Copyright Sociological Research Online, 1998

Smith, R. D. (1998) 'Social Structures and Chaos Theory'
Sociological Research Online, vol. 3, no. 1, <http://www.socresonline.org.uk/3/1/11.html>

Abstract

Up to this point many of the social-scientific discussions of the impact of Chaos theory have dealt with using chaos concepts to refine matters of prediction and control. Chaos theory, however, has far more fundamental consequences which must also be considered. The identification of chaotic events arise as consequences of the attempts to model systems mathematically. For social science this means we must not only evaluate the mathematics but also the assumptions underlying the systems themselves. This paper attempts to show that such social- structural concepts as class, race, gender and ethnicity produce analytic difficulties so serious that the concept of structuralism itself must be reconceptualised to make it adequate to the demands of Chaos theory. The most compelling mode of doing this is through the use of Connectionism. The paper will also attempt to show this effectively means the successful inclusion of Chaos theory into social sciences represents both a new paradigm and a new epistemology and not just a refinement to the existing structuralist models. Research using structuralist assumptions may require reconciliation with the new paradigm.

Keywords:

Chaos Theory, Complexity, Connectionism, Consciousness, Memes, Network Theory, Reductionism, Social Change, Soliton, Structuralism, Timelines, Transduction

Structures

1.1

One of the most enduring aspects of all social theories are those conceptual entities known as structures or groups. In psychology apparently repetitive behaviour patterns are assembled into traits of personality. In sociology people are combined into aggregates which are themselves described with terms such as citizen, resident of a municipality, ethnic group, religion, political conviction, sexual preference, race and so on.

1.2

Ways of assigning people to these aggregates vary. Initially people were placed in categories which themselves had attained the status of cultural validity. Castes, estates, tribes, sex are examples of these. Assignment of individuals to these groups was taken as an a priori and they were used to assemble definitive and clear-cut patterns of other attributes.

1.3

The International Encyclopaedia of the Social Sciences defines a 'structure' in the following way:

The term 'structure' may be defined as a pattern, ie. an observable uniformity, in terms of which action (or operations) takes place. ... Structure refers to an aspect of empirical phenomena that can be divorced from time. The patterns of action, qua patterns, do not exist as concrete objects in the same sense that sticks and stones do. The patterns of action in this sense are abstractions from concrete empirical phenomena, and they 'exist' and are empirically verifiable in the same sense that the squareness of a box exists and is empirically verifiable. (Levy, 1968)

1.4

Despite the fact this definition is nearly 30 years old it is sufficiently expressive of what most social scientists actually believe to be deemed 'intuitively clear'. Most researchers talk of such social-structural phenomena as 'class', 'ses', 'family', 'household', 'babyboomers', 'Xers', 'ethnic groups', 'minorities', 'women', 'homosexuals', 'races', 'castes' and so on without much further thought.

1.5

Structuralism has two problems. First is the practical matter of determining what the structures are and how our perceptions are to be classified. Second is the problem of 'change' - of how one structure becomes another.

1.6

Structuralism has the problem of being a very fluid term. It could be almost any pattern in this definition. The issue here was 'where do the lines get drawn?' or what are the boundaries of the structure? The history of social thought refers to this as the 'taxonomy' problem. It is a debate which goes back at least to Plato and Aristotle.

1.7

Drawing the boundaries forces on us two decisions. One is 'what categories shall we create?' and the other is 'how shall we understand and interrelate these categories?'

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'I shall call this figure the duck-rabbit. It can be seen as a rabbit's head or as a duck's. And I must distinguish between the 'continuous seeing' of an aspect and the 'dawning' of an aspect. The picture might have been shown me, and I have never seen anything but a rabbit in it.' - Ludwig Wittgenstein (after Fuglesang, 1982).

Figure 1: Duck-Rabbit

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1.8

To return to the first definition given, structural entities or patterns were also to 'be divorced from time' even as their very existence is recognised as a temporary phase of a time-dependent process: evolution. Little wonder then, that one of the most intractable theoretical debates of the 'structural- functionalist' era was 'the problem of change'. How one can hope to discuss 'change' in a set of conceptual entities which have been specifically 'divorced from time' presents a considerable intellectual challenge.^[1]

1.9

A visit to any library will reveal shelves of books all published in the post-war period in which systems theory was applied to social science. Most notable in these titles are those that attempt to incorporate change, or flux, into the array of boxes and arrows which represented the formal paradigm of Systems Theory. Almost all these books start with general flow charts and map out the various paths influences can and must take in order to be able to influence the systems' 'functioning'.

1.10

The problem of change also has two features. The first is related to the taxonomic boundaries. Boundaries change when, say, one ethnic group is assimilated or absorbs another. This phenomenon is well known but the formalism of social structuralism has never been able to deal with it. People in ethnic groups can be seen to follow assimilation patterns through time. Ethnic groups are not defined so much in terms of the individuals who are in them but in the behaviour patterns the members exhibit.

1.11

For example, the illustration in Figure 1 can 'change' from a bird to a rabbit as an act of cognition divorced from any physical alteration to the picture itself.

1.12

This is still bracketed, however, when we realise that while some people may see a duck and others a rabbit, no-one is likely to see a cup of coffee. Moreover, in all instances, what one is doing is seeing marks on a page and being led to remember a bird or a bunny. Duckrabbit is in this sense neither duck nor rabbit. It is a visual stimulus made of ink and paper (or of pixels). The conundrum of Duckrabbit contains both objective and subjective aspects. Objectively virtually all will agree it is a picture. At the other end of the continuum it is probably as futile to argue this pattern symbolises a 'duck' and not a 'rabbit' as it is to dispute over opening eggs at the little end as opposed to the big end.

1.13

Between these poles is a range of things the pattern might also be symbolising. It is almost certainly not a cup of coffee, a tree, an igloo or the eruption of Krakatoa. In short, we do not have the luxury of using this or any other ambiguity of our symbol system as evidence for the adoption of pure relativism. Some interpretations are simply less believable than others. In this case we may be forced to resolve our interpretation of the symbol into two, not one, defensible meanings. Perhaps there are three such interpretations. But there are certainly not an infinite number of them.

1.14

The fact we cannot have perfect congruence between our cognitive symbols and the reality they purport to represent is not evidence of our inability ever to know what that reality is. Those who make such assertions are arguing not logically but ideologically.

1.15

Many ideas which seem clear at the level of abstract theory concepts become less clear at the operational level. One can have a reasonably good idea of what a bird is but when asked a companion question 'rate the following birds regarding how good they are as examples of birds' the matter becomes less clear. McNeill and Freiberger (1993: pp. 84 - 85) report that when respondents were asked to rate birds on a scale from 1 (excellent) to 4 (poor) it was found that robins scored 1.1, eagles 1.2, wrens 1.4, ostriches 3.3 and chickens 3.8. Bats scored 5.8.

1.16

Our categories themselves (such as 'birds') are not hermetically sealed. Human understanding can take a creature which is known not to be a bird at all and still classify it along a 'birdness' dimension. If a bat can be a rather inadequate example of birdness then what kinds of potential equivocations or misunderstandings can and doubtless do arise with terminology which is not so formally clearcut as that of biological taxonomy? What is the price in lost clarity and perhaps in wrong analytic decisions when the existing ideas of social structure are employed?

1.17

To take a common example of a social structure consider ethnicity^[2] and the change often associated with it: assimilation. Figure 2 shows the classic signposts which are generally taken to represent assimilation but even as we observe people making the transition we have some nagging questions. For one thing, how do we conceptualise in terms of a structural category those people who are 'in-between' or hybrids of one group or set of traits and another? Should we assign a label to them now? derive labels from logical categories? waiting until a certain percentage are able to be so classified? wait for them to demand their rights? something else?

