"When the solution is simple, God is answering" (Einstein).
"God always takes the simplest path" (Einstein).
"The mathematical activity of man is a spiritual activity" (Luitzen Egbertus Jan Brower).
"To feel the world as a limited whole is the mystical" (Wittgenstein, Tractatus 6.45).
Mathematics and Spirituality
Mathematics is the rational science par excellence, since it allows us to draw conclusions from logical rules. However, mathematics has also been associated with subjects or aspects superior to reason: with intuition, the ideal, the perfect, the pure, the eternal, the immutable, consciousness, the imaginary, the transcendent, the infinite, supreme knowledge, the metaphysical and even mysticism.
Mathematics internalizes us, takes us to the depths of things, to their essence, to the maximum abstraction. To know mathematics is an initiation to a superior and transcendent world. The mathematical world is considered the closest to the spiritual world. God manifests Himself in the mathematical laws that govern everything, and the understanding of those laws, and even simply the exercise of mathematics brings us closer to God. God is beyond reason; he is something prior to and superior to reason.
God draws the universe with a compass. Bible Moraliseé (c. 1250)
Here are some opinions in this regard:
"Mathematics is the link between God and the world" (St. Augustine).
"God created the integers, everything else is the work of man" (Kronecker).
"Step by step, by the ladder of mathematics, the soul makes the sublime ascent to the highest orbs of heaven" (Marsilio Ficino).
"Mathematics is one of the quintessential human activities that make us fully human and thereby lead us to transcendence" (Donal O'Shea).
"With mathematics a transcendence is achieved from a given state of vision to a new vision beyond appearance." (Geoge Spencer Brown).
"All geometries are ultimately manifestations of the sacred order" (José Argüelles).
"Every geometrical appearance contains a metaphysical meaning" (Pythagoras).
"Mathematics is the science of infinity" (Hermann Weyl).
"Mathematics elevates the human mind to a closer proximity to the divine than can be attained by any other means" (Herman Weyl).
"It is impossible to do anything in mathematics, even in the most elementary arithmetic, without implicitly referring to infinity" (Rebecca Goldstein).
"Mathematics is the logic of imagination" (Leibniz).
"Number is in another space, not in this space" (George Spencer Brown).
"Mathematics is not an end in itself, without a key to a world beyond the reach of ordinary description" (George Spencer Brown).
"Mathematics is a bridge between relative and transcendent states of consciousness" (Franklin Merrel-Wolff).
For René Guénon, the true function of mathematics is to serve as a support for a higher consciousness.
For Brower, mathematics is closer to the realm of art, beauty and mysticism than to the realm of the sciences. "The edifice of intuitionistic mathematics is an art, not a science" (Brower).
Charles Hermite believed that mathematical existence was a kind of supernatural state to which mortal mathematicians could only fleetingly catch a glimpse.
Descartes believed that the existence of God could be demonstrated by rational thought, that is, by logic, the essence of which is mathematical. And he actually tried to do so. His attempt he reflected in "Metaphysical Meditations, Meditation III".
Grothendieck linked metamathematics with mysticism, which he reflected in his autobiography "Harvests and Sowings". What there is no doubt is that Grothendieck had deeply intuitive and creative ideas, which led to the creation of new mathematical concepts.
The Mandelbrot set (The Buddha).
However, modern society has trivialized mathematics, has given it a profane, superficial meaning (to count, measure, calculate, relate, etc.), ignoring its true essence which is sacred, even mystical, a vehicle of union with the higher and transcendent. Today, this fundamental spiritual aspect has been completely forgotten. But still today a small number who have not forgotten this aspect.
Pitágoras
Pythagoras −the first to call himself a philosopher− is the figure who best embodies the relationship between mathematics and spirituality, because for him, mathematics was more than a science, it was fundamentally a spiritual activity, a way of approaching the divine, the eternal and immutable. Because of his high level achieved, many considered him a mediator between God and men.
Pythagoras in "The School of Athens" (Raphael).
For Pythagoras, mathematics has an initiatory character, since they are in a higher dimension than the observable empirical world, and their knowledge elevates the human being spiritually. For Pythagoras, numbers have a mystical character and constitute the essence of reality. Numbers reveal the unfolding of the unmanifested unity in the multiplicity of manifestations. Pythagoras was more interested in the pattern, the profound, rather than the superficial.
