8.1 Problems of comprehension: It is appropriate to note that work in the well-defined field of "multi-valued logic" does not seem to have had any impact on these concerns37. Nor does that on the "theory of numbers"33. It is only more recently in studies which face up to non-quantitative considerations with propositions for 3 or 4-valued logics that the nature of the link begins to emerge (48, 49). The reason for the lack of progress would appear to be that in both fields named above the problems of comprehension, and the status of the observer, are ignored despite the early efforts of Korzybski on general semantics (54). It is here that the questions of self-reference (see above) and the wider implications of complementarily are now significant in legitimating further investigation  (55).
Comprehension may be considered purely as a problem of "pattern recognition" in non-verbal data. This is now receiving considerable attention in some branches of information science concerned with the man-machine interface. It is a quantitative response to complexity and is of limited relevance here (although the illusion that the conventional quantitative mode is neutral and "value-free" is now being widely attacked (56, 57). A much more subtle problem is associated with the comprehension of qualities, and as such necessarily involves the observer actively to a greater or lesser degree.
The question is how qualitative distinctions can be comprehended and communicated. Clearly the problem does not even become apparent until differences in interpretation create difficulties. Even then it may be disguised by explaining differences as characteristic of different schools of thought, social backgrounds, educational levels, or cultures. The effect of such perceived differences on the medium used to portray the quality in question may even be the focus of appreciation, where the preoccupation is primarily aesthetic (painting, music, poetry, etc.), thus again disguising the problem. Where deliberate efforts are made to use words to define the meanings to be conveyed by other words, obvious discrepancies can be resolved whilst subtler ones remain. The systematic approaches currently explored by COCTA and INTERCONCEPT (58) may further reduce the problem.
Nevertheless, even when the ideal has been achieved of an agreed definition for a qualitative attribute (available in a universally understood language), a core problem still remains. The word-ensemble constituting the definition will be comprehended in different ways according to the capacity and inclination of the reader--even if the definition triggers a single gestalt perception of the quality rather than a serial perception of its multiple facets. It would be naive to expect that the ultimate definition of "beauty", "justice" or any other quality could be formulated in 1979--thus depriving the future of any possibility of comprehending, describing or expressing them more appropriately than is now possible. Similarly both a child and an adult may share a verbal definition of 'peace"--but their comprehension of it is likely to differ, especially if one has experienced the realities of war. Further elaboration of a verbal definition does not eliminate the difficulty and is quickly counterproductive.
The nature of the challenge to comprehension can be illustrated by the simple sequence of numbers: 1, 2, 3, 4, etc.:
(a) where 1 denotes any single entity isolated from its context, no demand on comprehension is made; higher numbers merely provide an arithmetic total. The totality is never more than an aggregate and the relationships between the elements are irrelevant.
(b) where 1 is used to denote a totality within which no element has been isolated, then use of higher numbers indicates successive degrees of subdivision of the original totality . With each higher number the total pattern becomes increasingly difficult to comprehend.
(c) where 1 is used to denote the totality encompassing the universe as experienced, then the comprehension demanded is associated in traditional cultures with a supreme being. Higher numbers then reflect hierarchies of such gods, each governing qualities of an appropriately lower level of abstraction .
(c) where 1 is used to denote the totality encompassing the universe as experienced and including the experiencer, the state or level of consciousness of the observer is necessarily affected. Higher numbers then denote successively more multi-faceted levels of consciousness at increasingly lower levels of abstraction. The focus of this paper is on complete sets which in some way aim to encompass a structured totality. These may raise problems of comprehension of type (b), (c) or (d) depending on the level of abstraction of the set elements and the degree of "insulation" of the observer.
There is currently great faith that when verbal descriptors are used as identifiers for such set elements, they will carry some universally understood meaning (e.g. peace, justice, human rights, development, democracy, etc.). As argued above, and as ongoing investigations are demonstrating (59, 60), this is far from the case. Such fundamental characteristics elude complete or even adequate definition by any particular set of words . Clearly the definition or label merely points towards a comprehensible experience. It is not the comprehension of that experience. ("The map is not the territory.")
