Rudy's Diamond Strategies

This complementary Blog to the Chinese Challenge Blog is presenting studies to a mathematical theory of Diamonds. Diamond theory is studying for the first time, tabular categories as an interaction of categories and saltatories.

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Monday, April 6, 2009

Triadic Diamonds

Robertson’s algebra of triadic relations, Gunther’s founding relation, Toth's semiotics and diamond triads

FULL TEXT Diamonds/Triadic Diamonds.html Diamonds/Triadic Diamonds.pdf

Some further thematizations and formalizations of diamond topics, especially triads, are presented. Triads, and founded triads, are presented in the context of Gunther’s epistemology, Toth’s semiotics with the help of Robertson’s “Algebra for triadic relations”. It is proposed that founding relations had been thematized externally only. An implementation of founding strategies into the system to be founded by the diamond approach is realizing the simultaneity of construction and verification of the triad.

Chinese Ontology and Diamonds
A new attempt to formalize the idea of founding relations is proposed by the diamond approach which takes into account the simultaneity of the model and its foundation. It also reflects the fact, that a foundation of an operation is localized on a different level of abstraction. The activity of modeling and the activity of founding are complementary activities demanding different kinds of abstractions. Hence, any applicative iteration of the model on itself is not fulfilling the criteria of foundation. "The idea of in-sourcing the matching conditions into the definition of diamonds tries to realize the two postulates of "Chinese Ontology", the permanent change of things and the endness (finitness) or closeness of situations. That is, diamonds should be designed as structural explications of the happenstance of compositions and not as a succession of events (morphisms).

More exactly, diamond are contemplating the interplay of acceptional and rejectional thematizations. Thus, morphisms with their matching conditions and composability are in fact of secondary order for the understanding of diamonds.

The complementarity of construction and verification, which is happening at once and not in a temporal delay, is a consequence of the finiteness and dynamics postulate of polycontextural "ontology". This simultaneous interplay is based on the insight that a delayed verification (or testing in programming) would not necessarily verify the construction in question because, at least, the context will have changed in-between. Delayed verification is possible only in the very special case of frozen dynamics.

In other words, in a changing open/closed world, the activities of construction and verification (of correctness and relevance) have to happen at once. Otherwise, because the conditions might have changed, the relevancy of the construction to be verified would have to be verified itself, again, and this ad nauseam.

Obviously, the statement is not about/against the stability of the construction (program, system, agreement, contract), this might be rock solid, but about the relevancy of the rock solid construction.

(In therapy or coaching, even by constructivists, this delayed checks are called “reality check”. Nearly always, such a reality to be checked has escaped any relevance.)

In-sourcing the matching conditions

Diamond strategies are offering a fundamentally different approach.
Each step in a diamond world has simultaneously its counter-step. Hence, each operation has an environment in which a legitimation of it can be stated. The legitimation is not happening before or after the step is realized but immediately in parallel to it.

Morphisms are representing mappings between objects, seen as domains and codomains of the mapping function. Hetero-morphisms are representing the conditions of the possibility (Bedingungen der Möglichkeit) of the composition of morphisms. That is, the conditions, expressed by the matching conditions, are reflected at the place of the heteromorphisms.

Hetero-morphisms as reflections of the matching conditions of composition are therefore second-order concepts realized "inside" the diamond system. 
Morphisms and their composition are first-order concepts, which have to match the matching conditions defined by the axiomatics of the categorical composition of morphisms. But these matching conditions are not explicit in the composition of morphism but implicit, defined "outside" of the compositional system. Hence, in diamonds, the matching conditions of categories are explicit, and moved from the "outside" to the inside of the system. In this sense, the rejectional system of hetero-morphisms is a reflectional system, reflecting the interactions of the compositions of the acceptional system. Heteromorphisms are, thus, the "morphisms" of the matching conditions for morphisms.

FULL TEXT Diamonds/Triadic Diamonds.html Diamonds/Triadic Diamonds.pdf