CYCLIC MECHANICS: THE PRINCIPLE OF CYCLICITY

Cyclic mechanic is intended as a suitable generalization both of quantum mechanics and general relativity apt to unify them. It is founded on a few principles, which can be enumerated approximately as follows:
1. Actual infinity or the universe can be considered as a physical and experimentally verifiable entity. It allows of mechanical motion to exist.
2. A new law of conservation has to be involved to generalize and comprise the separate laws of conservation of classical and relativistic mechanics, and especially that of conservation of energy: This is the conservation of action or information.
3. Time is not a uniformly flowing time in general. It can have some speed, acceleration, more than one dimension, to be discrete.
4. The following principle of cyclicity: The universe returns in any point of it. The return can be only kinematic, i.e. per a unit of energy (or mass), and thermodynamic, i.e. considering the universe as a thermodynamic whole.
5. The kinematic return, which is per a unit of energy (or mass), is the counterpart of conservation of energy, which can be interpreted as the particular case of conservation of action “per a unit of time”. The kinematic return per a unit of energy (or mass) can be interpreted in turn as another particular law of conservation in the framework of conservation of action (or information), namely conservation of wave period (or time). These two counterpart laws of conservation correspond exactly to the particle “half” and to the wave “half” of wave-particle duality.
6. The principle of quantum invariance is introduced. It means that all physical laws have to be invariant to discrete and continuous (smooth) morphisms (motions) or mathematically, to the axiom of choice.
The list is not intended to be exhausted or disjunctive, but only to give an introductory idea about the text, which follows:

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Free Will in Human Behavior and Physics

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 a certain goal, and the choice is only the mean, by which the aim can be achieved or not by the one who determines the target. 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. Quantum mechanics introduces the choice in the fundament of physical world involving 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 2006; 2009). Any quantum system either a human or an electron or whatever else has always a choice: Its behavior is not predetermined by its past. This is a physical law. It implies that a form of information, the quantum information underlies all existing for the unit of the quantity of information is an elementary choice: either a bit or a quantum bit (qubit).

Key words: axiom of choice, choice, free will, free will theorems, goal, hidden variables, quantum information, quantum mechanics

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The published paper (Vasil Penchev. Free will in human behavior and physics [Свободная воля в поведении человека и физике]. Labour and Social Relations. 2020.Vol. 30 Issue 6. P. 185-196. DOI 10.20410/2073-7815-2020-30-3-185-196) as a PDF, @ PhilPapers, @ SSRN, or @ JournalSite

“Схизмата във физиката” и ... неравенствата на Бел

Квантовата механика предлага множество "скандални" за здравия разум твърдения. Сред тях е и нарушаването на "неравенствата на Бел": достатъчно, но не необходимо усложие за "квантово сдвояване" - загадъчното действие на разстояние, "призрачно" по думите на Айншайн. Той, заедно с Подолски Розен го извежда още в 1935 г.като аргумент, доказващ математически "непълнотата на квантовата механика". "Схизмата във физиката" е прочута книга на Сър Карл Попър, посветена на философията и методологията на квантовата механика, в която той поддържа опозицията на Айншайн срешу квантовата механика. Обаче днес нарушаването на неравенствата на Бел е общоприет и многократно потвърден експериментален факт.

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Fleeting thoughts: A physical interpretation of Heidegger’s distinction of “existence” and ‘being’ by neutrinos

Neutrino exists only within the framework of energy conservation. (Wolfgang Pauli arguing with Niels Bohr forecast their necessary existence due to energy conservation.) If so, what would be representation of "neutrino" out of that framework? If one answers the latter, what would be the physical meaning of existence out of (energy) conservation? Does it correspond to Heidegger's distinction of "existence" ("conservation", "metaphysics") and "being" (generalizing "conservation", "ontology", eventually "fundamental")?

Well, neutrino is the difference between any "path of Feynman" and the privileged one of energy conservation. However, if the neutrino is observed experimentally, this implies that Feynman's paths

are not (only) virtual or a merely thought construction, but actual ones therefore generating all "immortal" neutrinos penetrating the universe.

So, one should search for neutrinos in the actual (rather than Born's probabilistic) viewpoint of Hugh Everett III to quantum mechanics postulating all "paths" as real "worlds": then neutrino would be a certain difference between a given "world" ("path") and the privileged "zero" one of energy conservation.

Furthermore, that privileged "zero" world of energy conservation can be identified as a privileged reference frame consistent to general rather than to special relativity and determined unambiguously by the "cosmological constant" even if its value is zero. That viewpoint of general relativity interprets

the neutrinos as direct physical equivalents of the certain curvatures of space-time in each point of it.

Thus, the mysterious neutrinos however interpreted both within and out of the framework of energy conservation are able to bridge over the gap between quantum mechanics and general relativity. Thus, the neutrinos offer a physical and very fruitful interpretation of Heidegger's distinction between "existence" and 'being’:

Existence corresponds to the “zero” reference frame of energy conservation, and ‘being’, to all possible variations about it (illustrable by Feynman’s paths) or, to all possible curvatures of pseudo-Riemannian space to the “zero” curvature of the cosmological constant.

Well, that being as a single one as the totality implies the equivalence of the set of all paths of Feynman (quantum mechanics) to all curvatures of space-time (general relativity) under the condition of that totality or “fundamentality” (in Heidegger’s word).

The physical meaning of the condition is the totality of the universe: its externality (as far as the universe is the physical totality) is described by quantum mechanics within it. Thus, it corresponds necessarily to its internality as it is described by general relativity.

An illustration: all worlds (in the interpretation of Hugh Everett III) or all possible discretely separated “universes” out of ours should be represented equivalently within it and thus describable by general relativity as all possible reference frames arbitrarily accelerated. The discrete transition between any two separated “universes” turns out to equivalent to smooth transition between any two reference frames!

Key words: Heidegger, fundamental ontology, being, existence, neutrino

Gravity as entanglement, entanglement as gravity

A generalized and unifying viewpoint to both general relativity and quantum mechanics and information is investigated. It may be described as a generaliztion of the concept of reference frame from mechanics to thermodynamics, or from a reference frame linked to an element of a system, and thus, within it, to another reference frame linked to the whole of the system or to any of other similar systems, and thus, out of it. Furthermore, the former is the viewpoint of general relativity, the latter is that of quantum mechanics and information.
Ciclicity in the manner of Nicolas Cusanus (Nicolas of Cusa) is complemented as a fundamental and definitive property of any totality, e.g. physically, that of the universe. It has to contain its externality within it somehow being namely the totality. This implies a seemingly paradoxical (in fact, only to common sense rather logically and mathematically) viewpoint for the universe to be repesented within it as each one quant of action according to the fundamental Planck constant.
That approach implies the unification of gravity and entanglement correspondiing to the former or latter class of reference frames. An invariance, more general than Einstein's general covariance is to be involved as to both classes of reference frames unifying them. Its essence is the unification of the discrete and cotnitinuous (smooth). That idea underlies implicitly quantum mechanics for Bohr's principle that it study the system of quantum microscopic entities and the macroscopic apparatus desribed uniformly by the smmoth equations of classical physics.

