He uses an entire book - Shadows of the Mind (the "sequel" to The Emperor's New Mind) - to discuss how consciousness can't arise from a consistent mathematical system. He begins, essentially, with E=mc2: a number of superposed quantum states in the brain "work" until there is a gravitational difference between their energy and mass. The gravitational significance of this difference causes the states to collapse - or unfold - into one. This single conglomerate state then becomes one observable in the gross physical world - as an action potential in a neuron, perhaps.
At this point, Penrose would seem to agree with David Bohm's implicate order: actuality is the result of probabilistic collapses/unfolding of quantum and subquantum states. We observe light similarly. When medium of a laser absorbs particular wavelenths of light, the electrons of its atoms elevate to their highest state of energy. When so many electrons are excited to this high energy state (population inversion), they collapse together to a lower energy level which results in the emission of light, an observable condition created from a quantum conglomeration.
I have always been somewhat fearful of humanity's discovering the substrate of consciousness and applying it to artificial intelligence. It is comforting that Penrose agrees with me (!) that we will not be able to design A.I. with consciousness in the foreseeable future because it's something we are not nearly close enough to understaning, not to mention being able to coalesce and manipulate.
It is possible to suggest, then, that the connect between consciousness and brain is a physiological exploitation of the vast magnitude of activity in collapsed quantum states. This "non-algorithmic ingredient", as Penrose coins it, also jives with Bohm's suggestions of probability's role in the playing-out of the quantum universe in the gross or actual universe. Could we give our fatty brains such credit as to be the medium by which quantum states become consciousness? This is a question Penrose explores in The Emperor's New Mind, and which I will not dare attempt to disect.
Not at the moment, anyway.
More to come, then.