Mathematical Metaphysics: Tangent Spaces and Evolutionary Lines

of points $p$ of $D$ which, by virtue of the restrictions placed on $D$, allow us to predict the evolution of a being $E$ along an evolutionary line effectively realized in $D$, when one of these influences is internally realized. To meditate effectively on what has just been asserted, it is necessary to make within ourselves a clear distinction between the points of $D$, the true and proper possible states of existence in $D$, and the virtualities that are present in $p$, by the very fact of considering such a state within the domain $D$ alone. The set of such virtualities in $p$ can be symbolized through a form that has contact with $D$ only at point $p$, but which otherwise evolves freely, in correspondence with the possible directions present in $D$; we will call this form $T_pD$. The notion of tangent space to a surface of our ordinary three-dimensional space is a possible initial basis of meditation to approach the form $T_pD$ that we are proposing here, but the symbolic possibilities opened by simultaneous meditation on $D$ and on the set of all virtualities present in the points of domain $D$, due to the fact that these points are considered, precisely, only in $D$, has a much deeper scope than such an initial meditative basis would suggest. Let us now perform a meditative act of unification of all the virtualities in $D$ present in all points of $D$. The first result of this single generative act is the generation of a new space that we will call $TD$ and which is simply the union of all forms $T_pD$; in the linguistic formalism with which the results of this interior generation are communicated, this result is denoted in the following way: $$TD := \bigcup_{p \in D} T_pD$$ Simultaneously with the generation of $TD$, the general action of $TD$ over $D$ is generated. Indeed, a virtuality in $p$, that is, an element of $T_pD$, is the datum of a pair $(p,v)$ where $p \in D$ is the point of $D$ in which the virtuality $v$ acts. Therefore, we can generate within ourselves the general action $\pi: TD \to D$ that links each element $(p,v)$ of $TD$ to the application point $p$ of the virtuality $v$; in the linguistic formalism usually used in current mathematical writing, this is written in the following way $(p,v) \mapsto p$ or: $\pi(p,v) \mapsto p$. Let us now see some contemplative possibilities that this symbolic generation allows us to realize. To integrate a set of virtualities and realize them effectively, we specialize our meditation to the case in which a specific influence $X$ is activated in a state $p \in D$. Such influence $X$ can be symbolized with a single virtuality $X(p)$ for each point $p$. In synthesis, we can collect all the virtualities animated by the influence $X$ through the law $X: p \mapsto X(p)$, which associates to each $p \in D$ the corresponding virtuality $X(p)$; in simplified language it is written: $$D \ni p \mapsto X(p) \in T_pD$$ It is important to emphasize that knowledge of the general action of influence $X$ allows us to predict, by integration, the evolutionary curve of a being $E$ initially living in state $p$ and subject to influence $X$, but it is also true that if one knows the evolutionary line of such being $E$ then it is possible to derive, by differentiation, the knowledge of $X$; the two great principles, the one underlying integration and the one underlying differentiation are unified, synthesized in the evolutionary line. It is in this synthesis that many predictions find their theoretical basis, both in the various sciences and in divinatory practices. A fundamental meditative step consists in visualizing simultaneously, in a single act of the mind, all the evolutionary lines associated with the field of influence $X$. It is as if in $D$ flow directions were polarized, a flow current, determined by the activation of influence $X$. Let us begin by following one of these flow lines. Given the form of $D$ that we have generated previously, there exists and is unique the evolutionary line that passes through $p$ associated with $X$. Let us now consider the entire family of modifications undergone by a being $E$ along its evolutionary line. Such modifications are states of $D$ and therefore it is metaphysically founded to symbolize them with points $q \in D$. In the metaphysical person who simultaneously realizes such states, where everything, in truth, is simultaneous, the family is a single instantaneous global presence. The manifestation of such global presence occurs through what we can symbolize as an additional dimension to be added to those present in domain $D$; in mathematical physics, for example, such additional dimension is often the one associated with the variable given by time. In the metaphysical reality of p...