Tuesday, June 26, 2007

Potential Infinity?

Clearly whole numbers are objective abstract objects, whose properties are independent of our theories (judging by the obvious objectivity of, for example, these three words). And the natural numbers clearly comprise, collectively, a definite totality (which is therefore another thing, one collection) because a very simple, finite rule (of adding 1 to the previous number, starting with 1) completely determines what they are. Consequently their totality has qualities that are in many ways finitesque (i.e. it has properties that are paradigmatically possessed by the finite collections that we more easily envisage). We can, for example, consider how there are as many rational numbers as natural numbers, by considering a bijection between those two totalities.
......Nonetheless such simply infinite collections differ from finite collections in going on and on, in being generated by their finite rules endlessly. So although mathematical realists (such as myself) do think that they form actual collections (objective structures), if their properties happen not to be as finitesque as is commonly believed (which must be discovered, not stipulated), then their totality will be more like the potential infinity of the constructivists, in many ways, than the actual infinity of the set theorists. The underlying cause of such infinitesque behaviour (if it exists) would not be that our numbers are mental constructions (which is the usual reason for thinking of the natural numbers as potentially infinite) but that they are given by an endless reiteration. That obvious fact about the natural numbers (which all can agree upon) might naturally lead some collections of natural numbers (some sub-collections of that definite totality) to be relatively indefinite—it might, or it might not (and for a realist, such obscure facts must be discovered, not stipulated).
......If we were repeatedly tossing a coin forever, with heads meaning membership (of N in the collection, when obtained on the Nth toss) and tails non-membership, then clearly we would never obtain a definite sub-collection—and whether or not we would in principle be able to obtain one (e.g. by tossing ever faster, or by simultaneously tossing aleph-null coins, etc.) is an open question. It is generally assumed (e.g. by most mathematicians) that we could, but why? Even if set theory is consistent, that would not make it true; and there are indications that we could not (e.g. my forthcoming paper). That the potentially infinite does not depend upon a metaphysically prior constructivism is also indicated by the infinitesque behaviour of the proper classes that inevitably accompany the usual sets.
......But I shall end with a devilish detail, another way in which my notion of potential infinity departs from the norm. I see no compelling reason why lines should not exist (as objective structures) and be full of points, but then a point on a line might determine (in conjunction with two points labelled ‘0’ and ‘1’) an endless sequence of binary places. Then even a random point would determine a sub-collection of the natural numbers. Still, even then the totality of such sub-collections would not exist (so saving my resolution of Cantor’s paradox) because even such pre-determined sequences would be relatively incomplete. They would not be unities in their own right but would depend, for their definition, upon their generating points—in themselves they just go on and on, endlessly, without the intrinsic definition that the existence of their totality would require (cf. Russell's paradox). That is, each generating point would contain far more information (in its precise position relative to the points 0 and 1) than would fit into such a potentially infinite sequence (and while those points would of course form a totality, a line, that would not be a set of points, but rather as described in my last paper).

No comments: