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Open Physics

What If Quantum Theory Violates All Mathematics?

Elemér Elad Rosinger

Published Jan 1, 2017• 0 citations
Publisher

Abstract

Abstract It is shown by using a rather elementary argument in Mathematical Logic that if indeed, quantum theory does violate the famous Bell Inequalities, then quantum theory must inevitably also violate all valid mathematical statements, and in particular, such basic algebraic relations like 0 = 0, 1 = 1, 2 = 2, 3 = 3, … and so on … An interest in that result is due to the following three alternatives which it imposes upon both Physics and Mathematics: Quantum Theory is inconsistent. Quantum Theory together with Mathematics are inconsistent. Mathematics is inconsistent. In this regard one should recall that, up until now, it is not known whether Mathematics is indeed consistent.

DOI
10.1515/phys-2017-0069

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Abstract It is shown by using a rather elementary argument in Mathematical Logic that if indeed, quantum theory does violate the famous Bell Inequalities, then quantum theory must inevitably also violate all valid mathematical statements, and in particular, such basic algebraic relations like 0 = 0, 1 = 1, 2 = 2, 3 = 3, … and so on … An interest in that result is due to the following three alternatives which it imposes upon both Physics and Mathematics: Quantum Theory is inconsistent. Quantum Theory together with Mathematics are inconsistent. Mathematics is inconsistent. In this regard one should recall that, up until now, it is not known whether Mathematics is indeed consistent.
Ever since the celebrated 1964 paper of John Bell, the statement that " Quantum systems violate the Bell inequalities " , [1,2], has a very large support among quantum physicists as well as others claiming some knowledge about quanta. Amusingly, it has so far escaped the general notice that, if indeed, quanta do violate that Bell inequalities, then-due to elementary facts of Logic-they must also violate all other valid mathematical relations, thus among them, the equation 0 = 0. Here the respective elementary facts of Logic are presented.
Logical foundation for quantum theory is considered. I claim that quantum theory correctly represents Nature when mathematical physics embraces and indeed features, logical anomalies inherent in pure mathematics. This approach links undecidability in arithmetic with the logic of quantum experiments. The undecidablity occupies an algebraic environment which is the missing foundation for the 3-valued logic predicted by Hans Reichenbach, shown by him to resolve `causal anomalies' of quantum mechanics, such as: inconsistency between prepared and measured states, complementarity between pairs of observables, and the EPR-paradox of action at a distance. Arithmetic basic to mathematical physics, is presented formally as a logical system consisting of axioms and propositions. Of special interest are all propositions asserting the existence of particular numbers. All numbers satisfying the axioms permeate the arithmetic indistinguishably, but these logically partition into two distinct sets: numbers whose existence the axioms determine by proof, and numbers whose existence axioms cannot determine, being neither provable nor negatable. Failure of mathematical physics to incorporate this logical distinction is seen as reason for quantum theory being logically at odds with quantum experiments. Nature is interpreted as having rules isomorphic to the abovementioned axioms with these governing arithmetical combinations of necessary and possible values or effects in experiments. Soundness and Completeness theorems from mathematical logic emerge as profoundly fundamental principles for quantum theory, making good intuitive sense of the subject.
Different level sources of possible inconsistencies of Quantum Physics are studied. The point of view is defended that inconsistencies may arise due to improper philosophical foundations of Quantum Physics but that there is a way out to each of the up to now suggested purely mathematical inconsistencies of each of the main mathematical formulations of Quantum mechanics. An example is worked out explicitly. The point of view is also supported that even if the above statement may be correct, i.e., that no purely mathematical inconsistencies exist in each of the main mathematical formulations, it is highly worthwhile to study alternatives as those offered by the Lie-Admissible Formulations of Quantum Mechanics.
Bell's inequality itself is usually considered to belong to mathematics and not quantum mechanics.We think that this is making our understanding of Bell' theory be confused.Thus in this paper, contrary to Bell's spirit (which inherits Einstein's spirit), we try to discuss Bell's inequality in the framework of quantum theory with the linguistic Copenhagen interpretation.And we clarify that the violation of Bell's inequality (i.e., whether or not Bell's inequality holds) does not depend on whether classical systems or quantum systems, but depend on whether a combined measurement exists or not.And further we conclude that our argument (based on the linguistic Copenhagen interpretation) should be regarded as a scientific representation of Bell's philosophical argument (based on Einstein's spirit).
A factual inconsistency of the nonlocality theory is demonstrated via a careful analysis of the conceptual and formal contents of Bell's theorem and the inequalities associated with it. The conceptual inconsistency of the theory follows from the consideration that it is, in principle, impossible to formulate quantum mechanics in terms of local hidden variables. The formal inconsistency of the nonlocality theory follows from the facts that: (i) nonlocality is not a physical property that can be represented by a function or an operator; (ii) it is impossible to derive Bell's inequality in a mathematically consistent way; and (iii) Bell's definition of locality is incorrect. Therefore, Bell's theorem cannot serve as the foundation for the introduction of the nonlocality notion in quantum mechanics.
Some notes about quantum physics, an interpretation if one wishes, are put forward, insisting on `closely following the mathematics/formalism, the `nuts and bolts of what quantum physics says'. These, basically well-known, issues seem to highlight some rather bold points about the `logic' aspect in quantum physics, necessarily restricting when and which logic may be admissible. And one may understand why that path is hardly followed in the literature. The mathematics/formalism of quantum, compared with classical, physics, may be fairly basically characterized by non-commutative algebras replacing commutative. These classically appearing, in fact, in dealing with systems of possibilities (say, all possible planetary motions under gravity of which one is the actual one). In particular, contrary to too common usage, the quantum non-commutativity should make it impossible to simply `transcend' the `system of possibilities' aspect into a `yes-no' logic essential for an `actual world'. One may have the latter only in a `haven' of approximately commutative algebras of `quasi-classical macroscopic observables', and moreover that `yes-no actual world' would plainly be an `extra ingredient' to the base quantum theory itself.