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Article 4676 of comp.ai.philosophy:
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>From: clarke@acme.ucf.edu (Thomas Clarke)
Newsgroups: comp.ai.philosophy
Subject: Re: Infinite Minds?
Message-ID: <1992Mar23.171539.27586@cs.ucf.edu>
Date: 23 Mar 92 17:15:39 GMT
References: <1992Mar21.024804.10085@husc3.harvard.edu>
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In article <1992Mar21.024804.10085@husc3.harvard.edu>  
zeleny@zariski.harvard.edu (Mikhail Zeleny) writes:
| 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?  
|  
You raise several other interesting points in your commentary.  Before  
clarifying them, let me try to be very precise about how I see Church's thesis  
fits into the problem of mind.

Consider the propositions:

A.   Effective (precisely specifiable mechanical procedure, guaranteed to work,  
computations) are just those that can be described as recursive functions (or  
can be carried out by a Turing machine).

B.  Mind is physical.  That is mind can be reduced to physical phenomena that  
are subject to prediction by physics.

C.   Physics can be reduced to an effective procedure.  That is an effective  
method exists for computing the results of a given experiment to arbitrarily 
small accuracy.  

Then A, B and C then the mind can be simulated by a Turing machine.  
    If not, then the mind is a physical process which cannot be effectively  
computed by a Turing machine contradicting C.

Note A, B, and C are necessary, but not sufficient.
   If A is false, then the mind may just be an example of non-recursive  
physics.
   If B is false, then C does not hold, and the soul can be either effectively  
computable or not.
  If  C is false, then the mind may just be an example of physics which is not  
effectively computable.  

There thus seems to be a connection between finite (that is recursive) physics  
and the question of whether a digital computer can implement a mind.
| 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.
| 
To my ear effective computations is coming to be synonymous with partial  
recursive/Turing computable.  My phrase "precisely specifiable" was an attempt  
to avoid this identification.  My choice of language is or course defeated by  
Fermat's theorem and other marginalia.
| 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? 
I don't know whether physics is mechanizable by a Turing machine or not.   
Actually I suspect and hope not.  This is just hypothesis C.
| 
| 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.
| 


