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From: news@ccncsu.ColoState.EDU (USENET News)
Newsgroups: sci.math,comp.ai
Subject: musings on Godel's theorem
Message-ID: <11351@ccncsu.ColoState.EDU>
Date: 23 Nov 90 07:05:57 GMT
Organization: Colorado State University, Ft. Collins  80523
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the chance to absorb "Godel's Proof" by Nagel and Newman.  IMHO, the major
achievement in Godel's paper was the invention of the expressions for
provability and self-substitution in the formal system, the form of which was
mostly unspecified in the book, so naturally I speculated!  Now, apart from
the immediate requirement that they be constructed exclusively with the
symbols of the formal system, what are other demands on their form?
What about these:
 
* They have to be a finite size.
 
(We need to compute the Godel number of expressions with the provability and
substitution expressions embedded within, and if either did not have a
finite size the finite transcribed Godel number of the overall expressions
could not be determined.)
 
* They have to be invariant.
 
(The proof hinges on the idea that the provability and self-substitution
expressions can be paralleled in separate formulas "H" and "G".  It seems
to me that if either of the two expressions (proving and substitution) varied
between the H and G formulas the conclusion may not be valid.)
 
Can anybody outline the method by which Godel accomplished this?
The "provability" formula must take a number that can represent symbol
or string of symbols of arbitrary length, parse it, and check each step
for provability.  How is this feat possible within the confines of an
expression of finite length?  The "substitution" expression similarly must
be able to deal with expressions (represented by the Godel number) of
arbitary length.  At the very least, it seems the formal system requires
an operator "the number such that..."
 
* * *
 
On a different topic, I think it is presumptuous at best and irresponsible
at worst for claims about machine intelligence to be made from a supposed
basis in Godel's proof, as Nagel and Newman do in their work (and also as
in the article on the subject in Scientific American):
 
``Godel's conclusions bear on the question whether a calculating machine
can be constructed that would match the human brain in mathematical
intelligence.  Today's calculating machines have a fixed set of directives
built into them; these directives correspond to the fixed rules of inference
of formalized axiomatic procedure.  But, as Godel showed in his incompleteness
theorem, there are innumerable problems in elementary number theory that fall
outside the scope of a fixed axiomatic method; and that such engines are
incapable of answering, however intricate and ingenious their built-in
mechanisms may be and however rapid their operations.''
 
This link between an axiomatic system and computation seems rather ill-defined
and tenuous!  (The philosopher Searle may be taking advantage of this with his
"Chinese Room" argument, claiming that computation is merely axiomatic or
"syntactic" in his word.)
 
How is it that one single self-referential statement in a system has been
construed to hold so much importance?  Nagel and Newman state above that
Godel's proof showed
 
``there are innumerable problems in elementary number theory that fall
outside the scope of a fixed axiomatic method...'' (?!)
 
How is it that one "truth" that cannot be demonstrated in the system has
been extended to conjectures in number theory?  Evoking Godel seems like a
weak evasion of their challenge.
 

ld231782@longs.LANCE.ColoState.EDU