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The following scheme, formulated by a philosopher-mathematician who was a director of industrial research, is of interest because of the special effort made to clarify the significance of number as a means of ordering phenomena and values in complex systems. The extracts, made for an earlier paper, are based on information in:
The series below was developed by J. G. Bennett to replace the Aristotelian and Kantian categories, with their dualistic characteristic. His definitions of systematic fea- tures are given . . The characteristics given here summarize the extensive descriptions of Bennett (vol. I, p.31 48, vol.3,p. 14 -75).
Comprehension of the systems proceeds in a definite sequence, given their order of emergence into awareness and the minimum number of terms required to exempli- fy their attributes. Only 12 systems are identified here, although systems of any number of terms may be con- sidered in order to encompass whatever degree of con- creteness one is capable of grasping. The limitation is one of understanding.
A particular system never exhausts the possibility of description and comprehension for, whatever number of terms is reached, some degree of abstraction remains and additional terms must be admitted in order to move towards a greater concreteness. Growth in understanding requires recognition of the representational power of successive systems and a deepening appreciation of their significance. As implied here and as stressed in the main text, Bennett's word labels and comments are only indi- cative and do not encompass or exhaust the meanings to which they refer. Their indicative power may be severely eroded by irrelevant polysemantic associations and in- creasingly so for the 3-term case and above. Conversely the richness of meaning in a given case is indicated by the symbol complexes which cultures produce to exem- plify such systems. The symbols may facilitate a better intuitive grasp of each system as a whole, in contrast to the fragmented comprehension resulting from the follow- ing descriptions presented as linear text.
I-term representation and comprehension ("Whole- ness")
Systemic attribute: universality. Term designation: totality. Term character: diversity in unity.
Any situation to which we direct our attention is a monad, but some exemplify the systemic attribute of universality more strongly than others. The monadic character of the universe as a totality is present in all its parts. Wholeness is universal and omnipresent but rela- tive; it may be transformed into identity. The combina- tion of confused immediacy and the expectation of find- ing an organized structure gives the monad a progressive character; it is what it is, but it holds the promise of being more than it appears to be.
Aspects of wholeness: unity, coherence, togetherness, completeness, order, organization.
2.-term representation and comprehension ("Polarity") Systemic attribute: complementarity. Term designation: poles. Term characters: positive, negative.Connectivity: force.
Any pair of terms between which both connection and disjunction are recognized, although few pairs stand in more than weak opposition to one another or with more than insignificant connection. Through polarity, everything is in a state of strain which polarity itself can do nothing to relieve. If gives rise to force which may be transformed into direction. It can neither show how oppositions arise nor how they may be resolved. Its closure is not that of completeness.
Aspects of polarity: active/passive; pleasant/unplea- sant, like/dislike, etc.
3-term representation and comprehension ("Related- ness")
Systemic attribute: dynamism. Term designation: impul- ses. Term characters: 1, affirmation; 2, receptivity; 3, reconciliation. Connectivities (1st order): acts (1-2, generation, 2-3, consent; 3-1, decision). Connectivities (2nd order): actions (1 --2-3, expansion; 1-3-2, inter- action; 3--2 --1, freedom; 2-1 -3, concentration; 2-3 --I, identitiy; 3-1 -2,order).
livery dynamic structure has the form of a triad and the three independent impulses found are those to which all relationships are reducible. Such relatedness may be. transformed into interaction. The triad shows how acts enter into the structure of the world and resolve contradictions.
4-term representation and comprehension ("Subsis- tence")
Systemic attribute: activity. Term designation: source. Term characters: motivational (1, ground; 2, goal); operational (3, direction; 4, instrument). Connectivities (1st order): interplays
Subsistence is the limitation of existence within a framework and may be transformed into maintenance. The tetrad specifies an event. It is the form of alt acti- vities that lead to a change of order and as such is in- herently inflexible. Its very nature is to be an activity of transformation. Its lack of central emphasis allows activity to be studied as ordered diversity, but prevents the association of the activity with a particular entity. Indeed it does not allow for the existence of separate entities.
5-term representation and comprehension ("Potential- ity")
Systemic attribute; significance. Subsidiary attributes: potentiality and meaning. Term designation: limit. Term characters: 1, intrinsic; internal limits (2, lower; 3, up- per); external limits (4, upper; 5, lower). Connectivities (1st order): mutualities (10 dyads). Connectivities: (2nd order): 10 triads. Connectivities (3rd order); 5 tetrads.
