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John et al,
. I have had considerable problems with trying to tie down the nature of pure structure, theory, etc. I tried to apply the term "concept" to this, but was ruled out of order on the basis that a concept had to have been conceived by someone, and by implication, at a given point in time. The latest draft of ISO/IEC 111793 accordingly defines "Concept" as
· "a unit of thought constituted through abstraction based on characteristics common to a set of objects".
. I fear the same attitude may be taken to "Abstract", since someone had to perform the (mental) task of "drawing away" the pattern or structure from the original observation or thought.
. Personally, I feel there is something in the nature of pure structure or pattern that can exist independently of whether it exists in physical reality, and also independently of whether any intelligent being has conceived of or observed it.
. By way of explanation, I would suggest that both Newton's law of gravity and Kepler's laws of planetary motion are not precisely accurate, and so do not exist in physical reality, but are rather mere approximations of physical reality. Yet, I would suggest that they are both timeless and without location or mass ("zerodimensional" in the physical sense), and hence existed infinitely before either Newton or Kepler were born, and will continue to do so after all records and other traces of them have been obliterated.
. The problem is I don't know what term I can use to apply to such pure structures or patterns that one can abstract or conceive.
. Neither does "information" cover this. One can observe and document the spread of information, whether be it physical, such as through the spread of genes, etc., or purely conceptual, such as with nemes, by means of, say, semaphore communication.
. The reason I ask this question is that there are times when I wish to consider these "zerodimensional" patterns / structures.
. Incidentally, t have a related problem about the nature of "dimensionality". I wish to apply the term beyond the traditional physics scope of mass, length and time, into far more abstract areas. This is because it can be extremely useful in information management and information based sciences to apply to a far broader range of concepts. Statisticians psychologists and pharmacists do this when they are looking to ascertain which factors are relevant, and which irrelevant, for particular avenues of research and endeavour. The concept of "dimensionality", along with others like "mutual exclusivity" or "orthogonality" also come into play here, and can greatly facilitate conceptual design and analysis.
. Can anyone suggest a rigorous definition that covers this broader scope?
Cheers Graham Horn
National Data Standards Unit
Australian Institute of Health and Welfare
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Phone: 02.6244.1094
Fax: 02.6244.1199
Email: Graham.Horn@aihw.gov.au
Knowledgebase: www.aihw.gov.au/knowledgebase/
-----Original Message-----
From: John F. Sowa [mailto:sowa@bestweb.net]
Sent: Wednesday, 2 January 2002 14:28
To: kif@philebus.tamu.edu
Cc: standard-upper-ontology@ieee.org
Subject: 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:
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