3, to the synthesis from which every development has departed. Finally, exponentiation and the nth root are two modes of an external projection to the considered agent and an internal reabsorption of something previously externalized. Thus, for example, $2^2 = 4$ expresses a projection or externalization of Dualism in Quaternary form, while $\sqrt{4} = 2$ constitutes its reabsorption and return to the original Dualism.
Reduction, Triangular and Decomposition
Alongside the previous binary operations, some classic unary operations are added: reduction, the triangular of a number, and prime factorization. The reduction of a Number serves to reduce any Number to one of the first 9 Numbers. It applies to Numbers of at least two digits and consists in summing the digits, repeating this operation until obtaining a Number between 1 and 9. For example $\oplus(111) = 1 + 1 + 1 = 3$.
The triangular of a Number instead consists in the sum of all natural numbers up to that Number. The triangular of a Number highlights a connection between two natural Numbers where one is the unfolding of the other. For example $\triangle(4) = 1 + 2 + 3 + 4 = 10$ indicates Number 10 as the logical development contained in Number 4.
Finally, since multiplication corresponds to the development of a specific principle, the decomposition into prime factors of a Number is a very useful tool for determining the numerical family to which a Number belongs. Prime Numbers, in this sense, represent family heads, from which entire numerical groups depart. For example, to the Ternary group belong 3, 6, 12 and 24, while to the Quinary group belong 5, 10, 20 etc. For the study of the proper characteristics of a Number, the decomposition into prime factors that compose it is therefore of primary importance for determining the numerical families to which it belongs.
How Many Numbers Are There?
How many numbers are there? From earliest childhood we believe we can answer this question by saying "infinite." This answer, in its deceptive simplicity, expresses the idea that there are more numbers than it is possible to count. But if from a quantitative point of view, such an answer may even be justifiable, from a qualitative point of view such an answer is not only useless, but also misleading. After all, are we not accustomed to qualitatively classify the immense spectrum of colors into only Seven 7 fundamental colors? Are we not accustomed to reducing all the infinite possible musical note frequencies to only the Twelve 12 notes of the chromatic scale, or even to only the Seven 7 notes of the natural diatonic scale? Is it therefore possible to reduce the indefinite qualitative numerical nuances to a finite number?
To this question various traditions have answered affirmatively by proposing a different number that would be sufficiently complete to classify all numbers, but sufficiently small to still be easily manageable. The Pythagoreans, for example, indicated Number 4 and Number 10 as possible representatives of a complete qualitative cycle. The Hebrew tradition refers to Number 10 with its sephiroth or "enumerations" of which 7 are of "construction." The Iranian Zoroastrian tradition refers to Number 7 identified with the Unique 1 solar Verb Ahura Mazda together with his 6 Spirits, the Amesha Spenta. The Christian tradition instead uses Number 12, chosen by Christ himself for the 12 Apostles, in analogy with the 12 Zodiacal Signs, etc. As far as we are concerned, we will refer to the Christian and archeosophical Tradition, which we consider the most complete.
Beyond the justifications of traditional character, there is also a mathematical demonstration that indicates the adequacy of Number 12 in representing the qualitative totality of Numbers. This consists in considering the divergent summation of Natural Numbers $\mathbb{N}$, i.e. $1 + 2 + 3 + 4 + \ldots$. Such summation, according to the usual summation criteria typical of Real Numbers $\mathbb{R}$, leads to an indefinite result indicated with the symbol $\infty$. However, if we extend the domain of definition to Complex Numbers $\mathbb{C}$, it is possible, with simple analytical techniques, to reduce this summation to a specific finite number notoriously identified by Ramanujan, i.e. $1 + 2 + 3 + 4 + 5 + \ldots = -\frac{1}{12}$.
