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Article 2854 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: <1992Jan17.174414.7730@husc3.harvard.edu>
Date: 17 Jan 92 22:44:11 GMT
References: <1992Jan16.123103.2429@arizona.edu> <1992Jan16.233232.7674@husc3.harvard.edu> <1992Jan17.165308.12243@galois.mit.edu>
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In article <1992Jan17.165308.12243@galois.mit.edu> 
jbaez@nevanlinna.mit.edu (John C. Baez) writes:

>In article <1992Jan16.233232.7674@husc3.harvard.edu> 
>zeleny@zariski.harvard.edu (Mikhail Zeleny) writes:

Mikhail Zeleny:
>>>>To sum up, my position is that, were I a Turing machine, there would exist
>>>>some formal mathematical theory whose meaning I couldn't understand.
>>>>Personally, I find this implausible; feel free to judge to the contrary.

BS:
>>>Well, okay, but come on!  Suppose I were to take a simple theorem
>>>of algebra, such as the statement "\pi is transcendental", and
>>>translate it into the language of Principia -- it would go on
>>>for pages and pages, but I could do it.  Suppose I were then to
>>>invert every quantifier, replacing "there exists" with "for every"
>>>and vice versa.  The resulting statement would be well formed, and
>>>you would have not a chance in hell of understanding what it means.

MZ:
>>You misunderstand me, Bill: I said that I'll understand it, provided that
>>there is anything to understand.  In other words, I trust that you will
>>make it meaningful, rather than merely well-formed. 

JB:
>I think your point boils down to this: you'll be able to understand it,
>if the guy who wrote it down understands it.  This is just a claim that
>your mathematical ability is >= to his.

Yes on the first, no on the second.  Do you seriously believe that
Dedekind's accomplishment in understanding set theory was greater than
Cantor's in discovering it?  Even though I believe mathematical
understanding to be essentially creative, I certainly don't take it to be
nearly as creative as mathematical discovery.

BS:
>>>Have you actually worked with mathematical logic?  I can hardly
>>>believe that you're disputing this.

MZ:
>> [various erudite ramblings deleted.]
>>
>>Perhaps you still haven't figured out my point.  The epistemological claims
>>of strong AI entail that we can produce a mathematical theory of such
>>complexity, that we would be unable to reflect on its interpretation.  I
>>contend that this is absurd, since in order to formulate such a theory, we
>>would already need to understand its meaning.  Prove me wrong.

>>>	-- Bill

JB:
>BS back there has already shown how to formulate a theory that we can't
>understand: take a long formal proof of something, chop off the first
>200 lines, invert the quantifiers in the remaining sentences, and call
>them axioms.  There's a theory you can't understand, though you can
>easily deduce some consequences.  No doubt you will say that this is not
>an honest theory, because it was concocted for cynical reasons.  Well,
>okay: we usually make theories for which we *can* "reflect on their
>interpretation," or, to use the technical term, which we can grok.  Just
>like we try to build tools we can use.

This is not an honest theory, because it was concocted through purely
syntactic means.  From the standpoint of lexical semantics, which assigns
meaning to words according to their morphological structure, even Lewis
Carroll's "Jabberwocky" is perfectly meaningful, albeit in a nonstandard
way, insofar as it is embedded in a corpus of meaningful linguistic
phenomena, the "theory" of Victorian English.  Not so with Bill's little
concoction, which is presumably to be taken in isolation from other,
meaningful mathemathical theories.

`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'
: Qu'est-ce qui est bien?  Qu'est-ce qui est laid?         Harvard   :
: Qu'est-ce qui est grand, fort, faible...                 doesn't   :
: Connais pas! Connais pas!                                 think    :
:                                                             so     :
: Mikhail Zeleny                                                     :
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