8.2 Comprehension, remembering and mnemonic aids: The special problem in comprehending complete sets lies in the relationships between the interdependent elements. These are seldom explored in any verbal definition, thus detracting from its adequacy. But even if the member elements can be comprehended singly or in groups in serial fashion, remembering them is increasingly difficult and their relationships are lost as is any grasp of the totality.
It is at this point that various mnemonic aids are used in describing such sets in order to provide some reinforcement to memory. The crudest and most prevalent is a simple numbering of elements. At the other extreme are sophisticated diagrams showing their relationships. Attention has even been drawn to the advantages of interactive computer graphics as an aid to maintaining creative "thinking momentum" and obtaining a grasp of a total pattern . But as noted in Part 1 (Section 5), it is the mandala-type representations which constantly stress this mnemonic function. It is the manner in which they are designed to be used which recognizes the challenge to comprehension and causes attention to be focused (as with an optical lens) through the member elements disposed in an appropriate configuration.
Significantly it is from the continent of the mandate that have come papers on the relevance of mnemonics to classification (62, 63) .
"The basic idea underlying seminal mnemonics is that concepts of objects or phenomena which are apparently unrelated at the phenomenal level, may be seen to be related to each other at a deeper level of perception. Seminal mnemonics consists of assigning the same notational digits to such "seminally equivalent" concepts, regardless of their verbal denotation or class context. But the perception of seminal equivalence of concepts is a difficult process, and demands a high degree of intuitive ability in the classificationist." (62, p. 16)
The last sentence and other comments in the paper link back to the discussion of the previous section. But the technique is said to have been used only "intuitively and almost unconsciously" by S. R. Ranganathan in developing the Colon Classification.
8.3 The quagmire of number symbolism: the past: As a contrast, those preoccupied with number symbolism over the centuries have been quite deliberate in their attempts at seeking to associate numbers and concepts . Such investigations have from time to time been very fashionable, whether from Pythagoras onwards in the West , or in the East, as Ranganathan writes in his Prolegomena:
"In the mystic tradition of Chaldea and India, many such equivalences are believed to have been recognized. It gives seminal mnemonic significance to letters as well as numerals. A correct knowledge of it will make the use of digits conform with seminal mnemonics. The forgotten tradition needs to be recaptured. As the deep region of seminal equivalences transcends expression in words alone, communication through the written or printed word is difficult. Seminal equivalences are ineffable, but they get permeated by personal association and communication in a school." (67)
But although there have been many investigations and the literature is vast, the result is a veritable quagmire into which many have ventured and from which few have returned unsullied. This is not to deny that many of the eminent intellects who have been attracted to this question have not come up with valuable insights, but rather that it is now difficult to filter the significant insights from a rich smorgasbord of culture-bound speculations and outright nonsense which have accumulated over many centuries.
The investigations of von Franz and Bennett of the qualitative attributes of the simple systems are therefore to be welcomed because they successfully disassociate themselves from number mysticism in its more unfortunate traditional forms. Indeed, working independently, they provide the necessary complementary perspectives of psychologist and physicist which is von Franz's objective (see above). She herself explores material concerning the first four integers. Bennett ambitiously explores up to 12-term systems  - thus inviting misunderstanding, however due to the ever present problems of comprehension to which he himself draws attention.
The nature of the danger is illustrated by his results which are summarized in Annex 2. Although it is the most systematic and disciplined attempt (in the West, at least), its main weakness lies in the verbal descriptors. These are really only useful as tentative signposts lacking any indication as to how the referent is to be experienced. The problem, as with all number symbolism in the past, is that it is only too easy for a reader to assume that his own comprehension of the descriptor as defined is as complete as that which is intended (leaving aside questions of Bennett's own limitations and difficulties of comprehension). Although more limited in scope, the same reservations must apply to the verbal descriptors of seminal mnemonics .
