MENTAL, A METAPHORIC LANGUAGE

"Metaphorical language is what brings us closest to the unattainable and essential reality" (Nietzsche).

"Our ordinary conceptual system, in terms of which we think and act, is fundamentally metaphorical in nature." (George Lakoff & Mark Johnson)

"If I were a bold thinker [...], I would say that there are only a dozen metaphors, and that all other metaphors are only arbitrary games" (Borges).

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The Metaphor and its Characteristics
The word "metaphor" comes from the Greek "meta" (beyond) and "phorein" (to move), that is, it literally means "to move beyond". A metaphor is a figure of speech that consists of describing something or referring to something by means of its resemblance or analogy to something else. The opposite of metaphorical is literal. The study of metaphors belongs to semantics and pragmatics.

There are many types of metaphors and countless examples. Here are a few:

• Biology.
  "The tree of life" (Darwin), "Life is an information process" (John von Neumann), "Life is a virtual checkerboard. The squares are called cells" (John Conway).
  
  
• Sports.
  "Throwing in the Towel," "The Red Lantern," "The Thick of the Pack," "Playing the Lock," "The 12th Player," "The Spanish Armada."
  
  
• Esotericism,
  "The Third Eye", "The Green Language".
  
  
• Spirituality.
  "The wise man knows that he himself is the way" (Paul Twitchell), "The eye is the lamp of the body" (Matthew 6:22-23), "God is a sphere whose center is everywhere and whose circumference is nowhere" (attributed to Hermes Trimesto).
  
  
• Philosophy.
  "The educator is a sower" (Plato), "Ignorance is a disease" (Plato), "A man cannot bathe twice in the same river" (Heraclitus).
  
  
• Physics.
  "The atom is a miniature solar system" (Niels Bohr), "The creation of the universe began with the Big Bang" (Fred Hoyle), "God does not play dice with the universe" (Einstein), "Time is like a circle that spins infinitely" (Borges), "The Universe is much more like a great thought than a great machine (Sir James Jeans).
  
  
• Computer science.
  "The Information Highways" (Al Gore), "The World Wide Web" (Tim Berners-Lee), "The Information Society", "Virtual Reality", "Augmented Reality".
  
  
• Linguistics.
  "Language is the house of being" (Heidegger), "The limits of my language mean the limits of my world" (Wittgenstein), "Language is a channel" (Michael Reddy), "Language is a social art" (Quine).
  
  
• Literature.
  "Life is a dream" (Calderón de la Barca), "We are made of the same stuff as dreams" (Shakespeare), "Fear has many eyes" (Cervantes).
  
  
• Mathematics.
  "The book of nature is written in the language of mathematics" (Galileo), "Algebra is the cornerstone of modern mathematics" (Tobias Dantzig).
  
  
• Poetry.
  "Our lives are the rivers that run into the sea that is to die..." (Jorge Manrique), "The spring of life", "The sea of your eyes", "Life is a journey", "Your hair is golden", "Your eyes are two stars", "You are the light of my life".
  
  
• Policy.
  "The Cold War", "The Iron Curtain", "The Iron Lady".
  
  
• Popular.
  "Throwing the house out the window", "Time is money", "Going for the kill", "Killing two birds with one stone", "Greed breaks the sack", "All roads lead to Rome".
  
  
• Psychology.
  "The unconscious is structured like a language" (Jacques Lacan), "Common sense is the instinct of truth" (Max Jacob), "Consciousness is the operating system" (Gregg Braden).
  
  
• Cognitive Psychology.
  "The mind is to the brain as software is to hardware."
  
  
• Sociology.
  "We live still in the infancy of humanity" (John Wheeler).

The issues

Several questions may be raised on the subject of metaphors, including the following:

• Are there universal or absolute metaphors?
  
  
• Are there higher order metaphors (metaphors of metaphors)?
  
  
• Are there primitive or essential metaphors on which other metaphors are based?
  
  
• What is the relation between metaphor and consciousness (or conscience)?
  
  
• Is knowledge based on metaphors? Is our knowledge metaphorical?
  
  
• Are there abstract metaphors?
  
  
• Are there self-metaphors, metaphors that refer to themselves?
  
  
• Is it possible to classify metaphors into categories?
  
  
• Are there levels of metaphors? Are there surface metaphors and deep metaphors?
  
  
• Is metaphor an anomaly or deviation of language?
  
  
• What is the role that metaphors play in language? Is it an auxiliary or a fundamental role?
  
  
• Are the world, life and history manifestations of a universal metaphor?
  
  
• What is the relation of metaphor to the structure of reality?
  
  
• Are there negative metaphors?
  
  
• What is the relationship between metaphor and creativity? Are there creative metaphors?
  
  
• Do metaphors bring us closer to the truth?
  
  
• Are there metaphors that defy rationality and bring us closer to intuition?
  
  
• Are there visual metaphors?
  
  
• Are there antimetaphors or negative metaphors?

Structure of metaphors

A metaphor consists of 3 elements:

1. The tenor or general abstract term. It is the subject to which attributes are assigned. It is also referred to as the source element.
   
   
2. The vehicle or concrete or specific term that resembles the tenor. It is the object from which attributes are taken or borrowed. It is also called the target element.
   
   
3. The foundation, which are the attributes of the vehicle, which establish the relationship of similarity or analogy between the two previous elements.

Metaphor is a relationship between two concepts, one abstract or generic and the other concrete or particular. The relationship that is not explicit, but arises as an idea when confronting these two concepts. This relationship is a tension or interaction between the two poles, between two meanings. Actually, the vehicle condenses the metaphor because of its intuitive and imaginative power, which is why sometimes the metaphor is associated only to the vehicle and is called "metaphorical concept". The same tenor can have several vehicles, and the same vehicle can have several tenors.

The general and most common form of a metaphor is "Tenor is Vehicle". For example, in the metaphor "Time is a river," the tenor is "time," the vehicle is "a river," and the ground is the attribute of the river, which is flowing, continuous motion. The literal phrase implied would be "Time is a flowing, continuous motion."

When the foundation has several attributes, the metaphor is open to multiple interpretations. For example, "The sea of your eyes" may refer to the color blue or to large eyes. The metaphorical relationship may be somewhat trivial or establish a surprising and creative analogy or correspondence by relating previously unrelated concepts.

A metaphorical concept (vehicle) is concrete or less abstract than tenor. However, it has in itself a "metaphorical value" or "metaphorical power" because of its symbolism, because it hides behind its superficial appearance general or universal characteristics, such as: river, sea, war, sun, earth, net, etc. That is why the vehicle can be identified with the metaphor. For example, it is said that the river is the metaphor of flowing, of continuous movement.


Metaphor as trope

A metaphor is a type of trope. A trope −from the Greek, "tropos" (direction)− is the substitution of one expression for another whose meaning is figurative. It is the change of direction of an expression that deviates from its original content to adopt another content. Metaphor is the most characteristic expression of rhetoric, the rhetorical resource par excellence. The metaphor has been more discussed by philosophers than all the other tropes put together.

The number and identity of tropes has varied throughout the history of rhetoric. Among those most commonly contemplated are the following:

• The simile establishes the relation "A is like B", e.g., "Time is like a river." A typical metaphor states the relation "A is B", i.e., it associates A with the essence of the term B.
  
  A simile is something superficial, which explains. The metaphor is deep, internal to the sentence, part of it, and describes. The simile is a symmetrical or balanced relationship, while the metaphor is an asymmetrical, unbalanced and descending relationship: from the abstract or general to the concrete or particular. A good (creative) metaphor is one that highlights the unbalanced characteristics between tenor and vehicle. A metaphor is more powerful than a simile.
  
  
• The allegory is a figure of speech that pretends to represent an idea using images so that it can be better understood, to make visible what is only conceptual. For example, a blind woman with a scale is an allegory of justice; a skeleton with a scythe is an allegory of death. Plato's allegory of the cave (Republic VII) is the most famous allegory in the history of philosophy, in which he tries to represent the problem of knowledge of the metaphysical by means of the sensible.
  
  
• The parable is a narrative which, by resemblance, conveys a truth or moral teaching. For example, Aesop's fables.
  
