|Finn Sivert Nielsen||Anthropologist|
In Worlds of Mirrors
Models of Complex Societies and Anthropological Complexity
by Finn Sivert Nielsen
Chapter 6. Pyramid and Cloud
Chaos Theory as a Metaphor in Social Science (1989)
This chapter will bring together several themes, but is perhaps most fundamentally a statement about choice. It is clear to start with, that choice must have a central place in the social sciences. Society is nothing but human action and all action is somehow linked to choice. This would seem a self-evident truth. Nonetheless, like so many other truths it is often forgotten – or if not, lost among a multitude of other interesting and important issues. This occurred in Chapter 5, in which a variation of historical determinism was implicitly evoked, and the options remaining to the individual were seriously curtailed. The following programmatic quote from Fredrik Barth (1966) – a theoretician very explicitly concerned with choice – illustrates a similar dilemma:
"[T]he behaviour of any one particular person [cannot] be firmly predicted – such human conditions as inattentiveness, stupidity or contrariness will... be unpredictably distributed in the population... Indeed, once one admits that what we empirically observe is not 'customs', but 'cases' of human behavior, it seems to me that we cannot escape the concept of choice in our analysis: our central problem becomes what are the constraints and incentives that canalize choice." (p.1)
The unnoticeable way in which this statement evades the issue it purports to address is symptomatic. Human beings are unpredictable, Barth says, so choice must be important. His conclusion however, is not that we should study choice itself, but the "constraints and incentives that canalize" it.
It is easy to sympathize with this stance. Without "constraints and incentives" on the individual, our image of society would be fragmented and chaotic. Naturally this would complicate matters for social scientists, or even – if taken in full earnestness – make our vocation entirely futile. Without assumptions of this kind it is hard indeed to evade the common-sense sociologist bashing that a critic like C. S. Lewis indulges in:
"That's what happens when you study men: you find mare's nests. I happen to believe that you can't study men; you can only get to know them, which is quite a different thing." (1946, p.71)
Social science has traditionally defined its scope of inquiry narrowly, a self-limitation or even humility, which is in many ways admirable. Barth and many with him might reply to Lewis that sociologists and anthropologists do not study "men", but "men's" acts; or not even their acts, strictly speaking, but the "observable frequency patterns, or regularities" of their behavior (Barth, op.cit. p.1). It is not that the existential or psychological problems faced by individuals do not concern us, but we do not pretend to understand them better than other human beings.
But the question remains: can we study choice if we do not know what choice is? And can we investigate the nature of choice without being driven inexorably into the existential "mare's nests" we wish to avoid? Consider for example the dilemma formulated by Dag Østerberg:
"Every act is purposeful, but not in the sense that it is determined by some definite, clearly preconceived purpose. The act makes manifest the purpose, which was not 'at hand' as a finished plan prior to the act. As far as spontaneous action is concerned we must instead say that the plan changes, indeed, that it only acquires form in the process of execution, so the act makes known to us what the purpose was." (1963, p.15; my translation from the Norwegian.)
If this is true, what is then choice? How can we catalogue the "constraints and incentives" influencing choices if there are no delimited motives "behind" them? Indeed Østerberg's statement suggests that acts are often not primarily concerned with "getting things done" at all, but with "learning what things are". The act is a "tool", but a tool of perception as much as an instrument of change. Or perhaps even this (which was the central preoccupation of Chapter Three) is insufficient. Perhaps – and more importantly – the act may best be defined as a means of connecting us with the outside world and enabling us to handle this connection (as suggested in Chapter Four). As Simone Weil tersely puts it:
"Tool: a balance between man and the universe... [T]he sailor in his boat balances equally against the infinite forces of the ocean. (Remember that a boat is a lever.) At every moment the helmsman – by the weak, but directed, power of his muscles on tiller and oar – maintains an equilibrium with that enormous mass of air and water. There is nothing more beautiful than a boat." (1970, p.20)
If the act is a balance, a mode of connection between the very great and the very small, does it not follow that choice is the same as skill? If so, Barth's "constraints and incentives" take on a new meaning. They are that which is balanced, the forces within and the forces without. Indeed, every choice then implies a problem of locating a point of equilibrium, a point where balance, and hence action, become possible.
