, Measurement.- 1.2 Theory, Metatheory, Protophysics.- 2. Two Classics.- 2.1 Classical Mechanics.- 2.2 Statistical Mechanics.- 3. Two Relativities.- 3.1 Special Relativity.- 3.2 General Relativity.- 4. Quantons.- 4.1 Classons and Quantons.- 4.2 The State Function and its Referents.- 5. Chance.- 5.1 Probability.- 5.2 Double Slit and Double Logic.- 6. Realism and Classicism.- 6.1 Measurement and Projection.- 6.2 Hidden Variables, Separability, and Realism.- 7. Chemistry.- 7.1 Philosophy and Chemistry.- 7.2 Is Chemistry Reducible to Physics?.- 8. Megaphysics.- 8.1 Earth Sciences.- 8.2 Cosmology.- 9. Concluding Remarks.- Index of Names.- Index of Subjects.
Volume 1: Introduction P. Humphreys. Part I: Probability. Some Contributions to Formal Theory of Probability K. Popper, D. Miller. Elementary Non-Archimedean Representations of Probability for Decision Theory and Games P.J. Hammond. Random Sequences and Hypotheses Tests R. Chuaqui. Changing Probability Judgements I. Levi. Upper and Lower Probability T.L. Fine. Some Connections between Epistemic Logic and the Theory of Nonadditive Probability P. Mongin. On the Properties of Conditional Independence W. Spohn. Qualitative Probabilities Revisited Z. Domotor. The Monks' Vote: a Dialogue on Unidimensional Probabilistic Geometry J.-C. Falmagne. Part II: Probabilistic Causality. Probabilistic Causation without Probability P.W. Holland. Causal Tendency, Necessitivity and Sufficientivity: an Updated Review I.J. Good. Practical Causal Generalizations E.W. Adams. In Place of Regression C. Glymour, P. Spirtes, R. Scheines. Testing Probabilistic Causality D. Costantini. Psychologistic Aspects of Suppes' Definition of Causality P. Legrenzi, M. Sonino. Name Index. Subject Index. Volume 2: Part III: Philosophy of Physics. Probability and Quantum Theory B. Loewer. Schrodinger's Version of EPR, and its Problems A. Fine. Classical Field Magnitudes J. Vuillemin. Quantity, Representation and Geometry B. Mundy. Numerical Experimentation P. Humphreys. Part IV: Theory Structure. Theories and Theoretical Models R. Wojcicki. Suppes Predicates and the Construction of Unsolvable Problems in the Axiomatized Sciences N.C.A. da Costa, F.A. Doria. Structural Explanation J.D. Sneed. Part V: Measurement Theory. Fifteen Problems concerning the Representational Theory of Measurement R.D. Luce, L. Narens. The Meaningfulness of Ordinal Comparisons for General Order Relational Systems F.S. Roberts, Z.S. Rosenbaum. Theories as Nets: the Case of Combinatorial Measurement Theory C.U. Moulines, J.A. Diez. Name Index. Subject Index. Volume 3: Part VI: Philosophy of Language and Logic. Patrick Suppes' Contribution to the Philosophy of Language D. Follesdal. Open Problems in Relational Grammar M. Bottner. A Variable-Free Logic for Anaphora W.C. Purdy. Is Snow White? J. Moravcsik. Can there be Reasons for Putting Limitations on Classical Logic? P. Weingartner. Quantum Logic as a Logic of Identification J. Hintikka, I. Halonen. Logic and Probability in Quantum Mechanics M.L. dalla Chiara, R. Giuntini. Part VII: Learning Theory, Action Theory, and Robotics. From Stimulus-Sampling to Array-Similarity Theory W.K. Estes. Action as Seeing to it that Something is the Case R. Tuomela, G. Sandu. Command Satisfaction and the Acquisition of Habits C. Crangle. Part VIII: General Philosophy of Science. Some Observations on Patrick Suppes' Philosophy of Science M.C. Galavotti. Epilogue. Postscript P. Suppes. Chronological and Topical Bibliography of Patrick Suppes' Publications. Name Index. Subject Index.
