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Article 2498 of comp.ai.philosophy:
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>From: ian@cambridge.oracorp.com (The Floating Man)
Subject: Re: Grasping concepts... is it polite?
Message-ID: <1992Jan5.004957.15767@cambridge.oracorp.com>
Keywords: grasping, representations, numerals, ultra-intuitionism
Organization: ORA Corp, 675 Mass Ave, Cambridge, MA 02139
References: <1992Jan2.221158.17575@cambridge.oracorp.com> <1992Jan4.173301.3331@csc.canterbury.ac.nz>
Date: Sun, 5 Jan 92 00:49:57 GMT

In article <1992Jan4.173301.3331@csc.canterbury.ac.nz> chisnall@cosc.canterbury.ac.nz writes:
>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.

I don't think there's much point in debating whether the answers are
"no" for all possible definitions that are "relevant".  Let's just
make some definitions, and proceed from there.  If somebody wants to
make other definitions that he or she thinks are interesting, that's
fine too.  All I ask is that we (1) don't continue to discuss this
question without defining what the hell we're talking about, and (2)
don't start arguing about what the "right" definition is.  If somebody
proposes a definition and somebody else thinks it's wrong, let the
latter person just hit "j".

>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.

You say this as if there were some God-given definition which was
superior to others.  I'm sure there are many admissible definitions
for many different purposes.  If we don't fix on some, however, this
will just become another idiotic exchange of blather like so many
other discussions on this newsgroup.

>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.

I would suggest that the meaning of a particular entity "grasping" a
mathematical object should be defined relative to what questions the
entity can answer about the object with a certain collection of
resources (actually, you also need to make precise exactly what's
meant by a question, etc.).  This kind of "relative" definition makes
it possible for a given object to be graspable or not, depending on
the particular kinds of questions, allowable resources, etc. that are
relevant to a particular context.

If, to conclude that I "grasp" a number, you require me to be able to
say whether a given formula of set theory holds of a number or not,
using only my unaided brain, then (modulo making everything fuzzy in
this example precise) I probably can't be said to "grasp" the number
"0".  Having said that, I'll turn the question back to Mr. Prophet
and Mr. Throw Up: is there some collection of questions, set of
resources, etc., that you can come up with under which the numbers
you can "grasp" form a finite initial segment of "little" numbers
like "0", "1", "2", etc.?

>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.

I suggest that the proper collection of properties (which are
essentially what I called "questions" above) will capture your
informal notion.  By the way, I wonder if you really mean (3) in any
precise sense.  Could you, for instance, tell me off the top of your
head (that's a restriction on resources :-) how many apples you could
actually SEE in a pile of a million apples of a certain shape?  I
can't.

>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.

I think that reduces the problem to something far less interesting.
If you talk about grasping REPRESENTATIONS of numbers, then you may
as well say that you can grasp a representation if you can say it out
loud without making a mistake (presumably with some degree of
reliability).
-- 
Ian Sutherland                          ian@cambridge.oracorp.com

Sans peur


