Representation and Reality by Language

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 innite 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 nite 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 = A8B) 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 sucient condition of that representation 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 dened rigorously as an exact representation but rather as some simile and the vehicle between the two twins can be only metaphor. Furthermore the metaphor can be anyway dened 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 dierent interpretation of the external \twin" so that a dierent metaphor is generated in each case. The representation seems to be vague, defocused, after which the image is bifurcate and necessary described by some metaphors within the language.
Consequently reality is in an indenite, 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. Furthermore, language can be formally dened by representation after the latter is in turn dened as a one-to-one mapping between two innite sets, one of which is dened as reality and the other, as its image. Language is namely the natural interpretation of that image.
The advantage of that approach is to link the representation of the human being supplied by language to the representation by a machine (e.g. a computer), which should
be formally modeled to be constructed. Another point of interest is the following: That mathematics, which is underlain by the mapping between sets, can be related to language by link of representation.

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Choice, Infinity, and Negation: Both Set-Theory and Quantum-Information Viewpoints to Negation

The concepts of choice, negation, and infinity are considered jointly. The link is the quantity of information interpreted as the quantity of choices measured in units of elementary choice: a bit is an elementary choice between two equally probable alternatives. “Negation” supposes a choice between it and confirmation. Thus quantity of information can be also interpreted as quantity of negations. The disjunctive choice between confirmation and negation as to infinity can be chosen or not in turn: This corresponds to set-theory or intuitionist approach to the foundation of mathematics and to Peano or Heyting arithmetic. Quantum mechanics can be reformulated in terms of information introducing the concept and quantity of quantum information. A qubit can be equivalently interpreted as that generalization of “bit” where the choice is among an infinite set or series of alternatives. The complex Hilbert space can be represented as both series of qubits and value of quantum information. The complex Hilbert space is that generalization of Peano arithmetic where any natural number is substituted by a qubit. “Negation”, “choice”, and “infinity” can be inherently linked to each other both in the foundation of mathematics and quantum mechanics by the meditation of “information” and “quantum information”

Key words: choice, entanglement, infinity, information, negation, quantum correlations, quantum information


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Quantum Occasionalism

Both transition and transformation link the ideal and material into a whole. Future is what “causes” the present, and the latter in turn is what “causes” the past. That kind of “reverse causality” needs free choice and free will in the present in order to be able to be realized unlike classical causality. A few properties feature the concept of “quantum occasionalism” as follows. Some hypothetical entity generates successively a series of well-ordered states. That hypothetical entity is called “coherent state” in quantum mechanics and defined as a superposition of all possible states of the quantum system. The already generated well-ordered series can be interpreted as a causal sequence. Thus the generating cause remains hidden behind the visible well-ordering of the series and hides itself behind the perfect visible order created by it. That visible order only seems to cause itself by itself.

Key words: causality, choice, freewill, occasionalism, quantum occasionalism, reverse causality

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A new reading and comparative interpretation of Gödel’s completeness (1930) and incompleteness (1931) theorems

Abstract. Peano arithmetic cannot serve as the ground of mathematics for it is inconsistent to infinity, and infinity is necessary for its foundation. Though Peano arithmetic cannot be complemented by any axiom of infinity, there exists at least one (logical) axiomatics consistent to infinity. That is nothing else than a new reading at issue and comparative interpretation of Gödel’s papers (1930; 1931) meant here. Peano arithmetic admits anyway generalizations consistent to infinity and thus to some addable axiom(s) of infinity. The most utilized example of those generalizations is the complex Hilbert space. Any generalization of Peano arithmetic consistent to infinity, e.g. the complex Hilbert space, can serve as a foundation for mathematics to found itself and by itself.

Key words: axiom of choice, foundations of mathematics, Gödel's completeness and incompleteness theorems, Hilbert space, Peano arithmetic, infinity, quantum information, qubit, set theory, Skolem's paradox

Prehistory and background: There are some statements referring the exceptionally famous papers of Gödel [1-2] and their interpretation, which turn out to be rather misleading. Their essence consists in linking (the logical) completeness [1] to finiteness, and correspondingly (the arithmetical or equivalent) incompleteness (or alternatively, inconsistency, [2]) to infinity. Their historical background and problematics have been reconstructed nowadays by means of the crisis in the foundation of mathematics, being due to actual infinity in the “naïve” set theory, its axiomatiozations, reducing the problem of their completeness and consistence to that of their model in Peano arithmetic, Hilbert’s program for the arithmetical foundation of mathematic, and Russell’s construction in Principia. Skolem ([3])’s conception (called also paradox) about the “relativity of the concept of set”, once the axiom of choice is utilized, should be specially added to that background to be grounded the present viewpoint. 

A sketch of the present viewpoint: 

(S1) One can trivially demonstrate that Peano arithmetic excludes infinity from its scope fundamentally: Indeed, 1 is finite; adding 1 to any natural number, one obtains a finite natural number again; consequently, all natural numbers are finite according to the axiom of induction. 

