Reality both within and out of language:

The vehicle of metaphor and representation

Vasil Penchev

• Bulgarian Academy of Sciences: Institute for the Study of Societies of Knowledge

• vasildinev@gmail.com

Lublin, 23 -26 June, 2014“Language, Culture and Mind VI”

Reality as if is doubled in relation to language

• Language and reality are referred to each other

• Their relation can be discussed as a set of mappings between them

• Depending on those mappings, reality and language can be considered either as two identical copies (or “monozygotic twins”) or as two similes (or “fraternal twins”)

• Representation is the former case (“copies”), and metaphor is the latter one (“similes”): So, representation and metaphor are correspondingly “image and simile” between reality and language

“ ” means ‘language’, “ ” ‘reality’, “ ” the ‘exact image of reality in language’

Defining :𝑨 ∪ 𝑩

(𝟏:𝟏)

A set-theory model:

“ ” means‘ ’

Two twins of reality: the one within language

• One can think of the same as two twins of reality: The one is outside of language, i.e. the so-called reality by itself, but the other is within it or even coincides with 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

• Thus the relation between the two twins is represented as two motions directed inversely: into/ outside of language

The two “twins”: in and out

The two “twins”: whether copies or similes?

• Both representation and metaphor are called to support the correspondence between the two twins as the “image and simile”

• Representation can be formalized as one-to-one mapping between the two twins: “the reality by itself and the reality reflected into language”. The motto of that mapping is a little bit “totalitarian”: One thing – one word – one reality – one language!

• Then metaphor can be formalized as any other mapping (i.e. which is not one-to-one and thus it is not a representation)

Metaphor and representation

The correspondence between the “twins”

If the correspondence between reality and language is formalized as mappings, therefore the reality and language themselves should be preliminarily representable as sets, which are perhaps infinite, so that mappings between them can exist

The elements of the ‘reality set’ are any “things” and those of the ‘language set’ are any “units of meaning” such as words, terms, propositions, etc.

The mechanism of that correspondence and its formal conditions can be investigated by the construction, which follows:

Defining :𝑨 ∪ 𝑩

(𝟏:𝟏)

A set-theory model:

“ ” means‘ ’

Language as an infinite set of meanings

• Language is reduced to an infinite countable set (A) of its units of meaning, either words or propositions, or whatever others

• Their exact form does not matter and consequently they can be equivalently representable by any other objects such as “numbers”, “things”, or whatever else, which can be generalized as the elements of a set

• However, the natural condition seems to be for the set to be countable though it can be as finite as infinite

Language:

T

An infinite set of“things” or “words”or “numbers” orwhatever else

elementsAs an actualwhole

Time and meanings

The language should include all possible meanings, which can be ever expressed in it, rather than the existing till now, which would always a finite set

After that is the case, the language can be represented as a well-ordered series of finite sets of meanings, each set of which corresponds to a certain moment in time

Thus the meanings being elements of a mapping between “things” and “words” link time and language

...

Language and Time

Language as a series of meanings well-ordered in time

... ...𝑳𝒕𝟏 𝑳𝒕𝟐 𝑳𝒕𝒏

Time𝑡1 𝑡2 𝑡𝑛

The other “twin”: reality by itself

The set-theory construction is as follows:

The external twin of reality is introduced by another set (B) such that its intersection with the set of language to be empty

That condition of an empty intersection between the language and reality is preliminary: It can be removed further once the metaphor is introduced as an equivalent of entanglement in language

Metaphor as converging of “the twins”

Metaphor means:language is reality

The correspondence between the twins as a mapping

• Both “twins” are formalized as sets, which are infinite in general

• 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. That condition is necessary in general as the “twins” can be infinite

• However if one removes the condition for the axiom of choice to be valid, mappings, which are not yet one-to-one, between language and reality can anyway exist: At least more than one of one-to-one mappings can equivalently substitute them

The role of the axiom of choice in the construction

The axiom of choice can be interpreted in two ways in the case:

As the set of all constructive ways, in which a mapping between the two sets at issue can be built

As all ways that mapping to exist independent of whether it can be constructed in any way somehow or not

The complement and its image

• One designates the image of B into A through f by “B(f)” so that B(f) is a true subset of A

