ResumoIn this article I intend to show that certain aspects of A.N. Whitehead’s philosophy of organism and especially his epochal theory of time, as mainly exposed in his well-known work Process and Reality, can serve in clarify the underlying assumptions that shape nonstandard mathematical theories as such and also as metatheories of quantum mechanics. Concerning the latter issue, I point to an already significant research on nonstandard versions of quantum mechanics; two of these approaches are chosen to be critically presented in relation to the scope of this work. The main point of the paper is that, insofar as we can refer a nonstandard mathematical entity to a kind of axiomatical formalization essentially ‘codifying’ an underlying mental process indescribable as such by analytic means, we can possibly apply certain principles of Whitehead’s metaphysical scheme focused on the key notion of process which is generally conceived as the becoming of actual entities. This is done in the sense of a unifying approach to provide an interpretation of nonstandard mathematical theories as such and also, in their metatheoretical status, as a formalization of the empirical-experimental context of quantum mechanics.
ALBEVERIO, S. et al. Nonstandard Methods in Stochastic Analysis and Mathematical Physics, Orlando: Academic Press, 1986.
ANDREEV, P., GORDON, E. “An Axiomatics for Nonstandard Set Theory, Based on Von Neumann-Bernays-Gödel Theory". The Journal of Symbolic Logic, 66, 3, pp. 1321-1341, 2001.. “A Theory of Hyperfinite Sets". Annals of Pure and Applied Logic, 143, 1-3, pp. 3-19, 2006. dalla CHIARA, M.L. “Logical Self Reference, Set Theoretical Paradoxes and the Measurement Problem in Quantum Mechanics". Journal of Philosophical Logic, 6, pp. 331-347, 1977.
EPPERSON, M. Quantum Mechanics and the Philosophy of A.N. Whitehead. New York: Fordham University Press, 2004.
FARRUKH, O.M. “Application of nonstandard analysis to quantum mechanics". Journal of Mathematical Physics, 16, 2, pp. 177- 200, 1975.
FESENKO, I. “Several nonstandard remarks". AMS/IP Advances in the Mathematical Sciences, AMS Transl. Series 2, 217, pp. 37-50, 2006.
FOLSE, H. “The Copenhagen Interpretation of Quantum Theory and Whitehead’s Philosophy of Organism". Tulane Studies in Philosophy, 23, 33, 1974.
FRANCIS, E.C. “Application of non-standard analysis to relativistic quantum mechanics". Journal of Physics A: Mathematical and General, 14, 10, pp. 2539-2551, 1981.
FRIEDMAN, A. “Nonstandard extension of quantum logic and Dirac’s bra-ket formalism of quantum mechanics". International Journal of Theoretical Physics, 33, 2, pp. 307-38, 1994.
GOLDBLATT, R. Lectures on the Hyperreals: An Introduction to Nonstandard Analysis, New York: Springer-Verlag, 1998.
GRIB, A.A. “Quantum Logical Interpretation of Quantum Mechanics: The Role of Time". Int. Jour. of Theoretical Physics, 32, 12, pp. 2389-2400, 1993.
HEISENBERG, W. The Physical Principles of the Quantum Theory, Chicago: The University of Chicago Press, 1930.
HUSSERL, E. Ideen zu einer reinen Phänomenologie und phänomenologischen Philosophie, Erstes Buch, Hua Band III/I, hsgb. K. Schuhmann, Den Haag: M. Nijhoff, 1977. Formale und Transzendentale Logik, Hua Band XVII, hsgb. P. Janssen, Den Haag: M. Nijhoff, 1974. Zur Phanomenologie des Inneren Zeibewusstseins, Hua Band X, hsgb. R. Boehm, Den Haag: M. Nijhoff, 1966.
NELSON, E. Predicative Arithmetic. Mathematical notes, Princeton: Princeton Univ. Press, 1986.
RAAB, A. : “An approach to nonstandard quantum Mechanics". Journal of Mathematical Physics, 45, 12, pp. 4791-4809, 2004.
ROBINSON, A. Non-standard Analysis, Amsterdam: North-Holland Pub, 1966.
ROBINSON, A. & ZAKON, E. A set-theoretical characterization of enlargements, in: Applications of Model Theory to Algebra, Analysis and Probability, (ed. W.A.J. Luxembourg), pp. 109-122, New York: Holt, Rinehart & Winston, 1969.
SHIMONY, A. Quantum Physics and the Philosophy of Whitehead, in: Boston Studies in the Philosophy of Science, (eds. R. Cohen & M. Wartofsky), 2:308, New York: Humanities Press, 1965.
TANAKA, Y. The Individuality of a Quantum Event: Whitehead’s Epochal Theory of Time and Bohr’s Framework of Complementarity in: Physics an Whitehead: Quantum, Process and Experience, (eds. Timothy Eastman & Hank Keeton), pp. 164-179, New York: State University of New York Press, 2004. van DALEN, D. Logic and Structure, Berlin: Springer-Verlag, 2004.
VOPENKA, P. Mathematics in the Alternative Set Theory, TeubnerTexte zur Mathematik, Leipzig: Teubner Verlag, 1979.
WHITEHEAD, N.A. Process and Reality. An Essay in Cosmology, New York: The Free Press, 1978. An Enquiry Concerning the Principles of Natural Knowledge, (originally published in 1919), New York: Cosimo, Inc, 2007. The Concept of Nature, (originally published in 1920), New York: Cosimo, Inc, 2007.
YAMASHITA, H. Nonstandard methods in quantum field theory I: a hyperfinite formalism of scalar fields, Intern. J. Theor. Physics, 41, pp. 511-527, 2002.