SUO: Re: Motion for a Joint SUO Project (Follow-Up)(Repost)
[Reposting after 20 hours]
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Matthew, Patrick,
Some continuing thoughts ...
If you are looking for a structural model that does somewhat address
these issues of variant perspective -- to me it's the (onto-)logical
analogue of projective geometry, there are some suggestive precedents
in the theory of manifolds, more generally in topology, and even more
generally in the theory of categories itself.
Manifold theory goes back to Riemann -- it's a big piece of the math that
made it possible to think about relativity in physics to any useful purpose.
I have started an "excerpt system" that can be followed from this location:
http://suo.ieee.org/ontology/msg04769.html
The prevailing metaphor in manifold theory is an "atlas of charts",
which is an apt image for how different observers overlay different
templates on their objective reality and view it in different ways.
You can get a good picture of what it's all about by starting with
the diagram of a "translation map" or a "communication situation":
http://suo.ieee.org/ontology/msg04782.html
This stuff is hyper-generalized in category theory under the
headings of "fiber bundles" and "descent theory", but the
basic ideas are really quite intuitive, and what we need
most for the task before us is to develop the logical
analogues a bit better than they have been so far.
Topology is very largely about viewing the
same sets under different organizational
schemes that say what goes next to what,
or what is in the neighborhood of what.
Excerpts from a classic text are here:
http://suo.ieee.org/ontology/msg04739.html
Other notions from category theory are especially pertinent here.
For instance, the idea of different views (for different pews?)
is formalized in the notion of a "functor" between categories,
and the idea of translating between POV's is captured by the
notion of a "natural transformation".
None of this contravenes the utility of lattices and other orders
as pegboards on which to hang our panoply of parts and tools, but
provides us with the extra dimension that is needed to reflect on
the actual application of these components to an outside reality.
Jon Awbrey
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Jon Awbrey wrote:
>
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> Matthew, Patrick,
>
> Lattices of models, libraries of axiom books, and other sorts
> of generalization hierarchies, useful as they undoubtedly are,
> do not quite address the kinds problems that we are talking
> about here, which have to do with the different ways that
> experience may be construed along the lines of logically
> incompatible models, and the fact that we often have to
> choose a model an act upon it long before we know for
> sure whether it is adequate to the objective res or
> even consistent within itself.
>
> Consider the three main classical geometries, which share all of
> their axioms but split upon the issue of how many lines can be
> drawn parallel to another line through a non-incident point.
>
> ? ? ? ? ? o ? ? ? ? ?
>
> ---------------------
>
> 0. none.
> 1. one.
> 2. many.
>
> In classical geometry, you cannot have a single
> axiom set with more than one choice for this value.
>
> Mathematically speaking, this is no problem, you are merely
> considering three different classes of mathematical objects.
>
> But if you assume that physical space is a coherent object at all,
> then you can ask: Of which theory is it the model? Trick question.
> Most likely none of them. It appears that physical space is really
> much weirder than any of these, if we take the question strictly.
> But if we take the question loosely, we can ask whether one theory
> forms a better approximation to physical space, at a certain scale,
> according to a specific panoply of measuring instruments, and so on,
> but all of these answers will depend on approximate (non-deductive)
> reasoning, and various types of judgement calls.
>
> Jon Awbrey
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> Patrick Cassidy wrote:
> >
> > Matthew West raises an important point regarding "consistency"
> > that may require more careful definition of terms.
> >
> > > {Matthew West]
> > > There is a flaw in your reasoning.
> > >
> > > You are (I think) assuming that it is desirable that all theories in some
> > > SUO product should be consistent with each other. This is not at all the
> > > case. What is true is that you would want the subset you were using to
> > > solve some problem to be consistent. However, there are many theories
> > > that might be inconsistent with each other, but which might be useful
> > > for different purposes. 3D vs 4D is one that immediately springs
> > > to mind.
> >
> > PC: It wasn't the intention of the revised motion to create any preference for
> > logically consistent modules, but to provide a mechanism for recognizing which
> > are inconsistent with each other. The proposed motion recognizes that there
> > can be logically inconsistent modules within a "generalization hierarchy".
> > I agree with Matthew on this point:
> >
> > MW: What is true is that you would want the subset you
> > were using to solve some problem to be consistent.
> >
> > PC: One of my main motivations in looking for the maximum degree
> > of "consistency" (see below for a definition) between ontologies
> > is the problem of communication between them. For a user that
> > wants to create a solipsistic ontology-driven application that
> > never talks to another cognitive system, any self-consistent set
> > of modules from the standard library could be used. When one
> > wants one's application to interoperate with others, choosing
> > sets that are consistent with each other would be not merely
> > "desirable", but perhaps essential. To some extent, the
> > sharing of research results on systems using representations
> > of complex concepts should also be more effective when consistent
> > representations are used. Nevertheless, the comments within
> > this group make it clear that there will always be some
> > users who feel that it is necessary to use some ontological
> > theories that are logically inconsistent with those of others.
> > That's fine. For me, one purpose of trying to discover the
> > maximum degree of consistency between ontologies would be
> > to allow such users to make an informed choice, to be
> > able to be certain that the benefits of using one ontology
> > inconsistent with another outweigh the benefits of
> > increased interoperability between systems.
> >
> > PC: With regard to the follow-up sentence:
> >
> > MW: However, there are many theories that might be inconsistent with each other,
> > but which might be useful for different purposes. 3D vs 4D is one that
> > immediately springs to mind.
> >
> > PC: Matthew considers 3D and 4D views to be inconsistent. What I
> > had in mind was that 3D and 4D might *not* be "inconsistent" in the
> > sense that an assertion about a 4D object might not have any
> > inferences that are logically contradictory to assertions in a 3D
> > ontology, since there are no assertions about 4D objects in a 3D
> > ontology. Perhaps "consistent" is not the proper term for such a
> > relation between ontologies. We may need another term --
> > "orthogonal" would convey some of the notion that these deal with
> > different logical things, but is too strong in suggesting that
> > there is nothing in common.
> > What I am hoping is that different ontologies that treat the
> > real world differently in some respect, but do not generate inferences
> > that are logically contradictory, can be translated into each other
> > accurately and without loss of information. Depending on the
> > way they are axiomatized, I think that 3D and 4D ontologies might
> > be translatable in that way. If they can be translated into
> > each other, "interconvertible" or "intertranslatable" might
> > be proper terms. Other suggestions?
> > I have considered "inconsistent" ontologies as those which
> > genuinely lead to logically contradictory inferences. These would
> > have to be treated differently from ontologies that view the world
> > from a different perspective, but don't create logically contradictory
> > assertions.
> > So "consistent" does not appear to be the proper term to refer
> > to "non-contradictory" ontologies. I'm open to suggestions about
> > how to refer to these distinctions.
> >
> > Pat
> >
> > =============================================
> > Patrick Cassidy
> >
> > MICRA, Inc. || (908) 561-3416
> > 735 Belvidere Ave. || (908) 668-5252 (if no answer)
> > Plainfield, NJ 07062-2054 || (908) 668-5904 (fax)
> >
> > internet: cassidy@micra.com
> > =============================================
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