Quantum Computer: Quantum Model and Reality

There are a few most essential questions about the philosophical interpretation of quantum computer:

1. Can a quantum model unlike a classical model coincide with reality?

2. Is reality interpretable as a quantum computer?

3. Can physical processes be understood better and more generally as computations of quantum computer?

4. Is quantum information the real fundament of the world?

5. Does the conception of quantum computer unify physics and mathematics and thus the material and the ideal world?

6. Is quantum computer a non-Turing machine in principle?

7. Can a quantum computation be interpreted as an infinite classical computational process of a Turing machine?

8. Does quantum computer introduce the notion of “actually infinite computational process”?  

Any computer can create a model of reality. The hypothesis that quantum computer can generate such a model designated as quantum, which coincides with the modeled reality, is discussed. Its reasons are the theorems about the absence of “hidden variables” in quantum mechanics. The quantum modeling requires the axiom of choice. The following conclusions are deduced from the hypothesis:

A quantum model unlike a classical model can coincide with reality. Reality can be interpreted as a quantum computer. The physical processes represent computations of the quantum computer. Quantum information is the real fundament of the world. The conception of quantum computer unifies physics and mathematics and thus the material and the ideal world. Quantum computer is a non-Turing machine in principle. Any quantum computing can be interpreted as an infinite classical computational process of a Turing machine. Quantum computer introduces the notion of “actually infinite computational process”.

The hypothesis is consistent with quantum mechanics. The conclusions address a form of neo-Pythagoreanism. Unifying the mathematical and physical, quantum computer is situated in an intermediate domain of their mutual transformations.

References:

1. Kochen, Simon and Ernst Specker {1968) “The problem of hidden variables in quantum mechanics,” Journal of Mathematics and Mechanics 17 (1):   59-87.

2. Neumann, Johan von (1932) Mathematische Grundlagen der Quantenmechanik, Berlin: Verlag von Julius Springer.

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Appendix:

The Metaphysics of Quantum Computing

There are a few closely connected and most essential questions about the philosophical interpretation of quantum computing: 1. Can quantum model unlike a classical model coincide with reality and thus: 2. Can reality be interpreted as a quantum computer? 3. Can physical processes be understood better and more generally as computations of quantum computer? 4. Is quantum information the real fundament of the world? 5. Does the conception of quantum computer unify physics and mathematics and thus the material and the ideal world? 6. Is quantum computer a non-Turing machine in principle? 7. Can quantum computing be interpreted as an infinite classical computational process of a Turing machine? 8. Does quantum computer introduce the notion of “actually infinite computational process”?  

The answers of these questions could be searched in the following directions correspondingly:

1. The theorems about the absence of hidden variables in quantum mechanics (Neumann 1932; Kochen, Specker 1967) can be interpreted as that coincidence. Those alleged hidden variables would be situated in the complement of the quantum model to reality, but that complement has to be an empty set, which means the coincidence of quantum model and reality.

Quantum model requires the axiom of choice in the following sense: It maps any coherent quantum state before measurement with a well-ordered set of measured values. This means that the well-ordering theorem has to have been utilized, and it in turn is equivalent to the axiom of choice. 

2. The answer depends on whether the computation of quantum computer is a quantum model. If the former is a computation in Hilbert space and the latter is a model in Hilbert space, the answer would be positive. The computation in Hilbert space is a generalized computation where the integers are represented as qubits. This can be visualized geometrically thus: any integer is like a point, which can be thought as a degenerate unit ball (with a point on its surface), which is equivalent to a qubit.   

3. The fundamental duality of quantum mechanics or its complementarity in Bohr’s sense generates an analogical and derivative interrelation between any physical process and its computational counterpart of a quantum computer. This admits an intermediate domain between physics and mathematics. Any pair of wave functions corresponds both a physical process and to a quantum computation representable by replacing all bits in a Turing machine with qubits. Wave function represents a completed whole of an infinite number of qubits while quantum computation is the same set of qubits as a successive process: Thus actual and potential infinity turn out to be mapped between each other one-to-one.

4. If quantum information can be defined as a relation or even ratio corresponding to quantities of any physical process and of its quantum-computational counterpart, it can offer a better and more general viewpoint. Its physical sense is to be a quantity for the transformation of energy into some probability distribution like entropy. Energy and probability distribution should refer to one and the same quantum system. So the quantum system can be seen as a “Janus” with a physical and a mathematical “face”, which are complementary. Then quantum information as a quantity unifies both “faces” and their mutual mapping into each other. The physical “face” is depicted as a successive process point after point while the mathematical one is given immediately as a completed whole though different from the former in general.

5. The conception of quantum computer can unify physics and mathematics since it adds a computational and thus mathematical counterpart of any physical process and allows of discussing the energetic value of information or the informational value of energy. It creates a bridge cherished long ago between the material as the physical and the ideal as the mathematical. The mathematical represents the global result, which directs locally the physical process as a quantum computation.

6. Since any result obtained by a Turing machine keeps the fundamental difference between that result and the reality modeled by it, quantum computer under the above conditions should be a non-Turing machine in principle. If a Turing machine can choose between finitely many of alternatives, the quantum computer can do it from an infinite set.  The latter choice directs the former one to be able to reach it as a limit.

7. If the quantum-computational process be projected in a Turing machine, it has to be represented by an infinite Turing computation. The kind of that projection is the same as measurement in quantum mechanics: A coherent whole has to be represented as a finite time series. If that projection is on a finite Turing computation, it will turn out randomly chosen. Nevertheless the result of any quantum computation is representable as that limit, to which an infinite computational process of a Turing machine converges. If that process does not converge, it has at least one quantum counterpart, which converges.

8. Quantum computer introduces necessarily the notion of actual infinity after being projected on a Turing machine as it requires an infinite computational process to be reckoned as completed and as a whole. The actual infinity is representable as a single number, to which the infinite computational process converges. This number can be embedded in the process to direct it to converge.

