1. The paper discusses quantum measure from a philosophical viewpoint. Quantum measure is a proposed three-dimensional measure that can measure both continuous and discrete phenomena uniformly.
2. Quantum measure is proposed as a more universal measure than Lebesgue or Borel measures. It is constructed to be invariant to the axiom of choice and resolves issues like how probability can become physical quantity.
3. The relationship between quantum mechanics and general relativity can be understood through this framework. They are complementary aspects of the quantum state of a system like the universe. However, general relativity is built assuming the axiom of choice while quantum mechanics is invariant to it, posing challenges for a unified theory of quantum gravity.
Vasil Penchev. Theory of Quantum Мeasure & ProbabilityVasil Penchev
Notes about “quantum gravity” for dummies (philosophers) written down by another dummy (philosopher)
Contents:
1. The Lebesgue and Borel measure
2. The quantum measure
3. The construction of quantum measure
4. Quantum measure vs. both the Lebesgue and the Borel measure
5. The origin of quantum measure
6. Physical quantity measured by quantum measure
7. Quantum measure and quantum quantity in terms of the Bekenstein bound
8. Speculation on generalized quantum measure:
The Gödel incompleteness can be modeled on the alleged incompleteness of quantum mechanics
Then the proved and even experimentally confirmed completeness of quantum mechanics can be reversely interpreted as a strategy of completeness as to the foundation of mathematics
Infinity is equivalent to a second and independent finiteness
Two independent Peano arithmetics as well as one single Hilbert space as an unification of geometry and arithmetic are sufficient to the self-foundation of mathematics
Quantum mechanics is inseparable from the foundation of mathematics and thus from set theory particularly
David hume and the limits of mathematical reasonBrendan Larvor
David Hume devoted sections of his works to refuting the indivisibility of space and time and ridiculing the doctrine of infinitesimals and the paradox of the angle of contact between a circle and tangent. Following the mathematical references Hume cites reveals that these paradoxes were used by philosophers not well-versed in mathematics, like Hume, to argue that rational mathematical inquiry had reached its limits and must surrender to faith or skepticism. Hume appears to have been out of date on developments in mathematics that had resolved these supposed paradoxes.
Quantum information as the information of infinite seriesVasil Penchev
The quantum information introduced by quantum mechanics is equivalent to that generalization of the classical information from finite to infinite series or collections. The quantity of information is the quantity of choices measured in the units of elementary choice. The qubit, can be interpreted as that generalization of bit, which is a choice among a continuum of alternatives. The axiom of choice is necessary for quantum information. The coherent state is transformed into a well-ordered series of results in time after measurement. The quantity of quantum information is the ordinal corresponding to the infinity series in question.
Action as the common measure of randomness and free willVasil Penchev
One needs a common measure of randomness and free well:
Both cause results, which are comparable
Given any fact without being known its origin, the question whether its origin is natural (i.e. random, occasional) or freewill action (intentional, constructive) arises
The result in both cases is one and the same as well as action and even the physical quantity of action
However the natural process being just random would require much more time in comparison with any intentional action for one and the same result
The thesis is: Einstein, Podolsky and Rosen’s argument (1935, Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? ) is another interpretation of the famous Gödel incompleteness argument (1931, Über formal unentscheidbare Sätze der Principia mathematica und verwandter Systeme I ) in terms of quantum mechanics
PRINCETON'S SPIRIT
Contemporary neopythagoreanism – The lodged at Princeton refugees – On quantum information as a mathematical doctrine – “The sixth problem” of Hilbert– Axiomatic logics, geometries, but why not also “physicses”? – The axiomatizing both of the theory of probability and of mechanics – The coincidence of model and reality as a solution of “the sixth problem” of Hilbert – The theorem about the absence of hidden parameters as a proof for the coincidence of model and reality – Bell’s inequalities as a generalization of von Neumann’s theorem – “The second problem” of Hilbert – Why “arithmetization”? – Arithmetization vs. geometrization? – Meta-mathematics: the foundation or self-foundation of mathematics – The problem of actual infinity – Actual infinity as a derivative of wholeness – The theory of Hilbert space as that domain of mathematics, which is able to found itself – Mathematical existence and existence in general – Mathematics as ontology: Pythagoreanism – Completeness, consistency … and additivity – The quantum nostrum of non-additivity – Transfinite induction: Peano or Gentzen arithmetic – A dual foundation of arithmetic: the “geometrization” of arithmetic – Gödel and Hilbert mathematics – The Kochen and Specker theorem – “Hidden parameter” does not “the element of reality” – The theorem of Kochen and Specker as a generalization of von Neumann’s – Duality, holism, and numberness (numericality) – Of I Ching generating Yin and Yang – The cyclic and holistic paradigm of dualistic Pythagoreanism versus the classical bipolar episteme – Any complete and consistent structure is non-additive − The incompleteness both of quantum mechanics and arithmetic? – Choice, number, and probability − Ψ-function in a generalized notation – The sense of Einstein’s “common covariance” – “Princeton” also for gauge theories – More about “dualistic pythagoreanism” – Quantity and property – Projection operator as statement (à la von Neumann)− Simultaneous undecidability – Does the notion of physical quantity imply the invariance of time moments? – Commuting and non-commuting operators – Perfecting the notion for simultaneous immeasurability – Quantum mechanics in Procrustean bed – The world is also a mathematical structure for its essence
Indeterminism in quantum mechanics can be interpreted as a form of occasionalism, which is called “quantum occasionalism”
It in turn implies the option for any probabilistic determinism and thus for causal determinism as a particular case to be extended equivalently to that form of occasionalism
Vasil Penchev. Theory of Quantum Мeasure & ProbabilityVasil Penchev
Notes about “quantum gravity” for dummies (philosophers) written down by another dummy (philosopher)
Contents:
1. The Lebesgue and Borel measure
2. The quantum measure
3. The construction of quantum measure
4. Quantum measure vs. both the Lebesgue and the Borel measure
5. The origin of quantum measure
6. Physical quantity measured by quantum measure
7. Quantum measure and quantum quantity in terms of the Bekenstein bound
8. Speculation on generalized quantum measure:
The Gödel incompleteness can be modeled on the alleged incompleteness of quantum mechanics
Then the proved and even experimentally confirmed completeness of quantum mechanics can be reversely interpreted as a strategy of completeness as to the foundation of mathematics
Infinity is equivalent to a second and independent finiteness
Two independent Peano arithmetics as well as one single Hilbert space as an unification of geometry and arithmetic are sufficient to the self-foundation of mathematics
Quantum mechanics is inseparable from the foundation of mathematics and thus from set theory particularly
David hume and the limits of mathematical reasonBrendan Larvor
David Hume devoted sections of his works to refuting the indivisibility of space and time and ridiculing the doctrine of infinitesimals and the paradox of the angle of contact between a circle and tangent. Following the mathematical references Hume cites reveals that these paradoxes were used by philosophers not well-versed in mathematics, like Hume, to argue that rational mathematical inquiry had reached its limits and must surrender to faith or skepticism. Hume appears to have been out of date on developments in mathematics that had resolved these supposed paradoxes.
Quantum information as the information of infinite seriesVasil Penchev
The quantum information introduced by quantum mechanics is equivalent to that generalization of the classical information from finite to infinite series or collections. The quantity of information is the quantity of choices measured in the units of elementary choice. The qubit, can be interpreted as that generalization of bit, which is a choice among a continuum of alternatives. The axiom of choice is necessary for quantum information. The coherent state is transformed into a well-ordered series of results in time after measurement. The quantity of quantum information is the ordinal corresponding to the infinity series in question.
Action as the common measure of randomness and free willVasil Penchev
One needs a common measure of randomness and free well:
Both cause results, which are comparable
Given any fact without being known its origin, the question whether its origin is natural (i.e. random, occasional) or freewill action (intentional, constructive) arises
The result in both cases is one and the same as well as action and even the physical quantity of action
However the natural process being just random would require much more time in comparison with any intentional action for one and the same result
The thesis is: Einstein, Podolsky and Rosen’s argument (1935, Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? ) is another interpretation of the famous Gödel incompleteness argument (1931, Über formal unentscheidbare Sätze der Principia mathematica und verwandter Systeme I ) in terms of quantum mechanics
PRINCETON'S SPIRIT
Contemporary neopythagoreanism – The lodged at Princeton refugees – On quantum information as a mathematical doctrine – “The sixth problem” of Hilbert– Axiomatic logics, geometries, but why not also “physicses”? – The axiomatizing both of the theory of probability and of mechanics – The coincidence of model and reality as a solution of “the sixth problem” of Hilbert – The theorem about the absence of hidden parameters as a proof for the coincidence of model and reality – Bell’s inequalities as a generalization of von Neumann’s theorem – “The second problem” of Hilbert – Why “arithmetization”? – Arithmetization vs. geometrization? – Meta-mathematics: the foundation or self-foundation of mathematics – The problem of actual infinity – Actual infinity as a derivative of wholeness – The theory of Hilbert space as that domain of mathematics, which is able to found itself – Mathematical existence and existence in general – Mathematics as ontology: Pythagoreanism – Completeness, consistency … and additivity – The quantum nostrum of non-additivity – Transfinite induction: Peano or Gentzen arithmetic – A dual foundation of arithmetic: the “geometrization” of arithmetic – Gödel and Hilbert mathematics – The Kochen and Specker theorem – “Hidden parameter” does not “the element of reality” – The theorem of Kochen and Specker as a generalization of von Neumann’s – Duality, holism, and numberness (numericality) – Of I Ching generating Yin and Yang – The cyclic and holistic paradigm of dualistic Pythagoreanism versus the classical bipolar episteme – Any complete and consistent structure is non-additive − The incompleteness both of quantum mechanics and arithmetic? – Choice, number, and probability − Ψ-function in a generalized notation – The sense of Einstein’s “common covariance” – “Princeton” also for gauge theories – More about “dualistic pythagoreanism” – Quantity and property – Projection operator as statement (à la von Neumann)− Simultaneous undecidability – Does the notion of physical quantity imply the invariance of time moments? – Commuting and non-commuting operators – Perfecting the notion for simultaneous immeasurability – Quantum mechanics in Procrustean bed – The world is also a mathematical structure for its essence
Indeterminism in quantum mechanics can be interpreted as a form of occasionalism, which is called “quantum occasionalism”
It in turn implies the option for any probabilistic determinism and thus for causal determinism as a particular case to be extended equivalently to that form of occasionalism
History as the ontology of time requires to be understood what time is
Time is the transformation of future into the past by the choices in the present
History should be grounded on that understanding of historical time, which would include the present and future rather than only the past
Representation & Realityby Language (How to make a home quantum computer) Vasil Penchev
Reality as if is doubled in relation to language
We will model this doubling by two Turing machines (i.e. by usual computers) in a kind of “dialog”: the one for reality, the other for its image in language
The two ones have to reach the state of equilibrium to each other
At last, one can demonstrate that the pair of them is equivalent to a quantum computer
One can construct a model of two independent Turing machines allowing of a series of relevant interpretations:
Language
Quantum computer
Representation and metaphor
Reality and ontology
In turn that model is based on the concepts of choice and information
Vasil Penchev. Continuity and Continuum in Nonstandard UniversumVasil Penchev
1. The document discusses infinity and the axiom of choice in mathematics. It describes how mathematics approaches infinity through axioms, negation of axioms, and postulating properties of infinite sets by analogy to finite sets.
