"Every archetype is always an abstraction" (Pierre Janet).
"Archetypes are metaphors that drive us and serve as models" (Carlos Alberto Churba).
Numerical Archetypes
In mathematics there has always been a more or less conscious search for first principles, for primary or fundamental categories or archetypes. To found mathematics in this way is to do it from consciousness, with the principle of descending causality, that is, from the general or universal to the particular. These primary categories or archetypes must necessarily be of the highest level of abstraction and simplicity.
For physicists Pauli and Heisenberg, archetypal forces are "primary mathematical intuitions", and that the essential foundation of nature are mathematical abstractions of order and symmetry.
As we have mentioned above, historically there have been attempts to ground mathematics by considering the concepts of number, set, category, structure, and function. But these last three are not really archetypes because they are composite concepts; they are not pure, primary concepts. Therefore, let us consider sets and numbers.
Pure sets and sequences
"Pure" sets are those that are recursively constructed from the empty set as follows:
Although Set and Sequence are primary archetypes, pure sets and sequences are secondary archetypes because there is a pattern and not concrete elements, as in the concept of number.
Natural numbers as archetypes
The natural numbers were born from the harmonization or overcoming of duality. From 1 and 2, 3 was born, and with 3, the infinite natural numbers were born.
With the natural numbers all mathematical concepts were born, among them: arithmetic, zero, negative numbers, fractional numbers, real numbers, imaginary numbers, algebra (generalization of arithmetic), order, set, sequence, cardinality, symbols, variables, the decimal numbering system, differential and integral calculus, algorithm, function, the principle of induction, analytic geometry, matrix, vector, geometric algebra, etc. "God created the integers, everything else is the work of man" (Kronecker).
For Pythagoras, number is the beginning of all things, the essence and foundation of all that exists. And that to understand the universe it was necessary to delve into the secret of the harmony of numbers.
For Jung and Pauli, numbers are primary archetypes, so they should be part of the "neutral language" they both sought, to express the physical and the psychic. For Jung, numbers are archetypes that have become conscious.
Numbers are archetypes, intermediaries between the inner and the outer world, between the deep and the superficial. We cannot "see" numbers, but we can intuit them, like all archetypes. They reside in the deep and we can only see their manifestations in the physical world.
According to numerology, every number has a meaning, a vibration (like a musical note), an associated symbolism and evokes a state of consciousness.
A number is the deeper (or more synthetic) the more manifestations it has. A superficial number is more analytical (or more synthetic) and has fewer manifestations.
Among all natural numbers, the first ones are the most important, synthetic and deep:
0. This number is archetypal for two reasons. First, because it unites opposites, since every positive number has its opposite (the negative number), and both add up to zero. Secondly, because of its multiple manifestations in the various fields of mathematics, in philosophy and even in mysticism.
In ancient times, negative numbers were considered imaginary numbers. In this sense, zero connects the real with the imaginary.
0 symbolizes pure consciousness, the Self, transcendence, the absolute and the undifferentiated: emptiness and also completeness. Its symbol is a circle. In Eastern philosophy, zero symbolizes infinite possibilities. The 0 derives from the Hindu concept of sunya, which means "emptiness".
The 1. This number is an archetype because it is the origin and source of all numbers and at the same time is present in all of them. All numbers emanate from the 1. The 1 represents the essential unity of all things. For the Pythagoreans, the 1 was not really considered a number, but the source of all numbers. The 1 is the source of all numbers in a double sense: 1) because it is part of all numbers; 2) because every number is a number (2 is a number, 3 is a number, etc.).
The 1 is the divine number par excellence. Heraclitus identified it with God, Plotinus with universal intelligence, and the Yajur-Veda with the Self. "The Tao begets One. One begets Two. Two begets Three. Three begets the ten thousand things" (Tao Te King).
The further we move away from the 1, the deeper we go into the superficial and particular. And the closer we get to the 1, the closer we get to the profound. The 1 is the foundation and consciousness of all numbers. The 1 symbolizes consciousness because it is in all numbers. All numbers are connected to the deep through the 1.