All these extraordinary features of quantum computing reveal its importance in philosophy at all rather than only in philosophy of mathematics, information, computation. They can be summarized as a form of Pythagoreanism. Furthermore the quantum computer can be interpreted as an infinite series of Turing machines and thus to be investigated the conditions, under which it converges. Consequently that circle of fundamental mathematical and philosophical problems outlined in the beginning allows of being tested in physical or quantum-computational experiments at least in principle.

References:

Kochen, Simon and Ernst Specker 1968. “The problem of hidden variables in quantum mechanics,” Journal of Mathematics and Mechanics.  17 (1):   59-87.

Neumann, Johan von 1932. Mathematische Grundlagen der Quantenmechanik, Berlin: Verlag von Julius Springer.

Atoms, quanta, and qubits: Atomism in quantum mechanics and quantum information

The original conception of atomism suggests “atoms”, which cannot be divided more into composing parts. However, the name “atom” in physics is reserved for entities, which can be divided into electrons, protons, neutrons and other “elementary particles”, some of which are in turn compounded by other, “more elementary” ones. Instead of this, quantum mechanics is grounded on the actually indivisible quanta of action limited by the fundamental Planck constant. It resolves the problem of how both discrete and continuous (even smooth) to be described uniformly and invariantly in thus. Quantum mechanics can be interpreted in terms of quantum information. Qubit is the indivisible unit (“atom”) of quantum information. The imagery of atomism in modern physics moves from atoms of matter (or energy) via “atoms” (quanta) of action to “atoms” (qubits) of quantum information. This is a conceptual shift in the cognition of reality to terms of information, choice, and time.      

Key words: atom, quantum, qubit, quantum information, choice

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Fleeting thoughts: Human enhancement "by technics" versus "by genome editing": a philosophical comparison

Prehistory and background:

Human enhancement has been realized by two independent ways: by natural evolution leaded to the contemporary human: homo sapience sapience before the beginning of the human history properly; by technics in a broad sense including all human innovations such as language, religion, science, social organization, and technics in a narrow sense.

The comparison between those two ways demonstrates that human enhancement by technics has been much faster thus replacing natural evolution in the period of human history: evolution has been restricted very much by the alternative evolution by technics because of the following causes:

1. The evolution by technics is much faster and more universal being accomplished even in the lifespan in a single generation. Any evolutionary change is impossible in the same period in principle needing many generations to appear and establish as well as the constancy of a new essential factor of environment.

2. Technics neutralizes the advantages of those individua with relevant mutations and thus prevent their attachment and dissemination, since any one possessing or not a relevant mutation is able to use the corresponding technical device realizing alternatively the same functions.

Now, after decoding human genome, technics is able to intervene human evolution directly by editing human genome. A new distinction between technics and evolution in human enhancement becomes obvious in that background. Technics has been relatively independent of the nature of human being until now. Both ways of human enhancement are able to entangle from now on.

Problem:

Which are the advantages and disadvantages of human enhancement by technics in comparison to genome editing?

A practical dimension of the problem:

Given a certain human enhancement. How to decide which way is better for its realization: whether by technics as until now or by human genome editing?

Those kinds of decisions need a general comparison between the advantages and disadvantages of both methods for human enhancement:

1. The "switch-off" problem: any technical device in both narrow and broad sense can be switched off or not used. It may be switched on only if need be. Will the genome edited enhancements be able to be analogically switched on or off?

2. The "who-decides" problem. Who needs a certain technical device decides to use it or not. If the decision of genome editing is before one's birth, that one is forced to use the corresponding enhancement and that one's free will is deprived of choice.

3. The "unforeseen side effects" problem. The human genome is very, very complex structure. Its decoding does not include yet clearing the interrelations between its different parts. If a part of it is changed, the possible unforeseen side effects seem to be too many and eventually dangerous.

All those problems implies for the human genome edition to start by very, very tiny repayments obviously and immediately threatening the life or normal existence of human babies. Accumulating more and more experience, the application will expand with caution.

Fleeting thoughts: Transcendentalism as a philosophical method to totality

The viewpoint: it will be that of metaphilosophy rather than that of history of philosophy studying Kant’s or Hegel’s philosophy or real ways of the transformation of the former to the latter. Thus, a special subject of philosophy is postulated: totality. It is independent of different possible interpretations in different philosophical systems during the milenia of Western philosophy and theology.

“Totality”: one means that kind of wholeness which has to contain its externality within its internality, and thus, within it in definition.

“Transcendentalism”: a properly philosophical method directed to totality. It represents its externality within its internality in a way conserving the difference between external and internal elements within totality.

“Transcendental dialectics”: one means the correspondence (“synthesis”) of an internal element (“thesis”) to an external element (“antithesis”), as an element of totality.

“Transcendental metaphilosophy”: transcendentalism being a method definable within metaphilosophy to a series of philosophical systems can be applied in turn to metaphilosophy as an external viewpoint to philosophy. Thus, transcendentalism allows for metaphilosophy to be considered as a part of philosophy as a corollary from the special subject of philosophy, namely totality, and the “bad infinity” of a series of metaphilosophies to be escaped.

“Transcendental information”: the sketched approach to transcendentalism allows and needs a series of new concepts linking philosophy to mathematics rather than only to logic, and even to physics and science of nature. The first in it is “transcendental information” meaning the reverse correspondence to that “transcendental dialectics” above. It possesses the formal structure of the unit of information, a bit, and thus, that of information.  

“Transcendental time”: both “transcendental information” and “transcendental dialectics” generate opposite series of members, which can be considered as opposite directions of the philosophically defined “time”. 

Hilbert Space and pseudo-Riemannian Space: The Common Base of Quantum Information

Thesis:

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.

