After a couple of rather unsatisfactory philosophical chapters, Berto does quite a bit better in the next two. (Which is good, because these comments were getting unseasonably grumpy!)
Ch. 10 is called “Mathematical Faith”, and discusses what we should learn from the unprovability of consistency of a theory (a consistent theory containing enough arithmetic) within that theory. Does it show that the mathematician has to rely on blind faith in some worrying sense? To which the right answer is “no!”. Here Berto pretty closely follows a good discussion by Franzen. There’s nothing that adds much to Franzen’s similarly introductory discussion, but equally Berto doesn’t go astray.
Ch. 11 is on the Lucas/Penrose argument. Again Berto’s discussion is sane and sensible. The ur-Lucas argument is sabotaged by the familiar Putnam riposte. Souped up versions are sabotaged by souped up versions of that riposte. But it remains that something can be learnt from Gödel incompleteness about the nature of the mind — namely the disjunctive conclusion of Gödel’s Gibbs lecture (prefigured also in Benacerraf’s old discussion). This is rapidly done, though: for example, what on earth will the beginner with the thin background Berto is officially presupposing make of the invocation of transfinite ordinals on p. 183? Still, this chapter could make for helpful introductory reading for some students working towards on an essay on this topic.