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>From: jbaez@nevanlinna.mit.edu (John C. Baez)
Newsgroups: comp.ai.philosophy,sci.philosophy.tech,sci.logic
Subject: Re: Penrose on Man vs. Machine
Message-ID: <1992Jan21.001814.25035@galois.mit.edu>
Date: 21 Jan 92 00:18:14 GMT
References: <1992Jan15.140324.27354@oracorp.com> <1992Jan18.131648.7770@husc3.harvard.edu>
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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:
>
>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.

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.  