Arguments “pro” the thesis:

1. Hilbert space is introduced as the fundamental space of the quantum formalism for it is the simplest one, which can contain the solution of any case of the equivalence of a discrete motion (quantum leap) and a smooth motion (any motion according to classical physics). Consequently, any motion described as a linear automorphism of Hilbert space can be interpreted equally well both as a quantum and as classical motion. Any quantity featuring that automorphism such as any physical quantity definable according to quantum mechanics as a selfadjoint operator in Hlibert space is referable both to a classic and to a quantum motion.

2. However, the probabilistic interpretation of Max Born demonstrates even more: Hilbert space can unify furthermore the description of a possible and an actual state of a quantum system rather than only those of a discrete actual physical motion and of a smooth actual one. Thus it can guarantee the uniform description of a physical process in the future, present, and past, though absolute dissimilarity of these temporal “media”: The future is unordearble in principle corresponding to a coherent state of a quantum system containing all possible state as a “superposition”. On the contrary, the past is always well-ordered being absolutely unchangeable. The present is forced to mediate and agree these two temporal “poles”. Mathematically, this implies the well-ordering theorem equivalent to the axiom of choice. The present is the only temporal “media”, in which the harmonization of the “no any ordering” state of the future and the well-ordered state in the past can be realized as a relevant series of choices exhaustedly describing any physical process and motion.

3. The quantity of information can be described as the quantity of elementary choices necessary for an unordered state to be transformed into an ordered one or for an ordered state to be transformed into another also ordered but otherwise. A bit (i.e. a “binary digit”) is the unit of an elementary choice between two equiprobable alternatives (e.g. “0” or “1”). A qubit (i.e. a quantum bit) is analogically interpretable as the unit of an elementary choice between infinitely many alternatives if it is defined as

usual: A qubit is the normed superposition of two orthogonal subspaces of Hilbert space. It is isomorphic to a unit ball with two points chosen in it: the one can be any within the ball, and the other should be only on its surface. Hilbert space can be equivalently represented as an ordered series of qubits, and any point in it (i.e. any wave function or state of any quantum system), as just one value of this series.

4. Thus Hilbert space and Minkowski space can be discussed as equivalent or as Fourier “twins” in terms of quantum information for both represent ordered series of qubits being a discrete series in the case of separable Hilbert space and a continuous but discretizable one in the case of Minkowski space.

5. Pseudo-Riemannian space is smooth. Thus it possesses a tangent Minkowski space in any point of it. Gravity according the Einstein field equations can be defined only as a relation between two or more points (i.e. tangent Minkowski spaces) of pseudo-Riemannian space, but not in a single one (i.e. in one tangent Minkowski space). As Minkowski space and Hilbert space are equivalent in the sense of quantum information as above any tangent Minkowski space can be substituted by the corresponding Hilbert space and therefore one can demonstrate that gravity is nonlocal in the sense of quantum information.

6. According to the Standard model, the electromagnetic, weak, and strong interaction can be unified as the following composite symmetry of a single Hilbert space: [U(1)]X[SU(2)]X[SU(3)]. Consequently, these three fundamental physical interactions are local in the sense of quantum information.

7. One can discuss that pseudo-Riemannian space, in which the tangent Minkowski spaces are replaced by equivalent Hilbert spaces being even isomorphic in the sense of quantum information, as Banach space. Any two or more points of that Banach space possessing one tangential Hilbert space in each of them define an entanglement between those quantum systems, which are describable each in each of those Hilbert spaces. Consequently, entanglement is also nonlocal in terms of quantum information and can be considered as a counterpart of gravity after substituting the pseudo-Riemannian space with Banach space, and the tangent Minkowski spaces with the corresponding tangent Hilbert spaces.

Arguments “contra” the thesis are not known till now.



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Metaphor as entanglement

The concept of entanglement is coined by the theory of quantum information to designate that special correlation of two or more quantum entities. Furthermore, it means an exactly defined mathematical structure grounded on Hilbert spaces and underlying all phenomena of entanglement studied by quantum mechanics.

That same structure can be utilized for a mathematical model of metaphor as a special kind correlation between the meanings and senses of two or more words. The philosophical core of the model can be described so: Metaphor restricts the meaning of a term by the meaning of another term in a probabilistic, loose way calling for interpretation.

The introduction of that underlying mathematical structure allows of establishing unambiguous correspondence between metaphor and entanglement in an absolutely exact, mathematical way, after which measurement in quantum mechanics corresponds to interpretation in language: This determines some interpretations of a given metaphor as more probable, but no one can be excluded.

The term utilized as metaphor restricts the area of meaning of its object to a small true subset of it. That set can ground the essential features, properties or relations of the object of the metaphor pioneering the scientific or even formal definition of the term serving as the object of the metaphor at issue. Thus some metaphor founds any scientific notion therefore “erasing” the grounding metaphor and the rest interpretations except one of them. The corresponding phenomena in quantum information is the process of de-coherence, after which the interacted object is cut off from its environment just as a rigorously defined notion is cut off from its context to designate one and the same in any context.

The opposed process can be observed both by the theory of metaphor and that of quantum information: A notion begins to lose its clear outlines coined in everyday speech and media accumulating new and new interpretations and uses. A quantum entity analogically starts to lose the measured values of the quantities as if dissolving in the common and inseparable whole of the universe. The suggested mathematical structure describes equally well both processes representing its interpretations.    

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.

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Intention and attention: Intension, extension, and “attension” of a notion or set

Prehistory and background:

Philosophical phenomenology starting from Brentano and Husserl introduced (or restored from scholastic philosophy) the conception about intentionality of consciousness. Especially Husserl being a mathematician in education and early carrier linked that fundamental and definitive property of consciousness to the essence of mathematical cognition by means of the concept of “epoché”: Indeed, mathematical cognition remains open the problem whether the described and investigated objects exist or not. In other words, mathematical cognition is invariant to and thus independent of the existence (“reality”) or non-existence of its objects.

Thus attention turns out to be dual to the phenomenological “intention” in a sense: It postulates its objects as real independently of whether they exist or not. So, the attention and intention constitutes a dual pair in dependence whether the objects at issue are declared as real or not (here “not” does not mean for them to be declared as unreal or nonexisting, but that they might be real or unreal).

