Sunday, November 10, 2013

What is Proof?

Over a hundred years ago, Cantor proved that the natural numbers are temporal: Assume, with Plato and against Aristotle, that they are not temporal, so that they all coexist, insofar as numbers do exist (the main thing is that we can count them, e.g. {1, 2, 3} are three numbers). Since they all exist atemporally, so do all their subsets (e.g. {1, 2, 3}), and so there is a set (an atemporal collection) of all the subsets of that set. By a simple diagonal argument (which you can google) that set of subsets is of larger cardinality than the original set. (Two sets have the same cardinality when the elements of one of them can be put into some one-to-one correspondence with the elements of the other.) And the set of all of the subsets of that set of subsets is of even larger cardinality, because the diagonal argument applies quite generally to any set and the set of all its subsets. (This gives us three equivalence classes of infinite sets, associated with three infinite cardinalities.) So, we can get cardinally bigger and bigger sets in that way. And when there is an endless sequence of such sets, the union of all of them will also be an atemporal collection, because each of those sets was implicit in the previous set, and it will have a cardinality larger than any of those sets, because each is followed in that sequence by sets of larger cardinality. And from that union we can again consider the set of all its subsets, and so on. Now, all these atemporal collections are implicit with the set of natural numbers, so they all exist (insofar as such things do exist) atemporally; but, Cantor proved that they cannot all exist atemporally: Suppose they do. Then there is a set of them all. But implicit in them is the collection of all of their subsets, which would be cardinally more of them, whereas we have assumed that we had the set of them all. So, we have assumed that the natural numbers are not temporal, and obtained a contradiction; that is a classical mathematical proof of the temporality of the natural numbers. However, most people assume that numbers are timeless, and so Cantor took himself to have proved that the totality of the numbers was indeed contradictory (akin to human reasoning being inferior to religious insight), while most of his peers replaced the natural numbers with axiomatic structures that had not been shown to be contradictory. Axiomatic set theory has been the foundation of mathematics for nearly a hundred years, but why do mathematicians throw numbers away (why take number-words to be referring to axiomatic sets) just because of an inconvenient proof? We expect others to accept the conclusions of our proofs, when we have proofs...