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On the Present State of the Philosophy of Quantum Mathematics

Published online by Cambridge University Press:  28 February 2022

Howard Stein*
Affiliation:
The University of Chicago

Extract

It was with some trepidation that I agreed to speak today, because of a strong doubt that I could say anything substantial not already to be found in the literature of the subject. I cannot say that this trepidation has been subsequently relieved: all I can claim to offer in this paper is a review of certain basic characteristics or themes in the quantum-mechanical situation (which by now should, I think, be thoroughly understood by everyone engaged with the matter), supplemented by some rather general reflections on our philosophical predicament. In aid of these more general reflections, I shall indulge a proclivity for calling on historical matters—some fairly recent, some older, some ancient—which I hope may serve to place current issues in a useful perspective; and I ask your forgiveness for allowing myself to quote certain previous, but hitherto unpublished, remarks of my own.

Type
Part XII. Philosophy of Quantum Mechanics Today
Copyright
Copyright © 1983 Philosophy of Science Association

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References

Bell, John S. (1964). “On the Einstein-Podolsky-Rosen Paradox.” Physics 1: 195-200.10.1103/PhysicsPhysiqueFizika.1.195CrossRefGoogle Scholar
Bell, John S. (1966). “On the Problem of Hidden Variables in Quantum Mechanics.” Reviews of Modern Physics 38: 447-452.CrossRefGoogle Scholar
Bell, John S. (1977). “Free Variables and Local Causality.” Epistemologioal Letters 15: 79-84.Google Scholar
Burnet, John. (1930). Early Greek Philosophy. 4th ed. London: Macmillan Company.Google Scholar
Clauser, John F.; Horne, Michael A.; Shimony, Abner; and Holt, Richard A. (1969). “Proposed Experiment to Test Local Hidden-Variable Theories.” Physical Review Letters 23: 880-884.CrossRefGoogle Scholar
Einstein, Albert. (1905). “Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt.” Annalen der Physik (series 4) 17: 132-148.CrossRefGoogle Scholar
Everett, Hugh, III. (1957). “‘Relative State’ Formulation of Quantum Mechanics.” Reviews of Modern Physics 29: 454-465.CrossRefGoogle Scholar
Gleason, Andrew M. (1957). “Measures on the Closed Subspaces of a Hilbert Space.” Journal of Mathematics and Mechanics 6: 855-893.Google Scholar
Heisenberg, Werner. (1958). Physics and Philosophy. New York: Harper & Brothers.Google Scholar
Hellman, Geoffrey. (1982). “Stochastic Einstein-locality and the Bell Theorems.” Synthèse 53: 461-504.CrossRefGoogle Scholar
Jarrett, Jon P. (1983). Bell's Theorem. Quantum Mechanics. and Local Realism. Unpublished Ph.D. Dissertation, University of Chicago.Google Scholar
Kochen, Simon. (1979). “The Interpretation of Quantum Mechanics.” Unpublished typescript; forthcoming in Advances in Mathematics.Google Scholar
Kochen, Simon and Speaker, Ernst. (1967). “The Problem of Hidden Variables in Quantum Mechanics.” Journal of Mathematics and Mechanics 17: 59-87.Google Scholar
Lakatos, Imre. (1970). “Falsification and the Methodology of Scientific Research Programs.” In Criticism and the Growth of Knowledge. Edited by Lakatos, Imre and Musgrave, Alan. Cambridge: Cambridge University Press. Pages 91-196.CrossRefGoogle Scholar
Ernst, Mach. (1883). Die Mechanik in ihrer Entwicklung historischkritisch dargestellt. Leipzig: F.A. Brockhaus. (Reprinted as The Science of Mechanics: A Critical and Historical Account of Its Development. (trans.) McCormack, T.J.. La Salle, IL: Open Court, 1960.)Google Scholar
Mackey, George W. (1978). Unitary Group Representations In Physics, Probability, and Number Theory. Reading, Mass.: Benjamin/Cummings Publishing Co.Google Scholar
Maxwell, James Clerk. (1876). “On the Proof of the Equations of Motion of a Connected System.” Proceedings of the Cambridge Philosophical Society 2: 292-294. (As reprinted in Niven, W.D. (ed.). The Scientific Papers of James Clerk Maxwell, Volume II, Cambridge: Cambridge University Press, 1890. Pages 308-309.Google Scholar