For example, we cannot "see" the true number 37, that is, the number at the deep level, beyond its concrete manifestations. Nor can we "see" the true and perfect equilateral triangle, beyond its imperfect concrete manifestations. Mathematics is "pure" and its study elevates us above the imperfections of the physical world. Numbers and geometric figures are universal, and their truth can only be revealed through pure thought, untainted by the physical senses.
Pythagoras created a philosophical-mystical school of initiatory character, which had several peculiarities:
Its members were vegetarians, practiced the community of goods and led an ascetic, almost spiritual life. Each member was obliged to spend a period of five years in contemplation, maintaining complete silence.
They believed in the transmigration of souls. Souls are immortal entities that undergo a process of successive incarnations, a process that is overcome by purification (catharsis), culminating in the return of the soul to its place of origin.
They held secret meetings and rituals. The teaching was structured in levels. It was necessary to accredit to be prepared to accede to a superior level. The Pythagorean teachings were divided into arithmetic, musical harmony, geometry and astronomy.
It was not allowed to record the teachings in writing. No one violated this rule until after the death of Pythagoras and the dispersion of the Pythagorean school.
His philosophy was universalistic and was based on the motto "Number is everything," i.e., number is the foundation of the universe, the principle of all things. Everything could be explained by the intrinsic properties of numbers. Numbers possessed magical and transcendent properties. Physical bodies had geometry, and behind the geometrical designs were numbers.
The Pythagoreans discovered that harmony in music corresponded to simple relationships between numbers.
The Pythagorean Tetraktys
The Tetraktys is a triangular figure consisting of 10 points placed in four rows of 1, 2, 3 and 4 points, respectively. The Pythagoreans considered it a mystical, sacred structure, symbolizing the order, harmony and perfection of the universe, the symbol of universal manifestation, of the cosmic plan in the unfolding of its infinite possibilities. For the Pythagoreans, 10 was considered the most sacred number, for it symbolized totality, the source of the eternal and universal, the beginning of all things, the supreme knowledge.
The Pythagorean Tetraktys
The 4 rows of the Tetraktys symbolize, respectively:
The Unity (monad), the divine, the origin of all that exists, the unmanifest.
The Dada, dualism, the splitting of the Unity. It also symbolizes the feminine principle.
The Triad, which transcends the opposites and participates in both Unity and duality. It also symbolizes the masculine principle.
The Quaternary, which symbolizes harmony and the universe as a manifestation of the 4 elements (earth, air, fire and water).
The set of all the above constitutes the Decade, the totality of the universe and the return to unity: 1+2+3+4 = 10 = 1+0 = 1.
Plato
Plato was influenced by the ideas of Pythagoras and the Pythagoreans. He was persuaded that the universe is essentially mathematical and that the study of mathematics was the key to knowledge. Plato believed that all matter was based on different combinations of triangular atoms. In the Timaeus he explains how the different kinds of triangles combine to form the 4 elements.
Mathematical ideas are the fundamental ideas, the most important of all, since they constitute the essence of reality.
Mathematics elevates the spirit towards the abstract and towards truth in order to escape from the sensible realm. It elevates the soul from sensible things to intelligible truth, cognizable by the rational way.
Mathematics leads us beyond ordinary existence. And it shows us the underlying structure of all creation. Mathematics is a key to a transcendent world.
Mathematics is connected with the divine: "God always does geometry". We must try to discover the laws that govern the universe in order to approach the supreme reality of God.
The fundamental objective of mathematics is the contemplation of Being.
In the depths of the world of ideas, in its fundamental core, is the mathematical world.
The physical world is a projection or manifestation of the mathematical world. Therefore, there is a correspondence between the laws of nature and those of mathematics. Physical laws are mathematical.
Mathematical enlightenment" is the highest state of mathematical Platonism, a state of mental coherence, unified, of maximum consciousness, where everything is contemplated as a unity and where everything is connected.
Mathematical Platonism
Also called "mathematical realism", it is the application of Platonic idealism to the mathematical world. This term was first coined by Paul Bernays [1935], to refer to the doctrine that mathematical concepts have an objective reality independent of the cognitive subject.
The principles of mathematical realism, as it is conceived today, are as follows:
Existencia. Mathematical objects exist, they are real.
Abstracción. Mathematical objects are abstract. They do not belong to the physical world. Therefore, they have no spatial location.
Independencia. Mathematical objects exist by themselves, and are independent of the physical world and human beings.
Eternidad. Mathematical objects are eternal or timeless. They have existed since the beginning of time and before humans first perceived them.
Inalterabilidad. Mathematical objects cannot be modified or destroyed.