  
• The hyperbole is a figure of speech that consists of intentionally exaggerating an idea with the aim of making it more important to the receiver, making a greater impact and provoking an image that is difficult to forget. For example, "Érase un hombre a una nariz pegado" (Quevedo).
  
  
• The metonymy consists of designating one thing with the name of another, making use of some semantic relation existing between both. For example, "To pledge allegiance to the flag", where "country" is replaced by "flag".
  
  
• The synecdoche is a trope in which a part of something is used to represent the whole. For example, "Man is a rational being" ("man" refers to the entire human species); "The police are coming" (refers to some members of the police, not the entire institution).
  
  
• The antonomasia is a synecdoche that consists of substituting a name for a representative description. Antonomasia is a particular form of metonymy. For example, "The eternal city" (Rome), "The city of light" (Paris).
  
  
• The catacresis is a figure of speech that consists of metaphorically using a word to designate a reality that lacks a specific term. In this case, the metaphorical acts directly as a literal element. Originally metaphors, they ceased to be metaphors and were incorporated into the language. For example, "hydrant", "black hole", "skirt of the mountain", "foot of the mountain", "legs of a piece of furniture", "arms of a chair", "computer mouse", "sheet of paper", "head of garlic", "bottle neck", etc.
  
  
• A diaphor is a type of metaphor in which between the associated terms there is no similarity, but its objective is to activate the imagination to perceive, conceive or generate a new reality, new ideas or a new way of contemplating things.

Characteristics of metaphors

• Intuition.
  Metaphors work by intuition, they appeal to intuition. In contrast, the literal appeals to rationality. Translating metaphorical language into literal language is not always possible and, if it is done, it is usually to degrade it, because metaphorical language is deep and literal language is superficial. It is the same thing that happens with the symbol, that to explain it is to degrade it.
  
  
• Imagination.
  Metaphors awaken the imagination, especially metaphors of the poetic kind. Metaphors play a constitutive, essential role in poetry.
  
  
• Consciousness.
  Intuition and imagination are closely related to consciousness. Metaphorical language is associated with the consciousness of the right hemisphere (HD) of the brain. And literal language to the left hemisphere (HI). Metaphors, when confronting or relating concepts, activate imagination, consciousness and intuition. Consciousness links the concepts of tenor and vehicle, in the form of abstract-concrete or generic-specific.
  
  
• Quality.
  A metaphorical concept is of a qualitative type. Any qualitative concept can be metaphorical. Concepts that have more salient qualities are best suited to be metaphorical.
  
  
• Ambiguity.
  Metaphor is a fuzzy concept, difficult to define and explain exactly, as is the case with consciousness. Metaphor has no precise limits. That is why metaphorical language has been criticized or rejected, for example, by the empiricists (Hume, logical positivism) and the nominalists (Hobbes).
  
  
• Reification.
  A metaphor is a reification. To reify is to make abstract ideas concrete. And a metaphor reifies the tenor in the vehicle.
  
  
• Abstraction.
  Metaphor is the way to understand abstract concepts. Abstraction is a mechanism of simplification of reality that makes it easier for us to understand the world. Concepts are abstractions, they eliminate irrelevant details to keep only the essential. The more abstract a metaphor is, the closer it is to the truth.
  
  
• Generalization.
  A metaphorical concept can also be of a general type. In this case, the relationship between tenor and vehicle is considered as an instantiation of the tenor in the vehicle, i.e., going from the general to the particular.
  
  
• Interpretation.
  A metaphor can have different interpretations depending on the cultural context and the receiving person.
  
  
• Analogies.
  Metaphors are also generated in the transfer of a meaning in one conceptual domain to another by analogy, when we have the structure of two dualities A-B and C-D:
  
  A/B = C/D (A is to B as C is to D). For example, old age/life = evening/day.
  
  From this we obtain 3 metaphors that correspond to the relations:
  
  
  1. Pair A-D: The old age of the day.
  2. Par C-B: The evening of life.
  3. Par A-C: Old age is the afternoon.
  
  "Old age is to life as evening is to day; therefore evening may be called 'the old age of day'" (Aristotle. Poetics).
  
  The fundamental metaphor of cognitive science has the form.
  
  
  mind/brain = software/hardware
  
  (mind is to brain as software is to hardware).
  
  
• Correspondence.
  According to Lakoff's [1993] principle of invariance, there is a coherent correspondence between the tenor and the vehicle, such that the structure is maintained. For example, if the tenor is a person and the vehicle is a tree (or vice versa), the top corresponds with the head, the trunk with the body, the branches with the arms, and the roots with the legs and feet.
  
  
• Synthesis.
  A metaphor is an analogy characterized by synthesis, condensation, and simplification.
  
  
• The unknown.
  A metaphor also serves to connect or relate the inexpressible or unknown to the known. There are concepts that do not belong to the material realm such as love, time, morality, happiness, etc., so they can only be defined metaphorically. Metaphor is the only resource to represent and imagine the unrepresentable, the unknown, the imperceptible, the intangible and the infinite. But how can we imagine what is not perceptible? According to Wittgenstein, we can only imagine non-existent combinations of existing elements, such as, for example, a centaur, which is a combination of man and horse.
  
  
• Cognition.
  Metaphor is a key element of human cognition. It is a natural, spontaneous and intuitive resource that we frequently employ in everyday life. Metaphors are fundamental to understanding new concepts. A metaphor does not stand alone; it is part of a conceptual system. "Our conceptual system, in terms of which we think and act, is fundamentally metaphorical in nature" [Lakoff & Johnson, 2005].
  
  Metaphors are not ornaments or figures of speech. If they were, they would only add vividness, emotional impact, and so on. But they are not: they contribute to the cognitive meaning of discourse and are indispensable in all discourse, whether ordinary, literary, scientific, etc.
  
  
• Beyond formal language.
  A metaphor goes beyond the strict limits of formal language. A metaphor is a special, informal use of language because it cannot be interpreted literally. If this were done, it would cause anomalies or semantic distortions. A metaphor is, by nature, inaccurate if interpreted literally. A metaphorical language is open, flexible, deep and with tension. Literal language, on the other hand, is closed, rigid, superficial and without tension.
  
  
• Degrees of generalization.
  Metaphorical concepts have degrees of generalization. A universal metaphorical concept is one that can be referenced by every particular concept.
  
  An example of a universal metaphorical concept is "being", since everything that exists or everything that we can reference has existence at some level (physical, mental, imaginative, etc.).
  
  
• Creativity.
  The creativity of a metaphor appears when two disparate concepts, with no apparent relationship, are related and a new meaning is created or a new vision emerges. The more disparate, the greater the creativity. According to Steven Johnson [1999], "What makes metaphor powerful is the separation between the two poles of the equation. Metaphors create relationships between things that are not directly equivalent."
  
  
• Levels of metaphors.
  There are higher order metaphors (metaphors of metaphors). A metaphor is terminal or borderline when it is self-metaphor, metaphor of itself, when it is not possible to establish a higher abstraction.
  
  
• Transcendence of the rational.
  A metaphor allows us to transcend dual and rational thinking. In Buddhism and Taoism, metaphors are used to transcend this type of thinking in order to access higher levels of consciousness. A koan, in the tradition of Zen Buddhism, is an apparently absurd, illogical or paradoxical question that a master asks his disciple so that the novice detaches himself from rational thought and contacts the deep levels of the mind, the non-verbal, the intuitive and ineffable. The ultimate goal is for the disciple to reach enlightenment, the satori where unity is perceived, the transcendent, that which is beyond duality. Famous Koans are: "What is the sound produced by clapping with one hand?", "What was your true face before you were born?", "All things are reduced to unity, but what is unity reduced to?" and "What is the sound of a tree falling in the forest when no one is there?".

Types of metaphors

According to Lakoff & Johnson [2005], there are 3 types of metaphors:

1. Orientational.
   These are those that relate a concept to a spatial orientation: up-down, top-bottom, top-bottom, left-right, front-back, inside-outside, deep-surface, central-peripheral. For example: more is up, less is down; generic is up, particular (detail) is down; a switch in the on position is up, in the off position is down; consciousness is up, unconsciousness is down.
   