Furthermore, as the last quote shows, balance is not always – or perhaps it is never – a question of the relative strength of the forces involved. Choice is not primarily an assessment of powers but of placements and patterns. It is where the oar is applied, not the force wielding it, which determines the sailor's skill. What then of the aggregate consequences of choice? How can we be so sure that "the regularities in social life... have to do with the repetitive nature of acts... [i.e.] frequency distributions around a mode" (Barth op.cit., p.1)? Perhaps similar constraints do not produce repetitive acts, perhaps every choice is new, every act original, and the "equilibrium" of the total system constantly changing its emphasis and nature?
The concepts here broached – pattern, balance, repetition and choice – have lately acquired a new dimension through the combined efforts of researchers in seemingly unrelated fields such as climatology, population dynamics, fluid physics, cardiology, macro-economics and mathematics. In fact James Gleick (1987) suggests that we may be seeing the birth of a new trans-disciplinary science – "Chaos". In a narrow sense Chaos is a mathematical theory whose merits (as a tactful mathematician reminded me) I am in no position to assess. But social scientists have a long-standing tradition of appropriating ideas and concepts from the "hard" sciences and using them for their own purposes. Nineteenth century anthropological notions of social evolution were based on a parallel to biology. Functionalism, which supplanted evolutionism in the inter-war period, replaced the metaphor of society as a species with that of society as an organism. Biology resurfaces again in modern ecological analyses, but other disciplines are perhaps more common sources of inspiration – mathematics (in Cybernetics and Games Theory), linguistics (Structuralism, Semiotics), law (norms, rules), even dramaturgy (Role Theory). Mathematical language has had particular attraction. Terms such as "level", "equilibrium", "feed-back", "aggregate", "logical hierarchy" are by now so ingrained in the anthropological vocabulary that we would be hard pressed to think productively without them. This eclecticism has had profound and contradictory effects on the way social scientists think. Societies are unpredictable and immensely complex, and terminology derived directly from analysis of their complexity tends to be either vague or unsuitable for generalization. Imposing the purified theoretical structures of "hard" science on this material has undoubtedly had a disciplining and clarifying effect. But society is simply too complex to fit the concepts in question. Rigorously defined terms must be used loosely and intuitively or not at all – they indicate rather than define, suggest rather than prove. Strictly speaking, they no longer function as concepts at all, but as metaphors.
Personally I think this process is inevitable, though it is naturally a source of irritation to many mathematicians. But at the same time it is true that social scientists often lose sight of the constraints inherent in the metaphors they absorb. Mathematics are concerned with ideal types and pure states, which do not even have approximate correlates in social reality. To apply mathematical language usefully we must therefore eliminate complexity and compress variation wherever possible, and this of course implies a radical distortion of the reality under study.
In this perspective the metaphors of Chaos may have a liberating effect. Chaos is concerned with extremely complex systems – the infinite variability of weather, the irregular beauty of mountain-ranges and leaves – "turbulent" systems which the mainstream "hard" sciences have hitherto been unable to describe satisfactorily. The formal language in which these phenomena are discussed has little direct relevance for my argument – it is the "shape" of the geometrical patterns they produce that is at issue. We should take note however, of the complexity of these images themselves. Chaos theory is still dealing with idealized worlds, but its idealizations are subtler, and hence closer to social reality, than those of classical mathematics. In effect, Chaos theory allows us to see our own established mathematical imagery in a new light. If the purity of classical mathematics misleads us to assume that society has a similar purity – that our adopted metaphors relate to reality as a one-dimensional line to a pencil scrawl – then Chaos prompts us to use the scrawl itself as a model. As a magnifying glass shows, the pencil leaves a mark, which is neither one-dimensional nor clearly bounded, but rough, patchy and irregular. Until recently such "fuzzy", "natural" shapes were not considered amenable to formal description. In classical physics they were represented by geometrical "equivalents" and the resulting deviation was corrected for as experimental error, random deviation or "noise". Lately however, one has realized that such deviations are highly cumulative in many complex, interactive systems, and that even the most meticulous corrective procedures therefore distort our understanding of their dynamics fatally. Chaos theory springs from this realization. Climatology was one of its original applications.