Key words: entanglement, gravity, invariance to the discrete and continuous, quantum gravity, reference frame




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Prolegomena: The quantum-information link between the Schrödinger equation (QM) and the Einstein filed equation (GR)

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Quantum Mechanics as a Measure Theory: the Theory of Quantum Measure

Quantum mechanics is well representable as a theory of a new kind of measure: quantum measure. It can be interpreted as a covariant counterpart of its usual consideration. The unit of quantum measure is a qubitand thus all physical quantities share a common fundament, that of quantum information, and all physical processes can be interpreted as computational ones. That interpretation of quantum mechanics is fruitful and heuristic: This can be visualized by an explanation of the Aharonov–Bohm effect.

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Prolegomena: Quantum measurement

The concept of measurement in quantum mechanics is seen from a few philosophical and mathematical viewpoints further: quantum invariance, the foundation of mathematics and the axiom of choice, and quantum measure. Then a few general and more particular philosophical problems can be guided in a new direction. Those are:

1. Quantum gravity: Can general relativity be considered as a theory of quantum gravity?
2. The laws of conservation: Can quantum mechanics offer a more general law than that of conservation of energy (energy-momentum) so that to be consistent with general relativity where energy-momentum is conserved only locally?
3. The principle of relativity: Can the Einstein relativity principle be generalized in a way to comprise quantum movements as well?
4. Invariance to discreteness or continuity: Can quantum mechanics offer a more general viewpoint to unify continuous (or smooth) and discrete (quantum) motions?
5. The alternatives of the “Big Bang”: Can the universe arise necessarily from nothing, in mathematical laws?
6. Nothing and anything: Can nothing and anything have a common measure?
7. Quantum information: Is quantum information that quantity both physical and mathematical one, which uses that measure?
8. Mathematics and physics: Is there a smooth transition between them, in which a mathematical structure or its element like a number or a probability distribution can transform into a physical entity like a particle?
9. The foundation of mathematics: Can set theory be “repaired” in a way to include directly quantum correlations or entanglement between the elements of a primary mathematical structure like ‘set’?
10. The interpretation of Hilbert space: Can one think Hilbert space as that structure in the frameworks of the set-theory based mathematics, which is both the simplest and may involve quantum correlations or entanglement?

The list can be continued rather. One should demonstrate that quantum measurement is relevant to it after all:

1. Quantum measurement and the axiom of choice:

The theorem of Kochen – Specker (1968) and yet John von Neumann’s (1932) one before it require the absence of any hidden variables in any quantum state before measurement and thus they exclude any well-ordering in it. In fact any well-ordering before measurement is excluded by any theory utilizing differential equations to describe the states in the studied area. It can happen only if the axiom of choice is added externally for the smooth continuum necessary for any differential equation cannot be well-ordered otherwise than by it. However the difference between a point of the continuum and a corresponding point in a well-ordered subset of it can converge to zero. That way out is forbidden to quantum mechanics because of the Plank constant for it imposes a finite difference between the states before and after measurement. Furthermore any set of measured results is well-ordered, e.g. by the exact time of recording in a central computer.
If one combines the Kochen – Specker theorem with the well-ordering after measurement, the well-ordering theorem equivalent to the axiom of choice is unavoidable: So quantum measurement involves necessarily the axiom of choice. One can object that the sets before and after measurement are different so that the well-ordering theorem is irrelevant. In fact even then the mapping between a set, which cannot be well-ordered in principle, and that, which is always well-ordered, requires the axiom of choice for the Cartesian product between them to exist, of which the mapping is a subset. So the measurement of any quantum system or state designated shortly as ‘quantum measurement’ cannot be free of the axiom of choice.
Since the Kochen – Specker theorem does not admit the axiom of choice in any quantum system or state by itself, i.e. before measurement, and quantum measurement requires it as above, quantum mechanics has to be invariant to the axiom of choice: Any statement or equation in it has to be equally valid both if the axiom of choice is accepted or not.

2. Quantum measurement and quantum invariance:

This unique relation of quantum mechanics to the axiom of choice is due to the true fundament of it, wave-particle duality and thus, to involving quantum leaps in mechanical motion. One can speak of quantum invariance underlying the above noticed invariance to the axiom of choice for quantum measurement. The wave-particle duality has another side, which can be designated as the discrete-smooth duality of mechanical motion, since there is an exact correspondence of a wave function to a quantum leap (a discrete motion) as well as of a particle to a smooth trajectory (a world line). Thus that discrete-smooth duality is a kind of generalization as to Einstein’s (1918) principle of relativity, which requires the invariance of all physical laws only to smooth relative motions (reference frames), so that it comprises already also quantum leaps.
That new and more general invariance is quantum invariance: It means that all physical laws have to be invariant to any, both discrete and continuous (smooth), transformation between two or more reference frames. The generalization pioneers the simplest pathway between general relativity and quantum mechanics, i.e. that of quantum gravity.
Any discrete motion does not allow defining a finite value of relative speed. If yet it is defined, this excludes to determine the distance of the leap. Both complementary restrictions constitute together the Heisenberg principle of uncertainty. This uncertainty is a new and unique physical variable, a free variable unboundable in principle, by a natural law such as uncertainty. Its physical dimension is action: The physical quantity of action is exceptional and singular since it is as the dimension of that unboundable free variable (in quantum mechanics) as the dimension of the corresponding bound variable (in general relativity) of the same name and dimension. Consequently what is conserved passing from each to the other theory is action in a rather extraordinary way of conservation: A dimensionless physical quantity like entropy is transformed in its reciprocal also dimensionless physical quantity like information and vice versa. Philosophically said the disorder of entropy is transformed in the order of information and vice versa conserving action as in each of both theories as between them: The theory of quantum gravity turns out to be the theory of quantum information.
Furthermore the mutual transformation between entropy and information or between disorder and order in the framework of conservation of action describes well quantum measurement and quantum invariance, and even the invariance to the axiom of choice:  Indeed quantum measurement transforms an initial fundamental disorder of coherent state into the order of a well-ordering of measured results, and quantum invariance means the equivalence of the disorder of probable values in a quantum leap and of the determined order of a smooth trajectory.

3. Quantum measurement and the foundation of mathematics:

The mutual transformation of order and disorder as the invariance to the axiom of choice deserves to be independently described for it can refer to the foundation of mathematics and thus to a new understanding how mathematics and physics are connected in reality and in quantum measurement. In fact the invariance to the axiom of choice is well known in the foundation of mathematics for a long time as the so-called paradox of Skolem (1922). He introduced the “relativity of the notion of set” meaning that any infinite set can be enumerated by the axiom of choice or even to be interpreted as a finite one. So a bridge exists from quantum measurement to a new, quantum foundation of mathematics. Its main idea is to borrow from nature the way how mathematics is founded in it generalizing the well-ordered set of natural numbers to that of qubits, which is equivalent to Hilbert space:
In other words, the idea is the founding set of natural numbers, which is always countable and thus it is noninvariant to the axiom of choice, to be replaced by the simplest one, which is invariant to the axiom of choice being uncountable and countable as a quantum coherent state before and after measurement. Such a one is Hilbert space, or the set of all well-ordered series of qubits. A point in it (or a sequence of qubits, or a wave function) can represent equally well both a coherent state before measurement and its corresponding statistical ensemble after measurement. That point is invariant to the transformation between entropy (E) and information (I) if and only if the definition of entropy and information is modified in a way to be invariant to the reciprocal transformation of their variables: . The Shannon type definition is not invariant so. However, the definition of physical quantity or observable in quantum mechanics by a selfadjoint operator is invariant just so. Consequently the latter is to be accepted as the relevant definition of information at least as to the quantum foundation of mathematics.
That mathematics founded in this way cannot involve undecidable statements since decidability can be generalized as the invariance of entropy and information to the reciprocal transformation of their variables as above. Furthermore any statement Gödel codable can be coded as a series of qubits and thus undecidable statements cannot exist in that mathematics. Furthermore it cannot be divided from physics in the bridge of quantum mechanics and thus from reality. The conception of quantum measurement serves as the base of total decidability. It leads to some kind of quantum Pythagoreanism.

