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Article 2482 of comp.ai.philosophy:
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>From: jbaez@riesz.mit.edu (John C. Baez)
Subject: Re: Penrose on Man vs. Machine
Message-ID: <1992Jan2.205947.5740@galois.mit.edu>
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Organization: MIT Department of Mathematics, Cambridge, MA
References: <1991Dec23.112144.6884@husc3.harvard.edu> <1991Dec30.172852.3305@csc.canterbury.ac.nz> <1992Jan2.174436.26054@cherokee.uswest.com>
Date: Thu, 2 Jan 92 20:59:47 GMT
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In article <1992Jan2.174436.26054@cherokee.uswest.com> ken@dakota (Kenny Chaffin) writes:
(Someone writes:)
>>But "10^10^10" is an exceedingly compact representation for the number
>>it denotes.  It is a mere 8 ascii characters, and the function "^" can
>>be defined fairly succintly.  Can your mind grasp, say, the base 10
>>representation of "10^10^10"? 
>>
>	But they are not the same thing (are they?). 10^10^10 is simply a
>pointer or a place holder for the concept of the actual number, but so is
>the number as a whole number. They are just two different ways of expressing
>a particular idea. Does the thing pointed to by 10^10^10 really exist? No,
>other than as a concept. The concepts exists, but there is no physical object
>other than the patterns of neuron firings which represent it differently in
>every brain that "grasps" it.

It's important to note the ways in which various representations are, or
are not, interchangeable - and with what *ease* they are
interchangeable.  10^10^10 is a nice representation of this number for
many purposes, and it's certainly unfair to say that the base 10
representation is more "for real" -- even the base 10 representation
being a terse abbreviation of its base 1 representation (a list of 10^10^10
scratch marks).   More generally, different representations are good for
different things -- this is I think more important than the issue of
"neuron firings".  For most purposes, having a billion digits of pi
isn't as good as saying "Pi", but if you want to know the 23497923th
digit, it has its points.  (This kind of desire - to know the
23497923th digit of some irrational number - is so rare that it's
ridiculous to say, as some do, that rational numbers are better known
than irrationals.  For the purpose of taking cosines, for example, pi is much
"better known" than 355/113.)   It's purely a foundationalist prejudice
to think that there must be a "primary" or fundamental representation of
a mathematical object.  (Think of expressing ordered pairs in terms of
sets -- it's just a hack, not a deep fact.)




