Mathematical Metaphysics: Tangent Spaces and Evolutionary Curves

True possible states of existence in $D$, and the virtualities that are present in $p$, due to the very fact of considering such state within the domain $D$ alone. The set of such virtualities in $p$ is symbolizable through a form that has contact with $D$ only at point $p$, but that for the rest evolves freely, in correspondence with the possible directions present in $D$; we will call such 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 the 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 what such first meditative basis lets us suppose. We now perform a meditative act of unification of all the virtualities in $D$ present in all points of $D$. The first result of such unique generative act is the generation of a new space that we will call $TD$ and that is simply the union of all the forms $T_pD$; in the linguistic formalism with which the results of this interior generation are communicated, such result is denoted in the following way: $$TD := \bigcup_{p \in D} T_pD$$ Simultaneously to the generation of $TD$, the general action of $TD$ over $D$ is generated. In effect, 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 in us 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: $\pi(p,v) \mapsto p$. We now see some contemplative possibilities that such symbolic generation permits to realize. To integrate a set of virtualities and effectively realize them, we specialize our meditation to the case in which in a state $p \in D$ a specific influence $X$ is activated. Such influence $X$ is symbolizable with a unique 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 one writes: $D \ni p \mapsto X(p) \in T_pD$. It is important to underline that the knowledge of the general action of the influence $X$ permits to predict, by integration, the evolutionary curve of a being $E$ initially living in state $p$ and subject to the 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, that underlying integration and that 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 to the field of influence $X$. It is as if in $D$ there were polarized directions of flow, a current of flow, determined by the activation of the influence $X$. We begin meanwhile to follow 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 to $X$. We now consider the whole family of the 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 that simultaneously realizes such states, where everything, in truth, is simultaneous, the family is a unique global instantaneous presence. The manifestation of such global presence happens through what we can symbolize as an additional dimension to add to those present in the domain $D$; in mathematical physics, for example, such additional dimension is often that associated to the variable given by time. In the metaphysical reality of the metaphysical person such global integrality is the immobile unity of all the changes that are traversed in apparent succession in the manifestation of $E$ in $D$. The mobile image of eternity is therefore the law $E_p: T_p \to D$ that to each instant $t \in T_p$ associates the point $q$ of the evolutionary line of $E$, or exactly that unique state in which a being would find itself living.