The 1 is the opposite of nothingness. The 1 is the first manifestation of consciousness. And it only makes sense in relation to 0, to emptiness. The 0 and the 1 constitute the fundamental duality. When we perceive an object, we are implicitly connecting with the archetype 1. The 1 connects unity and diversity.
Every number has a quantitative and qualitative property, but in the 1 both properties coincide, they cannot be separated: the 1 is both quantitative and qualitative.
The 2 symbolizes duality, confrontation, division between opposing forces. After 1, the most important number is 2, since it is present in half of the numbers (the even numbers).
3 is the number of consciousness, the number that harmonizes and transcends the opposites, which considers them complementary from a higher level. It symbolizes Heaven, balance, peace and consciousness. It is represented by a triangle. In Freemasonry, it is the symbol of the triangle with the all-seeing eye.
The 4 symbolizes the Earth. It is the archetype of the manifested or created. It is represented as a square.
The 5 is the ether, the quintessence of the alchemists. It is symbolized by the 5-pointed star, in which the human figure is inscribed, making it the symbol of man. It refers to itself, as the golden ratio that is implicit in the 5-pointed star. It is the point of intersection of the cross.
Prime numbers (those that are divisible only by themselves and unity) are mathematical archetypes because they refer to unity, to themselves and to all numbers.
Prime numbers are deep or primitive numbers because they are part of all other numbers. Compound numbers (non-prime numbers) are shallower, more analytic, less synthetic numbers.
Prime numbers have brought into communication many areas of mathematics (number theory, geometry, analysis, logic, probability theory) and even physics (quantum physics).
The imaginary unity (i). According to Jung, it unifies conscious and unconscious, the inner and the outer world. The imaginary unit appears in many fields: geometry, electromagnetism, quantum physics, relativity, etc.
Perfect numbers and friendly numbers
A perfect number is a number that satisfies the property that the sum of its proper divisors (i.e., excluding itself) is the number itself. A perfect number is self-referential, it refers to itself. Pythagoras discovered that all perfect numbers are the result of the sum of consecutive numbers starting from one. The first perfect numbers are:
Two numbers are "friends" if the sum of the proper divisors of one is the other and vice versa. For example, 220 and 284 are friends because
220 = 1 + 2 + 4 + 5 + 10 + 10 + 11 +20 + 22 + 44 + 55 + 110 (sum of the divisors of 284)
284 = 1+2+4+71+142 (sum of the divisors of 220)
By this definition, a perfect number is a friend of itself.
The number 6 is the only number that is both perfect and factorial:
6 = 1+2+3 = 3! = 1·2·3
For the Pythagoreans, perfect numbers possess mystical properties. And so it is, because these numbers are more profound, as two properties converge in them: they are sum numbers and at the same time they are the sum of their proper divisors.
The factorials and the summands
The factorial of a natural number n (symbolized by n!) is the product of the first n numbers. The first factorials are:
Factorials can be considered "anti-primes" in the sense that they are the most shallow, as opposed to primes that are deep and essential.
Additive numbers are like factorial numbers, but with the operation of addition, rather than with that of product. They are triangular numbers, numbers that by their shape connect with geometry. The first addends are:
1
1+2 = 3
1+2+3 = 6
1+2+3+4 = 10 (this is the number of the Pythagorean Tetraktys)
1+2+3+4+5 = 15
1+2+3+4+5+6 = 21
1+2+3+4+5+6+7 = 28
The general formula is:
1+...+n = (1 + n)n/2
All perfect numbers are additive, i.e., triangular. A triangular number of n rows contains n triangular numbers:
1, 1+2, 1+2+3, ... , 1+...+n
It is a fractal or recursive number.
The scale of number consciousness
The traditional division of irrational numbers is: algebraic and transcendental, according to whether or not they are roots of polynomial equations with rational coefficients. But from the point of view of downward causality and archetypes, it is better to classify the real numbers into:
Conscious.