1.18

In the vernacular of many societies these individuals are (usually derisively) called apples, bananas, oreos, half-breeds and so on. Do they form 'structural groups' or not? In a more direct example: what of the people of Hong Kong? Are they British? Chinese? Something else? And how does one decide?

1.19

The arbitrariness of assigning people to categories the did not seem even to know existed led some theorists to embrace a 'negotiated order' approach to all social reality. Given that the signal uniqueness of the human species seems to be self-consciousness the work of Blumer (1969) and others sought to seek its patterns there.

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Subprocess or Condition	Types or Stages of Assimilation
Changing of cultural patterns to those of host society	Cultural or behavioural assimilation
Large-scale entrance into cliques, clubs, and institutions of host society, on primary group level	Structural assimilation
Large-scale intermarriage	Marital assimilation
Development of sense of people-hood based exclusively on host society	Identificational assimilation
Absence of prejudice	Attitude receptional assimilation
Absence of discrimination	Behaviour receptional assimilation
Absence of value and power conflict	Civic assimilation

Figure 2: Ethnic Assimilation Variables (source: Gordon, 1964)

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1.20

This trend became so disturbing that George Homans dedicated his 1964 presidential address to the American Sociological Association (ASA) to the call for bringing individual human beings 'back in' to social theories. In his address he observed that social structural theories did not really seem to have any need for human beings (Homans, 1964).

Memes

2.1

As with Duckrabbit ethnic identity and assimilation is, at the level of the individual person, some kind of interaction between sense data and cognition. Thus ideas, too, may be thought of as being 'structured' (as when they are assembled into a paradigm). In this guise an ethnic group may be understood as a collection of people who both accept one-another as members and define membership in terms of attitudes, attributes, behaviours or understandings which comprise identifiable patterns.

2.2

Structuralism also leads us into another complication. Paradigms which represent reality as a set of mutually exclusive categories have difficulties other than the problems of change and categorisation. They also lead into debates on 'reductionism'. The onset of sociology was marked by the efforts of Emile Durkheim to insist that sociological laws existed and could be studied independently from the study of the psychological laws for individuals who made up society. Sociology did not need to be reduced to psychology any more than biology had to be explained in terms of physics. Social processes could be studied in their own right. In the time since then some critics have observed social theories were so abstract people were not really able to influence the long-term outcomes that macro laws would predict.

2.3

Structuralism, in short, could not easily deal either with change or with how or to what extent the actions of individuals could influence the future state of their community, their culture, the world.

2.4

Ideas (or as Dawkins calls them, memes^[3] (Lynch, 1998)) can flow through time, being advocated by different individuals but retain highly stable intellectual content. In this instance the membership of the group changes but presumably the 'ideas' remain the same or, if they do not, they change slowly enough so that the actual members of the ethnic group do not experience the destabilising emotional consequences of changes in understanding that are taking place too rapidly.^[4]

2.5

Consider, for example, the matters of political affiliation, political ideology and political campaign promises. In the 1996 US Presidential election there was some discussion as to whether incumbent William J. Clinton was a 'liberal' or a 'moderate conservative'. Comparisons of his policies and principles with fellow-Democrat Mario Cuomo suggested to many people that Clinton was not a liberal at all. The issue of classification is this: both of these men were indisputably members of the Democratic Party. This is a legal status and cannot easily be rendered ambiguous.

2.6

We might also note that what passed for liberalism in the 19th century was a devotion to laissez-faire economics, the restriction of domestic government to the role of guarantor of contracts, and an aversion to almost all forms of taxation. The late 20th century contents of liberal ideology are rather different. The matter before us is to render this apparent fluctuation or evolution in label and meaning (terminology and membership) in a way consistent with a conceptual formalism. This rendering will hopefully be relatively free of what is known generically as 'hand waving'.

2.7

Within the current corpus of human intellectual life there is at least one example which could prove to be of use. The conceptual paradigm of quantum mechanics contains within its formalism a well articulated approach to structure which allows easily for situations of this type to be discussed.^[5]

2.8

One could, for example, suggest that two memes (ideas) such as individual liberty and collective responsibility might combine as in Figure 3. In this way we can think of an idea as a soliton, a solitary wave, moving through history.

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Solitons are described mathematically as Kortweg-de Vries (or KdV) equations. They are of the form:

u_t - 6uu_x - u_xxx = 0

where: u (x,t) = exp{-k³t + ik(x-t)}; k R, k 0

and: u_x = u/x

(from Drazin and Johnson, 1989)

The figure shows the computed dynamics of a two-soliton interaction. It is essential to realise this soliton is a wave travelling through a medium or some other matrix capable of transmission. Water waves, for example, pulse through the medium (oceans or lakes or rivers) but at every stage of their travel they are made up of different collections of water molecules. The water molecules which make up the wave at any given time are largely distinct. As complex as these waves are they are not infinitely complex (within a given dimensional space) unless the dimensions themselves are continuous. Water, coming in molecules, is quantised. The issue of infinitely variable waves must then rely to some extent on whether space (or space-time) itself is amenable to being infinitely subdivided or it is also quantised. (see for example: Misner et al (1973). This consideration will appear later when the specific case of the Lorenz attractor is examined.

Figure 3: Computer Rendering of a Two-Soliton Interaction

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2.9

Are these solitons 'structures'? Is the region of the two-soliton interaction pattern also a structure? Those of us who do survey research know how difficult it is to guess at what the underlying three waves are if all we have is time-insensitive data collected from different people with non-standard questions at irregular intervals. After gathering what data we can we could employ such statistical techniques as Factor Analysis, Cluster Analysis and Principle Components Analysis before using LISREL to 'analyse' the structures we found. Even using approximations to deal with temporal fluctuations by employing lagged variables and autocorrelation analysis, long standing approaches in the more mathematically sophisticated areas of macro-economics, is increasingly being seen as having limitations which our computer technology no longer requires we continue to accept (Rosser, 1991).

2.10

Viewing structural entities as waves forces us to revisit the question of rigid taxonomic boundaries. In survey research it is common to regard certain collections of individuals as 'cohorts'. If one extends this then one possible visualisation for a soliton is that of a cohort (such as 'Babyboomers' or 'Generation X-ers' who owe their designation to the historical period in which they were born) moving through temporal space and interacting with other cohorts. Structural entities can also, of course, be based on attitudinal as well as demographic attributions so that people who arguably qualify as boomers by virtue of their age may have life experiences more akin to those of Generation X and thus occupy some hybrid (or in the language of wave mechanics a 'resonance state').^[6]

Chaos

3.1

At the intuitive level, at any rate, chaos theory seems to provide a means of escaping structuralism's two signal flaws: it is inherently dynamic and time-sensitive and it permits a definition of social structural entities in such a way that if real, living, unique human beings vanished then the structures of society would also vanish.^[7] This also provides, by means of its formal inclusion of human cognition, an analytic paradigm which might allow us to see how providing a set of democratic laws to a totalitarian culture does not necessarily transform it into a democratic society, or how it is that despite having a constitution written by American soldiers after World War II, Japan still boasts a decidedly Asian culture. Things can change and individual people can make a difference at the 'macro' level.

