Pat, (01)
I agree that all sets are mathematical. (02)
JFS>> For mathematical sets... (03)
PH> There is no such thing as a 'mathematical' set.
> Sets are sets, whatever they are sets of. Set theory
> is ontologically neutral. (04)
Yes. I was using a convenient, but possibly confusing feature
of English, which allows an adjective that modifies a noun to
be used with the relationship left unspecified. (05)
The longer phrase I had in mind was (06)
"For sets composed of mathematically defined entities" (07)
as contrasted with (08)
"For sets composed of physical entities" (09)
JFS>> ... the "simple" criterion usually involves a proof of
>> equivalence: for anything other than small finite sets, it
>> is necessary to prove that the two specifications determine
>> exactly the same elements. (010)
PH> That is needed in order to SHOW that two sets are identical, yes. (011)
Yes indeed. And for any sets composed of physical entities, it is
necessary to SHOW that a variable S that represents a set at one
instant of time can be assumed to represent "the same" set at
a different time: (012)
1. The set theoretical operations are ontologically neutral. (013)
2. That neutrality lulls people into thinking that the precisely
defined operations on sets composed of mathematical entities
are just as reliable when carried out on sets composed of
physical entities. (014)
3. But as Whitehead pointed out, "The baby in the cradle and the
grown man in middle age are in some senses identical and in
other senses diverse. Is the train of argument in its conclusions
substantiated by the identity or vitiated by the diversity?" (015)
4. Therefore, any use of set theory for representing physical entities
must recognize that a set at one instant of time is not "exactly
the same" set of supposedly "exactly the same" things at any
other instant. (016)
5. We can often say that they are "approximately" the same for
one particular application, but the differences might be
critical for a different application. (017)
Mathematicians never think about such issues in abstract set theory,
but any application of set theory to the physical world must address
them. (018)
John (019)
_________________________________________________________________
Message Archives: http://ontolog.cim3.net/forum/uom-ontology-std/
Subscribe: mailto:uom-ontology-std-join@xxxxxxxxxxxxxxxx
Config/Unsubscribe: http://ontolog.cim3.net/mailman/listinfo/uom-ontology-std/
Shared Files: http://ontolog.cim3.net/file/work/UoM/
Wiki: http://ontolog.cim3.net/cgi-bin/wiki.pl?UoM_Ontology_Standard (020)
|