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25th April 2006 | Draft
Comprehension of Requisite Variety for Sustainable Psychosocial DynamicsTransforming a matrix classification onto intertwined tori- / -Introduction From matrix to torus Systemic sub-systems vs Preferred modes: "Columns" vs "Rows" Generic matrix Distribution of significance on torus surface Interlocking tori: combining the two alternative representations Dynamics of interlocked tori Distribution of significance "within" a torus: using a third dimension for "Engagement with reality" Contiguity of paired circular cross-sections "Empty" centres and four-dimensionality Psychosocial relevance of torus-related dynamics Coherence, instability and failure of psychosocial systems Resonant associations and comprehension of feedback vital to sustainability Beyond the plane: form and medium in terms of the calculus of indications Resonant association of psychosocial identity with intertwined tori Memorability: musical clues to psychosocial system sustainability Possible immediate applications of relevance to sustainability Choosing the dimensionality of living Potentially problematic consequences References IntroductionSystems of classification, whether described as nested thesauri or ontologies, necessarily take the form of nested lists. This approach is facilitated by the ease of representation in printed texts and the menu structure of standard software applications. Additional requirements and constraints are imposed on any classification that is organized as a matrix. This is a format typical of many academic papers where a list structure is viewed as inadequately modest in its dimensionality. The emphasis here is however on comprehension of psychosocial systems rather than descriptive explanation. In the light of some use of matrix organizations of categories, the exploration here is extended to the projection of a matrix pattern of categories onto a single torus and onto several intertwined tori. This implies that some distinctions explicit there are only implicit or conflated in any representation in a simpler matrix or list. The assumption here is that purely descriptive articulations, no matter how sophisticated the language used, lack the requisite variety to encompass the differences characteristic of psychosocial dynamics. However, efforts to "grasp" and "possess" higher-dimensional processes through their lower-dimensional "containers" misunderstand the nature of those processes and their relationship to the understanding vital to sustainability. From matrix to torusIngetraut Dahlberg founded the journal International Classification (1974), which later became Knowledge Organisation (1993) and was the instigator in 1989 of the International Society for Knowledge Organization. In 1981 Dahlberg proposed a matrix-structured Information Coding Classification (ICC) as a universal classification scheme -- and represented it in 1996 (see discussion by Joseph T. Tennis, Layers of meaning: disentangling subject access interoperability, 2001). Dahlberg's early approach had been adapted for the classification of international organizations, the problems perceived by them, and the strategies advocated by them (cf Functional Classification in an Integrative Matrix of Human Preoccupations, 1982). This adaptation continues to be used, implicitly, as the basis for online access to the associated databases. Questioning the limitations of a matrix representation, the possibility of using projection onto centro-symmetrical polyhedra was envisaged in 1980 (cf Needs Communication: viable need patterns and their identification, 1980) and explored in relation to the issues and strategies of the 1992 Earth Summit (Configuring Globally and Contending Locally: shaping the global network of local bargains, 1992; Spherical Configuration of Categories to Reflect Systemic Patterns of Environmental Checks and Balances, 1994). The approach taken in what follows advocates a different step in integration by transforming the matrix into a torus, as notably fundamental to the organization of computer memory for parallel processing, especially in supercomputers [more | more] . The process is described in MathWorld as follows:
The projection of a matrix onto a torus is achieved in two distinct ways:
The two approaches are of course geometrically equivalent. The form raises interesting questions regarding:
The general concern here is with the challenge to richer comprehension and how it needs to match match the complexity of policy-making -- especially when different factions prefer to order their understanding of complex systems in different ways. Systemic sub-systems vs Preferred modes: "Columns" vs "Rows"The design of the matrix -- used notably for the databases of the Encyclopedia of World Problems and Human Potential -- benefitted from the insights of Erich Jantsch (The Self-organizing Universe: scientific and human implications of the emerging paradigm of evolution, 1980). Like that of Ingetraut Dahlberg, the adaptation highlighted the "levels" of human preoccupation from the extremely concrete to the extremely subtle -- represented by successive rows in the matrix. The various preferred approaches to reality (discussed below) constituted the columns of the matrix. Engagement with reality: There is always a case for looking more attentively at the manner in which humans attend to their reality -- the manner of their cognitive engagement with it. This can of course vary from extremely concrete modalities to those of the subtlest, existential forms of identification with the environment. A distinction is however made here between:
