Friday, March 02, 2018

an inconvenient Proof ?

Do I have a proof that there is a transcendent Creator?

Well, the essence of Cantor's paradox is a logical argument for a contradiction, with an obscure lacuna: there need be no contradiction if arithmetic (that is Millian, or ordinary, common or garden arithmetic) is constructed forever. Were it not for that lacuna, we might have to throw logic away and replace it with some formal logic (symbolic calculi called "logics") or other, whilst being unable to choose logically between them. But, there is that lacuna, and so we do have a proof of the existence of a transcendent constructor of arithmetic; and of course, Millian arithmetic could only be constructed by the Creator of all other things.
      Note that a purely logical existence proof would be the appropriate signature of the Creator of homo sapiens. And consider some other kinds of proof, by way of comparison; here are a couple of examples:

Suppose there was a serious crime. Fortunately you have a suspect, and a good case against him. The defense says that your case means nothing, because you are only human, that to err is human. She goes on to detail how nice your suspect is. You point out that she should therefore doubt that opinion of him, since she is human, whilst you need not entertain any such doubts because it is not your argument; indeed, you have convicted lots of criminals on less evidence, so you ask her if they should all go free? Of course not, she says, it is only your case against this man that is thrown into doubt by our common humanity, because he is so very nice indeed! She has simply ignored your observation about her own humanity; maybe she erred in doing so! But what should we conclude from an assumption that we cannot trust our conclusions?!
      Let us suppose that your case is exceptionally water-tight: there is lots of physical evidence, and everyone else has cast-iron alibis, while your suspect has no alibi at all; and this crime is just the sort of thing that he would do. There really is no reasonable way that your suspect is innocent. He even bragged about his guilt to you in private. Since your case is so water-tight, hence all her talk about your humanity is just that: talk. It is, if anything, further evidence of his guilt, that she feels that she has to resort to such meaningless talk.

Or, suppose I say that 2 + 2 = 4. Someone says that he would prefer it to be 5, and tries to show that it really can be 5 by saying that: "If we measure two lengths and put them together, then we could find that two point three five units plus two point three five units equals four point seven units; and if we round all those measurements to the nearest integer, so that it is arithmetic, then we get two plus two equals five." Even so, there are such proofs as this (due to Frege): 2 + 2 = 2 + (1 + 1) = (2 + 1) + 1 = 3 + 1 = 4 (via definitions and associativity). And note that he only wants 5 because it really is bigger than 4 = 2 + 2. My interlocutor retorts that we have to have axiomatic arithmetic, on pain of paradox (e.g. Cantor's paradox), and that he likes those axioms that let him have 5. Could some paraconsistent logic not give arithmetical axioms the power to give him his 5 as well as us our 4, he wonders; but no, that is not really logic, and axioms that give him 5 are not arithmetical. My interlocutor will not give up though, and he has lots of friends. Even so.

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