Defining :𝑨 ∪ 𝑩

(𝟏:𝟏)

The necessity of an image of openness into universality

• The set-theory construction only makes visible a more general and philosophical idea:

• Infinity should reconcile two properties seeming contradictory and inconsistent to each other: universality and openness

• Indeed universality means completeness as any entity should be within the universality in a sense

• However openness means incompleteness as some entities should be outside it to be able to be open to them

Image and SimileIf the axiom of choice holds, there is always an internal and equivalent image as “B(f)“ for any external set as “B“

However the relation between “B” and “B(f)” is ambiguous in a sense:

According to “f”, or the universality of infinity, “B” and “B(f)” should be identical, indistinguishable. Then “B(f)” is an exact image of “B”

However according to the openness of infinity, they should be only similar, distinguishable, or “B(f)” is a simile of “B”

Mapping the image of the latter twin within the former

o The mapping (f) produces an image (B (f)) of the latter set (B) within the former set (A)

o The set, which is the complement of B(f) to A, is nonempty in general. Its elements cannot correspond to any element of the set of reality (A). It can include both false statements and non-existing objects as both cannot corresponds to reality

o The construction shows in terms of set theory how the language can generate fictions, literature

False and nonexisting:within language but out of reality

Reality and its model into language

• However, one needs the axiom of choice in general to be able to distinguish a fiction from that reality reflected in language:

• 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)

• Its complement to A “∁𝐴(𝐵(𝑓)” serves as the image of language fictions in the utilized terms of set theory

False and :both need the axiom of choice

The condition of choice

→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:

→The axiom of choice means that the choice is guaranteed always. In fact that omnipotence choice constitutes both disjunctive areas: that of reality by itself and that of reality reflected in language

→Thus the choice is what supplies the exact correspondence between the two “twins” therefore distinguishing disjunctively truth from false in language

In the beginning was the Word ...

In the beginning was the Choice ...

means that

The axiom of choice: to be or not?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

Then the vehicle between the two “fraternal twins” of reality can be only metaphor

Even more, the case for the axiom of choice to be valid and the opposite case for it not be valid should coincide for and as far as reality is a single one:

Indeed if the condition of choice does not hold language and reality converge (however as well as truth and false)

In the beginning was the Word ...

In the beginning was the Word ...

only means that

In the beginning was the Word ...

In the beginning was the Word ...

only means that

If the choice is, the word is the representationof reality

If the choice is not,the word isthe metaphor of reality

Choice?Yes! No!

Either metaphoror representation,the Word is

Quantum invariance: the axiom of choice in quantum mechanics

A few theorems (Neumann 1932; Kochen, Specker1968) deduce from the mathematical formalism of Hilbert space that no hidden variable and thus no well-ordering is allowed for any coherent state in quantum mechanics. However, the latter is well-ordered after measurement and thus needs the well-ordering theorem equivalent to the axiom of choice

The epistemological equivalence of a quantum system before and after measurement forces the invariance to the axiom of choice. That invariance is shared by the Hilbert space formalism. This fact can be called quantum invariance

Metaphor as a set of realities

Furthermore, the metaphor can be anyway defined by 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 representation is generated in each case

In other words metaphor can be interpreted as a set of representations as well as a single but fussy representation, a defocused image of reality

Metaphor as a set of representations

I1

Languagein a narrowsense

Ontology

Language as the totality

A thing

Differentrepresentationsof the thing

A metaphorof the thing

Differentinterpretationsof the metaphor

I2

I3

In

Metaphor as a defocused representation

The representation seems to be vague, defocussed, after which the image is bifurcate and necessary describable by more than one metaphor within the language

Even more, each representation in that set of representations representing in turn a separate metaphor defocuses and thus it is substituted by secondary metaphors, after that by tertiary ones, etc.

