Universal Flow and Evolution Lines in Mathematical Physics
If a being $E$ had been found in $q_1$ at time 0 instead of in $p$, then at time $t_2$ such being would have had to be found in $q$, that is $L_{t_2}(L_{t_1}(p)) = L_{t_1+t_2}(p)$. This is a symbolization of the law of concatenation of effects. Therefore, fixing $p$, the uniqueness of the evolutionary line demonstrates that there is a unique maximal evolutionary line $L_p: T_p \to D$ that realizes the influence $X$ at all its points and passes through $p$. What we have meditated upon regarding point $p$ is extendable to any other point $p'$ of $D$, where the meditation is supported by the totality of a domain over which the influence $X$ acts entirely. In the fully realized Metaphysical Personality, such distinct punctual instantiations define, in correspondence with each point $p'$ of $D$, a global datum $T_{p'}$. Through the uniqueness of the Universal Intellect, there is simultaneous integration of such global data $T_p$ into a single foundational act, which we shall call $T$. The nature of $T$ is what regulates the action and global integration of influence $X$ in metaphysical Persons. In $T$ there is the integration in itself of all global instantiations $T_p$ but still localized in various and different points $p$ of $D$. We recall that a being $E$ manifested at point $p$ on which influence $X(p)$ is exercised evolves into a state $q$ and metaphysically there is the law $E_p: T_p \to D$. The general integration of possibilities given by $X$, from the point of view of substance, manifests as a general flow obtained by joining together all laws $E_p$. From the point of view of essence, there is a unique law $L$ that instantiates as the union of laws $L_t: D \to D$ of instantaneous change. In mathematical physics, $T$ is often time. The metaphysical integration in its essential reality dominates manifestation in $D$ through the law of Universal Flow: $$\Phi_X((t,p)) = L_p(t) = E_t(p)$$ The Flow is the law that links the evolution of a being $E$, which evolves from a state $p$ to one $q$ realized in $t$: $q = E_p(t)$, with the activation of state $q$ in the series of states of $D$: $q = L_t(p)$. We use the Greek letter $\Phi$ to denote the flow law and put $X$ to remind that we have the law in the case of a single influence $X: D \to TD$. The flow law is founded on the law of the identical insofar as there is the initial identity, in symbols $\Phi_X(0,p) = p$, which expresses the potential identity of $D$ in the absence of change. The flow law manifests, even for the case where one considers a single influence $X: D \to TD$, the law of universal concatenation: in symbols $\Phi_X(p, t_1 + t_2) = \Phi_X(\Phi_X(p, t_1), t_2)$. If domain $D$ symbolizes a complete integrality of states of existence, comprehensive of the indefinite prolongations inherent to it, then the flow law assumes a particularly suitable form as support for meditation. In this case, the simultaneous coexistence of prolongations of states of existence and their completeness permits symbolizing the integrated completeness of $X$ in the universal personality. In this case all global data localized at $p$ are in absolute essential simultaneity coincident with the entire $T$. Having $D$ in itself its indefinite prolongations, it follows that the Flow law becomes $\Phi: T \times D \to D$. In the case of a one-dimensional continuous indefinity for $T$, it is possible to take the field of all real numbers $\mathbb{R}$. We shall see in other work the symbolization of the case where $T$ can be taken as the field of complex numbers or another complete field. Setting $T = \mathbb{R}$ and fixing $p \in D$, the entire evolutionary line $t \mapsto \Phi_X(t,p) = E_p(t)$ resurges in us, which can be visualized as a curve effectively traversed within domain $D$. We note that by differentiation: $$\frac{d}{dt}\Phi_X(t,p)\bigg|_{t=0} = X(p)$$ We recover the influence $X$ at point $p$. Knowledge of the Flow Law therefore corresponds to the total intelligibility of the action of $X$ on $D$. The great law of flow can be synthesized in the following metaphysical affirmation: every acting influence determines a unique evolutionary line starting from every state of existence of the considered domain. The collection of all possible evolutionary lines is the manifestation of a single great flow in the domain. There is then a second law: an influence acting on a domain complete in its indefinite prolongations is such that all evolutionary lines are indefinitely prolongable. If we meditate with greater intensity on what we have explicated, we realize that an evolutionary line is realized through a juxtaposition of infinitesimal linear displacements where the direction and length of each displacement is governed by influence $X$ at the point corresponding to the arrival point of the infinitesimal displacement. This intuition of a succession of small rectilinear displacements is an initial base for meditation that comes...