From: jqb@netcom.com (Jim Balter)
Date: Wed, 13 Nov 1996 22:03:17 GMT

In article <328A2BEC.70E1@ix.netcom.com>,
Phil Roberts, Jr.  wrote:
>You're right.  Lucas is _trivially_ wrong, in that he referred to his
>attempt to extend Godel to the material realm as a _proof_ of the falsity
>of mechanism (last paragraph).  What he offered was merely an argument,
>but a pretty damn good one, if I'm not mistaken:

You are very mistaken.

>	Godel's theorem states that in any consistent system 
>	which is strong enough to produce simple arithmetic
>	there are formulae which cannot be proved-in-the-system,
>	but which we [standing outside the system] can _see_ to
>	be true.

First, for any such system, there is another consistent system in which the
Godel sentence of the first system can be proven.  The "we" here is a red
herring.  In addition, inconsistent systems of the same strength can also
"prove" that Godel sentence.

Second, although we can "see" that the Godel sentence of any such system must
be true, we cannot necessarily see the Godel sentence *as* true.  That is, if
presented with the Godel sentence, there is no reason to think that we will be
able to see that it is true, unless we can show that it *is* the Godel
sentence for that system.  And, just as we can "see" that Godel's theorem is
true (after all, it is a theorem), there are, contrary to popular belief,
formal consistent systems in which Godel's theorem can be derived (after all,
it is a theorem).

It may be that "we" could be modeled as a sufficiently strong consistent
system, in which case, per Godel, we could not see that our own Godel
statement is true, even though we can see that our Godel sentence must be
true.  This is not a contradiction; rather it implies that we would not be
able to recognize or derive our own Godel sentence.  This is not too terribly
subtle, but it is too much for Lucas and many of his readers.

>	Godel's theorem must apply to cybernetical machines,
>	because it is of the essence of being a machine, that
>	it should be a concrete instantiation of a formal
>	system.  It follows that given any machine which is
>	consistent and capable of doing simple arithmetic,

This is confused.  The system *implemented* by a Turing machine need not
be consistent, nor strong enough for Godel's purposes.

>	there is a formula which it is incapable of producing
>	as being true -- but which we can _see_ to be true.

Again, a red herring; Godel sentences are *per system*.  Each system fails to
be able to derive its *own* Godel sentence.  Even if we can see the Godel
sentence of a system to be true, that says nothing about our *own* GS, if we
have one.  In addition, we *cannot* see the truth of the vast majority systems'
GS's; they are way too complicated.  We *can* see the truth of Godel's
theorem, but *that's a different matter*.

>	It follows that no machine can be a complete or adequate
>	model of the mind, that minds are essentially different
>	from machines.

Totally wrong.  There are consistent systems that can derive Godel's theorem.
Nonetheless, they cannot derive their own Godel sentences.
Lucas does not show that we cannot be such a system.

Now, those are a host of Lucas' more subtle failures.  But the easy, trivial,
refutation, as I and Turing before me stated, is that both machines and minds
can implement inconsistent systems, to which Godel does not apply.

The bottom line is, Lucas gets almost everything wrong, and it is a pathetic
waste of time to build anything upon such ill-informed error.
--