Newsgroups: comp.ai.philosophy
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From: daryl@oracorp.com (Daryl McCullough)
Subject: Re: Penrose's new book
Message-ID: <1994Oct21.130352.19564@oracorp.com>
Organization: Odyssey Research Associates, Inc.
Date: Fri, 21 Oct 1994 13:03:52 GMT
Lines: 54

boroson@spot.Colorado.EDU (BOROSON BRAM S) writes:

>But it does seem to me
>
>1) "True" arithmetic, by Godel's Theorem, can not be derived from a formal
>axiomatic system.  So a Turing-machine computer could not derive all true
>statements of arithmetic, which Penrose claims human beings could know.
>
>You've pointed to a limitation in a system a computer could not encompass;
>this does no harm to Penrose's claims.

What it shows is that Penrose' arguments for why human reasoning
cannot be completely captured by a Turing machine can equally well (or
equally badly) be used to show that human reasoning cannot be
described by *any* mathematics whatsoever:

(1) Assume that there exists a mathematical description of a human
brain.

(2) Using that description, describe in set theory the collection T of
all statements (in the language of set theory) that a human could ever
discover to be "unassailably true". (T may not be Turing computable,
but that doesn't matter.)

 (3) Since we know that we are consistent, we conclude that T is
consistent.

(4) The statement "T is consistent" is formalizable in set theory.

(5) Since we "know" T is consistent, and by assumption, T exactly
expresses those statements we can come to know, if follows that the
statement "T is consistent" is a theorem of T.

(6) By the extended Godel's theorem, T must be inconsistent.

Therefore, the assumption that human brains are describable by
mathematics at all, together with the assumption that we know we are
consistent leads to the conclusion that we are inconsistent. But
if we are inconsistent, that invalidates Penrose' argument that
our reasoning cannot be described by a Turing machine (inconsistent
theories can prove their own consistency).

So the only way to keep Penrose' arguments intact is to suppose that
human brains are beyond the descriptive capacity of any mathematics,
whatsoever. This would seem to doom from the start Penrose' program
of discovering the physics behind the operation of the brain.

Personally, I think the problem is (3): we don't know that we are
consistent. (Actually, some people, such as Penrose, believes that he
is consistent, but he's wrong.)

Daryl McCullough
ORA Corp.
Ithaca, NY