Chaos is a geometry of the bewilderingly complex "order" of natural shapes and processes such as coast-lines, weather, swirling water or clouds. As in the case of social "forms", we grasp intuitively that such phenomena are not entirely arbitrary – yet all attempts at describing them in statistical or quantitative terms have failed. Chaos does not succeed at this either, or at most succeeds only partially and in exceptional cases. It has however, supplied mathematical expressions that "imitate nature" in a surprisingly accurate fashion – creating synthetic videos of "real" landscapes, predicting mal-functions of the heart, modelling the greenhouse effect, the arms race or artificial intelligence. These attempts depend on a two-sided process. On the one hand one seeks insight into the "shape" of the empirical reality one wants to imitate, on the other one looks for a mathematical expression which "resembles" this shape, and which may then be assumed to "predict" aspects of it that are not open to empirical investigation. In the task of coordinating these two lines of inquiry – of "fitting" mathematics to reality – one has recourse to more or less clear-cut guide-lines. Still, one depends greatly on trial and error. It is a fact, for example, that Chaos research is dependent on high-power modern computers with the capacity to calculate and represent graphically many alternative hypothetical "imitations", each of which, perhaps, involves billions of calculations.
One of the earliest empirical systems to be recognized as chaotic was the climate, and as an example it may teach us something of the nature of such systems in general. Several things are noteworthy about weather. First, it obviously follows certain cycles, and stays, even in the long term, within more or less definite boundaries. Phenomena such as trade-winds and monsoons, seasons and ice-ages, day and night, cold spots and arid belts clearly demonstrate that weather is not random. These regularities may be treated statistically, and there is clearly a lot to learn from this. However, as any victim of weather-forecasters can testify, even the most sophisticated measurements and analyses do not enable us to predict what the weather will be like more than (at most) a few days ahead of time. On the face of it this seems strange. The physical laws governing weather are simple and well known, as are the main factors contributing to climatic variation. But in spite of this meteorologists have recently realized that they will never be able to increase the reliability of prediction significantly with the methods currently in use. The reason is referred to as "sensitive dependence on initial conditions", or – in more colorful language – the "Butterfly Effect".
"Sensitive dependence on initial conditions" may be illustrated in the following way. In any system of interacting elements, each interaction will (directly or indirectly) influence and be influenced by all others. The cumulative effect of all interactions determines the state of the system as a whole – e.g. (in a rather minimalist model of weather) the pressure of an enclosed gas against the walls containing it is an aggregate effect of billions of impacts between the molecules of which the gas consists. Individual molecules may move around in the container with vastly differing speeds, but on the aggregate level their differences cancel each other out, so the pressure of the gas as a whole may be measured as a unitary value. This is only true however, as long as the gas is enclosed. The walls of the container act as a "simplifying assumption" which overrides all random variability. They eliminate the complexity of the aggregate, by so to speak focusing all its energies onto themselves. To give a more realistic picture of weather the walls must be removed. We may observe the smoke rising from a cigarette to appreciate the result. At first it acts in a way analogous to the enclosed gas. Its heat relative to the surrounding air overrides all other facts about the smoke, and as a result we see it ascending in a more or less straight column. Then suddenly it starts eddying and swirling around itself in intricate and ever-changing patterns. The many forces acting on each particle of smoke are now, on the average, more or less equal. No single factor predominates, and for this reason the simplified description of the smoke's behavior no longer applies. The interactions between individual particles accumulate or neutralize each other at random, sometimes forming a bit of one kind of order (a left-handed swirl), sometimes another (a right-handed swirl), sometimes no order at all. No influence is too small to determine the boundary between such states. The movement of a single particle, or even a single electron, may have enough leverage to change the entire smoke pattern. Consequently, (meteorologists say) the flap of a butterfly's wings in Singapore today may determine the weather in New York tomorrow.