Meaning and potentiality must he added to activity, if the significance of a structure for itself (and for the tota- lity that contains it) is to be specified. Only then docs a structure become a bounded significant entity. Such entities have limits of significant connectedness with the outer world and limits of connectedness with their inner range of meaningful potentialities. Everything that exists has potentialities for actualization that outstrip the rela- tionships that it can sustain within any concrete situa- tion.
6-term representation and comprehension ("Repetition")
Systemic attribute: coalescence. Subsidiary attributes: recurrence, progress and self-realization, independence, form of events. Term designation: law (governing the coalescence of events). Term characters: 1, order; 2, ex- pansion; 3, identity; 4, freedom; 5, concentration; 6, interaction. Connectivities (1st order): steps.
Coalescence is understood as the property of struc- ture, whereby significance acquires depth and enrichment and yet retains the unique character associated with a particular event. The hexad, as progressive cyclicity, is the system most appropriate for studying structures in a step-by-step process of realizing their significance as events. It expresses the two-fold character of creation and counter-creation and also the movement of the entire process towards a goal. Although potential energy can be stored up indefinitely, it can only renew itself through the repetitive two-fold action of a disturbing and a restoring force. Success in action requires a balance be- tween attention to what actually is and what potentially might be; events continue to transform themselves even when their actualization is completed. However the hexad does tend to emphasize the separateness and iso- lation of such events from one another.
7-term representation and comprehension ("Structure")
Systemic attribute: transformation. Subsidiary attributes: structure, history. Term designation: state. Term charac- ters: 1, initiation; 2, involvement; 3, separation; 4, har- monization; 5, insight; 6, renunciation; 7, completion. Connectivities (1st order): intervals. Connectivities (,2nd order): harmonies.
A structure is a self-regulating system capable of rela- tively independent existence. Such a system is no longer closed and changes in the environment accompany chan- ges in the entity. A transformational superstructure is therefore provided by the heptad to reconcile the self- realization requirement of the well-defined entity (namely the acquisition of new properties that were previously neither potential nor possible) and the dissolution of identity required for integration as a part within a whole. A heptadic system is required whenever there is change involving a real gain or loss in significance. By such trans- formation, significant events are integrated into the stream of universal history.
8-term representation and comprehension ("Individua- lity")
Systemic attribute: completedness, organized totalities. Term designation: element. Term characters: active (1, summit; 3, atom; 5, base; 7, totality); structural (2, states; 4, functions; 6, necessities; 8, ideals). Connectivities (1st order): components (dyads). Connectivities (2nd order): initiations (triads). Connectivities (3rd order): fields (tetrads). Connectivities (4th order): signi- ficant substructures (pentads).
Individuality (whether actualized or potential) is the source of initiative residing in organized structures; it may be transformed into endurance, and is also a unique centre of conscious subjective experience. The octad is able to represent organized structures and historical pro- cesses ranging in scale from unity to totality. Us value is classificatory, interpretative, heuristic and predictive. It is however only applicable to structures organized in depth.
9-term representation and comprehension ("Pattern") Systemic attribute: harmonization. Term designation: sources (3). steps (6).
Experience would lose all coherence if there were not alwaysactive sources of order residing in the patterns of organized structures. The ideal completion of the octad does not take into account the uncertainty and hazard encountered in actual experience. The ennead permits the representation of everyday working structu- res (disturbed by unpredictable environmental factors) in which harmony is established and maintained. The harmonization is dynamic and indeterminate.
10-term representation and comprehension ("Creativity") Systemic attribute: integrative complementarity.
In all experience there is evidence of a creative (pat- tern generating) activity that is not only the source of order but also the vehicle of disorder - a polarity exem- plified by the decad. At this level several sets of processes are able to compensate for one another's defects and produce an overall harmony that reacts on, and sustains, the individual structures.
11-term representation and comprehension ("Domina- tion") Systemic attribute: synergism
This is the highest form of relatedness and is the power, subject only to the law of necessity, that recon- ciles order and disorder through the agency of creativity. It provides the conditions for mutual completion of structures of different kinds.
12-term representation and comprehension ("Autocracy")
Systemic attribute: perfection.
The dodecad is significant as a master pattern for understanding all total structures of the universe, be- cause it is the first system in which the main elements of experience can all be represented. It combines dyna- mism and diversity, or relativity and relatedness. It is the culmination of the transformations whereby the struc- ture of existence is first disordered, then corrected, then redeemed and finally perfected. Autocracy is the primary affirmation by which all possible experience is brought into existence whether as potential pattern or as the ac- tual process of the universe. It is the element that acts without dominating, wills without reacting, and unifies all possibilities.
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