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From: mcdermott-drew@cs.yale.edu (Drew McDermott)
Newsgroups: comp.ai.philosophy
Subject: Re: Minds, machines, and Godel
Message-ID: <28087@cs.yale.edu>
Date: 16 Jan 91 21:58:31 GMT
References: <1991Jan16.035058.7465@bronze.ucs.indiana.edu>
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   In article <1991Jan16.035058.7465@bronze.ucs.indiana.edu> chalmers@bronze.ucs.indiana.edu (David Chalmers) writes:
   >Dull around here.  How about everybody tries to give the decisive refutation
   >of the Lucas/Penrose arguments that use Godel's theorem to "show" that human
   >beings are not computational (or more precisely, to "show" that human beings
   >are not computationally simulable)?
   >
   >Just to refresh your memory, the argument goes like this: if I were a
   >particular Turing Machine T, there would be a mathematical sentence G (the
   >"Godel sentence" of T) that I could not prove.  But in fact I can see that G
   >must be true.  Therefore I cannot be T.  This holds for all T, therefore I am
   >not a Turing machine.
   >
   >-- 
   >Dave Chalmers                            (dave@cogsci.indiana.edu)      
   >Center for Research on Concepts and Cognition, Indiana University.
   >"It is not the least charm of a theory that it is refutable."

Now that you've demanded a high level of precision, I realize I don't
quite understand the argument.  The claim is that 

  If T is an arbitrary Turing machine, then there is a sentence G that
  T cannot prove.

What does it mean for an arbitrary Turing machine to prove something?
The argument is usually made with respect to a formal logical system,
where I can clearly see that the system "proves" P iff P is a theorem
of the system.  Granted, there is an isomorphism between Turing
machines and formal systems, but then the claim translates into 

  If T is an arbitrary Turing machine, then there is a state G that
  T cannot reach

which, let us suppose, can be made true by restricting the class of
Turing machines a bit.  But then, so what?

If you want to back up and rephrase the argument in terms of Peano
arithmetic or the like, then *of course* people are not equivalent to
any system of arithemtic.  For starters, they're not consistent.

[I pursue the argument along these lines in my reply to Penrose in BBS.]

                                             -- Drew McDermott