In this paper Lucas suggests that many of his critics have not read carefully neither his exposition nor Penrose’s one, so they seek to refute arguments they never proposed. Therefore he offers a brief history of the Gödelian argument put forward by Gödel, Penrose and Lucas itself: Gödel argued indeed that either mathematics is incompletable – that is axioms can never be comprised in a finite rule and so human mind surpasses the power of any finite machine – or there exist absolutely unsolvable diophantine problems, and he suggest that the second disjunct is untenable; on the other side, Penrose proposed an argument similar to Lucas’ one but making use of Turing’s theorem. Finally Lucas exposes again his argument and considers some of the most important objections to it.