This is the basic difficulty with verbal descriptors and their definitions however much effort (à la Academie Française) is made to govern their usage and significance. It is worth considering the possibility that those such as Pythagoras were aware of this problem, as well as of the others indicated above: level of comprehension, superseding dualistic logic, the need for mnemonics, and self-reference (including the distinguisher's relationship to the denotative mark). What better way could they use to embody the subtlety of their insights than through numbers and their interplay, specially since the above questions are all number-related? The use of numbers does at least continually confront each user with his responsibility for any verbal descriptors he chooses to associate (temporarily and according to circumstance) with the concept. More important, it continually challenges him to greater levels of understanding of that concept and the manner in which it relates to others. It also leaves the future free to reinterpret the concepts in other ways or to greater depths--a process which is full of pitfalls and discontinuities when culture-bound verbal descriptors are used as at present.
However numbers are sufficient only to the very few. They do not provide a concrete image for those incapable of sustained thought at that level of abstraction. They can however be readily associated with archetypal figures (divinities, etc.) which each constitute a highly complex composite of qualities comprehensible as a whole at many levels of understanding and according to ability (9). Such figures are characteristic of many cultures for which their nature is powerfully clarified by many vivid tales and myths concerning their relationships. Their value lies in their ability to clarify or intensify ideas or emotions through appeal to sense experience. By such symbols, "the abstract may be brought into the realm of the concrete, where it is immediately recognizable and meaningful" (66, p. vii). And indeed studies of the symbolic nature of medieval thought and expression "reveal in the medieval mind a web-like structure of abstract ideas and concrete realities so closely interwoven and interdependent that no serious gap was felt to exist between them." (66, p. vii)
8.4 The quagmire of number symbolism: the past resurgent: This is of course essentially a sympathetic assessment and it cannot be denied that much that was produced within this context, if not most, now appears at best as fascinating nonsense--but this is a predictable consequence of using the culture-bound verbal descriptors of interpreters feeding endlessly on one another's products. But it would be a great mistake to believe that modern society has completely resolved the issues to which number symbolism responded.
On the one hand very many scholarly or administrative papers now enumerate lists of fundamental issues, principles, values, problems, etc. (to be compared with the medieval predilection for N virtues, sins, principles, etc.) as discussed in Part 1. It is then the task of the classifier to prescribe some meaningful order. But for various reasons, society now faces a crisis of meaning which the plethora of studies is instrumental in aggravating rather than alleviating. And, despite the efforts of classifiers (who themselves have various preferences for number-governed ordering systems), such studies tend to achieve quicker oblivion than their medieval counterparts, and are just as meaningless to the uninitiated. Meaningful synthesis is rare and of limited relevance to societal problems. Comprehension and integration tend to result in spite of current enumerations and classifications and not because of them.
On the other hand, in an effort to render meaningful the nature of the complex issues which face society and the importance of the values by which changes should be guided, government agencies, social-change movements and educational authorities are now obliged to resort to symbols which can be satisfactorily communicated through the available media .
This requires that such symbols be easily comprehensible, coherent, and that they bear only a simple meaning (irrespective of the complexity of the issue). Because of the low status of "public relations", such symbols are only indirectly linked to the weakly ordered substantive items enumerated in agency programmes or in the scholarly studies on which they may be based. Where complex abstract notions must be communicated (e.g. concerning ecological systems), cartoon "personalities" are often used, appropriately adapted to each culture. Were it necessary to embody the characteristics of justice, beauty, love, etc. which were a preoccupation of the past, it is not unlikely that public relations would need to resort to symbols of updated versions of the superhuman beings on which that period so successfully projected its beliefs. (As it is, cartoon characters and film stars define the limits of our subtlety.) Were it considered necessary to show the relationships between the concerns of the different United Nations agencies (e.g. education, justice, agriculture, health, etc.), it is not unlikely that public relations would have to resort to an interplay of such personalized symbols in a modern series of "myths". The quagmire of the past has not been avoided. It is in process of being re-evoked, because the problems from which it arises have not been recognized.