In systems such as rising smoke or weather one of the most fundamental assumptions of applied physics is thus negated. In the enclosed gas it was meaningful to speak of energy existing on different levels of scale. What happened on the micro-level was irrelevant to the macro-level – where all individual variation was evened out. This is no longer true in the case of the rising smoke, where individual events on the level of quantum physics may "cascade" up to aggregate levels and determine their state directly. I shall refer to empirical systems of this kind as "non-scalar". A non-scalar system is an aggregate, which is not dominated by any single overriding factor or consideration. For this reason the action of any one of its component parts may influence the state of the whole directly and unpredictably. Many real systems – in particular biological and social systems – are obviously to some extent scalar. The human body would disintegrate if random variations in the functioning of individual molecules could affect its total state. A society, which accorded unlimited influence to each individual act, would dissolve into incoherent babbling. Overriding considerations, which focus and neutralize variation at certain points, are clearly necessary to survival. But living systems are equally dependent on flexible adaptation to variation, on assimilating and utilizing randomness creatively. If the mind, for example, were a purely scalar system, it would be as rigid and compartmentalized as a filing cabinet. In living systems I shall therefore assume that scalar and non-scalar organization coexist, creating aggregates which are at the same time functionally specific and flexible, uniquely themselves and open to adaptive change.
The key factor in mathematical simulations of non-scalar, empirical systems is the notion of "self-similarity". This trait is exhibited strikingly by the geometrical figures called fractals, which "look the same" on every level of scale. A simple example is a cross, with crosspieces on all four arms, which again have crosspieces, etc. Far more complex is the so-called Mandelbrot set, a weird shape which may be magnified indefinitely to reveal infinite vistas of detail and beauty. While the fractal cross replicates itself exactly on each level of magnification the Mandelbrot set is not "self-similar" in this trivial sense. It is transformed unpredictably, but nonetheless systematically, as we move from level to level.
Self-similarity seems at first sight little more than a decorative and useless trick of geometry. It becomes interesting however, when we realize that many phenomena in nature resemble these exotic shapes. In swirling cigarette-smoke, the visible curls and eddies are repeated on every level down to that of the individual particle. A coastline is indented on every level – there are inlets and promontories down to microscopic dimensions. Clouds consist of billows within billows. The blood vessels in the body split and re-split in a similar way – as do the roots of trees. In a different area, prices on the world market exhibit fluctuations within fluctuations – yearly, daily, hourly. Static on telephone wires, population-levels among tadpoles, fluttering heartbeats, the orbits of galaxies... self-similarity is endemic in the most diverse areas.
The significance of this observation becomes apparent when we consider that the infinite complexity of fractal geometry is produced by extremely simple mathematical manipulations. Fractals are generated when certain functions are allowed to act "recursively": An initial input is processed, the resulting output treated as input and re-processed and so on, in an interminable chain of iterations. An example is a figure developed by the astronomer Michel Hénon. His formula is simple: xnew = y + 1 – 1.4x² and ynew = 0.3x. The expression is recursive – each time values for xnew and ynew are calculated, they are inserted as x and y in the next iteration of the function. Consequently a long string of values for x and y is accumulated which may be mapped as points on a two-dimensional graph. The result is seen in Figure One. As a whole this figure rather resembles a distorted ellipsoid. But if we zoom in on any part of it we discover finer and finer detail – every bit of it, no matter how small, is as complex as the whole, the same stretched and folded structures recur on all levels of magnification. What looks like a simple line on one level breaks up into multitudes of curved and folded lines on levels below and so on – quite literally – ad infinitum, as the section in Figure Two suggests. What makes this incredibly fine-grained self-similarity all the more striking is that it contains absolutely no repetition. No two points ever occur in the same position, no sequence of points ever recurs, no matter how long the function is iterated. When the figure builds up in front of you on a computer screen the effect is eerie. The points are deposited on the screen completely at random. A sequence of say 9 points (as those numbered on Figure One) seems totally haphazard, and gives not the slightest indication of the structure of which it is a part. It goes without saying that linear analysis of the recursive function reveals nothing of its pattern.