Then one can speak of “attention” as the reverse operation to “epoché”: The latter takes or removes reality, and the former gives or adds reality. Thus attention being inherently linked to the problem of reality turns out to be a fundamental philosophical concept rather than only a psychological one. For example, if the operation of that philosophical “attention” is applied to any intention, one would obtain the corresponding “idea” or “eidos” (i.e. appearance as a whole) in a Platonic sense, i.e. as “reell”.     

Furthermore, “intention” has another counterpart, “intension” in logic, mathematics, epistemology, and cognitive science. Intension is what is able to constitutes unambiguously a separated unit such as a notion, set, image, or any unit of cognition by a finite definition, i.e. by a finite set of bound variables interpretable as the logical constant of that unit. An extension as the collection of objects, each of which satisfies the definition at issue, corresponds to any intension possibly as an empty one if the definition is contradictory. The collection may include as existing as nonexisting individuals. 

Thesis:

One can introduce the concept of “attension” as to any unit enumerated above, e.g. as to a notion. It means both all individuals of the extension as existing and their wholeness as existing, too. Thus “attension” is relative to “intension” and “extension”, on the one hand, and to the Platonic “idea” and “eidos”, on the other hand. Furthermore, “attension” can be defined as the application of the “philosophical attention” to any explicit or implicit (e.g. contextual) intension.

Attension complements intension to the pair of both biggest and least element of the mathematical structure of lattice extended from the intention of consciousness to the idea therefore giving both logical and ontological structure of the notion or whatever else unit. That structure orders

the extension in question in a potential taxonomy (i.e. classification of genera and species), the biggest element of which, i.e. the idea of the thing defined by the extension or even that thing itself or by itself, is generated just by the philosophical attention as the corresponding attension. 

On the contrary, if the notion or unit is supplied as usual by any logical or ontological structure, thus its attension is implicitly certain, too.   

A few main arguments:

1 Philosophical phenomenology establishes an inherent link between: (a) logic and mathematics; (b) philosophy; (c) psychology: The link relates the three by means a kind of transcendental idealism in the German philosophical tradition, which Husserl called “solipsistic” in some his works. Thus a bridge for transfer and reinterpretation between notions of psychology, logic and mathematics is created under the necessary condition for those concepts to be considered as philosophical as referred to that kind of transcendental subject.

2 The initial research of Husserl about The psychological Foundation of Arithmetic (1890) leaded him to opposite conclusion in the later Logical Investigations (1900-1901), namely that psychology (and further philosophy) should be underlain rather by logic and mathematics. In fact, the initial base of that synthesis can be found even in Ancient Greece in Pythagoreanism, in the origin itself of philosophy, and a little later, in Plato’s doctrine and Euclid’s geometry. The German idealism including the subject and mind as a fundamental philosophical category had been what allowed of Husserl to add psychology in that huge synthesis.

3 The link in question is grounded in the way of cognition in logic and mathematics, philosophy, and the seen in thus psychology rather than in any reference to reality, to experienced or experimental data for the reality itself should be inferred in particular by the new approach of phenomenology: This suggests for reality not to be presupposed, but to be “bracketed” initially.

4 Indeed, logic and mathematics do not connect the concept of truth, in their framework, to any confirmation by external reality. Therefore, they do not presuppose any reality, and their cognition is independent of reality as a hypothesis or premise. As to philosophy, it ought not to presuppose reality for the reality itself is its main problem (Heidegger underlay the problem of being as a deeper one). At last, psychology should not be referred to reality as far as its object of research is just that being which seems to be opposed to and thus separated from reality, namely mind and psychics (Heidegger refuted this, the latter, and Husserl blamed him for “naturalization”).

5 Thus logic & mathematics, philosophy, and psychology need and would share a relevant method of research, which should be independent of the hypothesis (or axiom) of reality. In particular, that method cannot be experimental or ground on any experience in reality.

6 Logic and mathematics as the most advanced ones in that kind of cognition can suggest its extended model and interpretation where “intension” would correspond to “intention”, and “extension” to some area of reality relevant to that intension at issue.

7 Then “attension” is the “extension” with reality added secondarily as far as reality cannot be presupposed in phenomenological research.   

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Beyond and across space: entanglement

Einstein, Podolsky, and Rosen (1935) suggested a thought experiment in order to demonstrate that quantum mechanics was ostensibly incomplete. Furthermore, they showed that the mathematical formalism of quantum mechanics had been granted as complete, it would imply some action at a distance beyond and across space, “spooky” in words of Einstein. Since that kind of action contradicted the principle of physics, quantum mechanics should be incomplete in their opinion. Edwin Schrödinger (1935) also pointed out that quantum mechanics implies some special kind of interaction between quantum systems by means of “vershränkten zustände” using his term.  John Bell (1964) suggested a real experiment apt to distinguish quantitatively and observably between the classical case without that “spooky action at a distance” and the quantum one involving a special kind of correlation between physical systems, which can exceed the maximally possible limit of correlation in classical physics. The experiments of Aspect, Grangier, and Roger (1981, 1982) as well as all later ones show unambiguously that the forecast quantum correlations are observable phenomena. Thus that “spooky action at a distance” exists and quantum mechanics should be complete. The new phenomenon was called “entanglement” and a separate branch of quantum mechanics, the theory of quantum mechanics, studying that kind of phenomena has appeared and blossomed out since the 90^th of the past century.   

The theory of quantum information showed that the phenomena of entanglement are underlain by the necessary restriction of the concept of space in relation to the coherent states in quantum mechanics. Space is a well-ordered set of points in relation to any observer or reference frame in it while coherent state in quantum mechanics is a whole of those points, which is inseparable and thus unorderable in principle. Both space and coherent state are initial elements of cognition mutually restricting their applicability. However, space refers to our everyday experience while the concept of coherent state or entanglement to scientific cognition in an area inaccessible to our senses. The concept of space should be limited to the relations between physical bodies of commeasurable mass. If a human is granted as an observer in space, the range of masses comparable with that mass (or energy) determines fussily the domain, in which the concept of space is applicable.

