Introduction of the Aristotle's final causation in CAST concept and method of incursion and hyperincursion

Abstract

This paper will analyse the concept and method of incursion and hyperincursion firstly applied to the Fractal Machine, an hyperincursive cellular automata with sequential computations where time plays a central role. This computation is incursive, for inclusive recursion, in the sense that an automaton is computed at the future time t+1 in function of its neighbour automata at the present and/or past time steps but also at the future time t+1. The hyperincursion is an incursion when several values can be generated at each time step. The incursive systems may be transformed to recursive ones. But the incursive inputs, defined at the future time step, cannot always be transformed to recursive inputs. This is possible by self-reference. A self-reference Fractal Machine gives rise to A non deterministic hyperincursive field rises in a self-reference Fractal Machine. This can be related to the Final Cause of Aristotle. Simulations will show the generation of fractal patterns from incursive equations with interference effects like holography. The incursion is also a tool to control systems. The Pearl-Verhulst chaotic map will be considered. Incursive stabilisation of the numerical instabilities of discrete linear and non-linear oscillators based on Lotka-Volterra equation systems will be simulated. Finally the incursive discrete diffusion equation is considered.