Re: representing patterns and email@example.com (Fritz Lehmann)
Date: Mon, 21 Nov 94 18:05:35 CST
From: firstname.lastname@example.org (Fritz Lehmann)
To: email@example.com, firstname.lastname@example.org, email@example.com
Subject: Re: representing patterns and structures
Cc: firstname.lastname@example.org, email@example.com, firstname.lastname@example.org
Pat Hayes said:
Years ago, complicated
interactive CAD systems allowed notions of 'grouping', so you could specify
eg. a pattern of holes all of a certain diameter* a certain spacing* apart
in a hexagonal* grid filling a certain area* on a sheet of metal, just by
specifying the starred parameters. Then it was possible to say that they
should be all moved a leeeetle bit closer together, and have the system
draw the result for you on the screen. Very pretty, until you wanted to
fill in just one of these holes. This was impossible if the thing was
specified this way, since in a real sense that particular hole-token didnt
actually exist in the internal representation.
This issue is a very deep one, that of individual identity.
Those holes are _intensional_ in the initial description, yet you want
to "tag" one of them as an individual to state an exception. The problem
is, what does "one of them" mean? If the holes are moved "a leeeetle bit
closer together", are they "the same" holes? What if a circle of eleven
holes is made into a circle of ten holes? Which of the eleven evaporates?
The only ways I see to approach this are to A. Be very careful about
intensional descriptions --- you can (carefully) _describe_ the hole to
be filled-in by inspecting all holes to see if any qualify under the
description (you don't know the cardinality of the set of qualifying
holes is one). B. Identify individuals only in the real physical world
(as opposed to the abstract geometrical world), where we assume
Leibniz's Identity of Indiscernibles and intuitively feel we have
notions of single, distinct individuals; this means wait until the
piece is machined before you refer to a single, particular hole.
In knowledge Representation, it is often deemed a trivial matter
to decide what the (extensional) "individuals" in a domain are. It
isn't trivial. In manufacturing, an "individual part" in a design might mean
the abstract individual in the design, or it might mean a particular
physical embodiment. In more abstract areas the same question arises.
How many convex Platonic polyhedra are there? Is there but one "cube"
or do many cubes exist? I like to say "No two graphs are isomorphic
in pure graph theory -- if so they are one graph, not two. Only
particular emodiments, or labelled versions, can be isomorphic."
This matter should be Handled With Care.
Yours truly, Fritz Lehmann
GRANDAI Software, 4282 Sandburg Way, Irvine, CA 92715, U.S.A.
Tel:(714)-733-0566 Fax:(714)-733-0506 email@example.com