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Article 2549 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: <1992Jan8.120438.7226@husc3.harvard.edu>
Date: 8 Jan 92 17:04:36 GMT
References: <1992Jan7.031553.24886@oracorp.com> <1992Jan7.105117.7193@husc3.harvard.edu> <OZ.92Jan8051539@ursa.sis.yorku.ca>
Organization: Dept. of Math, Harvard Univ.
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In article <OZ.92Jan8051539@ursa.sis.yorku.ca> 
oz@ursa.sis.yorku.ca (Ozan Yigit) writes:

>M. 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.

OY:
>Since you offer no useful explanation of your claim regarding
>our ability to "understand ... a program halting on a given input",
>one presumes you are still waving the mystical lucas/penrose banner.
>If not, perhaps you care to produce your very own explanation and/or
>proof of this claim.

Avec plaisir, mon vieux, though I was hoping for an audience capable of
putting 2 and 2 together...  The notion of a program halting on a given
input is dependent on the concept of finitude, which, as I have rather
laboriously explained, is second-order, and hence non-algorithmic.  More
information can be found in Church's logic book, and any standard text on
recursion theory.

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

OY:
>dream on.

`Arguably' means that there exists an argument, so your oneiric invitation
is quite unnecessary.  If our understanding of any sentence S (belonging to
a language L, which includes PA) expressing a proposition P were
algorithmic, then, by Church's thesis, the relation between S and P would
be recursive; hence we could use G\"odel's arithmetization trick on the
semantic relation of expression with predictable results.

If you don't like this predicament, it's incumbent upon you to develop a
semantic theory that wouldn't quantify over abstract objects.  As notes
Putnam, this includes even such extensional abstract objects as the
truth-values.  Good luck, and let Paul Churchland know is you succeed;
he'll be green with envy.

>oz


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: Mikhail Zeleny                                                     :
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