   
2. Ontological.
   Ontological metaphors refer to things in our everyday experience as objects, entities, or containers to help explain concepts. They serve many different purposes: to refer, to quantify, to identify aspects, to set goals, and so on. For example, inflation as a physical entity: "We have to fight inflation", "Inflation damages our standard of living".
   
   Examples of computer-like ontological metaphors: "Capture data" (data as substance); "Data flow" (data as stream); "Kill or abort a process" (process as object); "Save the current state of a process" (state as object); "The program has an error" (error as object).
   
   
3. Structural.
   These are the most popular and familiar metaphors, the ones we use in everyday life. They involve a vehicle that is a known and familiar concrete object or concept from the real world. Structural metaphors are more specific than ontological metaphors. The more specific a vehicle is, the greater its structure. The greater the difference between tenor and vehicle, the better the metaphor is detected and the greater its impact. Examples: "An argument is a war"; "He behaved like a pig"; "Your eyes are the sea".
   
   Structural metaphors are always partial, incomplete. There are characteristics of the vehicle that cannot be applied to the tenor. And features of the tenor that are not explained by the vehicle.
   
   Structural metaphors are the most specific of the three types. And they are the most susceptible to cultural influences. The more specific, the more they reflect cultural influence.

Between these 3 types of metaphorical categories there are relationships:

• Levels of abstraction increase from structural metaphors to ontological metaphors to orientational metaphors.
  
  
• Structural metaphors are more specific ontological metaphors (or particularizations of ontological metaphors) according to some aspect. For example, "A datum is an object" (an ontological metaphor) and "A datum is a document" (a structural metaphor), since a document is a specific type of object.
  
  
• Not all structural metaphors have an ontological foundation. And not every ontology can be associated with a structural metaphor.
  
  
• There are metaphorical entailments (metaphorical entailments). These are expressions referring to some aspect of the vehicle that applies to the tenor of a metaphor. For example, in the metaphor of discussion as war, metaphorical entailments are: "Positions that can be attacked", "Discussions are won or lost", etc. Metaphorical entailments can be thought of as informal descriptions of the meaning of a metaphorical concept or vehicle. Different people might assign different sets of metaphorical entailments to the same metaphorical concept.

Other types of metaphors are:

• A visual metaphor is a resource that allows, in a quick and simple way, to make an idea, a thing, a place, an event, a person, etc. understandable by means of a stereotyped image. For example, "Light bulb on → Idea", "Heart → Love", "Two crossed tibias and a skull → Poison", "Little stars around the head → Blow".
  
  There can also be more elaborate visual metaphors to convey more complex ideas. A well-known example of such a visual metaphor is the so-called "Flammarion engraving," an anonymous illustration first published in Paris in 1888 in a meteorology text (L'Atmosphère) by Camille Flammarion. The caption of the illustration reads: "A medieval missionary tells that he had found the place where Heaven and Earth meet". This engraving illustrates the human desire to transcend the limits of the everyday and access the gears of the world.
  
  

  Flammarion engraving

  
  
• An absolute or universal metaphor is one that relates universal or absolute concepts. For example, "Light is knowledge." For Hans Blumenberg [1995], absolute metaphors are pure truths that allow us to perceive reality as a whole, such as "Truth is light".
  
  
• A paralogical metaphor (or antimetaphor) is a metaphor in which tenor and vehicle bear no relation to each other.
  
  
• Joseph Grady [1997] distinguishes between primary metaphors and complex metaphors. Primary metaphors are based on correlations or correspondences between subjective and objective experiences. Complex metaphors (e.g., "Theories are buildings") are composed of primary metaphors (e.g., "Logical organization is physical structure").
  
  
• For Karl Popper, there are "root metaphors" (root metaphors) underlying every philosophical doctrine.

Metaphors, paradigms, archetypes, myths and symbols

Metaphors are related to paradigms, archetypes, myths and symbols:

• Paradigms.
  In a given field or domain, a paradigm is a conceptual framework or set of general ideas that allow one to approach, understand, or formalize that domain. A paradigm is a general metaphor or set of interrelated metaphors that provide us with an overall view of a domain. A change of paradigm implies a change of metaphors.
  
  
• Archetypes.
  Archetypes are universal or absolute metaphors. Primary archetypes are abstract in nature.
  
  
• Myths.
  Myths can be considered supreme metaphors. According to Stephen Jay Gould, "Myths are eternal metaphors."
  
  
• Symbols.
  A symbol is a visual metaphor, of an abstract type, that activates the intuitive awareness and imagination in a more direct way than a linguistic metaphor. A symbolic metaphor has more power than a linguistic one. To interpret a symbol requires recourse to metaphorical thinking.

Metaphor and the philosophy of science

Before the metaphor, the philosophy of science, went through 3 stages:

1. Initially it rejected the metaphor, considering that science is the territory of the particular, concrete, specific and literal. The clearest rejection came from positivism and neopositivism, rejecting anything non-physical or metaphysical.
   
   
2. In a second stage, science considered that metaphors could play auxiliary functions, although removed from the central core of science. These functions were: a) didactic and informative, since metaphors help to understand abstract concepts; b) as a guide or imaginative vehicle at the beginning of an investigation; c) as a factor favoring creativity, by relating ideas from different fields. But in the end, the researcher had to get rid of metaphors and return to the literal, to the "serious".
   
   
3. In a final stage, metaphors were finally, not only accepted, but came to play a central role in science because of their great advantages:
   
   
   • They provide a framework or conceptual model of a general type from which the particular emerges.
     
     
   • They are very useful strategies or heuristics to achieve knowledge or truths.
     
     
   • They are associated with the mode of consciousness of the right hemisphere (HD), that is, with the generic and deep. The particular is associated with the left hemisphere (LH) mode of consciousness, that is, with the particular and superficial. A science must be based on the general and metaphorical. A science based exclusively on the literal is meaningless.
     
     
   • They provide clarity and simplicity. Metaphors lift us up or take us deep where everything is better understood and everything is seen more clearly.
     
     
   • They are related to consciousness, in the connection between the two modes of consciousness, that is, the connection between the general and the particular. In reality, nothing is literal, since everything is based on metaphor, on the general. Actually, consciousness would be the universal metaphor or mother metaphor, since it is the foundation of everything. This universal metaphor is difficult to grasp and to define, because it is not hidden, it is visible to all, but it is so obvious that we do not perceive it.
     
     
   • They provide the imaginative foundation necessary to give unity to scattered particular knowledge.
     
     
   • They reside at the very core of all scientific theory. They underpin scientific theories. A scientist must verify that any metaphor used is correct and, if it is not, must modify or replace it.
     
     
   • They are the best vehicle to approach the reality of things. Reality is not the superficial and particular but resides in the deep and general, which is metaphorical in nature.
     
     
   • They constitute the creative force that provides vitality and energy to science.

The conception of metaphors by authors

• Plato.
  Plato did not define metaphor, but he used many metaphors in the Dialogues, such as, "The educator is a sower" and "Ignorance is a disease." He associated the 4 elements (air, earth, fire and water) with geometric solids (the famous "Platonic solids") and the universe with a dodecahedron. This theory seems somewhat naive to us today, but it can be considered metaphorical in nature as it establishes a relationship between reality and geometry. Plato was able to point out the antecedent of one of the most important discoveries of modern science: the impossibility of expressing the ultimate constitution of matter in a language other than mathematical.
  
  
• Aristotle.
  He describes his conception of metaphor": "Metaphor consists in giving to one thing a name which corresponds to another thing, there being a transference from genus to species, or from species to genus, or from species to species, or according to relations of analogy" (Poetics).
  
  
• Darwin.
  He used metaphors in his theory of evolution and to describe nature: 1) nature as a "tangled web" (alluding to its complexity); 2) nature as "the tree of life" (genealogically interconnecting all living things); 3) nature as a being with two faces, one luminous (of balance and harmony) and the other dark (of struggle and suffering). According to Stephen Jay Gould, Darwin was "the master of metaphor".
  
