Concerning interoperability of mathematical theories: (01)
> Pat's exact words were (Feb. 1):
>
> "If you doubt that incompatible theories can be described in terms of
> common, more basic elements, try presenting some incompatible theories
> (and show how they are logically incompatible) and I will show how it
> can be done."
>
> I'm happy you agree the axiomatic set theories of mathematics are such
> incompatible theories.
>
> If Pat will now include the caveat, "not mathematics" (or anything
> based on manipulations of symbols?) I suppose that is progress of a
> sort.
> (02)
Since you suggested that I try showing how incompatible mathematical
theories can be expressed using the same primitives, I assumed (not being
familiar with how the term "primitive" is used in mathematics) that there
was such a thing as interoperability of mathematical theories. But Pat
Hayes has very emphatically denied that any different mathematical theories
(seemingly contradictory or not) can be related to each other in any way: (03)
[PH]: " Each theory nails down ONE set of concepts. And they are ALL
'primitive' in that theory, and they are not primitive or non primitive in
any other theory, because they aren't in any other theory AT ALL." (04)
It seems to me clear from Pat Hayes's comment that "interoperability" of
mathematical theories is a meaningless concept. If you consider that a
"caveat", so be it.
The contradictory theories I was referring to as being describable using
semantic primitives were the ontologies (or parts of ontologies) that are
intended to model realworld entities (including abstract ones like
feelings). So perhaps the better term would be "ontology" rather than
"theory" to avoid confusion with abstract pure mathematics. The difference
is that an ontology is directly linked to entities outside the set of
symbols, and when two different ontologies refer to the same external
entity, *then* they can be evaluated as consistent or contradictory. The
ontologies I am concerned with interoperating are not merely closed sets of
symbols, they represent entities outside the set of symbols and the logical
operations have consequences outside the set of symbols.
Now I am well aware that in logicbased ontologies logical consistency
will be evaluated based on logical operations of the same type that are used
in pure mathematical theories. But there is still a very big difference, in
that when used in a practical application, the models interact with the real
world  through the interpretations of ontologists, programmers, or database
developers, and potentially by direct operations such a sensor or robotic
actions or internet access, as well as user input. I am happy that PatH
and JohnS have clarified the interpretations that mathematicians put on the
terms "meaning" and "primitive" in pure mathematics. In a practical
ontology, the meanings as I interpret them are quite different. Unlike the
case described by PatH, the symbols in two different ontologies that refer
to some identical entities (have identical intended meanings) have referents
that are **very much** "in" both ontologies. (05)
[RF]: > >> By the way, my alternative is to work directly with observations
of
> >> different kinds, perhaps indexed by labels, and implement
> >> interoperability based on overlaps between sets of these, as the
> task demands.
> >
[CM] > > Your alternative to *what*?
>
[RH] > FO, Pat C, Feb 14:
>
> Pat C: "In these discussions of the principles of an FO and a proposed
> FO project,
> not only has there been no technical objection to the feasibility of an
> FO
> to serve its purpose (just gut skepticism), but there has also been a
> notable lack of suggestions for alternative approaches that would
> achieve
> the goal of general accurate semantic interoperability"
> (06)
Perhaps you could provide more detail for your alternative method of
achieving general accurate semantic interoperability? The sentence above
conjures up nothing concrete in my imagination. How does one "implement
interoperability based on overlaps between sets [of observations]"??? (07)
Pat (08)
Patrick Cassidy
MICRA, Inc.
9085613416
cell: 9085654053
cassidy@xxxxxxxxx (09)
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