Descubrimiento. Mathematical truths are not created, they are discovered. They are not constructions of the mind. We discover mathematical entities that already exist.
Apriorismo. Mathematical truth is a priori knowledge, prior to all experience.
Intuición. Intuition is the faculty that allows us to know mathematical reality.
Perfection and order. The mathematical world is a perfect and ordered world.
According to Davis and Hersh [1982], Platonism is the faith of (almost) all mathematicians, but that it is an occult religion exercised in private.
Ramanujan and mathematical enlightenment
India's Srinivasa Ramanujan, one of the greatest geniuses of all time, is perhaps the most representative example of mathematical enlightenment. He had a very powerful mathematical intuition. According to him, mathematical concepts, theorems and formulas were transmitted to him in dreams by the Hindu goddess Namagiri. Ramanujan, as soon as he woke up, wrote down everything that the goddess had revealed to him in his dreams. This goddess is venerated especially in the Namakkal district of the Indian state of Tamil Nadu, in the south of India, and is often represented as the lotus flower. The name "Namagiri" translated from Sanskrit into Tamil sounds like "Namakkal". Namagiri is revered as the consort of Narasimha, an incarnation of Vishnu.
One of his great contributions was an elliptic modular function, named "Ramanajan" in his honor, which forms the basis of modern string theory in quantum physics. According to this theory, there are more than four dimensions (three spatial and one temporal). According to M-theory (from "mother") there are 12 dimensions (11 spatial and 1 temporal). According to the F theory (from "father"), there are 10 spatial and 2 temporal dimensions. The different modes of vibration of the string produce the whole variety of elementary particles.
Transcendental Mathematical Topics
In mathematics there are transcendental themes, which go beyond reason or defy reason, and which link with the higher or spiritual. Among these themes, we can highlight the following:
The infinite.
The infinite has always been associated with God, with the indefinable and indescribable. It cannot be expressed, but it can be intuited. Plotinus, the founder of Neoplatonism, associated infinity with God (the One), and that meditating on infinity brought a certain knowledge of God.
Infinity defies logic. Galileo said that there were as many even numbers as natural numbers. Zeno's paradoxes or antinomies led to conclusions outside of logic (Achilles never catches up with the turtle).
For the ancient Greeks, the infinite was called "apeiron" (that which cannot be defined, that which has no limits, the indeterminate), and was based on 4 main ideas: 1) The non-rational aspect; 2) The unlimited, the indeterminate, that which has no form or quality; 3) The impossibility of defining and describing the infinite; 4) The ultimate principle of all things.
The great promoter of the subject of infinity was Cantor. He proved that there are infinite types of infinity. To avoid trouble with the Catholic Church, he used the term "transfinite."
0 and 1.
Leibniz discovered (in 1679) the power and mystical elegance of binary arithmetic, since with only two digits (0 and 1) and their combinatorial power it was possible to construct the entire universe. The 1 represents God, being, existence. And the 0 represents the void, the non-existent: Omnibus ex nihil ducendis sufficit unum (One is sufficient to derive everything from nothing). For Leibniz, 0 and 1 constitute the essence of reality.
The freedom of creation.
For Cantor, the essence of mathematics is freedom. It is the freedom to create or define new mathematical entities.
Among the possibilities of freedom is the possibility of assigning names to mathematical entities. The action of naming has been considered a creative act that has been associated with existence and individuality. "To name is to have individuality" (Nikolai Luzin). "When I name an object with a word I affirm its existence" (Andrei Bely).
The idea that a name is something transcendent is very old and goes back to Plato's Cratylus. And the idea of naming as a creative act has a long history in religious and mythological thought. It is said that Ptah, the creator god of Egyptian mythology, created by naming with his tongue what his thought conceived. According to Genesis "God said 'Let there be light' and there was light". Naming and creating were the same thing.
The natural numbers.
Numbers are the symbols and paradigms of divine principles. Numbers contain and manifest the laws of God's universe. According to Kronecker "God created the integers, everything else is the work of man". Natural numbers are the origin of mathematics.
MENTAL and Spirituality
MENTAL is a fundamentally mental language, but it transcends the mental because the primary archetypes are present in all things. It is not a spiritual language, for if it were, it would be a contradiction, since the spiritual is a higher dimension than the mental. Nevertheless, it is an approach to the spiritual world and presents certain parallels, analogies or connections with the spiritual world:
Hierarchy.