  
• Nietzsche.
  For Nietzsche, scientific concepts are metaphorical residues, metaphors that have become conventions. We have forgotten the origin of our concepts. We believe that they come from experience and logical reasoning. However, they come from fantasy, from imagination in the form of metaphors. All discourse is metaphorical.
  
  "Truth is a mobile army of metaphors, metonymies, anthropomorphisms, in short, a sum of human relations [...]; truths are illusions of which we have forgotten that they are illusions, metaphors that have been worn out by frequent use and have lost all sensual vigor" (Nietzsche. On truth and lies in an extramoral sense).
  
  
• Borges.
  Metaphor is for Borges the foundation of much of his reflections, which he reflected in various articles such as "La metáfora", "Examen de metáforas", "Después de las imágenes", "Las Kenningar", "Menoscabo y grandeza de Quevedo" and "Otra vez la metáfora".
  
  Borges considered metaphors to have unlimited possibilities. They constitute a way of dreaming other worlds and of overcoming the linguistic limits of concepts to construct spaces of fantasy, expand the limits of the real and access impossible realities.
  
  "If I were a bold thinker (but I am not; I am a very timid thinker, and I am groping my way forward), I would say that there are only a dozen metaphors and that all other metaphors are only arbitrary games" (Borges. La metáfora).
  
  "Perhaps universal history is the history of a few metaphors. Perhaps universal history is the history of the diverse intonation of a few metaphors" (Borges. Pascal's Sphere. Other Inquisitions).
  
  "The best metaphors are those that have no author, those that are born of the people, those that are created in the markets" [Borges' lecture on metaphor at the headquarters of the Instituto de España, Madrid, 22-04-1980].
  
  Borges resorts to mysterious and cryptic metaphors to recreate the universe of his fictions. The metaphors used in "The Aleph" (1945) and in "The Library of Babel" (1941), which we discuss in detail below, stand out.
  
  
• Emmanuel Lizcano.
  Lizcano [2006] states that metaphorical language "thinks us" and determines our conception of the world: "One of the main functions of analogy, and of that contraction of it which is metaphor, is the cognitive function. By means of it, what is problematic or unknown is assimilated to something close or familiar in order to better be able to handle it or model it." "Metaphor is that tension between two meanings, perceiving one as if it were the other but without actually being the other." "It is in metaphor that the imaginary is faithfully reflected."
  
  
• Other authors.
  "Metaphor is the rhetorical process by which discourse liberates the power that certain fictions have to rediscover reality" (Paul Ricoeur).
  
  "The greatest breakthroughs will come not from making what we are doing bigger and faster, but by finding new metaphors, new starting points" (Terry Winograd).
  
  "Metaphor is finding the similar in diverse things" (Emmanuele Tesauro).
  
  "Metaphor is condensed analogy" (Chaim Perelman).
  
  "Metaphor is an abbreviated form of simile, condensed into one word" (Cicero).
  
  "All truly creative and non-mythical thought, whether in the arts, sciences, religion, or metaphysics, is necessarily metaphorical" (Douglas Berggren).
  
  "Metaphor, in short, is not a disease of science. It is the creative force that gives vitality, and perhaps also the best vehicle for approaching the reality of things" (Alfredo Marcos).
  
  "Every word is a dead metaphor" (Leopoldo Lugones).

"The Aleph" and "The Zahir", by Borges

"The Aleph" is a collection of 17 short stories by Borges, published in 1949. One of them is "El Aleph", which gives the book its title. In it he refers to a point in space that contains all points, a place where are, without confusion (without overlapping or transparency), and simultaneously, all the places of the universe seen from all angles.

"What eternity is to time, the Aleph is to space. In eternity, all time-past, present, future-coexists simultaneously. In the Aleph, the sum total of the spatial universe is contained in a small glowing sphere of less than an inch."

The Aleph represents the unified consciousness of all spatial reality. Actually the infinity represented by the Aleph is of order two (infinite infinities), for all the infinite places in space are in the Aleph, and every thing is also infinite things, because they are contemplated from infinite points of view.

Aleph is the first letter of the Hebrew alphabet, the sacred language of the Kabbalah. For this doctrine, the Aleph is the spiritual root of all letters and the bearer, in its essence, of the whole alphabet. It symbolizes both the universe and the divinity, unlimited and absolute, whose apparent fragmentation gives rise to all things. It symbolizes the universal man, who points to both Heaven and Earth, to indicate that the lower world is the reflection of the upper world. It represents the unity of nature and all that is expressible. In one version of the story of the Golem, in Hebrew mythology, writing the letter Aleph on the forehead of the Golem brings it to life.

Aleph

The letters of the Hebrew alphabet are states of consciousness, fundamental principles or potencies of being. Aleph is the supreme, subtle, living, primal energy that is in everything and everything is in Aleph. It is beyond definition. It is the primal consciousness. The 22 letters are 22 names originally used to designate different states or structures of cosmic energy, which is the essence and manifestation of everything. Each letter-energy-consciousness has two aspects: material and spiritual, qualitative (letter) and quantitative (number).

The letter yod (which symbolizes the equivalent of a mustard seed), corresponds to the plane of emanations or principles. It is the smallest letter of the Hebrew alphabet. From it the other letters are formed. The Aleph is composed of 4 yods.

"The Zahir" is the title of another Borges short story. The Zahir is a magical object that traps our consciousness, creating a fixed idea that is impossible to remove from our thoughts and prevents us from thinking about anything else. For the one who is trapped, the universe is the Zahir, which becomes absolute.

In the story, many diverse objects are said to have been a Zahir: "a tiger in Guzerat in the 18th century, an astrolabe in Persia, a compass in the 19th century, a vein in the marble of a pillar in the aljama of Cordoba, the bottom of a well in Tetouan," and so on. In the case of Borges' story, the author stumbles upon a common object that is a Zahir: a 20-cent coin.

The Zahir is, in a sense, the opposite of the Aleph, Whereas with the Aleph all things are seen from all angles, the Zahir is seen as the only existing object.

For the Qur'an, Zahir is the apparent, external or superficial meaning of things, although it also has a deep, esoteric meaning called Batin. In Arabic, Zahir means notorious, visible. It is also one of the 99 names of God.

Borges used three religious symbols to represent the universal microcosm: the Aleph of Judaism, the Zahir of Islam (the apparent or exoteric meaning of the Koran) and the Bhavacakra of Tibetan Buddhism (a representation of Samsara, the wheel of reincarnations or wheel of life).

The Aleph and the Zahir are inspired by "The Crystal Egg" by H.G. Wells (1897).


"The Library of Babel", by Borges

"The Library of Babel" is a short story by Borges that appeared in the collection of short stories "The Garden of Forking Paths" (1941), a collection that was later included in "Fictions" (1944). The Library is a metaphor for the universe, the human mind and all that is possible. It is a universal metaphor in which everything is represented. The story begins like this:

The universe (which others call the Library) is composed of an indefinite, and perhaps infinite, number of hexagonal galleries, with vast ventilation shafts in the middle, enclosed by very low railings. From any hexagon you can see the lower and upper floors: endlessly. The distribution of the galleries is invariable. Twenty shelves, at five long shelves per side, cover all but two sides; their height, which is the height of the floors, barely exceeds that of a normal bookcase. One of the free sides faces a narrow hallway, which leads to another gallery, identical to the first and all. To the left and right of the hallway are two tiny cabinets. One allows you to sleep standing up; the other, to satisfy your final needs. Through there passes the spiral staircase, which abysses and rises into the remote. In the hallway there is a mirror, which faithfully duplicates appearances. Men usually infer from that mirror that the Library is not infinite (if it really were, why this illusory duplication?); I prefer to dream that the burnished surfaces appear and promise infinity... The light comes from spherical fruits called lamps. There are two in each hexagon: transverse. The light they emit is insufficient, incessant.

The Library has the following characteristics:

• It consists of all possible books of 410 pages, 40 lines per page, 80 characters per line. The "fundamental law of the Library" is that it uses an alphabet of 22 symbols (the same number as the Hebrew alphabet) and 3 special characters (the blank space, the comma and the period). No two books are identical.
  
  
• The number of books in the Library is enormous, but finite. But the Library is unlimited because it repeats itself indefinitely.
  
