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Article 5455 of comp.ai.philosophy:
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>From: ramsay@unixg.ubc.ca (Keith Ramsay)
Newsgroups: comp.ai.philosophy
Subject: Re: penrose
Message-ID: <1992May6.220605.26774@unixg.ubc.ca>
Date: 6 May 92 22:06:05 GMT
References: <2524@ucl-cs.uucp> <1992May1.025230.8835@news.media.mit.edu>
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In article <1992May1.025230.8835@news.media.mit.edu>
minsky@media.mit.edu (Marvin Minsky) writes:
> The math seems generally OK, but the stuff on universal Turing machines seems
>amateurish.  He either did not know, or neglected to point out that
>there are known to be very small Universal Turing Machines (e.g, 4
>symbols, 7 states).  

Is there some special significance to this fact (so that one would
make a special point of including it)?

>  So far as I can see, Penrose's discussion about Godel's theorem
>depends on making peculiar assumptions about (1) that humans have
>magical abilities to recognize mathematical truths

I think it is a little more subtle than that. Penrose (incorrectly,
IMO) concludes the non-computable abilities from our ability to apply
Godel's reasoning to arbitrary formal systems. It is more a mistake
than an untoward assumption.

>and (2) that the
>Turing machines aren't allowed to generate new Turing machines that
>use different sets of axioms.

I'm not sure I see the relevance of this. One can emulate a
deterministically evolving "community" of Turing machines using just
one Turing machine. The theorems of the community are then just
theorems generated by one machine, but tagged by "source". How does
this help?
-- 
Keith Ramsay           Let the discourse of the mathematicians cease.
ramsay@unixg.ubc.ca    ...both the teacher and the taught shall be
                       sentenced to death.  - The Theodosian Code, c.365


