Stewart Shapiro has had two shots at exploring the troubles with Lucas/Penrose-style arguments, first in his well-known paper ‘Incompleteness, Mechanism and Optimism’ Bull. Symb. Logic (1998), and then — expanding his treatment of Penrose’s efforts in Shadows of the Mind (1994) — in ‘Mechanism, Truth, and Penrose’s New Argument’ Jnl. of Philosophical Logic (2003). As you’d predict, Shapiro’s discussions are eminently lucid and very sharp; and his treatment of the Penrose argument in particular is extraordinarily patient and constructive, trying to get something out of the argument, and finding some interesting lines (though nothing that gives Penrose what he wants). He concludes with a
challenge to the anti-mechanist to articulate the new Penrose argument in a way that blocks the Gödel–Kreisel–Benacerraf ploy [i.e. the move of saying that perhaps we can be simulated by a computer but if so we can’t, with mathematical certainty, know which] but does not invoke unrestricted truth and knowability predicates [as apparently, but problematically, required by the Penrose argument, when the wraps are off].
If you don’t know the papers, they are terrific. And Shapiro’s insightful exploration surely has become the necessary starting point for any subsequent discussion here.
It is disappointing to have to report, then, that Penrose’s contribution to KGFM is written as if Shapiro had never made the effort to try to sort things out.
Well, that isn’t quite true: there’s a footnote which has a reference to Shapiro 2003. But otherwise, as far as I can see, Penrose just gives a (too brief to be useful) thumbnail sketch of his 1994 argument, and doesn’t address at all the technical problems that Shapiro explores. In so far as he does respond to critics, Penrose just offers some rather thin remarks about the sort of worries concerning idealization and vagueness that we noted that Putnam rehearses. But of course, the interesting thing about Shapiro’s discussion is that, for the sake of the argument, he gives the game to Penrose on those matters, allows Penrose’s anti-mechanist argument at least to get to the starting point, but then still finds trouble. Lots of trouble. And there’s nothing in Penrose’s paper here which offers any reponses. So I can’t say that this is a useful contribution to the debate on the impact of Gödelian arguments.