  
• The Library is a sphere whose center is any hexagon whose circumference is inaccessible.
  
  
• The Library exists from eternity and is immutable.
  
  
• Everything that can be written in any language is already written in some book of the Library.
  
  
• Possibility is equivalent to existence. "It is enough for a book to be possible for it to exist."
  
  
• There is no problem (particular or general) whose solution is not in some book.
  
  
• Some book must be the perfect compendium of all the others, which includes the Fundamental Law or Theory of the Library.
  
  
• The content of the books is meaningless and can have multiple interpretations. The Library is of a superficial, formal or syntactic type. There is no semantics.

Theories of Metaphors
The Cognitive View

The literal is the linguistic, the formal and direct use of language, which is supposed to be adequate to represent our knowledge and experiences. About the metaphor it is discussed whether it is linguistic in nature or not. According to the linguistic and rhetorical tradition, metaphor is something artificial, unnatural of language, a linguistic artifice, an informal, indirect and unnatural way of using language that moves away from "serious" language, which is the literal one.

But according to the cognitive perspective, metaphor is no anomaly of language. On the contrary, it constitutes the foundation of natural language, for natural language is essentially metaphorical.

The classical conceptualization of metaphor is based on linguistic, superficial aspects. The cognitive revolution brought a new perspective on metaphor. From being conceived as a linguistic phenomenon (external) it was conceived as a mental phenomenon, a deep phenomenon (internal). Under the general cognitive conception, metaphor has the following characteristics:

• Metaphor has to do with the processes by which we organize our knowledge of reality.
  
  
• Metaphorical linguistic phenomena are manifestations of underlying cognitive phenomena.
  
  
• There is no clear separation between the literal and the metaphorical. The boundary is fuzzy, as there are cognitive processes that produce both types.
  
  
• To explain metaphors one must focus, not on superficial or particular metaphors, but on deep or generic metaphors. Superficial metaphors can be considered particularizations of generic metaphors.

There are two main cognitive theories of metaphor: relevance theory and conceptual metaphor theory.


The theory of relevance

The concept of "relevance" is generic −meaning "important" or "significant"−, so it is studied in different fields, including cognitive science, logic, information and communication science, but fundamentally it is studied in epistemology (the theory of knowledge). Different theories of knowledge consider different concepts relevant. In logic, a premise (or antecedent) is relevant to the conclusion (or consequent).

The characteristics of relevance are:

• The meaning of "relevant" is difficult or impossible to capture within conventional logical systems. In general, something (A) is said to be relevant to a task (T) if it increases the probability of achieving the goal (O) implied by T.
  
  
• Relevance is key, as every problem requires the identification of relevant elements that can contribute to building a solution.
  
  
• Relevant can be anything: a concept, a piece of data, a document, a relation, etc.
  
  
• There are many types of relevance: individual and collective, subjective and objective, intrinsic and extrinsic, etc.
  
  
• Relevance is related to possible worlds. A thing is relevant with respect to a possible world.
  
  
• The relevances of a given subject are not isolated; they form a system, which can be stable or change according to context or circumstances. According to Aron Gurwitsch [1978], the relevant elements of a certain topic form a unity or system, so that "The unity of context is the unity of relevance."

The theory of relevance, by Dan Sperber and Deirdre Wilson, presented in their work "Relevance. Communication and Cognition [1986], is a new paradigm for pragmatics and a new theory of communication:

• An alternative to the code model.
  The code model is based on the transmission of a message from a sender to a receiver using a code previously known to both. This model is superficial, formal, purely informational.
  
  In the relevance model, a sender supplies evidence of his intention to convey a certain meaning. It is the receiver or recipient of the communication who must search for meaning and interpret the message through an inferential process based on the evidence provided. This model of communication, which is also called "inferential," is a deep model, the model that produces understanding.
  
  Relevance is to the receiver as rhetoric is to the sender.
  
  
• Communication is governed by a single cognitive principle: the principle of relevance. It is the receiver who selects or pays attention only to information that seems relevant to him.
  
  
• The recognition of intentions. Successful communication is based on the receiver's recognition of intentions. This is the theory of Paul Grice [1989], who laid the foundations of an inferential model of communication as an alternative to the classical code model. Communication with non-explicit meanings admit different interpretations, inferences or −according to Grice's denomination− "implications" or "conversational implicatures".
  
  There are two types of intentions: informational intention (the sender informs the receiver of something) and communicative intention (the sender conveys an intention).
  
  The context (or environment) is relevant, a context that includes both physical and cognitive aspects. The context is chosen by the receiver to interpret the sender's message.
  
  The receiver seeks to maximize the relevance of the message so that it provides as much cognitive value as possible. A message might not provide any value when, for example, the message is encoded in a language unknown to the receiver, when it is too technical, when it is ambiguous, etc.
  
  
• Metaphor is not something "special". Nothing different is required to interpret it than what is required for the interpretation of the literal. Metaphors serve for what cannot be expressed literally. They have greater communicative power than the literal, they are more relevant. The principle of relevance of metaphor is the expression that metaphor is more cognitively efficient than the literal.
  
  
• Accepts the representational theory of mind, i.e., that mental contents have an internal representation of a linguistic type. Language is the essential common element between the external and the internal world.
  
  
• The literal consists in the identity between thought (the internal language) and the external formal language, between the representation and the represented. It follows the idea of the first Wittgenstein (that of the Tractatus) that reality, language and thought have the same logical form or structure.

The theory of conceptual metaphor

The conceptual metaphor theory (or cognitive metaphor theory) was proposed by the linguist George Lakoff and the philosopher Mark Johnson [Lakoff & Johnson, 2005], [Lakoff, 1993]. According to these authors, the traditional concept of metaphor has 3 characteristics: 1) Metaphor is a linguistic phenomenon; 2) Metaphor is a deviation from the literal use of language that is used as a rhetorical device; 3) Metaphor is based on similarities or analogies between two concepts.

The theory of conceptual metaphor departs from this traditional conception and offers a new vision. It refers to the understanding of a conceptual domain A in terms of another conceptual domain B, such that:

• The domain A is the source or origin domain, and its concepts are abstract or generic. The domain B is the target or goal domain and its concepts are concrete (or less abstract than the concepts of A) and close to physical experience. The general scheme is "The source domain is the target domain" or "The domain A is the domain B". In the case of a concrete metaphor, the schema is "Tenor is vehicle", e.g., "Time is money" (time is the abstract and money is the concrete). Another example is "Life is a journey".
  
  
• To each concept of the domain A corresponds a concept of the domain B. Metaphors are sets of correspondences between the two conceptual domains.
  
  
• The correspondence, projection, function or transformation of one conceptual domain into another is subject to the formal principle of invariance that ensures congruence between the two domains, so that the structure or topology of the source domain is preserved in the target domain.

Examples of conceptual metaphors are:

• Domain A: time. Domain B: space.
  "Next week" (the future as an entity moving forward in space toward us); "Last week" (the past as an entity moving away in space from us); "We are approaching the end of the year" (time as a space within which we move); "We are in the midst of difficult times" (time as a spatial field).
  
  
• Domain A: time. Domain B: money.
  "Time is money"; "Gaining time; "Saving time"; "Stealing time"; "Wasting time".
  
  
• Domain A: quantity. Domain B: the vertical.
  "Prices are rising," "The stock market fell sharply yesterday."
  
  
• Domain A: a discussion. Domain B: war.
  "A discussion is a war." In a discussion we think in terms of war: there are winners and losers, positions to defend, use of strategies, and so on.

This theory goes in the opposite direction to relevance theory. Its aim is to establish a general cognitive model based on metaphor. Metaphor is considered the central mechanism of abstract thought. It is a theory of the up-down type, from the general (the model) to the particular.

Lakoff and Johnson's [2005] "Metaphors we live by", published in 1980 is a text that has been highly influential in emphasizing metaphor as a key to human cognition and in which the notion of "conceptual metaphor" was introduced. The theory of conceptual metaphor has helped to establish the importance of metaphor as a foundation of language, and is today one of the central areas of research in the field of cognitive linguistics.

