Quantum Mathematics

PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1980:512 - 531 (1980)
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Abstract

This paper explores the development of mathematics on a quantum logical base when mathematical postulates are taken as necessary truths. First it is shown that first-order Peano arithmetic formulated with quantum logic has the same theorems as classical first-order Peano arithmetic. Distribution for first-order arithmetical formulas is a theorem not of quantum logic but rather of arithmetic. Second, it is shown that distribution fails for second-order Peano arithmetic without extensionality. Third, it is shown that distribution holds for second-order Peano arithmetic (second-order quantum logic) with extensionality. Some remarks about extensions to quantum set theory are made.

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Jon Michael Dunn
PhD: University of Pittsburgh; Last affiliation: Indiana University, Bloomington

Citations of this work

Alternative logics and applied mathematics.Timothy Williamson - 2018 - Philosophical Issues 28 (1):399-424.
Distribution in the logic of meaning containment and in quantum mechanics.Ross T. Brady & Andrea Meinander - 2013 - In Francesco Berto, Edwin Mares, Koji Tanaka & Francesco Paoli (eds.), Paraconsistency: Logic and Applications. Springer. pp. 223--255.
Wittgenstein on Incompleteness Makes Paraconsistent Sense.Francesco Berto - 2013 - In Francesco Berto, Edwin Mares, Koji Tanaka & Francesco Paoli (eds.), Paraconsistency: Logic and Applications. Springer. pp. 257--276.

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