"Symbolism is an immediate datum of total consciousness" (Mircea Eliade).
"The power of a symbol resides in the condensed idea behind it" (George Spencer-Brown).
"To new ideas necessarily correspond new signs" (David Hilbert).
"In the symbol is hidden the revelation" (Thomas Carlyle).
The Symbols and their Properties
Symbols are simple graphic representations that do not require external interpretation, because by themselves they evoke archetypes of the collective unconscious and appeal to the intuition. Symbols precede conscious understanding.
According to Ernst Cassirer, man does not live in a physical universe, but in a symbolic universe. Man relates to things by means of a complex symbolic network that serves to order and make reality comprehensible. The mind abstracts reality in the form of symbols.
For Erich Fromm, symbolic language is universal. It is the true and only common language, which humanity forgot, so it had to resort to conventional, particular, sign-based languages.
Properties of symbols
Transcendence.
A symbol refers, evokes and refers to something invisible, inexpressible and ineffable, something that is beyond what is represented and that cannot be expressed in any other way. This something transcends the opposites and is found in a higher dimension. Jung precisely calls "transcendent function" the property of symbols to unite opposing forces and overcome them to open the way to the progress of consciousness. This transcendence of opposites can be interpreted as a universal connection or solidarity.
Intuition.
A symbol appeals to our intuition, to the synthetic, to the consciousness of the right hemisphere of the brain. It is beyond our analytical, sequential and discursive thinking, for a symbol escapes all definition of discursive language. "Synthetic symbolism opens up truly unlimited possibilities of understanding" (René Guénon). It also activates the imagination, the faculty of the soul connected with our intuition.
Analogical power.
The symbol gives free course to analogies. A symbol, from the deep, allows us to establish analogies between apparently different entities on the surface.
Union of opposites.
As manifestations of archetypes, symbols unite the conscious with the unconscious, the temporal with the timeless, the particular with the universal, the differentiated with the undifferentiated, the known and expressible with the unknown and inexpressible, the visible with the invisible. A symbol makes us perceive the double relationship between the unity underlying all things and the manifested multiplicity. A symbol allows us to go beyond the opposites and connect with the dimension of intuition.
Truth.
When contemplating a symbol and intuiting the unrepresented, we experience an elevation of consciousness, we are transported to another dimension which is the dimension of Being, our true inner being, which is timeless. The deep knowledge provoked by the symbol is an experience that refers to a profound reality, an experience of knowledge or totalizing consciousness, of synthesis, of harmony, of union, of connection with all that exists. The symbol opens the mind to the unknown and infinite, to universal consciousness, to undifferentiated unity, to the source of all. The symbol opens doors to consciousness and transforms us internally. Symbolism is a way for communication or access to truths of a higher order.
Universality.
A symbol is universal because it is beyond particular cultures. The language of symbols is universal. Symbols are universal but hide behind the particular. A symbol connects us with the universal. It stimulates our deep consciousness, the one that connects us with all things.
Pluridimensionality.
Symbols are multidimensional or polyvalent. Because of their profound character, they express multiple relations in different planes or levels of reality, so that a symbol admits, superficially, multiple interpretations or meanings. It is the passage from the synthetic mode of consciousness (perception) to the analytical mode of consciousness (interpretation).
Energy.
A symbol is charged with energy, by holding the tension of opposites, which is at the basis of our psychic life. Symbols are ideas-force that, when contemplated, awaken our inner energies, producing in us a resonance, that is, an activation of multiple aspects or related forces. The energy of a symbol is never exhausted, since it remains indefinitely suggestive. In the symbol operates a centripetal force, a force towards the Center of all, where the essential unity resides. The symbol is a unifying force. By connecting or attuning us to consciousness and harmony, symbols have been used as talismans for physical, psychic and spiritual healing.
Geometry.
The symbols are geometric in nature. Geometry is associated with synthetic consciousness and constitutes a bridge between the unmanifest or absolute world and the manifest or relative world.
Signs vs. Symbols
Sign and symbol are located at the two poles of consciousness.
A sign is a mere representation of a simple concept.
It is of a superficial, analytical and rational type.
