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Article 2430 of comp.ai.philosophy:
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>From: weemba@libra.wistar.upenn.edu (Matthew P Wiener)
Newsgroups: comp.ai.philosophy
Subject: Pour-El and Richards say some things, don't say other things
Message-ID: <61146@netnews.upenn.edu>
Date: 29 Dec 91 17:32:59 GMT
References: <1991Dec23.135321.6894@husc3.harvard.edu> <1991Dec27.051804.6985@cambridge.oracorp.com> <1991Dec27.184248.6939@husc3.harvard.edu> <1991Dec28.194855.16543@galois.mit.edu>
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Reply-To: weemba@libra.wistar.upenn.edu (Matthew P Wiener)
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In-reply-to: jbaez@nevanlinna.mit.edu (John C. Baez)

I refuse to join in Zeleny's crossposting orgies.

In article <1991Dec28.194855.16543@galois.mit.edu>, jbaez@nevanlinna (John C. Baez) writes:
>>Why shouldn't a neuron be capable of infinitely many distinguishable states?

>Well, having a neuron whose functioning made essential use of
>infinitely many distinguishable states in order to repeatably compute
>unrecursive functions goes against what we know about physics.  See
>for example my paper "Recursivity in Quantum Mechanics," Trans.
>A.M.S. 280, p. 339, or Pour-El and Richards' book on computability in
>physics.

Indeed.  The second part of Pour-El and Richards makes it clear that there
are self-adjoint operators with non-computable spectra.  Were you reading
the abridged edition, or what?

>							  Unless the
>normal functioning of a neuron makes use of laws of physics other than
>the ones we know about,

Or uses it in ways that we haven't caught on to yet....

>			 it seems most likely that its *expected*
>behavior is recursive.  I emphasize *expected* because quantum mechanics
>is nondeterministic, so that if you want to "compute a nonrecursive
>function" it appears that all you have to do is keep track of the ticks
>of a Geiger counter, or any other quantum process.

Well, yes.  Exactly.  But this is not the issue.

>						     One can't find a
>quantum system following laws of the sort we know that *repeatedly,* or
>even "on average" computes a fixed nonrecursive function.

But the question is not whether the mind computes non-recursive functions,
but of whether it is a state that can be achieved by recursive functions.
This is of much lesser unsolvability (and greater subtlety) than that of
non-recursive computation, and the work of Pour-El and Richards et al does
not address this question.

As Mikhail Zeleny stated in the article you replied to:

>>The point is not whether human beings can solve all instances of the
>>halting problem, or tell whether an arbitrary collection of axioms is
>>consistent, but that each time they do so in any particular case, their
>>reasoning is essentially non-algorithmic, as claims Penrose.

Where *does* this mathematical intuition of ours come from?  I don't see
any TM model that accounts for this.  Yet imagine if the Pour-El & Richards
non-computable observables were in our heads--say one for the truth of ZFC--
interacting with neural nets or pumped phonon condensates or whatnot.  Then
of course we have this unprovable intuition, and of course mathematics
proceeds using a direct introspective observation process that has a high
degree of unpredictability.

Nowhere do we need to suppose the ability to actually compute non-recursive
functions.

>If anyone believes that somehow quantum mechanics, or the fact that
>biological systems are continuous rather than discrete, gives the brain
>powers over and above that of a universal Turing machine with a random
>number generator oracle, then they have some explaining to do concerning
>HOW this is supposed to work.

Indeed.  Yet for some ludicrous reason it is not as obvious to some people
that if someone believes that the brain can be simulated via a UTM with RNG
oracle, they too have some explaining to do concerning HOW this is supposed
to work.
-- 
-Matthew P Wiener (weemba@libra.wistar.upenn.edu)


