SUO: Re: Consensus?
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
Robert,
It may take me a couple of days to work through all of this material.
I started some threads on "Category Theory & Model Theory Unplugged"
because I am going to dedicate myself to making this stuff as second
natural as it actually is in practice. Just between you and me and
the lambda-post, category theory proper quit being general enough
for me about 23 years ago, but I'll be willing to settle for this
much progress as a start. So my plan is to just keep on talking
about things of type f : X -> B until the step to topoi seems
about as commonplace a move as falling off a logos. But I
did find a new book that seems to give good coverage of
some of the gaps between here and there:
| Lawvere, F.W. & Rosebrugh, R.,
|'Sets For Mathematics',
| Cambridge University Press,
| Cambridge, UK, 2003.
Robert E. Kent wrote:
>
> 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 is completed) withdrawn in favor of a more bottom-up approach aimed at trying the model the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 contributor 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.
> >
> > Mathematicians 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
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o