Toward quantum mathematics. I. From quantum set theory to universal quantum mechanics

1999 ◽  
Vol 40 (3) ◽  
pp. 1344-1358 ◽  
Author(s):  
Karl-Georg Schlesinger
Keyword(s):  
Quantum Mechanics ◽  
Set Theory ◽  
Universal Quantum ◽  
Quantum Mathematics

Indeterminacy

2020 ◽  
pp. 172-184
Author(s):  
Alastair Wilson
Keyword(s):  
Quantum Mechanics ◽  
Quantum State ◽  
Modal Realism ◽  
Many Worlds ◽  
Philosophical Literature ◽  
Universal Quantum ◽  
Quantum Indeterminacy

In Everettian quantum mechanics, the universal quantum state is fundamental, non-contingent, and wholly determinate. By contrast, the parallel worlds of diverging EQM, and the contingency constituted by self-location amongst those worlds, are emergent and partly indeterminate. In particular, it is indeterminate both how many worlds there are, and what microscopic qualitative features those worlds have. This chapter discusses various ways to understand indeterminacy in the Everettian multiverse, and argues that the indeterminacies of EQM present no obstacle to the analytic ambitions of quantum modal realism. Everettians can understand quantum indeterminacy using models of indeterminacy that are familiar from the philosophical literature on vagueness.

scholarly journals Indistinguishability and the origins of contextuality in physics

2019 ◽  
Vol 377 (2157) ◽  
pp. 20190150 ◽  
Author(s):  
J. Acacio de Barros ◽  
Federico Holik ◽  
Décio Krause
Keyword(s):  
Quantum Mechanics ◽  
Set Theory ◽  
Physical Quantity ◽  
Theoretical Framework ◽  
Quantum Systems ◽  
Strong Sense ◽  
Theme Issue ◽  
Identity Of Indiscernibles ◽  
Theoretical Structure

In this work, we discuss a formal way of dealing with the properties of contextual systems. Our approach is to assume that properties describing the same physical quantity, but belonging to different measurement contexts, are indistinguishable in a strong sense. To construct the formal theoretical structure, we develop a description using quasi-set theory, which is a set-theoretical framework built to describe collections of elements that violate Leibnitz's principle of identity of indiscernibles. This framework allows us to consider a new ontology in order to study the properties of quantum systems. This article is part of the theme issue ‘Contextuality and probability in quantum mechanics and beyond’.

scholarly journals Two strategies to infinity: completeness and incompleteness. The completeness of quantum mechanics

2020 ◽  
Author(s):  
Vasil Dinev Penchev
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Set Theory ◽  
Quantum Computer ◽  
Special Kind ◽  
Albert Einstein ◽  
Internal Position ◽  
Essential Properties ◽  
Completeness Of Quantum Mechanics ◽  
The Axiom Of Choice

Two strategies to infinity are equally relevant for it is as universal and thus complete as open and thus incomplete. Quantum mechanics is forced to introduce infinity implicitly by Hilbert space, on which is founded its formalism. One can demonstrate that essential properties of quantum information, entanglement, and quantum computer originate directly from infinity once it is involved in quantum mechanics. Thus, thеse phenomena can be elucidated as both complete and incomplete, after which choice is the border between them. A special kind of invariance to the axiom of choice shared by quantum mechanics is discussed to be involved that border between the completeness and incompleteness of infinity in a consistent way. The so-called paradox of Albert Einstein, Boris Podolsky, and Nathan Rosen is interpreted entirely in the same terms only of set theory. Quantum computer can demonstrate especially clearly the privilege of the internal position, or “observer”, or “user” to infinity implied by Henkin’s proposition as the only consistent ones as to infinity.

scholarly journals All science as rigorous science: the principle of constructive mathematizability of any theory

2020 ◽  
Author(s):  
Vasil Dinev Penchev
Keyword(s):  
Quantum Mechanics ◽  
Set Theory ◽  
Liberal Arts ◽  
Scientific Theory ◽  
Mathematical Structure ◽  
Peano Arithmetic ◽  
Formal Proof ◽  
Transfinite Induction ◽  
Information Interpretation ◽  
Future Work

