SUO: RE: Re: Consensus?
Dear Robert,
Your point below brings out the main reason to date why I am
attracted to Category Theory. I like to understand what the
bedrock is that I am relying on. I have not been able to
find anything convincing in tuples. To me they seem to be
just a structure that requires interpretation. On the other
hand the functional approach of Category Theory seems to
give some real meaning.
Matthew West
Principal Consultant
Shell Information Technology International Limited
Shell Centre, London SE1 7NA, United Kingdom
Tel: +44 20 7934 4490 Other Tel: +44 7796 336538
Email: matthew.west@shell.com
Internet: http://www.shell.com
> -----Original Message-----
> From: Robert E. Kent [mailto:rekent@ontologos.org]
> Sent: 16 May 2003 17:00
> To: Jon Awbrey
> Cc: SUO
> Subject: SUO: Re: Consensus?
>
>
>
> Jon and others,
>
> Initially, the IFF axiomatization included the axioms for
> topos theory.
> Quoting Goldblatt in the prospectus of his book "Topoi" ["set theory
> provides a general conceptual framework for mathematics. Now
> since category
> theory, through the notion of topos, has succeeded in axiomatizing set
> theory, the outcome is an entirely new *categorical foundation of
> mathematics*! The category-theorists attitude that "function"
> rather than
> "set membership" can be seen as the fundamental mathematical
> concept has
> been entirely vindicated."] However, since the topos axioms
> were thought too
> heavy by various SUO WG members, these have been (at least
> temporarily until
> absolutely needed, which means probably when the
> axiomatization for the IFF
> lower set namespace is completed) withdrawn in favor of a
> more bottom-up
> approach aimed at trying the model the mathematical intuitions of the
> "working ontologist". BTW, the IFF follows the spirit mentioned in
> Goldblatt's note, that the notion of function is fundamental,
> whereas the
> notion of relation is secondary. The IFF metalevel only uses
> the notion of
> binary relation -- multivalent relations are modeled at the
> IFF metalevel,
> but actually appear and are used only in the object-level
> ontologies. This
> contrasts with the Common Logic KIFish approach that the
> notion of relation
> is fundamental. In fact, the IFF has a _very_ elaborate, flexible and
> intuitive axiomatization for multivalent relations that is
> centered in the
> IFF Ontology meta-Ontology in the lower IFF metalevel.
>
> The IFF is based upon a three-tiered set-theoretic backbone of small
> collections (called sets), large collections (called
> classes), and generic
> collections. The category denoted by *Set* of all small sets and all
> functions between small sets is a case in point. The objects in this
> category are intuitively small sets. Now in general, a
> category consist of
> two components -- its collection of objects and its
> collection of morphisms
> (aka arrows). The collection of all small sets, which
> constitutes the object
> collection of *Set*, is a class. So also is the collection of
> all functions
> between small sets. A (normal) category is a category whose object and
> morphism collections are set-theoretic classes
> (set-theoretically larger
> categories are called "quasi-categories" in the IFF and
> elsewhere). The
> collection of all (normal) categories is a generic collection
> -- neither
> small nor large. These comments indicate some of the
> intuitive notions in
> category theory. And this is what the IFF axiomatization aims
> to represent,
> at least at first cut. The three-tiered set-theoretic
> backbone of generic
> notions, large notions, and small notions is reflected in the three
> metalevels of the IFF: top, upper and lower.
>
> From Robert "The IFF radical"
>
> ----- Original Message -----
> From: "Jon Awbrey" <jawbrey@oakland.edu>
> To: "Adam Pease" <apease@ks.teknowledge.com>
> Cc: "Robert E. Kent" <rekent@ontologos.org>; "John F. Sowa"
> <sowa@bestweb.net>; "SUO" <standard-upper-ontology@ieee.org>
> Sent: Friday, May 16, 2003 6:50 AM
> Subject: Re: Consensus?
>
>
> > o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
> >
> > SUO WG,
> >
> > I have been looking at all three proposals again,
> > and I find the incompatibilities too radical for
> > words. I would like to suggest a strategy for
> > moving the project forward, independently of
> > whatever official status these or any other
> > potentials have at a given moment in time.
> >
> > Let each conributor to the possible merger
> > first index each of their terms with the
> > name of the group, for example, giving
> > "Set_IFF", "Set_OC", "Set_SUMO", etc.
> > for their corresponding concepts of
> > a set. This would facilitate the
> > formation of "disagreement sets"
> > as used in standard unification
> > algorithms, though, of course,
> > only in a rough way at first.
> > The axioms or comments for
> > the several concepts could
> > then be hung on the lattice
> > with glee, and if any of them
> > turn out to be coincident, that
> > would be just wonderful, even if
> > purely coincidental, but hanging on
> > to the indices would also leave room
> > for future extensions by other parties.
> >
> > Mathematicinas and others have coped with a similar situation
> > for many years now, for instance, implicitly dealing with
> > Set_Naive, Set_Axiomatic, and within the latter class,
> > Set_HBvNG, Set_ZF, and several others. Of course,
> > this was handeled informally, in a context-wise,
> > discourse-wise way, and that is not a dodge
> > that we can get away with as we formalize
> > concepts in a computable fashion.
> >
> > Jon Awbrey
> >
> > o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
> >
> >
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