I imagine that whenever the mind perceives a mathematical idea, it makes contact with Plato's world of mathematical concepts... When mathematicians communicate, this is made possible by each one having a direct route to truth, the consciousness of each being in a position to perceive mathematical truths directly, through the process of 'seeing'.
Roger PenroseIntelligence cannot be present without understanding. No computer has any awareness of what it does.
Roger PenroseWell, I donโt know if I can comment on Kant or Hegel because Iโm no real philosopher in the sense of knowing what these people have said in any detail so let me not comment on that too much.
Roger PenroseNo doubt there are some who, when confronted with a line of mathematical symbols, however simply presented, can only see the face of a stern parent or teacher who tried to force into them a non-comprehending parrot-like apparent competence--a duty and a duty alone--and no hint of magic or beauty of the subject might be allowed to come through.
Roger Penrose