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Quantum Mathematics

Published online by Cambridge University Press:  21 March 2022

J. Michael Dunn
J. Michael Dunn*
Affiliation:
Indiana University
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Extract

It has been argued, most notably by Putnam (1969, 1974) that socalled quantum logic (which we shall often refer to as. orthomodular logic (OML)) is the one true logic,-as has been shown by quantum mechanical experiments. Putnam also has strong universalist tendencies, which would require him to use the same logic in all domains of reasoning, including mathematics, and it has been urged, recently (as one horn of a dilemma) by Hellman (1981), that there might be some problems in working out the classical mathematics of the Hilbertspace foundations of.quantum mechanics in a quantum logic framework.

In this paper we show (given one natural framework)(§3) first that if the first-order Peano arithmetic is formulated with quantum logic that it has the same theorems as the classical first-order Peano arithmetic. Distribution (for first-order arithmetical formulas) is a theorem not of quantum logic but rather of arithmetic.

Type
Part IX. Quantum Logic and Quantum Mathematics
Information
PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association , Volume 1980 , Issue 2: Volume Two: Symposia and Invited Papers 1980 , 1980 , pp. 512 - 531
Copyright
Copyright © 1981 by the Philosophy of Science Association

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Footnotes

1

I have benefited from helpful discussions with Gary Hardegree, Saul Kripke, H. Putnam, William Tait, Richmond Thomason, and especially with my colleague Geoffrey Hellman. Also I should remark that Robert Meyer's (1976), investigations into arithmetic founded on relevance logic had strong suggestive influence. There too some theorems of classical logic turn up as theorems not of logic but of arithmetic. Cf., also Dunn (1979).

References

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