From sbelknap@UIC.EDU Wed Mar 1 08:42:12 2000
X-Digest-Num: 380
Message-ID: <44114.380.2110.959273826@eGroups.com>
Date: Wed, 1 Mar 2000 10:42:12 -0600
From: Steven Belknap <sbelknap@UIC.EDU>
Subject: Re: Sets etc.

In high school geometry class, I was taught the importance of having 
undefined terms as the basis for the development of mathematics. The 
four undefined terms I was taught for geometry were line, point, set, 
and "betweenness"

One can not avoid circular definition, which I fear is at the root of 
the problem pycyn is exploring.

>From: pycyn@aol.com
>
>"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
>
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Steven Belknap, M.D.
Assistant Professor of Clinical Pharmacology and Medicine
University of Illinois College of Medicine at Peoria