Mathematical Symbolism of the Cross and Initiation Theory

to $p \in D$ which measures the possibility or impossibility of symbolizing the overcoming of the law of universal Necessity. The non-uniqueness of the integral curve passing through a point $p \in D$ constitutes an indication of which geometric forms can better symbolize aspects that guide a being toward Liberation.

The mathematical theory of singular varieties is very vast. In it, the symbolism of the geometry of the Cone and that of the Cross find a natural place, and these geometric forms, in turn, constitute fundamental examples of it.

Before embarking on this meditation, it is appropriate to reflect on some aspects of initiation that we would like to always be present in the heart of those who approach to meditate, together with us and with us, on the symbolism of the Cross. We remember that the Cross, formed by a horizontal arm and a vertical one, identifies Initiation in the highest possible sense, in which human initiation, represented by the horizontal arm, meets the divine one, represented by the vertical arm.

Figure 12: The two branches of the cross in analytical geometry are given by the set $\{(x,y) \in \mathbb{R}^2 | xy = 0\}$.

The first aspect of initiation is the remarkable absence of elements proper to differential geometry associated with solutions of stationary-type equation systems that can well symbolize the spiritual event that goes under the name of initiation: there exist no stationary natural laws, such as those governed by systems of partial differential equations with unique solution and continuously dependent on initial data, that can be taken as symbolic support to orient oneself toward the heights on which a true initiatic path occurs.

The second aspect is given by the uniqueness of initiation and its absolute non-repeatability. The authentic initiatic act is not a rational, repeatable act, aimed at improving, perhaps through successive approximations, who knows what social performances or at making specific and egotistic individual interests mature. We know, on our part, that one of the most common criticisms against the Cross, or rather against initiatic realization, consists precisely in attributing strongly egotistic desires and intentions to those who seek initiation. On the contrary, whoever goes against the initiatic way and the initiatic domain, whether conscious of it or not, favors individualizing egoism that leads, in the most favorable case, to realizing a path constrained within the necessitating limits of the Law of Causality alone.

The third aspect of initiation is constituted by its social aspect, centered in a social network that manifests through individual encounters and knowledge that structure its visible aspects, the only ones measurable even by a profane person; the heart that does not tremble of initiation is the direct experience of a condensed Reality, super-exceeding the limits of the considered domain of existence: the social aspect is a necessary limitation, forced by the limiting conditions of existence that must be respected by one who manifests as a man of flesh and blood.

The fourth aspect of initiation is that it is one with full awareness, intimately already realized, of the radical uncertainty at the base of phenomenal existence. The radical uncertainty of existence is what men fear most. We can write here that for a pragmatic mind, scientific and technological progress is an attempt to domesticate the radical uncertainty of existence. This type of mentality expresses itself in research, sometimes very misleading, oriented toward studying every phenomenon according to scientific-mathematical laws of stationary nature, which can put Humanity in a position to use the Law of Causality. The example given by certain pretended applications of mathematics to economic science represents an example that would be comic, if it did not drip with the blood and pain of billions of human beings.

Unfortunately, this mental attitude is not expressed only in some sectors of academies evidently deprived of the Light that emanates from the Initiatic Way. The conciliatory and (pseudo) solidaristic laicism and atheism of much of modern mentality have become a complex of notions uncritically accepted even in many initiatic ways, which, in their perennial struggle to illuminate and keep illuminated the Way, have ended up mistaking means, useful and valid in their very restricted applicative spheres, for ends. The programmatic laicism with which the writings and speeches pronounced in some ways, with vain initiatic pretenses, are impregnated, has ended up feeding a morbid negation both of the radical difference existing between individual beings, and of the radical uncertainty on which every single existence is founded. Having realized in themselves the existence of incommensurable differences between individual beings and the founding nature of the radical uncertainty that envelops our life are the starting base of the Way. It is not surprising that those who strive to erase their intimate immanence within the heart are no longer able even to conceive the aspect or...