The argument here is that any adequate descriptive modelling of a complex psychosocial system will include both sets of processes as sub-systems. On the other hand the two are readily conflated so that vital distinctions are lost -- reinforcing patterns of confusion. This typically occurs when the subjective dimension is considered irrelevant or denied. However it is also the case that the objective dimension may be denied, considered irrelevant, or treated as secondary, in the light of other understandings. Preliminary objective approach: The emphasis in the first matrix / torus to be discussed initially is therefore on the sub-systems that lend themselves to more objective description, potentially acceptable even to the behavioural sciences. More subjective distinctions are discussed below in relation to a third dimension.. The emphasis here is however not so much on what the distinctions are (in terms of content), nor on how many should be made, but rather on the need to make a sufficient number in order to reflect diversity to an adequate degree (in accord with Ashby's Law of Requisite Variety). It is difficult to make rich music with a monochord unless it is --- It is equally difficult to make music with an instrument with a wide range of variants (as on an organ). The art is to select a sufficient number to be useful and comprehensible -- recognizing that the distinctions are being made in a continuum. The same applies to the variety of modes (columns) through which reality is engaged to whatever degree (as represented by the rows). Visually the point is made by the fret structure of a stringed instrument such as a guitar or a violin. The complexity of the structure needs to be sufficient for the music but not too challenging for the player or the audience. A sitar represents an extreme of complexity open only to the few.Preferred modes ("columns") : Here the approach advocated is to take account of various conventionally preferred "ways" of making such distinctions. These could be understood as corresponding to different worldviews -- each understood to be complete in its own right. Without at this stage endeavouring to reconcile the differences between sets of approaches, the following sets are indicative of the ways in variety is distinguished:
Systemic sub-systems ("rows"): Here the distinctions relate to the way in which the psychosocial system is understood to work. Given any preference for a "mode" or "way" of understanding, the systemic sub-systems distinguished may be labelled in quite different ways. From a systemic or cybernetic perspective the sub-systems must necessarily work together as a system and their distinct functions are necessary for the viability of that system. The distinctions in question are however made in various ways, whether or not those distinctions are conflated under certain circumstances. An obvious example is the distinction between the basic computer operating systems (Macintosh, Windows, and Linux), and the associated worldviews, notably as highlighted by Eric S. Raymond (The Cathedral and the Bazaar, 1997):
Generic matrixIn the light of the above, it is possible to point to the existence of what might be termed a generic matrix
From an abstract cognitive systemic perspective, it could be argued that distinct modal preferences ("columns") and the distinction of sub-systems ("rows") could be conflated together into a single set of more abstract distinctions -- an archetypal list of complementary insights (typical of pantheons, etc). Splitting them into two sets interwoven as a matrix however makes them more accessible to comprehension. A matrix cell at any point of intersection then holds insight that is more specifically and readily comprehensible. It also draws attention to aspects which might be readily neglected in any exposition of a particular preferred mode ("column") or in discussion of a particular systemic sub-system ("row"). Note that in the generic presentation above, the number of rows or columns may be decided by convention or preference. Distribution of significance on torus surfaceCorresponding to the two modes noted above, significance may be distributed in either of two ways:
Earlier examples of the possibility of knowledge organization on a torus include:
In considering the question of the number of distinctions it is useful to make with regard to columns and rows, it is appropriate to note the mathematical constraints relating to the colouring of a torus. The torus coloring of an ordinary (one-holed) torus requires 7 colours (consistent with the Heawood conjecture). Interlocking tori: combining the two alternative representationsThe two possible modes of representation may be represented together in a manner which holds a higher degree of complexity, raising interesting questions and possibilities. This is achieved by projecting each matrix (Mode A and Mode B) onto a separate torus in the manner discussed. The two tori are then allowed to interlock as illustrated by the following
In the above representations the interlocked tori are of equal dimensions and have contiguous surfaces. This condition is the subject of further discussion below. However such interlocking may involve tori of quite unequal dimensions such that the surfaces do not touch or that one tube is of smaller radius than the other (whether or not they touch). Mathematica provides numerous examples. These point to the use of such differences to hold other kinds of significance concerning the relative emphasis placed on one representation or the other. Dynamics of interlocked toriRather than assume that the two tori are simply static representations of sets of categories, there is a strong case for taking account of the dynamics associated with many occurrences of a torus in natural phenomena -- from smoke rings, through plasma containment in fusion reactors, to the environment of black holes. Anything defining the surface of a torus may then have two dynamics: (a) around the tube of the torus *** rotation***, (b) along the tube of the torus. Give that such dynamics may occur in both tori when interlocked, interlocking is most smoothly achieved when the direction of the dynamics in each is such as to be mutually reinforcing. In effect the dynamics around the tube of one reinforces the dynamics along the tube of the other. If the tori are understood to be serrated, as in the aesthetic representation above, then the serrations effectively function like the teeth of gears at the interface between the two tori. With respect to the cognitive content of the two matrices interrelated in this way, it can then be recognized how in a dynamic situation the "preferred modes of knowing" (potentially emergent within any psychosocial system) interact with the "sub-systems" variously perceived as important (to understanding and ensuring the operation of any psychosocial system) -- through the dynamics of the changing contact at the interface of the two tori. When, as noted above, the dimensions of the torus are different, their relationship may reflect cognitive challenges corresponding to those the Kama Sutra delights in articulating through euphemism -- regarding the relationship between partners of different dimension: males (as rabbit, bull or horse) and females (as doe, cow or she-elephant) [more].
