Or. another angle. Proofs represent discoveries about the invented grammar they use, with the proviso of "so far as we can see"? The way we define grammars changes to suite our intentions occasionally, but we're generally trying to identify things inherent in nature, for grammars drawn as conclusively as we know how to make them. They might not show us about the aspects of nature that are inconclusive, of course, but we still would like to know if our constructs are at least pointing to something real. What I find interesting is that every proof seems to imply "and therefore I can't think of anything else" a conclusion based on a lack of imagination. That point to proof as an acceptance of adding a branch to a constructed tree, I think? If the 'tree' itself at least reflects something that exists in nature when the grammar surely didn't is the puzzle.
From: [EMAIL PROTECTED] [mailto:[EMAIL PROTECTED] On Behalf Of John F. Kennison Sent: Tuesday, October 07, 2008 1:01 PM To: [email protected] Subject: Re: [FRIAM] Wittgenstein I would like to respond to Wittgenstein's idea that a mathematical proof should be called an invention rather than a discovery. When solving a Suduko puzzle, I often produce a logical deduction that the solution is unique. It seems clear to me that I discovered that there is only one solution. I don't see how to make any sense of the idea that I "invented" the fact that there is only one solution. "Wittgensteins technique was not to reinterpret certain particular proofs, but, rather, to redescribe the whole of mathematics in such a way that mathematical logic would appear as the philosophical aberration he believed it to be, and in a way that dissolved entirely the picture of mathematics as a science which discovers facts about mathematical objects . I shall try again and again, he said, to show that what is called a mathematical discovery had much better be called a mathematical invention. There was, on his view, nothing for the mathematician to discover. A proof in mathematics does not establish the truth of a conclusion; if fixes, rather, the meaning of certain signs. The inexorability of mathematics, therefore, does not consist in certain knowledge of mathematical truths, but in the fact that mathematical propositions are grammatical. To deny, for example, that two plus two equals four is not to disagree with a widely held view about a matter of fact; it is to show ignorance of the meanings of the terms involved. Wittgenstein presumably thought that if he could persuade Turing to see mathematics in this light, he could persuade anybody." Turing apparently gave up on W. a few lectures later. I have to admit the distinction that W. is making here does not move me particularly. It seems to me as much of a discovery to find out what is implied by the premises of a logical system as to find out how many electrons there are in an iron atom, and since logic is always at work behind empirical work, I cannot get very excited about the difference. Perhaps because I am dim witted. No response necessary. Nick Nicholas S. Thompson Emeritus Professor of Psychology and Ethology, Clark University ([EMAIL PROTECTED])
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