Newsgroups: comp.ai.philosophy
From: Lupton@luptonpj.demon.co.uk (Peter Lupton)
Path: cantaloupe.srv.cs.cmu.edu!das-news2.harvard.edu!news2.near.net!howland.reston.ans.net!pipex!demon!luptonpj.demon.co.uk!Lupton
Subject: Re: Godel, Lucas, Penrose, and Putnam
References: <3ddp99$tc@usenet.ucs.indiana.edu>
Distribution: world
Organization: No Organisation
Reply-To: Lupton@luptonpj.demon.co.uk
X-Newsreader: Newswin Alpha 0.6
Lines:  93
X-Posting-Host: luptonpj.demon.co.uk
Date: Wed, 28 Dec 1994 13:00:22 +0000
Message-ID: <686752099wnr@luptonpj.demon.co.uk>
Sender: usenet@demon.co.uk

In article: <3ddp99$tc@usenet.ucs.indiana.edu>  
chalmers@bronze.ucs.indiana.edu (David Chalmers) writes:

>However, while the argument just described takes up most of Penrose's
>discussion, he has a quite separate and more interesting argument up his
>sleeve toward the end of Chapter 3 (summarized in 3.23).  Unfortunately it
>is easy to miss this argument on just skimming the book, although in my
>discussions with Penrose he has been quick to point to it as carrying the
>book's main burden.  (In discussion he has conceded the loophole in his
>previous argument, although he says he finds it hard to believe that the
>algorithm underlying mathematical thought could be so complex.)  Putnam
>appears to have missed the new argument entirely, which is a pity, and
>which significantly weakens his review.

Yes, this argument is presented in detail on pages 165-166 of 
'Shaman of the Mind' and is quite interesting in its own right.

>Penrose's second argument proceeds, in essence, by saying "suppose I *am*
>formal system F", and then arguing that with this assumption in place, he
>can see the truth of statements that are beyond F's powers, even if F is
>supplemented by the assumption that it is F.  Let F' be the system derived
>by supplementing F with the assumption that it is F.  Then for the usual
>reasons F' cannot see the truth of its Godel sentence G(F').  But Penrose
>argues that (under the assumption that he is F) he can see the truth of
>G(F'): he knows that he is consistent, so (under the assumption that he is
>F) he knows that F is consistent, and indeed that F' is consistent, so he
>knows that G(F') is true.  This contradicts the initial assumption that he
>is F.  Similar reasoning shows that he cannot be any formal system.

This argument by Penrose seems to be unsound and unsound in a
very specific way. Clearly, F' is not the same formal system as F.
That is plain enough, since the formal system F is not supposed
to know that "I am F", but F' has, as an axiom, that "I am F".

However, since F is not equal to F', what is this axiom "I am F" doing
here? If F' is *not* F, then how can it have, as an axiom, that it *is*
F? Look again at the statement "I am F". It asserts, in effect, that
"This statement is not an axiom" (Since F does not have that statement 
as an axiom!). Now add it as an axiom. Viola! Unsoundness. Of course adding 
a statement that has the effect of denying its own axiom-hood as an axiom 
is unsound. Who would ever have thought otherwise?

What arguments does Penrose offer to persuade us that F' is sound?
The argument is that F' would be sound on the basis of "I am F"
because:

   "F' *does* encapsulate what it unassailably believes concerning
    its ability to derive PI-1 sentences on the basis of M" 
    [where M  = "I am F", in our notation]

This won't do, since it just considers the question of further 
deductions. What we want is an argument about adding "I am F"
to F in the first place. This Penrose does not argue for.

The positive conclusion I infer from this is the following:

 If system P contains the formal representation of Q
 and P uses Q to derive the Goedel statement G(Q) (and
 others of the same ilk), and if P and Q are both sound, then 
 P is not equal to Q.

>This argument avoids the obvious flaws with the Lucas argument and the
>first Penrose argument, as it nowhere requires that he is directly able to
>determine the soundness of F.  It simply requires that he can know that he
>himself is sound.  This is a much weaker claim, and indeed one that Putnam
>is in effect willing to grant.  I think there are still some places at
>which the argument is vulnerable -- Daryl McCullough has argued
>persuasively in correspondence that Penrose's assumption that he knows he
>is sound already leads to a contradiction, independently of the assumption
>that he is computational -- but the flaws are less obvious.  

I, too, have been involved in private communication with Daryl. However,
I am less persuaded of Daryl's argument. In effect, Daryl accepts the
soundness of the transition from F to F' (which I reject). Daryl thus
argues from the manifest unsoundness of F' (something I don't deny)
to the unsoundness of F. This leaves Daryl asserting that the very idea
of believing oneself to be sound and using it ala "the Goedelian insight"
is irredeemably unsound. My own position is that I can see no reason for
Daryl's conclusion - there is a very specific unsound move made by Penrose,
the transition from F to F'. On my account the unsoundeness of F' leaves
untouched the question of whether F is or is not unsound.

>So I think
>this is a more interesting and original argument than the previous one; it
>is one of the more significant contributions in Penrose's book.  It would
>be nice if discussion focused on this argument rather than on the
>hackneyed previous version.

Glad to oblige.

Cheers,
Pete Lupton
