Patterns of Conceptual Integration

Qualitative characteristics of sets

Anthony Judge

9.1. Characteristics of multi-term systems: The remarks of the previous section provide a context within which efforts at establishing the characteristics of multi-term systems can be considered as defined in Annex 1 . This question cannot be explored here. It serves as an indication only therefore, that the results of J. G. Bennett's exercise are summarized in Annex 2. This suffers from the disadvantage of not establishing explicit links to the rich variety of cultural and mathematical material reviewed by von Franz in her study of the first four integers. Such material should be used to interpret and broaden the meanings, otherwise Bennett's (or any other) particular orientation is too easily assumed to exhaust the meaning associated with each system--thus subjecting the approach to the difficulties raised in the previous sections.

Bennett points out that "no one system taken alone can exemplify the organized complexity of real structures. We usually need to take more than one system into account in order to gain the insights needed for understanding any existing structure that we find. According to the aspect of structure that happens to be relevant to a given purpose, a system of one order may be more useful than another." (45, vol.3, p. 11-12).

Also (bearing in mind the limited value of label words for the system attributes identified in Annex 1): "The series of multi-term systems is a progression such that each system implies all the earlier ones and requires those that follow. We cannot understand the triad unless we already group the notions of universality and complementarity, and the dynamism of the triad is not realized without the activity of the tetrad. The later systems are not only more complex and more highly organized than the earlier ones; they embody an understanding of reality that is more comprehensive and practical. The progression is from abstractness to concreteness." (45, vol.3, p. 12).

But: "Not all structures exemplify all stages of the progression to the same degree. A given structure may exemplify one attribute strongly and others weakly.... One other general property of systems remains to be considered. This we shall refer to as term-adequacy. If the terms of a system cannot be clearly discerned in a given structure, the required characters will be lacking and the system in question is then inadequately represented." (45, vol. 3, p. 13). Namely the set is weak in that attribute.

In the light of this argument, attempts should be made to explore a 3-term set re-interpreted as a 4-term system or more, particularly in the case of fundamental sets. In Bennett's study of systematics [61], he finds that: "for purposes of practical utility, the systems fall naturally in groups of four. The first four from the monad (1-term) to the tetrad (4-term) help us to see how structures work. The systems from pentad (5-term) to octad (8-term) show why they work and how they enter into the pattern of reality. The third group from the ennead (9-term) to the dodecad (12-term) is mainly concerned with the harmony of structures: that is, the conditions that enable them to fulfill tinier destined purpose." (45, vol.3, p.12)

9.2. Clarification of specific sets: Two procedures are outlined (in Annex 3) for the clarification of material on complete sets. Both procedures ensure that any given set is embedded in a context. In the first case, this is in relation to alternative (or more superficial) possibilities. In the second, it is in relation to more fundamental possibilities.

By such procedures the set is being tested and refined in a manner which should establish the constraints on its meaningfulness and communicability to those who--in contrast to its vigorous advocates - may be sensitive to other aspects of the context in which it is embedded [62]. The procedures necessarily highlight the extremely limited value of dependence on the univocal, unambiguous meaning of any words (in definitions) used to label such sets or their elements.

It should be stressed that, in contrast to the usual competitive preoccupation, the concern is not with establishing any particular set as the most valid. Rather it is to give some understanding of the probability that any such set will be advocated, perceived as valid, or widely comprehended and communicated. At the same time it supplies a context for elucidating the meaning underlying whatever marks (words, numbers, codes, etc.) are used to identify a set and its elements.