On the Question of Whether the Mind Can Be Mechanized, II: Penrose’s New Argument

On the Depth of Gödel’s Incompleteness Theorems

“…In this paper, I will not examine the broad question of whether the mind can be mechanized, which has been extensively discussed in the literature; 45 instead I will only examine the question of whether G1 implies that the mind cannot be mechanized in the sense that the mathematical outputs of the idealized human mind outstrip the mathematical outputs of any Turing machine. In the following, I give a concise overview of the current progress on this question based on Koellner's work in [54], [55] and [56]. For more detailed discussion of the question of whether the mind can be mechanized, we refer to Koellner's recent nice survey in [55] and [56].…”

Proving that the Mind Is Not a Machine?

“…In three recent papers, Koellner (, , ) uses this strategy for resolving the intensional paradoxes to investigate arguments against mechanism. Koellner raises the general challenge for the antimechanist to provide a philosophically motivated theory of truth in which a convincing argument for the claim that the mind is not a machine can be carried out.…”

“…In this case the only assumption we are allowed to make is that the theorems of the explicit system of axioms and rules are recursively enumerable, that is, are the output of a Turing machine. This is a more general formulation of mechanism and it is the one Koellner (, , ) adopts in his discussion. This would be some form of nonorthodoxy about logic.…”

“…Gödel's disjunction is actually a conditional claim: it asserts that if the human mind is a machine, then there are sentences the human mind can neither prove nor refute, that is, there are absolutely undecidable sentences. If we accept Gödel's terminology his disjunctive conclusion indeed follows from the incompleteness theorems (Koellner , , ). But, as Gödel argued and we have also pointed out, this by no means establishes the first disjunct, namely, that “ the human mind infinitely surpasses the powers of any finite machine ”.…”

“…The strategy of resolving the intensional paradoxes by connecting the intensional notions to a non‐naive notion of truth has been independently developed by Stern (, , ) and Koellner (, ).…”