4. Quantum measurement and quantum measure:

The unit of quantum measurement is specific, quite different from the classical one even as the mathematical notion of measure: It is not a Lebesgue or Borel measure. This is quantum measure, which is complete as the former and universal as the latter. It is neither arbitrarily dimensional nor one-dimensional. It is three-dimensional. Its universal unit is a qubit isomorphic to a unit ball. This is the unit of quantum information: So quantum measure serves as the quantity of quantum information of any measured. Indeed it is really universal. It can measure equally well both the disorder of entropy and the order of information, both anything, which exists being actual, and nothing, which does not exist, being virtual, only possible and probable. Consequently, quantum measure can describe the “genesis from nothing” as a process, of course not as a process in time, because that is the genesis of time: For example, the “Big Bang” is not more than a visualization of quantum measure onto the “screen” of the usual understanding of measure.

„Верблюдът Радичков“: въображението като реалност

Целта на този научен доклад е, разбира се, магическа: да оповести пред заклети посветени заклинанието, което ще стори Радичков на верблюд. Средствата, както науката изисква, са чисто логически: идват от Логоса, думата и нейната способност да общува с човешката душа.
В наше време светът е вече „глокален“: хората мислят глобално, но действат, както от незапомнени времена, локално; с една дума – „глокално“. Дали мислят или си въобразяват (Хайдегер твърди, че „ние още не мислим“) винаги е глобално, а виж: реалността е локална. Така, ако човек е в София, реалността е една, а ако отиде – като Радичков ­– в Сибир, друга. Уверявам ви, че в Йерусалим реалността е трета, в Рио де Жанейро – четвърта, а пък в Ню Йорк – пета. В Чикаго, обаче, не е шеста, а пак пета. Като се върна от Санкт Петербург, ще мога да ви кажа дали там е шестата реалност или е пак четвъртата, както в Рио де Жанейро.
Колкото места, толкова и реалности, дори и да са понякога еднакви на разни топоси, че дори и хроноси. А въображението е едно. Радичков като отиде в Калиманица, не може да остави въображението си в София; нито никой от нас, където и да се премести да живее или само на гурбет. Затова въображението е глобално: човек първо си въобразява еднакво, а после само действа различно в някой от всичките възможни светове, в който се е случил по рождение или с нарочно намерение. Това e вярно за всички нас, не и за Стив Джобс: той мисли различно и дори се изразява различно: think different.
Think different: мисли различен. Някой си Дас Ман се опитва да поправи Джобс: не се казва „мисли различен“, а „мисли различно“: think differently. Стив обяснява и настоява: „мисли различен“ като „бъди различен“: be different.
Каква е разликата? Защо Джобс настоява и според характера си, се налага?
Невъзможно е да мислиш различно. Мисленето като въображението е едно. Но за да мислиш, трябва да си различен. Иначе просто не мислиш. Не може да се мисли различно.
Das Man не може да мисли. Това ни казва Хайдегер с „още-не-мисленето“ на хората. Когато се прави нещо, както се прави, не се мисли. Мисленето е единственото, което Das Man не може да прави. Ала живеем в общество, покорено от този безлик Das Man. И следователно “още не мислим”.
„Мисля, затова съм“. И обратното: “Аз съм, затова мисля“. А “мисли се” е безсмислица. Както и: „мисли се различно“; грешка по определение.
Бъди различен, бъди в друга реалност и ще започнеш да мислиш. И наистина Стив е пробвал реалности: например индийската или онази на ЛСД. Иска същото от сътрудниците и всички хора: think different. Бъди различен и ще мислиш.
Радичков е различен, затова мисли. Неговото въображение е една философска школа по мислене. Въображението е реалността, в която Дас Ман е безсилен.
Единствената, една единствената реалност, това е реалността на всесилния Дас Ман и тази, в която „ние още не мислим“, защото Дас Ман ни го е забранил, а всички ние покорно и унило следваме повелите му.
Единствената реалност е реалността без въображение: реалността, в която въображението е забранено. И дори може да има “философия” на единствената реалност, “реализъм”: “философията” на Дас Ман; също грешка по определение, защото философията е мислене, онова, което Дас Ман не може да прави, а ние му се подчиняваме...
Не и Радичков!
Учи ни на истина, която знаем: всеки е различен. Ала за разлика от него, почти никога нямаме куража да я отстояваме.
Така Радичков не е магически реалист, а човек. Затова пък всички хора сме магически реалисти. Обитаваме всеки своята реалност, обща с колеги и приятели, семейството и децата, с „грижа“, ала „поетично все пак“ както твърди Хайдегер, цитирайки Хьолдерлин.
Въображението е едно и глобално, реалностите – много и до една локални. Навярно щастливите семейства си въобразяват и затова се приличат, както Толстой започва „Ана Каренина“, а нещастните семейства са нещастни, всяко по своему, защото са изолирани в собствена и несподелима, нещастна реалност.
Айнщайн също като Радичков не е магически реалист. Така той си въобразява мислени експерименти, които нарича Gedankenexperimenten, за да разбере, че всяка реалност барабар с нейното време и пространство са относителни: ако човек е на едно място по едно време нещата, сиреч реалността, изглеждат едни, а същите неща от други място и време – съвсем различни. След Айнщайн много хора си въобразяват неговите Gedankenexperimenten и веднага разбират съвсем същото като него, защото въображението е едно и с него става ясно, че реалностите са много и относителни.
Всяка книга ни въвлича в своя „хронотоп“, както го нарича Бахтин след като чува за Айнщайновата теория на относителността, за да прибави необезпокоявана там солидна порция художествена измислица, задължителна за да получим чувството, че „всичко е реално“.
Радичков пък, защото е по-скоро човек и така магически реалист, а не писател, прави обратното: посипва своя „хронотоп“ с толкова реално въображение, че никой не може да се усъмни: „нищо не е измислено“. И всеки, като повтори неговите Gedankenexperimenten, се уверява в силата на въображението, разкрило ни истината за относителността на многото възможни реалности.
Така един ден, когато от околностите на Калиманица, стартира звездолет, изпращан от духова музика, то екипажът му ще забележи удивителни неща в своето космошествие (за малко да сбъркам и да напиша „пътешествие“) към далечните звезди, каквито те изглеждат от нашата земна реалност. Както всички нас, и звездолетците първоначално ще мислят, че звездите са си звезди, където и да са, а не само да изглеждат като звезди от земята. Те ще мислят, че и самите те ще са хора, където и да са:
Но какво се оказва?!
Както пердашат светлинна  година за минути, за да сколасат да се върнат преди да са се оженили за булките си (или мъжорята си), техният звездолет, без ни най-малко да го допускат, неусетно се превръща в кожен пъпеш, а самите те - в хлебарки, които пъплят по страниците на Радичковите книги и като открият съмнително място, правят челна стойка върху обезпокоителния пасаж, размахвайки и пляскайки със задни крачета, за да привлекат вниманието на останалите.
Ще попитате: „Как така?!“, досущ както онези жаби, които издават такива звуци според Радичков.
Енрико Ферми, един от многото предполагаеми бащи на атомната бомба, го е обяснил съвсем просто, след като посмятал наум с вероятности не по-малко точно отколкото как атомната бомба да гръмне: ако реалността навсякъде беше навсякъде една и съща, пришълците от звездите отдавна щяха да са тук. Тях обаче ги няма (освен ако не се прикриват изкусно както инопланетяните, отмъкнали една опъната свинска кожа и куче, хванато с клюси, след като са разкрити), ergo, реалността се променя от звезда към звезда, както се променя от мечта към мечта. Така звездите само ни изглеждат звезди от земята, но ако можехме да отидем там, щяхме да се уверим, че са нещо съвсем различно.
И дори още по пътя звездолетът заедно с екипажа, без да може да го осъзнае, ще започне да губи своя облик. (Това, че ще се превърнат съответно в кожен пъпеш и хлебарки, изучаващи недобронамерено произведенията на Радичков, е само реалистичен детайл, добавен в качеството на художествена измислица.) Всяка отправна система във вселената си има свой Стандартен модел, а той е това, което определя кое какво е.
На земята и дори в слънчевата система всичко остава едно и също, защото те са като една точка във вселената. Ако обаче разстоянията са големи, така че да не изглеждат като точка спрямо вселената, промените биха били видими.
Така разбираме, че магическият реализъм е дори физическа теория, според която всяко място, ако се подчинява едновременно на законите на квантовата механика и теорията на относителността, генерира една своя собствена реалност, един от многото паралелни светове, които квантовата механика предсказва, които обаче не могат да са разделени с пропаст, ако и теорията на относителността е вярна, а плавно преминават един в друг, за да може първата и втората производна от разстоянието по времето да са винаги дефинирани.
Според квантовата механика звездолетът може скокообразно да се превърне в кожен пъпеш, а според теорията на относителността това ще стане гладко, без никой нищо да може да усети, а разликите ще станат видими само на огромни разстояния.
И тъкмо тази физическа теория можем да открием в Радичковия „Верблюд“: там едно и също нещо, което руснаците според тамошната реалност биха разпознали като камила, се  оказва коридор, надгробно слово, хорската завист и какво ли не още. Радичков със звездолета на своето въображение е успял да обходи множество светове с тяхната собствена и неповторима реалност, нещо което знаем е съвсем невъзможно за реален звездолет  поради парадокса на Ферми.
Можем ли да отречем, че сега, в този момент, онзи верблюд, който се подава иззад лявото ви рамо, докато още четете, не е Радичков?!