It refers to something external, it requires an external interpretation, that is to say, it is necessary to provide semantics.
Its meaning is conventional. It can be changed by convention.
Its meaning is specific to a particular domain or context.
It is not self-sufficient. Its value is conditional. It depends on the relationship with other signs.
It is expressible, explicable.
A symbol transcends the sign:
It is a simple graphic representation, that is to say, it is made with one element or a few elements.
It does not require external interpretation, since they refer to themselves (they are self-referential).
It is not conventional. Its meaning is absolute.
It is deep and synthetic.
Its scope is universal. It covers all domains.
It is not conditional. It is self-sufficient.
It evokes an archetype of the collective unconscious.
It is intuitive. A symbol appeals to intuition and totality. Symbols are components of the language of intuition.
It is used by mythical languages.
It is ineffable. It is not susceptible of description. To speak of the symbol is to degrade it, to impoverish it because it is to bring to the surface what is profound.
Some Milestones of Mathematical Notation
The digits 0 to 9, and the decimal representation system, are of Indian origin, but were spread by the Arabs.
Simon Stevin (1548-1620) is considered the father of negative numbers, since he was the first to accept them as the result of algebraic equations. The horizontal bar was already used by Fibonacci in the 13th century, although it was not generalized until the 16th century.
Francois Viète (1540-1603) introduced modern algebraic notation. He was the first to represent the constants of an equation by letters.
Descartes (1596-1650) used the initial letters of the alphabet to indicate known quantities, and the final letters (x, y, z) to designate variables. Descartes also introduced the (two-dimensional) exponential notation xn. In computer science, the notation x^n, with "^" being the circumflex accent.
Robert Recorde (1510-1558) invented the "=" sign in his 1557 work "The whetstone of witte".
The sign "+" (abbreviation of the Latin copulative conjunction "et", "and") was first used by Nicole Oresme (1323-1382) in "Proportionum Algorismus".
The signs "+" and "−" were first used by Johannes Widmann (1499-1545) in "Aritmetica Mercantil" to refer to surpluses and deficits in commercial problems.
To Thomas Harriot (1560-1621) we owe the signs "<" and ">".
The symbol "√" was introduced by Christoph Rudolff, reminiscent of a lowercase "r", the initial of the Latin term "radix" (radical).
In 1631, William Oughtred introduced the multiplication sign (×), St. Andrew's cross. Leibniz did not like this symbol because it was confused with the x. He preferred simply a dot. In the end it has been imposed not to include any sign, as when we write xy to indicate "x for y". In computer languages, the symbol "*" (asterisk) is often used.
The infinity symbol (∞) was introduced by John Wallis in 1655. 40 years later, Bernouilli named it "lemniscus", lemniscata.
In 1659, Johann Heinrich Rahn invented the sign "÷" for division. It only became widespread in England and the USA.
In 1652, William Oughtred used the Greek letter π to indicate the perimeter of a circumference. In 1706, William Jones gave it its present meaning.
In 1694, Leibniz invented the symbol ":" for division.
In 1777, Euler used the symbol "i" to refer to the imaginary unit.
The productorio symbol (∏, π capital π for "product") came into use as early as 1812.
In 1849, De Morgan introduced the slash (/) for division.
In 1879, Frege invented the concepts of universal and existential quantifier. Modernly, the symbols ∀ and ∃ are used.
The symbol "⊂" (inclusion) was introduced by Ernst Schröder in 1890.
The summation (∑) is due to Euler in 1895.
The symbol "∈" is the stylized Greek letter epsilon. It is the initial of the word "element". It was invented by Peano in 1895.
The symbol "≡" is used for equivalence and also for congruence (it was used by Gauss).
The symbol for integral (∫) was used by Leibniz, who took it from the initial of the Latin word "summa".
The symbols of set theory were introduced by Cantor. They are very expressive.
The sign "e" is due to Euler. It is not known if it is the initial of his name or if it comes from exponential.
The tilde (∼) for logical negation was used by Peano in 1897.
The arrow symbol (→) for logical implication was first used in 1922 by David Hilbert.
The symbol ∅ for the empty set was invented by the Bourbaki group.
