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“Fundamental randomness” by the “apophatic” Kochen - Specker theorem: Toward a universal method in Hilbert arithmetic’s number theory

10 June 2025, Version 1
This content is an early or alternative research output and has not been peer-reviewed by Cambridge University Press at the time of posting.

Abstract

One of the most fundamental theoretical results in quantum mechanics, the theorem of Simon Kochen and Ernst Specker (1967), is investigated from a rather mathematical and philosophical than physical viewpoint (i.e. unlike as usual). The absence of hidden variables is interpreted philosophically and ontomathematically: as the identity of the mathematical model by the separable complex Hilbert space (equivalent to the qubit Hilbert space) and physical reality. It implies the completeness of just that model to physical reality including in the sense of its cognitive finality. In other words, “hidden variables” are also interpreted as any mismatch between model and reality implied by the theorem not to be. That property of the qubit Hilbert space unique among all other mathematical theories (i.e., all first-order logics) is deduced as a corollary from the theorem as well. The approach at issue to the theorem implies its inherent contextuality to be deduced also in an “apophatic” or holistic version stating that omitting even a single “axis” of Hilbert space or a single value of probability density break its contextuality thus being inconsistent to the theorem. The concept of “fundamental randomness” is newly introduced as originating from the absence of hidden variables in any probability (density or not) distribution, the characteristic function of which is interpreted to be a (qubit) wave function. Then, the Kochen - Specker theorem implies no deviation from that “fundamental randomness” exists once its preconditions have been satisfied.

Keywords

characteristic function of probability distribution
degrees of freedom of a variable
fundamental randomness
Kochen - Specker theorem and its “apophatic reformulation”
quantum model and reality
qubit Hilbett space
qubit wave function

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