From newshub.ccs.yorku.ca!ists!helios.physics.utoronto.ca!news-server.csri.toronto.edu!rutgers!hsdndev!husc-news.harvard.edu!zariski!zeleny Thu Dec 26 23:58:17 EST 1991
Article 2378 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
Summary: meaningfulness = cognitive grasp; potential realizability = existence
Keywords: analytic arguments, reflection principle, standard model
Message-ID: <1991Dec23.112144.6884@husc3.harvard.edu>
Date: 23 Dec 91 16:21:42 GMT
References: <1991Dec19.041945.27038@oracorp.com> <1991Dec22.131401.6869@husc3.harvard.edu> <1991Dec23.042312.10049@cambridge.oracorp.com>
Organization: Dept. of Math, Harvard Univ.
Lines: 102
Nntp-Posting-Host: zariski.harvard.edu

In article <1991Dec23.042312.10049@cambridge.oracorp.com> 
ian@cambridge.oracorp.com (Ian Sutherland) writes:

>In article <1991Dec22.131401.6869@husc3.harvard.edu> 
>zeleny@zariski.harvard.edu (Mikhail Zeleny) writes:

MZ:
>>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.

IS:
>Well, I'm glad we got THAT settled.

Your contentment has been noted.

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.

IS:
>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.

Don't be taken in by terminology.  Markov, who is, in effect defining his
point of disagreement with Yessenin-Volpin (even though as a Politically
Correct toady, he daren't speak the name of his dissident colleague), does
not, qua constructivist, say that integers like 10^10^10 exist only
potentially; for him, potential realizability = mathematical existence.
Likewise, for anyone but a Yessenin-Volpinite, potential realizability =
semantic grasp.  For accepting compositionality in semantics is tantamount
to stipulating potential realizability of arbitrarily large semantic objects.

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

IS:
>Perhaps I misinterpret your remarks (couched as they are in what
>SEEMS TO BE quite heavy sarcasm) but I think not, not quite anyway.

My apologies for not having used the Internet idiot crutch, the ubiquitous,
reprehensible "smiley" punctuation.  Yes, I am outraged at the fact that
the man who is not only arguably the most important living constructivist
thinker, but also a hero of the struggle against the Communist political
oppression, can find himself much lauded when persecuted, and without a job
when released.  It seems that the West likes its freedom fighters
conveniently dead and enshrined.  I should add that I am not personally
acquainted with Yessenin-Volpin, knowing him and his story only from
publications and the account by Gabriel Stolzenberg.

IS:
>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.

Semantic grasp is identical with (the realization of) meaningfulness.  
To use the idiom due to David Kaplan, this is both incontrovertible and
uncontroversial.  Incidentally, my mind has just grasped 10^10^10.

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

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