From: "Phil Roberts, Jr." 
Date: Thu, 21 Nov 1996 19:07:23 -0500

On Wed, 13 Nov 1996 22:03:17 GMT
Jim Balter wrote:
>
>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.
>

Its important to understand that this is not Lucas's idiosyncratic view of
Godel.  This _is_ what Godel's theorem amounts to when translated into
simple English.

        Godel's Theorem says that in any consistent axiom system rich
        enough to generate the arithmetic of the natural numbers, there
        are statements we cannot prove in the system, but which can be
        _seen_ by other means to be true (Brainstorms, p. 257).

        ...what Godel proved, _beyond any doubt_, is that when it comes
        to axiomatizing simple arithmetic, there are truths that "we
        can see" (Godel) to be true but that can _never_ be formally
        proved to be true (Darwin's Dangerous Idea, p. 429).

Its also important to understand that Godel's theorem (and the above simple
English statement of that theorem) is all but universally regarded as 
valid or "true" by mathematicians:

	The technical details of Godel's proof itself need not concern us;
	*no mathematician doubts its soundness* (ibid p. 429).

>First, for any such system, there is another consistent system in which the
>Godel sentence of the first system can be proven.

Which in turn has its Godel sentence which can not be proven and on and on.

> The "we" here is a red herring.

The point is that we can "see" truths that neither we nor our computers can
prove.  You need a "we" to do the seeing, at least for the time being, since
"we" are the only systems we know of at the moment that can accomplish this
feat.

>In addition, inconsistent systems of the same strength can also
>"prove" that Godel sentence.

Formal systems which prove theorems employing inconsistent logic?  That's
a new one on me.  I'm all ears.

>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.

In other words, the truth we can "see" is not some universal truth, but one
which, in this case, is context dependent and limited in its scope.  I don't
have a problem with that.  But it still leaves the argument untouched.  We
can just "see" a truth (albeit context dependent and limited in scope) that
neither we nor our machines can prove.

>And, just as we can "see" that Godel's theorem is
>true (after all, it is a theorem),

I'm not so sure we can.  But its irrelevant.  The point is that Godel
has _proven_ his theorem, and what it says about the nature of cognition
appears to be fairly startling, that we can "see" truths that can't be
proven.

>there are, contrary to popular belief,
>formal consistent systems in which Godel's theorem can be derived (after all,
>it is a theorem).
>

Correct but irrelevant.

>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.

I think you mean we could not prove our own Godel sentence, even though we
can see that it must be true.  But that's just a repetition of Godel itself,
since it merely refers to truths which can not be proven, by humans _or_
machines.

>This is not a contradiction; rather it implies that we would not be
>able to recognize or derive our own Godel sentence.

We might well be able to _recognize_ it, but we would not be able to
derive (prove) it, since that's precisely what Godel's theorem says.

>This is not too terribly
>subtle, but it is too much for Lucas and many of his readers.

If by that you mean those who agree with Lucas, they include both Godel and
Penrose, a couple of pretty dim bulbs in your book, eh?

        Godel himself thought that the implication of his theorem was
        that human beings -- at least the mathematicians among us --
        cannot, then, be just machines, because they can do things no
        machine can do (ibid., p 430).

>>      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.

If consistent systems strong enough for Godel's purposes can't prove the
Godel sentence, then all the worse for inconsistent systems or systems
insufficient for simple arithmetic.  Unless, of course, you can show
us how to prove theorems using inconsistent logic, which you seem to
think possible (above).

>
>>      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.

Yes it does.  It says we can "see" it even though we can't prove it.

>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.
>

Its not Lucas, but Godel that concerns us here.  And he has actually _proven_
that, although we can prove Godel's theorem, neither we nor our machines
can _prove_ our Godel sentences.  So why would Lucas want to show that we
ourselves can do something (prove our G sentence) that the theorem from
which he derives his argument says we cannot do?

>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.
>

True.  Nor can they prove theorems.  Unless, of course, you plan to show us
how.

>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.
>--
>

Why then do you think two of the century's most eminent mathematicians
have, indeed, wasted so much time on it?  What truth is it that you can
just "see" that seems to have evaded both Godel and Penrose, and me,
for that matter (at least I'm in pretty good company)?


I think its important to keep in mind the fact that there are actually
three issues at play here:

1. The validity of Godel's theorem (including what it says, I suppose).
2. The validity of Lucas' argument that it applies to material systems,
   and therefore that minds are different from theorem proving machines.
3. The testability of Lucas' argument.

Its actually 3 which is where Dennett thinks Lucas and Penrose are most 
vulnerable (ibid, Chapt 15), and which is why I have assumed my own 
theory and the _empirical_ _evidence_ I have presented should be of 
some small interest to thoseworking in strong A.I.
        
>
>--
>
>Phil Roberts, Jr.
>
>Feelings of Worthlessness from the Perspective of
>So-Called Cognitive Science
>
>http://www.geocities.com/Athens/5476
>