8.5 Encyclopedic memory systems: Strangely enough it is only through the recent remarkably insightful work of historian Frances Yates (68) that the role of memory in relation to knowledge classification has been subject to a preliminary investigation. She demonstrates that: "The history of the organization of memory touches at vital points on the history of religion and ethics, of philosophy and psychology, of art and literature, of scientific method" (p. 374). She makes it clear that not only did the Greeks (and possibly the Egyptians) possess an art of memory, but that this art was widely practised, extensively developed up to the Renaissance, and instrumental in the growth of the scientific method. Briefly, the art involved the memorization of an ordered set of "places" (loci or topoi ) which effectively constituted a permanent system of filing locations--whether based on a building, a town, or a set of divinities . Onto these the user "impressed" images (imagines agentes, "corporeal similitudes") which would trigger access to the things or ideas to be remembered--a technique reminiscent of that described by memory and calculating prodigies in recent years. It is somewhat disconcerting that this lost art permitted its exponents to use over 100,000 filing locations (p. 120) and that these could be explored in any sequence. The importance of the art for orators, scholars, administrators, merchants, etc. is clear at a time when text reproduction was difficult and paper expensive.
What is even more disturbing is that with the Ramist educational reform in the 16th century, and increasing reliance on the printed medium, it is clear how the seeds were sown for the problems and dichotomy identified in the previous section. This reform explicitly rejected (for religious reasons) the use of memory triggering images, which seem to have been essential to the art, in favour of the present approach and its associated forms of classification. Memory is now considered as a "mechanical" facility only to be tested during examinations and otherwise to be enhanced by data banks processing information in serial order.
Whilst it would be foolish to deny the need for the reform, it seems clear that this cuts off some lines of investigation which could have proved fruitful (irrespective of the "nonsense" from which it is difficult to disentangle them). Despite the subsequent interest in memory of Bacon, Descartes and Leibniz, it does appear that insights were lost (or driven underground) with the rejection of the highly complex memory systems developed by Raymond Lull, Giordano Bruno and Robert Fludd. Although Yates acknowledges that, with the available information, their full scope eludes her, she makes it clear that they at least had more or less explicit concerns for:
- maintaining the individual's encyclopaedic relationship to the separate categories of knowledge
- discovering better ways of fitting or packing such knowledge into memory, of reflecting the world in memory
- improving sensitivity to the possible systems or patterns of relationships between such categories of knowledge.
- developing appreciation of the qualitative aspects of knowledge
- discovering methods for the organic unification of knowledge in memory and understanding
- intensifying and focusing the individual's exposure to knowledge as a means of provoking a beneficial change in level of awareness.
8.6 Augmented comprehension: Having assembled much evidence, Yates bequeaths to others the problem of whether the Renaissance did in fact possess a secret memory technique for stimulating the human psyche to a wider range of creative achievement than ever before (p. 354). That the described techniques claim to provoke memory to retain the interrelationship between many elements in a whole pattern is clear. That this involved a preoccupation with complete and ordered sets in also clear, as is their relationship to number (in the light of von Franz's review of the same authors). Proportion, harmony and connection in the representation of such sets are considered vital to success in empowering this new comprehensive grasp which is consequently intimately related to artistic expression: poetry, painting, music, architecture, theatre . Current research on computer augmentation of intellect lacks this artistic dimension although it facilitates manipulation of categories (69).
The concern with personally meaningful vivid image ry is echoed in recent studies of symbols as signs charged with meaning (70), (71), (72). To exert their psycho-dynamic organizing effect such symbols presuppose homogeneity of signifiant and signifé (70, p. 20 - 21) Whether and how, such "charging" can be accomplishes is presumably the key to the question. Mircea Eliade has studied a primitive approach to this (73).
Contemporary interest is reflected in research on altered states of consciousness (74), and psychotropic research (75) although the process by which advertising and propaganda impart significance to isolated commercial or political signs is also of great importance. Other factors, such as iconicity, merit attention . But for the special significance of the configuration approach characteristic of the Renaissance representation of sets of categories-cum-symbols, insight can perhaps best be gained from contemporary use of the mandate as mentioned above (38). This preoccupation has been absent from western thinking until the work of Jung--it was with the rotoe of Lull, Bruno and company that development ceased. The technique must therefore have aimed to dissolve the dichotomy identified in the previous section, and to move beyond the neatly disciplined relationships of the concept triangle to a condition in which the "uninsulated" observer was impelled to move, change or create by exposure to symbols. That information no longer moves people to act, is a major preoccupation of those attempting to mobilize resources against world problems (76). The perspectives that Yates opens up suggests that the "quagmire" discussed above may conceal some valuable insights.