(S2) Utilizing the axiom of choice equivalent to the well-ordering principle (theorem), any set can be one-to-one mapped in some subset of the natural numbers. As Skolem emphasized expressively, this means that any set even being infinite (in the sense of set theory) admits an (“nonintrinsic” or “unproper”) one-to-one model by some subset of the natural numbers, which should be finite, rather than only by a countably infinite model. 

(S3) In fact, the so-called countable power of a set is introduced in the (Cantor, or “naïve”) set theory as the power equivalent to that of all natural numbers and different (and bigger) than that of any finite number. However, the number of all natural numbers should be a natural number and thus finite in Peano arithmetic as a corollary from (1) above. 

(S4) Consequently, if one compares Peano arithmetic and set theory (e.g. in ZFC axiomatization), a discrepancy about (countable) infinity is notable: 

(S4.1) Peano arithmetic is incomplete to set theory for that arithmetic does not contain any infinity (including the countable one). 

(S4.2) Furthermore, Peano arithmetic cannot be complemented by any “axiom of infinity” because it contains only finite numbers according (1) above. In other words, if it is complemented to become “complete” in the sense of S4.1, it would become inconsistent furthermore. Those statements (S4.1 ̶ S4.2) reconstruct Gödel’s incompleteness (1931) argument in essence, but in a trivial way. 

(S5) If one considers a logical axiomatization in the sense of Principia (as in Gödel, 1930) without any mapping and even correspondence to Peano arithmetic, some axiom of infinity is implicitly allowed, and thus completeness provable, but only nonconstrucively for whether explicit or implicit reference to infinity does not admit any constructiveness in a constructive way1 in principle. 1 One can mean some constructiveness in a nonconstructive way, i.e. as “pure existing” constructiveness by virtue of the axiom of choice. 

Thesis: 

(T1) Peano arithmetic cannot serve as the ground of mathematics for it is inconsistent to infinity, and infinity is necessary for its foundation. Though Peano arithmetic cannot be complemented by any axiom of infinity, there exists at least one (logical) axiomatics consistent to infinity. That is nothing else than a new reading at issue and comparative interpretation of Gödel’s papers meant here. 

(T2) Peano arithmetic admits anyway generalizations consistent to infinity and thus to some addable axiom(s) of infinity. The most utilized example of those generalizations is the complex Hilbert space. (T3) Any generalization of Peano arithmetic consistent to infinity, e.g. the complex Hilbert space, can serve as a foundation for mathematics to found itself and by itself. 

A few main arguments: 

(A1) Skolem’s relativeness of ‘set’ 

(A2) The viewpoint to Gödel’s papers sketched above (in S1 ̶ S5). 

(A3) The complex Hilbert space can be considered as a generalization of Peano arithmetic as follows. Hilbert space is an infinite series of qubits. A qubit is defined as usual and thus isomorphic to a unit ball in which two points are chosen: the one from the ball, the other from its surface. Any point in that space would representable as some choice (record) of values in each qubit. If the radiuses of all those unit balls are degenerate to 0, the complex Hilbert space is reduced to Peano arithmetic. On the contrary, if two choices, each one among a limited uncountable set and thus representable as a pair of complex numbers, α, β: |𝛼𝛼|2 + |𝛽𝛽|2=1, are juxtaposed to any natural number, one obtains the complex Hilbert space as a series of qubits and as a generalization of Peano arithmetic. The essential property of the complex Hilbert space as that model is that the set of all natural numbers is mapped one-to-one to a series of infinite sets. 

(A4) The theorems of the absence of hidden variables in quantum mechanics [4-5] can be interpreted as a completeness proof of the above model based on the complex Hilbert space. Indeed, the complex Hilbert space is sufficient for the proof of those theorems, and the absence of hidden variables corresponds unambiguously to completeness. Any hidden variable would mean the incompleteness of the complex Hilbert space in the above sense.

References: 

1. Gödel, K. (1930) "Die Vollständigkeit der Axiome des logischen Funktionenkalküls," Monatshefte der Mathematik und Physik 37 (1): pp. 349-360. 

2. Gödel, K. (1931) "Über formal unentscheidbare Sätze der Principia mathematica und verwandter Systeme I," Monatshefte der Mathematik und Physik 38 (1): 173-198. 

3. Skolem, T. (1922) "Einige Bemerkungen zur axiomatischen Begründung der Mengenlehre," in Selected works in logic of Thoralf Skolem (ed. E. Fenstad), Oslo, Univforlaget, pp. 137-152. 

4. Neumann, J. (1932) Mathematische Grundlagen der Quantenmechanik, Berlin, Springer, pp. 167-173. 

5. Kochen, S., Specker, E. (1968) "The problem of hidden variables in quantum mechanics," Journal of Mathematics and Mechanics 17 (1):. 59-87.

The published paper (2016): Логико-философские штудии 13 (2): 187-188.