These are the natural numbers, the integers and the rational numbers. They are all expressible and computable by means of a finite number of arithmetic expressions. They are archetypes that have become conscious, as Jung says.
Unconscious.
They are the irrational numbers, which are inexpressible and incomputable. They cannot be expressed by a finite expression. The vast majority of real numbers are of this type. Conscious numbers are a tiny part.
The word irrational here has a double meaning: 1) an irrational number cannot be expressed as the ratio (division) between two integers; 2) an irrational number as not accessible by reason, but by intuition.
The Pythagoreans discovered irrational numbers and called them "alogos", i.e., inexpressible.
Level
Number Category
Conscious (expressible)
Naturals
Integers
Rationals
Conscious-Unconscious
Expressible Irrationals
Unconscious (inexpressible)
Inexpressible Irrationals
Between expressible and inexpressible numbers are numbers that are expressible (describable) by an infinite number of arithmetic expressions because they have a pattern, or are expressed by a recursive function (which implicitly implies infinity as well).
These numbers are not accessible (like all irrationals) but we can systematically and formally approximate them as much as we want (e.g., of the number have been calculated to the order of 1013 decimal places). They are second-order (or secondary) archetypes, since they are expressed by the primary archetypes of addition and subtraction and their arithmetic derivatives.
Examples of second order archetypes
The number π is an archetype because, besides being a transcendent number, it connects the linear and the circular.
In this formula, the sign of each term is only negative when the denominator of the fraction is a prime of the form 4k+1. If the number is composite, the signs of each prime are multiplied.
It can also be represented as continued fractions:
The number π appears, not only in geometry and arithmetic. It also appears in other fields, such as probability theory and quantum physics.
The number e, defined as the limit of (1+1/n)n when n tends to infinity.
It can also be expressed as a continued fraction: e = [2; 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 1, 6, 1, 1,.... ], of pattern 1, 2k, 1, being [a0; a1, a2, a3,. ..] the continued fraction
The number e appears in multiple fields: geometry, finance, population dynamics, growth processes, heat diffusion, radioactive decay, aerodynamics, music, etc.
In geometry it appears in the logarithmic spiral, which symbolizes the union of the internal and the external. It also appears in the catenary, the hanging chain, the shape we can observe in power lines. The catenary connects geometry and gravity.
The number e is the basis of neperian logarithms. Thanks to logarithms, products become additions, powers become products, which in turn become additions, and so on for higher-order arithmetic operations. Thus, logarithms link addition (in short, the number archetype) with its manifestations (the derived arithmetic operations).
The function ex is a function that refers to itself, since it is its own derivative.
The square root of 2. It is expressed recursively and as a continued fraction.
√2 = 1 + 1/(1 + √2)
In general, any square root of a number that is not a perfect square can be expressed in infinitely many ways as a recursive function or as a continued fraction. It follows from the following formulas:
√n = k + x n = k2 + x2 + 2nx x = (n − k2)/(x + 2k)
x = √n − k
√n = k + (n − k2)/(k + √n)
The number Φ, the golden ratio or golden number, is present in geometry (pentagon, pentacle, dodecahedron), in arithmetic (the Fibonacci series) and in the forms of nature.
Φ = 1 + 1/(1 + Φ)
Φ = √(1 + Φ)
Euler's formula
The formula that best represents the interrelationship between these numerical archetypes is the famous Euler formula:
exi = cos(x) + i·sin(x)
from which it follows: e2πi = 1 and eπi + 1 = 0
The latter expression has interesting properties:
It connects the 5 most important constants of mathematics: 0, 1, π, e, i.
Through these constants, he connects the 4 main branches of classical mathematics: arithmetic (0 and 1), algebra (i), geometry (π) and analysis (e).
Connect the 3 basic arithmetic operations: addition, product and power.