MENTAL is based on universal principles that are at the top of the hierarchy of primary concepts. Immutable principles that underlie all others, which follow the principle of descending causality. This principle is of a spiritual type, since God is the principle from which everything emanates and sustains everything.
Simplicity.
The principle of economy or simplicity of MENTAL is a universal principle that manifests itself at all levels of reality. It is also a spiritual principle, for simplicity is associated with truth and consciousness.
According to the parable of the mustard seed, the mustard is the simplest and smallest of seeds, but when it grows it manifests itself as a large tree that unites Heaven and Earth and on which the birds, the intermediaries between the two worlds, perch.
Truth.
MENTAL is the truth, the true reality, which resides in the deep. It is the necessary, the universal, the common to all possible worlds, the immutable and eternal. The superficial is not the true reality. They are only manifestations of the deep, of the true reality. Truth implies beauty, harmony, order, perfection and consciousness. Truth is simple and is found in the deep.
"Truth is simple. Why doesn't everyone know it? Because it is too simple. Because we have our preconceived notions about what truth is or what it should be" (Harold Klemp).
"Truth is not in the heights but at the bottom of all things" (Paul Twitchell).
"What is true is simple" (René Mey).
"Truth is the whole" (Hegel).
Ineffability.
MENTAL, as such, is ineffable, inexpressible. Only its manifestations, concrete expressions, can be expressed.
"What is the truth?" Pilate asked Jesus (John 18:38). And Jesus remained silent because truth is inexpressible.
"The Tao that can be expressed is not the eternal Tao" (Lao-Tse).
Consciousness.
MENTAL is a language of consciousness because it is based on archetypes of consciousness and on the union of opposites: mathematics and metamathematics, the rational and the intuitive, science and humanism, the inexpressible and the expressible, etc.
"In God is found the coincidence of opposites" (Nicholas of Cusa).
"Consciousness is linguistic" (Vigotsky).
Imagination.
Imagination is a faculty of the soul and is always present in all thought. One cannot think without imagining. For Jung, archetypes are the categories of imagination. In MENTAL we can speak of "abstract imagination".
In MENTAL, imagination is present in every concrete expression and especially in imaginary expressions.
"Imagination is the language of the soul" (Jung).
"Imagination is more important than knowledge. Knowledge is limited to what we know and understand, while imagination encompasses the whole world" (Einstein).
Intuition.
MENTAL is both an intuitive and a rational language. But intuition is what is fundamental, what brings us closer to the soul.
"Intuitions are messages from the soul" (Gary Zukav and Harold Klemp).
"Ultimate reality is a vital impulse comprehensible only by intuition" (Bergson).
"I have faith in my intuition" (Einstein).
"I am definitely on the side of intuition" (Gregory Chaitin).
"Intuition is the clear conception of everything at once" (Johann Lavater).
"The object of mathematical rigor is to sanction and legitimize the conquest of intuition, and there is no other object for it" (Jacques Hadamard).
"Ultimate reality is a vital impulse comprehensible only by intuition" (Bergson).
"Mathematics is a free creation, independent of experience, developing from a single Primordial Intuition a priori" (Brower).
Totality.
MENTAL provides a unifying point of view, from which the essence of reality and totality is contemplated. It is reminiscent of the vision of the mystics.
"The only way to catch the totality of life is through the spiritual senses" (Paul Twitchell).
Depth.
MENTAL is a deep language with superficial manifestations. One must first look for the deep. From that place the superficial is easily contemplated, because everything superficial is a manifestation of the deep. MENTAL connects us with the deep to see the universal in the particular. MENTAL is a higher level of consciousness from which problems are diluted, resolved or clarified. Spiritual masters also insist on this theme: from a higher level of consciousness, everything is resolved or clarified.
"Seek first the kingdom of God and his righteousness and all else will be added to you" (Matthew 6:33).
Possibility.
MENTAL is a language of possible worlds. Primitives are the foundation of all possibilities.
According to Leibniz, all possible worlds exist in the understanding of God. And from the infinite possibilities he had, he selected this world as the best of the possible worlds: the most perfect, the simplest in hypothesis and the richest in phenomena. It is his principle of economy: the "maximin". It is also the most balanced world between variety and homogeneity, and the world where the combinations are the best possible.
Freedom.
MENTAL is a language whose dimensions or degrees of freedom are the primary archetypes. From the depths, where truth dwells, one experiences freedom, the world of all possibilities. Being in several places at once, instantaneous transfer to another place, etc. are common characteristics of the spiritual world and the mental world.
