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Article 2592 of comp.ai.philosophy:
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>From: zeleny@zariski.harvard.edu (Mikhail Zeleny)
Newsgroups: comp.ai.philosophy,sci.philosophy.tech,sci.logic
Subject: Re: Penrose on Man vs. Machine
Message-ID: <1992Jan9.062913.7262@husc3.harvard.edu>
Date: 9 Jan 92 11:29:11 GMT
References: <1992Jan8.133305.22630@oracorp.com>
Organization: Dept. of Math, Harvard Univ.
Lines: 80
Nntp-Posting-Host: zariski.harvard.edu

Before making your reply, kindly reflect on your past responses.  I have a
feeling that we have been going around and around on this one point.

In article <1992Jan8.133305.22630@oracorp.com> 
daryl@oracorp.com writes:

>Mikhail Zeleny writes:

MZ:
>> Since the notions of a program halting on a given input, or a theory
>> being consistent are fundamentally second-order, i.e. non-recursive,
>> our ability to understand them is sufficient evidence of our ability
>> to perform non-algorithmic tasks.

DMC:
>If your basic assumption is that understanding is non-algorithmic,
>then Penrose' argument is unnecessary, and certainly doesn't add any
>additional plausibility to your assumption. The fact that you can talk
>about second-order notions doesn't imply that you are using second-order
>reasoning; ZFC can talk about second-order properties, as well.

Please give me a break, Daryl.  My *argument* is that understanding is
non-algorithmic; my assumption is only that I am capable of understanding.
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.

Incidentally, first-order ZFC can only "talk about" second-order properties
like finitude and denumerability, as related to the standard model.  But we
have already discussed that, too.

MZ:
>> Indeed, it is arguably true that all understanding is fundamentally
>> non-algorithmic;

DMC:
>The question of this thread is whether the Penrose argument is
>evidence in favor of this claim. Bringing in additional arguments that
>our understanding must be non-algorithmic doesn't help to show that
>the Penrose argument is correct; for an argument to be correct it is
>not sufficient for the conclusion to be true, as you know.

Agreed.

MZ:
>> however, in view of our past disagreements, I shan't repeat an
>> argument to that effect, limiting myself to the claim that it is
>> intuitively obvious to me that I am capable of understanding.

DMC:
>The argument is not whether you understand, but whether machines are
>also capable of understanding.

Quite so.  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.

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


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: Mikhail Zeleny                                                     :
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