From pycyn@aol.com Wed Mar 1 07:05:24 2000
X-Digest-Num: 380
Message-ID: <44114.380.2108.959273826@eGroups.com>
Date: Wed, 1 Mar 2000 10:05:24 EST
From: pycyn@aol.com
Subject: Sets etc.
"Wherefore all this strife there be / 'twixt Tweedle Dumm and Tweedle Dee?"
A class is any collection of things conceived as together. Usually we think
of it as all the things satisfying some formula: is a cow, is a root of
equation..., etc., but in the full horrors of mathematics there are provably
classes for which there is not formula (denumerably many formulae,
non-denumerably many classes). A set is a class satisfying certain further
conditions, amounting to its being able to be a member of other sets (though
not so circularly -- actually recursively -- defined). A mass (in the Lojban
sense) is a class considered in a certain way, additively rather than
collectively or distributively. Almost all Lojban descriptors, LE, are about
classes; they differ in how the properties ascribed to the class are related
to the properties of the individuals that make it up. In the simplest cases,
le and the like, the property of the class is that of some or all of its
members (which is specified by the quantifier, explicit or implicit, used).
For masses, the property is the sum (in some often quite inexplicit, even
metaphorical, sense) of those of the members: the weight of a mass is
literally the sum of the weights of the members, the triumph of the mass is
the result of the combined efforts of the members (even including
some that had a negative impact on that triumph -- the crowd stormed the
Bastille despite some who ran away and some who aided the Ancien Regime), the
performance of the school is some kind of average of the performances of the
students, and so on (you have quite a bit of freedom here, but need to be
able to explain if push comes to shove). And at some point, the whole can
come down to the proeprty of one member, the logical summation of an "or,"
and thus collapse back toward the first sort of usage. Finally, a class may
be viewed collectively, and then the properties attributed to it have little
to do with the properties of the individual but rather with matters like how
many there are of them or (more related to their proerties) what toher
classes they belong to -- cardinality, inclusion, and the like -- set
theoretic properties, in short, which only rarely have value in ordinary
discourse.
For the most part, then, the use of the set markers is, like all of MEX, in
the system
because someday we may want to talk mathematics, the most recognizable special
language system within our (and every) language. so far we haven't been
inclined to try that, but we should not be prevented from it for lack in the
language (and, of course, we should not have help up the development of the
language just to get it in -- and Lojban did not hold up....much).
As for JCB's lo -- it was a muddle and everyone -- even JCB -- knew it was a
muddle of half a dozen different ideas floating around in his head. I think
we now have most of them sorted out in Lojban, though we still seem to get
into fights over a few from time to time (and pretty generally, having
forgetten how we solved it the last time, come up with the opposite solution
the next).
pc