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Article 2382 of comp.ai.philosophy:
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>From: daryl@oracorp.com
Subject: Re: Penrose on Man vs. Machine
Message-ID: <1991Dec23.165606.5935@oracorp.com>
Organization: ORA Corporation
Date: Mon, 23 Dec 1991 16:56:06 GMT

DMC:
>>Therefore, in these cases, Penrose' arguments amount to the following:
>>
>>    (1) Assuming that we can tell which Turing machines H are partially
>>        correct for solving the halting problem, then our reasoning
>>        is nonalgorithmic.
>>
>>    (2) Assuming that we can tell which theories are sound, then our
>>        reasoning is nonalgorithmic.
>>
>>In other words, assuming that we can do things that no machine can do,
>>then we can do things that no machine can do.

MZ:

>Quite so.  As I stated earlier, all analytic arguments can be
>construed as a form of begging the question by an opponent who doesn't
>accept the intuitive validity of their premises.

Agreed, the point of such an argument is to reduce the truth of some
questionable statement to the truth of some statements that we find
more intuitively valid. However, I think that Penrose' premises, that
we can solve all instances of the halting problem, and that we can
tell whether an arbitrary collection of axioms is consistent, is much
*less* plausible than his conclusion, that our minds are not
algorithmic.

It seems to me that there is no empirical evidence suggesting the
truth of the premises. As far as I know, every rigorous proof in
mathematics (at least in the limited domains of number theory and
analysis) has proved a theorem of ZFC, so there is no example of a
mathematical result that is beyond purely formal methods.

MZ:
> I hope that you would understand that your claim about the purely
> formal nature of reflexion principles as illustrated by Penrose was
> made in error.

Let me be more precise. While I might agree that the meaning of a
reflection principle may be beyond what is formalizable in first-order
logic (just as the meaning of arithmetic is beyond such
formalization), there is a way to formalize a fragment of that meaning
in set theory. My claim is that there is no mathematical result that
depends on a reflection principle that isn't also a consequence of the
fragment of the reflection principle that ZFC is capable of.
Therefore, there is no evidence that reflection principles enable us
to go beyond what is provable in ZFC.

Another point that makes Penrose' claim about the power of reflection
principles dubious: Penrose claims that, given an a syntactic theory,
by reflecting on the *meaning* of the symbols, we can arrive at truths
that are not derivable in that theory. This implies that our minds
are, in a sense, more powerful than any syntactic theory whose meaning
we can understand. However, to go from this claim to the claim that
our minds are more powerful than *any* syntactic theory, it's
necessary to assume that we can figure out a meaning for an
*arbitrary* syntactic theory, not simply variants of first-order
arithmetic and set-theory!

I find that assumption quite unbelievable: it would imply that given
an arbitrary consistent r.e. set of axioms, we can figure out a
meaning for the symbols that makes all the axioms true, and that
allows us to use reflection to get a new, true but unprovable
statement. Penrose only considered theories like PA that come with a
ready-made interpretation (because they were invented by humans, with
a particular human-understandable meaning in mind).

If you don't buy the fact that we can make sense out of an arbitrary
consistent collection of axioms, then Penrose' argument leads not to
his desired conclusion, but to the conclusion that:

     If I were given a first-order axiomatization of my own mind, I
     wouldn't be able to make sense of it.

You might be interested in a brief conversation that I had with
Penrose following his lecture at Cornell on The Emperor's New Mind.

Daryl: Professor Penrose, it seems to me that there is no mathematical
       result that human mathematicians can rigorously derive that is
       not a consequence of ZFC, a purely formal system.

Penrose: We know that ZFC is consistent, which is something that ZFC cannot
         prove.

Daryl: Wait a minute! What's your argument that ZFC is consistent?

Penrose: If it's not consistent, why would anybody bother using it?
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Daryl McCullough
ORA Corp.
Ithaca, NY