The way, in which the concept of space is being diluted gradually to and the beyond the limits of comparability in mass, can be visualized as follows: A de Broglie wave (@) can be attached to any physical entity according to quantum mechanics. Its period is reciprocal to its mass (or energy). One can interpret this period as the length of the present moment specific to the corresponding physical entity of this mass (energy). If the masses (energies) of the interacting physical entities are commeasurable, they can share approximately a common enough present. If their masses (energies) are incommeasurable, what is the case in quantum mechanics studying the system of a macroscopic device, which measures one or more microscopic quantum systems, and that or those systems, the lengths of their present moments are also incommeasurable: The present of the entity of much bigger mass (energy) can be idealized as a point on the segment representing the length of the present of the entity with much less mass (energy).  Furthermore, the present of the measured quantum systems being an approximately common segment will include as the past as the future rather than the present of the device. The past of the device will be represented as all points of the segment, which are before the point of the present of the device, and its future as those after this point.

Consequently, the concept of coherent state in quantum mechanics refers both to the future and past as well as to the present of the investigated system while that of space only to the present, because of which the condition for the present of all discussed entities to be commeasurable is necessary in the latter case. Indeed the future of any entity is unorderable in principle and just this property of it is rigorously represented by the concept of coherent state. However, the past of any entity is always well-ordered as the series of all past moments in time. Therefor the description in quantum mechanics has to provide the invariance both to the unorderable future and to the well-ordered past. In mathematical terms, this means that the so-called well-ordering theorem equivalent to the axiom of choice is necessarily involved. Furthermore, the present always situating and intermediating between the past and the future is just what any choice transforming future into past shares. Space makes possible choice and thus the transformation of future into past.

Entanglement transcending space should be defined as temporal interaction involving the future and past of the macroscopic devices displaying quantum correlations. While any classical correlation should refer only to the present of the correlating entities and thus to the space, in which they are and which they share, any quantum correlation transcends the present and space involving the future and past in order to be able to exceed the maximal possible bound of all classical correlations. Furthermore, the entanglement involves the concept of quantum information. It is a generalization of the classical concept of information in relation to the choice among an infinite set of alternatives.

All those studies in quantum mechanics and the theory of quantum information reflect on the philosophy of space and its cognition. Space should discuss as a “transcendental screen” (a necessary condition of visualization or objectification), on which all phenomena are represented by masses comparable with those of observers granted as human beings. Our sensual experience as well as classical physics observes and studies only phenomena within the framework of space and therefore it cannot transcend it. However, quantum theories can do this allowing of interpreting space newly as the domain of interaction of bodies of commeasurable mass or of physical entities of commeasurable energy and thus as that area of choice, which is able to transform future into past.

References:
Aspect, A, Grangier, R., Roger, G. (1981) "Experimental tests of realistic local theories via Bell’s theorem," Physical Review Letters 47 (7): 460-463.
Aspect, A, Grangier, R., Roger, G. (1982) "Experimental Realization of Einstein-Podolsky-Rosen-Bohm Gedanken Experiment: A New Violation of Bell’s Inequalities," Physical Review Letters 49 (2): 91-94.
Bell, J. (1964) "On the Einstein ‒ Podolsky ‒ Rosen paradox," Physics (New York) 1 (3): 195-200.
Broglie, L. de (1925) "Recherches sur la théorie des quanta (Researches on the quantum theory), Thesis (Paris), 1924," Annales de Physique (Paris, 10-ème série) 3: 22-128.
Einstein, A., Podolsky, B., Rosen, N. (1935) "Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?" Physical Review 47 (10): 777-780.
Schrödinger, E. (1935) "Die gegenwärtige situation in der Quantenmechanik,"
Die Naturwissenschaften  23 (48): 807-812; 23 (49): 823-828; 23 (50): 844-849.

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Quantum information as the information of infinite series

Abstract:

The quantum information introduced by quantum mechanics is equivalent to that generalization of he classical information from finite to infinite series or collections. The quantity of information is the uantity 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 wellordered series of results in time after measurement. The quantity of quantum information is the ordinal corresponding to the infinity series in question.

Key words: axiom of choice, bit, choice, information, quantum information, qubit, set, well-ordering

Extended abstract (with references to sources cited in the paper):

The quantum information introduced by quantum mechanics is equivalent to that generalization of the classical information from finite to infinite series or collections. 

The conception of quantum information was introduced in the theory of quantum information studying the phenomena of entanglement in quantum mechanics. The entanglement was theoretically forecast in the famous papers of Einstein, Podolsky, and Rosen (1935) and independently by Shrödinger (1935) deducing it from Hilbert space, the basic mathematical formalism of quantum mechanics. However, the former three demonstrated the forecast phenomenon as the proof of the alleged “incompleteness of quantum mechanics”. John Bell (1964) deduced a sufficient condition as an experimentally verifiable criterion in order to distinguish classical from quantum correlation (entanglement). Aspect, Grangier, and Roger (1981, 1982) confirmed experimentally the existence of quantum correlations exceeding the upper limit of the possible classical correlations. The theory of quantum information has thrived since the end of the last century in the areas of quantum computer, quantum communication, and quantum cryptography.

The fundament of quantum information is the concept of ‘quantum bit’, “qubit” definable as the normed superposition of any two orthogonal subspaces of complex Hilbert space as follows:

‘Qubit’ is the normed superposition of any two orthonormal vectors (e.g. the orthonormal bases of any two subspaces) in any vector space (e.g. Hilbert space, Euclidean space, etc.). Thus, Hilbert space underlying quantum mechanics is representable as the quantity of quantum information and any wave function, i.e. any state of any quantum system being a point in it can be seen as a value of that quantity. Consequently, all physical processes turn out to be quantum-informational, and nature or the universe is a quantum computer processing quantum information.