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“Fifth Estate” and “Fifth Power”: Removing the Quotation Marks

“The Fifth Estate” is a movie (2013) devoted to “WikiLeaks” and Assange, translated in Bulgarian “Петата власт” (“The Fifth Power”): a metaphor for the social influence of Internet commensurable and overcoming that of the traditional three powers and classical media. This melts the terms “estate” (“Lords spiritual”, “Lords temporal”, and “Commons”) and “power” (“legislative”, “executive”, and “judiciary”). The media and networks are called literally “estate”, but interpreted as “power”.
Furthermore, the term of the “fifth power” addresses the research of its specification to the rest “powers”.
The classical three powers, electable in democracy, coordinate society within the state: “legislative power” creates the written framework, which is detailed in two ways: as instructions and execution by the “executive” one, and as the estimation of relevance or irrelevance of actions to the legislative framework by the “judiciary” one.
Media are specifiable as the “broker” of public opinion and thus, of direct democracy. While the mandates of the three powers’ representatives are a few years, the more and more dynamical society needs information and feedback to the powers daily and more often.
Internet sites and networks unlike media are featured by:
- Right of initiative and organization (public protests, etc.)
- Information and feedback in “real time“
- Transcending national borders to globality
- Execution of rights by discriminate people: young; in dictatorship, in traditional or legal restrictions; female; citizens of any state to events in any stat
- The booming power.

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Why anything rather than nothing? The answer of quantum mechanics

Many researchers determine the question “Why anything rather than nothing?” as the most ancient and fundamental philosophical problem. Furthermore, it is very close to the idea of Creation shared by religion, science, and philosophy, e.g. as 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 fact, can be interpreted also philosophically, and even religiously. However, only the philosophical interpretation is the topic of the text.
The essence of the answer of quantum mechanics is:
1. The creation is necessary in a rigorous mathematical sense. Thus, it does not need any choice, free will, subject, God, etc. to appear. The world exists in virtue of mathematical necessity, e.g. as any mathematical truth such as 2+2=4.
2. The being is less than nothing rather than more than nothing. So, the creation is not an increase of nothing, but the decrease of nothing: it is a deficiency in relation of nothing. Time and its “arrow” are the way of that diminishing or incompleteness to nothing.

Key words: Creation, dark energy, dark matter, entanglement, quantum information, the “Big Bang”, the Standard model

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The square of opposition: Four “colors” enough for the “map” of logic

Intrigue (a few questions):
Why square? Why four? What is the common in the following facts?
1) The square of opposition.
2) The “letters” of DNA.
3) The number of colors enough for any map.
4) The minimal number of points, which allows of them not be always well-ordered.

The number of entities in each of the above cases is four though the nature of each entity seems to be quite different in each one.

Prehistory:

The first three share (1-3) being great problems and thus generating scientific traditions correspondingly in logic, genetics and mathematical topology. However, the fourth one (4) is obvious: triangle has not diagonals, quadrangle is just what allows of its vertices not to be well-ordered in general just for its diagonals. Thus the limit of three as well as its transcendence by four seems to be privileged philosophically, ontologically, and even theologically: It is sufficient to mention Hegel’s triad, Peirce’s or Saussure’s sign, Trinity in Christianity, or Carl Gustav Jung’s discussion about the transition from Three to Four in the archetypes in “the collective unconscious” in our age.

Thesis:

The base of all cited absolutely different problems and scientific traditions is just (4). Thus, the square of opposition can be related to those problems and interpreted both ontologically and differently in terms of the cited scientific areas and in a few others.

Arguments in favor of the thesis:

(1) Logic can be discussed as a formal doctrine about correct conclusion, which is necessarily a well-ordering from premise(s) to conclusion(s). To be meaningful, that, to which logic is applied, should not be initially well-ordered just for being able to be well-ordered as a result of the application of logical tools. 

(2) Consequently the initial “map”, to which logic is to be applied, should be “colored” at least by four different types of propositions, e.g. those kinds in the square of opposition. They are generated by two absolutely independent binary oppositions: “are – are not” and “all – some” thus resulting exactly in the four types of the “square”. 

(3) Five or more types of propositions would be redundant from the discussed viewpoint since they would necessary iff the set of four entities would be always well-orderable, which is not true in general. 

(4) Logic can be discussed as a special kind of encoding namely that by a single “word” thus representing a well-ordered sequence of its elementary symbols, i.e. the letters in its alphabet. The absence of well-ordering needs at least four letters to be relevantly encoded just as many (namely four) as the “letters” in DNA or the minimal number of colors necessary for a geographical map. 

