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

1. Reality both within and out of
language:
The vehicle of metaphor and
representation

2. 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”

3. 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

4. “ ” means ‘language’, “ ” ‘reality’,
“ ” the ‘exact image of reality in language’
Defining :
𝑨 ∪ 𝑩
(𝟏:𝟏)
A set-theory model:
“ ” means
‘ ’

5. 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

6. The two “twins”: in and out

7. 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)

8. Metaphor and representation

9. 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:

10. Defining :
𝑨 ∪ 𝑩
(𝟏:𝟏)
A set-theory model:
“ ” means
‘ ’

11. 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

12. Language:
T
An infinite set of
“things” or “words”
or “numbers” or
whatever else
elementsAs an actual
whole

13. 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

14. ...
Language and Time
Language as
a series of meanings
well-ordered in time
... ...𝑳 𝒕 𝟏
𝑳 𝒕 𝟐
𝑳 𝒕 𝒏
Time𝑡1 𝑡2 𝑡 𝑛

15. 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

16. Metaphor as converging of “the twins”
Metaphor means:
language is reality

17. 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

18. 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

19. 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)

20. 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

21. Image and Simile
If 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”

22. 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

23. False and nonexisting:
within language but out of reality

24. 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

25. False and :
both need the axiom of choice

26. 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

27. In the beginning was the Word ...
In the beginning was the Choice ...
means that

28. 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)

29. In the beginning was the Word ...
In the beginning was the Word ...
only means that

30. In the beginning was the Word ...
In the beginning was the Word ...
only means that
If the choice is,
the word is
the representation
of reality
If the choice is not,
the word is
the metaphor of reality
Choice?
Yes! No!
Either metaphor
or representation,
the Word is

31. Quantum invariance: the axiom of
choice in quantum mechanics
A few theorems (Neumann 1932; Kochen, Specker
1968) 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

32. 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

33. Metaphor as a set of representations
I1
Language
in a narrow
sense
Ontology
Language as the totality
A thing
Different
representations
of the thing
A metaphor
of the thing
Different
interpretations
of the metaphor
I2
I3
In

34. 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

35. The totality as the defocusing of metaphor
Language
in a narrow
sense
Ontology
Language as the totality
A thingI2
I3
In
I1
The metaphor
as a defocused
representation

36. 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”)

37. Reality and ontology
in depending on choice
Language Reality
Choice?
Yes!
No!
Language as the totality = ontology including reality

38. 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

39. Representations or metaphors
in depending on choice
Language Reality
Choice?
Yes!
No!
Language as the totality
Metaphors
Representations

40. 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

41. 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 entanglement
means that the representation
is not the exact copy of reality:
i.e. the “space” of reality
and that of its representation
are not identical

42. 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

43. Time and entanglement
A coherent
state
A few entangled
states
A few well-ordered series in time

44. 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

45. The entangled Hilbert spaces
The Hilbert space
of a part
The Hilbert space
of another part
The Hilbert space
of the system
11
1
2
2
2
nn
n∞
∞
∞

46. 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”

47. 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

48. 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

49. Hilbert space and infinity
1
2
∞
𝑛
A point
in
Hilbert space
(a wave function)
A transfinite
ordinal
number
∞
𝑛
2
1
...
...

50. 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)

51. 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

52. Metaphor as a modification of meaning
Object
Vehicle
Metaphor
Object
Vehicle
+

53. 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:

54. Measurement and interpretation
Language as ontology
Measurement Interpretation
UnderstandingReality

55. 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

56. 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

57. 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

58. The interaction of meanings of a metaphor
Probability
“Meanings”
“Meanings”
of the object
“Meanings”
of the vehicle
“Meanings” of the metaphor as a whole

59. 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

60. The notion and metaphor as twins
“Meanings”
“Probability”
An ideal
notion
The corresponding
metaphor
A pair of an ideal
notion and
a corresponding
metaphor
as quantum “twins”

61. 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)

62. The notion and metaphor as twins
“Meanings”
“Probability”
The metaphor
centers the term
as a base of
a notion

63. 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

64. From a wave function to a measured value
“Values”
“Probability”

65. 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

66. Back: from a notion to a metaphor
“Meanings”
“Probability”

67. 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

68. A notion
From a rigorously defined notion to the coherence
of a “widely uncertain” metaphor

69. 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

70. A particle
From a “particle” rigorously outlined in space-time
to its wave function in the universe

71. A common mathematical structure and
its 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

72. Hilbert space:
A general
mathematical
structure
underlying
the world
The world
as
language
for linguistic
The world
as
the universe
for physics

73. 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

74. A neo-
Pythagoreanism

75. 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”

76. 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

77. 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

78. 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

79. 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.

80. 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 Grundlagen
der 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 zur
axiomatischen Begründung der Mengenlehre. ‒ In: T.
Skolem,” in Selected works in logic (ed. E. Fenstad), Oslo:
Univforlaget (1970).

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