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Information and Communication: Mathematical Models

Shannon’s mathematical theory of communication, Wiener’s cybernetics, and even Habermas’s theory of communicative action share the European approach of rationality to communication Its essence consists in dividing the rational from the irrational, and then “bracketing” the latter as a “black box". There exist the practice of separating rationality from irrationality disjunctively, representing irrationality as some “black box”, e.g. that of choice, postulating that “black box” as a an initial element of rationalityox” and initial element for rationality. It tends to all different from the rational and ordered to be generalized and ignored as the “irrational”. hen, the irrational sides of communication can be studied only as obstacles and “informational noise” or as far they admit some rationalization by means of any more or less relevant model preferably mathematical.

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Skepticism about the Skepticism

The question is:
How should skepticism refer to itself?

The classical example might be the doubt of Descartes, which led him to the necessary obviousness of who doubts. The formal logical structure is the same as the “antinomy of the Liar”. That new interpretation of it can be called “antinomy of the Skeptic”. Descartes resolved it by introducing the meta-position and furthermore defining the “Self” just as that meta-position allows of any other position including that of doubt.

Rather unexpectedly, one can reveal the sceptic position to itself in Heisenberg’s uncertainty in quantum mechanics. It generates the necessity of the observer in final analysis as what allows of that fundamental principle of uncertainty. As the Self of Descartes, the observer in quantum mechanics is necessary logically for one to be able to state certainly the uncertainly of all, which is not the observer.

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Appendix: Reflective Skepticism

The question is: How skepticism should refer to itself? The classical example might be the doubt of Descartes, which led him to the necessary obviousness of who doubts. The formal logical structure is the same as the “antinomy of the Liar”. That new interpretation of it can be called “antinomy of the Skeptic”.
Descartes resolved it by introducing the meta-position and furthermore defining the “Self” just as that
meta-position allows of any other position including that of doubt. Rather unexpectedly, one can reveal the sceptic position to itself in Heisenberg’s uncertainty in quantum mechanics. It generates the necessity of the observer in final analysis as what allows of that fundamental principle of uncertainty. As the Self of Descartes, the observer in quantum mechanics is necessary logically for one to be able to state certainly the uncertainly of all, which is not the observer.
Not less unexpectedly, Gödel’s conception of incompleteness of any infinite logical structure can be interpreted as another logical and mathematical form of skepticism about the skepticism. He himself revealed the structure of the “Liar” (consequently also that of the “Skeptic”) in that statement stating that it is untrue, which underlies his theorems of incompleteness. However following the solution of Descartes, that statement should be in a meta-position to the theory.. The infiniteness, which is necessary for the construction, is what forces incompleteness mixing the meta-position and position. Consequently, the fundamental result of Gödel can be interpreted as alternative or complementary to
the solution of Descartes: If the position and meta-position in skepticism are not absolutely divided from each other, skepticism about the skepticism requires infinity and involves incompleteness
One can abstract those two solutions, which can be ascribed correspondingly to Descartes and to Gödel, as to skepticism about the skepticism at all. Might some other, i.e. third solution exist? In other words, can one reproduce the sceptic position both to skepticism about the skepticism and to the possible solutions of the antinomy, which appears? This seems to be possible utilizing the following construction:
These two solutions can be considered as two poles. The one consist in the absolute separation of the
position from the meta-position, the other in the absolute convergence of them. They generate a continuum between them consisting in turn in the gradual transition from the absoluteness of separation to the other, opposite absoluteness of convergence. The skepticism is what generates that continuum for it doubts the absoluteness both of separation and of convergence. Furthermore, that continuum allows of an interpretation in terms of quantum mechanics and quantum information as “entanglement”.
The examples show that the skeptic position at all as well as the skepticism about the skepticism is rather productive in modern philosophy. It can be revealed in a series of problems and their solutions in physics, mathematics, and logic, some of which are enumerated above. Skepticism about the skepticism can be investigated furthermore in other branches of science as well as in contemporary art, religion politics, etc.

The cinematographic method of thought in Bergson

Henri Bergson (1907) utilized a metaphor borrowed from cinematograph to represent the usual way of human thought about motion and evolution in comparison with his original approach to them grounded on his concept of time as durée (duration). The analogy consists in restoring the motion from a series of immovable pictures (frames) only as a subjective illusion. On the contrary, durée is that understanding of time, in which motion and evolution are primarily given rather than secondarily and as an auxiliary or even illusion by a series of static states. In the latter case, static underlies kinematic reducing it only to many static states therefore canceling the creative essence of motion and evolution according to Bergson.

Bergson’s views influenced essentially Luis de Broglie (1925), who offered in his thesis (1924) the wave interpretation of any particle and its motion in quantum mechanics. His work received the Nobel Prize in Physics (1929) “for his discovery of the wave nature of electrons”. The wave-particle duality continues to be one of the most fundamental principles in quantum mechanics nowadays.  

The cinematograph embodied and thus made visible and obvious the fundamental way of human thought of motion therefore allows of reflecting it and generalizing it in Bergson’s philosophy and quantum mechanics.  (https://zkm.de/de/vasil-penchev)    

References:

Bergson, Henri (1907) L'Évolution créatrice, Paris: PUF, 298-307.

Broglie, Louis de (1925) “Recherches sur la théorie des quanta,” Thesis (Paris), 1924, Annales de Physique (Paris, 10-ème série) 3: 22-128.

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