(5) The alphabet of four letters is able to encode any set, which is neither well-ordered nor even well-orderable in general, just to be well-ordered as a result eventually involving the axiom of choice in the form of the well-ordering principle (theorem). It can encode the absence of well-ordering as the gap between two bits, i.e. the independence of two fundamental binary oppositions (such as both “are – are not” and “all – some” in the square of opposition). [Fifth World Congress on the Square of Opposition, November 11-15, 2016, Easter Island, Handbook of abstracts (edited by Jean-Yves Beziau, Arthur Buchsbaum, Manuel Correia), pp. 35-36]

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From the four-colors theorem to a generalizing “four-letters theorem”: A sketch for “human proof” and the philosophical interpretation

Abstract. The “four-color” theorem seems to be generalizable as follows. The four-letters alphabet is sufficient to encode unambiguously any set of well-orderings including a geographical map or the “map” of any logic and thus that of all logics or the DNA (RNA) plan(s) of any (all) alive being(s).
Then the corresponding maximally generalizing conjecture would state: anything in the universe or mind can be encoded unambiguously by four letters.
That admits to be formulated as a “four-letters theorem”, and thus one can search for a properly mathematical proof of the statement.
It would imply the “four colour theorem”, the proof of which many philosophers and mathematicians believe not to be entirely satisfactory for it is not a “human proof”, but intermediated by computers unavoidably since the necessary calculations exceed the human capabilities fundamentally. It is furthermore rather unsatisfactory because it consists in enumerating and proving all cases one by one.
Sometimes, a more general theorem turns out to be much easier for proving including a general “human” method, and the particular and too difficult for proving theorem to be implied as a corollary in certain simple conditions.
The same approach will be followed as to the four colour theorem, i.e. to be deduced more or less trivially from the “four-letters theorem” if the latter is proved. References are only classical and thus very well-known papers: their complete bibliographic description is omitted.

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God, Logic, and Quantum Information

The thesis is:
The concept of quantum information introduced by quantum mechanics allows of an interpretation of the world as conscious, and of logic as the result of the action of that conscious medium.

As usual, quantum mechanics and the theory of quantum information call that interpretation “quantum computer” or “the universe (world) as a quantum computer”. However, one can show that this “quantum computer” possesses the capability of free choice and some kind of natural teleology. The link between the former and latter generates one phenomenon, which can be investigated by science: It can be called the free will of the universe and interpreted as a scientific conception of God or as a hypostasis of Him studied by theology.
The same viewpoint includes all logics or the conception of universal logic in a natural way. Any logic “of anything” can be seen as a partial ordering and thus as a stage of the universal and single well-ordering of the universe going to the past and accomplished by the “universe as a quantum computer”. Thus, any that should be a partial result in the ordering in the course of time from future to past by the meditation of the present and of the choices made in the present.

Arguments:
(I) Course of time can be described in terms of quantum mechanics as follows: The absolutely coherent state of the future de-coheres gradually into less and less entangled quantum systems by means of choices (or “measurements”) made in the present. Thus, those entangled quantum systems are being transformed in mechanical systems absolutely separated to each other in and after the limit from the present to the past.

(II) Universal logic can be considered as the series of partial orderings of some universal class, e.g. that of all sets. Then, any given logic will be exactly one member of that series and can be defined (1) by the set, to which the partial orderings refer, and (2) by the rule, which can generate just the partial ordering, i.e. by the property, which describes the set of all well-orderings representing the partial ordering in question. The definition (1) determines the logic as the “logic of something” where that “something” is the set, which has to be ordered and its “logic” means the way and degree of the ordering. The definition (1) includes both (1.1) any scientific theory as the logic of the object of the theory, and properly (1.2) the “logics of something” where that “something” is some set more interesting by the rule (2), which can generate rather than by itself. The definition (2) includes both (2.1) the case of the explicit property generating (all) well-orderings on any set independently of the interpretation of its elements and (2.2) the “topological representation” of the logic as the description of all well-orderings one by one rather than a common property determining unambiguously all well-orderings as it is in the former case.

(III) The collaboration of quantum mechanics by the conception of quantum information allows of a natural ontological interpretation of universal logic. There is a natural process of ordering in the course of time independent of what is ordered. What is ordered can be e.g. the world, i.e. the universe as a whole, or any part of it, i.e. any quantum system. So, universal logic can be interpreted as the successive partial results in the process of ordering independent of what is ordered. That ordering is a well-ordering and it originates from the course of time. According to quantum mechanics, the general course of time can be described as a (well-) ordering in thus: The coherent state of future is being ordered into the single well-ordering of the past here by the meditation of the all choices in the present accomplishing the ordering. That universal well-ordering in turn orders well all partial results, each of which is some logic. Consequently, the series of all logics turns out to tbe well-ordered if that is the conception of universal logic. The distance between any two logics can be measured by the quantity of (quantum) information. Any logic is unambiguously determined by the information distance from the coherent state (the “absolute future”), on the one hand, and from the single well-ordering (the “absolute past”), on the other hand. (Handbook of the 1st World Congress on Logic and Religion, April 1 - 5, 2015, João Pessoa, Brazil, pp. 171-173)

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"No hidden variables!": From Neumann's to Kochen & Specker's theorem in quantum mechanics

The talk addresses a philosophical comparison and thus interpretation of both theorems having one and the same subject: the absence of the other half of variables, called “hidden” for that, to the analogical set of variables in classical mechanics This implies the existence of quantum correlations, which can exceed any classical correlations (e.g., violating Bell’s inequalities), thus quantum information and is essential for the interpretation of quantum mechanics. The theorem and proof of John von Neumann (1932) are formulated in the context of his fundamental treatise devoted to quantum mechanics (Mathematische Grundlagen der Quantenmechanik, pp. 167–173). He deduced the absence of hidden variables from the availability of non-commuting operators in Hilbert space corresponding to conjugate physical variables in quantum mechanics. The unification of wave mechanics (1926) and matrix mechanics (1925) as well as of the representation by Ψ-functions (1930) implies the introduction of Hilbert space. The theorem of Simon Kochen and Ernst Specker (The problem of Hidden Variables in Quantum Mechanics, 1968) generalizes von Neumann’s result: Once Hilbert space has introduced, this implies immediately the absence of hidden variables even if the quantities are non-conjugate and thus their corresponding selfadjoint operators in Hilbert space commute. The proof of Kochen and Specker is founded on the interpretation of the commeasurable quantities in quantum mechanics  s mathematically commeasurable sets sharing a common measure It introduces implicitly quantum measure unifying quantum leaps and smooth changes thus deducing entanglement and the absence of hidden variables from the core principle of quantum mechanics: wave-particle duality.

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Universal History and the Problem of Time

The establishment of universal history requires to be understood what time is. Time is the transformation of the future into 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 refers to the past in tradition, i.e. to a limited and finite part of time, which is past. Thus history refers immediately both to time and more exactly to the past. •What is past can be even neglected speaking of the history of anything as some genus. Universal history can be understood as that genus.

History as the ontology of the past time turns out to be a set of histories. In fact, all entities such as states, nations, civilizations and all the rest is unified by their common present and future and distinguished by unique and single past. All this does not allow of other universal viewpoint than the logical one as in Hegel.

