Re: Wittgenstein
Posted by
Phil Henshaw-2 on
URL: http://friam.383.s1.nabble.com/Wittgenstein-tp1133169p1304610.html
Re: [FRIAM] Wittgenstein
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.
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 ([hidden email])
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