What does this figure "mean"? The question may be answered in several ways. First, we might ask how the fractal relates to empirical reality. Hénon was searching for regularities in the orbits globular star clusters. Such clusters may be envisioned as cosmic parallels to swirling tobacco-smoke – each star's orbit is constantly influenced by all other stars in the cluster, and no overriding factor focuses their motions into a unitary aggregate. The cluster as a whole is "shapeless". Hénon looked for a mathematical analogy to this situation. His starting point was a formula expressing an ellipsoid – the prototype of any orbit. He then inserted factors in the formula, which distorted the ellipsoid by twisting and folding it in a manner similar to the distortion of a real orbit by other stars. When the formula is now iterated the ellipsoid thus goes through an infinite series of deformations, replicating all possible orbital patterns on a graph. The very approximate "fit" between mathematics and astronomy in this case are typical of Chaos research. The fractal imitations of nature are – as mentioned above – produced by guesswork, and trial and error. Nevertheless, Hénon's function does give a better approximation of reality than can be obtained by other means.
On an abstract level Hénon's figure illustrates a general principle, which seems to emerge from Chaos research. Self-similar mathematical configurations may be used to model a great variety of objects and processes. What is almost unbelievable is that no matter what is modelled – economy, biology, astrophysics, weather – all self-similar systems have definite formal traits in common. One striking indication of this is that the "fit" between empirical reality and mathematical formulae has in a number of instances been shown to be almost perfect. The Mandelbrot Set is thus derived from a formula as simple as Hénon's but without even a minimal connection with "reality". Nevertheless, the exact same shape has emerged in empirical experiments, e.g in the detailed filigree work of magnetized and non-magnetized fields in metals. One of the most suggestive results of chaos research has been the discovery of such universality. It indicates that we may have to acknowledge the existence of patterns – indeed highly complex and intrinsically ornamental patterns – which not only evade statistical and quantitative analysis, but are independent of the content and functions of the systems they organize. It seems, in other words, that our common-sense distinctions between physical, biological and social systems are challenged. Non-scalarity thus has a wider meaning than the erasure of distinct levels of scale. We are dealing with systems, which have no inherent qualities at all except that of pattern. The ice crystals forming on a window pane have the same shape as leaves and ferns.
The above discussion supplies us with a number of elements from which we may construct a metaphor applicable to social science. Let us start with the notion of self-similarity, which is the basis of fractal geometry and may be recognized in many empirical systems as well. In real systems however, self-similarity is not merely a matter of shape, but an indication of non-scalarity. If an object "looks the same" on all levels of magnification it cannot be thought of in terms of discrete levels of scale. "Sensitive dependence on initial conditions" will prevail in such systems. In social situations "cascading" from micro- to macro-levels will be common. An individual choice may change the face of society completely. Reinterpretation of a single concept may transform entire structures of meaning.
What empirical circumstances are conductive to such conditions? To answer this question I must indulge in some rather far-fetched analogies. Indeed, from this point on our discussion gradually leaves even the pretense of mathematical verisimilitude behind, and enters the realm of pure metaphorical speculation. A comparison of mathematical and empirical Chaos is a good starting point. Fractal figures are generated by certain types of recursive functions. In real systems recursiveness may be thought of as a parallel to the interaction between the system's elements. In Hénon's example recursive formulae model the mutual influence of stars. In cigarette smoke self-similar swirls are produced by interaction between individual particles. But cumulative interaction does not in itself produce Chaos. If an overriding factor focuses the dynamics of the aggregate Chaos is avoided – as in the enclosed gas. In social systems we should therefore be prepared to find non-scalarity in situations with intensive interaction between individual actors or elements of meaning, in the absence of overriding principles, which might focus this interaction (e.g. effective power structures or a consistent moral codex).
If parts of society may indeed be thought of in these terms, we should be prepared to draw a further analogy to the systems examined above. Empirical cases of non-scalarity are governed by the same organizational principles regardless of what they organize. If there is chaotic order in society it would therefore have to be envisioned as a kind of "ornament" – an aesthetic rather than functional order. It would be pointless to analyze this order in terms of "constraints and incentives that canalize choice", since there is no functional relationship between individual acts and the whole of which they are part. The impact of this conclusion is momentous, particularly since it is precisely in such chaotic and non-functional situations that we would expect individual acts to exert the greatest and most direct influence on the system as a whole. We might, for the sake of illustration, think of the power-vacuum of inter-war Germany as conductive to non-scalarity. The rise of Hitler might in that case have to be explained as a result of aesthetic rather than functional processes. The class-interests of his supporters, the economic and political aspects of his policy would be secondary factors, while his utilization of traditional European symbolism for rhetorical and architectural purposes would be primary.