The qubit is also isomorphic to a ball in Euclidean space, in which two points are chosen: A qubit is equivalently representable as a unit ball in Euclidean space and two points, the one chosen within the ball, and the other being the orthogonal projection on its surface, i.e. as a mapping of a unit ball onto its surface (or any other unit sphere).
Quantum information is equivalent to the generalization of information from finite to infinite series.

Indeed information can be interpreted as the number of choices necessary to be reached an ordering of some item from another ordering of the same item or from the absence of ordering. Then, the quantity of information is the quantity of choices measured in the units of elementary choice. A bit is that unit of elementary choice: It represents the choice between two equally probable alternatives. Furthermore, the unit of quantum information, the qubit, can be interpreted as that generalization of bit, which is a choice among a continuum of alternatives.

Thus it is able to measure the quantity of information as to infinite sets. The axiom of choice is necessary for quantum information in two ways: (1) in order to guarantee the choice even if any constructive approach to be chosen an element of the continuum does not exist; (2) to equate the definition in terms of Hilbert space and that as a choice among a continuum of alternatives:

Indeed, the theorems about the absence of hidden variables in quantum mechanics (Neumann 1932; Kochen, Specker 1968) demonstrate that the mathematical formalism of quantum mechanics implies that no well-ordering of any coherent state might exist before measurement. However, the same coherent state is transformed into a well-ordered series of results in time after measurement. In order to be equated the state before and after measurement, the well-ordering theorem equivalent to the axiom of choice is necessary. The measurement mediating between them should be interpreted as an absolutely random choice of an element of the coherent state, for which no constructive way (equivalent to some “hidden variable”) can exist in principle. Thus the quantity of quantum information can describe uniformly the state before and after measurement (equivalent to a choice among an infinite set). Thus, Hilbert space can be understood as the free variable of quantum information. Then any wave function, being a given value of it, “bounds” an unorderable and a well-ordered state as the quantity of qubits (i.e. the “infinite choices”) necessary for the latter to be obtained from the former.

The quantity of quantum information is the ordinal corresponding to the infinity series in question. Both definitions of ‘ordinal’ (Cantor 1897; Neumann 1923) are applicable as the ordinals are small. The ordinal defined in Cantor – Russell (Russell, Whitehead any edition) generates a statistical ensemble while that in Neumann, a well-ordering. Both correspond one-toone to a coherent state as the one and same quantity of quantum information containing in it.

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A Formal Model of Metaphor in Frame Semantics

1A formal model of metaphor is introduced. It models metaphor, first, as an interaction of “frames” according to the frame semantics, and then, as a wave function in Hilbert space. The practical way for a probability distribution and a corresponding wave function to be assigned to a given metaphor in a given language is considered. A series of formal definitions is deduced from this for: “representation”, “reality”, “language”, “ontology”, etc. All are based on Hilbert space. A few statements about a quantum computer are implied: The so-defined reality is inherent and internal to it. It can report a result only “metaphorically”. It will demolish transmitting the result “literally”, i.e. absolutely exactly. A new and different formal definition of metaphor is introduced as a few entangled wave functions corresponding to different “signs” in different language formally defined as above. The change of frames as the change from the one to the other formal definition of metaphor is interpreted as a formal definition of thought. Four areas of cognition are unified as different but isomorphic interpretations of the mathematical model based on Hilbert space. These are: quantum mechanics, frame semantics, formal semantics by means of quantum computer, and the theory of metaphor in linguistics.

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The paper is published in Proceedings of the 41st Annual Convention of the Society for the Study of Artificial Intelligence and the Simulation of Behaviour, ISBN  978-1-5108-0386-2, pp. 187-194. The published paper @  Sematic Scholar or as a PDF

Appendix:

A Model of Causal and Probabilistic Reasoning in Frame Semantics

The thesis of the talk is fourfold: (1) Probabilistic reasoning can be seen as the interaction of at least two frames in a sense of frame semantics. (2) Then causal reasoning can be interpreted as the particular case of zero interaction between the frames. (3) In turn, this allows of the frames to be interpreted formally as correspondingly “reality” and the “image of reality”, and language as an (even one-to-one) mapping between those two universal and formal frames of “reality” and its “image”. (4) Probabilistic reasoning can be further represented formally as the “entanglement” of two or more frames and thus in terms of quantum information.

A few terms need some specification, namely: “frame semantics”, “frame” “formal semantics”, “entanglement”, “quantum information”, and “quantum computer”: “Frame semantics” is meant in the sense of Charles J. Fillmore: “Frame semantics offers a particular way of looking at word meanings, as well as a way of characterizing principles for creating new words and phrases, for adding new meanings to words, and for assembling the meanings of elements in a text into the total meaning of the text” [1: 111].

“Frame”: “The idea is that people have in memory an inventory of schemata for structuring, classifying and interpreting experiences, and that they have various ways of accessing these schemata and various procedures for performing operations on them” [2: 25]. “By the term ‘frame’ I have in mind any system of concepts related in such a way that to understand any one of them you have to understand the whole structure in which it fits ...” [1: 111]. The “frame” already linked to formal semantics is specified as a set of well-orderings referring to something as its “logic”, in which any property, relation, part or feature of that something can be understood by somebody or by a group. Consequently, that formal and semantic “frame” means the relation between the wholeness of that something and the “logic” of it as a collection of well-orderings.

“Formal semantics” is a term used both in logic and in linguistics but in partially different meanings.

The common is the utilization of mathematical and logical models. However, the logical “formal

semantics” addresses the natural entailment in language in terms of logical sequence while the linguistic “formal semantics” discusses rather the correspondence both of linguistic units and the wholeness of texts to reality in terms of mathematical mappings, set theory, and logic.

“Entanglement” is a term in quantum mechanics, meaning the information interaction between two
or more quantum systems and thus being fundamental for the theory of quantum information.