Characteristics of the theory:

• Metaphor has a rigorously cognitive consideration. The linguistic phenomena linked to metaphors are the manifestation of underlying cognitive processes. "Our ordinary conceptual system, in terms of which we think and act, is fundamentally metaphorical in nature" [Lakoff & Johnson, 2005].
  
  
• We must distinguish between conceptual metaphor and metaphorical linguistic expression. Conceptual metaphors are abstract schemes of thought: associations between elements (ontological correspondences). These metaphors are manifested in metaphorical language.
  
  
• Conceptual metaphors are present in the mind (in the ways we think), in everyday language and in the ways we act. Language is impregnated with metaphors, although we are not always aware of them. We perceive and act according to conceptual metaphors. Conceptual metaphors are also used to understand theories and models.
  
  
• Metaphor is not a linguistic matter, but a mental matter, of thinking, of correspondence (mapping) between conceptual domains. Language is secondary. "The generalizations governing metaphorical expressions are not in language, but in thought represented as correspondences in conceptual domains" [Lakoff, 1993].
  
  
• Many of the basic concepts we commonly use (such as: time, quantity, action, cause, category, change, etc.) are usually understood through metaphors. The metaphors of everyday life form a huge system or network of correspondences between conceptual domains.
  
  
• The metaphors of our conceptual system are based almost entirely on a set of basic or fundamental metaphors.
  
  
• Metaphors allow us to understand new concepts in terms of already known concepts, establishing correspondences.
  
  
• "Metaphor is the main mechanism by which we understand abstract concepts and perform abstract reasoning [Lakoff, 1993].
  
  The higher the level of abstraction, the more layers of metaphors are required to express it. Metaphors create the ideas rather than reflect them. Metaphors constitute the universe of abstract ideas.
  
  
• The literal occurs in the realm of concrete physical experiences. Abstraction or generalization is accomplished by metaphorical processes. Non-metaphorical thinking is only possible when we speak of a physical reality.
  
  
• The metaphorical is the product of cognitive imagination, which allows us to relate different conceptual domains. Abstract concepts derive from metaphorical images. These relations are neither representational nor have to do with the internal organization of our knowledge of external reality.
  
  
• The evolution of human thought has been the process of developing better and better metaphors.

Characteristics of conceptual metaphors:

• Automaticity.
  Conceptual metaphors are constantly used in an automatic, effortless way, without our noticing.
  
  
• Unidirectionality.
  The structure of the source domain is projected onto the target domain, but not the other way around. For example, "Time is money," but money is not conceptualized as time. Unidirectionality goes from the abstract to the concrete, i.e., the abstract is explained in terms of the concrete.
  
  
• Incompleteness.
  The associations between the two domains are not complete.
  
  
• Multiplicity.
  The same domain can act as a source for several goals. For example, time as space and as money. And the same goal domain can have different source domains, e.g., a trip can apply to a college career, a marriage, an advertising campaign, and life itself.
  
  
• Hierarchy.
  Metaphors are usually particular cases of more general metaphors, from which they inherit their structure. That is, there are hierarchies of metaphors. For example, a journey is a particular case of "long-term activity".

The pioneer of the conception of metaphor as thought rather than language was Michael Reddy in his celebrated essay "The Conduit Metaphor" [Reddy, 1979]. This metaphor refers to the fact that "language is a conduit", a channel for the transmission of mental content between people. Reddy discovered the importance of metaphorical language in the conceptualization of the world and in communication in everyday language; that the source of metaphor is not language but thought. Lakoff acknowledges that his theory of conceptual metaphor has Reddy's work as its antecedent.


Metaforología

Metaphorology is a theory of metaphor developed by Hans Blumenberg [1995, 2000, 2001, 2003]. Its fundamental ideas are as follows:

• What lies beneath the metaphors and idioms of language is closest to the truth, and farthest from the truth are ideologies.
  
  
• Metaphors and involuntary and intuitive expressions allow us to apprehend or contact true reality.
  
  
• Philosophy is metaphorical in character. Philosophy would be nothing without metaphor.
  
  
• Metaphors have a separate existence from what we consider "real world", they transcend the real world, although they are used to illustrate or understand the real world. The real world is not understood without metaphors.
  
  
• Metaphor is not part of any pre-scientific thinking, but it is the instrument that allows us to give meaning to the inconceivable, something that logical, rational or scientific thinking, which pretends to conceptualize everything, cannot provide.
  
  
• Metaphor appeals to the intuitive and the imaginative. The language of metaphor is imaginative, capable of grasping what is not conceivable. Metaphorology points to the limits of the philosophy of language as it confronts an inconceivable reality.
  
  
• Metaphor is the foundation of our understanding of the world. Metaphor allows us to better understand the world, man, history, language, philosophy and life.
  
  
• Metaphors are fundamental representations of human orientation in the world. They are sources of meaning, units superior to the concept, although they are manifested through concepts. A metaphor is a totality of (conceptual) interpretations.
  
  
• There are "absolute metaphors", which are pure truths that allow us to perceive reality as a whole. They are fundamental elements of philosophical language and cannot be replaced by rational or logical concepts. They are necessary prerequisites for human thought, action and orientation and underlie all culture. Science also needs absolute metaphors. Examples of absolute metaphors are: "Man is not the center of the universe", "The world is a book; to read that book is to understand the world", "Truth is light".
  
  The truth of absolute metaphors is pragmatic because they guide human behavior. Metaphoralogy is an approach to a theory of absolute metaphor.
  
  
• An explosive metaphor is one that relates contradictory elements that make imagination and intuition of the object impossible.
  
  
• Concepts emerge from metaphors. But it may happen that metaphors emerge from concepts. The most illustrative example is that of Copernican cosmology: the absolute metaphor of the dethronement of the human being as the center of the universe.
  
  
• There is a difference between myth and absolute metaphor. Myth has an ancient and unfathomable origin and makes the world intelligible, it gives meaning to reality. Meaning can be explained but not defined. Metaphor is a fiction that makes understanding possible. Metaphors can be remnants on the way from myth to logos.

Metaphors in Mathematics and Computer Science
Conceptual metaphors take us to a higher conceptual ground, from which everything is better understood and from which the underlying complexity is hidden. They also help to demystify mathematical and computer concepts.


Mathematical metaphors

Mathematical concepts are metaphorical in nature. Mathematical thinking has been built from metaphorical thinking. Mathematical language is an object-based metaphorical language. In fact, the names of mathematical entities have their roots in objects in the physical world. For example, we use the word "existence" even though we refer to abstract objects.

Metaphors play a basic role in the teaching of mathematics. The trend is toward creating images that help us understand concepts better. The deep understanding of mathematics (the so-called "mathematical enlightenment") occurs through intuitive and synthetic leaps of consciousness, associated with the right hemisphere (HD) of the brain. Geometry, because of its visual character and therefore associated with the HD), plays an essential role in mathematical metaphors. It is the source of many metaphorical concepts.

George Lakoff (linguist) and Rafael Núñez (psychologist),in their work "Where Mathematics Come From" [2001], attempt to ground mathematics by means of conceptual metaphors, in the basic cognitive resources common to all human beings. Lakoff builds on his earlier works: Lakoff [1993] and Lakoff & Johnson [1999, 2005]:

• Mathematics is a development of the human cognitive apparatus and, therefore, must be understood in cognitive terms, through cognitive mechanisms such as human experiences, metaphors, and generalizations. The problem is to identify these cognitive resources and relate them to the philosophy of mathematics.
  
  
• Mathematics consists of a large number of metaphorical constructions.
  
  
• They reject the Platonist philosophy of mathematics. All we know and can know is "human mathematics," the mathematics that arises from the human intellect. The question of whether or not there is a transcendent mathematics (independent of human thought) is meaningless.

Among the mathematical concepts of a metaphorical type we can identify the following:

• Natural numbers.
  Since the concept of number is indefinable, that is, it is not reducible to simpler concepts, we must necessarily appeal to metaphor. Numbers are abstract entities that we conceptualize and give meaning to by means of metaphors.
  
  Euclid, in the Elements, defines number as "A multitude composed of units" (Book VII, Definition 2). For Euclid, the unit is not a number, but something of which numbers are composed.
  
