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Thoughts on the Edge of Chaos

M.M.Taylor and R.A. Pigeau


1. Overview

2. Basic Ideas: Information and structure, Attractors and Repellors

3. Basic Ideas: Catastrophe

4. Structure and Chaos

5. Six Kinds of Replication

6. Categories and Logic

7. Surprise and importing structure

8. Replication



Replication is the central concept of our view of cognition. It is through replication that tenuous concepts attain stability, that concepts evolve to match environmental influences, that learning happens and that concepts fuse to create novel insights. It is also through replication that the chaotic regime necessary for attentive perception can be developed.

It is not possible for an isolated system to replicate. Hence replication is a property that occurs only in part of an isolated system; moreover it is in a part through which there is an energy flow, so that the sub-system (or module) can acquire negative entropy from the input and pass positive entropy to the output. A replicating module maintains itself at a particular energy level, averaged over an entire replication cycle.

The replication of a module does not mean that after replication there exist two such modules; the act of replication may involve the destruction of the original module. What it means is that the existence of the module in its particular environment will result in the later emergence of a more or less faithful copy of the original module. The original module may still exist, in which case the environment for both is different from that which permitted the replication. The new environment may or may not permit the further replication of the two modules. Replication is a function not only of the module, but of the environment in which the module is found. The congeniality of the environment is critical.

Replication does not always mean duplication, though if the process is to be stable, the replicated version should be very like the original. The term "replication" is used in the sense that x n = f ( xn-1) is a replicating function. f need not be the identity function, but if it is to be useful in a somewhat stable system, successive applications of f should not move x n too far from x n-p after an initial transient. In other words, the attractor of a useful replicating function should not be at infinity.

If the attractor of f for a particular x0 is a point, then replication eventually becomes duplication. If the attractor is a cycle, xn is identical to xn-p for some value of p. This condition is like the generational cycle of living organisms. The baby is unlike the child it becomes, which is unlike the adolescent, which is unlike the adult, which is unlike the baby born of the adult, although the new baby is quite like the original one that became the adult. (Of course, the new baby's genetic structure is different, but that is beside the point for this example.) The replications concerned with the functions of life generally approximate cyclic attractors rather than points. We shall argue, however, that the attractors may well be strange, looking sometimes as if they were cyclic, and at other times not. Such nearly periodic behaviour is typical of strange attractors. Long periods of apparently cyclic behaviour are separated by short periods of rapidly changing behaviour. The non-quasi-cyclic intervals perhaps correspond to the moments of species shift in punctate evolution, to moments of insight in cognition, or to paradigm shifts in the social structure of science.

Replication implies that so long as the environment is conducive, the same (or similar) patterns will recur. The provision of an environment favourable to replication is essential to the stability of forms, since it is through replication that a damaged form may be repaired: somewhere there exists a template derived from the original form or from some part of the cycle that produced the damaged form, and from this template the damaged parts can be replaced by correct parts. In DNA replication, the template for each half of the double helix is the other half, and because the halves are complements of each other rather than duplicates, they are unlikely to be damaged similarly by external events. It is probably a fairly general statement that an effective replicating environment does not involve the use of copies of the thing being replicated; rather it may use complementary copies, algorithms that produce elements of the thing, micro-structures that favour the recreation of the original in its absence, and so forth. Such environmental structures are unlikely to be subject to exactly the same damaging influences as the thing being replicated, and thus form a more stable memorial milieu than simply the provision of multiple copies of everything.

Illustrative examples of replicative stabilitization

To illustrate the possible action of replication, consider the Whorfian hypothesis of linguistic relativity: that the language one speaks will affect the way one views the world. Operationally, this statement is probably indistiguishable from one that says that the way a culture views the world will affect the language it speaks. Each may be true, and each may reinforce the other, or one may be true without the other, or each may be false. For some reason, tests of the Whorfian hypothesis have concentrated on the perception of colour (summarized by Berlin & Kay, 1969; Kay & McDaniel, 1978, Kay & Kempton, 1984). Colour is seen as a suitable field in which to test the hypothesis even though it is probably more closely tied to physiological processes than any other cognitive function that could have been studied. A successful test in colour would be strong evidence that the hypothesis of linguistic relativity has some validity, whereas a negative result would be only weak evidence against it.

