Reply to Edwinia's View of Categories Re: Q1-6 comments

To All:

Edwinia writes:

>
>So, I don't agree that category theory is the 'way to go'. I consider
>that category theory operates within a dyadic rather than triadic
>framework.

Let's look more carefully at the logical structure of a mathematical category.

A category is not merely a monadic, dyadic or triadic mathematical object.

A category is a tetradic framework.

Why is category theory a tetradic framework?
 
A category operates, at a *minimum*, on four objects (domains and
codomains) and three morphisms linking them to one another. Roughly
speaking, a category operates concomitantly on three dyadic
frameworks, two triadic frameworks and one tetradic framework.

For example,

*IF* one chose to define Pierce's "firstness", "secondness" and
"thirdness" as morphisms ,
*then* one could (possibly) construct a correspondence relation
between the semantics of firstness, secondness and thirdness
*and* the nature of the morphisms of a category.

Secondly, at issue is the nature of the four objects as related to
the three morphisms. A precise logical relation within a category
exists between the four objects and the three morphisms such that a
logic coherence is generated among the four objects and three
morphisms. (For an introductory text on categories, see the book by
Lawvere, Conceptual Mathematics.)

For example, in certain cases, the objects of a category can be
logical propositions.
The case where the objects are propositions has been explored
extensively and has led to the a deeper understanding of logic itself
as well as a branch of mathematics known as "model theory" which
seeks to integrate mathematical concepts over a wide range of
mathematical species.

In another example, I have recently succeeded in developing a model
of chemical species directly from the metaphysical primitives of
matter, identity, space and time. Thus it became possible to
construct an organic theory of communication based on an organic
species as a mathematical species as alluded to within my abstract.
This allows one to contrast it with the Shannon engineering approach
(which is purely a formal syntactical structure without meaning).

Tensions between the concepts of semiotics, semantics and syntax are
related to the question of structure and the relations among such
structures.

Within category theory, one has substantial "logical space" to
associate semiotic concepts with semantic concepts and with
syntactical concepts. The above example illustrates an approach to
resolving the tensions between semiotics and mathematical syntax in
terms of associated structures of a category.

In contrast with other mathematical species (such as vector spaces or
fields), a category is grounded on the associative relations among
the four objects and three morphisms. (See Stephan Foldes,
Structural Mathematics; this may not be the exact title?)

Let me close with a note of caution.

Category theory is a theory of mathematics. Application of category
theory to scientific questions is a very challenging undertaking
because of its abstractness and the complex origin of its coherence.

Cheers to All

Jerry LR Chandler
Received on Tue Jul 9 18:21:35 2002