The formal and mathematical definition of “entanglement” as that Hilbert space, which cannot be factorized to any tensor product of the Hilbert spaces of subsystems, allows of the term to be generalized to any model utilizing Hilbert spaces. For the formal and semantic model used here is based on Hilbert space(s), the concept of entanglement is applicable. It is the mathematical base for the model of probabilistic reasoning.“Quantum information” is a term initially coined by quantum mechanics to describe the base of a generalized kind of information underlying all quantum mechanics. Quantum information can be interpreted both as transfinite series of bits and as finite or infinite series of qubits. A bit is the elementary choice between two equally probable alternatives, and a qubit (i.e. quantum bit) can be interpreted as the elementary choice among an infinite set of alternatives though it is initially defined in quantum mechanics as the normed superposition of two orthogonal subspaces of Hilbert space. The quantity of information whether classical or quantum is the quantity of the corresponding elementary choices (whether bits or qubits) necessary for transforming a well-ordering to another (both, whether finite or transfinite). Thus quantum information can be interpreted as the quantity of elementary choices necessary to transform a frame into another and consequently the information of a probabilistic reasoning formalized as above.

“Quantum computer” [3, 4, 5] is a mathematical model involved by quantum mechanics to interpret

its formalism as a generalized kind of calculation, processing quantum information. Thus all physical
states and processes may be also seen as computational.
The argumentation for the thesis:
(1) Probabilistic reasoning can be understood as the appearance of a new frame by interaction of two

or more initial frames for some essential part of each of them is shared by all. Thus the understanding of each of them separately generates immediately the understanding of the probabilistic reasoning as a new whole demonstrating therefore the appearance of a new frame, which is not the simple additivity of the sub-frames composing it. The set of well-orderings formalizing semantically a frame can be substituted by a point of Hilbert space, and interpreted as a wave function of a quantum system. Any possible frame is measurable as a single value of quantum information. Then the probabilistic reasoning will be interpretable as the entanglement of the quantum systems corresponding to each sub-frame composing it.

(2) Causal reasoning can be interpreted after that as a particular and borderline case of probabilistic

reasoning, a “zero” probabilistic reasoning, or just the simple additivity of the sub-frames composing

them. The corresponding “wave functions” are orthogonal to each other and there is no entanglement
between them.
(3) Language is reduced to an infinite countable set (A) of its units of meaning, either words or

propositions, or whatever others. It includes all possible meanings, which can be ever expressed in the language rather than the existing till now, which would always a finite set. The external twin of reality is introduced by another set (B) such that its intersection with the above set of language to be empty. The union of them (C=A∪B) exists always so that a one-to-one mapping (f: C↔A) should exist under the condition of the axiom of choice. The mapping (f) produces an image (B (f)) of the latter set (B) within the former set (A). That image (B (f)) serves as the other twin of reality to model the reality within the language as the exact causal reasoning of the reality out of language (modelled as the set B). In the model, the necessity and sufficient condition of that causal reasoning between reality both within and out of the language is just the axiom of choice: If the axiom of choice does not hold, the relation between the sets B (f) and B cannot be defined rigorously as an exact causal reasoning but rather as some simile and the vehicle between the two twins of reality can be only probabilistic reasoning.

(4) Probabilistic reasoning formalized as above is representable as the wave function of the frame

compounded by two or more sub-frames, which interact between each other by means of the shared

nonzero intersection. The quantity of quantum information of a probabilistic reasoning is different from that quantity of the corresponding causal reasoning. Thus the probabilistic reasoning demonstrates the entanglement of the composing sub-frames after they have been formalized as points in Hilbert space.

One can utilize the picture of the maximal frame, in which are chosen two positions as two points.

Furthermore, the proposition connects them by a single “classical trajectory” while, the metaphor does the same by all possible trajectories, each of which is differently probable. Any understanding chooses only one of them. The analogy to the Feynman interpretation of quantum mechanics [6, 7, and 8] is obvious. It addresses further the idea for the mathematical formalism of quantum mechanics to be only adapted to the relevant terms of frame semantics. Indeed any measurement in quantum mechanics corresponds to a given understanding of what the metaphor mean. The metaphor unlike any proposition does not predetermine how it should be understood, however it defines implicitly a “wave function” of all possible understandings as the set of pathways, in any of which it can be interpreted equally justifiably.

References:

[1] C. Fillmore  (1982) "Frame semantics," in: Linguistics in the Morning Calm. Linguistic Society of Korea (Ed.) Hanshin, Soeul, Korea, pp. 111-137.

[2] C. Fillmore (1976)  "Frame semantics and the nature of language," in: Origins and Evolution of Language and Speech (S. R. Harnad, H. D. Steklis, J. B. Lancaster, eds.) (Annals of the NY Academy of Sciences, Vol. 280) New York Academy of Sciences, New York, USA, pp. 20-32.

[3] R. Feynman   (1982) Simulating Physics with Computers," Innternational Journal of Theoretical Physics 21 (6/7): 467-488.

[4] R. Feynman (1986) "Quantum Mechanical Computers," Foundations of Physics 16 (6): 507-531.

[5] D. Deutsch. (1985) "Quantum Theory, the Church-Turing Principle and the Universal Quantum Computer," Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 400 (1818): 97-117 .

[6] R. Feynman. Space-Time Approach to Non-Relativistic Quantum Mechanics. Reviews of Modern Physics, 20(2): 367-387 (1948).

[7] R. Feynman and F. L. Vernon, Jr. (1963) "The theory of a general quantum system interacting with a linear dissipative system," Annals of Physics (N.Y) 24: 118-173.

[8] R. Feynman; A. R. Hibbs. (1965) Quantum mechanics and path integrals. New York: McGraw-Hill .

Inductive Logic from the Viewpoint of Quantum Information

The resolving of the main problem of quantum mechanics about how a quantum leap and a smooth motion can be uniformly described resolves also the problem of how a distribution of reliable data and a sequence of deductive conclusions can be uniformly described by means of a relevant wave function “Ψdata”.