3.2

The fundamental observation of chaos theory is that small changes can have large consequences. Butterflies might cause tornadoes, saving a life in 1930 might mean that Earth's political future is that of being a squalid dictatorship never to join the UFP and build the starship Enterprise, and so on. Even as this staggering realisation started to sink into the consciousness of philosophers and scientists other more rustic souls were reminding us that 'for want of a nail a kingdom was lost'. This was not new news.^[8]

3.3

Historians have known for years that 'plausible' different or future worlds (Hawthorn, 1991) are the stuff of worthwhile thought experiments. This exercise forces us to distinguish between what our theories tell us could not have happened and what our history tells us did not happen.

3.4

Pipes (1995) argues that the Bolshevik Revolution in Russia was not inevitable but that given its success the emergence of Stalin was all but assured. He also maintains that '[i]t is my considered judgement that had it not been for the Russian Revolution, there would very likely not have been National Socialism, probably no Second World War and no decolonisation, and certainly no Cold War...' (Pipes, 1995: p. 1) He also points out that had Fannie Kaplan's poor eyesight not spoiled her aim she might very well have successfully assassinated Vladimir Lenin in August of 1918. Without Lenin the Revolution would almost certainly have failed.

3.5

When confronted with such 'if only' scenarios it is not hard to see why many historians take the quest for a Grand Theory of society less than seriously. But even if Fannie Kaplan had succeeded, what would our present world look like? The truth is we do not know.

3.6

Skocpol (1984) reveals the difficulty in using the very limited tools of social theory to deal analytically with historical events. What, then, to do with an approach to structuralism or the 'structuring of society' that muddies the distinction between the plausible possible and in plausible impossible? How are we to differentiate the utterly impossible from that which is merely formidably difficult? Answering these questions will allow us to deal more directly with the question 'are all worldviews equally arbitrary?'

3.7

The answer is equivocal. Pipes' earlier noted comments contain both chaotic and non-chaotic elements. Fannie Kaplan's attempt to kill Lenin and the consequences of her failure are presented in a manner consistent with Chaos Theory: a very small event can have extreme ramifications. Pipes also argues, though, that Kaplan's failure all but assured the eventual emergence of Joseph Stalin and by extension National Socialism, WWII and so on. Is he really trying to have it both ways? Did no other opportunities emerge which, had they been seized, would also have altered this outline of events?

3.8

Pipes' attitude is not atypical of many historians. There is a sense of human history being played out on various levels or planes and mostly these levels do not interact or interfere with one another. Sometimes, however, they do. Sometimes a distant butterfly wingflap causes storms, property damage, and stock market fluctuations half a world away. Sometimes it doesn't.^[9]

3.9

This being the case it becomes all the more essential to be clear about the formal and the informal uses of the chaos concept in social science. Chaos, like Relativity, Darwinism and Mechanism before it, has the potential to be transformed into a metaphor and to have its terminology misunderstood and misapplied. Indeed numerous new titles currently available indicate this trend has already begun. (eg. Savage, 1988). The best defence against this is a well-defined and accepted formalism

3.10

In order to have a formal notion of structure which is consistent with the chaos theory paradigm we now need to return to a requirement so far unelaborated. In its most precise rendering chaos can only arise when the possibility of any given state repeating itself is potentially zero. To take the illustration of a strange attractor such as the Lorenz attractor (as in Figure 4) what is needed is a situation in which the orbital pathway of a flow or flux can continue for an indefinitely long period of time (for eternity) without ever passing through the same point twice. If this condition is not met then the orbit is not in fact chaotic but periodic even though highly convoluted. What this in turn means is that the phase-space in which the flux is propagated should be continuous and not quantised. A quantised space, however large, is effectively finite and thus cannot provide for truly chaotic behaviour.^[10]

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For true chaos to be present the orbital pathway must be a line of infinite length. If it passes through the same point twice then the deterministic nature of chaos informs us the orbit is of fixed periodicity and length. In order for true chaos to exist in nature, then, it may be necessary for space-time to be continuous and not quantised. If it is quantised then the number of possible positions is, while incomprehensibly large, finite. Conservation of energy means that the phase-space for such an orbit would likewise be limited. Lattice-models in which both chaos and solitons arise, however, have been proposed (Sayadi and Pouget, 1991).

Note that in this plot the lines clearly appear to intersect. They only appear to do so because this a two-dimensional rendering of a three-dimensional process. Chaos in a phase-space depends on the dimensions of the phase-space. If the actual dimensions of the connections (the lattice through which the signal or influence propagates) is of evolving or growing dimensionality then apparently finite restrictions may be ephemeral. See, however, for opposing views on themes related to this, Dyson (1988) and Gell-Mann (1994).

Figure 4: The Lorenz Attractor

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Complexity

4.1

Even the briefest glance at the graph of the Lorenz attractor will produce the sense that a pattern is clearly present. The fact that 'chaos' is involved in producing it does not remove the feeling that something 'macro' is going on.

4.2

This is the position taken by complexity theorists (eg. Kauffman, 1995) who insist (as Durkheim did a century ago) that large-scale stochastic regularities can be empirically shown to 'exist' and can be analysed as such.

4.3

Ever since social sciences first began to analyse groups of people as if they comprised a single entity or 'structural component' a constant objection has been raised: social structural entities to do not really exist save as heuristics. People exist. Classes do not wage war on each other. People do. This much is patently obvious. But we are also confronted with what we deem to be irrefutable (or at least highly persuasive) evidence to the contrary. Ethnic groups, socioeconomic classes, demographic cohorts and the like are useful empirical generalisations and, with the correct data analysis tools, can be used to group individuals together in categories the people themselves may not even suspect exist.

4.4

So while human consciousness is clearly necessary for many of the social patterns we observe, the patterns may not be identified by the actual people who participate directly in them, or they may be labelled differently from the way relatively external observers classify them (sometimes leading one group to accuse the other of possessing false consciousness). This, though, does not argue in favour of relativism as much as for the need of a formalism with the ability to be able to translate (or to 'map' in the terminology of mathematics) one perspective into another. Einstein's theory of relativity not only stipulated that the way certain phenomena were perceived was attributable to the frame of reference of the observer (where one sits determines what one sees) but it also demanded that for such a pronouncement to make any sense at all the underlying laws must be invariant and must hold in the same way everywhere. Relativism thus depends on absolutism.^[11]

4.5

Obviously the world does not present itself to us as being as complicated as it could possibly be. Despite the theoretical potential for unbridled chaos we live in a condition sufficiently ordered to permit us to exist and concoct theories. Solitons are complex and largely non-chaotic entities within the life-span of their existence. Does this fact mean that our world is made up of events which are effectively sealed off from one-another's influences after all? Or is this division of phenomena into such levels of abstraction as 'micro' and 'macro' just another heuristic whose time has gone? If we accept this latter position then our chaos- informed formalism must be able to preserve or at least incorporate the apparent existence of these various levels of analysis and our human notion of relative stability in these things.

4.6

We must now go back to deal with Pipes' assurance that, given Lenin, Stalin would prevail.

4.7

Just as the argument in the 19th century raged over whether laws of history were reducible to laws of psychology, this debate now has been transposed into the joust between chaos theorists (those who insist on starting at the bottom and working up) and those complexity theorists who claim this is both hopelessly complicated and in any event not needed. Starting with the analysis of sandpiles, complexity theorists discovered, after spending many arduous afternoons on Long Island beaches, that when sand was piled up it eventually tended towards a critical angle of 51° 52'.

4.8

The process whereby sand piles up effectively at random but sooner or later a part of it achieves the needed critical angle is called 'self-organised criticality' (Bak et al, 1988).