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Problem of the direct quantum-information transformation of chemical substance

Arthur Clark and Michael Kube–McDowell (“The Triger”, 2000) suggested the sci-fi idea about the direct transformation from a chemical substance to another by the action of a newly physical, “Trigger” field. Karl Brohier, a Nobel Prize winner, who is a dramatic persona in the novel, elaborates a new theory, re-reading and re-writing Pauling’s “The Nature of the Chemical Bond”; according to whom: “Information organizes and differentiates energy. It regularizes and stabilizes matter. Information propagates through matter-energy and mediates the interactions of matter-energy.” Dr Horton, his collaborator in the novel replies: “If the universe consists of energy and information, then the Trigger somehow alters the information envelope of certain substances –“.
“Alters it, scrambles it, overwhelms it, destabilizes it” Brohier adds.
There is a scientific debate whether or how far chemistry is fundamentally reducible to quantum mechanics. Nevertheless, the fact that many essential chemical properties and reactions are at least partly representable in terms of quantum mechanics is doubtless. For the quantum mechanics itself has been reformulated as a theory of a special kind of information, quantum information, chemistry might be in turn interpreted in the same terms.
Wave function, the fundamental concept of quantum mechanics, can be equivalently defined as a series of qubits, eventually infinite. A qubit, being defined as the normed superposition of the two orthogonal subspaces of the complex Hilbert space, can be interpreted as a generalization of the standard bit of information as to infinite sets or series. All “forces” in the Standard model, which are furthermore essential for chemical transformations, are groups [U(1),SU(2),SU(3)] of the transformations of the complex Hilbert space and thus, of series of qubits.
One can suggest that any chemical substances and changes are fundamentally representable as quantum information and its transformations. If entanglement is interpreted as a physical field, though any group above seems to be unattachable to it, it might be identified as the “Triger field”. It might cause a direct transformation of any chemical substance by from a remote distance. Is this possible in principle?

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Reality in a few thermodynamic reference frames: Statistical thermodynamics from Boltzmann via Gibbs to Einstein

The thesis is: The starting point of initial reality is changed as follows:

1. (Carnot) Classical thermodynamics describes laws in terms of quantities of that reality, which is as macroscopic as empirically and experimentally observable.
2. (Boltzmann) The mechanical motions of the huge number of microscopic elements of a statistical ensemble results into the thermodynamic quantities of any macroscopic physical object averagely. The empirically and experimentally observable quantities are deduced as derivative from a hidden theoretical reality of microscopic elements such as atoms and molecules.
3. (Gibbs) The mechanical motions of the huge number of microscopic elements are substituted by different possible states of a macroscopic physical object equivalently and mathematically. The empirically and experimentally observable thermodynamic quantities are deduced as derivative from a hidden theoretical reality of different possible macroscopic states of the physical object as a whole.
4. (Einstein) The mechanically and experimentally observable thermodynamic quantities are some function of the Gibbs ensemble of all possible states (and thus some relation to it). They can be furthermore also referred to the Boltzmann ensemble of microscopic elements. Reality includes both the observable object and the hidden theoretical model as whether a Gibbs or a Boltzmann ensemble as well as the function or relation between the object and that model.

Conclusion: Reality in those reference frames can be identified in the following oppositions: macroscopic – microscopic; elements – states; relational – non-relational; observable – theoretical:

1. (Carnot): Macroscopic, both observable and theoretical.
2. (Boltzmann): Microscopic, elements, non-relational, theoretical.
3. (Gibbs): Macroscopic, states, non-relational, theoretical.
4. (Einstein): Both macroscopic and microscopic, both elements and states, relational, both observable and theoretical.

One can forecast that one synthesis is still forthcoming as to that reality, which can be utilized in a statistical and thermodynamic theory: both relational and non-relational. All other syntheses, which are implicit in the development of the concept of statistic and thermodynamic reality before it, are already completed in the Einstein theory.
One possible hypothesis might be that quantum statistical thermodynamics is what accomplished that last synthesis along that it involves still one dimension of another opposition as to reality: continuous (smooth) – discrete (quantum). All four theories mentioned above mean the thermodynamic and mechanical quantities implicitly only as continuous (smooth) though some of them introduce discrete elements.
Summarizing: The examples of a few statistical thermodynamic theories demonstrate that the concept of “reality” is changed or generalized, or even exemplified (i.e. “de-generalized”) from a theory to another. The change can be described as the explicit introduction of some new opposition as a still one and new dimension of relevant reality, and the generalization as a synthesis to some already involved opposition so that the theory is invariant to the relevant dimension of reality. The exemplification can also be observed being a condition for introducing a few new dimensions of reality. Thus, that exemplification simplifies reality in a dimension (“a step back”) complicating it in a few others (“two steps forward”). 

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Both necessity and arbitrariness of the sign: information

Introduction:
There exists a fundamental problem about the relation of information and the sign as it is defined in Saussure. A creative contradiction to the sign penetrated his main work therefore generating in turn the duality of information in any sign:
The sign meant internally or actually is both necessary and isomorphic to a single bit of information. Indeed, any sign is interpretable either as a signifier or the signified just as in an empty cell of information can be recorded either “0” or “1”. Seen in thus, i.e. inside, the sign is a totality, in which the link between the signifier and signified is necessary.
On the contrary, the sign considered outside, is uncertainly arbitrary. It is the potential for the sign meant actually only as some signified to assign (a-sign) any signifier therefore completing the structure of the sign as actual, described above. Then the sign needs the non-sign outside of it, in which only it might find a corresponding signifier. The choice of a signifier is often restricted to a finite set of elements such as an alphabet or a vocabulary. The quantity of information depends on the number of elements of that set being arbitrary and more than a bit in general.
The problem:
Information in a sign is unambiguous: it is necessarily a single bit inside, but quite uncertain outside (depending on the utilized alphabet or vocabulary or even on all texts written by that alphabet or vocabulary).
Saussure’s implicit creative intuition penetrating his Course:
The concept of sign needs and therefore generates a space between the necessity and unity of the sign and its arbitrariness and uncertainness among the elements of alphabet or vocabulary depending furthermore on all their uses (all words or texts recorded by means of them). The sign being always and moving in that space can be only partial, motivated by the unrealizable aspiration to complete ultimately the infinite process of signifying. Even much more: Saussure’s semiology is an implicit ontology as the being of all is what appears in that infinite process of signifying.
The resolving of the above problem in quantum mechanics and information in relation to semiology:
Quantum mechanics had to resolve the problem of how to describe uniformly both quantum leaps and smooth motion, namely by the Schrödinger equation. It was reformulated thoroughly in terms of quantum information in the end of the 20th century. Though involved differently in quantum mechanics, quantum information can be equated unambiguously to the generalization of information to infinite sets and series. The Schrödinger equation itself can be also exhaustedly interpreted in terms of quantum information.
That latter interpretation links it to Saussure’s tension of the sign generating an implicit ontology as semiology. Then the Schrödinger equation can be seen as a solution of the problem above about the relation of information and Saussure’s sign: both “arbitrary sign” outside and corresponding quantum information are equated to both “necessary sign” inside and corresponding information.

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From the principle of least action to the conservation of quantum information in chemistry..Can one generalize the periodic table?

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 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. That generalization implies a generalization of the periodic table including any continuous and smooth transformation between two chemical elements.

Key words: conservation, Emmy Noether’s theorems of conservation, quantum information, periodic table, quantum chemistry

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Appendix: Quantum information chemistry: the shift of viewpoint (From quantum chemistry to quantum information chemistry) 

Quantum information chemistry investigates how entanglement influences chemical substances, properties, and reactions. Many or all of them can be reduced to quantum physical interactions, first of all, the electromagnetic one among them. As far as entanglement is a quantum phenomenon very well confirmed experimentally, especially as to electromagnetic interaction, it is relevant to chemistry: quantum information chemistry appears.

The relation of entanglement and the Standard model (quantum electrodynamics, first of all) underlies fundamentally the relation of quantum information chemistry and quantum chemistry. As far as the former two are complementary (in Bohr’s extended sense) to each other rather than competitive, the latter two as well.

