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>From: zeleny@zariski.harvard.edu (Mikhail Zeleny)
Newsgroups: comp.ai.philosophy,sci.philosophy.tech
Subject: Re: Infinite Minds?
Message-ID: <1992Mar21.024804.10085@husc3.harvard.edu>
Date: 21 Mar 92 07:48:03 GMT
Article-I.D.: husc3.1992Mar21.024804.10085
References: <1992Mar19.100550.10019@husc3.harvard.edu> <1992Mar20.133924.19215@cs.ucf.edu>
Organization: Dept. of Math, Harvard Univ.
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Nntp-Posting-Host: zariski.harvard.edu

In article <1992Mar20.133924.19215@cs.ucf.edu> 
clarke@acme.ucf.edu (Thomas Clarke) writes:

>In article <1992Mar19.100550.10019@husc3.harvard.edu>  
>zeleny@zariski.harvard.edu (Mikhail Zeleny) writes:

>| In article <1992Mar18.183651.26822@cs.ucf.edu> 
>| clarke@acme.ucf.edu (Thomas Clarke) writes:

TC:
>| >Wild conclusions?  I merely try to make the point that it is plausible
>| >that the brain may not function digitally.  Until the Church-Turing thesis
>| >is rigorously proved or disproved, the question remains open.   

MZ:
>| Pardon my intrusion, but what the fuck does Church's thesis (Turing, a
>| student of his, was rather a latecomer) have to do with the claim of
>| digital brain functioning?  

TC:
>I'll see if I can make the connection clear.
>
>Church's thesis, as amplified by Turing, says that any precisely specifiable  
>computation is equivalent to a Turing machine calculation.

That's "effective computation".  A computation involving a step like "let
x=x+1 if $(\exists n > 2)(\exists x, y, z) x^n + y^n = z^n$, else let
x=x+2" is precisely specifiable, yet not effective.

TC:
>							    Given a complete  
>physics and reducibility, all future observations in the world can be  
>calculated in a precisely specifiable manner.  

Maybe so, but why would this manner have to be effective?  Suppose that the
calculation involves transfinite recursion; now derive a contradiction with
your own assumption.  Are you perchance assuming digital functioning of the
universe? 

TC:
>					      Church's thesis thus applies to  
>these calculations of physical computations [A little difficulty with reals,  
>but rationals are dense in the reals, and rationals are pairs of integers ...  
>see "Computability in Analysis and Physics", Pour-El and Richards, Springer,  
>1987]

That's a big book.  Please explain how you propose to reduce the reals to
finite representations by the rationals.  No approximations, please.

TC:
>Therefore, the future actions of organisms, and networks in particular, are  
>computable. May be some problem with finding a complete physics, or with the  
>program of reducing all to physics...

You clearly seem to be making additional assumptions.  As illustrated
above, not every precisely specified computation is computable.  A complete
physics may very well be of such a nature.  Please elaborate.

TC:
>With regard to networks simulating Turing machines see the article "On the  
>Computational Power of Neural Nets", by Siegelman and Sontag (available as  
>siegelman.turing.ps.Z on the neuroprose archive site  
>archive.cis.ohio-state.edu).  They outline the construction of a neural net  
>with <1000 nodes that implements a universal Turing machine.  No tape is used,  
>rather an unbounded tape is simulated by using what amounts to a Bernoulli  
>shift operation on a selected network to implement a stack.  A pair of stacks  
>makes a tape.  That is, if 0<=q<=2 is the variable then b+q/2->q pushes the bit  
>b into q, and {int(q)->b, q/2->q} pops the variable b from q.  
>
>This approach is not physically realistic, noise and quantum effects would  
>rapidly spoil the information stored in the rational number q, but I think it  
>is an interesting construction.  Siegelman and Sontage claim that when real  
>numbers are used, they can get a network that computes the uncomputable.  They  
>promise publication real soon now.  I am very interested in how this works  
>since such a net would cause problems for the Turing modified Church thesis if  
>you believe in physics and reductionism.  My guess is it may have to do with  
>Chaitin's uncomputable real number which codes the halting probablility of the  
>universal Turing machine. 

That's very interesting, but quite beside the point.  Church's thesis has
to do with effective computability; it places no restrictions whatsoever on
its complement with respect to computability by an organism, fortuitous
computability.  In other words, the thesis in no way presupposes digital
functioning of the mind.


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