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 (modelled 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 just the axiom of choice:
If the axiom of choice does not hold, the relation between the sets B (f) and B cannot be defined rigorously as an exact representation but rather as some simile and the vehicle between the two twins 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.
Furthermore, language can be formally defined by representation after the latter is in turn defined as a one-to-one mapping between two infinite sets, one of which is defined 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 modelled 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.
A formal approach to reality by means of completeness is explored. The problem of completeness in mathematics and even in an experimental science such as quantum mechanics has been well investigated. Furthermore, reality seems to be definable just by being complete and opposed to any representation of it for representation remains incomplete always as a principle.
Thus the pair of completeness and incompleteness in turn seems to be able to underlie the formal idea of language.
Another and practical viewpoint links reality, representation, and language to calculation by the concept of Turing (1936) machine being the mathematical model of our computers. Any real calculation is finite unlike that reality meant or modeled by it. Indeed after the Gödel incompleteness theorems (1931) or the insolvability of the so-called halting problem (Turing 1936; Church 1936) as to a classical machine of Turing, one of the simplest hypotheses is completeness to be suggested for two ones. That is consistent with the provability of completeness by means of two independent Peano arithmetics.
Many modifications of Turing machines cum quantum ones are researched for the Halting problem and completeness. The model of two independent Turing machines seems to generalize them in that relation.
Then, that pair of two independent Turing machines can be postulated as the formal definition of reality therefore being complete unlike any of them standalone, remaining incomplete without its complementary counterpart. Representation is formal defined as a one-to-one mapping between the two Turing machines, and the set of all those mappings can be considered as “language” therefore including metaphors as mappings different than representation. One can investigate that formal relation of “reality”, “representation”, and “language” modeled by (at least two) Turing machines.
The independence of (two) Turing machines can be further interpreted by means of game theory and especially of the Nash equilibrium.
The idea of Turing machine can be seen as a fundamental philosophical idea. The formal concepts of reality, language and representation need to transcend somehow any finiteness and calculation: The simplest way for this doubles each of them to an independent pair therefore involving choice and information as the quantity of choices. That approach seems to be equivalent to that based on set theory and the concept of actual infinity in mathematics, on the one hand, and allowing of practical implementations.
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