4.9

If only sandpiles or Egyptian pyramids acted this way social scientists might never have thought of structures. Many things, though, exhibit very similar patterns in nature and in society. One area of extensive study is that of diffusion of innovation. As early as 1971, Fisher and Pry noticed that capital markets adopted financial products with a very predictable form: the logistic equation. Ten years later, Marquette (1981) applied a logistic model to political mobilisation and Cesare Marchetti (1980) showed that the logistic equation not only predicted very well the ability of a child to learn a language but also the general cycles of discovery, invention and innovation. He concluded that society acted like one huge 'learning system'. Societies learn to use technology in patterns almost identical to those which are seen in children learning language. Aggregates of human brains seem to adapt or learn in ways parallel to the ways in which those aggregates of nuerons known as human brains do.^[12] Power laws can also be used to predict earthquakes and asteroid strikes on planets including earth.

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Market penetration of four processes or machines in the United States. The time constant is about 50 years for all four cases. Here the process is one way there are no cycles involved. The assumption is that the rate of these penetration processes is the primary clock of the economy. This rate may be closely linked to human behaviour and may be the deep reason for the stability of the cycles' length. (Marchetti, 1980)

Note that the power law here is based on a logistic relationship.

Marchetti's suspicion that a 'deep reason' exists for why these relationships all conform to a power-law relationship is arguably a manifestation of statistical self-similarity being detected by a researcher. Self-similarity results from two or more apparently distinct phenomena exhibiting commonalties. These commonalties apparently pique human curiosity and cause us to ask "is this merely a coincidence or is something else going on here?" Human psychology seems to be distrustful of assigning too many coincidences to the reason of random chance.

Self-similarity is also evident in the lattice approach of connectionism when people are in some ways networked together in patterns highly reminiscent of those which connect neurons in the brain. This is by no means either necessary or sufficient proof of the superiority of a connectionist paradigm but it is indeed adequate to justify further research and speculation in this area.

Figure 5: Power-Law Curves Exhibiting Market Penetration

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4.10

This property, if it really is there, is an example of 'self-similarity'. Self-similarity is a general concept which is intended to convey the fact that a similar (or identical) pattern can be detected at various levels of abstraction or observation. It is the manifestation of self-similarity at various levels of chaotic systems which formally differentiates chaos from randomness.

4.11

Truly random systems cannot reveal patterns but only coincidences which humans mistake for patterns (Knuth, 1969; Tezuka, 1995). When real patterns are revealed then chaos is said to be present. Chaos, the apparent state of disorder or irregularity, is in fact a state in which strict determinism reigns but the flow of the patterns is formidably hard to predict owing to the primitive nature of contemporary mathematical models and computer technology.

4.12

The techniques for identifying patterns within this investigative paradigm is to speculate deductively about what the rules governing a system might be (usually these rules are expressed as differential equations) and then to carry out exhaustive computer simulations which require the researcher to examine carefully the patterns which do emerge and try to decide if they look like anything resembling (similar to) situations of interest. Langton (1995) has applied this technique to his study of artificial life. Danielson (1992) and others have turned their attention to artificial morality.

4.13

Other more deductive approaches to large-scale pattern identification can also be used. Recent hypothetical studies by Rinaldi et al (1994) have produced the following application of 'strange attractor' concepts to the flux of political popularity.

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A Trajectory
P' Æ P''
close to the degenerate heteroclinic loop.

where:
x(t)= the public support (popularity) of politicians at time t.
y(t) = the hidden assets that the corrupt politicians hold at time t.
z(t) = the investigation effort at time t (ie. the sum of activities by different institutions such as police, courts, and the press).
and:
A = A(x); A representing positive actions taken
A*(x(t)); A* is the level of positive expectations held by the public
B = B(x,y); B representing bribes and all other moneys earned illegally by politicians
C = C(B,y); C representing the private consumption of the politician as constrained by bribes and hidden assets.
D = D(y,z); D representing the discovery of the corruption. (Here: the amount of money confiscated by the courts).
And this leads to the following model:

1. x' = (+ (A*(x(t)) - A(x)) - (- xD(y,x)
2. y' = xy + B(x,y) - C(B(x,y),y) - D(y,z)
3. z' = (D(y,z) - (z

Where ( represents the positive or negative reactions of the public, ( the persistence of investigations and ( represents the relative importance of manpower expended to dollars recovered.

Figure 6: Strange Attractor dynamics for the popularity of a corrupt politician (Rinaldi et al, 1994)

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4.14

Two points can be made here. Marchetti's work has demonstrated yet again the formal self-similarity of learning systems to learning individuals. Rinaldi's work shows additional similarities which we can assume would be equally true of every individual in the population. Like all real paradigm shifts, the epistemological consequences go beyond mere redefinition of some terms.

4.15

What, then, are we to do with the attempts to analyse social change using structuralist terminology? Clearly the success of those who advocate the adoption of the chaos/complexity paradigm cannot expect to be taken seriously if all they can offer is an analytic framework which states something like 'what we call 'structure A' and what we call 'structure B' are simply different places on a complex pathway'. This is obviously true and almost equally obviously useless as an analytic statement. Taking a more specific variant such as: 'The collapse of the USSR was a period of great turmoil and instability which saw the reassertion of nation states and statelets, the boundaries of which were much more closely aligned with traditional ethnic territories than the Imperium they supplanted' will do little formally to aid our analytic quest.

4.16

What is also true is that the formalism of traditional structuralist paradigms cannot accommodate the examples above. By effectively isolating at a formal level the micro from the macro forces, structuralism all but makes the short-range and micro level all but epiphenomenal. This confining further means that any attempt to illustrate the effects of the micro on the macro are to be made anecdotally or not at all. There remains the suspicion, however, that the social world may not have an inevitable single final stage. There may not be a social form which corresponds to the Natural world's Big Crunch (or entropic death). There may be other options. The world - our world - is revealed in the chaos/complexity paradigm as will and representation within the constraints of natural law. What the chaos/complexity paradigm adds is the possibility of human cognition playing a role within these constraints.

4.17

We also need to resist the temptation to stray from the requirements of a formalism. Influences must propagate through a medium such as a lattice. Memes can, of course, lie dormant in books or carved on granite blocks or in computer files. They lose any relevance at all if they are lost or become, like Linear B, indecipherable. Memes, like genes, must be passed by contact along the relevant dimensions. In the case of memes it is required that they be passed from one living being to another and that these beings be sentient.

4.18

It is therefore to the discussion of the matrix or medium within which chaos arises that we must now turn our attention. Let me now propose what I hold to be the underlying architecture of our epistemology and how it can accommodate both our perception of chaos and of order simultaneously. This architecture has already been dubbed in the literature with the name 'Connectionism'.

Connectionism

5.1

The second point of interest is the role of connectivity. Complexity theorists have realised that only when all the sandgrains can 'communicate globally' with one-another do the phenomena associated with sandpiles manifest themselves. Connectionism thus appears to be essential. It also goes without saying that the specific dimensions of communication (or connectivity) dictate the global attributes observed. The sandgrains must be in physical contact and within a gravitational field of significantly greater magnitude than their mass alone could generate. The formation of the pile depends centrally on an 'external' gravitational field providing meaning to the term 'up'. Only then can avalanches cascade 'down'. We also need to recognise the rather arbitrary distinction between the sandpile and the beach - a beach made up of more sandgrains - upon which it is built.

5.2

Connectionism refers to a paradigm or model in which particular attention is given to the ways in which the actors, units or elements of the theory are related to (connected to) each other. The basis of these interrelations involves additional concepts like transmitter and receptor.

