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. The deviations at issue are seen to be any excluded degrees of freedom in a variable describing whatever. Just this statement is interpreted to be the theorem's "apophatic interpretation". A powerful method for proving statements in Hilbert arithmetic's number theory is offered as a conjecture to be reasoned and proved in a forthcoming paper.
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. The deviations at issue are seen to be any excluded degrees of freedom in a variable describing whatever. Just this statement is interpreted to be the theorem's "apophatic interpretation". A powerful method for proving statements in Hilbert arithmetic's number theory is offered as a conjecture to be reasoned and proved in a forthcoming paper.
There is ongoing discourse in philosophical and physics literature about whether the probabilistic essence of quantum mechanics indicates that deterministic interpretations of nature must be false. In 1967 two mathematicians, Simon Kochen and Ernst Specker, developed a mathematical theorem to the effect that particles do not have well-defined property values at a given time. A logical consequence of this is that there can be no well-defined hidden variables responsible for the outcomes of mysterious quantum behaviors. The theorem is thus fatal to the many deterministic interpretations of quantum mechanics that assume hidden variables, including pilot wave mechanics, a quantum theory developed by Louis de Broglie and David Bohm a few decades prior to the formulation of the Kochen-Specker Theorem. The purpose of this essay is to evaluate the legitimacy of the logical basis of Kochen’s and Specker’s argument considering its severe implications. It is concluded that Kochen and Specker do not provide sufficient evidence that deterministic theories must fail, due to their misinterpretation of the axioms on which hidden variable theories rely. This paper also finds that pilot wave mechanics is mathematically successful while remaining consistent with the deterministic understanding of broader physics and deserving of more serious consideration as a quantum theory. Above all it is emphasized that the nonsensicality of absolute randomness and indeterminism should be acknowledged, and theorems purporting to prove their existence should be critically examined, for such a discovery would undermine the legitimacy of logic itself.
The question of a hidden variable interpretation of quantum contextuality in the Mermin-Peres square is considered. The Kochen-Specker theorem implies that quantum mechanics may be interpreted as a contextual hidden-variable theory. It is shown that such a hidden-variable description can be viewed as either contextual in the random variables mapping hidden states to observable outcomes or in the probability measure on the hidden state space. The latter view suggests that this apparent contextuality may be interpreted as a simple consequence of measurement disturbance, wherein the initial hidden state is altered through interaction with the measuring device, thereby giving rise to a possibly different final hidden-variable state from which the measurement outcome is obtained. In light of this observation, a less restrictive and, arguably, more reasonable definition of noncontextuality is suggested. To prove that such a description is possible, an explicit and, in this sense, noncontextual hidden-variable model is constructed, which reproduces all quantum theoretic predictions for the Mermin-Peres square. A critical analysis of some recent and proposed experimental tests of contextuality is also provided. Although the discussion is restricted to a four-dimensional Hilbert space, the approach and conclusions are expected to generalize to any Hilbert space.
The paper is a continuation of another paper (https://philpapers.org/rec/PENFLT-2) published as Part I. Now, the case of “n=3” is inferred as a corollary from the Kochen and Specker theorem (1967): the eventual solutions of Fermat’s equation for “n=3” would correspond to an admissible disjunctive division of qubit into two absolutely independent parts therefore versus the contextuality of any qubit, implied by the Kochen – Specker theorem. Incommensurability (implied by the absence of hidden variables) is considered as dual to quantum contextuality. The relevant mathematical structure is Hilbert arithmetic in a wide sense (https://dx.doi.org/10.2139/ssrn.3656179), in the framework of which Hilbert arithmetic in a narrow sense and the qubit Hilbert space are dual to each other. A few cases involving set theory are possible: (1) only within the case “n=3” and implicitly, within any next level of “n” in Fermat’s equation; (2) the identification of the case “n=3” and the general case utilizing the axiom of choice rather than the axiom of induction. If the former is the case, the application of set theory and arithmetic can remain disjunctively divided: set theory, “locally”, within any level; and arithmetic, “globally”, to all levels. If the latter is the case, the proof is thoroughly within set theory. Thus, the relevance of Yablo’s paradox to the statement of Fermat’s last theorem is avoided in both cases. The idea of “arithmetic mechanics” is sketched: it might deduce the basic physical dimensions of mechanics (mass, time, distance) from the axioms of arithmetic after a relevant generalization, Furthermore, a future Part III of the paper is suggested: FLT by mediation of Hilbert arithmetic in a wide sense can be considered as another expression of Gleason’s theorem in quantum mechanics: the exclusions about (n = 1, 2) in both theorems as well as the validity for all the rest values of “n” can be unified after the theory of quantum information. The availability (respectively, non-availability) of solutions of Fermat’s equation can be proved as equivalent to the non-availability (respectively, availability) of a single probabilistic measure as to Gleason’s theorem.