  Plato conceives of number as an abstract and pure unity. The essence of all numbers is unity, the knowledge or contemplation of which elevates the soul of man to a state in which he can contemplate the essence of being.
  
  But Plato was referring to the invisible, deep, abstract numbers, which reside in the higher world of Ideas, and which are not perceptible by the physical senses. Plato considered the concrete and practical numbers as superficial manifestations of the abstract numbers. Concrete numbers belong to the material world and are inferior entities. Even more inferior are fractional numbers, which are things of merchants, architects, builders, masons, farmers, etc.
  
  For Aristotle, number is "a definite plurality" and also "a combination of units."
  
  
• Negative numbers.
  When negative numbers that were initially considered imaginary appeared, more metaphors were needed to understand them because they do not appear in nature, like natural numbers; they belong to the realm of the mental. The metaphors that are often used are: the elevator metaphor (negative numbers as floors below ground level) and the debt metaphor (owing an amount of money to someone), etc.
  
  
• The zero and the one.
  The "0" is the metaphor for nothingness or emptiness. At first it seems contradictory to assign a symbol to nothingness, but it makes perfect sense when emptiness is positional in the numerical representation. Besides representing emptiness, paradoxically it also represents completeness, because it is the union or synthesis of all opposites (n−n = 0), so it contains all numbers and their opposites. Zero can be considered to be the field from which all numbers emerge. In Buddhism, emptiness is analogous or synonymous with fullness. The Mayans represented zero by a shell (evoking emptiness, hollowness). The "1" is the metaphor for something that is considered a unit, regardless of its particular characteristics. It is also the metaphor of totality. Both (0 and 1) are metaphors for universal duality: nothingness and everything.
  
  
• Fractional numbers.
  The metaphor "part-whole" is used. A n/m number is based on dividing a unit or whole into m parts and taking n.
  
  
• Irrational numbers. They were discovered by the metaphorical relationship between the diagonal and the side of a square (the √2 for side of length 1). The denomination "square root" of a number is a metaphor, since really "square root" is an abbreviation (or deformation) of the expression "root of the square" or "the root that generates the square", establishing the correspondence root/square = side/square.
  
  
• Real numbers.
  The metaphor of points on a straight line (the real line) is used.
  
  
• Infinity.
  The concept of infinity is elusive, difficult or impossible to define, so we must necessarily resort to metaphors. Paradoxically, this mysterious concept plays a central role in mathematics. There are two types of infinity: potential and actual (although the latter is often misnamed "actual").
  
  
  • Potential infinity is an infinity of an operative or quantitative type, characterized by a repetitive process without end, unattainable, without final result. The metaphor of potential infinity is a wheel that, starting from rest, turns indefinitely, without ever stopping. Natural infinity is the infinity generated by adding 1 indefinitely to a variable initialized to 0 and is the cardinality of the set of natural numbers.
    
    
  • The real infinity is a descriptive or qualitative type of infinity that corresponds to an infinite process that is considered to be already realized, completed. The metaphor of real infinity and the one that best illustrates this concept is that of a regular polygon inscribed in a circle. It starts with a polygon of 3 sides, then 4, etc. up to a polygon of infinite sides, which is identified with the circumference, which is the final result of the process.
  
  
• Transfinite numbers.
  Cantor used the metaphor of "one-to-one correspondence" (pairing or equiparability) to compare the relative (qualitative) size of two sets A and B, establishing pairs of elements, one of A and the other of B. If such a correspondence can be established between all the elements of the sets, then both sets have the same cardinality (number of elements) if both sets are finite. But in the case of infinite sets the cardinalities may not be equal. For example, the set of even numbers does not have the same cardinality (quantitative infinity) as the set of natural numbers, but they have the same qualitative infinity.
  
  Thanks to the metaphor of equipartition, Cantor was able to show that there exists an infinite hierarchy of infinite numbers, which he called "transfinites".
  
  
• Sets.
  The metaphor is that of a container of objects.
  
  
• Functions.
  There are two metaphors: 1) a machine transforming or processing some inputs and producing an output, a process carried out in a finite number of steps; 2) a cooking recipe indicating a series of steps to be carried out with some input ingredients to produce an output.
  
  
• Algebra.
  Cartesian graphical representation was not only a way to integrate algebra and geometry, but it provided algebra with a powerful metaphor: geometry.
  
  
• The fundamental constants of mathematics (e, Φ, π, i) have a geometric root (see figure).
  
  
  • The number π comes out of the circle as the universal ratio of circumference length to diameter.
    
    
  • The imaginary unit i is the unit of an imaginary axis perpendicular to the real line. With both axes the complex numbers are represented, those of type (r₁ + r₂*i). To multiply a complex number by i is to rotate it 90º counterclockwise: (−r₂ + r₁*i). The numbers 1, i, −1 and −i are the 4 solutions of the equation x⁴ = 1, which lie on a circle of radius 1.
    
    

    Real- imaginary space

    
    
  • The number e is in the logarithmic spiral, also called growth spiral or equiangular spiral. The radius vector grows exponentially with respect to the angle of rotation. The distance between two successive turns grows exponentially. It is called equiangular spiral because the angle that the spiral forms with a radius vector is always the same.
    
    

    Logarithmic Spiral

    
    
  • The golden ratio is in many geometric shapes, such as in the pentagon and the 5-pointed star.
    
    

    Golden Ratio (Φ)

The constants π, e, i, 0 and 1 are related by the famous Euler formula: e^πi + 1 = 0.


Computer metaphors

Computer slang is full of metaphors that express new concepts based on known concepts. Computer metaphors humanize, they approximate the human conceptual system. And they simplify; everything becomes easier to use, with minimal learning.

Every era has its metaphors, which usually begin as references to something specific, but progressively gain strength and expand to have a broad meaning, and even represent an entire culture. The current era is the era of information and communications technologies and, in particular, the Internet era.

The key metaphor of this era is the Internet, the World Wide Web, also called "the cloud", a decentralized space made up of nodes and links. The Web, a common digital space with which we can interact and communicate, has become the universal metaphor: the metaphor of the interconnection of all things. On the Net, physical space disappears, it is transcended and a new type of space appears: Cyberspace, a digital, electronic, virtual or telematic space through which we can "surf" or "surf".

Metaphors associated with the Internet era are: information highways (a term coined by Al Gore, former U.S. Vice President) and the information or knowledge society.

At the workplace level, the office supermetaphor is used, with various submetaphors such as:

• The desktop metaphor as a graphical user interface (GUI).
  The UGI metaphor is the most important one because it is precisely the boundary space where the human-machine interaction takes place and where the underlying complexity is hidden. All the GUIs have a metaphorical basis. The desktop metaphor is the most widespread and well-known UGI. It was introduced by the Xerox Alto and Xerox Star System workstations −developed at Xerox PARC (Palo Alto Research Center)−, imitated by the Apple Macintosh personal computer and pervasive today in almost all operating systems (including Windows, Microsoft's operating system).
  
  Xerox PARC was created in 1970 as an internal think tank for the development of new technologies. The Xerox Alto workstation was developed in 1973. It was the first computer to use the desktop metaphor. Xerox Star was introduced in 1981 and was an improvement on the Xerox Alto. It used an IGU based on windows, icons, folders, file servers, printer servers and e-mail. Xerox Star was an initiative conceived as "The Office of the Future," a metaphor for making office work easier and more intuitive.
  
  The design of these Xerox stations was influenced by the concepts of Smalltalk, the first object-oriented language (although it had a predecessor which was Simula). Smalltalk is an object-oriented operating system and programming language, also developed at Xerox PARC by a group of researchers led by Alan Key in the 1970s. Smalltalk objects communicate by sending messages. The motivation for creating Smalltalk was to provide a working environment for experimentation, research and creativity.
  
  A desktop is a two-dimensional (2D) space, but many have evolved to a space often referred to as "two-and-a-half-dimensional," in which the elements on the screen overlap. There is also the trend toward a 3D-type metaphor for reality, with physical metaphors (such as weight and size) and more natural interaction through fingers and hand movement. One such metaphor is that of rooms, also conceived by Xerox.
  