5.3

The approach to defining structure suggested here is quite close to that put forward a few years ago by network theorists. Network theorists defined structures as groups of elements which were linked together by ties (the density of ties, how often they were used, which way the communication or influence flowed, who initiated the communication and so on were all advanced as descriptors of the structures thus identified). (Rogers and Kincaid, 1981; Marsden and Lin, 1982; Wellman and Berkowitz, 1988)). Figure 7 shows an early approach to network analysis in which connectedness and directionality of influence are displayed.

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Two major connectivity dimensions are overlaid here. The 'mutual' and 'nonreciprocated' ties represent architectural features of this network. The flux patterns are a distinct type of structural category. The actual architecture only indicates what pathways the flux may follow and what qualitative kinds of ties they may be. Not all reasons for mutual choice are equal and so even though person 3 is mutually tied to both person 1 and person 2 this does not mean these two individuals have the same ability to influence person 3 to exploit any of his/her nonreciprocated ties. The idea of nonreciprocity is potentially subject to change as well. The architecture is thus not fixed permanently but only presumed to be constant for one or two cycles (iterations) of the flux structure. Excessive demands (or failures to engage in mutuality) could rupture some ties and create others. This network's architecture is itself capable of changing but it would likely do so at a slower rate than the actual transmission of information (or other communications) takes place.

The pioneering work of Hunter (1953) was one of the first systematic attempts to employ network concepts in the analysis of communities. Today's technology is still woefully inadequate to do carry out his project adequately.

Figure 7: Sociogram Showing Friendship Networks in a School Fraternity

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5.4

Second, network theorists in general never articulated a consistent and agreed-to conceptual base of their advanced ideas which would have allowed them to be treated quantitatively and dynamically.

5.5

This being said, however, the following two figures show patterns which allow something akin to 'structures' to be approximated in network theory. (Recently this has been put to very good use for tracking criminal conspiracies and other forms of organised crime).

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A network is saturated if all possible links are present. The density of the connections in (a) is almost saturated as only the A-D link is absent. In (b) the role of C is crucial and this status is sometimes called a 'sociometric star'. It could also be called a hub (as in hub-spoke airports) in airline industry terminology.

Figure 8: High and Low Density Networks

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This network has three subgroups each of which is either connectively saturated or only one link short of saturation. Taken as a whole the network is moderately saturated. If one redefines a network as a group of connected elements with more than one pathway able to connect any two elements then this figure represents three networks. Using such a bright-line definition is essential for some kind of conceptual simplicity but as even this simple illustration reveals it is not totally accurate. In this rendering we assume Links A and B are in the same dimensions as the dimensions which connect the elements of X, Y, and Z.

Figure 9: Fuzzy Structure: A Network with Three Distinct Sub-groups

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5.6

For all of the initial intuitive appeal network theory ran into two problems. First it was too complex for 1970s computer technology to manage. (Anyone who has tried to construct even a simple neural-net model of a social process has similar complaints about 1990s computer technology).

5.7

Connectionism arose in the field of computer science as a way of deal with parallel processes. Connectionist systems not only have rich interconnections (as all networks do) but the capacity to carry on more than one operation or set of instructions at a time (ie. in parallel). In connectionist systems the various processes are not only carried out simultaneously but often without reference to one-another. Another feature of connectionist systems is the tendency to regard 'information' not simply as whether a particular bit is on or off, but also as the pattern of network activation. In this second sense information can be understood as the pathway along which a signal flows.

5.8

A connectionist model allows two kinds of patterns to be studied. The first is the pattern of the signals, communications or other influences which flow through the lattice. This shall be referred to as the flux pattern. The Lorenz Attractor (Figure 4) has, for example, been used symbolically in clinical psychological practices to describe what might be happening in the brain of a person with a bi-polar disorder such as manic-depression. The two attractor foci represent the two mood-poles and the trajectory around these two poles is chaotic in two ways: first because the precise symptomatology from one episode to another is similar yet different since no point on the trajectory is encountered twice; second: because even though the transition from one mood-state to another is predetermined by the initial conditions the prediction of when these transitions will take place cannot be made with any accuracy.

5.9

The second kind of pattern which can be studied is defined by the architecture of the lattice itself: the pattern of connections among the various elements of the lattice and through which the signals propagate. This is known as the architecture, the matrix, or the lattice. The lattice places, for practical purposes, limits on what flux patterns are possible. The architecture allows us to differentiate plausible futures from implausible futures and, perhaps, from impossible futures.^[13]

5.10

Using a connectionist model of a process (eg. Eiser, 1993) requires that we treat the phase- state as the current network location of the influence under study. The propagation pathway is not infinite but confined to orbits which follow the pre-existing pathways between various elements.^[14]

5.11

The simplest kind of model allows only one kind of influence with one direction of flow. Each neuron in the human brain is susceptible to between 100 and 200 neurotransmitters, has multiple outputs and inputs, can create new connections, lose old ones, change their resistance to certain pathways, and die. The complexity of the human brain is significant (Harth, 1991) and one should at least be prepared to believe that human society is at least of equal complexity.

5.12

It is therefore the flux pattern chaos theorists explicitly study and it is in these patterns chaos theorists will ultimately be compelled to define their own notions of structures.

5.13

In other words they do not study structures qua structures themselves but their study of structures is geared to understanding the implications of the 'structures' which in the connectionist paradigm be understood as the flux patterns called social interactions.

5.14

Initially it is useful to draw a distinction between the connectivity pattern of the lattice (the architecture) and the communications which may or may not flow through this lattice. The lattice constrains what flow patterns might exist but does not require that they do. This constraining influences not only what channels and nodes but also what dimensions may be followed.

5.15

Using a more sociological description of these terms: the lattice may be likened to the actors, actresses, props and stage where a play is to be given. The flux or flow pattern is to be compared to the actual performance itself. Traditional uses of the term 'social structure' have - by paying insufficient attention to the time dimensions - made it rather more difficult than necessary to tell the difference between props and people.

5.16

The criticism Homans made can be seen not just as the requirement that sociological laws be rooted in physiological laws but that our conceptual paradigms should allow us to understand or express one set of abstractions in terms of another.^[15]

5.17

Connectionism however allows a different approach to theorising. Because connected networks can exhibit chaos the roles of individual elements cannot be discounted in advance. Because some networks can, under other conditions, exhibit highly predictable patterns, macro level analysis can be carried out and meaningful macro level generalities sought. Connectionist models lead us to consider the 'structure' of the architecture, the 'structure' of the flux, the temporal permanence of both. At this point we can join the debate over reductionism.

Reductionism

6.1

When reductionism first entered the vocabulary of the social sciences, it was employed by Durkheim in his arguments to legitimate sociology as an academic discipline. His definition implied that reductionism was to speak of wholes solely in terms of their parts - the wholes in this case being social aggregates while the parts individual human beings. While perhaps polemically justifiable this position has always faced a few problems. How can one reconcile an opposition to reduction with the widely held belief in evolution? What does it mean to say the observed patterns (laws) of living things have evolved or emerged from natural and inanimate processes but that life cannot be explained in terms of the laws of physics (or social behaviour in terms of psychology)? How can reductionism be rejected and evolution be retained? On the other hand, how can social structures and sociology itself be regarded as a legitimate discipline if only individual humans are really real?

