Mathematical Series and Symbolic Properties of Numbers
Consider the difference of reciprocals of two consecutive numbers: $\frac{1}{n(n+1)} = \frac{1}{n} - \frac{1}{n+1}$. Therefore we have $\sum_{n=1}^{\infty} \frac{1}{n} = \sum_{n=1}^{\infty} \frac{1}{n(n+1)} = \sum_{n=1}^{\infty} \left(\frac{1}{n} - \frac{1}{n+1}\right)$. Since the series is convergent, we can rearrange terms obtaining: $\sum_{n=1}^{\infty} \left(\frac{1}{n} - \frac{1}{n+1}\right) = \left(\frac{1}{1} - \frac{1}{2}\right) + \left(\frac{1}{2} - \frac{1}{3}\right) + \left(\frac{1}{3} - \frac{1}{4}\right) + \cdots = 1 + \left(-\frac{1}{2} + \frac{1}{2}\right) + \left(-\frac{1}{3} + \frac{1}{3}\right) + \left(-\frac{1}{4} + \frac{1}{4}\right) + \cdots = 1$.
Some Symbolic Meanings of the First 12 Numbers (Niko)
How many numbers are there? Since our earliest childhood, we believe we can answer with the word "infinite". This answer, in its deceptive simplicity, expresses the idea that there exist more numbers than all humans could ever count. While from a quantitative viewpoint, seen from the pole of substance, the previous answer goes in the right direction, from a symbolic perspective that takes more into account the pole of essence, it is justified to seek to symbolize the properties of numbers by meditating principally on a few Fundamental Numbers.
We should not be surprised by this statement: after all, are we not accustomed to qualitatively classifying the immense spectrum of colors into 7 fundamental colors? Are we not accustomed to reducing all the infinite possible frequencies or musical notes to only the 12 notes of the chromatic scale, or even to only the 7 of the natural diatonic scale?
Before beginning a study on Fundamental Numbers, let us concentrate our mind on the series of numbers in its entirety. The series of all natural numbers is an actual infinity, which, when realized in a single intuitive act, represents the foundation of what modern mathematicians call the set of all natural numbers and which they denote with the symbol $\mathbb{N}$. Such a view from above must nonetheless be able to account for what is operatively done with numbers, even in daily life, and which, if properly meditated upon, would allow even the most quotidian of our actions to be reconnected to eternal principles.
The property that is immediately grasped by our elementary intuition about number is the operation that consists in adding unity to a given number. The splitting $1 + 1 = 2$, and then all the successive acts that make us pass from one number to its immediate successor, show the most immediately visible characteristic of the series of numbers in its entirety, when viewed from the side of substance and, in particular, in its merely quantitative aspect.
Nevertheless, the series of numbers can be symbolized in two ways, which represent two distinct supports for metaphysical meditation. The first way is centered on the pole of essence. In using this way, one keeps alive in the intellect that light which made us realize that what everything originates from is Metaphysical Zero. In this path, Unity is originated from a determination of Metaphysical Zero. Particularly in this form of meditation it is necessary to admit two faces in Unity: that turned toward non-manifestation and that turned toward manifestation (this theological scandal is the essence of all metaphysics). In this symbolic path the generative operation is $0 + 1 = 1$.
Another symbolic path concentrates on the pole of substance without keeping track of the metaphysical origin of Unity. It is a path of inferior metaphysical level because in it the manifested unity constitutes and exhausts the entire speculative horizon. In particular, the number zero is not realized in it.
Nevertheless, in both paths it is possible to generate the entire series of remaining numbers by means of the same operation of passage to successor, which is the symbol of the action of essence upon substance. We will not develop here the two distinct methods of meditation and will limit ourselves to what they have in common.
Let us denote the action of passage to the successive number with the sign $S: \mathbb{N} \to \mathbb{N}$ where, for every element $n$ of the set $\mathbb{N}$, we understand: $S(n) := n + 1$. We note that the operation $S: \mathbb{N} \to \mathbb{N}$, at every step, generates a number different from that from which we started. Furthermore, given two distinct numbers $n$ and $m$, it never happens that $S(n) = S(m)$. An action that satisfies this property is called an injective function.
An important intuition is that which makes us see that the action $S: \mathbb{N} \to \mathbb{N}$ exhausts the entire series of positive numbers. This latter property, mathematically, is called the principle of induction. This principle of exhaustion of the entire series of numbers is of very simple formulation only when one has already grasped, with a single intuitive act, the entire series of numbers in its actual infinity. The formulation of the principle of induction is the following: if $A$ is a part of the series of all numbers (in formal language, if $A$ is a subset of $\mathbb{N}$) such that zero is an element of it and for every other element $n$ of it we have...