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Article 2367 of comp.ai.philosophy:
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>From: ian@cambridge.oracorp.com (Ian Sutherland)
Newsgroups: sci.philosophy.tech,sci.logic,sci.math,comp.ai.philosophy
Subject: Re: Penrose on Man vs. Machine
Keywords: analytic arguments, reflection principle, standard model
Message-ID: <1991Dec23.042312.10049@cambridge.oracorp.com>
Date: 23 Dec 91 04:23:12 GMT
Article-I.D.: cambridg.1991Dec23.042312.10049
References: <1991Dec19.041945.27038@oracorp.com> <1991Dec22.131401.6869@husc3.harvard.edu>
Organization: ORA Corp, 675 Mass Ave, Cambridge, MA 02139
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In article <1991Dec22.131401.6869@husc3.harvard.edu> zeleny@zariski.harvard.edu (Mikhail Zeleny) writes:
>Briefly, I appreciate his realism about propositional attitudes, as well as
>his rejection of functionalism; however I don't accept his apparent
>materialism, seeing it as incompatible with the former.  My own position is
>somewhere between the Popper-Eccles dualism and the O'Shaughnessy dual
>aspect theory, generally favoring the latter.

Well, I'm glad we got THAT settled.

>DMC:
>>I will take two examples:
>>
>>I. "How to outdo an algorithm", on pages 64-66 of my edition contains
>>the following paragraph:

MZ:

>As you undoubtedly are aware, all analytic arguments contain their
>conclusion in their premisses.  Penrose argues that we are capable of
>outdoing any algorithm; in order to do so he has to assume an equally
>powerful, but intuitively more plausible premiss.  It appears to me that he
>has succeeded in doing so, since the premiss that we can *potentially*
>determine the partial correctness of an arbitrary program by semantic
>reflection appears to me as intuitively unexceptionable as A.A.Markov's
>abstraction of potential realizability, which stipulates e.g. that the
>successor operation can be applied to an arbitrarily large integer.

The premise is not that we can "potentially" determine the partial
correctness of an arbitrary program, it is that we CAN determine said
correctness.  If there's even one program we CAN'T determine the
partial correctness of, it may be the very one which describes our
reflection process.

>In other words, by rejecting the assumption made by Penrose, you are
>implicitly committing yourself to ultra-intuitionism and its concept of
>feasible numbers.

Perhaps I misinterpret your remarks (couched as they are in what
SEEMS TO BE quite heavy sarcasm) but I think not, not quite anyway.
Intuitionism of the sort you're mentioning rejects the MEANINGFULNESS
of mathematical objects that are beyond the grasp of the human mind,
where "grasp" is interpreted extremely narrowly.  Daryl raises the
possibility that there may be certain things which we can describe
using ordinary mathematics which the human mind cannot, in a certain
well-defined sense, grasp.  That does not mean that he rejects the
MEANINGFULNESS of such objects.  It seems to me that even someone who
agreed with your position would admit that there are SOME mathematical
objects that the human mind cannot, in this same sense, grasp, without
asserting that such objects are therefore meaningless.
-- 
Ian Sutherland                          ian@cambridge.oracorp.com

Sans peur