6.2

In the 1930s philosophers of science addressed some of these issues. Nagel attempted to redefine reductionism by dealing with matters like 'generality of explanations', 'generality of laws' and relating these to the 'reduction of theories' (Nagel, 1961: p. 37). For Nagel, reductionism could mean extending the application of one theory to incorporate the full range of application of another but without reconceptualising anything (eg. extending the law of gravity as seen on Earth to include celestial objects as well). A second kind involves redefining some terms in order to expand the range of measurement. When temperature is measured with mercury thermometers then any temperature above which mercury vaporises (350°C) cannot be measured (Nagel, 1961: p. 341)^[16].

6.3

By 'reducing' a column of mercury to its parts (a collection of mercury atoms) and then measuring a different property of these parts (kinetic energy) the new notion of temperature which arises both provides the same value as the old within the original range of applicability and extends the quantitative measurement of temperature beyond the original limits. Reducing temperature to the atomic level made it possible to integrate temperature, volume and pressure and derive the ideal gas law from more fundamental assumptions.

6.4

When Durkheim used sociological analysis to identify patterns in suicide rates he embarked on a series of 'psychological' comments designed to answer the question 'why are these patterns seen?'. In the last 15 years the Philippines rejected the dictatorship of Ferdinand Marcos, the Shah of Iran was driven from power by an elderly cleric, and Communist regimes throughout Europe came to various ignominious ends.

6.5

At those periods in history when the structure of society is most unstable those professors who are charged with teaching courses on these societies discuss the personalities of the people who are making the headlines and speculate on the ways human psychology really works. Politics becomes local, the personal becomes political.

6.6

Inasmuch as reductionism plays a role as part of the reason for engaging in scientific research in the first place it seems the question is not whether to embrace reductionism but when.

6.7

Informally it appears the cognitive steps to reductionism involve at least the following:

1. Similar patterns imply similar laws are seen to be at work
2. The same pattern means the same law is at work
3. The smallest unit (time-scale, number of elements, physical size, etc.) in which the law is observed to hold will be taken as the fundamental unit
4. Reductionism (deductive) shows how the macro (larger) expression of the law can be derived from micro manifestations of it (with additional terms included to handle (dis)economies of scale if they prove to be needed)
5. The ability to reduce one law along a chain of levels says nothing about the ability to reduce any other law along the same chain or any other chain.

6.8

Some dangers which are clearly potential in reductionist attempts:

1. Hasty generalisation will lead to errors (especially as from #4 or #5 above)
2. Tendency to monocausalism (a special instance of the first of these but pragmatically of significant interest in that it can yield the kinds of oversimplifications which are main components of demagogic ideologies, cult worldviews, and so on).

6.9

Let us revisit the Fannie Kaplan saga once more. Ms Kaplan had the option of confronting Mr Lenin verbally and explaining her reasons for opposing Bolshevism. It is just barely within the grasp of one's imagination to suppose that had she done so, Lenin might have instantly been persuaded to of the futility of his cause and become a Trappist monk. Had this happened, the ideological structure the Western world would have been vastly different than the one our history reports to us.

6.10

Had she chosen this path she would have been attempting to influence the future memetic structure of her world by making a memetic intervention and hoping for the kind of disproportionate change in the system of the kind chaos theory represents. By electing to shoot him rather than reason with him she was attempting to influence the memetic dimension by an intervention not on Lenin's cognitive dimensions but on his physiological dimensions.

Transduction

7.1

Two kinds of chaotic situations present themselves to observers and the second kind involves transduction. The first kind occurs when the onset of chaos is fully predetermined by the initial conditions (as in the Lorenz equations) but almost impossible to predict owing to our technological inability to specify the initial conditions with infinite precision. This is the classic form of chaos. It arises from within the system and is not the result of outside forces.

7.2

The second form of chaos, one which is both much less interesting to mathematicians and much more common in the real world, involves the intervention of external shocks or factors which while minor in themselves have long-term and major consequences for the system under study. Examples of such external interventions could include Fannie Kaplan remembering her spectacles or the almost imperceptible kindness of a stranger transforming another's mood from despair to hope.

7.3

Transduction indicates an event has produced an effect which is qualitatively (dimensionally) different from itself. Loudspeakers are 'electro-mechanical transducers' because they convert electricity into soundwaves.

7.4

To illustrate briefly the way transduction can be viewed we can re-use Figures 8 and 9 and relabel the patterns of interconnectedness in them. This re-use of the same visual pattern also has the virtue of reinforcing our awareness of the capacity for different people to see the same things as different things as with the duckrabbit example earlier.

7.5

Figure 8 can be reinterpreted as follows: The same people can be connected in different ways at the same time. In this repetition of the linkage patterns we can envisage the same people are differentially connected along distinct dimensions. For example, (8-a) may represent the association patterns at a weekly golf game and (8-b)may illustrate the same individuals as they are interconnected according to their formal positions in the organisation which employs them all. The connections in one dimension (golf playing) may be employed to influence activities or decisions in the other (economic) setting. The possibility of explicitly and formally modelling the transduction of influence or communication from one dimension to another is an advantage of connectionist models.

7.6

When ideas lead to actions a generic form of transduction has taken place. It is worth remembering that the designation of the dimensions of the phenomena under study (as, indeed, the designation of a 'phenomenon' itself) is most likely a cognitive decision which can be located somewhere along the objective/subjective continuum and not at either pole.

7.7

We can also consider the re-labelling of Figure 9 in the following manner. X,Y, and Z can represent the same elements or individuals. Links A and B can represent transductions which transform an output of one type to an input of another type. People may belong to a religion, an ethnicity and a socio-economic status at the same time. The networks they possess as a result of each of these statuses may allow them to transfer their influence from one to another much as Weber suggested in his 'class, status and power' distinctions for social stratification.

7.8

The three sub-networks depicted can be composed either of the same elements represented in different connectivity dimensions or of different elements in all cases. The first of these possibilities can be used to represent potential 'conflict of interest' problems in corporate and governmental bodies and so on. The second is less likely to be representative of many modern and interlocking social situations but still arises between cultures where the linkage function is that of a broker (or perhaps this role could be rephrased as: a transducer of social relationships).

7.9

One major role of transduction links is to isolate influences and prevent their propagation throughout the network. Chaos in sub-system X might influence via Link A sub-system Y to some extent but have no measurable effect via Link B on sub-system Z. A person's family problems could disrupt his or her own working pattern but have no measurable impact on a large corporation at all. The chaos might be confined to specific dimensions.

7.10

Other kinds of transduction take place when psychological violence is repaid with physical violence, when visual stimuli result in the writing of a poem, and so on. These are examples known to us all but they are difficult if not impossible to encompass within most of the sociological formalisms put forward thus far. A formalism based on connectionism seems to hold the potential for providing this level of theoretical integration.^[17]

7.11

If the previous points on transduction are taken as feasible (and it goes without saying the theoretical justification for what constitutes a dimension is a serious loose end) then how are structures, finally, to be understood?

7.12

Initially we would be compelled to identify a structure as having the following attributes:

1. Qualitative Structural Aspects: confined to one or a known set of dimensions. The various weights each dimension has could be changed to identify subtypes. (This was first discussed in the pioneering work of Hunter, 1953)
2. Quantitative Structural Aspects: for an element to be taken as part of the structure a certain threshold of connectedness must be maintained along one or more of the structure's qualitative dimensions. There is no a priori requirement for reciprocity along the same dimension but some form of feedback must exist and some criteria must exist for determining if the frequency of interactions required for membership is being maintained.

7.13

The first of these requirements allows for transduction (the transformation of one kind of influence into another). The second of these allows statistical (ie. fuzzy) boundaries to be drawn around the structure and also allows it to change through time.

