Penrose's views and books have been the most popular topic for discussion on more than one internet newsgroup continuously since the first book was published. Only the first of these points concerns us here, since the motivation for considering extensions to quantum theory comes only from the limitations on what could otherwise be achieved. Penrose believes that Computers cannot, but human beings can, prove certain results. Godel's results apply to a narrowly specified set of formal logical systems, and are not applicable to arbitrary Turing machine computations. An important constraint, which certainly does not apply to people and need not apply to Turing machines, is that the system in question must, viewed as a logic, be consistent .
Some people achieve extraordinary intellectual feats, but no one is infallible.
One mistake and, viewed as a deductive system, you are likely to be inconsistent. If you don't have to be infallible then you can guess, and there is nothing which you might not suceed in guessing correctly. Genius is extremely good guesswork. It is not probable that so many should have debated the issues raised by Roger Penrose without casting more light than can be found in the above extraordinarily superficial dismissal. A search of the internet yields almost uncountably many additional sources of online material related to this controversy.