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Article 2683 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: <1992Jan13.200000.7489@husc3.harvard.edu>
Date: 14 Jan 92 00:59:57 GMT
Article-I.D.: husc3.1992Jan13.200000.7489
References: <1992Jan13.175936.2755@oracorp.com>
Organization: Dept. of Math, Harvard Univ.
Lines: 82
Nntp-Posting-Host: zariski.harvard.edu

In article <1992Jan13.175936.2755@oracorp.com> 
daryl@oracorp.com writes:

>Mikhail Zeleny writes:

MZ:
>> Our understanding of the results of first-order ZFC is related to our grasp
>> of its standard model.  I invite you to meditate on the difference between
>> the latter and its countable submodels before you emit any further
>> exclamations.

DMC:
>Okay. Jai Guru Deva Om. I'm finished meditating.
>
>I repeat: there is no standard mathematical result that is not in fact
>a theorem of ZFC. There is therefore no evidence that our
>understanding goes beyond what is captured in ZFC. Nonstandard models
>don't change this empirical fact.

Start with formalism, end up with strong AI.  One way to beat Searle,
Renrose, & Co., not to mention small fry like yours truly, is to deny that
we are capable of any understanding that transcends mindless symbol
manipulation.  Put another way, when Frege argues that mathematics differs
from a game of chess by having a cognitive content, your move would be to
ask: "What content?"  Given the evident infallibility of this move, I
wonder why you bother to keep this conversation going.  One of the main
purposes of a philosophical discussion is to bring to light the fundamental
assumptions of the parties; given that we have done so, I am ready to
declare this discussion over and done with.

MZ:
>> While you are at it, feel free to characterize "sufficiently
>> well" the predicate "... is finite" in a first-order language of your
>> choice. I eagerly await the results.

DMC:
>A set is finite if there is no function mapping it onto a proper
>subset of itself. That is definable in ZFC, if you interpret "function"
>as a set of ordered pairs.

Note that you are quantifying over the non-denumerable set of all functions
from N to N; in a countable submodel of V you would end up recognizing all
sort of interesting infinite sets as finite in virtue of the fact that the
requisite bijection would exist in a standard model of ZFC, but won't be
found in any of its countable submodels.

DMC:
>>> There is no evidence that humans can go beyond formalized reasoning, so
>>> it is not clear what you are demanding be explained.

MZ:
>> On the contrary, any semantic consideration goes beyond formalistic, purely
>> syntactical symbol manipulation.

DMC:
>I didn't ask whether you believed that our understanding goes beyond
>formal reasoning, I asked for evidence that our understanding goes
>beyond formal reasoning. Please don't take offense at the fact that
>I don't consider your beliefs to be evidence.

Do you consider *your* beliefs to be evidence?  If so, do you believe
yourself to be capable of reasoning about the standard model of ZFC, per my
discussion above?  If so, you have evidence that your understanding goes
beyond formal reasoning; if not, let's agree to disagree.


>Daryl McCullough
>ORA Corp.
>Ithaca, NY


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: Mikhail Zeleny                                                     :
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