I have several issues with that reasoning. The biggest one is that, even if a 'brain' can't be directly programmed, it can be simulated. Given enough processing power and enough knowledge about the way brains work, a human brain could be simulated, loaded with a set of -probably synthetic- memories and knowledge, and put to work on this kind of problems, and solve them. Even if we lack the technology necessary for this, it still works as a thought experiment.
Claiming that humans can solve a kind of logical problems that no computer can ever solve is, in my opinion, the same as saying that there is some 'magical sauce' in the human brain that prevents it from being understood.
Evidence of souls, Midiclorians or similar entities is really scarce, so there must be some other factor at play.
You know, I've always wondered if Turing's works would be different had he known the concept of 'fractal dimensions' from chaos theory.