SUO: Re: KIF: Re: SUMO axiomatization
Chris,
I would include all mathematical structures under the category Abstract
in my ontology. In fact, everything under that category is of the
same nature as Abstract; i.e., it is the category of every kind of
form that can be defined without contradiction.
The subtype Schema includes all the "static" structures, and the
subtype Script includes all the "dynamic" structures -- i.e., all
those that have a time-like succession of forms. As Nicola keeps
insisting, the category of abstractions is outside of space & time.
That is true, but you can have mathematical theories that include
a dimension labeled t, which just happens to be very time-like.
"Christopher A. Welty" wrote:
CW> Thanks for pointing this out, John. It brings up an interesting
> point for me - (sorry if this was discussed already during my
> SUO-absence) - do people find this to be "upper level"? Certainly if
> graphs is considered upper level, then this library demonstrates that
> there are over a hundred other things at the same "level".
At 1:55 PM -0500 12/22/01, John F. Sowa wrote:
JFS>There is a collection of over a hundred mathematical theories
> >(fully axiomatized) that were developed to run under IMPS
> >(Interactive Mathematical Proof System). The theorem-proving
> >programs that use these axioms run on most versions of LISP and
> >are available from Mitre as software that has been made freely
> >available. Following is the home page:
> >
> > http://imps.mcmaster.ca/
> >
> >The theories include graphs, groups, fields, automata,
> >and several versions of geometry, arithmetic, etc.
The category of all abstractions has forms for everything that exists
and everything that might exist without contradiction in any kind of
universe. So it is even richer than the category Physical, which only
includes those kinds of things that are physically possible in our
universe.
All those theories can be organized in a lattice, according to
the partial ordering defined by implication; i.e., if every axiom
of theory A is a theorem of theory B, then A is more general than B.
But not all mathematical theories are in the upper level. Chess,
for example, is a mathematical theory, but it would be farther down
the hierarchy.
And I would also include the mathematical forms of virtual reality
under the category of abstractions. For examples, see the following
web site, which has a couple of frames from a recent movie. It took
90 minutes of time on a supercomputer to do all the computations for
each frame:
http://a330.g.akamai.net/7/330/2540/e15dfb847b2b87/www.e-insite.net/ednmag/contents/images/185947f6.pdf
All the computer is doing is generating lots of colored polygons.
Those polygons would be fairly far down the hierarchy, but they're
still abstract.
John