The Statistics of Non-Boolean Event Structures.- Possibility and Probability.- Some Remarks on Hamiltonian Systems and Quantum Mechanics.- The Possibility Structure of Physical Systems.- Quantum Mechanical Physical Quantities as Random Variables.- On the Interference of Probabilities.- Classical and Quantum Probability and Set Theory.- Discussion.- A Generalized Measure and Probability Theory for the Physical Sciences.- Discussion.- Quantum Logic, Convexity, and a Necker-Cube Experiment.- On the Applicability of the Probability Concept to Quantum Theory.- Discussion.- A Mathematical Setting for Inductive Reasoning.- Discussion.- Classical Statistical Mechanics Versus Quantal Statistical Thermodynamics: A Study in Contrasts.- Discussion.- A Semantic Analysis of Niels Bohr's Philosophy of Quantum Theory.
The difficulties of the old quantum theory were largely dispelled by the development of quantum mechanics in the 1920s. First Heisenberg (1925) developed the rather abstract matrix mechanics and soon after Schro¨dinger (1926; see also Eckart, 1926) developed the physically more appealing wave mechanics. Subsequently he showed them to be mathematically equivalent. The physicists were greatly relieved; now they had once again a mathematical formalism that enabled the results of experiments to be calculated. It was possible to move forward. It is certainly a very elegant and indeed beautiful formalism, and does enable the results of many physical measurements to be calculated with great accuracy. It is used daily by atomic and nuclear physicists; there is no other way to make calculations. From the earliest days there has been intense debate about the interpretation of quantum mechanics, and it shows no signs of being settled; if anything it is becoming more intense over the years. As Penrose has remarked, no one understands quantum mechanics. Feynman (1967; see also Sykes, 1994) also admitted that ‘nobody understands quantum mechanics’, and Einstein once remarked: ‘I have thought a hundred times as much about the quantum problems as I have about general relativity theory’ (Pais, 1994, p. 57). It should therefore be evident that the most that can be done here is to try to show where the real difficulties lie, and how the failure to recognize that quantum mechanics is an essentially statistical theory has led to a series of confusing quantum paradoxes. It is just these paradoxes that have often been used to support theological speculations. It is useful to ask what ‘understanding quantum mechanics’ actually means. Certainly it means being able to calculate the results of many measurements. The same applies to Newton’s theory of gravitation, which we understand in the sense that we can use it as a basis of calculation, although we do not really understand the gravitational force itself. Very often, especially in modern physics, the mathematical formalism does not give us the sort of intuitive physical feeling for what is going on that we often obtain from classical mechanics. Perhaps we can never achieve more than this for quantum mechanics. In classical mechanics we can assume that a property that is measured objectively exists prior to the interaction of the measuring apparatus with the observed system. Quantum mechanics, however, ‘is incompatible with the proposition that measurement discovers some unknown but pre-existing property’ (Peres, 1997, p. 14). There is an important distinction between mysteries and absurdities. We may be forced to live with mysteries, but there is no reason why we should put up with absurdities such as the wave-particle duality, tunnelling through potential barriers, acausal events and the collapse of the wave function. These are the quantum paradoxes that will be discussed later on. They arise within the Copenhagen interpretation of quantum mechanics and are solved by other interpretations. These alternative interpretations enable us to make conjectures about what is going on, even though we cannot understand it in detail. This is acceptable and a stimulus to further research. It is certainly difficult enough to understand quantum mechanics, but there is no need to try to visualize logical absurdities in the effort to do so. Quantum mechanics is a mathematical formalism that makes it possible to calculate the results of some experiments. Anything more than this, such as questions about the interpretation of what is measured and whether it tells us what is actually going on in the system being studied, belongs to philosophy. Thus we are concerned here with what is essentially a philosophical debate; if it were just a matter of physics the debate would have been settled long ago. One of the difficulties is that it invokes the results of experiments and uses the language of quantum mechanics. Inevitably this language uses philosophical terms and then it is easy to give the impression that the philosophical propositions are in some way entailed by the physical results. These in turn can influence theological thinking. In order to be quite clear which arguments are justified and which are not, it is essential to separate the three levels of discourse: physical, philosophical and theological. Our task is to relate these together, and this immediately raises the traditional problems of the philosophy of science, including the relation of experiment to theory, and of theory to explanation. We begin by discussing experiments, distinguishing between those that are real and those that are imaginary.
