Friday, December 09, 2005
(new thread on Emergency Medical Ontology Project planning ŕ [home] )
Communication from Arun Majumdar With footnotes from Paul Prueitt also see elaboration ŕ [286]
Category Theory is a good foundation to
start with on these particular issues and having had some experience with the
subject and Analogical Reasoning in particular, my *belief* is that these may
hold shed light on a possible path forward.
[1]
As a starting point, for example, consider
metaphor and analogy as a means to unify between disparate ontologies --- where
an ontology is viewed as a theory over some model. In metaphor and analogy, a
problem in one field can be transformed into a problem statement that is
understandable in another field. In order to do this, however, you need an
invertible function which provides the LIFT to different algebra and this means
that the kinds of properties that you need will be found in the algebra of
toposes [2]
(since their are varieties that have pullback recovery) which is needed to
guarantee scope across a broad range of representations (here I use the term
for categories). [3]
The idea of "understanding by
changing perspective" deals with a general methodology of understanding
related to "change of basis" and states that in order to understand
an object, you just have to walk round the object to get another view (for
example, in mathematics you change from rectilinear to polar coordinates to solve
certain kinds of problems). This
trivial insight was mathematically defined by Nobuo Yoneda [Yoneda 1954] as the
Yoneda lemma which states that any mathematical object can be classified up to
isomorphisms by its "functor" which is the system of all
"views" or "perspectives" of the given object from all
other objects of the same category (ie. in structural form). This statement
even allows one to construct objects via their functors alone (ie. using only
the abstract definitions of inter-relatedness to recover an object into its
category or *structural environment*).
Hence, it is a deep an powerful concept, but, the mechanics of going
from a relation to its implied objects to which it applies in still non-trivial
to solve. [4]
However, these notions may be useful in
abstracting your understanding from the morass of linguistics (and its
attendant ambiguities) to mathematics, wherein we may specific domain specific
theories under which various models (providing the interpretations, and,
contexts) can be placed. As noted in
the short note on the insight of the Yoneda Lemma, and connecting this the Chu
Spaces mentioned by Rick Murphy (below), requires the development of theories
about functional dependency and a focus on relational and relational
derivatives, like information flow etc...
In terms of a concrete contribution beyond an email posting, I suggest
that there is a nice connection between Sowa's UF and one possible methodology
using Hirst's lexical chaining, by lifting the idea to "conceptual"
chaining - that may become something we can model in Chu Spaces perhaps more
effectively than within the rubric of the linguistically motivated lexical
chaining.
The challenge resides in mathematization
of the semiotics to the extent that we can take an object or term and make the
type/token distinction. Without a principled way to make the type/token
distinction, we will get inconsistencies or ambiguities. One possibility is by defining the
environment space in which the distinction is resolved (hence we create the objects
as embedded, or embeddings).
On a trivial level as a concrete example,
the phrase "I went to the bank" may not mean to a financial
institution, but if uttered near a rive, would imply a geographic feature. Hence, packaging the background environment,
I conjecture here, means that we need to work with *embeddings* as a
fundamental construction in ontological research and not only the
relations. Of course, embeddings are a
non-trivial subject.
Any reactions welcome - perhaps more
concrete examples may be needed, and, I would be happy to so do if
requested.
-Arun
[1] We share an interest in formalization of analogy, but feel that analogy cannot be fully formalized – because the formalization process itself must use induction to create a set of axioms. Quasi Axiomatic Theory approached this entire subject in a way that is similar to but with considerable differences to the work developed by Alexander Grothendiech, and later extended in your own work and in toposes and Chu spaces.
[2] Algebra of toposes: http://en.wikipedia.org/wiki/Topos
[3] Category theory is also developed by Robert Rosen and his mentor N. Rashensky (mathematical biology, theoretical biology). This type of category theory is present in a large percentage of the Grossberg type biologically feasible models of neural architecture. The category theory of Rosen focuses on anticipatory mechanisms (one of his books was “Anticipatory Systems”). The point is that this literature and discipline is quite different from the Chi Space and toposes constructions.
[4] The foundations to the notion of representational basis for knowledge representation can be traced to C S Peirce, but not in the way that Sowa interprets Peirce, rather in the way that the Soviet school headed by Pospelov and Finn interpreted Peirce. I meet both Pospelov and Finn several times, including during by visit to Moscow in 1997, where I gave a lecture on quasi axiomatic theory.