6.1. Whenever it is convenient, there is a widespread tendency to avoid consideration of the impact of those involved on research or on the policy-making process in which they participate. Researchers correct for bias in experiments and aim for reproducible results. Efforts are made to balance the interests represented at policy meetings. Consequently, when sets of basic values, problems, concepts, or principles are generated by either, they are conceived to be objective. The relationship between any such objectively determined category sets and the thinking processes of those involved (or on whom those categories are subsequently "inflicted") is not open to rational discussion in the same arenas and may well be perceived both as impolite and threatening. And yet it is recognized that:
"The categories in terms of which we group the events of the world around us are constructions or inventions. The class of prime numbers, animal species, the huge range of colours dumped into the category "blue", squares and circles: all of these are inventions and not "discoveries". They do not "exist" in the environment. The objects of the environment provide the cues or features on which our groupings may be based, but they provide cues that could serve for many groupings other than the ones we make. We select and utilize certain cues rather than others." (Jerome S. Bruner et al., (33), p. 232.)
And again:
"Nowadays we concede that the purpose of science is to invent workable descriptions of the universe. Workable by whom? By us. We invent logical systems such as logic and mathematics whose terms are used to denote discriminable aspects of nature and with these systems we formulate descriptions of the world as we see it and according to our convenience. We work in this fashion because there is no other way for us to work." (S S Stevens, (34), p. 93.)
In justifying their own work, Bruner et al. argue:
"Two consequences immediately become apparent... The characteristic forms of coding, if you will, now become a dependent variable worthy of study in their own right. It now becomes a matter of interest to inquire what affects the formation of equivalent classes or systems of equivalence coding. The second consequence is that one is now more tempted to ask about systematic individual and cultural difference in categorizing behavior." (33, p. 8).
This point was however made in 1956. Both in the research on which they report and in subsequent research, it would appear that the focus has been on categorization in the case of "laboratory problem" sets which are essentially trivial in comparison with the sets of fundamental concepts which are elaborated consciously in the course of research (or policy-formulation). The former are laboratory exercises requiring minutes or hours, the latter involve much reflection and a protracted "struggle" for the best "fit", possibly over a period of many months or years. In particular, to give the kind of "uncomfortable" example that is required, the research has not been applied to the sets and categories selected by those undertaking research in this very area, as an aid to explaining the differences of opinion which give rise to non-rational behavioural dynamics between the various schools of thought affected. Only "pointed", self-reflexive research of this kind, on the formulators of sets which are fundamental to social policy, can help to clarify the basis for the opposition between policies which tends to fragment society into hostile camps.
6.2 Laws of form: It is not sufficient simply to complain about the widespread tendency to avoid consideration of the impact of those involved in set formation on the sets which they formulate. The reason for such avoidance merits continuing study [26].
Part of the problem seems to lie in a missing link in the relation of mathematics to logic which has been provided, with the encouragement of Bertrand Russell, by G. Spencer Brown (18). Much of science (and that includes classification) makes explicit or implicit use of set theory based on Boolean algebra which was designed to fit logic--but in doing so detaches the observer from any involvement in the logical processes [27]. Spencer Brown argues that: "nobody hitherto appears to have made any sustained attempt to elucidate and to study the primary, non-numerical arithmetic of the algebra in everyday use which now bears Boole's name" (18), p. xi). And again: "That mathematics, in common with other art forms, can lead us beyond ordinary existence, and can show us something of the structure in which all creation hangs together, is no new idea. But mathematical texts generally begin the story somewhere in the middle, leaving the reader to pick up the thread as best he can. Here the story is traced from the beginning." (18, p.v) And, according to Francisco Varela: "By succeeding in going deeper than truth, to indication and the laws of its form, he has provided an account of the common ground in which both logic and the structure of any universe are cradled . . ." (42, p. 6).
The result of Spencer Brown's formal exercise to separate what are known as algebras of logic from the subject of logic, and to re-align them with mathematics is the explicit, and extremely elegant logical re-integration of the observer. His final chapter, entitled "reentry into the form" commences with: "The conception of the form lies in the desire to distinguish. Granted this desire, we cannot escape the form, although we can see it any way we please" (p. 69). It ends with:
"An observer, since he distinguishes the space he occupies, is also a mark . . . In this conception a distinction drawn in any space is a mark distinguishing the space. Equally and conversely, any mark in a space draws a distinction. We see now that the first distinction, the mark, and the observer are not only interchangeable, but, in the form, identical." (p. 76)
Spencer Brown shares the concern of Buckminster Fuller and Keith Critchlow (22, 36) with the initial conceptualisation of a whole and its subsequent subdivision. He explores this using a powerful logical notation (18), whereas Fuller and Critchlow explore the structural implications in 3-dimensions. The latter would appear to be fundamental to representation and hence to comprehension. Jay Kelley, in considering the connection between man and his knowledge and the requirements for an adequate information system, arrives at similar conclusions [28].
6.3 Logical "curvature": Spencer Brown may have effectively established a means of encompassing the "curvature" of the logical universe of our science-dominated culture. In Part I it was noted that our culture was weak in its ability to handle anything "above" the top of the hierarchies of categories we care to distinguish. His work seems to offer a remedy. For it would appear that there is a "curvature" in the more fundamental hierarchies back to the (otherwise detached) person's involvement:
(a) as an observer in the elaboration and subdivision of such ordered sets (whether conscious or tacit), and
(b) as a participant in the reality which such sets encode.
It is the observer/participant who links, through his own person, the top and the bottom of a hierarchy. Equally it is the observer/participant who links distinct hierarchies and is therefore challenged or fragmented by any conflict between competing coding systems to which his perception is subject.
Spencer Brown makes the point that "we cannot escape the form, although we can see it in any way we please" (p. 69). However all forms are not equally probable, as was argued above in the discussion of the numerical constraints on the subdivision of sets. His own work [29] explored the ordered emergence of certain forms. Rene Thom's (32) widely-acclaimed study is concerned with the stability of certain forms (in every domain of knowledge), of which the "islands of stability" encountered in the pattern of isotopes are a well-known example. His analysis extends to forms encountered in social systems and human thought [30, 31].
He argues that:
"It may seem difficult to accept the idea that a sequence of stable transformations of our space-time could be directed or programmed by an organizing canter consisting of an algebraic structure outside space-time itself. The important point here, as always, is to regard it as a language designed to aid the intuition of the global coordination of all the partial systems controlling these transformations." (32, p. 119)
This "algebraic structure" (which he expresses in geometric terms) would seem to play a role in the human psyche which is functionally equivalent to the Jungian "archetype" [32]. Although, even if this possible equivalence is invalid, this does not affect the argument below concerning such archetypes.
6.4 Self-reference and time: It is Francisco Varela (42) who has further developed the calculus of indications provided by Spencer Brown in order to deal with the many self-referential situations characteristic of our society.
"Stubbornly, these occurrences appear as outstanding in our experience. Particularly obvious is the case of living systems, where the self-producing nature of their entire dynamic is easy to observe, and it is this very fact that can be taken as the characterisation for the organization of living systems. Similarly the physiological and cognitive organization of a self-conscious system may be understood as arising from a circular and recursive neuronal network, containing its own description as a source of further descriptions" (p. 5).
In citing papers which address themselves directly to the self-referential nature of such systems, he notes that the topic is "normally avoided as undesirable difficulty (or circulus vitiosus)," and that such difficulties are rooted in language.
Consistent with the remarks of Rene Thom (above), and the preoccupations of von Franz (below), Varela argues that the duality of the producer and the produced (which embodies the producer, as in any category encompassing its user)
"can be pictured only when we represent for ourselves a sequence of processes of a circular nature in time. Apparently our cognition cannot hold both ends of a closing circle simultaneously; it must travel through the circle ceaselessly. Therefore we find a peculiar equivalence of self-reference and time, insofar as self-reference cannot be conceived outside time, and time comes in whenever self-reference is allowed." (42, p. 20)
In his own extended calculus based on a 3-valued system, "self-reference, time, and re-entry (into form) are seen as aspects of the same third value arising autonomously in the form of distinction" (42, p. 21). Use of a third value enables the system to explore self-referential situations which are the basis for the limitations examined by Gödel (43). In his conclusion Varela describes his achievement as follows:
"The starting point of this calculus, following the key line of the calculus of indications. is the act of indication. In this primordial act we separate forms which appear to us as the world itself. From this starting point, we thus assert the primacy of the role of the observer who draws distinctions wherever he pleases. Thus the distinctions made which engender our world reveal precisely that: the distinctions we make and these distinctions pertain more to a revelation of where the observer stands than to an intrinsic constitution of the world which appears, by this very mechanism of separation between observer and observed, always elusive. In finding the world as we do, we forget all we did to find it as such, and when we are reminded of it in retracing our steps back to indication, we find little more than a mirror-to mirror image of ourselves and the world. In contrast with what is commonly assumed, a description, when carefully inspected, reveals the properties of the observer. We, observers, distinguish ourselves precisely by distinguishing what we apparently are not, the world.
We then see that we stand in relation to the world by mutual negation, and that the union of us two has therefore an autonomous structure whereby the negation engenders a distinction which leads to its own negation in a ceaseless circular process which is, in fact, the symbol which tradition has chosen to represent the creation of everything since time immemorial.
Autonomy is seen in this light to engender the two stages of the form when this ceaseless process is broken into its constituents. By the introduction of a third autonomous state in the form, we do nothing but restore to our field of view that which was there at the beginning, and which we can only see now reflected as segments of the world or in language itself Conversely, by taking self-reference and time as our filum ariadnis through a succession of levels, we dwell upon the re-union of the constituents of these levels up to our own union with the world, and thus we find a way to retrieve the unity originally lost." (42, p. 22-3)
6.5 Number and time: Marie-Louise von Franz (of the C J Jung Institute, Zurich) has conducted an extensively documented study into the significance of number for mathematicians, in philosophy, and as symbols of psychological significance, in a deliberate effort to bridge the gap between psychology and physics. As she puts it, her remarks "balance to some extent on the razor's edge between philosophical-mathematical and numerical-symbolical statements" (ref. (9), p. 33 - 34). She deliberately bridges the gap between Western and other concepts of number, which is an aspect of a current debate into the wider interpretations of the concepts of science, space, and time, which have hitherto been supposed to conform conveniently to the Western versions (40) [33].
She notes that Niels Bohr has stressed that an important step had been taken toward realizing the ideal "of tracing the description of natural phenomena back to combinations of pure numbers, which far transcends the boldest dreams of the Pythagoreans" (9, p. I6). She argues that if we accept Wolfgang Pauli's contention that "certain mathematical structures rest on an archetypal basis, then their isomorphism with certain outer-world phenomena is not so surprising" (9, p. 19).
She sums up her argument as follows:
"To sum up: numbers appear to represent both an attribute of matter and the unconscious foundation of our mental processes. For this reason, number forms, according to Jung, that particular element that unites the realm of matter and psyche. It is "real" in a double sense, as an archetypal image and as a qualitative manifestation in the realm of outer-world experience. Number thereby throws a bridge across the gap between the physically knowable and the imaginary. In this manner it operates as a still largely unexplored mid-point between myth (the psychic) and reality (the physical), at the same time both quantitative and qualitative, representational and irrepresentational.
Consequently, it is not only the parallelism of concepts (to which Bohr and Pauli have both drawn attention) which nowadays draws physics and psychology together, but more significantly the psychic dynamics of the concept of number as an archetypal actuality appearing in its "transgressive" aspect in the realm of matter. It preconsciously orders both psychic thought processes and the manifestations of material reality. As the active ordering factor, it represents the essence of what we generally term 'mind'." (9, p. 52 --53)
She concludes that:
"Most probably the archetypes of natural integers form the simplest structural patterns in . . . (the common unknown confronting both physicist and psychologist) ... that manifest themselves on the threshold of perception." (9, p. 56)
In order to explore further, it is therefore necessary to return
"to the individual numbers themselves, and gather together the sum total of thought, both technical and mythological assertions, which they have called forth from humanity. Numbers, furthermore as archetypal structural constants of the collective unconscious, possess a dynamic, active aspect which is especially important to keep in mind. It is not what we can do with numbers but what they do to our consciousness that is essential." (9, p. 33)
Von Franz outlines the recommended programme as follows:
"When we take into account the individual characteristics of natural numbers, we can actually demonstrate that they produce the same ordering effects in the physical and psychic realms; they therefore appear to constitute the most basic constants of nature expressing unitary psycho-physical reality. Because of this I would conjecture that the task of future mathematicians will be to collect their characteristics and analyze. when possible, every number in its logical relationship to all others. This research should be undertaken in collaboration with physicists, musicians, and psychologists who are conversant with the empirical facts about the structural characteristics of numbers in different mediums." (9. p. 303)