Penrose's point, at least as I understand it, is that insofar human intuition can lead us to see the 'truth' of not-provable-within-the-formal-constraints-of-th

Penrose's point, at least as I understand it, is that insofar human intuition can lead us to see the 'truth' of not-provable-within-the-formal-constraints-of-the-system statements, we are doing something 'informal', aka non-computable, and that insofar if our brains were merely "computers made of meat" we would be unable to transcend the formalism, then we must have some non-computable process going on inside our brains somewhere.