2. It explains formulations of the axiom of choice like Zorn's lemma and defines paradoxes like Banach-Tarski that rely on the axiom of choice. It also discusses the continuum hypothesis.
3. The author argues that quantum mechanics provides empirical evidence supporting the axiom of choice since entanglement is analogous to the Banach-Tarski paradox. They also discuss negating the continuum hypothesis and axiom of foundation.
Problem of the direct quantum-information transformation of chemical substanceVasil Penchev
1. The document discusses the possibility of directly transforming one chemical substance into another through a "Trigger field" as proposed in a science fiction novel.
2. It explores how quantum mechanics, which underlies chemistry, can be interpreted in terms of quantum information and entanglement. Entanglement could theoretically allow the direct alteration of a substance's quantum information and transformation into another substance from a distance.
3. While a standalone "Trigger field" is not currently known to exist, the document argues that entanglement provides a theoretical framework for how a field could directly change a substance's quantum information and transform it into another, as envisioned in the science fiction story.
Gödel’s completeness (1930) nd incompleteness (1931) theorems: A new reading ...Vasil Penchev
(T2) Peano arithmetic admits anyway generalizations consistent to infinity and thus to some addable axiom(s) of inf(T1) Peano arithmetic cannot serve as the ground of mathematics for it is inconsistent to infinity, and infinity is necessary for its foundation
Though Peano arithmetic cannot be complemented by any axiom of infinity, there exists at least one (logical) axiomatics consistent to infinity
That is nothing else than right a new reading at issue and comparative interpretation of Gödel’s papers meant here
inity
The most utilized example of those generalizations is the separable complex Hilbert space: it is able to equate the possibility of pure existence to the probability of statistical ensemble
(T3) Any generalization of Peano arithmetic consistent to infinity, e.g. the separable complex Hilbert space, can serve as a foundation for mathematics to found itself and by itself
Математизирането на историята: число и битиеVasil Penchev
Книгата е първа част от по-широк замисъл − „Числа“. Така е назована на гръцки и от него на всички езици четвърта глава от Библията, посветена на преброяването на изра-илтяните и похода в пустинята. Стремежът е се да представи едно осъвременено пита-горейство. Предвиждат се още две части:
Синтактично-семантично интерпретиране на вълновата функция. Число и знак
Математизирането на историята. Число и история
Книгата е предназначена за научни работници в областта на философията, историята и математиката,
This document discusses quantum computers from a mathematical perspective by comparing them to Turing machines. It proposes that a quantum computer can be modeled as a Turing machine with an infinite tape of "qubits" rather than bits. This raises philosophical questions about the relationship between mathematical models and reality when dealing with infinity. The document also explores how concepts like information, choice, and measurement are understood differently in quantum as opposed to classical computation.
Quantum Mechanics as a Measure Theory: The Theory of Quantum MeasureVasil Penchev
This document discusses representing quantum mechanics as a measure theory, with the key points being:
1. Quantum measure can unify the measurement of discrete and continuous quantities by treating them as "much" and "many".
2. The unit of quantum measure is the qubit, allowing it to jointly measure probability, quantity, order, and disorder.
3. All physical processes can be interpreted as computations of a quantum computer, with the universal substance being quantum information.
4. Quantum measure can provide a nonlocal explanation for the Aharonov-Bohm effect by linking it to the electromagnetic nature of space-time itself.
The question is:
•
How should skepticism refer to itself?
•
The classical example might be the doubt of Descartes, which led him to the necessary obviousness of who doubts
•
The formal logical structure is the same as the “antinomy of the Liar”
•
That new interpretation of it can be called “antinomy of the Skeptic”
Is Mass at Rest One and the Same? A Philosophical Comment: on the Quantum I...Vasil Penchev
The way, in which quantum information can unify quantum mechanics (and therefore the standard
model) and general relativity, is investigated. Quantum information is defined as the generalization
of the concept of information as to the choice among infinite sets of alternatives. Relevantly, the
axiom of choice is necessary in general. The unit of quantum information, a qubit is interpreted
as a relevant elementary choice among an infinite set of alternatives generalizing that of a bit.
The invariance to the axiom of choice shared by quantum mechanics is introduced: It constitutes
quantum information as the relation of any state unorderable in principle (e.g. any coherent quantum
state before measurement) and the same state already well-ordered (e.g. the well-ordered statistical
ensemble of the measurement of the quantum system at issue). This allows of equating the classical and
quantum time correspondingly as the well-ordering of any physical quantity or quantities and their
coherent superposition. That equating is interpretable as the isomorphism of Minkowski space and
Hilbert space. Quantum information is the structure interpretable in both ways and thus underlying
their unification. Its deformation is representable correspondingly as gravitation in the deformed
pseudo-Riemannian space of general relativity and the entanglement of two or more quantum
systems. The standard model studies a single quantum system and thus privileges a single reference
frame turning out to be inertial for the generalized symmetry [U(1)]X[SU(2)]X[SU(3)] “gauging” the
standard model. As the standard model refers to a single quantum system, it is necessarily linear
and thus the corresponding privileged reference frame is necessary inertial. The Higgs mechanism
U(1) → [U(1)]X[SU(2)] confirmed enough already experimentally describes exactly the choice of the
initial position of a privileged reference frame as the corresponding breaking of the symmetry. The
standard model defines ‘mass at rest’ linearly and absolutely, but general relativity non-linearly
and relatively. The “Big Bang” hypothesis is additional interpreting that position as that of the
“Big Bang”. It serves also in order to reconcile the linear standard model in the singularity of the
“Big Bang” with the observed nonlinearity of the further expansion of the universe described very
well by general relativity. Quantum information links the standard model and general relativity in
another way by mediation of entanglement. The linearity and absoluteness of the former and the
nonlinearity and relativeness of the latter can be considered as the relation of a whole and the same
whole divided into parts entangled in general.
What is quantum information? Information symmetry and mechanical motionVasil Penchev
The concept of quantum information is introduced as both normed superposition of two orthogonal subspaces of the separable complex Hilbert space and invariance of Hamilton and Lagrange representation of any mechanical system. The base is the isomorphism of the standard introduction and the representation of a qubit to a 3D unit ball, in which two points are chosen.
The separable complex Hilbert space is considered as the free variable of quantum information and any point in it (a wave function describing a state of a quantum system) as its value as the bound variable.
A qubit is equivalent to the generalization of ‘bit’ from the set of two equally probable alternatives to an infinite set of alternatives. Then, that Hilbert space is considered as a generalization of Peano arithmetic where any unit is substituted by a qubit and thus the set of natural number is mappable within any qubit as the complex internal structure of the unit or a different state of it. Thus, any mathematical structure being reducible to set theory is representable as a set of wave functions and a subspace of the separable complex Hilbert space, and it can be identified as the category of all categories for any functor represents an operator transforming a set (or subspace) of the separable complex Hilbert space into another. Thus, category theory is isomorphic to the Hilbert-space representation of set theory & Peano arithmetic as above.
Given any value of quantum information, i.e. a point in the separable complex Hilbert space, it always admits two equally acceptable interpretations: the one is physical, the other is mathematical. The former is a wave function as the exhausted description of a certain state of a certain quantum system. The latter chooses a certain mathematical structure among a certain category. Thus there is no way to be distinguished a mathematical structure from a physical state for both are described exhaustedly as a value of quantum information. This statement in turn can be utilized to be defined quantum information by the identity of any mathematical structure to a physical state, and also vice versa. Further, that definition is equivalent to both standard definition as the normed superposition and invariance of Hamilton and Lagrange interpretation of mechanical motion introduced in the beginning of the paper.
Then, the concept of information symmetry can be involved as the symmetry between three elements or two pairs of elements: Lagrange representation and each counterpart of the pair of Hamilton representation. The sense and meaning of information symmetry may be visualized by a single (quantum) bit and its interpretation as both (privileged) reference frame and the symmetries of the Standard model.
Analogia entis as analogy universalized and formalized rigorously and mathema...Vasil Penchev
THE SECOND WORLD CONGRESS ON ANALOGY, POZNAŃ, MAY 24-26, 2017
(The Venue: Sala Lubrańskiego (Lubrański’s Hall at the Collegium Minus), Adam Mickiewicz University, Address: ul. Wieniawskiego 1) The presentation: 24 May, 15:30
Universal constants like Planck's constant h, the speed of light c, gravitational constant G, and Boltzmann's constant k can be used to structure theoretical physics. They lead to three main theories: quantum field theory (h and c), general relativity (c and G), and quantum statistics (h and k). While not fully unified, these theories underlie the standard models of particle physics and cosmology. Fundamental metrology provides reliable standards by determining the values of dimensionless constants that depend on h, c, k, and G. Metrology exists at the intersection of fundamental physics described by these theories and emergent physics involving statistical mechanics.
Quantum phenomena modeled by interactions between many classical worldsLex Pit
Michael J. W. Hall, Dirk-André Deckert, and Howard M. Wiseman
ABSTRACT
We investigate whether quantum theory can be understood as the continuum limit of a mechanical theory, in which there is a huge, but finite, number of classical “worlds,” and quantum effects arise solely from a universal interaction between these worlds, without reference to any wave function. Here, a “world” means an entire universe with well-defined properties, determined by the classical configuration of its particles and fields. In our approach, each world evolves deterministically, probabilities arise due to ignorance as to which world a given observer occupies, and we argue that in the limit of infinitely many worlds the wave function can be recovered (as a secondary object) from the motion of these worlds. We introduce a simple model of such a “many interacting worlds” approach and show that it can reproduce some generic quantum phenomena—such as Ehrenfest’s theorem, wave packet spreading, barrier tunneling, and zero-point energy—as a direct consequence of mutual repulsion between worlds. Finally, we perform numerical simulations using our approach. We demonstrate, first, that it can be used to calculate quantum ground states, and second, that it is capable of reproducing, at least qualitatively, the double-slit interference phenomenon.
DOI: http://dx.doi.org/10.1103/PhysRevX.4.041013
All those studies in quantum mechanics and the theory of quantum information reflect on the philosophy of space and its cognition
Space is the space of realizing choice
Space unlike Hilbert space is not able to represent the states before and after choice or their unification in information
However space unlike Hilbert space is:
The space of all our experience, and thus
The space of any possible empirical knowledge
Vasil Penchev. Cyclic mechanics. The principle of cyclicityVasil Penchev
1) The document discusses a theory of cyclic mechanics intended to unify quantum mechanics and general relativity.
2) It proposes several foundational principles, including that the universe can return to any point, time is not uniformly flowing, and all laws must be invariant to discrete and continuous transformations.
3) A key concept is introducing the notion of "quantum measure" and "quantum information" to provide a common measure to equate equations from different theories, with the goal of unifying them.
Vasil Penchev. Gravity as entanglement, and entanglement as gravityVasil Penchev
1. The document discusses interpreting gravity as entanglement by investigating the conditions under which general relativity and quantum mechanics can be mapped to each other mathematically.
2. It outlines a strategy to interpret entanglement as inertial mass and gravitational mass, and to view gravity as another interpretation of any quantum mechanical or mechanical movement.
3. This would allow gravity to be incorporated into the standard model by generalizing the concept of quantum field to include entanglement, represented by a cyclic Yin-Yang mathematical structure.
