Mathematical Approaches to Organic vs Mechanical System Evolution

The evolution of organic systems (a plant, an animal) could not be satisfactorily studied except when the high number of individual elements did not allow neglecting individual consciousness and enabled mechanical modeling. In our opinion, this situation is of contingent origin, in other words: we believe that mathematical tools exist that allow the study to understand the evolution of organic systems, and even of self-conscious systems. But the necessary mathematical tools are in themselves useless if a profound epistemological change does not occur in the interpretation of such tools and especially in the questions to which answers are sought. To clarify the argument, it is therefore necessary to proceed progressively, for this reason we will define some founding traits of the concept of evolution and subsequently distinguish two types of system evolution: the evolution of organic systems and the evolution of mechanical systems. The definitions of mechanical and organic system used here are much broader and more abstract than those usually considered and do not relate to the composition of the system, but to whether or not the evolutionary principle belongs to the system itself. This much more abstract definition than usual is a practical version that, in our opinion, captures both exoterically and esoterically the distinctive node between the two systems.

Fundamental characteristics of evolution: By evolution of a dynamic system we will consider not only its temporal evolution, understood as the variation of configuration with the flow of time, but more properly its evolution in a philosophical and real sense as well as purely formal. The evolution of every system occurs in a determined finite time. To be more precise, in this context time is that reality that allows the evolution of a system. What is outside of time does not evolve, but is. Moreover, if time is measured in relation to what locally repeats itself always identical to itself, such as the periodic rising of the sun, the oscillation of a pendulum or the vibration of an atom, it assumes a value only in relation to what does not repeat, but changes, transforms and evolves.

If everything that exists in the Universe were periodic and presented itself exactly identical to itself, there would not truly be evolution, nor would it be possible to determine an effective flow of time due to the presence of infinite identical periods that precede the current one and infinite identical periods that succeed it. Conversely, within a periodic phenomenon taken as a unit of temporal measurement, we notice that in an evolving system there is something that changes and thus breaks the perfect circularity of time giving it a beginning, an end and a duration.

What evolves within time is therefore an evolving substance that, in addition to an evolutionary principle which is the efficient cause of evolution, requires at least three other terms:

A beginning, or initial state that represents the base state of the substance that evolves;
An end, or final state that represents the completion of evolution;
A duration, or finite time that elapses between the initial state and final state of evolution.

It is indeed clear that without an initial state or substance there can be no evolution, since the starting point of the subject of evolution is missing. Without an end or final state to reach, one cannot speak of evolution, but only of a more or less random transformation. Finally, for the term evolution to make sense, one must suppose that this evolutionary end is reachable in finite time and is not instead an unreachable asymptote. If, in fact, the evolutionary process were infinite, that is without end and never realizable, there would be no evolution and there would be no progress exactly as there would be no progress in having walked some thousands of steps along a road that has no end.

However, the main distinction we will make regarding the dynamic evolution of systems will be that given by the relationship between substance and evolutionary principle. In the case where the evolving substance and the evolutionary principle are an integral part of the system itself, we will define such a system as organic and apply to it different laws compared to the mechanical case in which evolving substance and evolutionary principle belong to two separate systems.

Mechanical systems and organic systems: Let us therefore consider the systems previously classified: organic systems, in which the evolutionary principle is united to the evolving substance...