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From: bill@milford.UUCP (bill)
Newsgroups: net.math
Subject: Undecidability and Proof
Message-ID: <106@milford.UUCP>
Date: Thu, 10-Oct-85 08:49:25 EDT
Article-I.D.: milford.106
Posted: Thu Oct 10 08:49:25 1985
Date-Received: Sun, 13-Oct-85 04:05:15 EDT
Organization: Telecomp,Inc. , Milford Ct.
Lines: 27


In the recently published _The_Beauty_of_Doing_Mathematics_, Serge
Lang states (mini-book-review: in general this book is very
excellent, especially as a model of a class of the "Math-for-poets"-type)
"... if you succeeded in proving that Fermat's problem is unsolvable, 
then ipso facto you would have shown that the conjecture is true.
Because if there was a counterexample, then with some big computer,
some day someone would pull out the counterexample."

How common is this attitude toward undecidability questions? I
remember arguing some years ago with a distinguished Number Theorist
who "proved" that the Riemann Conjecture could never be proven
undecidable for the same reason (with almost identically the same
words). I tried to distinguish between "Truth" (capital T) and
algorithms establishing that Truth, to no avail; I did gain some
ground with the idea that undecidability is a higher order predicate
which could not be used in such Number Theoretic "proofs", but the
Professor continued to insist that undecidability results would
somehow "prove" these conjectures.

Its well know that in a certain sense the Godelian sentences must be
false, but this does not negate that they cannot be proven or
disproven within the theory under question.

Has anyone else encountered this type of argument? How do people
feel about this type of 'proof'? Also, what is the attitude toward
such a Platonic notion of ideal "Truth"?