History as the ontology of time requires to be understood what time is
Time is the transformation of future into the past by the choices in the present
History should be grounded on that understanding of historical time, which would include the present and future rather than only the past
Representation & Realityby Language (How to make a home quantum computer) Vasil Penchev
Reality as if is doubled in relation to language
We will model this doubling by two Turing machines (i.e. by usual computers) in a kind of “dialog”: the one for reality, the other for its image in language
The two ones have to reach the state of equilibrium to each other
At last, one can demonstrate that the pair of them is equivalent to a quantum computer
One can construct a model of two independent Turing machines allowing of a series of relevant interpretations:
Language
Quantum computer
Representation and metaphor
Reality and ontology
In turn that model is based on the concepts of choice and information
Vasil Penchev. Continuity and Continuum in Nonstandard UniversumVasil Penchev
1. The document discusses infinity and the axiom of choice in mathematics. It describes how mathematics approaches infinity through axioms, negation of axioms, and postulating properties of infinite sets by analogy to finite sets.
2. It explains formulations of the axiom of choice like Zorn's lemma and defines paradoxes like Banach-Tarski that rely on the axiom of choice. It also discusses the continuum hypothesis.
3. The author argues that quantum mechanics provides empirical evidence supporting the axiom of choice since entanglement is analogous to the Banach-Tarski paradox. They also discuss negating the continuum hypothesis and axiom of foundation.
Problem of the direct quantum-information transformation of chemical substanceVasil Penchev
1. The document discusses the possibility of directly transforming one chemical substance into another through a "Trigger field" as proposed in a science fiction novel.
2. It explores how quantum mechanics, which underlies chemistry, can be interpreted in terms of quantum information and entanglement. Entanglement could theoretically allow the direct alteration of a substance's quantum information and transformation into another substance from a distance.
3. While a standalone "Trigger field" is not currently known to exist, the document argues that entanglement provides a theoretical framework for how a field could directly change a substance's quantum information and transform it into another, as envisioned in the science fiction story.
Gödel’s completeness (1930) nd incompleteness (1931) theorems: A new reading ...Vasil Penchev
(T2) Peano arithmetic admits anyway generalizations consistent to infinity and thus to some addable axiom(s) of inf(T1) Peano arithmetic cannot serve as the ground of mathematics for it is inconsistent to infinity, and infinity is necessary for its foundation
Though Peano arithmetic cannot be complemented by any axiom of infinity, there exists at least one (logical) axiomatics consistent to infinity
That is nothing else than right a new reading at issue and comparative interpretation of Gödel’s papers meant here
inity
The most utilized example of those generalizations is the separable complex Hilbert space: it is able to equate the possibility of pure existence to the probability of statistical ensemble
(T3) Any generalization of Peano arithmetic consistent to infinity, e.g. the separable complex Hilbert space, can serve as a foundation for mathematics to found itself and by itself
Математизирането на историята: число и битиеVasil Penchev
Книгата е първа част от по-широк замисъл − „Числа“. Така е назована на гръцки и от него на всички езици четвърта глава от Библията, посветена на преброяването на изра-илтяните и похода в пустинята. Стремежът е се да представи едно осъвременено пита-горейство. Предвиждат се още две части:
Синтактично-семантично интерпретиране на вълновата функция. Число и знак
Математизирането на историята. Число и история
Книгата е предназначена за научни работници в областта на философията, историята и математиката,
This document discusses quantum computers from a mathematical perspective by comparing them to Turing machines. It proposes that a quantum computer can be modeled as a Turing machine with an infinite tape of "qubits" rather than bits. This raises philosophical questions about the relationship between mathematical models and reality when dealing with infinity. The document also explores how concepts like information, choice, and measurement are understood differently in quantum as opposed to classical computation.
Quantum Mechanics as a Measure Theory: The Theory of Quantum MeasureVasil Penchev
This document discusses representing quantum mechanics as a measure theory, with the key points being:
1. Quantum measure can unify the measurement of discrete and continuous quantities by treating them as "much" and "many".
2. The unit of quantum measure is the qubit, allowing it to jointly measure probability, quantity, order, and disorder.
3. All physical processes can be interpreted as computations of a quantum computer, with the universal substance being quantum information.
4. Quantum measure can provide a nonlocal explanation for the Aharonov-Bohm effect by linking it to the electromagnetic nature of space-time itself.
The question is:
•
How should skepticism refer to itself?
•
The classical example might be the doubt of Descartes, which led him to the necessary obviousness of who doubts
•
The formal logical structure is the same as the “antinomy of the Liar”
•
That new interpretation of it can be called “antinomy of the Skeptic”
Is Mass at Rest One and the Same? A Philosophical Comment: on the Quantum I...Vasil Penchev
The way, in which quantum information can unify quantum mechanics (and therefore the standard
model) and general relativity, is investigated. Quantum information is defined as the generalization
of the concept of information as to the choice among infinite sets of alternatives. Relevantly, the
axiom of choice is necessary in general. The unit of quantum information, a qubit is interpreted
as a relevant elementary choice among an infinite set of alternatives generalizing that of a bit.
The invariance to the axiom of choice shared by quantum mechanics is introduced: It constitutes
quantum information as the relation of any state unorderable in principle (e.g. any coherent quantum
state before measurement) and the same state already well-ordered (e.g. the well-ordered statistical
ensemble of the measurement of the quantum system at issue). This allows of equating the classical and
quantum time correspondingly as the well-ordering of any physical quantity or quantities and their
coherent superposition. That equating is interpretable as the isomorphism of Minkowski space and
Hilbert space. Quantum information is the structure interpretable in both ways and thus underlying
their unification. Its deformation is representable correspondingly as gravitation in the deformed
pseudo-Riemannian space of general relativity and the entanglement of two or more quantum
systems. The standard model studies a single quantum system and thus privileges a single reference
frame turning out to be inertial for the generalized symmetry [U(1)]X[SU(2)]X[SU(3)] “gauging” the
standard model. As the standard model refers to a single quantum system, it is necessarily linear
and thus the corresponding privileged reference frame is necessary inertial. The Higgs mechanism
U(1) → [U(1)]X[SU(2)] confirmed enough already experimentally describes exactly the choice of the
initial position of a privileged reference frame as the corresponding breaking of the symmetry. The
standard model defines ‘mass at rest’ linearly and absolutely, but general relativity non-linearly
and relatively. The “Big Bang” hypothesis is additional interpreting that position as that of the
“Big Bang”. It serves also in order to reconcile the linear standard model in the singularity of the
“Big Bang” with the observed nonlinearity of the further expansion of the universe described very
well by general relativity. Quantum information links the standard model and general relativity in
another way by mediation of entanglement. The linearity and absoluteness of the former and the
nonlinearity and relativeness of the latter can be considered as the relation of a whole and the same
whole divided into parts entangled in general.
What is quantum information? Information symmetry and mechanical motionVasil Penchev
The concept of quantum information is introduced as both normed superposition of two orthogonal subspaces of the separable complex Hilbert space and invariance of Hamilton and Lagrange representation of any mechanical system. The base is the isomorphism of the standard introduction and the representation of a qubit to a 3D unit ball, in which two points are chosen.
The separable complex Hilbert space is considered as the free variable of quantum information and any point in it (a wave function describing a state of a quantum system) as its value as the bound variable.
A qubit is equivalent to the generalization of ‘bit’ from the set of two equally probable alternatives to an infinite set of alternatives. Then, that Hilbert space is considered as a generalization of Peano arithmetic where any unit is substituted by a qubit and thus the set of natural number is mappable within any qubit as the complex internal structure of the unit or a different state of it. Thus, any mathematical structure being reducible to set theory is representable as a set of wave functions and a subspace of the separable complex Hilbert space, and it can be identified as the category of all categories for any functor represents an operator transforming a set (or subspace) of the separable complex Hilbert space into another. Thus, category theory is isomorphic to the Hilbert-space representation of set theory & Peano arithmetic as above.
Given any value of quantum information, i.e. a point in the separable complex Hilbert space, it always admits two equally acceptable interpretations: the one is physical, the other is mathematical. The former is a wave function as the exhausted description of a certain state of a certain quantum system. The latter chooses a certain mathematical structure among a certain category. Thus there is no way to be distinguished a mathematical structure from a physical state for both are described exhaustedly as a value of quantum information. This statement in turn can be utilized to be defined quantum information by the identity of any mathematical structure to a physical state, and also vice versa. Further, that definition is equivalent to both standard definition as the normed superposition and invariance of Hamilton and Lagrange interpretation of mechanical motion introduced in the beginning of the paper.
Then, the concept of information symmetry can be involved as the symmetry between three elements or two pairs of elements: Lagrange representation and each counterpart of the pair of Hamilton representation. The sense and meaning of information symmetry may be visualized by a single (quantum) bit and its interpretation as both (privileged) reference frame and the symmetries of the Standard model.
Analogia entis as analogy universalized and formalized rigorously and mathema...Vasil Penchev
THE SECOND WORLD CONGRESS ON ANALOGY, POZNAŃ, MAY 24-26, 2017
(The Venue: Sala Lubrańskiego (Lubrański’s Hall at the Collegium Minus), Adam Mickiewicz University, Address: ul. Wieniawskiego 1) The presentation: 24 May, 15:30
Universal constants like Planck's constant h, the speed of light c, gravitational constant G, and Boltzmann's constant k can be used to structure theoretical physics. They lead to three main theories: quantum field theory (h and c), general relativity (c and G), and quantum statistics (h and k). While not fully unified, these theories underlie the standard models of particle physics and cosmology. Fundamental metrology provides reliable standards by determining the values of dimensionless constants that depend on h, c, k, and G. Metrology exists at the intersection of fundamental physics described by these theories and emergent physics involving statistical mechanics.
Quantum phenomena modeled by interactions between many classical worldsLex Pit
Michael J. W. Hall, Dirk-André Deckert, and Howard M. Wiseman
ABSTRACT
We investigate whether quantum theory can be understood as the continuum limit of a mechanical theory, in which there is a huge, but finite, number of classical “worlds,” and quantum effects arise solely from a universal interaction between these worlds, without reference to any wave function. Here, a “world” means an entire universe with well-defined properties, determined by the classical configuration of its particles and fields. In our approach, each world evolves deterministically, probabilities arise due to ignorance as to which world a given observer occupies, and we argue that in the limit of infinitely many worlds the wave function can be recovered (as a secondary object) from the motion of these worlds. We introduce a simple model of such a “many interacting worlds” approach and show that it can reproduce some generic quantum phenomena—such as Ehrenfest’s theorem, wave packet spreading, barrier tunneling, and zero-point energy—as a direct consequence of mutual repulsion between worlds. Finally, we perform numerical simulations using our approach. We demonstrate, first, that it can be used to calculate quantum ground states, and second, that it is capable of reproducing, at least qualitatively, the double-slit interference phenomenon.
DOI: http://dx.doi.org/10.1103/PhysRevX.4.041013
All those studies in quantum mechanics and the theory of quantum information reflect on the philosophy of space and its cognition
Space is the space of realizing choice
Space unlike Hilbert space is not able to represent the states before and after choice or their unification in information
However space unlike Hilbert space is:
The space of all our experience, and thus
The space of any possible empirical knowledge
Vasil Penchev. Cyclic mechanics. The principle of cyclicityVasil Penchev
1) The document discusses a theory of cyclic mechanics intended to unify quantum mechanics and general relativity.
2) It proposes several foundational principles, including that the universe can return to any point, time is not uniformly flowing, and all laws must be invariant to discrete and continuous transformations.