From Euler's formula the trigonometric functions sine and cosine are deduced, so trigometry is a consequence or derivation of the exponential function ex and the imaginary unit i:
The geometric point is an archetype because it has no extension and yet it is present in space and in all geometric figures, that is, it connects non-space with space, the unmanifest and the manifest. The point is an intermediate element between the continuous and the discrete.
The point is the "geometric zero". As the zero, it symbolizes undifferentiated consciousness, the absolute and the void. Its primary manifestations are circles whose center is the geometric point itself. A circle can be considered as a manifested point.
The analogy can be established: Zero - Point - Empty Set - Empty Sequence. They are 4 entities that share the property of "empty".
The point is inexpressible, it belongs to a transcendent realm of non-space and non-time. However, but it can be represented as a number or sequence of numbers. A point is represented in Cartesian space of n dimensions by n ordered numbers (sequence). It is an archetype (the point) represented by one or more other archetypes (the numbers).
The point has no extension, so it has no parts, just as zero is indivisible. A point is an address, not a content because it has no extension.
The triangle
The triangle is the most essential and archetypal geometric figure. All other figures can be constructed from the triangle, including the circumference (it is a set of isosceles triangles with an infinitely small side.
Other archetypal geometric figures are the circle, the square, the cross, the spiral, the sphere, the torus, and the Platonic solids. An important geometric archetype is the Vesica Pisces (two rings intertwined at their centers) and its derivatives: the Borromean knot (3 interconnected rings), the Seed of Life (7 rings) and the Flower of Life (19 rings). In general, archetypes are the forms of the so-called "sacred geometry", whose essential characteristics are simplicity and symmetry or the union of opposites.
Analytic Geometry: The Union of the Archetypes of Number and Point
Descartes unified algebra and geometry by creating analytic geometry, although it appears that Fermat also invented it at the same time. The key concept was that of the "coordinate system" (which today we call "Cartesian"), which assigned coordinates (numbers) to points in geometric space. This was a major evolutionary leap in mathematics and in science in general. The principle is so simple that it is hard to believe that it took 3,000 years to discover it. All great ideas are simple, but difficult to find because of their obviousness.
With the Cartesian system, a point in the plane is an ordered pair of numbers. And geometric elements become algebraic expressions. For example, a line is ax+by+c = 0 and a circle is x2+y2 = 0.
Neither Descartes nor Fermat really realized the importance of the discovery, for the goal they both sought was the systematization of Euclidean geometry. But what they achieved was a new foundation of mathematics based on algebra as a general language. Descartes "algebrized" geometry, so that a problem of geometry could be reduced to mere manipulations of algebra. Analytic geometry unified two branches of mathematics that had previously appeared radically different: geometry (characterized by the continuous) and algebra (characterized by the discrete).
The use of coordinates was not new, as they had been used throughout history [see Addendum]. Descartes' breakthrough was, not only the coordinate system, but the use of generic patterns in the form of algebraic expressions to represent geometric elements.
Analytic geometry can be considered a meta-archetype, since it connects the number archetype with the geometric archetype of point. The connection of both archetypes produces the ultimate creativity.
The Pythagorean (or geometric) sum
From the point of view of number consciousness, let us define the Pythagorean sum of two numbers a and b as the diagonal of the right triangle formed with the legs of length those numbers: √(a2+b2). And the Pythagorean subtraction of a and b as √(a2−b2). These operations connect arithmetic with geometry.
For example, the Pythagorean sum of 1 and 2 is √(12+22) = √5 (its arithmetic sum is 3). In the Egyptian triangle 3-4-5, the 5 is the Pythagorean sum of 3 and 4 (their arithmetic sum is 7). The 5 symbolizes the union of the 4 (the Earth) and the 3 (the Heaven) and is the symbol of man. According to Plato, man dwells in the realm of Metaxy, an intermediate world between the temporal physical world and the transcendent world of the Forms.
