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>From: zeleny@zariski.harvard.edu (Mikhail Zeleny)
Newsgroups: comp.ai.philosophy,sci.philosophy.tech,sci.logic,sci.math
Subject: Re: Penrose on Man vs. Machine
Message-ID: <1992Jan20.231913.7847@husc3.harvard.edu>
Date: 21 Jan 92 04:19:11 GMT
References: <1992Jan15.140324.27354@oracorp.com> <1992Jan18.131648.7770@husc3.harvard.edu> <1992Jan21.001814.25035@galois.mit.edu>
Organization: Dept. of Math, Harvard Univ.
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In article <1992Jan21.001814.25035@galois.mit.edu> 
jbaez@nevanlinna.mit.edu (John C. Baez) writes:

JB:
>Okay, so Mike Z had no major objections to my last post (I agree, by the
>way, that understanding all of X's theories doesn't really mean your
>mathematical ability is >= X's), so let's try a different angle:

>In article <1992Jan18.131648.7770@husc3.harvard.edu> 
>zeleny@zariski.harvard.edu (Mikhail Zeleny) writes:

MZ:
>>The evidence to the contrary is provided by introspection: I am fully
>>confident that I am capable of characterizing the integers up to
>>isomorphism, so restricting the induction axiom to first-order definable
>>sets is clearly insufficient.  However, it seems that our intuitions differ
>>on this issue.  On the other hand, I don't know of any empirical evidence
>>to the effect that we can characterize non-feasible numbers, so perhaps by
>>your own criteria you are stuck with ultra-intuitionism after all.

JB:
>Okay, you have an intuition that you can do something called
>"characterizing the integers up to isomorphism."  Is this something you
>can actually DO in public?  Presumably you don't claim you can inspect
>any table of generators and relations and determine if the group so
>defined is isomorphic to the integers.  But what, if any, practical
>consequences are there of your ability?  Or is it sort of along the
>lines of "appreciating the beauty of Bach's harpsichord music" - which I
>imagine someone could claim to do even though illiterate in harmony
>theory and utterly unable to write music.  

Like everything else in philosophy, Platonism is a kind of story, which
only makes sense in its own terms.  Verificationism doesn't belong to the
story I'm telling; then again, many people on this net and elsewhere will
be happy to tell you that verificationism doesn't fit in with their stories
either.  As for appreciating the beauty of Bach's music, I confess to
satisfying both of your conditions.  On the other hand, it seems reasonable
to suppose that our noumenal intuition is as unreliable as our phenomenal
perception, as well as equally amenable to improvement through practice
(some day, I hope to have time and money for piano lessons).

As for characterizing the integers up to isomorphism in the privacy of my
world of ideas (cf. Dedekind's "Was sind und was sollen die Zahlen",
Proposition 66, and Russell's "Principles of Mathematics", 339), my claim
amounts to no more than having an understanding of the `...' part of the
sequence: 1, 2, 3, 4, 5, ..., which eo ipso excludes non-standard models of
the integers.  This amounts to claiming nothing less than the full-strength
second-order induction principle, and as I noted above, there is no way to
substantiate it by empirical means; on the other hand, given that my
chances of coming across a concrete collection of, say, 10^10^10 discrete
entities are rather slim, I don't know of any empirical means of justifying
far less controversial remaining Peano postulates, which, taken together,
imply the existence of arbitrarily large integers.  Whence my dilemma: you
must either choose Platonism, or ultra-intuitionism.

`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'
: Qu'est-ce qui est bien?  Qu'est-ce qui est laid?         Harvard   :
: Qu'est-ce qui est grand, fort, faible...                 doesn't   :
: Connais pas! Connais pas!                                 think    :
:                                                             so     :
: Mikhail Zeleny                                                     :
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