For a Naturalist Approach to Anticipation: from Catastrophe Theory to Hyperincursive Modelling

Living systems, in certain circumstances, try to predict future situations, and by that, begin to adapt in advance. The behaviour of adaptation of the sportsman, or of the predator, that begins before the properly told effort, show us remarkable anticipatory characteristics. It is impossible to understand the anticipatory behaviour, and the autonomous actions, of the individuals without having recourse to a dual control. We must distinguished a direct control, each element of the action is felt, and a dual control, only the aim of the target is consciously present (attended to), the other elements of the action are relegated to the periphery of the attention. The living being deals with its external middle to establish its internal coherence, but it takes this into account only to distinguish itself by the action. The relationship/separation between the living being and its environment leans both on an internal action (to adapt it), and on an external action (to adapt its middle). The importance of the dual action for the living being, holds in the fact that it exists by the means of its self-constituent activity, "connected to" and "distinct of' a no-self. We use a dynamic structure involving catastrophe theory, to model anticipative process. The dynamics of the predation, a good example of anticipating system, can be described by an attraction of the predator with regard to the prey. René Thom showed how to use the cusp catastrophe to model predation. The predation activity can be defined by a potential. This structure takes in account the duality of the living being, the substance which is a material organisation, and the goal, which is a relational abstraction. In this paper, a new interpretation of the catastrophe theory is given in the framework of hyperincursion: a hyperincursive system is an extension of recursive systems in which the state of the system is computed from a function of itself. A Hyperincursive Cusp Function can be modelled by a Heaviside Cusp Function. A Hyperincursive Boolean Table can be built and a hyperincursive algebraic linear function can model a cusp which represents an elementary flip-flop one bit memory. A recursive process defines the successive states from its initial conditions and a hyperincursive process defines the successive states from the path chosen in the control parameters space. The recursive process is related to an internal observer and the hyperincursive process is related to an extemal observer.