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Article 2495 of comp.ai.philosophy:
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>From: chisnall@cosc.canterbury.ac.nz (The Technicolour Throw-up)
Newsgroups: sci.philosophy.tech,sci.logic,comp.ai.philosophy
Subject: Re: Grasping concepts... is it polite?
Keywords: grasping, representations, numerals, ultra-intuitionism
Message-ID: <1992Jan4.173301.3331@csc.canterbury.ac.nz>
Date: 4 Jan 92 04:33:00 GMT
References: <1992Jan2.221158.17575@cambridge.oracorp.com>
Reply-To: chisnall@cosc.canterbury.ac.nz
Organization: Computer Science,University of Canterbury,New Zealand
Lines: 77
Nntp-Posting-Host: hihi.cosc.canterbury.ac.nz

>From article <1992Jan2.221158.17575@cambridge.oracorp.com>, by ian@cambridge.oracorp.com (Ian Sutherland):
> In article <1992Jan2.131048.18412@news.stolaf.edu> seebs@asgaard.acc.stolaf.edu (The Laughing Prophet) writes:
>>Can you *really* understand 10^10^10 *of* something? I.e., can you genuinely
>>understand what 10^10^10 pennies are, relative to a single penny? How about
>>just how much space they'd take up?
>
> Folks, I would hate to make a trivial remark, but I'm 100% certain
> that there are some reasonable senses of the words "grasp" and
> "understand" that make the answers to your questions "yes", and some
> other reasonable sense which make the answers "no".  Could you be a
> bit more precise about how you're using these words?

Offhand I'm not personally sure what senses of the  word  "grasp"  would
simultaneously  make  the  answers  "yes"  and  also  be relevant in the
philosophical context in which these questions are being asked.  You  do
have  a  point though since some light may be shed on these questions if
we knew more precisely  what  it  means  to  mentally  grasp  something.
Unfortunately  I'm  not exactly sure myself what I mean by "grasp" being
ignorant as I am of  the  detailed  workings  of  my  brain  when  I  do
mathematics.   I  have  a  vague  idea of what I mean by it, that it has
something to do with having some sort of mental  representation  of  the
object  under consideration, but there are problems with this definition
and I'm not sure how to tighten it up.  Maybe I can clear  things  up  a
bit by re-explaining my original point in more detail.

One  of  the essential properties of my notion of "grasping", one which,
moreover, I assume to be shared by other peoples' understanding  of  the
term,  is  that  mentally  grasping  a  thing is different from mentally
grasping properties of that thing.  For example my mind cannot  grasp  a
collection  of  one million apples.  Nor can it grasp, or build a mental
representation of, a set of one million objects.  But it can  grasp  (1)
how  many digits there are in the base 10 representation of one million,
(2) how many cubic meters would be occupied by an optimal packing of one
million  apples  into  cartons,  (3) what a pile of million apples would
*look* like, etc.

This is essentially what I was  getting  at  with  my  question  on  the
relative graspability of "10^10^10" versus the base 10 representation of
the same number.  I can grasp "10" and the operation denoted by "^"  and
that  "10^10^10"  is a definition using "10" and "^".  I know that there
are algorithms which will convert "10^10^10" into base 10 but  not  only
can  I  not  carry them out on paper I cannot carry them out in my mind.
The number that results has far too many digits for my mind to  be  able
to "represent" it.

Note  that  I  am  not talking about grasping numbers but about grasping
representations of those numbers.  I don't think its meaningful to  talk
about  grasping  numbers except in an elliptical sense.  As far as I can
see our minds don't have access to numbers per se but only to particular
representaions  of numbers (i.e. numerals).  When we talk about grasping
numbers we should, properly, be talking about  grasping  representations
of numbers.

Hopefully  this  goes  some  way to clarifying my stance.  The following
quote, due originally to P. Bernays, comes from the  start  of  Parikh's
1971 paper, and seems to echo my thoughts on this matter:

    "From two integers k,l one passes immediately to k^l; this process
    leads in a few steps to numbers which are far larger than any
    occuring in experience, e.g., 67^(257^729).

    Intuitionism, like ordinary mathematics, claims that this number can
    be represented by an arabic numeral.  Could not one press further
    the criticism which intuitionism makes of existential assertions
    and raise the question: What does it mean to claim the existence of
    an arabic numeral for the foregoing number, since in practice we are
    not in a position to obtain it?

    Brouwer appeals to intuition but one can doubt that the evidence for
    it really is intuitive.  Isn't this rather an application of the
    general method of analogy, consisting in extending to inaccessible
    numbers the relations which we can concretely verify for accessible
    numbers?"

--
Just my two rubber ningis worth.
Name: Michael Chisnall          email: chisnall@cosc.canterbury.ac.nz