Editorial. Part I: Epistemology and Philosophy of Language. A Realism, Idealism, and General Terms E. Bencivenga. From Normal Machine to Social Colony: Toward a Complex Dynamical Philosophy of Science W.E. Herfel, C.A. Hooker. Observation and Reliable Detection P. Humphreys. Conceptual Idealism and Stove's Gem A. Musgrave. Holism in Artificial Intelligence C. Penco. Part II: Philosophy of Probability. Model Selection for Causal Theories B. Desjardins. Nonstandard Analysis and a Classification of Probability Spaces S. Fajardo. Some Remarks on Objective Chance (F.P. Ramsey, K.R. Popper and N.R. Campbell) M.C. Galavotti. Misapprehensions about Significance Tests and Bayesianism M.G. Sandrini. Part III: Philosophy and Foundations of Physics. The Quantum Probabilistic Approach to the Foundations of Quantum Theory: Urns and Chameleons L. Accardi. The Creation-discovery-view: Towards a Possible Explanation of Quantum Reality D. Aerts, B. Coecke. Virtual Reality: Consequences of No-Go Theorems for the Modal Interpretation of Quantum Mechanics G. Bacciagaluppi, P. Vermaas. Is Chaos Indeterministic? R.C. Bishop, F.M. Kronz. Wormholes and Timelike Curves: Is There Room for the Grandfather Paradox? G. Boniolo. John von Neumann met Kurt Godel: Undecidable Statements in Quantum Mechanics T. Breuer. Set-theoretical Models for Quantum Systems N.C. da Costa, D. Krause. Non-Ideal Measurements and Physical Possibility in Quantum Mechanics M. Del Seta, M. Suarez. Time, Relativity, and the Spatiality of Mental Events M. Dorato. Bell-type Inequalities and Orthomodular Lattices A. Dvurecenskij. An Informal Presentation of Semantic Realism and Its Solution of Quantum Paradoxes C. Garola.Individualistic versus Statistical Interpretation of Quantum Mechanics P. Mittelstaedt. The Locality Scandal of Quantum Mechanics F.A. Muller. Uncertainty and Dependence in Classical and Quantum Logic -- the Role of Triangular Norms. Quantum Logics and Quantum Measurements M. Navara, P. Ptak. Quantum Logics and Quantum Measurements S. Pulmannova. Attempt at the Logical Explanation of the Wave-particle duality J. Pykacz. Part IV: History of Science. The Bohr-Einstein Photon Box Debate D. Dieks. Kantianism and Physics from the 19th to the 20th Century. Part V: Science and Music. Epimoric Ratios and Greek Musical Theory F. Bellissima. Music and Science G. Toraldo di Francia.
A large body of mathematics consists of facts that can be presented and described much like any other natural phenomenon. These facts, at times explicitly brought out as theorems, at other times concealed within a proof, make up most of the applications of mathematics, and are the most likely to survive changes of style and of interest. This ENCYCLOPEDIA will attempt to present the factual body of all mathematics. Clarity of exposition, accessibility to the non-specialist, and a thorough bibliography are required of each author. Volumes will appear in no particular order, but will be organized into sections, each one comprising a recognizable branch of present-day mathematics. Numbers of volumes and sections will be reconsidered as times and needs change. It is hoped that this enterprise will make mathematics more widely used where it is needed, and more accessible in fields in which it can be applied but where it has not yet penetrated because of insufficient information. The present volume is meant to be the first of a series on quantum mechanics and its applications. It deals with the foundations as well as the fascinating logic of quantum mechanics. This volume is more general and at places less factual than other volumes of the ENCYCLOPEDIA; nevertheless the thorough presentation will guide the reader to gain an overview of a theory that, together with relativity, is regarded as the greatest achievement in physics in this century.
The article discusses the problem of a correct description of reality in science and philosophy, taking into account the acceptance of the many-world hypothesis. Many-worlds are understood as modern physical hypotheses that allow the existence of other universes (many-world interpretation, chaotic inflation, string landscape, and others). It is shown that these models are likely to challenge the traditional intellectual intuition on which classical science is built: they require new methodological approaches, and perhaps a new epistemology in general. Some requirements of classical logic, based on principles such as the laws of non-contradiction or the excluded middle, may turn out to be incorrect and even counterintuitive under many-world hypotheses. It is suggested that some logical systems (paraconsistent logic, quantum logic) may in the future become an ontological and epistemological foundation for the concepts of many-world physical reality. It is concluded that the considered multi-world models in conjunction with the indicated logics are more intuitively acceptable, since they allow getting rid of not only a number of cosmological problems (for example, fine tuning), but also problems of a logical and mathematical nature arising from the considered traditional requirements for formal systems.