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From: morphy@nntp-server.caltech.edu (Jones Maxime Murphy)
Newsgroups: sci.math,comp.ai
Subject: Re: musings on Godel's theorem
Message-ID: <1990Nov25.204513.24169@nntp-server.caltech.edu>
Date: 25 Nov 90 20:45:13 GMT
References: <11351@ccncsu.ColoState.EDU>
Organization: California Institute of Technology, Pasadena
Lines: 37

news@ccncsu.ColoState.EDU (USENET News) writes:

>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.)
> 

The link is hardly tenuous. A machine is defined by it's changes of state in 
response to sentences from the language it accepts. The primitives of it's
language correspond to axioms, and syntax corresponds to an inferential
mechanism for extracting state changes from combinations of primitives.


>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.
> 

I presume the one "truth" being referred to here is the impossibility of 
a finistic proof of the consistency of arithmetic that can be embedded in the
language of arithmetic and the first-order predicate calculus. In addition to 
simple culture shock at the news that "logic" as we know it is no longer
omnipotent in it's domain of inquiry, the fact is that we must now be wary in
the quest for artificial intelligence.
 A great deal of human intelligence derives from self-reference, and weaknesses
in self-referential analysis in the inferential mechanisms of computing machines
which at the moment all rely on 1st order predicate calculus must be taken
seriously.
Jones