3) A key concept is introducing the notion of "quantum measure" and "quantum information" to provide a common measure to equate equations from different theories, with the goal of unifying them.
Vasil Penchev. Gravity as entanglement, and entanglement as gravityVasil Penchev
1. The document discusses interpreting gravity as entanglement by investigating the conditions under which general relativity and quantum mechanics can be mapped to each other mathematically.
2. It outlines a strategy to interpret entanglement as inertial mass and gravitational mass, and to view gravity as another interpretation of any quantum mechanical or mechanical movement.
3. This would allow gravity to be incorporated into the standard model by generalizing the concept of quantum field to include entanglement, represented by a cyclic Yin-Yang mathematical structure.
Hilbert Space and pseudo-Riemannian Space: The Common Base of Quantum Informa...Vasil Penchev
Hilbert space underlying quantum mechanics and pseudo-Riemannian space underlying general relativity share a common base of quantum information. Hilbert space can be interpreted as the free variable of quantum information, and any point in it, being equivalent to a wave function (and thus, to a state of a quantum system), as a value of that variable of quantum information. In turn, pseudo-Riemannian space can be interpreted as the interaction of two or more quantities of quantum information and thus, as two or more entangled quantum systems. Consequently, one can distinguish local physical interactions describable by a single Hilbert space (or by any factorizable tensor product of such ones) and non-local physical interactions describable only by means by that Hilbert space, which cannot be factorized as any tensor product of the Hilbert spaces, by means of which one can describe the interacting quantum subsystems separately. Any interaction, which can be exhaustedly described in a single Hilbert space, such as the weak, strong, and electromagnetic one, is local in terms of quantum information. Any interaction, which cannot be described thus, is nonlocal in terms of quantum information. Any interaction, which is exhaustedly describable by pseudo-Riemannian space, such as gravity, is nonlocal in this sense. Consequently all known physical interaction can be described by a single geometrical base interpreting it in terms of quantum information.
Gravity as entanglement, and entanglement as gravityVasil Penchev
1) The document discusses the relationship between gravity and quantum entanglement, exploring the possibility that they are equivalent or closely connected concepts.
2) It outlines an approach to interpret gravity in terms of a generalized quantum field theory that includes entanglement, which could explain why gravity cannot be quantized.
3) The key idea is that entanglement expressed "outside" of space-time points looks like gravity "inside", and vice versa, with gravity representing a smooth constraint on the quantum behavior of entities imposed by all others.
Dark matter modeled as a Bose Einstein gluon condensate with an energy density relative to baryonic energy density in agreement with observation (ArXiv: 1507.00460)
This document discusses dimensions in mathematics and physics. It begins by explaining one-dimensional, two-dimensional, and three-dimensional objects like lines, squares, cubes, and tesseracts. It then discusses higher dimensions posited by theories like string theory and M-theory. Key definitions of dimension discussed include topological dimension, Hausdorff dimension, and covering dimension. In physics, it discusses the three spatial dimensions and time as the fourth dimension, as well as theories proposing additional curled up dimensions to explain phenomena.
This document provides an introduction to complex systems theory by comparing and contrasting it to other fields like physics, biology, and social sciences. It discusses that complex systems theory draws upon aspects of all three disciplines. The document outlines that complex systems differ from traditional physics in that they often have unpredictable components and interactions that change over time, making them more algorithmic than analytic. Overall, complex systems theory studies systems with many interacting parts where small changes can lead to unpredictable consequences.
This document provides an introduction to complex systems theory by comparing and contrasting it to other fields like physics, biology, and social sciences. It discusses that complex systems theory draws upon aspects of all three disciplines. The document outlines that complex systems differ from traditional physics in that they often have unpredictable components and interactions that change over time, making them more algorithmic than analytic. Overall, complex systems theory studies systems with many interacting parts where small changes can lead to unpredictable consequences.
This document is Albert Einstein's book "Relativity: The Special and General Theory" which explores his theories of special and general relativity. The book has three parts, with the first part focusing on his special theory of relativity. It discusses topics like physical meaning of geometry, coordinate systems, classical mechanics vs relativity, and more. The book provides detailed explanations of Einstein's groundbreaking theories and the evidence supporting them.
Albert Einstein's book Relativity: The Special and General Theory explores both the special theory of relativity and the general theory of relativity. Part I focuses on the special theory of relativity, covering topics like the system of coordinates, space and time in classical mechanics, and the principle of relativity. Einstein aims to provide an exact insight into the theory of relativity for readers without an extensive mathematical background. He emphasizes developing a physical interpretation of concepts like distance and position.
NO ONE CAN FLY UNTILL LEARNED TO WALK.......
from monkeys to men ,,fro crawling to fly ..it took us centuries to take a small step..how we can evolve... and give our future what our ancestors given us ...try to learn the way THEY THINK... how beautifully they adore the nature and think the things .. differently.....
A relationship between mass as a geometric concept and motion associated with a closed curve in spacetime (a notion taken from differential geometry) is investigated. We show that the 4-dimensional exterior Schwarzschild solution of the General Theory of Relativity can be mapped to a 4-dimensional Euclidean spacetime manifold. As a consequence of this mapping, the quantity M in the exterior Schwarzschild solution which is usually attributed to a massive central object is shown to correspond to a geometric property of spacetime. An additional outcome of this analysis is the discovery that, because M is a property of spacetime geometry, an anisotropy with respect to its spacetime components measured in a Minkowski tangent space defined with respect to a spacetime event P by an observer O who is stationary with respect to the spacetime event P, may be a sensitive measure of an anisotropic cosmic accelerated expansion. The presence of anisotropy in the cosmic accelerated expansion may contribute to the reason that there are currently two prevailing measured estimates of this quantity
Similar to Quantum Measure from a Philosophical Viewpoint (20)
The generalization of the Periodic table. The "Periodic table" of "dark matter"Vasil Penchev
The thesis is: the “periodic table” of “dark matter” is equivalent to the standard periodic table of the visible matter being entangled. Thus, it is to consist of all possible entangled states of the atoms of chemical elements as quantum systems. In other words, an atom of any chemical element and as a quantum system, i.e. as a wave function, should be represented as a non-orthogonal in general (i.e. entangled) subspace of the separable complex Hilbert space relevant to the system to which the atom at issue is related as a true part of it. The paper follows previous publications of mine stating that “dark matter” and “dark energy” are projections of arbitrarily entangled states on the cognitive “screen” of Einstein’s “Mach’s principle” in general relativity postulating that gravitational field can be generated only by mass or energy.
Modal History versus Counterfactual History: History as IntentionVasil Penchev
The distinction of whether real or counterfactual history makes sense only post factum. However, modal history is to be defined only as ones’ intention and thus, ex-ante. Modal history is probable history, and its probability is subjective. One needs phenomenological “epoché” in relation to its reality (respectively, counterfactuality). Thus, modal history describes historical “phenomena” in Husserl’s sense and would need a specific application of phenomenological reduction, which can be called historical reduction. Modal history doubles history just as the recorded history of historiography does it. That doubling is a necessary condition of historical objectivity including one’s subjectivity: whether actors’, ex-anteor historians’ post factum. The objectivity doubled by ones’ subjectivity constitute “hermeneutical circle”.
Both classical and quantum information [autosaved]Vasil Penchev
Information can be considered a the most fundamental, philosophical, physical and mathematical concept originating from the totality by means of physical and mathematical transcendentalism (the counterpart of philosophical transcendentalism). Classical and quantum information. particularly by their units, bit and qubit, correspond and unify the finite and infinite:
As classical information is relevant to finite series and sets, as quantum information, to infinite ones. The separable complex Hilbert space of quantum mechanics can be represented equivalently as “qubit space”) as quantum information and doubled dually or “complimentary” by Hilbert arithmetic (classical information).
A CLASS OF EXEMPLES DEMONSTRATING THAT “푃푃≠푁푁푁 ” IN THE “P VS NP” PROBLEMVasil Penchev
The CMI Millennium “P vs NP Problem” can be resolved e.g. if one shows at least one counterexample to the “P=NP” conjecture. A certain class of problems being such counterexamples will be formulated. This implies the rejection of the hypothesis “P=NP” for any conditions satisfying the formulation of the problem. Thus, the solution “P≠NP” of the problem in general is proved. The class of counterexamples can be interpreted as any quantum superposition of any finite set of quantum states. The Kochen-Specker theorem is involved. Any fundamentally random choice among a finite set of alternatives belong to “NP’ but not to “P”. The conjecture that the set complement of “P” to “NP” can be described by that kind of choice exhaustively is formulated.
FERMAT’S LAST THEOREM PROVED BY INDUCTION (accompanied by a philosophical com...Vasil Penchev
A proof of Fermat’s last theorem is demonstrated. It is very brief, simple, elementary, and absolutely arithmetical. The necessary premises for the proof are only: the three definitive properties of the relation of equality (identity, symmetry, and transitivity), modus tollens, axiom of induction, the proof of Fermat’s last theorem in the case of n=3 as well as the premises necessary for the formulation of the theorem itself. It involves a modification of Fermat’s approach of infinite descent. The infinite descent is linked to induction starting from n=3 by modus tollens. An inductive series of modus tollens is constructed. The proof of the series by induction is equivalent to Fermat’s last theorem. As far as Fermat had been proved the theorem for n=4, one can suggest that the proof for n≥4 was accessible to him.
An idea for an elementary arithmetical proof of Fermat’s last theorem (FLT) by induction is suggested. It would be accessible to Fermat unlike Wiles’s proof (1995), and would justify Fermat’s claim (1637) for its proof. The inspiration for a simple proof would contradict to Descartes’s dualism for appealing to merge “mind” and “body”, “words” and “things”, “terms” and “propositions”, all orders of logic. A counterfactual course of history of mathematics and philosophy may be admitted. The bifurcation happened in Descartes and Fermat’s age. FLT is exceptionally difficult to be proved in our real branch rather than in the counterfactual one.
The space-time interpretation of Poincare’s conjecture proved by G. Perelman Vasil Penchev
This document discusses the generalization of Poincaré's conjecture to higher dimensions and its interpretation in terms of special relativity. It proposes that Poincaré's conjecture can be generalized to state that any 4-dimensional ball is topologically equivalent to 3D Euclidean space. This generalization has a physical interpretation in which our 3D space can be viewed as a "4-ball" closed in a fourth dimension. The document also outlines ideas for how one might prove this generalization by "unfolding" the problem into topological equivalences between Euclidean spaces.
FROM THE PRINCIPLE OF LEAST ACTION TO THE CONSERVATION OF QUANTUM INFORMATION...Vasil Penchev
In fact, the first law of conservation (that of mass) was found in chemistry and generalized to the conservation of energy in physics by means of Einstein’s famous “E=mc2”. Energy conservation is implied by the principle of least action from a variational viewpoint as in Emmy Noether’s theorems (1918): any chemical change in a conservative (i.e. “closed”) system can be accomplished only in the way conserving its total energy. Bohr’s innovation to found Mendeleev’s periodic table by quantum mechanics implies a certain generalization referring to
the quantum leaps as if accomplished in all possible trajectories (according to Feynman’s interpretation) and therefore generalizing the principle of least action and needing a certain generalization of energy conservation as to any quantum change.The transition from the first to the second theorem of Emmy Noether represents well the necessary generalization: its chemical meaning is the ge eralization of any chemical reaction to be accomplished as if any possible course of time rather than in the standard evenly running time (and equivalent to energy conservation according to the first theorem). The problem: If any quantum change is accomplished in al possible “variations (i.e. “violations) of energy conservation” (by different probabilities),
what (if any) is conserved? An answer: quantum information is what is conserved. Indeed, it can be particularly defined as the counterpart (e.g. in the sense of Emmy Noether’s theorems) to the physical quantity of action (e.g. as energy is the counterpart of time in them). It is valid in any course of time rather than in the evenly running one. That generalization implies a generalization of the periodic table including any continuous and smooth transformation between two chemical elements.