Entanglement is Einstein’s “spooky action at a distance” implied mathematically by the formalism of quantum mechanics. Thus, the “Holy Grail” of quantum information chemistry is the “chemical action at a distance” implied by entanglement. Though electromagnetic interaction (unlike the other two in the Standard model) is not space limited, it refers to to atoms in chemistry, and their stability needs strong interaction (mostly). Thus, all phenomena in chemistry (until now) are “here and now”, though those “here and now” might be very remote as in astrochemistry.

On the contrary, quantum information chemistry investigates remote chemical phenomena happening at a distance arbitrary in general.

Another exciting horizon promised by quantum information chemistry is the direct chemical transformation cherished by “alchemy”: entanglement might transform any given chemical substance into another in principle.

Quantum information mechanics allows for a new fundamental generalization and technics option: information (together with energy and matter) to be considered as a physical substance, even the most fundamental one among them as well as mutually transformable with them. This reflects on quantum information chemistry as determining relevant chemical substances, to which similar transformations would be verifiable experimentally and technically usable ever.

Quantum gravity as the unification of general relativity & quantum mechanics

A nonstandard viewpoint to quantum gravity is discussed. General relativity and quantum mechanics are to be related as two descriptions of the same, e.g. as Heisenberg’s matrix mechanics and Schrödinger’s wave mechanics merged in the contemporary quantum mechanics. From the viewpoint of general relativity one can search for that generalization of relativity implying the invariance “within – out of” of the same system.

Key words: cyclicality, quantum invariance, conservation of action



The paper as a PDF or @ SocArxiv, @ SSRN, @ PhilPapers

Fleeting thoughts: From Time to Quantum Time, or Conservation of Information

There is a viewpoint, according to which the time itself is the one “side” of any quantum representation of the world, and the other one is the well-known coherent state of any quantum system before measurement. If that is the case, the concept of quantum time is redundant in a sense. It can be anyway introduced in that case also addressing the concept of variable time in general relativity. 

Furthermore, that understanding of quantum time implies the generalization of the conservation of energy-momentum as conservation of action, which is consistent with general relativity, on the one hand, and with the conservation of information as a physical quantity, quantum information.

These statements will be discussed in detail:
1. The concept of quantum invariance

Time is the only physical quantity featuring by its “arrow”. However what is the sense of that arrow after the concept of quantum time should be introduced? The arrow-like time means well-ordering: Indeed the transition of any coherent quantum state being fundamentally unorderable according to the Kochen – Specker theorem (1968) to the always well-ordered statistical ensemble after measurement requires necessarily the well-ordering theorem, which is equivalent to the axiom of choice. Then the concept of quantum invariance can be introduced in thus: Quantum “ontology” before measurement excludes the axiom of choice just because of the cited theorem. Nevertheless, quantum “epistemology” of measurement requires the axiom of choice as above. Thus ‘quantum invariance’ is necessary to reconcile the quantum ontology and epistemology in a single unified quantum reality.

Furthermore, one can demonstrate that Hilbert space as the fundamental mathematical structure of quantum mechanics satisfies that condition of quantum invariance: Any point in it (a wave function, or a state of a quantum system) can be interpreted both as an infinitelydimensional vector and as the characteristic function of the probabilistic distribution of the values of that vector. The former is well-ordered, and the latter is fundamentally unorderable; the structure of Hilbert space does not distinguish between them.

Consequently, time being arrow-like can be interpreted as the well-ordered hypostasis of the quantum world being opposed to the other one of coherent state, and the concept of quantum invariance serves to unify both.
2. Quantum time and relativity time

Once time is involved as the one side of all and any quantum system, quantum time can be already introduced as an ordinary quantum quantity being reversible and supplied with a corresponding self-adjoint operator as any other one featuring a given quantum system. The physical meaning can be that of the period of the de Broglie wave associated with the system and specific to it.

Then a given measured and random value of quantum time will correspond to exactly one value of time according to general relativity corresponding to the curvature of pseudo-Riemannian space in a point of it. Both quantum and relativity time correlate unambiguously with the energy (mass) of the system. The set of probabilities of all given values of quantum time for the quantum system is mapped into the set of all pseudo-Riemannian spaces sharing a measured space-time point of the quantum system.

3. Conservation in general relativity as conservation of action

The well-ordered side of the quantum world, which is embodied in the quantity of time (not that of quantum one) implies conservation in general (Noether, 1918) and conservation of energy in particular for the well-ordering of time imposes the equality of all moments of time as all units after counting are equal. However, once quantum time is introduced, quantum (or relativity) moments of time are not equal and conservation is questioned. It can be anyway restored as the generalized conservation of action corresponding to some dimensionless uniform counting. This means that the product of energy-momentum and space-time volume should be constant for a conservative system in pseudo-Riemannian space.

4. Conservation of information

In turn the concept of quantum invariance suggests some conservation with the following interpretation: Any quantity in a coherent state is the same as that in the corresponding statistical ensemble, or: The well-ordering of a “much” into any “many” cannot change any quantity. The ordering addresses information and thus conservation of information in quantum invariance. Indeed both Shannon’s information and the dimensionless thermodynamic quantity of entropy can be interpreted as the product of a quantity “by itself” and it encoding in a vector in a space with an orthonormal basis: The quantity is invariant to any non-degenerating change of the orthonormal basis. Even more, if information is introduced as mutual entropy, it will conserve under any non-degenerating change of any complete basis (neither orthogonal nor normal in general). Conservation of action as above can be also represented in the same way addressing the quantity of action as the mechanical equivalent of information defined in thus.

5. Conservation of quantum information

The vector space to be defined information as above can be that as over an infinite as a finite field. The classical definition of information is the latter case. Quantum information is a case of the former where the field is that of all complex numbers. Thus any wave function can be associated with a value of quantum information:

Any self-adjoint operator acting on any wave function cannot change the quantity of quantum information though the change of the corresponding physical quantity associated with that operator changes the energy of the system in general. Furthermore, even the operator to be an arbitrary one rather that self-adjoint, which is equivalent to some change of the well-ordering of the vector and thus to quantum time rather than to time, the quantity of quantum information is conserved. This is another representation of the above conservation of action.

Conclusions:

The concept of quantum invariance allows of quantum time to be introduced in a consistent way distinguishing it from the arrow-like time.

Quantum time implies conservation of quantum information in general.

Reality and Ontology by Language: Representation and Metaphor

Reality as if is doubled in relation to language: The one counterpart of reality is within the language as the representation of the other counterpart of reality being outside the language and existing by itself. Both representation and metaphor are called to support the correspondence between the two twins as an “image and simile”. The mechanism of that correspondence and its formal conditions are investigated by the following construction:
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 representation of the reality out of language (modeled as the set B). In the model, the necessity and sufficient condition of that representation between reality both within and out of the language is jus 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 representation but rather as some simile and the vehicle between the two twins of reality can be only metaphor.
Furthermore the metaphor can be anyway defined to a set of one-to-one representations of the only similar external twin into a set of internal “twins”, each of which is a different interpretation of the external “twin” so that a different metaphor is generated in each case. The representation seems to be vague, defocussed, after which the image is bifurcate and necessary described by some metaphors within the language.
Consequently reality is in an indefinite, bifurcate position to language according to the choice formalized in the axiom of choice. If that choice is granted, the language generates an exact image of reality in itself; if not, only some simile can exist expressible within it only by metaphors.
If the axiom of choice does not hold, language and reality converge, e.g. as ‘ontology’: Ontology utilizing metaphors can describe being as an inseparable unity of language and reality within language abandoning representations and the conception of truth as the adequacy of language to reality. Furthermore, those metaphors should coincide with reality (and with physical reality in particular) in virtue of the ontological viewpoint.