The symbol Φ for the golden ratio comes from the initial of the Greek sculptor Phidias, as he used this ratio in his works.
The origin of the logical connectives (∧ and ∨) is not known. The first appeared in Alfred Tarski's "Introduction to Logic" (1940). The second appeared in "Principia Mathematica" (1910) by Russell and Whitehead.
Signs and Symbols in MENTAL
In the syntax of MENTAL primitives, no keywords are used. Instead, what we can call "sign-symbols" are used:
They are signs because they require an interpretation.
On the other hand, signs are also intended to be symbols, i.e., to be sufficiently intuitive so that they express semantics by themselves.
Since the universal semantic primitives are of a deep, synthetic and intuitive type, we should use pure symbols. But this goal is not possible to achieve, so we need an interpretation that helps to internalize the sign-symbols used. The same thing happens with derivatives.
The sign-symbols are of two types:
Delimiters. They are parentheses placed at the ends of an expression.
Operators. They act next to an argument (monadic operators) or between two arguments (dyadic operators).
Anyway, as already mentioned, this syntax should be considered only a proposal, and the user can modify or adapt it as he likes, even using keywords.
Table of primitives
The table of primitives with their signs-symbols used is as follows:
N°
Primitive
Syntax
1
Parameterized Generalization
⟨...⟩(1)
Non-Parameterized Generalization
⟨...⟩
2
Qualitative Particularization
/
Quantitative Particularization
\
3
Parallel Grouping (Set)
{...}
Series Grouping (Sequence)
(...)
4
Full Distribution
[…[…]…]
Linear Distribution
[…⌊…⌋…]
5
Potential Substitution
=:
Actual Substitution
=
Initial Substitution
:=
6
Equivalence
≡
Contrary Equivalence
≡'
7
Evaluation
No Operator
No Evaluation
°
8
Addition
+
Subtraction
−
9
Condition
←
Contrary Condition
←' or →
10
Upward Navigation
↑
Downward Navigation
↓
11
Start Execution
!
Finish Execution
¡
12
Continue Process
▶
Stop Process
■
(1) Parameters in bold
The "contrary" meta-operator
In the above table there is an operator acting on an operator, which is the contrary meta-operator ('). The equivalences are as follows:
(↓' ≡ ↑) (+' ≡ −) (!' ≡ ¡)
(■' ≡ ▶) (→' ≡ ←)
Addenda
Some quotes about symbols and signs
"Sign is that which by knowing it makes us know something else" (Peirce).
"We are an undeciphered symbol" (Hölderlin).
"The symbol is the primitive expression of the unconscious" (Jung).
"Only by means of the symbol can the unconscious be reached and expressed" (Jung).
"The whole of nature can be understood as a symbol of supernatural reality" (René Guénon).
"Symbolism is a means to elevate us to the knowledge of divine truths" (René Guénon).
"The true language of the world is that of symbols" (René Guénon).
"Philosophically, symbolism is considered 'the new key' of philosophy" (S.K. Langer).
"The symbol has a meaning that transcends the symbolized object: that is why it is not a mere formality" (Tobias Dantzig).
"Everything is symbol" (Leon Bloy).
"Every language is a tradition, every word a shared symbol" (Borges).
"Every ideal language must be based on the concept of 'expression,' a representation of signs and symbols" (Alfred North Whitehead).
"Circular symmetry is a metaphor of itself, a symbol of perfection, even of divinity" (Jorge Wagensberg).
"Numbers are symbols of divine realities" (Paul Twitchell).
"No symbol has meaning by itself but only in relation to other signs" (Ferdinand de Saussure).
"We only think in signs" (Jacques Derrida).
"Everything is sign" (Jacques Derrida).
"Every message is made of signs" (Ferdinand de Saussure).
"Every variable is a sign of a formal concept" (Wittgenstein. Tractatus, 4.1271).
"The simple signs employed in the proposition are called names (Wittgenstein. Tractatus, 3.202).
"Objects I can only name. Signs represent them" (Wittgenstein. Tractatus, 3.221).
"The sign is the part of the symbol perceptible by the senses" (Wittgenstein. Tractatus, 3.32).
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