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Article 2833 of comp.ai.philosophy:
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>From: jbaez@nevanlinna.mit.edu (John C. Baez)
Newsgroups: comp.ai.philosophy,sci.philosophy.tech,sci.logic
Subject: Re: Penrose on Man vs. Machine
Message-ID: <1992Jan17.165308.12243@galois.mit.edu>
Date: 17 Jan 92 16:53:08 GMT
References: <1992Jan16.112129.7632@husc3.harvard.edu> <1992Jan16.123103.2429@arizona.edu> <1992Jan16.233232.7674@husc3.harvard.edu>
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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.
>
>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. 

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.

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

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.