A principle, according to which any scientific theory can be mathematized, is investigated. Social science, liberal arts, history, and philosophy are meant first of all. That kind of theory is presupposed to be a consistent text, which can be exhaustedly represented by a certain mathematical structure constructively. In thus used, the term “theory” includes all hypotheses as yet unconfirmed as already rejected. The investigation of the sketch of a possible proof of the principle demonstrates that it should be accepted rather a metamathematical axiom about the relation of mathematics and reality.The main statement is formulated as follows: Any scientific theory admits isomorphism to some mathematical structure in a way constructive (that is not as a proof of “pure existence” in a mathematical sense).Its investigation needs philosophical means. Husserl’s phenomenology is what is used, and then the conception of “bracketing reality” is modelled to generalize Peano arithmetic in its relation to set theory in the foundation of mathematics. The obtained model is equivalent to the generalization of Peano arithmetic by means of replacing the axiom of induction with that of transfinite induction.The sketch of the proof is organized in five steps: (1) a generalization of epoché; (2) involving transfinite induction in the transition between Peano arithmetic and set theory; (3) discussing the finiteness of Peano arithmetic; (4) applying transfinite induction to Peano arithmetic; (5) discussing an arithmetical model of reality.Accepting or rejecting the principle, two kinds of mathematics appear differing from each other by its relation to reality. Accepting the principle, mathematics has to include reality within itself in a kind of Pythagoreanism. These two kinds are called in paper correspondingly Hilbert mathematics and Gödel mathematics. The sketch of the proof of the principle demonstrates that the generalization of Peano arithmetic as above can be interpreted as a model of Hilbert mathematics into Gödel mathematics therefore showing that the former is not less consistent than the latter, and the principle is an independent axiom.The present paper follows a pathway grounded on Husserl’s phenomenology and “bracketing reality” to achieve the generalized arithmetic necessary for the principle to be founded in alternative ontology, in which there is no reality external to mathematics: reality is included within mathematics. That latter mathematics is able to self-found itself and can be called Hilbert mathematics in honour of Hilbert’s program for self-founding mathematics on the base of arithmetic.The principle of universal mathematizability is consistent to Hilbert mathematics, but not to Gödel mathematics. Consequently, its validity or rejection would resolve the problem which mathematics refers to our being; and vice versa: the choice between them for different reasons would confirm or refuse the principle as to the being.An information interpretation of Hilbert mathematics is involved. It is a kind of ontology of information. The Schrödinger equation in quantum mechanics is involved to illustrate that ontology. Thus the problem which of the two mathematics is more relevant to our being (rather than reality for reality is external only to Gödel mathematics) is discussed again in a new wayA few directions for future work can be: a rigorous formal proof of the principle as an independent axiom; the further development of information ontology consistent to both kinds of mathematics, but much more natural for Hilbert mathematics; the development of the information interpretation of quantum mechanics as a mathematical one for information ontology and thus Hilbert mathematics; the description of consciousness in terms of information ontology.

scholarly journals Transfer principle in quantum set theory

2007 ◽  
Vol 72 (2) ◽  
pp. 625-648 ◽  
Author(s):  
Masanao Ozawa
Keyword(s):  
Quantum Mechanics ◽  
Quantum Logic ◽  
Set Theory ◽  
Von Neumann Algebra ◽  
Transfer Principle ◽  
Real Numbers ◽  
Von Neumann ◽  
Equality Relation ◽  
Adjoint Operators ◽  
Transfer Theorems

AbstractIn 1981, Takeuti introduced quantum set theory as the quantum counterpart of Boolean valued models of set theory by constructing a model of set theory based on quantum logic represented by the lattice of closed subspaces in a Hilbert space and showed that appropriate quantum counterparts of ZFC axioms hold in the model. Here, Takeuti's formulation is extended to construct a model of set theory based on the logic represented by the lattice of projections in an arbitrary von Neumann algebra. A transfer principle is established that enables us to transfer theorems of ZFC to their quantum counterparts holding in the model. The set of real numbers in the model is shown to be in one-to-one correspondence with the set of self-adjoint operators affiliated with the von Neumann algebra generated by the logic. Despite the difficulty pointed out by Takeuti that equality axioms do not generally hold in quantum set theory, it is shown that equality axioms hold for any real numbers in the model. It is also shown that any observational proposition in quantum mechanics can be represented by a corresponding statement for real numbers in the model with the truth value consistent with the standard formulation of quantum mechanics, and that the equality relation between two real numbers in the model is equivalent with the notion of perfect correlation between corresponding observables (self-adjoint operators) in quantum mechanics. The paper is concluded with some remarks on the relevance to quantum set theory of the choice of the implication connective in quantum logic.

scholarly journals Universal quantum mechanics

2008 ◽  
Vol 78 (8) ◽  
Author(s):  
Steven B. Giddings
Keyword(s):  
Quantum Mechanics ◽  
Universal Quantum