The Pythagorean sum is a sum that joins two numbers at the geometric level, one horizontal and the other vertical. It is a sum of consciousness, by joining two dual elements. The elementary Pythagorean sum is the diagonal of a square of side 1. Its value is √2. It is a fractal number, since it can be expressed as a function of itself.
Pythagorean addition is different from vector addition. Geometric addition operates with numbers and vector addition operates with vectors. However, there is a relationship between the two operations: the geometric sum of a and b is the modulus of the sum of the vectors (a, 0) and (0, b).
Addenda
The Pythagorean Tetraktys
The Pythagoreans used a very simple sacred symbol to represent totality, the principle of descending causality and the union of arithmetic and geometry: the Tetraktys ("fourfold" in Greek), a triangle formed by 10 points in 4 descending rows of 1, 2, 3 and 4 points. The name of this symbol was mentioned in their oath. According to the Pythagoreans, the sequence 1, 2, 3, 4 is the key to the universe.
Pythagorean Tetraktys
The 1 (the monad) is the origin and source of all things.
The 2 (the dyad) is the principle of duality. It symbolizes the straight line, with two ends (initial and final). Duality potentially exists in the 1.
The 3 (the triad) is the principle of balance and harmony of opposites. It symbolizes heaven and consciousness. It is represented by the triangle.
The 4 (the quaternary), symbolizes the material world and its 4 elements (fire, air, water and earth). It is represented by the tetrahedron.
The 10 is the sum of 1+2+3+4, the 4 numerical principles. The 10 symbolizes totality, the return to unity, closing the cycle, because 1+0 = 1.
In the Tetraktys are represented the first 4 triangular numbers: 1, 3, 6 and 10.
History of the use of coordinates
Egyptian surveyors probably used them.
Greek geometers, such as Hipparchus (200 BC), Menecmus (4th century BC) and Apollonius (262-190 BC) used coordinates.
Ptolemy (2nd century) used them in his maps.
The Romans divided their cities along two axes (east-west and north-south) and organized the streets with a rectangular coordinate system.
In the Middle Ages, Nicolas Oresme (14th century), Bishop of Lisieux, an original thinker and one of the founders of modern science, gave a remarkable impulse to coordinates. He had observed that curves could be defined by relationships between coordinates and obtained an equation form, but at that time algebra was not developed and failed to advance.
Fermat, in 1629, wrote some notes where he used coordinates to describe points and curves.
Descartes is believed to have been influenced by Oresme's ideas. In 1637, in his Discourse on Method, he included three appendices, one of them devoted to geometry, where he sets out his coordinate system that unified algebra and geometry.
According to these last two dates, Fermat was ahead of Descartes, so the "Cartesian system" could very well be called the "Fermatian system".
Bibliography
Aczel, Amir D. El cuaderno secreto de Descartes. Una historia verdadera sobre matemáticas, misticismo y el esfuerzo para entender el universo. Ediciones de Intervención Cultural, 2008.
Berlinski, David. Ascenso Infinito. Breve Historia de las Matemáticas. Debate, 2006.
Du Sautoy, Marcus. Simetría. Un viaje por los patrones de la naturaleza. Acantilado, 2009.
Ferrer, Eulalio. El lenguaje de las trilogías. Fondo de Cultura Económica, 2005.
Guénon, René. La gran triada. Obelisco, 1986.
Iambilichus. The Theology of Arithmetic. Red Wheel / Weiser, 1988.
Ifrah, Georges. Las cifras. Historia de una gran invención. Alianza Editorial, 1994.
Ifrah, Georges. Historia universal de las cifras. Espasa-Calpe, 2002.
Iglesias Janeiro, Jesús. La consciencia de los números. Kier, 1992.
Seife, Charles. Cero. La biografía de una idea peligrosa. Ellago Ediciones, 2006.
Schneider, Michael S. A Beginner´s Guide to Constructing the Universe. The Mathematical Archetypes of Nature, Art, and Science. Harper Collins, 1995.