Epistemology

8.1

Earlier versions of this paper prompted some to suggest that the epistemological position being advocated here was best described as 'naturalism' (Nagel, 1961). While it is true that naturalism is enjoying something of a renaissance at the present time (particularly in anthropology) there are a number of reasons why neither this nor any other extant approach to epistemology is being explicitly advocated in this paper.

8.2

First: Naturalism shares with other approaches (eg. materialism, realism, pragmatism, instrumentalism, empiricism, operationalism, positivism) allegiance to the 'scientific method'. This alone cannot be taken as a justification for privileging it.^[18]

8.3

Second: Naturalism's major project is to account for the existence of natural facts in terms of other natural facts (a goal it shares with materialism). The problem arises when no thoroughgoing definition of 'natural facts' is presented. Do they, for example, include ideas (memes)? Some contemporary exponents claim naturalism does but they do not deal with the question of how this claim avoids 'reductionism' which is something naturalism (in some of its guises) disavows.

8.4

Third: As implied in the first point and the second, naturalism shares with other doctrines the status of being an epistemological composite: a collection of positions on metaphysics, methodology, ontology and the role of (human) cognition and potentially others. Given the absence of a compelling a priori reason for accepting the unique blend expressed in one of the variants of naturalism (given that naturalism is itself a family of concepts), supporting or opposing it may be more of a political than a scholarly decision.^[19]

8.5

Fourth: The tenets of what appears to be neo-naturalism may be sufficient to carry out the kinds of investigations in this paper but they do not appear to be logically necessary.

8.6

The spirit of Duckrabbit can be recalled here. When analytic terms, originally created to convey a precise meaning, become part of the vernacular, there is an increasing tendency to use it as metaphor or to generalise it. This clearly happened to 'relativity'. The result tends to confuse much of social 'science' and allow the wooliness and fuzziness for which social sciences have so long been criticised. To some the sole alternative to this is to embrace what many contemporary theorists have called 'positivism'.^[20]

Summary

9.1

The preceding has been a brief overview of the connectionist approach . To determine if it might represent an alternative paradigm it is necessary to point to debates it has the potential to resolve and new areas of research it has the ability to open.

9.2

Chaos theory does not require that everything be chaotic or disordered but only that chaos and disorder must not be excluded in the formal statement of the paradigm. The complexity theorists can on the other hand point to equally compelling empirical observations which could be used to direct people to 'macro' studies. Only through the development of a formalism can we hope to imagine these two approaches will one day meet in the middle. Connectionism, the architecture proposed in this paper because it allowed for chaos to be present, carries with it the potential for extreme degrees of interrelatedness, also allows for complexity to emerge. The chaos/connectionism paradigm offers a technically difficult but intuitively satisfying approach to the analysis of structures.

9.3

By formally including chaos a means is provided for directly addressing the 'reductionist' debate which has been part of social sciences and history for many generations. Because chaos theory is intrinsically dynamic it's implementation also formally removes the need to regard either stability or change as problematic.

9.4

Another consequence would appear to be the need to address the matter of how human cognition is formally involved in social theory. Great advances in cognitive science have been made in the last few years (eg. Edelman, 1992) and any sociological paradigm which fails to appreciate (again, formally) the implications of these discoveries abandons the historical mandate of social theory itself.

9.5

What are we to make of the overworked 'subjective/objective' dichotomy as used by many contemporary writers on social issues? Objectivity seems only to be defined in terms of statistical reliability and hence restricted to those phenomena which are amenable to standardised measurement. All else qualifies as subjective: a matter of opinion.

9.6

By explicitly including human cognition into the formalism, chaos/complexity directly addresses this dichotomy and presents us with something else. Something which is at once both intellectually more appealing and methodologically more intractable. Subjectivity and Objectivity are not dichotomous opposites but are opposite ends of a continuum: the continuum we use to represent the interaction of human cognition with the world it is able to perceive. Chaos theory sends us back into Socrates' cave.

9.7

The need to include cognition resolves another problem. Structuralism has often been twinned with Functionalism to produce the hybrid known as Structural-Functionalism. The major justification for this approach was the dual questions of 'what it is' and 'what it does' when applied to the analysis of social institutions. This distinction, while still eminently practical for applied sociology, is now seen to be different ways of perceiving the same phenomenon (much as light can be either wave or particle) and contingent upon what the applied objectives (or teleology) of the analysis are. Within the connectionist paradigm the distinction between structure and function needs to be reconsidered.

Conclusions

10.1

Without Shakespeare there would be no Hamlet, but without Newton there would still be gravity. It is perhaps for this reason that Postmodernism has enjoyed its greatest success in literary criticism and none at all in physics. Even if one were to assume Newton's discovery of gravity was tainted with Eurocentric and Patriarchal influences, surely by now no sane person would choose to regard gravity as 'a matter of personal opinion' or being influenced by it the consequence of a 'false consciousness' imposed by some ideological hegemony.^[21] What we call gravity works whether or not people know the formula or believe it. This is not to suggest that Newton's specific formulation of how precisely gravity works is to be accepted as dogma. It is only to reaffirm the idea underlying all of science; that Truth exists and that the scientific method will, if followed, bring us asymptotically closer to grasping how it manifests itself in that part of the material world we are able to apprehend.

10.2

Given that the butterfly has flapped its wings that extra beat^[22] there are now 17 scenarios which will be possible, all of which include tornadoes in Kansas in 73 days time. Two of them will move Dorothy's house to Oz. Only one of them will happen.

10.3

The idea other timelines may exist is untestable and as far as we are concerned pure science fiction anyway. What chaos theory also does is force us to revisit the ideas of structure, the boundaries that separate the social science disciplines themselves, and the roles of psychology and human consciousness as central opportunities for theoretical advancement.

Notes

¹ The more conventional problem, the one that understands 'change' as a transformation which takes place in or through time, is very hard to reconcile with this definition of structure. Studying change, too, is greatly complicated since this definition of structure mandates the use of cross-sectional data and makes it necessary to 'bring time back in' much as humans had to be championed as theoretically meaningful a generation ago. If chaos theory did nothing else, its dynamical nature forces researchers to incorporate the passage of time directly into our formal models.

² Ethnicity is not the only area of sociological theorising which contains terminology which is, at best, overly general. Virtually everyone in society realises that what is called 'the family' has been in flux for several centuries. Attempts to articulate a formal sociological theory which logically (ie. in its 'formalism') allows change to be an integral component is seldom if ever seriously discussed.

³ meme: (pron. 'meem') A contagious idea that replicates like a virus, passed on from mind to mind. Memes function the same way genes and viruses do, propagating through communication networks and face-to-face contact between people. Root of the word 'memetics', a field of study which postulates that the meme is the basic unit of cultural evolution. Examples of memes range from melodies, icons, fashion statements and phrases to religious ideas, political ideologies, sexual mores, health beliefs and so on. The abstract entity manipulated by memory is known as a mnemon (Gabora, 1996).

⁴ The reliance of this on the 'subjective' or more accurately on the species-universal-but-relatively-subjective idea of what 'too rapidly' might mean is useful. It will allow us to begin formalising what is meant by the 'subjective/objective' distinction.

⁵ By 'easily' is meant 'intuitively obvious'. The actual mathematical models involved are extremely computation-intensive and computer technology well in advance of anything currently available would be minimally necessary to do any serious social modelling attempts.

⁶ Determining the dimensions for the soliton wave is crucial if this is to be more than just another metaphor misunderstood by and perhaps misleading to sociologists.

