The Ontology of Mathematics
If the objects of mathematics are not in nature and not in a “second plane of reality,” then where are they? Perhaps we can learn something from the physicists. Consider for example, the discussion of the “Anthropic Principle” . The advocates of this principle note that the values of certain critical constants are finely tuned to our very existence. Given even minor deviations, the consequence would be: no human race. It is not relevant here whether this principle is regarded as profound or merely tautological. What I find interesting in this discussion of alternate universes whose properties exclude the existence of us, is that no one worries about their ontology. There is simply a blithe confidence that the same reasoning faculty that serves physicists so well in studying the world that we actually do inhabit, will work just as well in deducing the properties of a somewhat different hypothetical world. A more mundane example is the ubiquitous use of idealization. When Newton calculated the motions of the planets assuming that each of the heavenly bodies is a perfect sphere of uniform density or even a mass particle, no one complained that the ontology of his idealized worlds was obscure. The evidence that our minds are up to the challenge of discovering the properties of alternative worlds is simply that we have successfully done so. Induction indeed! This reassurance is not at all absolute. Like all empirical knowledge it comes without a guarantee that it is certain.
My claim is that what mathematicians do is very much the same. We explore simple austere worlds that differ from the one we inhabit both by their stark simplicity and by their openness to the infinite. It is simply an empirical fact that we are able to obtain apparently reliable and objective information about such worlds. And, because of this, any illusion that this knowledge is certain must be abandoned. If, on a neoHumean morning, I were to awaken to the skies splitting open, hearing a loud voice bellowing, “This ends Phase 1; Phase 2 now begins,” I would of course be astonished. But I will not say that I know that this will not happen. If presented with a proof that PA is inconsistent or even that some huge natural number is not the sum of four squares, I would be very very skeptical. But I will not say that I know that such a proof must be wrong.