The process of defocusing continues and extends compromising all reality as a coherent and inseparable whole

The totality as the defocusing of metaphor

Languagein a narrowsense

Ontology

Language as the totality

A thingI2

I3

In

I1

The metaphoras a defocusedrepresentation

The relation between reality and language depending on choice

Consequently reality is in an indefinite, bifurcate position to language according to the choice formalized in the axiom of choice

Indeed:

If it holds reality is isolated as an independent area even primary to language, even more, generating it as its “image and simile”. Thus the concept of reality in turn grounds choice

If it does not hold language and reality converge, e.g. as ‘ontology’ (the etymology of the term is “logos (i.e. the Word) of the existing”)

Reality and ontologyin depending on choice

Language Reality

Choice?Yes!

No!

Language as the totality = ontology including reality

The relation between reality and language depending on choice

If choice (and thus reality, too) 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

However then reality cannot be isolated from language. There is a common universe containing indistinguishable elements, which can be called elements of being: both reality and language

If that is the case, the metaphors rather than scientific concepts (representations) are what is the relevant tool for the being to be studied

Representations or metaphorsin depending on choice

Language Reality

Choice?Yes!

No!

Language as the totality

Metaphors

Representations

Metaphor as entanglement

That the metaphor can be represented as entanglement is neither a metaphor nor false

Indeed if reality and language can converge into being, that convergence can be represented both in language (as metaphor) and in physical reality (as some phenomena, which one can recognize as those of entanglement in quantum mechanics)

The mathematical formalism of entanglement in quantum mechanics can be described in terms of the set-theory model of language

Metaphor as entanglement

Reality as a Hilbert space (i.e. as a quantum system)

Its representation as the dual Hilbert space (i.e. identical to reality)

Metaphor as entanglementmeans that the representationis not the exact copy of reality:

i.e. the “space” of realityand that of its representation

are not identical

The concept of entanglement

The concept of entanglement is coined by the theory of quantum information to designate that special correlation of two or more quantum entities:

An entity wherever in physical space can restrict the degrees of freedom of others just as a unit of meaning can fussily delimit others being used as a metaphor for them

The mathematical structure underlain both cases is one and the same: It is interpreted in terms of language as metaphors, and in those of reality (quantum mechanics) as entanglement

Time and entanglement

A coherentstate

A few entangledstates

A few well-ordered series in time

Entanglement as a relation of Hilbert spaces

Entanglement can be defined as an exactly determined mathematical structure grounded on Hilbert spaces and underlying all phenomena of entanglement studied by quantum mechanics:

Entanglement is the case when the Hilbert space of a common quantum systems cannot be factorized into a tensor product of the Hilbert spaces of its parts:

This means that the parts share some nonzero subspace of the common Hilbert space. They cannot be absolutely isolated as stand alone just because of that subspace

The entangled Hilbert spaces

The Hilbert spaceof a part

The Hilbert spaceof another part

The Hilbert spaceof the system

111

2

2

2

nn

n∞∞

∞

About quantum information

The conception of quantum information was introduced in the theory of quantum information studying the phenomena of entanglement in quantum mechanics:

The entanglement was theoretically forecast in the famous papers of Einstein, Podolsky, and Rosen (1935) and independently by Schrödinger (1935) deducing it from Hilbert space, the basic mathematical formalism of quantum mechanics

However, the former three demonstrated the forecast phenomenon as the proof of the alleged “incompleteness of quantum mechanics”

More about quantum information

John Bell (1964) deduced a sufficient condition as an experimentally verifiable criterion in order to distinguish classical from quantum correlation (entanglement)

Aspect, Grangier, and Roger (1981, 1982) confirmed experimentally the existence of quantum correlations exceeding the upper limit of all possible classical correlations

The theory of quantum information has thrived since the end of the last century in the areas of quantum computer, quantum communication, quantum cryptography, etcetera

Formalizing metaphor

That same structure can be utilized for a mathematical model of metaphor as a special kind of correlation between the meanings and senses of two or more units of meanings (e.g. such as words)

Its base is the set-theory model of language

The complex Hilbert space identical with that utilized in quantum mechanics is necessary to be involved for and as far as infinity is involved in the model of language

The complex Hilbert space allows of distinguishing between any two infinities in a rigorous and quantitative way

Hilbert space and infinity

1

2

∞

𝑛

A pointin Hilbert space(a wave function)

A transfiniteordinalnumber

∞

𝑛

2

1

...

...