From the principle of least action to the conservation of quantum information...Vasil Penchev
In fact, the first law of conservation (that of mass) was found in chemistry and generalized to the conservation of energy in physics by means of Einstein’s famous “E=mc2”. Energy conservation is implied by the principle of least action from a variational viewpoint as in Emmy Noether’s theorems (1918):any chemical change in a conservative (i.e. “closed”) system can be accomplished only in the way conserving its total energy. Bohr’s innovation to found Mendeleev’s periodic table by quantum mechanics implies a certain generalization referring to the quantum leaps as if accomplished in all possible trajectories (e.g. according to Feynman’s viewpoint) and therefore generalizing the principle of least action and needing a certain generalization of energy conservation as to any quantum change.
The transition from the first to the second theorem of Emmy Noether represents well the necessary generalization: its chemical meaning is the generalization of any chemical reaction to be accomplished as if any possible course of time rather than in the standard evenly running time (and equivalent to energy conservation according to the first theorem).
The problem: If any quantum change is accomplished in all possible “variations (i.e. “violations) of energy conservation” (by different probabilities), what (if any) is conserved?
An answer: quantum information is what is conserved. Indeed it can be particularly defined as the counterpart (e.g. in the sense of Emmy Noether’s theorems) to the physical quantity of action (e.g. as energy is the counterpart of time in them). It is valid in any course of time rather than in the evenly running one. (An illustration: if observers in arbitrarily accelerated reference frames exchange light signals about the course of a single chemical reaction observed by all of them, the universal viewpoint shareаble by all is that of quantum information).
That generalization implies a generalization of the periodic table including any continuous and smooth transformation between two chemical elements necessary conserving quantum information rather than energy: thus it can be called “alchemical periodic table”.
Poincaré’s conjecture proved by G. Perelman by the isomorphism of Minkowski s...Vasil Penchev
- The document discusses the relationship between separable complex Hilbert spaces (H) and sets of ordinals (H) and how they should not be equated if natural numbers are identified as finite.
- It presents two interpretations of H: as vectors in n-dimensional complex space or as squarely integrable functions, and discusses how the latter adds unitarity from energy conservation.
- It argues that Η rather than H should be used when not involving energy conservation, and discusses how the relation between H and HH generates spheres representing areas and can be interpreted physically in terms of energy and force.
Why anything rather than nothing? The answer of quantum mechnaicsVasil Penchev
Many researchers determine the question “Why anything
rather than nothing?” to be the most ancient and fundamental philosophical problem. It is closely related to the idea of Creation shared by religion, science, and philosophy, for example in the shape of the “Big Bang”, the doctrine of first cause or causa sui, the Creation in six days in the Bible, etc. Thus, the solution of quantum mechanics, being scientific in essence, can also be interpreted philosophically, and even religiously. This paper will only discuss the philosophical interpretation. The essence of the answer of quantum mechanics is: 1.) Creation is necessary in a rigorously mathematical sense. Thus, it does not need any hoice, free will, subject, God, etc. to appear. The world exists by virtue of mathematical necessity, e.g. as any mathematical truth such as 2+2=4; and 2.) Being is less than nothing rather than ore than nothing. Thus creation is not an increase of nothing, but the decrease of nothing: it is a deficiency in relation to nothing. Time and its “arrow” form the road from that diminishment or incompleteness to nothing.
The Square of Opposition & The Concept of Infinity: The shared information s...Vasil Penchev
The power of the square of opposition has been proved during millennia, It supplies logic by the ontological language of infinity for describing anything...
6th WORLD CONGRESS ON THE SQUARE OF OPPOSITION
http://www.square-of-opposition.org/square2018.html
Mamardashvili, an Observer of the Totality. About “Symbol and Consciousness”,...Vasil Penchev
The paper discusses a few tensions “crucifying” the works and even personality of the great Georgian philosopher Merab Mamardashvili: East and West; human being and thought, symbol and consciousness, infinity and finiteness, similarity and differences. The observer can be involved as the correlative counterpart of the totality: An observer opposed to the totality externalizes an internal part outside. Thus the phenomena of an observer and the totality turn out to converge to each other or to be one and the same. In other words, the phenomenon of an observer includes the singularity of the solipsistic Self, which (or “who”) is the same as that of the totality. Furthermore, observation can be thought as that primary and initial action underlain by the phenomenon of an observer. That action of observation consists in the externalization of the solipsistic Self outside as some external reality. It is both a zero action and the singularity of the phenomenon of action. The main conclusions are: Mamardashvili’s philosophy can be thought both as the suffering effort to be a human being again and again as well as the philosophical reflection on the genesis of thought from itself by the same effort. Thus it can be recognized as a powerful tension between signs anа symbol, between conscious structures and consciousness, between the syncretism of the East and the discursiveness of the West crucifying spiritually Georgia
Completeness: From henkin's Proposition to Quantum ComputerVasil Penchev
This document discusses how Leon Henkin's proposition relates to concepts in logic, set theory, information theory, and quantum mechanics. It argues that Henkin's proposition, which states the provability of a statement within a formal system, is equivalent to an internal and consistent position regarding infinity. The document then explores how this connects to Martin Lob's theorem, the Einstein-Podolsky-Rosen paradox in quantum mechanics, theorems about the absence of hidden variables, entanglement, quantum information, and ultimately quantum computers.
Why anything rather than nothing? The answer of quantum mechanicsVasil Penchev
This document discusses the philosophical question of why there is something rather than nothing from the perspective of quantum mechanics. It argues that quantum mechanics provides a solution where creation is permanent and due to the irreversibility of time. The creation in quantum mechanics represents a necessary loss of information as alternatives are rejected in the course of time, rather than being due to some external cause like God's will. This permanent creation process makes the universe mathematically necessary rather than requiring an initial singular event like the Big Bang.
The outlined approach allows a common philosophical viewpoint to the physical world, language and some mathematical structures therefore calling for the universe to be understood as a joint physical, linguistic and mathematical universum, in which physical motion and metaphor are one and the same rather than only similar in a sense.
This document discusses using Richard Feynman's interpretation of quantum mechanics as a way to formally summarize different explanations of quantum mechanics given to hypothetical children. It proposes that each child's understanding could be seen as one "pathway" or explanation, with the total set of explanations forming a distribution. The document then suggests that quantum mechanics itself could provide a meta-explanation that encompasses all the children's perspectives by describing phenomena probabilistically rather than deterministically. Finally, it gives some examples of how this approach could allow defining and experimentally studying the concept of God through quantum mechanics.
This document discusses whether artificial intelligence can have a soul from both scientific and religious perspectives. It begins by acknowledging that "soul" is a religious concept while AI is a scientific one. The document then examines how Christianity views creativity as a criterion for having a soul. It proposes formal scientific definitions of creativity involving learning rates and probabilities. An example is given comparing a master's creativity to an apprentice's. The document argues science can describe God's infinite creativity and human's finite creativity uniformly. It analyzes whether criteria for creativity can apply to AI like a Turing machine. Hypothetical examples involving infinite algorithms and self-learning machines are discussed.
Ontology as a formal one. The language of ontology as the ontology itself: th...Vasil Penchev
“Formal ontology” is introduced first to programing languages in different ways. The most relevant one as to philosophy is as a generalization of “nth-order logic” and “nth-level language” for n=0. Then, the “zero-level language” is a theoretical reflection on the naïve attitude to the world: the “things and words” coincide by themselves. That approach corresponds directly to the philosophical phenomenology of Husserl or fundamental ontology of Heidegger. Ontology as the 0-level language may be researched as a formal ontology
Both necessity and arbitrariness of the sign: informationVasil Penchev
There is a fundamental contradiction or rather tension in Sausure’d Course: between the necessity of the sign within itself and its arbitrariness within a system of signs. That tension penetrates the entire Course and generates its “plot”. It can be expressed by the quantity of information generalized to quantum information by quantum mechanics. Then the problem is how a bit to be expressed by a qubit or vice versa. The structure of the main problem of quantum mechanics is isomorphic. Thus its solution, namely the set of solutions of the Schrödinger equation, implies the solution of the above contradictionor tension.
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The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
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• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
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2. Vasil Penchev. Quantum Measure from a Philosophical Viewpoint
theorem1, and either can be distinguished only
unconstructively (AC; CH or NCH) as to nonBorel sets (NBS) or cannot be juxtaposed at all
(NAC; NCH). Then one can ascribe whatever
difference including no difference. That
incomparability is a typical situation in quantum
mechanics and it represents the proper content of
“complementarity”. Here the Lebesgue measure
is implied to be one-dimensional since the real
line is such.
Dimensionality, LM, and BM: One must
distinguish the dimensionality of the space
being measured from the dimensionality of
measure, by which the space is measured. The
idea of probability as well as that of number in
general is to be introduced a universal measure
(quantity), by which all (pears, apples, distances,
volumes and all the rest) can be measured as
separately, item (quality) by item, as together.
BM uses n-dimensional spheres, which it
compares in radius independently of the number
of dimensions. That radius is the Borel measure,
and if it is finite, admits the Kolmogorov
probability.
One can suppose, though counterintuitively,
the case, where the dimensionality of the space
being measured is lesser than that of the measure,
and that such a case may have a nonempty
intersection with NCH. The conjecture would not
make much sense while one does not point out a
universal measure of the dimensionality greater
than one.
2. Quantum measure (QM)
and its construction
QM is a three-dimensional universal one.
A motivation may be for it to be introduced an
as complete (like LM) as universal (like BM)
measure. It should resolve the problem for
completing BM in general (both AC & NAC):
An alternative, but equivalent approach is to
be measured empty intervals (without any points
in them), i.e. discrete or quantum leaps, in the
same way as complete intervals of continuum. In
fact quantum mechanics is what forced the rising
of such a measure (& probability).
Moreover, quantum measure is more
complete than LM in a sense or even is the most
complete measure known to mankind since it
can measure not only infinitely small empty,
but any finite and even infinite leaps. However
it postpones the question to complete them as no
need to do it initially, on the one hand, and the
general AC & NAC invariance even requires for
the complete and incomplete case to be equated
therefore rejecting the need of completion, on
the other. That rather strange state of affairs is
discussed in details below.
Given BM, the construction of QM is the
following:
The objective is to be measured all the NBS
as being reduced into some combination of the
following three types partially complete:
NBS complete in relative complement;
NBS complete in countable union;
NBS complete in countable intersection.
A partial measure (or a partial probability as
a finite measure) corresponds in each of the three
cases above.
If a NBS is incomplete in one or more,
or even in all of the three relation above, its
corresponding measure(s) [probability (-es)] is
(are) accepted as zero.
BM is the particular case where the three
measures (probabilities) coincide. If a NBS
is incomplete in any relation, it has a zero
BM anyway. That backdoor is substantive for
reconciling quantum theory based on QM and
general relativity grounded on LM or BM in fact.
