Divine Mathematics and Number Symbolism: Ramanujan Formulas and Theological Insights

In this manifestation, God himself successively returns to the creative-revelatory process, according to what is indicated in Proverbs 8:27: "When he established the heavens, I was there; when he drew a circle on the surface of the deep." The unique, incomprehensible and unnameable Conscious Power, in order to reveal itself, conceals itself, creating the All, circumscribing a zone of its being from which to withdraw, in order to then return with its creative and revelatory act. This circumscribed zone from which the Power withdraws is a region totally devoid of power and of any manifestable spiritual characteristic, of any determinable quality, of any activatable essence. It absolutely is not, because the divine Being, the truly existent, has concealed itself from it. This zone remains as a circumscribed spatial fluid. Adynamic, because devoid of all dynamics and all life. It is nothingness, zero. Being, the first determination of the One, that is, of Him who has in himself the source of Being, is the first garment of the Creative Power and is that by which beings have Being. Theologically, 1 symbolizes, therefore, the truly Existent. It is the first Center of gravity of the Spirit-Truth, which, perennially, renews its manifestation of Man and the Universe. The serious prejudice of those who hastily approach Shivaite formulations of Traditional Metaphysics consists in depotentiating the role of Divine Unity, thought in an a-generative fixity, almost like a place of mere initial conditions, which exhausts in this conditionality its unique reason for being, and in raising a supposed Self, universalizable through a limit passage prepared by a ritualized chain of actions and reactions activated with the assimilation of a supposed traditional science, possibly inaccessible and controlled by very distant and unknown initiates. We will certainly not deny the magical value of initiatory ritual, provided it is put into action by a pure heart, nor do we uncritically accept the stifling conditions of truth and verifiability dear to rampant profanity, both in the West and in the East. However, we consider incompatible the understanding of the symbolism of 0, of 1 and of $\infty$ with any pantheistic drift inherent in such alleged paths of liberation. Clarifications on zero and one "Serves to write many Numbers" - identity by sum and identity by multiplication. $k^0 = 1$ or rather, 1 is a projection of zero. Pi and natural numbers Those who find themselves constrained in a quantitative vision of Mathematics have great difficulty in understanding the profound meaning of some formulas of contemporary mathematics. In particular those that involve divergent series and products, which show apparently absurd values if approached according to a restricted quantitative mentality, but which, instead, not only have a meaning in pure mathematical theory, but are also penetrated into the reality made visible by Mathematical Physics. One of these results, about which we will certainly write in one of the next issues, is the well-known Ramanujan sum, relative to the triangular of infinity $\triangle(\infty)$, i.e. $$1 + 2 + 3 + 4 + 5 + \ldots = -\frac{1}{12} \quad (6)$$ This formula relative to the sum of all natural numbers clashes incredibly with the quantitative concept of numbers, to which many, incapable of going beyond the opportune limitations of elementary teaching, are anchored. We will dedicate a long article to formula (6) and therefore we will not dwell on it here. Here we want instead to focus on another formula, which decidedly deserves our attention, namely that relative, not to the sum of Natural Numbers, but to their product. The formula we want to become the basis for meditation among students of number symbolism is the following: $$1^2 \times 2^2 \times 3^2 \times 4^2 \times 5^2 \times \ldots = 2\pi \quad (7)$$ Brief hints at the demonstration of the formula Since the formula in question does not fall within the usual university training paths, we also want to give a hint of how this result is reached, which otherwise might appear somewhat arbitrary. The starting point is given by the additive property of the logarithm, thanks to which we have the opportunity to transform a product into a sum, through the formal equivalence: $$\log \left(\prod_{n=1}^{\infty} a_n\right) = \sum_{n=1}^{\infty} \log(a_n) \quad (8)$$ We now use a classical relation of differentiation: $\frac{d}{ds}(a^{-s}) = a^{-s}\log(a)$, and we can rewrite the previous in the following form: $$\sum_{n=1}^{\infty} \log(a_n) = -\frac{d}{ds}\sum_{n=1}^{\infty} a_n^{-s}\bigg|_{s=0} \quad (9)$$ and therefore from (9) we have that: $$\prod_{n=1}^{\infty} a_n = \exp(-\zeta_a'(0)) \quad (10)$$ where the function $\zeta$ is the generalized Riemann zeta function given by: $$\zeta_a(s) = \sum_{n=1}^{\infty} a_n^{-s} \quad (11)$$ The (10) constitutes the definition of regularized product through the zeta function. This new definition allows us to calculate the result of the product in many divergent cases. For example, Riemann had already discovered that: $$\zeta'(0) = -\frac{1}{2}\log(2\pi) \quad (12)$$ where $\zeta(s)$ is the very famous Riemann zeta function: $$\zeta(s) = 1 + 2^{-s} + 3^{-s} + 4^{-s} + \ldots \quad (13)$$ Applying (12) to (10), we obtain the admirable property: $$1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdots = \sqrt{2\pi} \quad (14)$$ In the case of regularized products the following operational properties hold: $$\prod_{n \in I \sqcup J} a_n = \prod_{n \in I} a_n \prod_{n \in J} a_n \quad (15)$$ $$\prod_{n \in I} a_n^k = \left(\prod_{n \in I} a_n\right)^k \quad (16)$$ $$\prod_{n \in I} \tau a_n = \tau^{\zeta_a(0)} \prod_{n \in I} a_n \quad (17)$$ which allow us to make (14) even more suggestive, i.e. $$1^2 \times 2^2 \times 3^2 \times 4^2 \times 5^2 \times \ldots = 2\pi \quad (18)$$ Analysis of the formula The synthetic character of Mathematics makes the discursive analysis of