The Mapping

The mapping is a discrete process. If we stand on a certain point of the evolution of the mapping (that point can be a pattern, a field, a numerical value, etc.; however, in this exposition we assume that this point is a numerical value), the next point of the mapping is a complex non-linear function of the present point and its history (which means, the previous points within the discrete process). If the process created by this mapping becomes stabilized, this means that the process continues creating almost the same numerical value after a certain number of steps. If we now denote the field that describes a segment of the overall pattern created by this mapping as A, then we could say that the operation of the mapping can be described by A going to the function of A. Further, if A became stabilized, this means that A became its own function (more formally, A=F(A)). In other words, we could say that the interaction between the present point and its history implies the next point of the mapping. The point and its history are the given, the if, whereas the function is the structure of the implication, which together with the next implied point, is the then. When the process becomes almost periodic, the system is going through its almost opposite and then returning to almost its initial point (which is the almost opposite of the almost opposite). This is the structure of second-degree paradoxes: if so and so, then almost its opposite. In figures A1 and A2, different processes of the mapping of certain generating mechanisms are illustrated. Different profiles can be different fields or the same field looked upon through different steps. As will be shown in the following pages, any of these different profiles can be used as a basic process to be destabilized and then stabilized in a new process.

The interaction of the process with its own history in order to create itself is actually the retromorphous looking, the stabilization factor of the process. The retromorphous interaction is a very important point to take note of because the structure of the interaction of the system with its own history is what creates the universal aspiration of the system to achieve stabilization and consistency. This structure is actually one of the main tenets of the theory.

It is not merely the function of the mapping that creates the evolution of processes, but rather, the function itself can evolve and change during the mapping. Stated differently, the function that generates the evolution of the mapping is itself a mapping. This again is the loop between "Structure" and "Significance". Stated differently, the generating principles that create the mapping are not unique. Furthermore, any generating principle contains (as an integral part of itself) the structure that can transform it within its own framework to become another generating principle.