The presentation also as a PDF or slides @ EasyChair

The shared information structure of the square of opposition and the concept of infinity

Prehistory and background:
One may demonstrate that the square of opposition shares the least possible structure of the ordering which cannot be well-ordered directly, namely two bits of information, absolutely independent of each other. That minimal possible information structure is a kind of the minimal alphabet of the universe, necessary for anything to be “written” unambiguously. That alphabet consists of four letters corresponding one-to-one to two independent bits and can be found as the four “letters” of human genome as the four “colors” enough to be colored any map as at a few other exemplifications.
The minimal possible alphabet consisting of only two letters, or a single bit of information, used on our contemporary computers, is enough for any well-ordering to be written exactly. However, it is insufficient to be written relevantly anything which is not well-ordered. The record of anything which is not well-ordered by only two letters implies the loss of information necessarily.
A exceptionally important exemplification of that loss of information is quantum measurement reducing a coherent state before measurement unorderable in principle to a statistical and thus well-ordered ensemble after measurement. Any record of something not well-ordered by the alphabet of two letters implies the well-ordering principle equivalent to the axiom of choice. In fact, the sense to that application of the axiom of choice is to add externally by external choices the information lost after recording by that insufficient alphabet of two letters. After assisting by means of the axiom of choice, the information equation before and after recording, respectively before and after measurement, is restored.
The power of the square of opposition proved in logic during millenia can be also utilized to be hinted the concept of God in the framework of logic, particularly the being of infinity.
A reasonable conjecture in the same course of thought is: the concept of infinity admits to be written by the same alphabet of four letters or two independent bits consistently or even that it might be equated to te the same elementary information structure of two independent bits. The talk intends the discussion of that hypothesis:
Thesis:
1. The concept of infinity can be written relevantly by the alphabet of four letters therefore sharing the universal alphabet enough for anything in the universe.
2. Even more, the concept of infinity is equivalent to that fundamental alphabet consisting of four letters or two independent bits.
3. Consequently, on can conclude metaphorically that anything in the universe, or in other words, anything existing is written by the language of infinity, however consisting necessarily of only four letters.
An idea for the proof of the thesis:
The structure of two independent bits is two-dimensional for it implies a gap between the two bits due right to their independence. The gap between any two dimensions both requires and impies the concept of infinity to be described explicitly or “cataphatically”. In other words, the gap between two dimensions such as two independent bits describes the same, however in an implicit, or “apophatic” way.
Thus, the minimal structure of two independent bits or the alphabet of four letters both can describe infinity and coincides with it.
Conclusion:
The power of the square of opposition, proved during milenia, can be explained by the ontological language of infinity for describing anything, by which it supplies logic.

The presentation also as a PDF, a video or as slides @ EasyChair

The Completeness: From Henkin's Proposition to Quantum Computer

The paper addresses Leon Hen.kin's proposition as a " lighthouse", which can elucidate a vast territory of knowledge uniformly: logic, set theory, information theory, and quantum mechanics: Two strategies to infinity are equally relevant for it is as universal and t hus complete as open and thus incomplete. Henkin's, Godel's, Robert Jeroslow's, and Hartley Rogers' proposition are reformulated so that both completeness and incompleteness to be unified and thus reduced as a joint property of infinity and of all infinite sets. However, only Henkin's proposition equivalent to an internal position to infinity is consistent . This can be retraced back to set theory and its axioms, where that of choice is a key. Quantum mechanics is forced to introduce infinity implicitly by Hilbert space, on which is founded its formalism. One can demonstrate that some essential properties of quantum information, entanglement, and quantum computer originate directly from infinity once it is involved in quantum mechanics. Thus, these phenomena can be elucidated as both complete and incomplete, after which choice is the border between them. A special kind of invariance to the axiom of choice shared by quantum mechanics is discussed to be involved that border between the completeness and incompleteness of infinity in a consistent way. The so-called paradox of Albert Einstein, Boris Podolsky, and Nathan Rosen is interpreted entirely in the same terms only of set theory. Quantum computer can demonstrate especially clearly the privilege of the internal position, or " observer'' , or "user" to infinity implied by Henkin's proposition as the only consistent ones as to infinity. An essential area of contemporary knowledge may be synthesized from a single viewpoint.

Keywords. entanglement, EPR, Henkin's propostion, Lob's theorem, quantum information, quantum computer, qubit.

The published paper: (1918) Логико-философские штудии 16 (1-2): 134-135.

The extended paper also as a PDF or a video

The quantum strategy of completeness

The thesis is:
The Gödel incompleteness can be modeled on the alleged incompleteness of quantum mechanics. Then the proved completeness of quantum mechanics can be reversely interpreted as a strategy of completeness as to the foundation of mathematics.
That argument supposes that the Gödel incompleteness originates from the deficiency of the mathematical structure, on which it is grounded. Furthermore, one can point out that generalized structure, on which completeness is provable and thus it can serve as a reliable fundament of mathematics.
Set theory and arithmetic were what was put as the base of mathematics. However, it is a random historical fact appealing to intuition or to intellectual authorities such as Cantor, Frege, Russell, Hilbert, "Nicolas Bourbaki", etc. rather than to a mathematical proof. Even more, the so-called Godel incompleteness theorems demonstrated that set theory and arithmetic are irrelevant as the ground of mathematics rather than no relevant branch of mathematics allowing of self-grounding though the orthodox view.
One can utilize an analogy to the so-called fundamental theorem of algebra:
It needs a more general structure than the real numbers, within which it can be proved.
Analogically, the self-foundation of mathematics needs some more general structure than the positive integers in order to be provable.
The key for a relevant structure is Einstein's failure to show that quantum mechanics is incomplete. The incompleteness of set theory and arithmetic and the alleged incompleteness of quantum mechanics can be linked. The close friendship of the Princeton refugees Godel and Einstein might address that fact. However, Godel came to Princeton in 1940 much after the beginning of Einstein's attempts to reveal that and how quantum mechanics was incomplete. In particular, the famous triple article of Einstein, Podolsky, and Rosen "Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?" pointed out as a kind of theoretical forecast as to the phenomena of entanglement and thus of quantum information was published in 1935. So, there should exist a common mathematical structure underlying both \incompletenesses" and in turn interpretable as each of them.
The mathematical formalism of quantum mechanics is based on the complex Hilbert space featuring by a few important properties relevant to that structure apt to underlie mathematics:
1. It is a generalization of positive integers: Thus it involves innity.
2. It is both discrete and continuous (even smooth): Thus it can unify arithmetic and geometry.
3. It is invariant to the axiom of choice: Thus it can unify as the externality and internality of an innite set as the probabilistic and deterministic consideration of the modeled reality as well as even model and reality in general.
The target of the presentation is:
I. Those three properties of the complex Hilbert space to be demonstrated.
II. A simple mathematical structure underlying both the Godel incompleteness and the alleged incompleteness of quantum mechanics to be described explicitly.
III. The undecidable statements according to the Godel incompleteness theorems to be demonstrated as decidable in that generalized structure of Hilbert space.
IV. The so-called Gödel first incompleteness theorem to be interpreted as allowing of the self-foundation of mathematics.

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Links to the paper @ Blogger or @ Wordpress