Be this as it may, a last element in a social model of Chaos must now be mentioned. When speaking of organic or social systems – i.e. systems, which are in some sense goal-seeking – I shall distinguish between non-scalar processes and non-scalar structures. Many physical processes – as weather, turbulence or static – are spontaneously non-scalar, and cultures and organisms must learn to adapt to their variability and utilize it. Non-scalar structures in our bodies such as neuronic or capillary networks are designed to contain, control and make use of such spontaneous non-scalarity. The same obviously goes for non-scalar structures in culture. For this reason organisms and societies are never uniformly non-scalar. A living thing has explicitly scalar aspects with a limited and precise range of functions, intertwined with patterns which are non-scalar and aesthetical: The flatness and thinness of a leaf are designed to afford maximum surface for photosynthesis, while the leaf's outline is pure pattern. Nevertheless, the two aspects are closely knit together and influence each other. For though a mind or culture may in certain respects be highly non-scalar it must always assign overriding priority to some functions and routines. Distinct scalar values are forced through in spite of the Butterfly Effect. In society scalar and non-scalar structures thus co-exist and limit each other.
In order to give substance to the idea of non-scalarity in social systems I shall now discuss a series of rough analogies between physical and mathematical representations of Chaos and phenomena related to human interaction and meaning. The reader may object that my parallels are too intuitive to give substantial knowledge about society. As I have emphasized above however, my point is not to apply Chaos theory to society, but to stimulate a different kind of thinking within social theory itself.
I shall start my discussion with the by now classical terms "analog" and "digital" information. These concepts derive from cybernetics and allude to different modes of encoding data – either along a continuous "more-or-less" scale (analog) or by means of discrete "on-and-off" states (digital). A wheel spinning faster or slower may be an analog e.g. of numeric quantities, as may a slide-rule, thermometer, or measuring cup. A light switch or typewriter, letters, numbers and outlines are all digital codes. Each of the two modes has its advantages and disadvantages. An analog code expresses more nuances and contains much more information. Digital codes are more exact, less liable to distortion by noise or static and easier to memorize. Most human computers are therefore digital, but there is general consensus that biological and social systems depend on both modes of coding. In the brain, for example, neurons respond to other neurons "if and only if" they fire (digital coding), but firing is only interpreted as such when certain chemical processes have built up "to a certain level" (analog coding). In social systems, as demonstrated by Roy Rappaport (1968), ritual may work as a digital "switch" regulating continuous processes of demographics and adaptation. In general it seems that digital coding is widely utilized to "govern" analog codes – e.g. to switch processes on and off or direct or "frame" them in specific ways (Wilden 1972, pp.155ff).
Let us now imagine some minimal unit of meaning – an "idea". This may be envisioned as a digital "outline" delimiting an analog "content" – a discrete unit born on a continuous substratum. Each time the outline is "applied" (by a culture or in the mind of an individual), it is "copied" onto a new bit of the analogic continuum. But as Anthony Wilden (p.169) points out this implies a paradox. The outline itself may be copied exactly, but the continuous segment it delimits is never exactly the same. Any real outline has a certain "width" and is therefore surrounded by an area of ambiguity as to what it includes and excludes. The idea will therefore in a digital sense always be "the same", but in an analog sense it will change – in fact, in creative thinking such change is not only inevitable but highly valued. Moreover, even if the change attendant on each "re-copying" is very small, it is cumulative. The ambiguity introduced by the first "copy" will increase the ambiguity of the next, and so on.
This description is roughly parallel to the non-scalar systems we have been discussing. The digital outline is similar to a mathematical function or physical law. Each time it is "re-copied" it defines a short-lived but static bit of meaning, analogous to one of the points plotted on Hénon's graph. The analogy is strengthened by the fact that each "re-copying" defines a point of departure for the next. It is, in other words, recursive. As long as no overriding factor restricts it we may thus expect any "idea" to start transforming itself into a non-scalar aggregate from the moment it is put to use. Each iteration inserts an unpredictable element of change, which may "cascade" up to the level of the "idea" as an aggregate whole and influence its meaning directly.