With the primary archetypes one can create concrete expressions and define new concepts that relate the archetypes to each other. The expressions do not refer to themselves, but to the deep indefinable archetypes. To use language is to create mental entities or to access an entity that already exists in a higher realm. "The truth shall set you free" (John 8:32).
Transcendence.
MENTAL transcends physical space and time. Space and time are united and abstract, where relationships between expressions can be local and non-local. It transcends mathematics itself. Mathematics is a manifestation of MENTAL.
The particular-universal relation.
The spiritual appears by connecting or relating the limited and the unlimited, the particular and the universal. "To feel the world as a limited whole is the mystical" (Wittgenstein, Tractatus 6.45).
Metalanguage.
MENTAL is the "mother" language of all particular formal languages. MENTAL is a language and also a metalanguage.
At Pentecost, the Spirit, manifested as tongues of fire, rested on the heads of the disciples and they began to speak in many languages. If you know the mother tongue you know all languages.
Paradoxes.
MENTAL unites opposites, where paradoxes occur, and also resolves paradoxes because it harmonizes opposites.
"Paradoxism is one of the common characteristics of mysticism" (Paul Twitchell).
"From paradox comes wisdom" (Masahiro Mori).
Infinity.
Infinity is associated with the limitless and the divine.
In MENTAL, infinite expressions can be described. For example, numerical infinity is expressible by a recursive expression: (∞ =: ∞+1).
Life.
Life is the manifestation of soul and consciousness in a body.
MENTAL can create "abstract life", a mental approach to life.
MENTAL expressions are inert, they have no life. What gives them life is the evaluation/execution engine.
"Archetypes come to life only when we try to discover their meaning" (Jung).
The law of karma.
The law of economy is connected with the law of cause and effect (or law of karma). We must act from our inner or deep "I", from the self, where truth, simplicity, maximum consciousness, freedom, harmony and creativity reign, in order to achieve maximum effectiveness with minimum effort. If this is not the case, one must pay the consequences: karma is generated and consciousness is lowered.
Nature is sustained by the law of love. When your actions are motivated by love, you are connecting with the depths of nature, with the self, then you act without effort, without waste of energy and without fear of any challenge. When your actions are motivated by power and control of people, you are strengthening the ego and wasting your energy and generating karma.
Symbolism.
MENTAL is a symbolic language.
"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).
Creativity.
MENTAL is a creative language, a reflection of the creativity of the soul.
"The soul is creative, infinitely creative" (Harold Klemp).
"What is creativity but another way of naming Spirit" (Ken Wilber).
Unity.
One of the characteristics of spiritual experience is to experience the profound, where the unity of all things is contemplated. Higher knowledge is achieved through a state of ineffable oneness with a transcendent reality.
MENTAL is union: of the primitives and their duals. MENTAL brings us closer to the unity of all things. MENTAL brings us closer to that experience of unity.
"God is before all things and in him all things hold together" (Colossians 1:18).
MENTAL produces an enlightenment of a mental and intuitive kind, a state of consciousness in which all things are seen as manifestations of the same deep and universal principles, where everything is contemplated as a unity.
With MENTAL, boundaries are blurred, for it offers a privileged point of view where everything is seen as manifestations of the same archetypes or principles that govern all things, a place where everything is connected.
With MENTAL the boundaries are blurred. It is the end of the "egos" of particular sciences and domains. As Alan Wolf says, "any object that possesses a boundary will possess an ego."
"The ultimate metaphysical secret, if we dare to put it that simply, is that there are no boundaries in the universe. Borders are illusions, products not of reality but of the way we represent and edit reality. And while it is fine to represent territory, it is fatal to confuse the two" (Ken Wilber. Consciousness Without Borders).
In short, MENTAL is not a spiritual language, but it helps us to better understand the spiritual world, and to contact through intuition a higher, transcendent realm.
Addenda
Ray Kurzweil's "Spiritual Machines"
Ray Kurzweil [1999] believes that, in the future, spiritual machines will exist. In principle, a machine belongs to a physical level, so it can never be spiritual; it would be a contradiction in terms. To say "spiritual machines" is an oxymoron, it is something like saying "honest thief". A machine can only bring us closer to the mental world.
It is not possible to create machines with consciousness, but it is possible to create machines based on archetypes of consciousness to create the maximum possible AI (artificial intelligence): between strong AI (the possibility of building an artificial mind) and weak AI (the simulation of the mind). The maximum possible approximation is when that machine uses primary archetypes.
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