  The desktop metaphor has its limitations: 1) It is based on intuition; the user has to intuit how a system works, but intuition can fail; 2) It is difficult to find metaphors for relationships, transformations, services, selections, attribute assignment, etc.; 3) It is limited to simple operations relating to individual elements, such as: manually selecting one or more elements (files, documents, folders, etc.), closing a given window, enlarging a window, deleting a window, etc.). But the metaphor does not work when it comes to generic operations, such as selecting elements with certain attributes, deleting them, changing an attribute, etc.
  
  The future of the IGU should be linguistic, because of the infinite possibilities provided by a language, as opposed to a small number of visual metaphors, which can nevertheless be used as an aid for very particular topics. The ideal in an IGU is that the user's external language is the same as the internal one, and that there is a "direct manipulation": what the user does at the external level must have its immediate reflection at the internal level.
  
  
• The metaphor of the filing cabinet, with folders and documents, with tree structure (hierarchical organization).
  
  
• The trash can metaphor, with operations such as: throwing a file into the trash can (to delete a file), emptying the trash can and taking a file from the trash can.
  
  
• The metaphor of the sheet of paper and the document in word processors, with operations such as scrolling through a multipage document (scroll).
  
  
• The clipboard metaphor, a working memory used to transfer data from one document or application to another, by copying (or cutting) and pasting.

Other metaphors are:

• The window metaphor, which allows us to see and access an internal aspect of the computer (a process, a data structure, etc.).
  
  
• The metaphor of the person in the Turing test: Is a computer capable of emulating the verbal behavior of a human being?
  
  
• A compiler as a translator.
  
  
• A variable as a box with different contents over time.
  
  
• Virtual reality and Augmented reality.

Computer science, as we see, is founded on many metaphors, but in turn the computer is the fundamental metaphor of cognitive science: "The mind is to the brain as software is to hardware."


MENTAL, Universal Metaphor
MENTAL is the return to the fundamental or primordial metaphors that we had forgotten. These metaphors are the support or foundation of every concept and of all knowledge. In fact, the whole MENTAL language is inspired by a central metaphor: the metaphor of the computer conceived as a universal language: a set of universal semantic instructions or primitives with which all computation and description can be expressed. Moreover, this set of primitives determines the limits of what can be expressed. MENTAL is the universal metaphor on which everything gravitates, since universal semantic primitives are present in all things. MENTAL is the absolute metaphor that makes us perceive reality as a whole.

Just as the computer has become a metaphor for the brain and the mind, MENTAL is a universal metaphor (or universal metaphorical concept), for everything can be associated with universal language. MENTAL is the new foundation of cognitive science.

• Archetypes vs. metaphors.
  Since the archetypal primitives of MENTAL are indefinable, one must appeal to metaphors to better understand them. To each primitive of MENTAL corresponds a metaphorical concept (vehicle). Archetypes are, at the same time, absolute metaphors, self-metaphors and radical metaphors.
  
  There are infinite possible metaphors, but there is only a reduced set of fundamental metaphors that allow us to understand the deep (abstract) structure of reality. Interestingly, Borges in "The Metaphor" speaks of only a dozen essential metaphors, the same number of primary metaphors of MENTAL.
  
  
• Primitives.
  MENTAL primitives are universal metaphorical vehicles or concepts. In the same way that symbols refer to themselves, MENTAL primitives are self-metaphors, for they refer to themselves.
  
  
• Universal Paradigm.
  MENTAL's metaphors cannot live in isolation and together they constitute a universal paradigm.
  
  
• The possible.
  MENTAL is a metaphor for the possible, embodied in the infinite possible expressions that can be formed with language. It is a metaphor analogous to that of Borges' Library of Babel, but restricted to well-formed expressions (in the Library of Babel one finds meaningless expressions). MENTAL allows to express imaginary and possible worlds.
  
  
• Mathematics and computer science.
  The cognitive resources of mathematics and computer science are the primitives of MENTAL. Being deep resources, their foundation is philosophical: they are philosophical categories.
  
  On the subject of user interface, the programming language (the internal one) and the end-user language (the external one) is the same. Everything is linguistic.
  
  
• El Aleph.
  Borges' Aleph is a metaphor for MENTAL, for this language is a privileged point of view from which everything is contemplated, for all that exists is a manifestation of the primary archetypes. Once installed in the Aleph, it also becomes Zahir, since everything is seen from this universal point of view and we can no longer detach ourselves from it when contemplating the essential unity of all things.
  
  
• Conceptual system.
  Our conceptual system is based on the primary archetypes, which are the essential and universal metaphors to which we necessarily refer, explicitly or implicitly.
  
  According to the theory of conceptual metaphor, metaphor is not a linguistic, but a mental matter. MENTAL connects the internal (mental) and the external (the real world) through the linguistic.
  
  
• The union of the metaphorical and the literal.
  In MENTAL the opposites, the metaphorical and the literal, the concrete and the abstract, are united through symbolic expressions. The metaphorical are the primitive (abstract) and the literal are the manifestations, the concrete expressions. The literal is the particular of the metaphorical. The union of the literal and the metaphorical is a particular case of union of opposites in MENTAL, which justifies its being considered the language of consciousness.
  
  
• Relevance.
  Relevant concepts are the primitives. MENTAL unites the code model and the relevance model. The set of relevant elements form a system, which is a language.

MENTAL language metaphors

• Abstract space and abstract time.
  Abstract space is the space where expressions "live". Abstract space and abstract time go together in sequences (linear abstract space and time) and in sets (concurrent abstract space and time).
  
  
• Dimensions.
  Universal semantic primitives can be thought of as dimensions or degrees of freedom. The metaphor is that of spatial dimensions.
  
  
• Expression.
  Metaphor is chemistry, the synthesis or combination of simple elements.
  
  
• Network (or web).
  The expressions of the environment are intertwined with each other forming a network. The space is created by the concrete expressions.
  
  
• The triad of meta-expressions "everything, nothing and something".
  They are represented, respectively, by the symbols "Ω", "θ" and "α".
  
  
• The empty set and the empty sequence.
  The empty set is symbolized by "{}" (or by "∅") and the empty sequence by "()". The content of both is "nothing", the null expression (θ):
  ({}↓ = θ)    (()↓ = θ).

The metaphors of primitives

1. The generic.
   When parameters are used, it is the category or class metaphor. Without parameters, it is the metaphor for permanent or necessary.
   
   
2. Particularization.
   The metaphor for the specific or one aspect of something.
   
   
3. Grouping (sequence and set).
   Sequence is the metaphor for linear, sequential, and ordered. The metaphorical image is a chain.
   
   The set is the metaphor for the parallel, the concurrent in space and time (in a set all the elements occupy the same space-time), for the non-ordered (but not disordered because all the elements of the set have the same status and because disordered implies that it was previously ordered). When there are groups (sequences or sets) of higher order, we have the metaphor of the tree.
   
   
4. Distribution.
   It is the metaphor of the expressible in a compressed form.
   
   
5. Substitution.
   It is the metaphor of transformation, of the dynamic.
   
   
6. Equivalence.
   It is the metaphor of what has the same meaning and different representation. The metaphor is a balance.
   
   
7. Evaluation (and non-evaluation).
   Evaluation is the new computational metaphor, which goes beyond mere calculation. Non-evaluation is the metaphor of the unchanging, the immutable.
   
   
8. The sum or the numerical.
   A number is a set of undifferentiated elements or units. Zero is the number of elements of the empty set and of the empty sequence:
   ({}# = 0)    (()# = 0).
   
   
9. Condition.
   It is the metaphor that everything can be conditional.
   
   
10. Hierarchical navigation.
    It is represented by the symbols "↓" (access to the content) and "↑" (access to the continent), which allow access to the different levels of the expressions. It is the metaphor of the "inside-outside" relationship.
    
    
11. Execution (beginning and end).
    The metaphor for the operation of a machine or device based on a series of sequential type operating steps, with several possible inputs and an output, which is the final result.
    
    
12. Stop and continue process.
    It is the metaphor of rest and movement.

In addition to the primitives, there is the "contrary" meta-operator, which is the metaphor for universal duality.

---

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