That kind of construction will be called
tricolor hereinafter. The tricolor has exact
correspondences in set theory and logic.
Let us now consider as an example of the
case of tricolor or quantum probability compared
–5–
3. Vasil Penchev. Quantum Measure from a Philosophical Viewpoint
with the classical one. One substitutes the unit
ball for the interval of [0, 1]:
The unit ball can be decomposed in a “spin”
way into two orthogonal circles.
The point of the unit ball generalizes that of
[0, 1].
The point of the unit ball can be represented
equivalently both as the two correlating complex
numbers (the two projections on the orthogonal
circles) and as three independent numbers (those
of the tricolor above).
As the interval of [0, 1] allows of introducing
the unit of classical information, a bit, as the unit ball
does the same for quantum information, a qubit:
Since a bit can be thought as the alternative
choice between two points: 0 or 1, a qubit might
be thought as the choice of a point of the ball, i.e.
as a choice among a continuum of alternatives in
final analysis.
The [0, 1] is the universal measuring unit
of all what can be classically measured. It can
be illustrated as a “tape measure” for anything
which is something, but not nothing. However
the unit ball is a more universal measuring
unit since it can measure as anything which is
something as nothing in a uniform way. In other
words, it can measure as the continuous as the
discrete without completing the latter with the
continuum of a continuous medium of points,
i.e. without transforming nothing in something.
Consequently, the unit ball is the perfect measure
for quantum mechanics since aids it in resolving
its main question, namely: How can nothing
(pure probability) become something (physical
quantity)?
Many philosophers reckon that the same
kind of question, why there is something rather
than nothing, is the beginning of philosophy,
too. Quantum mechanics gives an answer, which
is the single one that mankind has managed to
reach and which, fortunately or unfortunately, is
constructive besides.
Given LM, the construction of QM is the
following:
One builds a tricolor measure as the BM for
any dimension.
One might consider a “vector” measure,
which components are 3D balls. In fact, it is
equivalent both to Minkowski and to Hilbert
space. That unit-ball vector represents a unit
covariant vector, i.e. just a measure.
Any measure of the ball vector would be
QM on LM. If the measure is the usual one of the
vector length, the measured result would be a 3D
ball rather than a 1D length. The axiom of choice
does not use in that QM-on-LM construction.
Using the axiom of choice, a ball is equivalent
to any set of balls, which is known as the Banach
and Tarski paradox2. So, one need not construct
a ball-vector measure as above since it is directly
equal to a ball (i.e. QM) according to the axiom
of choice.
The last two paragraphs show the original
invariance of QM to the axiom of choice unlike
LM and BM. As to BM that invariance is an
undecidable statement. One might say that BM
even possesses anyway a specific invariance or
universality to the axiom of choice: the invariance
of incompleteness: BM is incomplete as with the
axiom of choice as without it. As to LM, it is
complete without AC, but incomplete with AC:
Indeed the construction of a Vitali set3, which is
immeasurable by LM, requires necessarily AC. At
the same time, the way of its construction shows
that any Vitali set is a subset of a null set such as
that of all the rational numbers within the interval
[0, 1] since there is a one-to-one constructing
mapping between the Vitali set and that set of
the rational numbers. Consequently LM under
the condition of AC is incomplete since there is
a subset of a null set, which is immeasurable: the
Vitali set.
The consideration shows that LM occupies
an intermediate position between the complete
–6–
4. Vasil Penchev. Quantum Measure from a Philosophical Viewpoint
QM and the incomplete BM being partly
complete (without AC) and partly incomplete
(with AC). Thus LM can also demonstrate AC
as the boundary between potential and actually
infinity. LM under condition of AC can measure
anything which is finite, but nothing which is
infinite. QM unlike it can measure both even
under AC.
Thus the invariance of QM to the axiom
of choice can be added to the motivation of QM
since quantum mechanics needs such invariance:
Really, quantum measuring requires the axiom of
choice, and any quantum state by itself rejects it
being due the “no hidden variables” theorems4,5.
Consequently, the epistemological “equation”,
which equates any state “by itself” and the result
of its measuring, needs that invariance in the case
of quantum mechanics.
A problem remains to be solved (as if): Is
there a BM or LM, with which no QM corresponds
after utilizing the aforesaid procedure? The finite
or infinite discrete leaps are described by QM
unlike LM and BM: Consequently QM can be
accepted as more general. However, are there
cases, too, which admit BM or LM, but not QM?
Unfortunately that question is not one of
abstract, purely mathematical interest since it is
an interpretation of the quantum-gravity problem
into the measure-theory language. General
relativity uses LM, while quantum mechanics QM.
If general relativity is true (as seems) and there is
a LM (BM) which is not QM (LM-no-QM), then
quantum gravity is an undecidable problem. Vice
versa: Quantum gravity is resolvable if and only
if QM is more general than (since it cannot be
equivalent with) LM (BM).
A try for a short answer might be as
follows:
The QM-on-LM construction excludes the
LM-no-QM conjecture. However it cannot serve
for refusing a nonconstructive proof of LM-noQM existence in general.
Any pure proof of that kind, which requires
necessarily the axiom of choice, can be neglected
because of the QM invariance to AC/ NAC.
No other proof of pure LM-no-QM existence
can be omitted, but whether there are such ones,
no one knows. That pure existence is not only a
question of abstract and theoretical interest. It
suggests that a more general measure than QM
can be ever found on the base of LM-no-QM.
One can suppose a new invariance to CH/
NCN similar to the QM invariance to AC/ NAC.
In fact, it would be equivalent to the existence of
a countable model for any mathematical structure
of first order: This is a well-known direct
corollary of the Löwenheim-Skolem theorem6.
Thus that alleged as a new invariance would not
expand out of QM, though. The reason is that CH
implies AC.
However one can continue the implication of
AC from CH in the following way: AC implies
Skolem’s “paradox”7: The latter implies the
impossibility to be compared infinite powers
and that CH/ NCH is undecidable for the sake
of that. That is: CH implies the undecidability
of CH/ NCH, but NCH does not imply that
undecidability since cannot imply AC. All this
is another argument in favor of QM and against
LM-no-QM.
Anyway “QM & an undecidability of QM/
LM” satisfies almost all combinations of AC, CH,
and their negations. Moreover it does not require
LM-no-QM since LM and QM are complementary
to each other where both AC & CH hold.
As to the problem of “quantum gravity”,
this means the following: Quantum gravity as
supposing QM is consistent as with NCH and
the AC/ NAC invariance as with CH & AC.
However it is not consistent with CH & NAC, in
the domain of which general relativity is built,
unfortunately.
What about LM-no-QM in “CN & NAC”?
Of course, one can construct QM on any LM
–7–
5. Vasil Penchev. Quantum Measure from a Philosophical Viewpoint
there, too. That construction implies AC, and
since NAC is valid there, the construction is
forbidden, though. This is a very amazing state of
affairs resembling the human rather than nature
laws: QM is possible, but forbidden where general
relativity is valid. After daring to construct QM
in its territory, anyone turns out to be expulsed
automatically in CH & AC where QM is admitted
since it is complementary to LM and does not
force LM to vanish.
What implies all that? Quantum gravity is
a question of choice. One can create the theory
as of quantum gravity as of general relativity,
however ought to choose preliminary which of
them. They should be equivalent to each other in
a sense and can be thought as one and the same.
Consequently general relativity can be reckoned
as the cherished quantum gravity.
That is the case though it is very strange,
even ridiculous. If and only if another and more
general than QM measure be discovered so that
the LM-no-QM be built constructively, then and
only then general relativity and quantum gravity
will be able to be distinguished effectively, i.e.
experimentally. Vice versa: if an experimental
refutation of general relativity be observed, a
generalization of QM (GQM) will be implied:
RIP both for Albert Einstein and for Niels Bohr
since general relativity (LM) and quantum
mechanics (QM) can be universal only together
and reconciled. GQM will be able to resolve the
dispute between them or will remove both when
it comes. However we have not got any idea about
GQM.
Finally, the example of BS can be used
to illustrate how the strange kind of as if
undecidability of CH to AC, and hence the relation
of general relativity and quantum mechanics in
terms of measure:
BS implies CH according to the
Alexandroff – Hausdorff theorem8: Any
uncountable BS has a perfect subset (and
any perfect set has the power of continuum).
However, CH implies AC in turn, and the latter
does Skolem’s “paradox”, i.e. the incomparability
(or more exactly, unorderability) of any two
infinite powers. Consequently, BS can be
consistent as with CH as with NCH since BS
and CH are complementary in a sense. If the
case is NCH, then AC is not implied and BS
remains consistent as with CH as with NCH.
Of course, this should be so as BM is a
particular case of QM, and the latter is consistent
with NCH (as well as CH & AC).
All illustrate how it is possible for BS and
BM to be consistent as with LM as with QM
even where CH & NAC hold. That is the domain
of general relativity, which should not exist if
CH implies AC. Really CH implies AC only that
AC implies the undecidabilty of CH or NCH,
which allows of existing the area of general
relativity.
One can abstract the logical relation of
general relativity and quantum mechanics by
means of the same one of LM and QM. Roughly
speaking, they are complementary because of a
similar complementarity of CH/ NCH and AC/
NAC rooted in the amazing or even paradoxical
properties of infinity: AC supposes a single
infinity which ought to be countable. However,
both CH and NCH suggest an infinite set of sets
which can be countable (CH) in turn, too.
That unordinary logical relation does not
generate any contradictions. In fact, it contravenes
only the prejudices. Anyway, we can attempt to
explain and elucidate the reason of our confusion
and misunderstanding:
Anything in our experience can be either an
indivisible whole (a much) or divided in parts (a
many): No “much” can be a “many” in the same
moment and vice versa.
The above postulate is not valid as to
infinity: It can be defined as that “much” which is
a “many” or as that “many” which is “much”.
–8–
6. Vasil Penchev. Quantum Measure from a Philosophical Viewpoint
Consequently it can be equally seen as a
single “much” consisting of a “many” of parts
(after AC) or as a many of indivisible wholes
(“much”-s) (after CH / NCH).
To reconcile the two viewpoints onto infinity
in a single illustration, one can utilize the image
of cyclicality.
While anything else consists of something
else and is not self-referential or cyclic, infinity
is what consists just of it self-referentially or
cyclically: Its “much” is forced to return back
into it as many units.
AC suggests that cycle while CH or NCH
unfold this cycle in a line. Consequently AC sees
infinity as a well-ordering (line) bounded as a
cycle while CH (or NCH) as many cycles wellordered in a line.
Any contradictions between them do not
arise since both are the same seen from opposite
perspectives.
3. QM compared with LM
and with BM
QM can be compared as with BM as with the
LM to stand out its essence and features:
QM vs. BM:
Similarities:
– Both are supposed to be universal.
– Both generate probabilities where they are
bounded.
– Both can be generated by BS.
– There is a common viewpoint, according
to which QM can be considered as a threedimensional or “tricolor” generalization of BM.
Differences:
– QM is complete, BM is not.
– QM is three-dimensional, BM is onedimensional.
– BM can be considered as the particular case
where the three dimensions of QM coincides.
QM vs. LM:
Similarities:
– Both are complete under NAC.
– QM and LM correspond to each other
“two-to-two”, i.e. “± to ±” or in other symbols
“square-to-square”.
– No one of QM and LM can be deduced
from the other or represented as a particular case
of the other.
The differences from each other (see above)
focus on a common 3D space where they vanish.
