Mitsch, Chris (2021) Hilbert-Style Axiomatic Completion: The (Not So) Hidden Contextuality of von Neumann's "No Hidden Variables" Theorem. [Preprint]

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Abstract

In this paper I provide a detailed history of von Neumann’s “No Hidden Variables” theorem, and I argue it is a demonstration that his axiomatization mathematically captures a salient feature of the statistical interpretation (namely, that hidden variables are incompatible). I show that this reading of von Neumann’s theorem is obvious once one recalls several contextual factors of his work. First, his axiomatization was what I call a Hilbert-style axiomatic completion; indeed, it developed from work initiated by Hilbert (and Nordheim). Second, it was responsive to specific mathematical and theoretical problems faced by Dirac and Jordan’s statistical transformation theory (then called ‘quantum mechanics’). Third, the axiomatization was essentially completed already in his 1927 papers, at least concerning the status of hidden variables, and this would have been obvious to the audience for those papers. Thus, the theorem’s statement and proof were only necessary when the material was presented for a general mathematical audience, i.e., in his 1932 Mathematical Foundations of Quantum Mechanics . With this reading in mind, his claim that quantum mechanics was in “compelling logical contradiction with causality” appears as a straightforward consequence of his theorem. I conclude by reassessing the theorem’s broader historical and scientific significance.

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Item Type:	Preprint
Creators:	Creators Email ORCID Mitsch, Chris cmitsch@uci.edu 0000-0003-2502-4343
Keywords:	von Neumann, quantum mechanics, hidden variables, Bohm, Bell, Hilbert, axiomatization
Subjects:	Specific Sciences > Mathematics > Methodology General Issues > History of Science Case Studies Specific Sciences > Physics > Quantum Mechanics
Depositing User:	Chris Mitsch
Date Deposited:	10 Sep 2021 18:00
Last Modified:	10 Sep 2021 18:00
Item ID:	19544
Subjects:	Specific Sciences > Mathematics > Methodology General Issues > History of Science Case Studies Specific Sciences > Physics > Quantum Mechanics
Date:	2021
URI:	https://philsci-archive.pitt.edu/id/eprint/19544

Available Versions of this Item

• Hilbert-Style Axiomatic Completion: The (Not So) Hidden Contextuality of von Neumann's "No Hidden Variables" Theorem. (deposited 10 Sep 2021 18:00) [Currently Displayed]
  • Hilbert-Style Axiomatic Completion: On von Neumann and Hidden Variables in Quantum Mechanics. (deposited 30 Jun 2022 17:12)

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