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Von Neumann's Impossibility Proof: Mathematics in the Service of Rhetorics

Dieks, Dennis (2016) Von Neumann's Impossibility Proof: Mathematics in the Service of Rhetorics. [Preprint]


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According to what has become a standard history of quantum mechanics, von Neumann in 1932 succeeded in convincing the physics community that he had proved that hidden variables were impossible as a matter of principle. Subsequently, leading proponents of the Copenhagen interpretation emphatically confirmed that von Neumann's proof showed the completeness of quantum mechanics. Then, the story continues, Bell in 1966 finally exposed the proof as seriously and obviously wrong; this rehabilitated hidden variables and made
serious foundational research possible. It is often added in recent accounts that von Neumann's error had been spotted almost immediately by Grete Hermann, but that her discovery was of no effect due to the dominant Copenhagen Zeitgeist.
We shall attempt to tell a more balanced story. Most importantly, von Neumann did not claim to have shown the impossibility of hidden variables tout court, but argued that hidden-variable theories must possess a structure that deviates fundamentally from that of quantum
mechanics. Both Hermann and Bell appear to have missed this point; moreover, both raised unjustified technical objections to the proof. Von Neumann's conclusion was basically that hidden-variables schemes must violate the "quantum principle" that all physical quantities are to be represented by operators in a Hilbert space. According
to this conclusion, hidden-variables schemes are possible in principle but necessarily exhibit a certain kind of contextuality.
As we shall illustrate, early reactions to Bohm's theory are in agreement with this account. Leading physicists pointed out that Bohm's theory has the strange feature that particle properties do not generally reveal themselves in measurements, in accordance with von Neumann's result. They did not conclude that the "impossible was done" and that von Neumann had been shown wrong.

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Item Type: Preprint
Dieks, Dennisd.dieks@uu.nl0000-0003-4755-3430
Additional Information: From a talk at HQ4, San Sebastian 2015; preprint for a special issue of SHPMP
Keywords: Hidden Variables, Von Neumann, Bell, Grete Hermann
Subjects: Specific Sciences > Physics > Quantum Mechanics
Depositing User: Dennis Dieks
Date Deposited: 17 Sep 2016 12:53
Last Modified: 17 Sep 2016 12:53
Item ID: 12443
Subjects: Specific Sciences > Physics > Quantum Mechanics
Date: August 2016

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