Gödelian arguments against AI

Abstract

This thesis gives a new spin to the debate on the Gödelian arguments that permeates the academic literature. In the first part of this thesis, I shall present Predicate Logic and the formal axiomatic method as a knowledge representation and reasoning framework emerging from a motivation to reflect upon an early idea of an artificial intelligence and at the same time laying the foundations of Gödel's Incompleteness Theorems; next I will highlight the role of Recursion Theory in generalizing the undecidability results and particularly of Turing Machines in providing an interface between formal systems and mechanical procedures. An informal exposition of the proof of Gödel's limitative results shall close this section. In the second part, I will reveal the weaknesses of the Gödelian arguments by building upon popular counter-arguments and also by means of a novel objection from the standpoint of mathematical cognition; finally, I will provide an analysis which to the best of my knowledge is missing in the classical discussions on the Gödelian arguments: the distinction between top-dpwn reasoning systems -such as automated-theorem provers and expert systems- and bottom-up reasoning systems-machine learning systems. This analysis is key to understand that even if the Gödelian objections showed that our mental resources for solving certain number-theoretic problems are beyond a Turing Machine, they are useless as a tool for refuting the possibility of an artificial intelligence.