Mathematical Symbolism and the Unity of Numbers

$2 + 2 = (2)$; $3 + 3 + 3 = (3)$; $4 + 4 + 4 + 4 = (4)$; $5 + 5 + 5 + 5 + 5 = (5)$; $6 + 6 + 6 + 6 + 6 + 6 = (6)$; $7 + 7 + 7 + 7 + 7 + 7 + 7 = (7)$; $8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 = (8)$; $9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 = (9)$: $\forall n \in \mathbb{N}$, $n^2 = 2\Delta(n)$ where $\Delta(n)$ denotes the triangular number, this implies that $n^2 = 1 + 2 + \cdots + (n-1) + n + (n-1) + \cdots + 2 + 1$, $\forall n \in \mathbb{N}$ Therefore, Number itself is not subject to any evolution; its force and spiritual reality remains unchanged as absolute, uncreated, free and therefore hierarchically superior to the Cosmos. Subject to becoming, instead, is the manifestation of 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 Unity. We have said every Number derives from Unity. In other terms, every Number has a relationship with Unity 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 an expression of a spiritual characteristic of Unity 1. Numbers, as a whole, constitute the Crown of Unity 1 and therefore have value only in function of this. At the base of Numbers lies Unity 1, or rather the idea of the Unity of Numbers. Numbers exist because they are united among themselves, or rather exist by virtue of the presence of Unity 1, while in the absence of this they would be disconnected from each other, devoid 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, each 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 played by the study of the reciprocal of a Number, or rather 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 sequence of these reciprocals is called the harmonic sequence. For example, the reciprocal of number 2 will be $\frac{1}{2}$, and of any number $n$ will be $\frac{1}{n}$: $$1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \ldots \quad (2)$$ whose subsequences still play a considerable importance in the study of prime numbers. The sum of the reciprocals of all Numbers, also called the 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 captures the symbolic aspect of the harmonic action of Numbers in the Cosmos and which we can call the harmonic square. This is an infinite square that gathers all the powers, and thus all the manifestations, of Numbers in relation to unity, i.e.:

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

We have here a 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 countless forms and the infinite succession of phenomena do not deny 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 to our Friends and future collaborators a mathematical element of reflection and meditation. **Double Duality: on the symbolism of yin-yang (Pentadecade)** **Operational symbolism of yin-yang** The symbol of Yin-Yang is a support for meditating on the universal flux and its two currents: the descending one and the ascending one. The universal flux consists of a double movement. The pole that stands above originates a descending current, which activates the pole that stands below, in which, by reflection, the current transforms into an ascending return current, which rises transformed toward the pole that stands above. The geometric center of the circle and the circumference that delimits the contour of the entire flux symbolize