Nash, John. (1956). “The Imbedding Problem for Riemannian Manifolds.” Annals of Mathematics 63: 20-63.CrossRefGoogle Scholar
Neugebauer, Otto. (1957). The Exact Sciences in Antiquity. 2nd ed. Providence: Brown University Press. (As reprinted 1969. New York: Dover Publications.Google Scholar
Neugebauer, Otto. (1975). A History of Ancient Mathematical Astronomy. Part One. New York: Springer Verlag.CrossRefGoogle Scholar
von Neumann, John. (1927a). “Mathematische Begrundung der Quantenmechanik.” Göttinger Nachrichten (1927): 1-57. (As reprinted in von Neumann (1961). Pages 151-207.Google Scholar
von Neumann, John. (1927b). “Wahrscheinlichkeitstheoretischer Aufbau der Quantenmechanik.” Göttinger Nachrichten (1927): 215-272. (As reprinted in von Neumann (1961). Pages 208-235.Google Scholar
von Neumann, John. (1932). Mathematische Grundlagen der Quantenmechanik. Berlin: Springer. (Reprinted as Mathematical Foundations of Quantum Mechanics. (trans.) Beyer, R.T.. Princeton: Princeton University Press, 1955.Google Scholar
von Neumann, John. (1961). Collected Works, Volume 1. (ed.) von Taub, A.H.. Oxford- London- New York- and Paris: Pergamon Press.Google Scholar
Newton, Isaac. (1726). Philosophiae Naturalis PrinoiPia Mathematica. 3rd ed. London: Royal Society.Google Scholar
Poincaré, Henri. (1890). Éléotricité et Optique. Volume 1. Paris: Georges Carré.Google Scholar
Poincaré, Henri. (1895). “A Propos de la Théorie de M. Larmor.” L'Éclairage Électrique 3: 5-13, 289-295; 5: 5-14, 385-392. (As reprinted in Poincaré. (1954). Pages 369-426.Google Scholar
Poincaré, Henri. (1900). “La Théorie de Lorentz et le Principe de Ré action.” Archives Néerlandaises des Sciences Exactes et Naturelles (2nd series) 5: 252-278. (As reprinted in Poincaré (1954). Pages 464-488.) .Google Scholar
Poincaré, Henri. (1902). La Science et l'Hypothèse. Paris: Ernest Flammarion. (As reprinted as Science and Hypothesis, (trans.) W.J. Greenstreet. New York: Dover Publications, 1952.)Google Scholar
Poincaré, Henri. (1906). “Sur la Dynamique de l'Électron.” Rendiconti del Circolo Matematico di Palermo 21: 129-176. (As reprinted in Poincaré (1954). Pages 494-550.CrossRefGoogle Scholar
Poincaré, Henri. (1908a). “La Dynamique de l'Électron.” Revue Générale des Sciences Pures et Appliques 19: 386-402. (As reprinted in Poincaré (1954). Pages 551-586. Translated as Part III of Poincaré (1908b). Pages 199-250.Google Scholar
Poincaré, Henri. (1908b). Science et méthode. Paris: E. Flammarion. (As reprinted as Science and Method, (trans.) Francis Maitland. London: T. Nelson and Sons, 1914.Google Scholar
Poincaré, Henri. (1954). Oeuvres de Henri Poincaré. Volume IX. Paris: Gauthier-Villars.Google Scholar
Schrödinger, Erwin. (1944). What is Life? Cambridge: Cambridge University Press.Google Scholar
Shimony, Abner. (1978). “Metaphysical Problems in the Foundations of Quantum Mechanics.” International Philosophical Quarterly 18: 3-17.CrossRefGoogle Scholar
Shimony, Abner. (1980). “The Point We Have Reached.” Epistemolϕgioal Letters 26: 1-7.Google Scholar
Shimony, Abner. (Forthcoming). “Contextual Hidden Variables Theories and Bell's Inequalities.” To be published in The British Journal for the Philosophy of Science.Google Scholar
Stein, Howard. (1970). “Is There a Problem of Interpreting Quantum Mechanics?” Noûs 4: 93-103.CrossRefGoogle Scholar
Stein, Howard. (1972). “On the Conceptual Structure of Quantum Mechanics.” In Paradigms and Paradoxes: the Philosophical Challenge of the Quantum Domain. (University of Pittsburgh Series in the Philosophy of Science. Volume V.) Edited by Colodny, Robert. Pittsburgh: University of Pittsburgh Press. Pages 367-438.Google Scholar
Stein, Howard. (1981). “‘Subtler Forms of Matter’ in the Period Following Maxwell.” In Conceptions of Ether: Studies in the History of Ether Theories, 1740-1900. Edited by Cantor, G.N. and Hodge, M.J.S.. Cambridge: Cambridge University Press. Pages 309-340.Google Scholar
van der Waerden, Bartel L. (ed.). (1967). Sources of Quantum Mechanics. New York: Dover Publications.Google Scholar