Analyzing the previous formula it is possible to reduce the expression of all Natural Numbers $\mathbb{N}$ to the ratio between Unity 1 and Twelve 12, which is thus characterized as "number of a perfect cycle." Thus, in the first twelve Numbers we can summarize the qualitative nuances of all other Numbers, even if, obviously, each of these maintains its own particularities.
Figurative Numbers and Evocative Numbers
Numbers, like all traditional symbols, operate on different planes of the Cosmos. Number 3 for example is an uncreated Number when it refers to the Divine Trinity, is always uncreated but relative to the order of emanations if it refers to divine Will, Wisdom and Intelligence, is created if it refers to Spirit, Soul and Eros, is personal if it refers to word, idea and action, etc. A Number can therefore simultaneously embrace all these planes linking one to another, or can refer to only one of these, which for simplicity we will limit to the divine, cosmic and individual.
In numerous talismans and pentacles, the Number, present through a geometric structure, is used as a ladder, surrounded by various Names or symbols harmonious among themselves that refer to various planes or orders of existence, thus allowing the ascent of the ascetic or the descent of the Spirit.
Keeping in mind the previous concepts, Numbers are divided into figurative and evocative. The figurative ones represent a definite conception, while the evocative ones evoke a conception that the number by itself does not express. For example, the One 1 evokes the ensemble conceived in its Unity, whatever it may be, abstract or concrete reality. From a practical point of view, the Circle, which can represent the One, must contain another symbol of the idea to be evoked, for example, a Name, a sign, a constellation of symbols. In this case, as an evocative number, it evokes its conception as a whole. In the absence of a specific symbol, of a polarization or other, Number One 1 does not represent a well-defined conception because it is not figurative. The same happens for Number Two 2. While different is the situation of Number 3 which instead is both evocative and figurative, that is mixed.
The evocative Numbers, or the mixed ones like 3, 5 and 9, are functional for evoking an idea, but this must be made explicit in some form: for example by repeating a formula 3 times as in the case of "Holy, Holy, Holy"; or through 3 hammer blows before or after an invocation formula; or through the repetition of an action 3 times; or through the use of 3 candles on an altar; or through a blessing gesture formed by 3 fingers; or through a triangle with something drawn in the center; etc.
Arithmetic Analysis of the First 12 Numbers
Below we present a simple arithmetic and geometric analysis of the first 12 Numbers. For each Number we will preface in italics a small passage taken from Palamidessi's text "The spirituality of Sacred Numbers," necessary to understand some aspects of the Number in question, and only afterwards will we present some lines of commentary and the arithmetic analysis of the Number.
1
"The 1 is not divisible; it is the number of God."
Number One is the origin of Numbers. It is the number of God. In the cosmos it is the Sun because the Word sits in the Spirit of the Sun. In Man it represents the Spirit and also the Self because it is the origin of all faculties.
The Pythagoreans called it "Simplicity," that is, what cannot be divided, but also "Intelligence," because it reunites all things in itself, as well as "Being" or "Cause of Truth." Other names given to Number 1 are those of "Ship" or "Chariot" because it "hosts everything," but also "Friend," because it reunites, "Life" because it is the source of life and "Happiness" because in it one is at home. When it is called "Darkness," "Mixture" and "Gloom" it means the fact that in it all Numbers are present, but not yet explicated, as in a kind of mixture, in a state still undifferentiated. It is the seed that contains everything. All numbers come from the One, in Him all Numbers are contained.
The graphic symbols of 1 One can be: the Circle, or the 1 One in a receptive or generative aspect; the Point, or the 1 One in a projective or creative aspect; the Circle with the Point, or the 1 One in a complete and total sense; the Line in the idea of reunion of opposites. To the numerical One corresponds the philosophical Monad and the geometric Point. This correspondence realizes the identity between Arithmetic, Geometry and Philosophy.
2
"The 2 symbolizes the Old and New Testament."
With two we have differentiation, distinction. It highlights a double moment of Unity: the external and internal, exotericism and esotericism, the old and the new, the profane and the sacred, the passive and active, the receptive and