Mathematical Induction and Natural Numbers Symbolism

that also the successor of $n$, namely $S(n)$, belongs to $A$, then $A$ is exactly the entire series of numbers; in formulas $A = \mathbb{N}$. The principle of induction in metaphysics is nothing other than the mathematical translation of the exhaustion of the potentialities of substance by the full actuality of essence. The metaphysical zero in equilibrium with itself vibrates both in the pole of essence, whose symbol is the number zero, and in that of substance, whose symbol is the empty set. The action of essence on substance originates manifestation, whose symbol is the unfolded series of numbers. The symbol of the fullness of such action is the principle of induction. All properties of natural numbers, such as the operations of addition and multiplication, as well as the order relation among natural numbers, are logically obtainable starting from $0$, from the injective operation $S: \mathbb{N} \to \mathbb{N}$ which, not by chance, defines $1 = S(0)$ and for which the only number it cannot generate is exactly the number $0$, and from the following three assertions: 1. Zero is a number; 2. The action $S: \mathbb{N} \to \mathbb{N}$ is injective; 3. The action $S: \mathbb{N} \to \mathbb{N}$ satisfies the principle of induction. In a forthcoming work we will present the result of a symbolization relative to the deduction of number properties starting from $0$, from the action $S: \mathbb{N} \to \mathbb{N}$ and from the three preceding axioms. An initiate knows that natural numbers are not the only vibration that resonates with the idea of number and indeed it would be appropriate for them to meditate at length on numbers such as pi, $\pi$, or the Golden Ratio $\varphi$, which are not integers. However, it is on Natural Numbers that all other meditations are founded. We note, in particular, that the meaning of each Natural Number originates from that of Number One. It is appropriate here to remember that the digit that often expresses One is given by a vertical stroke. Such a digit lends itself, in truth, more to a speculation centered on the pole of essence than on that of substance; it is not by chance that the Chinese tradition reserves two characters for One. In fact, in it One is written with a single horizontal stroke, while in large writing the character that identifies it means the act of unifying, of being identical, of being truthful. The importance of distinguishing cipher, name, and number in its intimate essence is, on the other hand, one of the simplest but fundamental supports for a Pythagorean-type meditation. The idea of Unity encompasses all manifestation and founds the Uniqueness of the manifestation of each being, without this implying an absolute separateness of each being from another. All our elementary sensible and psychic experience necessitates this notion of uniqueness even just to be able to orient ourselves both in exterior and interior life. In other words, it is the generativity of the idea of Unity that is the force that nourishes all our discernment and, ultimately, all Life. Number One in absolute sense is the foundation of the idea of Unity. As Ideal Number, It lives also in informal manifestation and One, in theological sense, is its manifestation. It is therefore metaphysically founded the necessity of understanding numbers through their relationship with Number 1. We will delimit our symbolic investigation to the symbolism of the first twelve numbers. However, according to some traditional ways, Number $1000$, $1000 = 10^3$, that is the triple development of number $10$, is considered as the symbolic limit of the most complete development possible of the series of numbers. It is necessary to understand this last assertion. Certainly such limit, $1000$, may appear arbitrary, induced by the use of the decimal positional system. A priori we could have considered as symbolic limit $1728$, considering that $1728 = 12^3$; and therefore refer to a triple development of $12$ instead of $10$. In reality, what is important is to be able to symbolize with a series of numbers that is sufficiently large to have a true qualitative completeness but sufficiently small not to give rise to qualitative repetitions. Traditionally the magus knows which series to evoke in each specific circumstance. Numbers that have often been adopted are: $4$, $7$, $10$ and $12$. We want to specify that, with what is written above, we do not want to assert that Numbers greater, for example, than $1000$, do not have a traditional symbolic value. Numbers are infinite and each has its individual characteristic, as Ramanujan also had occasion to point out to Hardy, according to the famous anecdote of taxi numbered $1729$! What we want to make understood is that to use Numbers operationally and effectively according to their symbolic value it is not necessary to meditate on each of them individually, also because what is obtainable with such meditation is a value