One can utilize the metaphor of the two eyes or
binocular sight for QM and LM.
Differences:
– QM is three-dimensional, while LM is of
an arbitrary even infinite dimensionality.
– The dimensionality of QM does not
correspond to that of the space measured, in
general. They can be interpreted differently even
in the particular case, where they coincide (three
dimensions). The dimensionality of LM always
coincides with that.
– QM is universal: It does not depend on
the dimensionality of the space measured. LM is
not universal: It does strictly correspond to the
dimensionality of the space measured.
If one uses the metaphor of binocular sight for
QM and LM, then their “global focus” is always
in the “plane” of QM, while LM can represent
the “local development or change” dimension by
dimension.
4. The origin of QM
QM arose for quantum mechanics when
Heisenberg’s (1925) matrix mechanics9 and
Schrödinger’s (1926a)10 wave mechanics were
united by the latter one11.
Though Hilbert space guaranteed as a
mathematical enough formalism, as John von
Neumann showed12, for the quantum mechanics,
the sense of that guarantee as well as its attitude
toward the two initial components, matrix
and wave mechanics accordingly, remained
misunderstood:
–9–
7. Vasil Penchev. Quantum Measure from a Philosophical Viewpoint
Heisenberg’s matrix mechanics represented
all quantum motions only as discrete rather than
continuous or smooth.
Schrödinger’s wave mechanics represented
all quantum motions only as smooth rather than
discrete, though.
Consequently the sense of quantum
mechanics, which unites both by means of Hilbert
space, is that, in fact, all quantum motions are
invariant to the transition between the discrete
and smooth.
However wave mechanics had advantage
that it could represent that invariance in terms
of the continuous and smooth, which terms were
dominating for classical mechanics, though
they were only prejudices, a legacy of the past,
needless or even harmful:
The determent consisted in that the invariance
of the discrete and smooth as to quantum motion
remained tightly hidden in the mathematical
apparatus of Hilbert space and accordingly
misunderstood in physical interpretation.
The real sense of QM is to suggest a
common measure both for the discrete and for
the continuous and smooth so that to offer a
suitable language for their invariance required by
quantum mechanics.
The case of a (discrete) quantum leap
measured by QM:
Any quantum leap can be decomposed in
harmonics by Fourier transform:
Then any of those harmonics can be
enumerated and considered as a QM for the nth
dimension of Hilbert space.
The nth dimension of Hilbert space can be
interpreted as a frequency or consequently, as an
energy corresponding one-to-one to it.
The above construction shows the transition
from real to complex Hilbert space and the
transition from LM to QM as well. By the way the
universality of QM is similar to that of complex
numbers.
The case of a continuous or smooth physical
motion measured by QM:
Since the continuous or smooth physical
movement means a motion in Euclidean space,
which is the usual three-dimensional one, it can
be decomposed in successive 3D spheres or balls
corresponding one-to-one as to all points of the
trajectory in time as to the successive spheres
or balls of the light cone in Minkowski space as
well as to the successive dimensions of Hilbert
space.
Consequently those points of the trajectory
can be enumerated and considered as a QM for
the nth dimension of Hilbert space in an analogical
way.
Now the nth dimension of Hilbert space can
be interpreted as a moment of time corresponding
one-to-one to it.
The two above constructions show why
QM is universal as well as the sense of that
universality. Since frequency (energy) and
time are reciprocal (or complementary in
terms of quantum mechanics), then they can
be juxtaposed as the two dual spaces of Hilbert
space connected and mapped one-to-one by
Fourier transform.
Max Born’s probabilistic mechanics:
Max Born suggested in 192613 that the
square of the modulus of wave function
represents a probability, namely that of the
state corresponding to that wave function.
However somehow it was called the “statistical
interpretation” of quantum mechanics. The
term of “interpretation” used by Max Born
himself as an expression of scientific modesty
and politeness should not mislead. Its utilizing
shows a complete misunderstanding of Max
Born’s conjecture and a yearning for its
understatement. In fact it was not and is not
an interpretation, but another, third form of
quantum mechanics among and with matrix and
wave mechanics. This is the cause for one to call
– 10 –
8. Vasil Penchev. Quantum Measure from a Philosophical Viewpoint
it probabilistic mechanics (after the expressions
of “wave mechanics” and “matrix mechanics”
are common) rather than an interpretation.
Probabilistic mechanics shares Hilbert
space with matrix and wave mechanics. However
wave function (i.e. a point in Hilbert space) does
not mean here a quantum leap decomposed in
energies, neither a trajectory decomposed in time
moments, but the characteristic function of a
complex random quantity (or of two conjugate
real quantity).
One should say a few words on the Fourier
transform of a complex random quantity and on
its characteristic function:
In fact their interrelation is quite symmetric
and simple: The Fourier transform and the
replacing of a complex random quantity by its
conjugate swap the two dual Hilbert space.
Consequently the characteristic function of
the conjugate of the complex random quantity is
just the complex random quantity itself.
The interpretation of a complex random
quantity and its conjugate is simple, too: Since a
complex random quantity can be interpreted as
two real conjugate (reciprocal) physical quantities
such as e.g. time and frequency (energy), then
the conjugate of the same random quantity
must represent merely swapping between the
corresponding physical quantities or the axes of
the complex plane, or its rotation of π/2.
Probabilistic vs. matrix mechanics: If one
compares them, the differences would be only
two: in interpretation and in choice between NAC
and AC.
However the wave function in both cases
and despite the differences would be the same and
the same point in Hilbert space. That sameness
inspires invariance as to probabilistic vs. matrix
“interpretation” as to NAC vs. AC.
Since the wave function is a sum of the
measured by QM, one can reduce completely that
invariance in terms of QM:
QM as quantum probability guarantees
the former members, and it decomposed in
dimensions (which are harmonics or energetic
levels in the case) supplies the latter ones.
A philosopher would emphasize the
extraordinary universality both of Hilbert space
and of QM contradicting common sense:
Why and where exactly? QM is so universal
that can measure both the unordered (and even
unorderable in principle) and the well-ordered
and thereof ordering it (them):
In our case it can measure and order quantum
probabilities (for the unorderable in principle) and
quantum leaps (for the well-ordered in harmonics
or energies), and therefore QM establishes a oneto-one mapping between quantum probabilities
and quantum leaps:
That one-to-one mapping is too shocking
to the prejudices. It shows that a level of energy
corresponds exactly to a quantum probability:
That is a physical quantity (what is the former)
can be equated with a real number being without
any physical dimensionality (what is the latter):
However this is what has been necessary
for the objectives declared in the beginning of
the paper: to demonstrate how QM allows of
becoming nothing to something or vice versa and
eo ipso creatio ex nihilo or reductio ad nihilum
(i.e. true creation or true annihilation).
Probabilistic vs. wave mechanics: All what
has been said above about the links between
probabilistic and matrix mechanics can be
almost literally repeated again in that case. The
immaterial differences are as follows:
– The dual Hilbert space replaces its dual
counterpart.
– The well-ordering in time replaces that in
frequency (energy).
The one-to-one mapping based on QM
establishes now a correspondence of a wave
function as quantum probability with a continuous
or smooth trajectory in time.
– 11 –
9. Vasil Penchev. Quantum Measure from a Philosophical Viewpoint
A threefold (even fourfold) one-to-one
mapping arises thereof: It states invariance or
equivalence in a sense between the quantum leaps
(for the discrete), the smooth trajectories in time
(for the continuous) and the quantum probabilities
(for the unorderable in principle).
That threefold mapping shows how pure
numbers even only the positive integers (for
“nothing”) can generate physical quantities
in pairs of (“reciprocal”) conjugates such as
frequency (energy) and time. The stages of that
generation are as follows:
– Nothing.
– The positive integers are given somehow,
maybe by God as Leopold Kronecker reckoned:
“Die ganzen Zahlen hat der liebe Gott gemacht,
alles andere ist Menschenwerk”14:
– Creation: Qubits (or QM) replaces each
of the positive integers generating Hilbert
space.
– The Hilbert space generates that threefold
mapping between quantum probability, energy
and time and thereof the physical world arises
already, too.
Quantum mechanics seen as the unification
of all three kinds of mechanics: probabilistic &
matrix & wave mechanics:
Quantum mechanics is better to be
understood as the unification of all the three
types of mechanics listed above instead only of
the last two.
The sense of that unification is the
extraordinary invariance (or equivalence in a
sense) of the discrete, continuous (smooth) and
the probabilistic in the common form of quantum
motion:
Quantum motion can be already thought
as a relation between two or more states despite
whether each of them is considered as a discrete,
continuous (smooth) or probabilistic one since
it is always represented by one the same wave
function in all the three cases:
This calls for far-reaching philosophical
conclusions, though:
The difference not only between the discrete
and continuous (smooth), but also that between
both and the probabilistic is only seeming
and accidental or even anthropomorphic in a
sense.
Quantum motion breaks down their barriers
and allows any transitions between them.
The probabilistic can be located “between”
the discrete and continuous (smooth) and can be
considered as something like a substance of that
kind of transition. Accordingly the discrete and
continuous can be supposed as the two extreme
or particular cases of the probabilistic, which are
opposite to each other, and that is not all:
What is the physically existing according to
common sense can be linked only to those two
extremes. Physical reality ostensibly consists just
of (and in) both since they are all the actual.
According to the same common sense the
probabilistic cannot be physically real since is
not actual: However quantum mechanics shows
that is the case, the probabilistic is physically
real: “So much the worse for quantum mechanics
because this means that it is incorrect or at least
incomplete”, declared the common sense then.
Quantum mechanics rather than that “common
sense” turns out to be the right again, though,
experimentally verified15.
If quantum mechanics is the right, what
does it mean about the philosophical interrelation
between reality and “virtuality”?
“Virtuality” is a term coined here to denote
just that new class required by quantum mechanics
and involving both reality, i.e. the discrete and
continuous (smooth), and “only” (ostensibly) the
possible so that to allow of any transfer between
them.
Consequently virtuality is a term for the
new constitution of being, according to which
the barriers between the actual and the possible
– 12 –
10. Vasil Penchev. Quantum Measure from a Philosophical Viewpoint
are broke down and all the kind of transitions
between them are unrestrained.
Thus virtuiality established by quantum
mechanics can resolve our properly philosophical
(and even theological) problem about creatio
ex nihilo or reductio ad nihilum: The area of
probability can describe very well both those
creatio and reductio as states and processes:
One can see the actual in creating or annihilating
rigorously mathematically, i.e. in the process
of creation or annihilation, as the change of
probability.
Not less striking is that the new “constitution”
of virtuality suggests for mathematics to be more
general than physics if the latter is defined and
restricted only to the actual; or in other words,
mathematics and a new and more general physics
can and even should coincide.
How to interpret the fermion and boson kind
of spin statistics in the light of that unification?
According to the so-called spin-statistics
theorem16 all the quantum particles can be
divided into two disjunctive classes after the
second quantization: fermions and bosons:
Since the second quantization maps the
wave functions of the quantum particles “twoto-two”, it admits two kinds of solving as to a
swap of the space-time positions of two quantum
particles: symmetric (++, −−) and antisymmetric
(+−, −+).
The bosons are supposed to be those of
symmetric swap, and the fermions are those of the
antisymmetric swap. Turns out the any number
of bosons can share one the same state and wave
function while if they are fermions, only two.