a general mathematical identity like (18) necessarily incomplete. In general, we can always symbolically apply a mathematical identity on three fundamental states: the state of individual being (microcosm), the state of being in general (macrocosm) and then there is the real metaphysics proper, which considers everything in its universality. Here we are interested in the analysis of (18) in its macrocosmic aspect. If we refer to this plane, the individual Numbers represent qualities or ideas and their square represents their phenomenal expression or materialization. The square in fact expresses in itself the idea of expansion as well as that of well-ordered manifestation. According to the doctrine of Tzim-Tzum, the Absolute, when it wanted to reveal itself, circumscribed a place from which to withdraw, in order to then return with the creative act through its uncreated Divine Energies. The product of squares that appears on the left of (18), from a geometric point of view, represents a hypervolume of an infinite-dimensional parallelepiped having as face the square of individual numbers. At the same time, from a symbolic point of view it represents the indefinite expression and phenomenal production given by the action of Numbers in the Cosmos. The individual qualities of Numbers manifest and materialize through individual squares and interact with each other through an indefinite product. On the other side of the identity we have $2\pi$ which is nothing other than the perimeter of the circumference. So if on one hand we have the infinite-dimensional volumetric realization given by the product, on the other we have the simple perimetric measure relative to an act of circumscription operated by Number 1. In this case the circumference represents the frontier, the limit, beyond which lies the true Infinite. The act of circumscription by 1, through which the Transcendent Being withdraws creating a portion of void in which to return with the creative act, is equivalent to the insertion of the indefinite manifestation of the qualities and powers that will be explicated and manifested in the phenomenal world. Insights The formula we have presented here opens the door to various formulas little investigated by academic mathematics and until now also by esoteric mathematics. We intend in upcoming articles to treat more thoroughly the question of divergent series, which together with divergent products constitute a still unexplored treasure of mathematical symbolism. However, anticipating a little, we want to present here some notable results. In the previous issue of Mathesis (cf. the article "The 3 and 4 in Judeo-Christian angelology" by Zadiq) it was said that, from a symbolic point of view, there are 3 fundamental binary operations that we must consider: sum, multiplication and exponentiation. We agree with Zadiq and are happy to present here for the first time for the use of students of Number Symbolism the result of the interaction of all Natural Numbers through these three operations: $$1 + 2 + 3 + 4 + 5 + \ldots = -\frac{1}{12} \quad (19)$$ $$1 \times 2 \times 3 \times 4 \times 5 \times \ldots = \sqrt{2\pi} \quad (20)$$ $$2^{3^{4^{5^{\ldots}}}} = \frac{2}{\sqrt{\pi}} \cdot 2 \quad (21)$$ Equation 21 is easily obtainable from 14 once the properties of the exponential are considered for which $a^{bc} = a^{b \cdot c}$. Clearly we have preferred to substitute the formula $1^{2^{3^{4^{\ldots}}}} = 1$ with the more significant $2^{3^{4^{5^{\ldots}}}} = \frac{2}{\sqrt{\pi}} \cdot 2$. The first would have been as misleading as writing $0 \times 1 \times 2 \times 3 \times 4 \times 5 \times \ldots = 0$. Finally we want to highlight how through the notable Archimedean approximation $\pi \approx \frac{22}{7}$ we have a significant approximation of the last equation, i.e. $$\frac{2}{\sqrt{\pi}} \cdot 2 \approx 2\sqrt{\frac{11}{7}}$$ We have wanted to highlight this approximation through the use of the ratio $\frac{11}{7}$ for the symbolic importance that this has and to which we will certainly dedicate a future article. Unity-totality and the birth of numbers All Numbers derive from the One ⊙ and all Numbers are contained in it. For this reason the One is also called "darkness", because it contains seminally all other Numbers. The passage from the One to the Numbers is equivalent to the passage from the divine essence, which absolutely "is" incommunicable, indivisible, inexperienceable and unparticipable, to its attributes, really distinct from the essence, and which are participable and communicable to creatures. Just as in the theological sphere the essence of the three Persons in God is considered the principle or cause from which the divine attributes or qualities emanate externally, in the same way the One is the principle or cause of Numbers from which all other Numbers emanate. The birth of Numbers The birth of Numbers does not occur through the insertion into the One of an element external to It, but through the explication of the qualities that live unified in it. The One is the unified Number and as such it still contains unexpressed all Numbers, which still live in a state of indetermination or "mixture". From this state, through a mode of interior polarization of the original One, all Numbers originate. Symbolically the explication of the One 1 occurs through its elevation to the square, or rather its projection. An evident example is obtained by considering the following chromatic explication: $$1111111112 = 12345678987654321 \quad (22)$$ For the perplexed, we note that symbolic equations similar to the previous one are found for any positional system, not only in the decimal one, and they effectively reside in the property of the One 1 of being an additive generator of any $(Z_n, +)$ and in particular of $Z_{10}$. In formula (22) we have the repetition of the One 1 nine times, nine interior centers, invisible, inscrutable, unexpressed, incommunicable because wrapped in the mystery of the divine essence, but which, once projected, emanated...