Laplace, Turing and the “Imitation Game” Impossible Geometry

Abstract

AbstractFrom the physico-mathematical view point, the imitation game between man and machine, proposed by Turing in his 1950 paper for the journal “Mind”, is a game between a discreteand a continuoussystem. Turing stresses several times the Laplacian nature of his discrete-state machine, yet he tries to show the undetectability of a functional imitation, by his machine, of a system (the brain) that, in his words, is not a discrete-state machine, as it is sensitive to limit conditions. We shortly compare this tentative imitation with Turing’s mathematical modeling of morphogenesis (his 1952 paper, focusing on continuous systems, as he calls nonlinear dynamics, which are sensitive to initial conditions). On the grounds of recent knowledge about dynamical systems, we show the detectability of a Turing Machine from many dynamical processes. Turing’s hinted distinction between imitation and modeling is developed, jointly to a discussion on the repeatability of computational processes in relation to physical systems. The main references are of a physico-mathematical nature, but the analysis is purely conceptual.KeywordsTuring Machineclassical determinismdynamical systemscomputational and dynamical hypothesesfunctional analyses of cognitioniterationLaplace