As opposed to the classical logic of true and false, viewed as an axiomatised theory, ordinary arithmetic conveys the three logical values: provable, negatable and logically independent. This research proposes the hypothesis that Axioms of Arithmetic are the fundamental foundation running arithmetical processes in Nature, upon which physical processes rest. And goes on to show, in detail, that under these axioms, quantum mathematics derives and initiates logical independence, agreeing with indeterminacy in quantum experiments. Supporting arguments begin by explaining logical independence in arithmetic, in particular, independence of the square root of minus one. The method traces all sources of information entering arithmetic, needed to write mathematics of the free particle. Wave packets, prior to measurement, are found to be the only part of theory logically independent of axioms; the rest of theory is logically dependent. Ingress of logical independence is via uncaused, unprevented self-reference, sustaining the wave packet, but implying unitarity. Quantum mathematics based on axiomatised arithmetic is established as foundation for the 3-valued logic of Hans Reichenbach, which reconciles quantum theory with experimental anomalies such as the Einstein, Podolsky & Rosen paradox.
As opposed to the classical logic of true and false, viewed as an axiomatised theory, ordinary arithmetic conveys the three logical values: provable, negatable and logically independent. This research proposes the hypothesis that Axioms of Arithmetic are the fundamental foundation running arithmetical processes in Nature, upon which physical processes rest. And goes on to show, in detail, that under these axioms, quantum mathematics derives and initiates logical independence, agreeing with indeterminacy in quantum experiments. Supporting arguments begin by explaining logical independence in arithmetic, in particular, independence of the square root of minus one. The method traces all sources of information entering arithmetic, needed to write mathematics of the free particle. Wave packets, prior to measurement, are found to be the only part of theory logically independent of axioms; the rest of theory is logically dependent. Ingress of logical independence is via uncaused, unprevented self-reference, sustaining the wave packet, but implying unitarity. Quantum mathematics based on axiomatised arithmetic is established as foundation for the 3-valued logic of Hans Reichenbach, which reconciles quantum theory with experimental anomalies such as the Einstein, Podolsky & Rosen paradox.
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.
Abstract As opposed to the classical logic of true and false, when elementary al-gebra is treated as a formal axiomatised system, formulae in that algebra are either provable, disprovable or otherwise, logically independent of axioms. This logical in-dependence is well-known to Mathematical Logic. The intention here is to cover the subject in a way accessible to physicists. This work is part of a project research-ing logical independence in quantum mathematics, for the purpose of advancing a complete theory of quantum randomness.
The purpose of this dissertation is to construct a radically new type of mathematics whose underlying logic differs from the ordinary classical logic used in standard mathematics, and which we feel may be more natural for applications in quantum mechanics. Specifically, we begin by constructing a first order quantum logic, the development of which closely parallels that of ordinary (classical) first order logic --- the essential differences are in the nature of the logical axioms, which, in our construction, are motivated by quantum theory. After showing that the axiomatic first order logic we develop is sound and complete (with respect to a particular class of models), this logic is then used as a foundation on which to build (axiomatic) mathematical systems --- and we refer to the resulting new mathematics as "quantum mathematics." As noted above, the hope is that this form of mathematics is more natural than classical mathematics for the description of quantum systems, and will enable us to address some foundational aspects of quantum theory which are still troublesome --- e.g. the measurement problem --- as well as possibly even inform our thinking about quantum gravity.\n\nAfter constructing the underlying logic, we investigate properties of several mathematical systems --- e.g. axiom systems for abstract algebras, group theory, linear algebra, etc. --- in the presence of this quantum logic. In the process, we demonstrate that the resulting quantum mathematical systems have some strange, but very interesting features, which indicates a richness in the structure of mathematics that is classically inaccessible. Moreover, some of these features do indeed suggest possible applications to foundational questions in quantum theory. \n\nWe continue our investigation of quantum mathematics by constructing an axiomatic quantum set theory, which we show satisfies certain desirable criteria. Ultimately, we hope that such a set theory will lead to a foundation for quantum mathematics in a sense which parallels the foundational role of classical set theory in classical mathematics. One immediate application of the quantum set theory we develop is to provide a foundation on which to construct quantum natural numbers, which are the quantum analog of the classical counting numbers. It turns out that in a special class of models, there exists a 1-1 correspondence between the quantum natural numbers and bounded observables in quantum theory whose eigenvalues are (ordinary) natural numbers. This 1-1 correspondence is remarkably satisfying, and not only gives us great confidence in our quantum set theory, but indicates the naturalness of such models for quantum theory itself. We go on to develop a Peano-like arithmetic for these new "numbers," as well as consider some of its consequences. Finally, we conclude by summarizing our results, and discussing directions for future work.
Abstract:
In 2008, Tomasz Paterek et al published ingenious research, proving that quantum randomness is the output of measurement experiments, whose input commands a logically independent response. Following up on that work, this paper develops a full mathematical theory of quantum indeterminacy. I explain how, the Paterek experiments imply, that the measurement of pure eigenstates, and the measurement of mixed states, cannot both be isomorphically and faithfully represented by the same single operator. Specifically, unitary representation of pure states is contradicted by the Paterek experiments. Profoundly, this denies the axiomatic status of Quantum Postulates, that state, symmetries are unitary, and observables Hermitian. Here, I show how indeterminacy is the information of transition, from pure states to mixed. I show that the machinery of that transition is unpreventable, logically circular, unitary-generating self-reference: all logically independent. Profoundly, this indeterminate system becomes apparent, as a visible feature of the mathematics, when unitarity --- imposed by Postulate --- is given up and abandoned.
Keywords:
foundations of quantum theory, quantum mechanics, quantum randomness, quantum indeterminacy, quantum information, prepared state, measured state, pure states, mixed states, unitary, redundant unitarity, orthogonal, scalar product, inner product, mathematical logic, logical independence, self-reference, logical circularity, mathematical undecidability.
The homogeneity symmetry is re-examined and shown to be non-unitary. This is motivated by the prospect that logical independence in elementary algebra, entering quantum mathematics, will constitute the basis for a theory explaining quantum randomness. Keywords: foundations of quantum theory, quantum physics, quantum mechanics, wave mechanics, Canonical Commutation Relation, symmetry, homogeneity of space, unitary, non-unitary, unitarity, mathematical logic, formal system, elementary algebra, information, axioms, mathematical propositions, logical independence, quantum indeterminacy, quantum randomness.
It has been argued, most notably by Putnam (1969, 1974) that socalled quantum logic (which we shall often refer to as. orthomodular logic (OML)) is the one true logic,-as has been shown by quantum mechanical experiments. Putnam also has strong universalist tendencies, which would require him to use the same logic in all domains of reasoning, including mathematics, and it has been urged, recently (as one horn of a dilemma) by Hellman (1981), that there might be some problems in working out the classical mathematics of the Hilbertspace foundations of.quantum mechanics in a quantum logic framework. In this paper we show (given one natural framework)(§3) first that if the first-order Peano arithmetic is formulated with quantum logic that it has the same theorems as the classical first-order Peano arithmetic. Distribution (for first-order arithmetical formulas) is a theorem not of quantum logic but rather of arithmetic.