Why Is the Lucas-Penrose Argument Invalid?

Abstract

It is difficult to prove that something is not possible in principle. Likewise it is often difficult to refute such arguments. The Lucas-Penrose argument tries to establish that machines can never achieve human-like intelligence. It is built on the fact that any reasoning program which is powerful enough to deal with arithmetic is necessarily incomplete and cannot derive a sentence that can be paraphrased as “This sentence is not provable.” Since humans would have the ability to see the truth of this sentence, humans and computers would have obviously different mental capacities. The traditional refutation of the argument typically involves attacking the assumptions of Gödel’s theorem, in particular the consistency of human thought. The matter is confused by the prima facie paradoxical fact that Gödel proved the truth of the sentence that “This sentence is not provable.”

Adopting Chaitin’s adaptation of Gödel’s proof which involves the statement that “some mathematical facts are true for no reason! They are true by accident” and comparing it to a much older incompleteness proof, namely the incompleteness of rational numbers, the paradox vanishes and clarifies that the task of establishing arbitrary mathematical truths on numbers by finitary methods is as infeasible to machines as it is to human beings.