At least two experiments seem to indicate that colour perception can be, but need not be, affected by the language in which colour is described. Languages differ in the number of basic colour terms--terms in which colours are commonly and consistently described without modification. For example, in English, red, yellow, orange, are examples of basic colour terms, but rose, turquoise, or bluish-purple are not. The latter terms are rarely or inconsistently used as compared with the former. Kay and McDaniel (1978) found a consistency in the way languages partition the space of observable colour (Fig 13). All languages have at least a two-way partition, dividing the dark-blue-green half of the space from the light-red-yellow half. Languages with more basic names subdivide these two half-spaces into successively finer subdivisions. For example, if a language has three terms, the red region will be split from the light-red-yellow part, and after that, either yellow is split out, or black is split from the dark-blue-green region, leaving blue and green to be represented with a single term. One of the experiments in question compare the perception of colours around the English blue-green division by speakers of a language (Tarahumara, Mexican) which has one basic colour term siyoname for this whole region of colour space (Kay & Kempton, 1984).

Kay and Kempton presented their subjects with triplets of colour chips varying in their blueness and greenness, and asked subjects to identify which pair were most alike and which were least alike. From the responses to this standard psychophysical technique, the experimenters could derive a scale of perceived differences among the chips. They found that distances were expanded for pairs whose chips were on opposite sides of the English blue-green boundary for English speakers but not for Tarahumara speakers. They had previously found that both groups had the same ability to discriminate absolute differences in greenness or blueness, even across the boundary, and subsequently found that neither group showed a boundary effect when asked to judge pairs as to their relative differences in greenness (leaf-likeness) or blueness (sky-likeness). Only when the judgement was of relative similarity did the linguistic boundary affect the judgments.

In a much earlier study, Kopf and Lane (1968) looked at the ability of subjects to make discriminations in an ABX paradigm\(emsubjects are presented with two standards, A and B, and then are asked to judge whether a test item, X, is like A or like B (usually, identity judgments are requested). In such an experiment, category boundary effects are often found: subjects are better able to discriminate when A and B belong to different categories (e.g. phonemes). Kopf and Lane found that the category boundaries of English speakers for colours around the red-yellow-green-blue circle were placed differently on the spectrum from those of speakers of Tsotsil (a Mexican language). The category boundary effect showed up for both kinds of boundary placement: subjects could discriminate better at those places where their language made a discrimination.

The Whorfian hypothesis is not the only situation in which replicative stability can be illustrated. It can be seen also in the problems some children have in learning to read. Many skills are involved in reading in an alphabetic script, among them the ability to identify the sounds of speech and the ability to isolate and recognize the letters of the script. Various studies (reviewed by Taylor & Taylor, 1983) have shown that the ability of readers of different scripts to segment the sound stream depends on the nature of their script. Readers of syllabaries easily segment the sound stream into syllables, but not into phonemes or words, whereas readers of alphabets more easily segment phonemes and words (which are designated by spaces in the script). Illiterates have more trouble with all kinds of segmentation of the sound stream than do literates, but they have particular difficulty with phoneme segmentation.

Taylor and Taylor hypothesized that reading the script enhanced the ability to segment the sound stream, and vice-versa, but only in respect of those aspects of the sound stream represented explicitly in the script. Bradley and Bryant (1983) studied children aged 4 or 5 to determine their phonetic segmentation ability, and tested their reading ability when the children reached about 8 or 9. There was a small but significant correlation between reading ability and the previously tested phonemic segmentation ability.

Having anticipated such a result, Bradley and Bryant selected some of the poorest segmenters for special training over a 2-year period. Some were trained on the semantic categorization of certain words, some on phonemic segmentation of the same words, and some on letter-sound correspondence using the same words. Each increment in training resulted in an increase of about 4 months in the finally tested reading age of the children (those trained in letter-sound correspondence wound up about 3 months behind the average of the normal group). The implication is that the training situation probably helped somewhat, perhaps by motivating the children; deliberately exposing them to the segmentation of phonemes helped more; but jointly exposing them to phoneme segmentation and giving them a non-phonemic (i.e. visual) pattern that they could link to the phonemes helped most of all. The letters supported the phonemes, and one may judge that the phonemes supported the letters (this latter point is attested by the relative success of reading instruction that includes phonics as opposed to instruction that does not).

In general it seems true that one can learn easily either if one has a good analogy to which the new things can be related, or if one is learning two quite different things that have a close analogic relation to one another.


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