8.7 Convergence of concept triangle elements: a limiting condition: Given the preceding remarks, the question is whether anything useful can be obtained from the vast amount of material available on number symbolism in its different forms (see (9)). The answer would seem to be positive in the light of von Franz's approach. But there is an immediate problem of how to handle the subtle differences between authors and the way attribute sets are shuffled into new configurations. As noted by Varela above, the distinctions selected are as much a description of the author as of the subject matter. This question has been studied by W. T. Jones (17). It would seem that authors can get caught in a subtle trap which does not deny the significance of their insights but rather limits the sectors of society (or period) within which their interpretations can be fully communicated and for which they will be valid and socially significant.
Aside from refining methods for sifting and testing complete sets, their relationship to one another must be clarified. This may be alluded to in terms of their relationship to sets of progressively greater "generality", here-and-now "concreteness" or "inclusiveness" [54, 55].
There is a qualitative convergence but its nature necessarily escapes verbal delimitation . It is a challenge to the comprehension of the observer and ultimately to the knower-known dichotomy. And, furthermore, whenever "fundamental" sets must be produced, they can only constitute aspects of a more fundamentally integrated understanding which must necessarily emerge progressively if future society is not to be deprived of all possibility of creative insight in this domain--to say nothing of any more mature insights on the part of the author. Clearly only the future can progressively identify and give content to more fundamental sets. Closure cannot be assumed --or, if comprehended, then not communicated. Closure in this domain cannot even be premature; it is impossible with maturing individuals in an evolving society (except for strictly limited purposes which carry the seeds of their own mortality). Any attempt at closure therefore merely sets the stage for production of "improved" versions, with all the resultant non-rational dynamics between the advocates of each and the hiatus as one version replaces another.
It is the argument of this paper that in such complete sets of a given number of elements, the latter are characterized by a qualitative pattern independent of the nature of the set elements. And as the set becomes more fundamental this qualitative characteristic predominates. For, as such sets become more fundamental or general, the characteristics of the elements (labelled by words) are increasingly affected (in their significance to the observer) by the semantic field associated with the numbers (whether explicit or implicit) characteristic of the representation of a given set. The label words therefore introduce increasing confusion, since the degree of precision they are expected to carry is severely eroded in comprehension by a cloud of polysemantic associations that are progressively more irrelevant to the elements distinguished. And, the more fundamental the set, the more probable it is that the numbers characteristic of the representation would more effectively label the set elements which in any case increasingly approximate the semantic fields of the numbers embodied in the representation . There is in fact a convergence or melding of the elements of the "concept triangle" (word, meaning, referent) with the observer, who is necessarily incorporated into the referent by the set, if it is fundamental. The 4 terms form a "concept tetrahedron" . Recent work has shown the link between the status of the observer and information systems viewed in the light of relativity theory (81).
This recalls Spencer Brown's point cited above that "We see now that the first distinction, the mark, and the observer are not only interchangeable, but, in the form, identical." (18, p. 76) This is clearly however a limiting condition and for less fundamental sets identity is necessarily not assumed, particularly since it is not experienced .
Even though the limiting condition may be ignored, the fundamental sets of interest here are sufficiently close to it, that any use of words must be viewed with great caution . What the words attempt to label is better coded by numbers with their qualitative associations. Differences in formulation of fundamental sets arise because each assumes it is containing fundamental elements but is effectively only containing those evident from an aspect of an even more fundamental domain (e.g. associated with a particular numeric quality). And each tends to focus on different aspects without being able to incorporate others even if the formulator is aware of them. At this level, however, there is a high degree of isomorphism between the numeric qualities characteristic of the sets centered on different aspects. This may be used to clarify the content of sets, and to identify more fundamental sets, without relying too heavily on the words used to carry meaning in any particular case.