The philosophical fundament

The philosophical core of the model can be described so:

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 (as above)

Metaphor as a relation between terms

Metaphor restricts the meaning of a term by the meaning of another term in a probabilistic, loose way calling for interpretation

A nonempty intersection of the meanings appears as the relation of the terms in question

That intersection is the essence of metaphor: It is that medium, by which the terms can interact therefore weakening some subdomains of meaning at the expense of emphasizing others

Thus metaphors serves to modify the meaning of the term, which is the target, by that, which is a vehicle of the metaphor

Metaphor as a modification of meaning

Object

Vehicle

Metaphor

Object

Vehicle+

The mathematical correspondence between metaphor and entanglement

The introduction of that underlying mathematical structure allows of establishing unambiguous correspondence between metaphor and entanglement in an absolutely exact, mathematical way, after which measurement in quantum mechanics corresponds to interpretation in language:

Measurement and interpretation

Language as ontology

Measurement Interpretation

UnderstandingReality

The probability wave of the meanings of a metaphor

Some interpretations of a given metaphor seems to be more probable, but no one can be absolutely excluded

The “field” of meanings undergoes certain deformation for the vehicle field of meanings

One can figure even any given term as a vehicle of metaphor concerning the being at all, God

Furthermore, any metaphor linking more than one usual term can be considered as an interaction between them as different metaphors of being or as different “names” of God

The result field of meanings of a metaphor

Probability

“Meanings”“Meanings”of the object

“Meanings”of the vehicle

“Meanings” of the metaphor as a whole

Term and metaphor

The term utilized as a vehicle of a metaphor restricts the area of meaning of its object to a small true subset of it

Thus any metaphor can be considered as a loose definition of that subset, which one means using the metaphor

Furthermore that subset can be also interpreted as a center determined namely by the metaphor, which serves to order the term about the center chosen in that way

The interaction of meanings of a metaphor

Probability

“Meanings”

“Meanings”of the object

“Meanings”of the vehicle

“Meanings” of the metaphor as a whole

From a metaphor via a term to a notion

That set can ground the essential features, properties or relations of the object of the metaphor pioneering the scientific or even formal definition of the term serving as the object of the metaphor at issue

Even more, the metaphor can be accepted as the exact continuous counterpart of the notion, which is discretely isolated and thus representing some external part of reality, some “thing”

This means, that any thought can be equivalently expressed both a metaphor and a concept, i.e. both poetically and conceptually

The notion and metaphor as twins

“Meanings”

“Probability”An ideal notion

The correspondingmetaphor

A pair of an idealnotion and a correspondingmetaphoras quantum “twins”

Any scientific concept is descended from metaphors

Thus some metaphor founds any scientific notion therefore “erasing” the grounding metaphor and the rest interpretations except one of them

Indeed the stages of that metamorphosis from a metaphor to a notion can be described so:

A is B (a metaphor)

A is like B (a simile) in relation to B1, B2, B3, ...

A is B1, B2, B3, ... (a definition of a notion)

The notion and metaphor as twins

“Meanings”

“Probability”

The metaphorcenters the termas a base of a notion

De-coherence and conceptualization

The corresponding phenomena in quantum information is the process of de-coherence:

The interacted object is cut off from its environment just as a rigorously defined notion is cut off from its context to designate one and the same in any context

The cut-off means that the Hilbert spaces of the environment (either a physical one or a context) and the entity (either a physical one or a piece of text) are orthogonal to each other

From a wave function to a measured value

“Values”

“Probability”

Coherence and generating a metaphor

The opposed process can be observed both by the theory of metaphor and that of quantum information:

The environment and entity whatever are begin interacting therefore their corresponding Hilbert spaces are not yet orthogonal to each other:

Thus each of them limits the degree of freedom of the other

Both merge into a single one in the other pole

So a continuum of gradually converging appears between the two poles

Back: from a notion to a metaphor

“Meanings”

“Probability”

From a notion to a metaphor

A notion begins to lose its clear outlines coined in everyday speech and media accumulating new and new interpretations and uses

So, a notion can pass all reverse pathway gradually losing the sharp distinguishability

The context of its use begins influencing more and more its meaning. The notion goes less and less informational as the most words having more than one meaning in a usual language:

In fact, almost all words in a natural language can be considered as metaphors

A notion

From a rigorously defined notion to the coherenceof a “widely uncertain” metaphor

From a particle to a wave function

A quantum entity analogically starts to lose the measured values of the quantities as if dissolving in the common and inseparable whole of the universe

Just as the metaphor is equivalent to the notion above, the particle is equivalent to a wave function:

That entity in question as a particle is isolated from its environment (and from the universe) just as a notion should be isolated from any possible context and thus remaining invariable. It as a wave function constitutes a whole with the universe and thus serves as a “metaphor” of the universe

A particle

From a “particle” rigorously outlined in space-timeto its wave function in the universe

A common mathematical structure andits interpretations in linguistics and physics

The suggested mathematical structure underlies and describes equally well both processes representing them as its interpretations:

There are two Hilbert spaces corresponding to an entity and its environment whatever be i.e. to a part and its complement to a whole

The module of the vector-angle between them increases according to the degree of entanglement between the part and whole whatever be

Hilbert space:A general

mathematicalstructure

underlyingthe world

The worldas

language for linguistic

The worldas

the universe for physics

The world & language underlain by mathematics: A common universum of

“things”, “words”, and “numbers”

The outlined approach allows a common philosophical viewpoint to the physical world, language and some mathematical structures:

Then the universe should be understood as a joint physical, linguistic and mathematical universum

The physical motion and metaphor are one and the same rather than only similar in a sense

A neo-Pythagoreanism

Conclusions:

Language allows a purely formal description involving and evolving only a few initial ideas concerning:

Semiotics as a set of mappings between “words” and “things” whatever they are

Ontology as a doctrine about infinity, which is represented as infinite sets and their mappings

Then, representation and metaphor can be considered as “twins” inside and outside of the language formalized as an infinite set of meanings (a mapping)

Language as the “twin” outside is called “reality”

Conclusions:

Language, reality and representation generate the conception of truth as adequacy:

That part of language, which represents reality is ‘true’. Its complement to language is ‘false’

Metaphor forces language and reality to converge to each other into ontology:

Thus the conception of truth as adequacy is inapplicable to ontology

Conclusions:

The infinity sets allow of models in terms of Hilbert space and thus easily interpretable as the wave functions of quantum systems (All physical objects are quantum systems)

Consequently, language by the mediation of set-theory models can be seen in terms physical reality

Then metaphor and representation being linguistic phenomena are not less interpretable as physical ones, more especially those of entanglement

Conclusions:

• Furthermore, entanglement allows of establishing a direct link between “things” (as quantum systems), “words” (as units of meanings in the set-theory model of language) and “numbers” (more exactly, fundamental mathematical structures as Hilbert spaces)

• That approach addresses the ontological unity of the world as a common linguistic, physical, and mathematical whole

References I:

Aspect, Alain & Grangier, Philippe & Roger, Gérard (1981) “Experimental Tests of Realistic Local Theories via Bell’s Theorem,” Physical Review Letters 47(7): 460-463.

Aspect, Alain & Grangier, Philippe & Roger, Gérard (1982) “Experimental Realization of Einstein-Podolsky-Rosen-Bohm Gedanken Experiment: A New Violation of Bell’s Inequalities,” Physical Review Letters 49(2): 91-94.

Bell, John (1964) “On the Einstein ‒ Podolsky ‒ Rosen paradox,” Physics (New York) 1(3): 195-200.

Einstein, Albert & Podolsky, Boris & Rosen, Nathan (1935) “Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?” Physical Review, 47(10): 777-780.

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

Neumann, Johan. von (1932) Mathematische Grundlagender Quantenmechanik. Berlin: Springer, pp. 167-173 (Chapter IV.2).

Schrödinger, E (1935) “Die gegenwärtige situation in der Quantenmechanik”, Die Naturwissenschaften 23(48), 807-812; 23(49), 823-828, 23(50), 844-849.

Skolem, Thoralf (1922) “Einige Bemerkungen zuraxiomatischen Begründung der Mengenlehre. ‒ In: T. Skolem,” in Selected works in logic (ed. E. Fenstad), Oslo: Univforlaget (1970).

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