Penrose is not arguing that the human mind is illogical. He is arguing that it is uncomputable, which is a big difference. In particular, he's saying that the

Penrose is not arguing that the human mind is illogical. He is arguing that it is uncomputable, which is a big difference. In particular, he's saying that the human mind is capable of outputting true statements that cannot be proven to be true via any bounded number of proof steps. I don't disagree with that. However, given that the human mind is capable of outputting false statements -- witness this conversation, wherein at least one of us is outputting false statements -- Penrose has failed to prove that the human mind is non-Gödelian, i.e. can output all the true statements that exist, while outputting no false ones.