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Article 2501 of comp.ai.philosophy:
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>From: daryl@oracorp.com
Newsgroups: comp.ai.philosophy,sci.logic,sci.philosophy.tech
Subject: Re: Penrose on Man vs. Machine
Message-ID: <1992Jan5.171147.27621@oracorp.com>
Date: 5 Jan 92 17:11:47 GMT
Organization: ORA Corporation
Lines: 52

I have been gone for two weeks, and while this thread continues, it
seems no longer to have anything to do with Penrose. Since our site
only keeps about 100 news articles for each newsgroup, I may have
missed the denouement of the discussion about whether Penrose'
arguments have any validity. At the risk of boring you readers with
repetition, let me reiterate my complaint about Penrose to see if
there are any responses:

Penrose claims to have demonstrated in his arguments in _The Emperor's
New Mind_ that human reasoning is not formalizable. He considers two
notions of formalization: as a Turing machine program, and as an
axiomatic, first-order theory. He has essentially the same argument for
both, so I will stick to the latter.

The major step in his demonstration is an informal argument that *seems*
to establish the following claim:

     A. For all axiomatic theories T extending Peano arithmetic,
        if T is consistent, then Roger Penrose knows some true statement
        of arithmetic that T does not prove.

>From this claim, the claim that human reasoning is not formalizable
follows easily (granting for the time being that Penrose is not
inconsistent himself).

However, Penrose does not establish claim A.; he establishes a much weaker
claim:

     B. For all axiomatic theories T extending Peano arithmetic,
        if Roger Penrose knows that T is consistent, then Roger
           ^^^^^^^^^^^^^^^^^^^^^^^^
        Penrose knows some true statement of arithmetic that T
        does not prove.

The addition of the phrase "Roger Penrose knows that" makes a subtle
but crucial difference that Penrose seems unaware of. While claim A.
has the desired consequence---Human reasoning is not
formalizable---claim B has the much weaker consequence---Human
reasoning cannot be captured by any formal system known (by us humans)
to be consistent.

This weakening leaves open the possibility of formalizing human
reasoning by a formal system (such as ZFC, or ZFC plus the existence
of a measurable cardinal), whose consistency is in question. I find
this completely plausible; given the strangeness of some human
reasoning, I believe that if we tried to capture it with a formal set
of rules, it would not be obviously consistent.

Daryl McCullough
ORA Corp.
301A Harris B. Dates Dr.
Ithaca, NY 14850-1313


