To: |
ONTAC-WG General Discussion <ontac-forum@xxxxxxxxxxxxxx> |
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From: |
arun majumdar <arun@xxxxxxxxxxxx> |

Date: |
Mon, 28 Nov 2005 14:28:58 -0500 |

Message-id: |
<438B5A7A.6010005@xxxxxxxxxxxx> |

Hi All, (01) 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. (02) 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 (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). 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 maths 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. (03) However, these notions may be useful in abstracting ur understanding from the morass of linguistics (and its attendant ambiguities) to mathematics, wherein we may specificy domain specifc 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. (04) 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 amgiduities. One possibility is by defining the environment space in which the distinction is resolved (hence we create the objects as embedded, or embeddings). (05) 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. (06) Any reactions welcome - perhaps more concrete examples may be needed, and, I would be happy to so do if requested. (07) -Arun (08) richard.murphy@xxxxxxx wrote: (09) >Barry, Pat & All: > > |

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