Distribution of significance "within" a torus: using a third dimension for "Engagement with reality"In the process of making the distinction above between "preferred modes" and "sub-systems", the possibility of a third dimension was noted. This concerns the degree of experiential ("subjective") engagement with "reality" as framed by the first two dimensions. It might be understood as the "maturity" or "depth" of response to objectively described systems and the degree of cognitive identification with them or "detachment" from them. This is important to any understanding of identity within such a system and for understanding the status of any perceived or conceived system in relation to the perceiver. The suggestion here is for a classification that ranges::
Some approaches to this range include:
The elements of such a range may then be understood as corresponding to positions along the radius of the tube of either torus such that concentric circles around the centre of the tube (centred on its axis) correspond to different levels of insight. The cross-section then corresponds to the radial, centro-symmetric organization characteristic of many traditional mandalas [more] The junction between the two interlocked tori may then be associated with a "deeper" or "more superficial" level of meshing. The surfaces of the tori may then be understood to "interpenetrate" corresponding to the level of insight in operation -- rather than interfacing at the "outer" surface as suggested by the above illustrations. One interestingly relevant representation of interlocking is based on an optical illusion by a random dot stereogram (see Two interlocking toruses). Contiguity of paired circular cross-sectionsThere are three unique points of particular interest at the interface between the two interlocked tori on the line linking the axial centres of their respective rings -- effectively the line representing the junction between the planes of the two tori. In each case, it is the point at which the circular cross-section of one torus tube is contiguous with the circular cross-section of the tube of the other. The circular cross-sections are however orthogonal to one another -- the planes of the torus tube cross-sections are at right angles to each other. The "twisted" surface connection bears some resemblance to a figure-of-eight or Möbius strip -- but with a second such loop, given that there are four such circles. "Empty" centres and four-dimensionalityA torus is characterized by two types of "centre" that may, in respect of any cognitive mapping, be understood as "empty":
From a cognitive perspective such "empty" centres may be related to zones of cognitive "emptiness" that are a typical focus of spiritual disciplines. John Fudjack and Patricia Dinkelaker (The Enneagram as Classic 'Double Mandala', 1999) clarify the manner in which mandalas wrap through their centre to form a torus, reconciling two incommensurable orders of awareness - the undifferentiated and the differentiated:
Also of interest is that in the two interlocked tori there are four such zones of "emptiness" (along the plane of intersection) -- two associated with the respective axial rings and two associated with the respective tubular centres. In four dimensions these may be understood as unseparated and potentially having a common focus. This points to the possibility of understanding life and identity as a dynamic in 4 dimensions. On this point note the work of Walter J Freeman (Strange Attractors that Govern Mammalian Brain Dynamics Shown by Trajectories of Electroencephalographic (EEG) Potential, 1988) indicating that the strange attractor may also have the structure of a 2-torus, that fluctuates in size and shape aperiodically -- described as "a torus that breathes". The dynamics along and around the tube of one torus, particularly as driven by the other, point to a particular condition at the circular cross-section where they are contiguous. This could be understood as constituting a third torus dynamic -- in relation to the "empty" centre at the axis of the tube at that point. This torus may be understood as arising from of curving the radial columns holding categories of "engagement with reality" to leave the axial centre free. The row dimensions then result from of the dimensions along the tube of the torus. There are therefore two such "inner" tori each located within one or other of the two original tori at their point of contact *****. Psychosocial relevance of torus-related dynamicsThe torus has emerged as a focus of attention in the study of the non-linear dynamics of complex systems. A torus attractor occurs when there is coupled motion of an oscillating pair. As a dynamical system increases in complexity, it shifts from fixed point, to limit cycle, to torus attractor. However, with further complexity, the system breaks into chaotic movement and a new attractor, the strange attractor, emerges. In a discussion of "integral awareness and the initial maturity of planetary civilization", Duane Elgin (Awakening Earth: exploring the evolution of human culture and consciousness, 1993) discusses an eighth dimension of societal development characterized by a "dynamically stable, self-referencing and self-organizing species-civilization" associated with a "wisdom-culture" within which polarities are effectively balanced "integrated continuously into a higher synthesis... comparable to a controlled chain reaction in a nuclear reactor...forever pulsing with creative life energy". He notes the unique value of the torus in modelling the associated process understanding:
With respect to the torus, Elgin adds:
Despite the elevated stage of consciousness that Elgin's eighth stage represents, he stresses its direct link to contemporary experience citing the Zen saying "After awakening, we sweep the floor", namely the recognition of the "extraordinary nature of the reality | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
