It is when one begins talking about infinite numbers that the trouble really begins. Cantor’s Continuum Problem is undecidable on the basis of our present-day theories of mathematics. For the formalists this means that the continuum question has no definite answer. But for a Platonist like Gödel, this means only that we have not yet “looked” at the continuum hard enough to see what the answer is.
In one of our conversations I pressed Gödel to explain what he meant by the “other relation to reality” by which he said one could directly see mathematical objects. He made the point that the same possibilities of thought are open to everyone, so that we can take the world of possible forms as objective and absolute. Possibility is observer-independent, and therefore real, because it is not subject to our will.
There is a hidden analogy here. Everyone believes that the Empire State Building is real, because it is possible for almost anyone to go and see it for himself. By the same token, anyone who takes the trouble to learn some mathematics can “see” the set of natural numbers for himself. So, Gödel reasoned, it must be that the set of natural numbers has an independent existence, an existence as a certain abstract possibility of thought.
I asked him how best to perceive pure abstract possibility. He said three things, i) First one must close off the other senses, for instance, by lying down in a quiet place. It is not enough, however, to perform this negative action, one must actively seek with the mind, ii) It is a mistake to let everyday reality condition possibility, and only to imagine the combinings and permutations of physical objects—the mind is capable of directly perceiving infinite sets, iii) The ultimate goal of such thought, and of all philosophy, is the perception of the Absolute.
