The Metaphysics of Numbers: Unity, Cosmos, and Harmonic Action

They appear not as distinct and fully explicit Numbers. The most attentive will have noticed the petitio principii of requiring a "number" of interior centers within the One in order to then explicate them externally to it. This from a symbolic point of view is absolutely coherent with the vision of Number that we expressed in the first volume of Mathesis. The Number as entity is transcendent, irreducible and indeducible (we will treat Peano's axioms in a separate article) and when therefore we deal with the "birth" of Numbers we only intend to describe their delivery from Unity and not the original insemination within it, for which there exists only one word: "Mystery," to be understood in a technical sense, namely of a truth that transcends the created intellect.

The action of Numbers in the Cosmos. We have symbolically represented the birth of the first nine Numbers through a squaring of the repeated One, now, if we want to analyze their action in the Cosmos, that is in space and time, we must again symbolically use a squaring that in this circumstance we denote with the symbol □ to highlight its geometric-symbolic aspect.

The entire doctrine of the Cosmos derives from Number 4, from the Tetraktys, source of inexhaustible Nature, which finds its analogue in the Tetragrammaton. Number 4 is the true guardian of Nature's secrets. Every reality in evolution, in space and time, can be ordered and classified through the "key" of Number 4. We thus have the four elements, the four seasons, the four aspects of the sun, the four cardinal points, the four ages of man, the four royal stars, the four guardians of heaven, the four angels that regulate the four universal powers, etc. Number 4 expresses the idea of Universality and manifestation.

Given these premises, the squaring of the first nine Numbers symbolizes the development of their action inserted in time, its manifestation. We have in fact the following relation valid for any natural number:

$$1^2 = 1 = 1 = □(1)$$ $$2^2 = 1+2+1 = 4 = □(2)$$ $$3^2 = 1+2+3+2+1 = 9 = □(3)$$ $$4^2 = 1+2+3+4+3+2+1 = 16 = □(4)$$ $$5^2 = 1+2+3+4+5+4+3+2+1 = 25 = □(5)$$ $$6^2 = 1+2+3+4+5+6+5+4+3+2+1 = 36 = □(6)$$ $$7^2 = 1+2+3+4+5+6+7+6+5+4+3+2+1 = 49 = □(7)$$ $$8^2 = 1+2+3+4+5+6+7+8+7+6+5+4+3+2+1 = 64 = □(8)$$ $$9^2 = 1+2+3+4+5+6+7+8+9+8+7+6+5+4+3+2+1 = 81 = □(9)$$

Symbolically, squaring hides within itself all possible expression of the number in time and space, therefore the action of the number in the Cosmos. Reading the preceding expressions, one might be led to believe that each number, inserted in time, temporarily loses its own individuality to acquire that of another number. This would be an erroneous interpretation: the sequential progression found in the preceding expression concerns only the phase of temporal development of the Number necessary for its full manifestation, it does not imply a change in the essence of the Number itself. The action of the Number in the cosmos does not change the Number itself which, in itself, remains immutable as the following numerical expressions demonstrate:

$$1 = □(1)$$ $$2 + 2 = □(2)$$ $$3 + 3 + 3 = □(3)$$ $$4 + 4 + 4 + 4 = □(4)$$ $$5 + 5 + 5 + 5 + 5 = □(5)$$ $$6 + 6 + 6 + 6 + 6 + 6 = □(6)$$ $$7 + 7 + 7 + 7 + 7 + 7 + 7 = □(7)$$ $$8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 = □(8)$$ $$9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 = □(9)$$

Therefore the Number in itself is not subject to any evolution; its force and spiritual reality remains unchanged insofar as it is absolute, uncreated, free and therefore hierarchically superior to the Cosmos. Subject to becoming, instead, is the manifestation of the Number in the cosmos, its shaping action on matter and consciousness. This can grow or diminish depending on ecological conditions.

The relationship of Numbers with the One. We have said "every Number derives from Unity." In other terms, every Number has a relationship with the One and, symbolically, this relationship with Unity is what maintains a Number as such, or rather the source of its force. To be more precise, every number is expression of a spiritual characteristic of the One 1. Numbers, in their totality constitute the Crown of the One 1 and therefore have value only in function of this.

At the base of Numbers there is the One 1 or rather the idea of the Unity of Numbers. Numbers exist because they are united among themselves, or rather they exist by virtue of the presence of the One 1, in the absence of this they would be disconnected from each other, deprived of any relation and therefore would remain as single empty shells. Conversely by virtue of their common origin and their substantial unity expressed by 1, behold that every Number is in relation to the other, indeed "is" the other by virtue of its co-presence in 1 and by virtue of the co-presence of 1 in all Numbers.

Given these premises it is evident how a privileged role in the study of symbolic mathematics is covered by the study of the reciprocal of a Number or rather of the relationship of 1 with the Number itself. The study of reciprocals of natural Numbers is very important in harmony, so much so that the succession of these reciprocals is called harmonic succession:

$$1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \ldots$$

whose sub-successions still cover considerable importance in the study of prime numbers.

The sum of reciprocals of all Numbers, also called harmonic series, is divergent, which means that the sum is not computable with traditional methods of summation (see our articles on the symbolism of divergent series). However, we can proceed with a construction that grasps the symbolic aspect of the harmonic action of Numbers in the Cosmos and which we can call harmonic square. This is an infinite square that gathers all powers and therefore all manifestations, of Numbers in relation to unity:

$$\begin{pmatrix} \frac{1}{2^2} & \frac{1}{2^3} & \frac{1}{2^4} & \frac{1}{2^5} & \frac{1}{2^6} & \cdots \\ \frac{1}{3^2} & \frac{1}{3^3} & \frac{1}{3^4} & \frac{1}{3^5} & \frac{1}{3^6} & \cdots \\ \frac{1}{4^2} & \frac{1}{4^3} & \frac{1}{4^4} & \frac{1}{4^5} & \frac{1}{4^6} & \cdots \\ \frac{1}{5^2} & \frac{1}{5^3} & \frac{1}{5^4} & \frac{1}{5^5} & \frac{1}{5^6} & \cdots \\ \frac{1}{6^2} & \frac{1}{6^3} & \frac{1}{6^4} & \frac{1}{6^5} & \frac{1}{6^6} & \cdots \\ \frac{1}{7^2} & \frac{1}{7^3} & \frac{1}{7^4} & \frac{1}{7^5} & \frac{1}{7^6} & \cdots \\ \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \end{pmatrix}$$

We have here a pure mathematical marvel. It is easy to demonstrate, and we will do so in an article in the next issue of Mathesis, that if we consider the sum of all these infinite terms we obtain nothing other than the number 1. Everything is expression of unity and included in unity. The innumerable forms and infinite succession of phenomena do not negate the unity of being, but affirm its spiritual fullness, expressing every quality and force to form a whole or rather the uni-totality. We will deepen the spiritual implications of this concept in the next Number. In the meantime it was our intention to leave our Friends and future collaborators a mathematical element for reflection and meditation.

Geometry without form