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Article 7317 of comp.ai.philosophy:
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>From: smaill@aisb.ed.ac.uk (Alan Smaill)
Newsgroups: comp.ai.philosophy
Subject: Re: Human intelligence vs. Machine intelligence
Message-ID: <SMAILL.92Oct17175821@hope.aisb.ed.ac.uk>
Date: 17 Oct 92 16:58:21 GMT
References: <1992Oct15.171636.10178@oracorp.com>
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In-Reply-To: daryl@oracorp.com's message of 15 Oct 92 17:16:36 GMT

In article <1992Oct15.171636.10178@oracorp.com> daryl@oracorp.com (Daryl McCullough) writes:


   The crucial part of the argument hinges on the question of whether
   humans can recognize arbitrary valid mathematical statements. If we
   can, then Penrose is right that we can do something that no machine
   can do. 

Penrose has explicitly said (eg in his reply in "Brain and Behavioral
Sciebce") that he does not claim that humans can recognise _arbitrary_
valid mathematical statements.  He just needs some instances.

   However, what kind of argument does Penrose advance that
   humans are capable of this? A ridiculous one, in my opinion. Just
   because mathematical theories are built up from "simple and obvious
   ingredients" does *not* imply that the consequences are simple and
   obvious. For example, nobody has the faintest idea whether Fermat's
   Last Theorem follows from the simple and obvious axioms of PA (Peano's
   axioms for arithmetic).

It is worth distinguishing between the _principles of reasoning_
involved and the theory in the sense of the true statements of some
domain.  It is only the former that Penrose claims to be clear (surely
no mathematician worthy of the name thinks that the consequences are
"simple and obvious").  


   You are assuming that there are only two possibilities: (a) NF is
   consistent and we can prove that it is consistent, or (b) NF is
   inconsistent.  There is a third possibility: (c) NF is consistent, but
   we can't *prove* that it is consistent. In cases (a) or (b), it does
   indeed follow that NF is inadequate for formalizing all of human
   mathematical reasoning. However, in case (c), there is no
   contradiction in assuming that NF captures all of human mathematical
   reasoning. For Penrose' trick to work, it is not enough that a theory
   be consistent--he must also *know* that it is consistent.  He must be
   able to rule out case (c), not only for NF, but also for *every*
   theory more powerful than Peano Arithmetic.

   There is nothing in any of Penrose' arguments that rules out the
   possibility of a theory T such that:

       1. T is consistent.
       2. T is more powerful than Peano Arithmetic.
       3. T captures all of human reasoning about arithmetic.
       4. It is impossible for human reasoning to prove that T is consistent.

I agree with all this (it wasn't clear to me that this was the point you
wanted to make in your ealier post).

   The mistake Penrose makes is in assuming that because something is
   true, then there exists a proof (or a convincing argument) that it is
   true. A theory T can be consistent without our being able to give a good
   argument for why it is consistent.

I'd be surprised if the sort of constructivist attitude you attribute to
Penrose here is one that he holds, given his total lack of sympathy for
Brouwer et al in his book.
--
Alan Smaill,                       JANET: A.Smaill@uk.ac.ed             
Department of Artificial           ARPA:  A.Smaill%uk.ac.ed@nsfnet-relay.ac.uk
       Intelligence,               UUCP:  ...!uknet!ed.ac.uk!A.Smaill
Edinburgh University. 


