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Article 5888 of comp.ai.philosophy:
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>From: peter@cs.vu.nl (Grunwald PD)
Newsgroups: comp.ai.philosophy
Subject: Re: Universe is a big place ,,,
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Date: 25 May 92 10:57:32 GMT
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dlh@cogsci.ed.ac.uk (Dominik Lukes) writes:

>,,, perhaps the biggest. Say! Why do I have to understand the mathematical
>part of the Goedel's incompleteness theorem to be able to use it for
>speculations about human cognitive ability. Well, I don't want to
>solve math problems, it bores me stiff. Anyway, as far as I can judge
>all the stuff only hangs on the pressupositions from ordinary life,
>supported by our natural language, that "true and unprovable" is
>somewhat more sound than "false and provable". How can the math part
>enhance my understanding of it. Is it banned to say sentence like "the
>natural language reasoning itself exhibits incompleteness,"( as it is
>known for some two and a half millenia) without providing mathematical
>proof? It is, by all means, more esthetical, but for whats sake?
>Kick me, shoot me, but tell me why!

Well, interestingly there are people who view the (original) construction
of the proof of Goedel's theorem as a proof in itself (or at least, an
indication) that the _power_ (!) of formal systems goes beyond our expectations

...these people therefore conclude from (the way) Goedel's theorem (was proven)
exactly the opposite than the Penrosians, Lucasians, Dreyfuzeniks etc. do,
 namely that 'semantics is an emergent quality of complex syntax' 
(quote from Hofstadter "Metamagical Themas", page 445)

Now Hofstadter is not taken very seriously by many in this group and otherwise,
but this view should surely be taken notice of...and it is only understandable
if you look, indeed, at the 'math part' of the proof!

To quote Hofstadter fully:

"Moreover, Goedel's construction revealed in a crystal-clear way that the line
between 'direct' and 'indirect' self-reference (indeed, between direct and
indirect _reference_, and that's even more important!) is completely blurry,
because his construction pinpoints the essential role played by _isomorphism_
(another name for coding) in the establishment of reference and meaning.
Goedel's work is, to me, the most beautiful possible demonstration of how any
notion of 'direct' meaning (ie codeless meaning) is incoherent. In brief, it
shows that _semantics is an emergent quality of complex syntax_, ...."

The problem (that is, MY problem) with many 'popularizations' of Goedel's
proof (e.g. Penrose!!!!!) is that they only say that Goedel was able to
construct a mathematical sentence that said 
  'I am not provable within the formal system for which I am constructed'

...they don't mention that this sentence has quite another meaning too...
it is a statement about number theory (or 'Aehnliche Systeme', if you want)
that is undecided by the axioms of that theory.



>Yours sincere & faithfull,
>Dominik.
>======================================|
>    My spELling iS wobbly.            |  
>It's goOd spelling bUt it wobbles     |  
>   and tHe letters get iN             | 
>      the wrOng plaCes.               |  
>,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,| .


Ciao,

Peter
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*								 	     *
* Peter Grunwald							     *
* peter@cs.vu.nl							     *
* tel. 0031206685792 / 0031205484149					     *
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* Charles de Gaulle _off the record_: "In fact, it is true: a country that   *
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