Turing: Uncalculability and Unpredictability

The objective of this paper is to highlight the co-presence, in Turing’s work, of two competing lines of research: one line the objective of which is to challenge the limits of calculability and then to consider the possibility of a thinking machine; the other line, relying upon the non coincidence of determinism and predictability of dynamical systems, that leads to the irreducibility of a human brain to a “discrete-state machine”.

The three oppositions discrete / continuous, logic / dynamics, and arithmetic / geometry make it possible to better understand the contradiction between these two lines of research, a contradiction which, in turn, commands a reassessment of the first of these three alternatives.