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From: mcdermott-drew@cs.yale.edu (Drew McDermott)
Newsgroups: comp.ai.philosophy
Subject: Re: Minds, machines, and Godel
Summary: Godel didn't prove what you think
Message-ID: <28145@cs.yale.edu>
Date: 18 Jan 91 16:06:00 GMT
References: <1991Jan16.035058.7465@bronze.ucs.indiana.edu> <28087@cs.yale.edu> <1991Jan18.012217.20029@news.cs.indiana.edu>
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   In article <1991Jan18.012217.20029@news.cs.indiana.edu> chalmers@iuvax.cs.indiana.edu (David Q. Chalmers) writes:

   [[ Quoting me -- ]
   >>What does it mean for an arbitrary Turing machine to prove something?
   >
   >OK, not quite an arbitrary Turing machine.  A Turing machine that takes
   >in mathematical statements as input, and produces "yes" or "no" (or
   >perhaps "don't know") as an output.  That's all we need: if mathematicians
   >are computationally simulable, we can create such a TM from a simulation
   >of them.
   >
   >(Alternatively, we could formulate it as a Turing machine that *generates*
   >statements, rather than judges them; we'd just require a special symbol to
   >be printed at the start and finish of every such statement.  I think the
   >judging TM is simpler, though.)
   >
   >Given such a judging TM, there is an corresponding formal system that produces
   >as theorems only those statements that the TM judges as true.  This we can
   >Godelize.
   ...
   >--
   >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."

I dispute that Godel ever proved anything about this broad class of
Turing machines.  This class includes systems that are *mistaken*
about mathematics to some degree, for instance.  Or systems that
change their minds.  If you rule such Turing machines out, then you've
simply begged the question whether human mathematicians are Turing
machines, because people do indeed change their minds, and can indeed
be mistaken about some parts of mathematics while being quite
competent at other parts.

I'm rather surprised at the course this discussion has taken.  I
expected Chalmers to elicit the usual woolly responses, and then give
prizes for the correct answers.  But his understanding of the debate
turns out to be about as woolly as the average.

At this point, I'd like to see what Chalmers considers the definitive
refutation of the "Mind, Machines, and Godel" argument.

                                             -- Drew