⁷ It is noted that almost all sociologists claim to believe this. The point is, though, that these phenomena must be accounted for by the formal logical relationships amongst the taxonomic entities of the paradigm under study (by 'the formalism' to adopt the usage from physics). In this terminology one might restate Homans' criticism as saying the formalism of structuralism has no real place for human consciousness.

⁸ There are at least two implications which result from this. With respect to classical 'structural-functional' sociological theory it means: the micro-macro distinction is without coherence. Within the realm of ethical theory it means any attempt to justify 'situation ethics' in terms of such arguments as 'the greatest good for the greatest number' (a form of consequentialist argument as discussed by Scheffler, 1982) is without any hope of being demonstrated empirically.

⁹ Pipes' theory is not alone in suggesting a so- called minor event leading to the Second World War and beyond. Calvin (1983) argues that President Woodrow Wilson's participation in the Paris peace conference of 1919 was drastically compromised by a bout of influenza. Prior to this incident his attitude towards Germany was to avoid humiliating it and indeed to include it as a member in the League of Nations. Following his illness he treated German delegates with contempt and was not a factor in preventing the imposition of the harsher features of the Treaty of Versailles.

Many feel that some of the consequences of Wilson's illness outlived his presidency ... and were to be seen in the events of the following decades - in the disastrous German inflation of the 1920s during the reparations exacted by the Allies, in the ensuing reaction to social disorder which led to the rise of the genocidal Nazis, and in the second world war. (Calvin, 1983: p. 124).

Juxtaposing the two thought experiments of Calvin and Pipes does two things: it illustrates how small changes in the initial conditions in an historical period might lead to radically different outcomes and, second, it draws us back to the question of system boundaries - of taxonomy - once again. For many reasons which are clearly legitimate one could in 1918 and again in 1998 regard Russia, Germany, and the USA as separate from each other. For other but equally legitimate reasons they must be seen as parts of larger entities. It is partly to deal with this exigency that 'connectionism' is introduced later in this paper. A final observation which can be made involves the realisation that despite the fact that historians often argue that 'laws of history' are demonstrably impossible as evidenced by the above complications, it is also the fact that some of their demonstrations do so by invidiously (in some cases) employing 'sociological laws' to reveal that more than one logically possible future scenario is possible. It is to be noted that the claim 'there are no laws of history' is epistemologically distinct from 'the laws of history do not yield a single and inescapable future scenario'. To discuss 'plausible futures' is to employ theory. For example: Had Wilson's health lasted a few more months and had Fannie Kaplan's memory been better how many of us alive today would ever have been born?

¹⁰ If the phase-space is sufficiently differentiated to allow for a single period longer than the effective of projected age of the universe then an interesting result is obtained. We may not be able to discriminate between a deterministic and a voluntaristic (free-will) condition for human existence.

¹¹ There is a similar apparent irony in the doctrine of free will. Free Will is meaningless in a non-deterministic universe because it is the very determinism of the universe which permits on of the most common ways to exercise free will: that of selecting a course of action because of its consequences. The statement 'I choose to do X because doing X will cause Y to happen' contains both voluntaristic and deterministic elements.

¹² The reductionism mandated by chaos theory derive from the requirement of a change or flux needing a matrix within which to manifest itself. The large-scale structures as studied by complexity theory are themselves rooted in the microstructure. One can at least speculate if the laws which may dictate the formation of 'social structures' are similar to 'brain structures'.

¹³ In more stilted language, the flux pattern is described in phase-space while the lattice exists in space alone. We leave to future time the discussion of the permanence of the lattice itself. People are just as transient as sticks and stones if divorces time from the system. (eg. Bohm and Hiley, 1993; Newton, 1993). For those who are unsure what is meant by 'phase spaces' and 'phase states' it is perhaps better to consult Gleick's (1988) extensive treatment of this idea. Schuster (1995) gives a more compressed discussion of the same ideas.

¹⁴ If the path is really finite then true chaos is impossible. If, however, the projected age of the universe is insufficiently great to allow even one iteration of the path then even though determinism was strictly present it might be impossible to detect this fact. This might, by extension, allow for the unassailable defence of a doctrine of free will, even if not one which is identical with the classical understanding.

¹⁵ The initial definition of structure makes the assertion that social structures do not exist in the same way concrete objects such as sticks and stones do. One is prompted to ask 'are human beings concrete objects or abstractions from concrete empirical phenomena'?

¹⁶ While Nagel's explanation of how the ideal gas law can be derived from the kinetic theory of gasses is logically satisfying it is less adequate as a sociology or psychological comment. Boyle's law relating temperature to pressure was postulated in 1662. Atoms had been proposed by Democritus (c460 B.C. - c370 B.C.) and some rudimentary concept of molecules also existed but at the same time both alchemy and astrology were considered scientific. The actual ideal gas law was generally accepted in the 1850s but this was almost one-half century after Avogadro published it in 1811. When Nagel wrote of the means by which the gas laws could be derived from the kinetic theory of gasses he lived in a world which accepted the existence of protons, neutrons and electrons (but not quarks).

¹⁷ Qualitative studies and their exponents (eg. Blumer) have often exhibited their greatest analytic strengths in documenting precisely these transitions. Those engaged in field research have long known of the limitations of formal structural theory.

¹⁸ All sciences must share the same methodological precepts even as they focus on different aspects of the natural world. Physics, biology, and chemistry therefore all share overarching notions of 'objectively collected data', of the formulation of testable hypotheses which are evaluated in accord to universal rules of logic and evidence, and the formulation of theories which generalise or extend observed patterns into areas which have not yet been observed. (This ability to extrapolate is intimately tied to the question of dimensions). 'Scientific method' at its core apparently means, reliance on logic, the use of empirical evidence, and the supremacy of evidence in case of conflict.

¹⁹ In a culturally more homogeneous age it made some statistical sense to refer to a 'family' as 'two adults of opposite sex, married to each other, and living with their common children'. While it may be politically advantageous (correct?) to extend the meaning of 'family' to include same-sex couples, single-parents, nannies, adoptive and step children, and so on, it is clear that this diversity plays havoc with sociological analysis. When a composite term is seen to be made up of elements which are varying essentially independently of one-another, scientific practice requires the composite term no longer be accorded primary theoretical, practical or research importance. After the tea kettle has boiled dry, using the term 'ice' or using predictions based on observations of ice are likely to be of diminished importance. The same can be argued about composite epistemological terms as well.

²⁰ Positivism though, like liberalism, came to mean different things to different people. When Auguste Comte first proposed the Positive Philosophy he sought to study society with the tools of the natural sciences, forgoing both metaphysics and theology, in the quest for ways to make the lives of people better. By the time the critical theorists of the current period had arrived, positivism had come to be associated with the disenchantment of the world and the imposition of Western Rationalism on all and sundry. Positivism, like so many other sociological doctrines, is both a research methodology and a political promise. Many modern scientists attempt to avoid the Janus-headed positivism of Comte and restrict their interpretation to it's methodological role only: 'That philosophical attitude which asserts that science is solely concerned with directly measurable quantities and that its aim is simply the harmonious reconciliation of observations without attempting to speak of an underlying physical reality.' (Polkinghorne, 1986: p. 108)

²¹ No insult to either members of the Natural Law Party or Post-Modernists is intended here.

²² This assumption is more deceptive than it might first appear. The model proposed here fundamentally deals with an attempt to grapple with how things change once they have been set in motion. It does not deal with why or how they were set in motion in the first place. It is therefore not a model of cause and effect. It is only an attempt to model the flux of the real.

Acknowledgements

This paper is based on one presented at Essex '96 Fourth International Social Science Methodology Conference, July 1 - 5 1996.

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