From newshub.ccs.yorku.ca!ists!helios.physics.utoronto.ca!news-server.csri.toronto.edu!rpi!usc!wupost!uunet!psinntp!scylla!daryl Thu Jan 16 17:19:55 EST 1992
Article 2666 of comp.ai.philosophy:
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>From: daryl@oracorp.com
Subject: Re: Penrose on Man vs. Machine
Message-ID: <1992Jan13.174008.1114@oracorp.com>
Organization: ORA Corporation
Date: Mon, 13 Jan 1992 17:40:08 GMT

Mikhail Zeleny writes:

> My *argument* is that understanding is non-algorithmic; my assumption
> is only that I am capable of understanding.

Fine. My point is that Penrose has no non-circular argument that
understanding is non-algorithmic.

> Refer to the infamous page 110, containing Penrose's discussion of
> reflection principles, with its hitherto unappreciated by you proviso:
> "by `reflecting' upon the _meaning_ of the axiom system and the rules
> of procedure"...  Yes, he says `meaning', and such reflection is
> arguably non-algorithmic; for even the concept of a formal proof is
> dependent on such second-order notions as those of finitude and
> effectiveness.  So kindly bag the charges of petitio principii: the
> argument is there, even if unappreciated by you.

I would never accuse someone of petitio principii! I might accuse
someone of circular reasoning, now and then, but I try to avoid Latin.
8^)

Anyway, I don't believe that "reflecting on the meaning" of a theory
will tell you whether a theory is consistent or not. Take Quine's New
Foundations; after all these years, we are no closer to knowing
whether it is consistent. Take ZFC plus there exists a measurable
cardinal. We still don't know whether it is consistent.
There is no evidence that "thinking about the meaning" of a
theory will tell us whether it is consistent or not. And certainly
Penrose has no argument showing that "reflecting about the meaning"
will *always* tell us whether a theory is consistent. He claims it,
but gives no justification, except for the single example of showing
that PA is consistent (which, as I said, is provable in ZFC).

The best we can do as far as showing consistency is the following:
*if* we can think of a model for a set of axioms, *then* we believe
that it is consistent. But there is no evidence that just because a
theory has a model, that we will be able to think of it.

> Kindly show me how a machine can understand a formal theory by
> reflecting upon the _meaning_ of its axiom system and its rules of
> procedure.  No fair using ZFC to reflect on PA, and such, -- I want
> it to reflect on the strongest theory it's presently using.

What do you mean, "no fair"? ZFC can prove the consistency of PA
because ZFC is a more powerful theory than PA. We believe the
consistency of PA because our understanding of arithmetic is more
powerful than PA, as well. That's all there is to it. The fact that
we are more powerful than one theory doesn't imply that we are more
powerful than *every* theory. There are theories whose consistency we
*can't* demonstrate; I've already given a few examples.

Daryl McCullough
ORA Corp.
Ithaca, NY