One of the statements one would hear pretty early in any quantum mechanics course is that for quantum particles there are properties which do not have definite values. For example, if we measure |+〉 in standard basis, the best we can do to describe the outcome is to say, that with probability 1/2 we will get 0 and with probability 1/2 we will get 1. Yet one could wonder, is this really the way things are or is it just because our description of this quantum particle is incomplete, but in fact there are some hidden variables inside every quantum system that contain enough information to tell the outcome of measurement (see Fig. 1). However, many people disdain the idea of hidden variables, so there has been rather long history of theorems that forbid theories involving hidden variables. Usually if one wants to prove some no hidden variables theorem he has to assume something about the hidden variables theories he is going to forbid. One of such assumptions is non-contextuality, which means that the outcome we get after measuring with an observable is the same regardless of what other observables we choose to measure at the same time. The first theorem forbidding hidden variables theories is due to von Neumann in 1932. Then for a third of a century everyone cited von Neumann’s work carelessly, till Bell in 1966 indicated that there was an unreasonable assumption in von Neumann’s proof and urged for further examination of the problem (see [6] and [1] for more historical details). Bell also constructed a hidden variables model for one qubit that could assign the outcome of any observable in a way consistent with quantum mechanics. He also proved two theorems that forbid non-contextual hidden variables theories using counterexamples with larger quantum systems. Yet surprisingly he himself thought that hidden variables do exists and considered these theorems only as characterizations of the hidden variables theory we should look for. In this essay we will discuss several versions of one of the two Bell’s theorems. The theorem we will consider was also independently proven by Kochen and Specker in 1967 (see [5]), hence it is called Bell-Kochen-Specker theorem. For shortness and in order not to mix this theorem with another Bell’s theorem from now on we will refer to it as Kochen-Specker theorem. We will be interested in proving several versions of this theorem and the following lemma will turn out to be of great use:
Non-commuting quantities and hidden parameters – Wave-corpuscular dualism and hidden parameters – Local or nonlocal hidden parameters – Phase space in quantum mechanics – Weyl, Wigner, and Moyal – Von Neumann’s theorem about the absence of hidden parameters in quantum mechanics and Hermann – Bell’s objection – Quantum-mechanical and mathematical incommeasurability – Kochen – Specker’s idea about their equivalence – The notion of partial algebra – Embeddability of a qubit into a bit – Quantum computer is not Turing machine – Is continuality universal? – Diffeomorphism and velocity – Einstein’s general principle of relativity – „Mach’s principle“ – The Skolemian relativity of the discrete and the continuous – The counterexample in § 6 of their paper – About the classical tautology which is untrue being replaced by the statements about commeasurable quantum-mechanical quantities – Logical hidden parameters – The undecidability of the hypothesis about hidden parameters – Wigner’s work and и Weyl’s previous one – Lie groups, representations, and psi-function – From a qualitative to a quantitative expression of relativity − psi-function, or the discrete by the random – Bartlett’s approach − psi-function as the characteristic function of random quantity – Discrete and/ or continual description – Quantity and its “digitalized projection“ – The idea of „velocity−probability“ – The notion of probability and the light speed postulate – Generalized probability and its physical interpretation – A quantum description of macro-world – The period of the as-sociated de Broglie wave and the length of now – Causality equivalently replaced by chance – The philosophy of quantum information and religion – Einstein’s thesis about “the consubstantiality of inertia ant weight“ – Again about the interpretation of complex velocity – The speed of time – Newton’s law of inertia and Lagrange’s formulation of mechanics – Force and effect – The theory of tachyons and general relativity – Riesz’s representation theorem – The notion of covariant world line – Encoding a world line by psi-function – Spacetime and qubit − psi-function by qubits – About the physical interpretation of both the complex axes of a qubit – The interpretation of the self-adjoint operators components – The world line of an arbitrary quantity – The invariance of the physical laws towards quantum object and apparatus – Hilbert space and that of Minkowski – The relationship between the coefficients of -function and the qubits – World line = psi-function + self-adjoint operator – Reality and description – Does a „curved“ Hilbert space exist? – The axiom of choice, or when is possible a flattening of Hilbert space? – But why not to flatten also pseudo-Riemannian space? – The commutator of conjugate quantities – Relative mass – The strokes of self-movement and its philosophical interpretation – The self-perfection of the universe – The generalization of quantity in quantum physics – An analogy of the Feynman formalism – Feynman and many-world interpretation – The psi-function of various objects – Countable and uncountable basis – Generalized continuum and arithmetization – Field and entanglement – Function as coding – The idea of „curved“ Descartes product – The environment of a function – Another view to the notion of velocity-probability – Reality and description – Hilbert space as a model both of object and description – The notion of holistic logic – Physical quantity as the information about it – Cross-temporal correlations – The forecasting of future – Description in separable and inseparable Hilbert space – „Forces“ or „miracles“ – Velocity or time – The notion of non-finite set – Dasein or Dazeit – The trajectory of the whole – Ontological and onto-theological difference – An analogy of the Feynman and many-world interpretation − psi-function as physical quantity – Things in the world and instances in time – The generation of the physi-cal by mathematical – The generalized notion of observer – Subjective or objective probability – Energy as the change of probability per the unite of time – The generalized principle of least action from a new view-point – The exception of two dimensions and Fermat’s last theorem
The paper is a continuation of Part I. The case of “n=3” is inferred as a corollary from the Kochen and Specker theorem (1967): the eventual solutions of Fermat’s equation for “n=3” would correspond to an admissible disjunctive division of qubit into two absolutely independent parts therefore versus the contextuality of any qubit, implied by the Kochen – Specker theorem. Incommensurability (implied by the absence of hidden variables) is considered as dual to quantum contextuality. The relevant mathematical structure is Hilbert arithmetic in a wide sense, in which Hilbert arithmetic in a narrow sense and the qubit Hilbert space are dual . A few cases involving set theory are possible: (1) only within the case “n=3” and implicitly, within any next level of “n” in Fermat’s equation; (2) the identification of the case “n=3” and the general case utilizing the axiom of choice rather than the axiom of induction.