Newsgroups: comp.ai.philosophy
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From: daryl@oracorp.com (Daryl McCullough)
Subject: Re: Penrose's Argument
Message-ID: <1994Nov17.141402.1811@oracorp.com>
Organization: Odyssey Research Associates, Inc.
Date: Thu, 17 Nov 1994 14:14:02 GMT
Lines: 89

Lupton@luptonpj.demon.co.uk (Peter Lupton) writes:

>[stuff deleted]
>
>Suppose we accept the thrust of Penrose's argument to the
>effect that 'human mathematicians are not using a knowably
>sound algorithm in order to ascertain mathematical truth'
>(page 76 SOTM).
>[I am aware that some will rail against this initial step,
>pointing out that the very idea of 'unassailable 
>mathematical truth' is incoherent, but bear with me.]
>Suppose we accept all this. What follows?
>
>Well, we can say that there may be an algorithm but that
>it is unknowable in the sense of 'know' required for Penrose's
>                 ---------------------------------------------
>use of Goedel's theorem. 
>------------------------

Peter, 

Actually, Penrose claims to have covered this possibility.
Let me try to make Penrose' response precise:

Let T be any algorithmic theory. Since Penrose finds it unassailably
true that his reasoning is sound, Penrose believes the following:
If T = Penrose (that is, if T is capable of reproducing all of
Penrose' reasoning abilities) then T is sound. (Meaning that all
the statements proved by T are in fact true.) Since adding any true
statement to a sound theory produces another sound theory, Penrose
concludes that:

1. If T = Penrose, then T + "T = Penrose" is sound.

Now, since Penrose believes statement 1, it must be the case that:

2. If T = Penrose, then T proves statement 1.

An obvious consequence of 2 is:

3. If T = Penrose, then T + "T = Penrose" proves statement 1.

But statement 1 is a conditional: If T = Penrose, then ...
And the theory T + "T = Penrose" certainly proves the hypothesis,
T = Penrose. Therefore, the theory proves the conclusion. So we
have:

4. If T = Penrose, then T + "T = Penrose" proves T + "T = Penrose" is
sound.

Since every sound theory must also be consistent, then it follows that:

5. If T = Penrose, then T + "T = Penrose" proves T + "T = Penrose" is
consistent.

But by Godel's theorem, no theory can prove itself to be consistent
unless it is actually inconsistent. Therefore,

6. If T = Penrose, then T + "T = Penrose" is inconsistent.

Now, if adding the statement "T = Penrose" to theory T makes T
inconsistent, it must be the case that T can prove "T is not equal to
Penrose". Therefore, if T = Penrose, then T can prove that T is not
Penrose, which is a false statement. Therefore:

7. If T = Penrose, then T is unsound.

An obvious consequence of this is:

8. If T = Penrose, then Penrose is unsound.

Which is logically equivalent to:

9. If Penrose is sound, then T is not equal to Penrose.

Since Penrose is convinced that his own unassailable beliefs are sound,
it follows by Modus Ponens that:

10. T is not equal to Penrose.

Therefore, since T was an arbitrary algorithmic theory, it follows
for Penrose that no theory T can be equal to his own reasoning.

Daryl McCullough
ORA Corp.
Ithaca, NY