The following can be easily spotted:
Quantum probabilistic mechanics explains very
well that property as to the bosons, and quantum
matrix-wave mechanics explains it not less
successfully as to the fermions:
Indeed the opportunity of sharing a
common state or wave function by the bosons
is due to the sharing of a common probability
by an arbitrary ensemble of quantum particles.
That ensemble, which possibly consists of an
infinite number of elements, is supposed not to
be well-ordered.
The same ensemble already well-ordered
can be distinguished in two kinds of wellordering corresponding to the two fermions
admitted in one the same state or wave function.
The one is well-ordered to, and the other from
infinity. If the ordering is in time and energy,
the one fermion as if corresponds to the
discrete “half” of wave-particle duality, and
the other accordingly to its continuous (smooth)
“half”.
Hence one can clearly see that the second
quantization giving rise to spin statistics
either is equivalent to, or is a particular case
of that quantum mechanics, which includes as
probabilistic as matrix and wave mechanics.
Indeed, the sense of the second quantization is to
be defined “quantum field”. In fact, this is done
by ascribing a wave function (i.e. a quantum
state) to each space-time point. That quantum
mechanics, which includes as probabilistic as
matrix and wave mechanics, ascribes a spacetime point to each wave function. Then:
If the quantum field is well-ordered, then the
mapping between all the wave functions (Hilbert
space) and all space-time points (Minkowski
space) is one-to-one, and the second quantization
is equivalent to that quantum mechanics, which
involves probabilistic mechanics. Then any
quantum particle must necessarily be either a
boson or a fermion.
If the quantum field is not well-ordered,
it admits two opposite options as well as both
together:
Two or more space-time points to share one
the same wave function, and thus the reverse
mapping not to be well-defined: It will not be a
standard function.
– 13 –
11. Vasil Penchev. Quantum Measure from a Philosophical Viewpoint
Two or more wave functions to share one
the same space-time point, and thus the straight
mapping not to be well-defined: It will not be a
standard function.
However the most general case is two or
more space-time points to share two or more wave
functions (i.e. together both above). If that is the
case, it can be equally described as some spacetime points, which share a part of some wave
functions (entanglement) or as wave functions,
which share a “part” of some space-time points
(“quantum” gravity). The interrelation or
equivalence of entanglement and gravity is being
studied.
If the quantum field is not well-ordered as
above, it can be represented in a few ways (as well
as in their combinations or mappings):
– As a curving of Hilbert to Banach space.
– As a curving of Minkowski to pseudoRiemainian space.
– As quantum particles with an arbitrary
spin: such which can be any real number.
The transitions between the “probabilistic”
wave function and “well-ordered” wave function
in any of the above ways describe in essence the
arising of “something from nothing and from
time” (“time” is for the axiom of choice) as a
continuous process as a quantum leap as well as a
purely informational event.
One can give examples of that arising in terms
of classical (gravity) or quantum (entanglement)
physics as a continuous process.
Quantum mechanics, QM and AC:
A question may have remained obscure:
Does quantum mechanics need AC?
Quantum mechanics is actually the
only experimental science, which requires
necessarily AC: Roughly speaking, the state
before measuring has not to be well-ordered, but
after that it has to. This means that measurement
supposes the well-ordering theorem, which is
equivalent to AC:
The “no hidden variables” theorems do not
allow of well-ordering before measuring.
However even only the record of the
measured results (which is after measuring, of
course) forces they to be well-ordered.
The basic epistemological postulates equate
the states before and after measurement and thus
imply AC.
Though measuring requires the AC,
it remains inapplicable before measuring.
Consequently quantum mechanics is ought to be
consistent both with AC and NAC in addition.
The only possible conclusion is too
extraordinary: Quantum mechanics is consistent
as with AC as with NAC. However, quantum
mechanics is not consistent with the absence as
of AC as of NAC.
We could see above that QM is linked to AC
in the same extraordinary way. This means that
quantum mechanics is consistent with QM as to
AC, which should expect.
Quantum mechanics, QM and the CH:
That extraordinary interrelation between
quantum mechanics, QM and AC goes on with
CH:
NCH is consistent as with AC as with NAC,
thus quantum mechanics is consistent with
NCH.
Reversely, CH should (ostensibly) imply
AC. However AC implies the undecidability
of CH and NCH, then CH implies by means of
AC the own undecidability. The only way out is
then to admit the complementarity of CH and
AC.
Both quantum mechanics and QM share
that extraordinary relation to CH by means of
AC.
Though the state of affairs is strange, it is
not logically contradictory. It messes up only
common sense. The cause of that ostensible
muddle is the intervention of infinity, of which
we try to think as of a finite entity.
– 14 –
12. Vasil Penchev. Quantum Measure from a Philosophical Viewpoint
5. Physical quantity measured
by QM or by LM, or by BM
The definition of physical quantity in
quantum mechanics involves measuring by QM.
It is a generalization of the corresponding notion
in classical physics and exact science.
One can see the quantum at all as that
generalization from LM and BM to QM. The
correspondence is the following:
A few conclusions can be drawn from that
correspondence:
The sense of a point of dual Hilbert space
is to be a “unit”, or something like a reference
frame, which can measure a point of Hilbert
space.
The measured value represents the distance
between the “unit” point (or its conjugate point,
too) and another point, which is for the measured
quantity. This distance can be thought of also as a
distance in the “reference frame” of that point.
Both cases are “flat”: They conserve
the measure under translation and rotation.
If the translation and rotation are understood
as usual as translation and rotation in spacetime, then that “flatness” implies the classical
laws of conservation complemented by Lorentz
invariance. One can especially emphasize the
time translation and energy conservation.
The “flatness” in general can be equated with
the axiom of choice. Indeed the well-ordering
requires that flatness since otherwise a second
dimension for ordering appears questioning the
well-ordering made only in the first dimension.
The above table (1) can be paraphrased in
terms of the “crooked” as the following table 2
asking how both tables can refer to each other:
The “crooked” case is that of general
relativity and gravity. The question for the
connection between the two tables addresses the
problems of quantum gravity in terms of general
relativity and measure theory.
To be together in front of the eyes, one can
combine the two tables as a new one (3):Table 3:
The above table (3) shows that the problem
of quantum gravity is a problem of measure: It
concerns the ostensibly contradictory properties
of infinity focused on how AC and CH should
refer to each other.
The gravity being as “crooked” as smooth
in general relativity supposes the “classical” case
of NAC and CH. However as CH implies AC, it
should not exist. After all it arises anyway since
AC in turn implies the undecidability between
HC and NHC. In last analysis it concerns the
property of infinity to be both cyclic and linear
unlike anything in our usual experience.
Table 1. From LM and BM to QM
Quantity
Unit
Value
The classical case
Real quantity
Unit
Real number
The quantum case
Wave function
Conjugate wave function
Self-adjoint operator
Quantity
Unit
Value
The classical case
Real quantity
Unit
Real number
The “crooked” case
Contravariant vector
Covariant vector
Metric tensor
Table 2. Table 1 paraphrased
– 15 –
13. Vasil Penchev. Quantum Measure from a Philosophical Viewpoint
Table 1 and Table 2 unified
Quantity
Unit
Value
Real quantity
Unit
Real number
The quantum case
Wave function
Conjugate wave function
Self-adjoint operator
The gravity case
Contravariant vector
Covariant vector
Metric tensor
Quantum gravity
???
???
???
The classical case
One can think of the quantum and gravity
case as complementary. In particular this means
that the values of a quantity as a self-adjoint
operator or as a metric tensor are complementary,
too, as well as QM and the “crooked” LM
therefore that QM and LM are even equivalent in
the distinctive way of quantum mechanics
Just this complementarity of QM and LM is
taken in account as to NCH, which is consistent
as with AC as with NAC, and for this, with the
curious invariance of AC and NAC.
For the above three as to the CH case, one
can be free to suggest that it exactly repeats the
NCH case in relation to AC and NAC because
of the undecidability between CH and NCH
after AC. This would means that any theory of
quantum gravity is not necessary since the pair
of quantum mechanics and general relativity can
represent whatever the case is.
The following should be highlighted in
background of the just said: In the same extent,
one can be free to admit the opposite: That is
the CH case does not repeat the NCH one just
because of the used undecidability between CH
and NCH after AC. This will say that a theory
of quantum gravity is possible though it will be
never necessary.
One can compare with the real state of
affairs: Indeed many theories of quantum gravity
appear constantly and supposedly some of them
3
4
1
2
do not contradict the experiments just because
they are possible. However they are not necessary
in principle since general relativity does not
contradict the experiments, too, and “Occam’s
razor” removes all of them remaining in hand
only general relativity.
The perspective
Is there any measure more general and
universal than QM? If there is, it could be
called “generalized quantum measure” (GKM).
According to current knowledge, we cannot even
figure what might cause such a measure to be
introduced or what would constitute.
One can postulate the absolute universality
of QM. This implies a series of philosophical
conclusions and new interpretations of wellknown facts. The most of them have been
already mentioned in a slightly different context
above. What is worth to emphasize here is the
following:
A universal measure as QM suggests that
all entities are not more than different forms
of a substance shared by all of them as their
fundament: It is quantum information and
represents a general quantity, which is both
mathematical and physical in its essence. The
longstanding philosophical idea of a single and
general substance can be already discussed in
terms of exact science.
Carathéodory, 1956, 149.
Banach and Tarski, 1924.
Vitali, 1905.
Neumann, 1932, 167-173.
– 16 –
14. Vasil Penchev. Quantum Measure from a Philosophical Viewpoint
7
8
9
5
6
12
13
14
15
16
10
11
Kochen and Specker 1968, 70.
Löwenheim, 1915; Skolem, 1919a; Skolem, 1919b.
Skolem, 1923.
Alexandroff, 1916; Hausdorff 1916; cf. Sierpiński 1924.
Heisenberg, 1925.
Schrödinger, 1926a.
Schrödinger, 1926b.
Neumann, 1932, 18-100.
Born 1926a; Born, 1926b; Born 1927a; Born 1927b; Born and Fock 1928; Born 1954.
Cited in: Weber, 1893, 19.
Bell 1964; Clauser and Horne 1974; Aspect et al, 1981; Aspect et al, 1982.
Fiertz, 1939; Pauli 1940.
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16. Vasil Penchev. Quantum Measure from a Philosophical Viewpoint
Квантовое измерение
с философской точки зрения
Васил Пенчев
Болгарская академия наук
Институт исследований обществ и знаний
Болгария, 1000 София, Неофит Рильски, 31
В данной статье представлены философские выводы о том, что квантовое измерение
предполагает взаимосвязь между квантовой механикой и общей теорией относительности.
Квантовое измерение является трехмерным, таким же универсальным, как мера Бореля, и
таким же полным, как мера Лебега. Единица квантового измерения – бит (кубит) – может
рассматриваться как генерализация единицы классической информации, бита. Квантовое
измерение позволяет интерпретировать квантовую механику в рамках квантовой информации,
и все физические процессы рассматриваются как информационные в обобщенном смысле, что
предполагает фундаментальную взаимосвязь между физическим и материальным, с одной
стороны, и математическим и идеальным – с другой. Квантовое измерение объединяет их
посредством одной объединенной информационной единицы.
Квантовая механика и теория общей относительности могут пониматься совместно
как целостный и временный аспект одного и того же, состояние квантовой системы, т.е.
состояние Вселенной в целом.
Ключевые слова: измерение, квантовая механика, теория общей относительности, квантовая
информация, перепутывание.