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From: jqb@netcom.com (Jim Balter)
Subject: Re: Open Letter to Professor Penrose
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Date: Wed, 3 Jul 1996 11:23:46 GMT
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In article <4r03ud$qmn@highway.leidenuniv.nl>,
Theo Vosse <vosse@ruls41.fsw.LeidenUniv.nl> wrote:
>Tom/Julie Haight-Curran (tomorjul@primenet.com) wrote:
>
>: Now this little self-referential item really doesn't have anything to do
>: with standard arithmetic or math.
>
>Arithmetics isn't all there is to math. If you would like to prove e.g.
>that there exist non-context-free languages, you won't get very far using
>arithmetical operations only. Or the four-colour theorem, or
>NP-completeness. Even more arithmetical theorems (there is no largest prime
>number, etc.) cannot be proven by only applying arithmetics. They all
>involve some higher order logic, using quantors etc. And that's were Goedel
>kicks in.

Hmmm, Euclid employed "quantors" to show that there is no greatest prime, eh?

>: What you're really saying is the proof-mechanism in question
>: can't handle nonmathematical self-reference when formalized in quantification 
>: theory. Big deal!
>
>Let's switch over then to Turing, shall we? His theorem shows that it is
>impossible to write an algorithm program that will always correctly decide
>whether or not a given program will terminate on certain input.
>
>This is approximately the same as Goedel's theorem, but it is easier to see
>its implications. It is e.g. impossible to write a program that computes
>the best optimization of a given program.

Are you intentionally equivocating here?  "a given program" is *any program*.
It is absurd to claim that humans can determine the best optimization for
*all* programs.

>It is also impossible to come up
>with an algorithm that will check whether two given programs do the same,

*any* two programs.  Obviously no human can either.

>and it is therefor impossible to do perfect automatic program verification
>or proof verification.

But humans employ computers to do program and proof verification at times
anyway, because we ain't so perfect ourselves.

>Or are you going to call this no big deal either? People obviously do these
>things...

Falsehoods are never obviously true; sorry.

Anything that a human does that is expressible in language can be done
by an FSM, let alone a TM.  This is basic.  Sadly, few here have a grasp
of the basics, and so these silly discussions go on interminably (or at least
for the > 10 years I've been reading c.a.p and it's predecessors).

>: What does this "follow" from?
>
>The fact that is impossible to learn certain tasks follows from Turing's,
>Church's or any else's incompleteness theorem. Read e.g. "Introduction to
>automata theory, languages and computation" by Hopcroft and Ullman.
>
>: If it does, like I said in an earlier post, it also shows that computers
>: are no computers.
>
>No, it shows computers cannot learn CFGs out of the blue.

Perhaps you would like to formalize "learn CFGs out of the blue".
At least Penrose tried to be *careful* in his demonstration.
-- 
<J Q B>