CONCEPTUAL FOUNDATIONS OF QUANTUM MECHANICS: THE ROLE OF EVIDENCE THEORY, QUANTUM SETS, AND MODAL LOGIC

1999 ◽  
Vol 10 (01) ◽  
pp. 29-62 ◽  
Author(s):  
GERMANO RESCONI ◽  
GEORGE J. KLIR ◽  
ELIANO PESSA
Keyword(s):  
Quantum Mechanics ◽  
Modal Logic ◽  
Set Theory ◽  
Possible Worlds ◽  
Evidence Theory ◽  
Foundations Of Quantum Mechanics ◽  
Interpretation Of Quantum Mechanics ◽  
Many Worlds ◽  
New Interpretation ◽  
Quantum Phenomena

Recognizing that syntactic and semantic structures of classical logic are not sufficient to understand the meaning of quantum phenomena, we propose in this paper a new interpretation of quantum mechanics based on evidence theory. The connection between these two theories is obtained through a new language, quantum set theory, built on a suggestion by J. Bell. Further, we give a modal logic interpretation of quantum mechanics and quantum set theory by using Kripke's semantics of modal logic based on the concept of possible worlds. This is grounded on previous work of a number of researchers (Resconi, Klir, Harmanec) who showed how to represent evidence theory and other uncertainty theories in terms of modal logic. Moreover, we also propose a reformulation of the many-worlds interpretation of quantum mechanics in terms of Kripke's semantics. We thus show how three different theories — quantum mechanics, evidence theory, and modal logic — are interrelated. This opens, on one hand, the way to new applications of quantum mechanics within domains different from the traditional ones, and, on the other hand, the possibility of building new generalizations of quantum mechanics itself.

scholarly journals From Boolean Valued Analysis to Quantum Set Theory: Mathematical Worldview of Gaisi Takeuti

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 397
Author(s):  
Masanao Ozawa
Keyword(s):  
Quantum Mechanics ◽  
Quantum Logic ◽  
Set Theory ◽  
World View ◽  
Open Problems ◽  
Boolean Valued Analysis ◽  
Mathematical World ◽  
Distributive Law ◽  
And Algebra ◽  
Adjoint Operators

Gaisi Takeuti introduced Boolean valued analysis around 1974 to provide systematic applications of the Boolean valued models of set theory to analysis. Later, his methods were further developed by his followers, leading to solving several open problems in analysis and algebra. Using the methods of Boolean valued analysis, he further stepped forward to construct set theory that is based on quantum logic, as the first step to construct "quantum mathematics", a mathematics based on quantum logic. While it is known that the distributive law does not apply to quantum logic, and the equality axiom turns out not to hold in quantum set theory, he showed that the real numbers in quantum set theory are in one-to-one correspondence with the self-adjoint operators on a Hilbert space, or equivalently the physical quantities of the corresponding quantum system. As quantum logic is intrinsic and empirical, the results of the quantum set theory can be experimentally verified by quantum mechanics. In this paper, we analyze Takeuti’s mathematical world view underlying his program from two perspectives: set theoretical foundations of modern mathematics and extending the notion of sets to multi-valued logic. We outlook the present status of his program, and envisage the further development of the program, by which we would be able to take a huge step forward toward unraveling the mysteries of quantum mechanics that have persisted for many years.

scholarly journals The Frontier of Time: The Concept of Quantum Information

2020 ◽  
Author(s):  
Vasil Dinev Penchev
Keyword(s):  
Quantum Mechanics ◽  
Quantum Information ◽  
Natural Number ◽  
Set Theory ◽  
Peano Arithmetic ◽  
Axiom Of Choice ◽  
The Axiom Of Choice ◽  
Infinite Set ◽  
In Virtue Of

A concept of formal transcendentalism is utilized. The fundamental and definitive property of the totality suggests for “the totality to be all”, thus, its externality (unlike any other entity) is contained within it. This generates a fundamental (or philosophical) “doubling” of anything being referred to the totality, i.e. considered philosophically. Thus, that doubling as well as transcendentalism underlying it can be interpreted formally as an elementary choice such as a bit of information and a quantity corresponding to the number of elementary choices to be defined. This is the quantity of information defined both transcendentally and formally and thus, philosophically and mathematically. If one defines information specifically, as an elementary choice between finiteness (or mathematically, as any natural number of Peano arithmetic) and infinity (i.e. an actually infinite set in the meaning of set theory), the quantity of quantum information is defined. One can demonstrate that the so-defined quantum information and quantum information standardly defined by quantum mechanics are equivalent to each other. The equivalence of the axiom of choice and the well-ordering “theorem” is involved. It can be justified transcendentally as well, in virtue of transcendental equivalence implied by the totality. Thus, all can be considered as temporal as far anything possesses such a temporal counterpart necessarily. Formally defined, the frontier of time is the current choice now, a bit of information, furthermore interpretable as a qubit of quantum information.

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