Key words: entanglement, hypothetical reasoning interpretation of wave function, inductive and probabilistic conclusion



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From the “Free Will Theorems” to the “Choice Ontology” of Quantum Mechanics

If the concept of “free will” is reduced to that of “choice” all physical world share the latter quality. Anyway the “free will” can be distinguished from the “choice”: The “free will” involves implicitly certain preliminary goal, and the choice is only the mean, by which it can be achieved or not by the one who determines the goal. Thus, for example, an electron has always a choice but not free will unlike a human possessing both. Consequently, and paradoxically, the determinism of classical physics is more subjective and more anthropomorphic than the indeterminism of quantum mechanics for the former presupposes certain deterministic goal implicitly following the model of human freewill behavior.

The choice is usually linked to very complicated systems such as human brain or society and even often associated with consciousness. In its background, the material world is deterministic and absolutely devoid of choice. However, quantum mechanics introduces the choice in the fundament of physical world, in the only way, in which it can exist: All exists in the “phase transition” of the present between the uncertain future and the well-ordered past. Thus the present is forced to choose in order to be able to transform the coherent state of future into the well-ordering of past. The concept of choice as if suggests that there is one who chooses. However quantum mechanics involves a generalized case of choice, which can be called “subjectless”: There is certain choice, which originates from the transition of the future into the past. Thus that kind of choice is shared of all existing and does not need any subject: It can be considered as a low of nature.

There are a few theorems in quantum mechanics directly relevant to the topic: two of them are called “free will theorems” by their authors, Conway and Kochen, and according to them: “Do we really have free will, or, as a few determined folk maintain, is it all an illusion? We don’t know, but will prove in this paper that if indeed there exist any experimenters with a modicum of free will, then elementary particles must have their own share of this valuable commodity” “The import of the free will theorem is that it is not only current quantum theory, but the world itself that is non-deterministic, so that no future theory can return us to a clockwork universe”

Those theorems can be considered as a continuation of the so-called theorems about the absence of “hidden variables” in quantum mechanics.

Key words: choice, freewill, freewill theorems, hidden variables in quantum mechanics



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History as the Ontology of Time

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:

• History rests immediately on what time is
• Time should be understood as a relation of the one and the same being in two mody: by itself and as an inseparable constituent of a relevant whole
• That approach stresses on the present and choice, which can be accomplished only in the present therefore reconciling absolutely uncertain future and the well-ordered past
• That understanding of time implies for history to be generalized from the historiography of the past facts to a relevant joint description of past facts, present choices, and future possibilities.

Key words: History, Ontology, Time, Choice, “Dazeit”

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Fleeting thoughts: Epistemology as metaphysics, or the case of quantum mechanics

Can knowledge coincide with its object? Can cognition coincide with its subject? Can epistemology coincide with metaphysics? Not in general. Nevertheless, there exists a famous particular case violating the rule of their distinction: that of quantum mechanics. Any quantum entity is our knowledge of it. This is due to the fundamental postulate of quantum mechanics: it studies the system of both quantum entity and macroscopic apparatus only by the readings of the latter.

Thus, quantum mechanics allows of studying the general case in philosophy: the coincidence of epistemology and metaphysics. A few of the most intriguing properties of quantum mechanics contradicting to all other sciences, in which epistemology and metaphysics are categorically distinguished, can be grounded right on that coincidence:

- The so-called Copenhagen interpretation

- Indeterminism and Max Born’s probabilistic interpretation

- The many-worlds interpretation

- The quantum correlations at a distance and the quantum-information interpretation

- The concept of quantum computer

The mathematical model of quantum mechanics, the separable complex Hilbert space being a mathematical structure is able to model the above extraordinary interpretations of quantum mechanics:

- It implies the theorems about the absence of hidden variables in quantum mechanics (Neumann 1932; Kochen and Specker 1968). Thus, it implies furthermore:

- The coincidence of model and reality

- The completeness of quantum mechanics furthermore reconfirmed experimentally (Clauser and Horne 1974; Aspect, Grangier, and Roger 1981; 1982)

- Any wave function can be interpreted as the characteristic function of the probabilistic distribution of a random physical quantity.

- It is the simplest mathematical structure allowing of the unification of discrete motion (sucCan knowledge coincide with its object? Can cognition coincide with its subject? Can epistemology coincide with metaphysics? Not in general. Nevertheless, there exists a famous particular case violating the rule of their distinction: that of quantum mechanics. Any quantum entity is our knowledge of it. This is due to the fundamental postulate of quantum mechanics: it studies the system of both quantum entity and macroscopic apparatus only by the readings of the latter.

Thus, quantum mechanics allows of studying the general case in philosophy: the coincidence of epistemology and metaphysics. A few of the most intriguing properties of quantum mechanics contradicting to all other sciences, in which epistemology and metaphysics are categorically distinguished, can be grounded right on that coincidence:

- The so-called Copenhagen interpretation

- Indeterminism and Max Born’s probabilistic interpretation

- The many-worlds interpretation

- The quantum correlations at a distance and the quantum-information interpretation

- The concept of quantum computer

The mathematical model of quantum mechanics, the separable complex Hilbert space being a mathematical structure is able to model the above extraordinary interpretations of quantum mechanics:

- It implies the theorems about the absence of hidden variables in quantum mechanics (Neumann 1932; Kochen and Specker 1968). Thus, it implies furthermore:

- The coincidence of model and reality

- The completeness of quantum mechanics furthermore reconfirmed experimentally (Clauser and Horne 1974; Aspect, Grangier, and Roger 1981; 1982)

- Any wave function can be interpreted as the characteristic function of the probabilistic distribution of a random physical quantity.

- It is the simplest mathematical structure allowing of the unification of discrete motion (such as a quantum leap) and smooth motion in classical mechanics

- It is that generalization of Peano arithmetic in which any positive integer is substituted by a qubit