Penrose says of the third, Platonic, world of mathematical objects: “What right do we have to say that the Platonic world is actually a ‘world,’ that can ‘exist’ in the same kind of sense in which the other two worlds exist? It may well seem to the reader to be just a rag-bag of abstract concepts that mathematicians have come up with from time to time. Yet its existence rests on the profound, timeless, and universal nature of these concepts, and on the fact that their laws are independent of those who discover them. The rag-bag – if indeed that is what it is – was not of our creation. The natural numbers were there before there were human beings, or indeed any other creature here on earth, and they will remain after all life has perished.” (Shadows of the Mind, p. 413)
Edward Everett, whose dedication speech at Gettysburg was so famously upstaged by Abraham Lincoln, put it more poetically: “In the pure mathematics we contemplate absolute truths, which existed in the Divine Mind before the morning stars sang together, and which will continue to exist there, when the last of their radiant host shall have fallen from heaven.” G. H. Hardy was ambivalent about the Divine, but like most mathematicians he believed “that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our ‘creations’, are simply our notes of our observations.”
This trilogy of worlds raises some questions, of course. The first is what Eugene Wigner called “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” As William Newton-Smith asks, “if mathematics is about this independently existing reality, how come it is useful for dealing with the world?” Why does the world follow the dictates of eternal Reason?
The second question is the mind-body problem. How are we conscious of the universe, and how do our decisions to act affect it? Does even our perception have strange quantum effects?
Finally, how do we become aware of the Platonic world? Elsewhere, Penrose says “When one ‘sees’ a mathematical truth, one’s consciousness breaks through into this world of ideas, and makes direct contact with it... When mathematicians communicate, this is made possible by each one having a direct route to truth.” (The Emperor’s New Mind, p. 554)
Plato, in the story of Socrates and Meno’s slave, tells how an uneducated slave is prompted to discover how to double a square. Plato sees this as evidence of memory from a past life, but it provides an example of mathematical intuition that all (successful) students of mathematics will recognise. As Augustine said, “The man who knows them [mathematical lines] does so without any cogitation of physical objects whatever, but intuits them within himself.” Yet Plato’s (and Augustine’s) belief in a soul makes the mind-body problem more acute. How do the three worlds tie together?