Gödel, Turing, the undecidability results and the nature of the human mind

In this paper Turing and Goedel’s standpoints toward the implications of the undecidability results are addressed. In the case of Turing, we show how his account on the issue was deeply connected to his project of actually building a computing machine showing an intelligent behaviour. Furthermore, we argue that his claim that the argument based on the halting problem offers no objection to that very project, is strenghtened by a general view on mathematical resoning and intelligence which has an anti-mechanistic flavour. As to Goedel’s position, we reformulate, by enhancing its modal character, an argument that is contained in an unublished paper belonging to the early 1950’s which ends in an open conclusion on minds and machines. We finally present Goedel’s interpretation of the solution of the famous P = NP problem contained in a letter to von Neumann as a further contribuition in this direction of working out the philosophical significance of certain mathematical achievements.