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Article 2422 of comp.ai.philosophy:
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>From: zeleny@zariski.harvard.edu (Mikhail Zeleny)
Newsgroups: sci.philosophy.tech,sci.logic,sci.math,comp.ai.philosophy
Subject: Re: Penrose on Man vs. Machine
Keywords: the limits of human understanding: no such thing
Message-ID: <1991Dec27.184248.6939@husc3.harvard.edu>
Date: 27 Dec 91 23:42:46 GMT
References: <1991Dec23.165606.5935@oracorp.com> <1991Dec23.135321.6894@husc3.harvard.edu> <1991Dec27.051804.6985@cambridge.oracorp.com>
Organization: Dept. of Math, Harvard Univ.
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Nntp-Posting-Host: zariski.harvard.edu

In article <1991Dec27.051804.6985@cambridge.oracorp.com> 
ian@cambridge.oracorp.com (Ian Sutherland) writes:

>DMC:
>>Therefore, in these cases, Penrose' arguments amount to the following:
>>
>>    (1) Assuming that we can tell which Turing machines H are partially
>>        correct for solving the halting problem, then our reasoning
>>        is nonalgorithmic.
>>
>>    (2) Assuming that we can tell which theories are sound, then our
>>        reasoning is nonalgorithmic.
>>
>>In other words, assuming that we can do things that no machine can do,
>>then we can do things that no machine can do.

MZ:
>>Quite so.  As I stated earlier, all analytic arguments can be
>>construed as a form of begging the question by an opponent who doesn't
>>accept the intuitive validity of their premises.

DMC:
>>Agreed, the point of such an argument is to reduce the truth of some
>>questionable statement to the truth of some statements that we find
>>more intuitively valid. However, I think that Penrose' premises, that
>>we can solve all instances of the halting problem, and that we can
>>tell whether an arbitrary collection of axioms is consistent, is much
>>*less* plausible than his conclusion, that our minds are not
>>algorithmic.

MZ:
>>Our intuitions differ.  This, in effect, is what your criticism of Penrose
>>amounts to.  Fine; this is how all philosophy, from Plato to Kripke, has
>>been done.  Just try not to delude yoursenf into thinking that you have
>>refuted his argument.

IS:
>It may be that this is all the response that can reasonably be made,
>Mr. Zeleny, but could you give some reasons why you feel that human
>beings can solve all instances of the halting problem, or tell
>whether an arbitrary collection of axioms is consistent?

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.

IS:
>                                                         I think
>that there are purely physical arguments that can be made that
>suggest that the human brain is the processing agent then a single
>human can't possibly do more than a single finite automaton (albeit
>with a very sophisticated program and a LARGE number of states).  What
>would you say to such arguments?  Do you believe that some of this
>processing goes on OUTSIDE the physical brain?  Do you believe that
>neurons in a person's brain can manifest infinitely many different
>states that can cause infinitely many different distinguishable
>behaviors?  Do you believe that there's some kind of physical
>mechanism in human brains that effectively acts as a nonrecursive
>oracle?

I am not about to address the mind-body issue; pending its resolution, I
see no reason to assume that the agency of mind supervenes on the physical
processes of the brain; indeed, I have argued to the contrary, under your
assumption of the brain's equivalence to a FSA.  In fact, I see no reason
to make this sort of assumption, independently of the legacy of the
McCulloch--Pitts proto-AI dogma.  Why shouldn't a neuron be capable of
infinitely many distinguishable states?

Finally, I believe that the human mind is fundamentally non-algorithmic, so
the question of an oracle is effectively moot.

IS:
>At the risk of being labeled ignorant, I'd appreciate it if any
>responses to the above would eschew references to various "isms" or
>existing philosophical schools of thought on these matters, and
>simply explain.  I am not a professional philosopher (merely an
>amateur lover of knowledge :-) and, judging from references in some of
>your recent postings, any such references will go right past me.

Simple explanation requires technical terminology.  Would you care to
"simply explain" the Turing degrees to an audience that demanded that you
eschew references to any standard mathematical notions, e.g. that of a
function?

MZ:
>>There is, however, ample *empirical* evidence of consistency of ZFC; surely
>>this result must be considered as part of mathematics.

IS:
>Surely it is NOT, since "empirical evidence" of the sort you mention
>has no weight in mathematics.

George Polya, in "Mathematics and Plausible Reasoning", argues otherwise.
I agree with him.

>-- 
>Ian Sutherland                          ian@cambridge.oracorp.com
>
>Sans peur


