We can see in terms of these developments why the question arose that I mentioned in the first part of the paper, namely, whether there are true but unknowable mathematical structures. In terms of the adequately, or perhaps "vividly," known, we have shown that there are structures that conform to an axiomatic system that cannot be proved from an axiomatic system. We know that there are models of the real number axioms which may never be explicitly formulated. Traditionally we have encompassed and understood an infinitude of structures by an axiomatic system. It has been the authority for our declaring that we know all of a certain type of structure. We cannot now make any comprehensive claim to know all structures for any complicated mathematical axiomatic system. Could there be compatible interpretations of a system that are somehow in principle impossible to know? Could there be true but unknowable mathematical facts?
I share Steen's and Robinson's skepticism about the existence of platonic mathematical structures that are true but unknowable. I find there is a certain presumption about affirming the existence of a platonic mathematical form that cannot be known -- either within a Platonic perspective or outside of it. In principle, one could never have any evidence of the form's positive existence. Also, I find the affirmation that there is a platonic realm of mathematical structures that are eternally fixed in their relationship to each other but never growing or diminishing in totality, to be also somewhat presumptuous. Our evidence historically, certainly in terms of what we know, is almost exclusively of a changing domain of mathematical structures, a domain that changes primarily by addition to itself. Of course, it may be claimed that this is simply a growth of our knowledge of a fixed domain, and I would certainly acknowledge the explosion of mathematical discoveries in this century. But it may be the case that there is an actual ontological addition to mathematical structures.
If one believes in any platonically understood realm of mathematical structures, it seems to me best to understand it as a loosely known multiplicity which is incapable of unification axiomatically and to which new relationships may be added. The addition of any new relationship would, of course, be compatible with some structures and logically incompatible with others. Instead of "true but unknowable" we might say "unknowable because not yet true."
In assuming that mathematical relationships have a kind of platonic reality at least in terms of being potentials for matters of fact as known by God, we recognize that these relationships may be structures of that which is known -- or part of the structures of knowing itself. The structures of knowing, at least the means by which one can know mathematics, have traditionally been known as logic. It is a well-known fact that these structures have been objectified and made epistemological objects whose nature can be examined mathematically as structures of the known. Gödel's theorem points out that the structures of knowing cannot all be formalized mathematically.
The new developments in mathematics seem to me to allow a better understanding of what it might mean for God to have the freedom to change the totality of potentials -- both in terms of the structures of knowing among human consciousness and in terms of the objects known. This would mean that not only could man's consciousness, as well as other structures of the world, evolve in ways hitherto unknown, and in ways impossible to know, but in ways that might be even a surprise to God -- a surprise in the sense that the potential mathematical structure that could characterize (in part) such consciousness might not even be at present. My viewpoint here is a departure from a strictly Whiteheadian process theology that could understand God's surprise at the way Beethoven's Fifth Symphony turned out, but a surprise because it turned out this way and not that way, or some other way, all ways being known as strict potentials. God may not be surprised, however, at new mathematical potentials that are added, for he may create and add them all himself. But we need not, in our knowledge, limit new potentials solely to God; they may come from God's interaction with the world or from the world itself, i.e., by the creative power given to the world by God.
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