Poseuses (1886-88) by Georges Seurat
THE LAW OF THREE IN COMPLEX PROCESS - A FRACTAL ESSAY
Anthony Blake, March 20, 1999
The main inspirations for this essay, besides those quoted in the text, are Steve Mitchell's novel 'Albion' and Spencer Brown's 'Laws of Form'. The essay is also offered as a commentary on William Pensinger's concept of 'identity transparency'.
The 'law of three' turns up in may fields. As a re-iterative form, it is common in complexity theory. In this guise, it is an operator of 3 steps or parts. Complexity results from such simple 'triads' applied as repeating operators. Such operators operate on themselves, generating complex patterns. Two examples may suffice: Mandlebrot's multifractals [see Scientific American, February 1999, 'A Multifractal Walk down Wall Street] and Kurzweil's evolution machine [see 'The Age of Spiritual Machines', Viking, New York, 1999]
Mandlebrot's multifractals apply to fluctuations of the price of stocks on the exchange market. The basic unit of such fluctuations is taken to be of the form:
A rise in price
A fall in price
A rise in price
A picture of this unity as a graph of price against time would show a zigzag line, first rising then falling then rising again. A little reflection reveals the Hegelian model lying behind this operator (Mandlebrot calls it a 'generator'). The Hegelian model is usually rendered as: thesis-antithesis-synthesis (this last being then a new thesis, so that the operation continues). Like Mandlebrot's operator there is, implicitly or otherwise, an assumption of overall progress (or price rise).
The operator can be altered - making a series of fractals - by changing the first part of the line, making it a 'faster' or 'slower' rise. The application of the model includes (a) making it more detailed and (b) using it predictively.
(a) is accomplished by self-similarity, the shape of the fractal (the 3 parts) applied to each part, scaling down and, in the case of the falling or second part, in reverse. This is done in successive stages to produce whatever degree of detail is required - in this case right down to the frequency of individual transactions.
(b) introduces a new aspect. The 3 parts of the operator are considered as in loose association and hence capable of 6 combinations in sequence. Here we have a parallel with the 'six laws' discussed in detail by John Bennett [see 'The Dramatic Universe' Vol. II]. The six permutations ate akin to faces of a die cast at each step. This random element introduces more complexity. It excludes any notion of 'guiding intelligence'.
Kurtzweil's evolution device involves what first seems to be an extremely vague concept: 'choose the next best step'. The context is important, such as a game board or an environment. Before one chooses the next best step, one has seen the results of the previous 'best' step. The sequence is therefore a triad of: best step-results-next best step and implies, as before, a progression. However, if the result is 'success' the process can stop there - e.g. if there is check-mate. If the result is 'failure' the process can also stop, such as in the case of the death of an organism or the extinction of a species. Otherwise, the process can continue, given the resources to continue. With only the most rudimentary conception of 'best' and enough time/energy for very many attempts, evolution becomes inevitable.
Such a procedure is similar to that applied in our 'trialogue' [see N-logue]. There is a succession of acts of speech - as question, answer and comment - the comment feeding into the next cycle. Here, there is not so much 'success' or 'failure' but a concern with 'making meaning' (or an addition of, or extension of, meaning). We can diagram the cyclic operator of trialogue in much the same way as a Mandlebrot fractal. Thus, we have these correspondences:
price rise - question - opening up
price fall - answer - closing down
price rise - comment - reflecting further
We have set price rise as equivalent to 'moving towards the unknown' and vice versa.
In actual conversation, where the rules of trialogue do not prevail, it is proposed that there is some mechanism such as permutation to give rise to fluctuations rather than to a progressive trend. However, conversations might be typified and predicted. This has not yet been done because of the problems of providing quantitative and objectively observable measures to the 3 parts.
The 3 steps or parts of the triadic operator are also found in symbolism such as the enneagram. Bennett's theory of the enneagram suggests a process subject to modifying corrections in 3 parts as: process-correction-synthesis. The notion of a correction is analogous to that of 'elimination of error' implied in an evolutionary model such as Darwin's 'natural selection'.
[In Bennett's traditionalist interpretation, evolution takes the form 2-1-3 where the second term (1) applies a selective intelligence to the given process (2). Such selection means that there is a waste product, or death. For example, thinking requires a step of elimination to reach insight; or, a group requires some exclusions in order to come together or 'gel']
This then suggests a remarkable conception, that biological process such as reproduction is subject to a corrective elimination by natural selection from the environment while proceeding on to a further phase in which the system gains intelligence for itself through a synthesis of process and correction. In other words, there is a third intrinsic phase of intelligence based on making the process of correction faster or more efficient. Strategies evolve as methods of gaining advantage. Sheldrake's postulate of a morphogenic field addresses the same issue but takes us away from the deeper conception of a triadic operator of intelligent process.
There seems to be a field of research here that covers many subjects and interests. An intriguing prospect might be to compute a novel in terms of frequencies of peak moments - such as sexual encounters or killings or revelations - out of a single triadic operator. Similar explorations might be feasible in music where something of this nature is already recognised. Undoubtedly, such things can also be explored in terms of the life incidents, previously the provenance of astrology and biorhythms.
The classical triad associated with Gurdjieff's ideas has suffered from being treated as a static form rather than as a form of action. With Bennett, I came to suggest the term 'dynamism' as most appropriate for the triad. But we had not then known of the significance of re-iteration. The triad as operator can enter into large numbers of computations of itself, producing a complex 'shape of intent' or 'path of intelligence' - or even something so complex that it appears utterly random. Common ideas of destiny and fate may reflect a primary intuition of a triad or 'logos' endlessly re-calculating itself behind the chaotic appearances of our life's journey.
The aspect of progressiveness, however simplistic our model might seem, draws our attention to questions of success and failure in life, and to the obscure issues of 'spiritual advance'. The representation made in only two dimensions is powerful just because of its limitations. We have only one consideration, then: of what is 'better', 'more', etc, This may explain the power of such models as those of Edward Matchett, who often applied the single concept of 'making more meaning'. Such a concept associates with his 3-M equation [media + matter = meaning] these three being, of course, the 3 parts of the basic operator. In this light, 'media', 'matter' and 'meaning' need not be distinguished as different kinds of thing at all.
Similar considerations apply to theology and the study of consciousness. In theology we need only think of something like Rudolf Otto's 'numinous' in relation to the trinity of Father, Son and Holy Ghost, another expression of the basic operator.
In terms of the study of consciousness, we might consider an operator of the understanding as: mind-matter-meaning. The materialist step is essential, otherwise we spin off into unbridled fantasy (as the stock market would flash into melt-down). The sequence here is like a fragment of dialogue, or one cycle of trialogue.
Q. what is consciousness?
A. a function of these brain cells
C. our consciousness is now changed
The one consideration that might apply is 'intentionality'.
The cycle of trialogue is akin to Hegel's 'reason'. Cast in the form of question and answer, it presupposes that the answer does not fully 'kill' the question, but leaves an unanswered residue which is akin to discovery. There is an element of play. So, we might represent, with playful seriousness, the 'study of consciousness' as the triad mind-brain-play and, starting with play, could find ourselves in such fields as language, art and sex.
There is a certain pragmatic core to the triadic operator. In the first example, we can think of arriving at the 'realistic price' of the stock. In the case of the evolution machine, at the solution of the problem, or the new species. Even in the case of the theological trinity, we are in effect seeking the best religion. In all the examples, there is a question of real progress, that is of a domain of fact subject to value, in which measurement itself becomes the fruit of the action.