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From: chalmers@bronze.ucs.indiana.edu (David Chalmers)
Newsgroups: comp.ai.philosophy
Subject: Re: Minds, machines, and Godel
Message-ID: <1991Jan18.184116.5299@bronze.ucs.indiana.edu>
Date: 18 Jan 91 18:41:16 GMT
References: <28087@cs.yale.edu> <1991Jan18.012217.20029@news.cs.indiana.edu> <28145@cs.yale.edu>
Organization: Indiana University, Bloomington
Lines: 58

In article <28145@cs.yale.edu> mcdermott-drew@cs.yale.edu (Drew McDermott) writes:

>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.

As I've said a number of times, I'm idealizing away from inconsistency,
because I don't think that inconsistency is the root of the problem.  You
may disagree.

Given that step, then take our TM that simulates the judgment of a
mathematician (or an army of mathematicians).  Convert it straightforwardly
to an axiomatic system that generates precisely those statements of
first-order arithmetic that the mathematicians judge to be true.  Then this
system has the following properties:

(1) It is consistent (by assumption).
(2) It has expressive power as great as the usual first-order systems
   of arithmetic, and has deductive power as least as great as such systems.
   (Assuming that human mathematicians have as a subset of their ability the
   capacity to apply the usual "axioms" of arithmetic straightforwardly, even
   consciously if necessary.  This should not be controversial.)

This is all that is required to apply the Godel theorem.

>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.

Hmm.  Of course the initial statement of the problem requires some
clarification before it really makes sense, and to avoid some basic
counterarguments.  I didn't make these clarifications at the beginning
(though I was tempted to) because Lucas and Penrose didn't either, and it
seemed more interesting to let them emerge through dialectic.  It is
interesting how many of the "standard" responses can be finessed by refining
the argument.

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

I think that the correct refutation is tied up with the question of how we
"see" that certain systems are consistent, and the arguments by Torkel
Franzen are coming close to that point, although even there the argument has
at least temporary defenses.  Note that this kind of refutation is completely
independent of the empirical fact of human inconsistency, which I firmly
believe to be a red herring as far as this argument is concerned (although
this is a fascinating point of issue in its own right).

-- 
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."