Even elementary units of meaning may thus be thought of as having chaotic properties. This affects our way of thinking about communication and choice profoundly. We often imagine a mind or culture as an archive of ideas from which we may retrieve any item we choose. Each idea is a kind of "magnet" at which our attention is arrested if we "search" for it in the right way. But how do we focus on an idea if it is not a fixed point, but as intangible as Hénon's graph? How do we single it out among others?
A simple physical experiment may clarify this point. It demands three magnets, placed at the points of an equilateral triangle, and an iron pendulum freely suspended above the triangle's exact center. We may think of our attention as the pendulum and one of the magnets as the idea we wish to arrive at. Each magnet is an "attractor" – a mathematical concept describing the state towards which some cumulative process ultimately tends. The attractor of a pendulum without magnets is simply a point – after a while, the pendulum stops. If the pendulum is kept going indefinitely with the same amplitude, the attractor is a circle – a motion first in one direction, then in reverse back to its origin. What is then the shape of the attractor of each of the three magnets in the experiment? We might determine this by dropping the pendulum from different points in the area around the magnets, waiting to see which magnet it ends up at, and plotting this point in a color specific to that magnet. Repeated experiments will then reveal the shape of each attractor as a different colored field – e.g. red, blue and green. The result of the experiment is rather unexpected: The area immediately surrounding the "red" magnet is – as we might suspect – uniformly red. But the closer we get to the mid-way point between two magnets the more complex and interwoven the colors become. There are uniform fields in the boundary zone too, but these are bordered on all sides by smaller fields of all three colors, which are again surrounded by still smaller fields etc. The attractors' boundaries are fractal and non-scalar – they have infinite complexity on all levels of magnification. When the pendulum is dropped from somewhere in the boundary zone, even the smallest imaginable displacement of its point of release will thus "cascade" – i.e. lead it a different magnet than the one we thought it was going to.
We may now return to the pendulum as a metaphor of our attention, trying to arrive at a choice in a moronic mind containing only three ideas. Several situations may be envisioned:
1. Our attention may start out at a point "close" to one of the ideas, or – to translate into more relevant terms – we may know exactly what we are looking for and where to find it. In this case we will most likely arrive at the intended point. Our choice will be predictable and "functional".
2. Alternatively, there may be some overriding factor in the mind, which neutralizes or weakens all but one idea. One "magnet" may be stronger than the others, or an external force may force the pendulum towards it. In this case we choose under (external or internal) constraint.
3. Finally, we may start in the border-zone between more or less equal "magnets", i.e. our ideas of "what we are looking for" may be ambiguous – as is often the case. Even the slightest displacement of a magnet – the subtlest change in the ideas we are drawn between – will then lead to unpredictable choice. And since as we saw above, ideas are not like magnets at all, but themselves non-scalar and constantly changing, such displacement is unavoidable. Our attention is therefore suspended between "strange attractors" – neither points nor circles, but infinitely complex, non-repetitive fractals like Hénon's graph. Choice then becomes a non-scalar and chaotic process, and the smallest imbalance in the "constraints and incentives" determining it will affect the outcome unpredictably.
Thus, under certain circumstances (perhaps often, since the fractal border-zone between magnets is wide), we may be assumed to make chaotic choices. Yes, ideas are not magnets and attention not a swinging ball. But we may safely assume that the mind is no less complex than this model. It is therefore intriguing to note that in still simpler systems – with two magnets and a pendulum confined to a two-dimensional plane, the fractal boundary disappears. A theory of culture based on dichotomies (cf. Chapter Two) thus evades the problem we are discussing entirely, and grossly distorts our image of how choice commonly functions.
Most real choices are suspended between a multitude of attractors, some simpler, some "stranger" than others. We may gain a rough appreciation of the implications of this by investigating the metaphorical implications of another mathematical model, developed by the ecologist Robert May to express the dynamics of a population of fish. May's model is based on a highly simplified function expressing population growth: xnew = rx(1 – x), where x is population size in one year, xnew in the next, r a parameter expressing the population's growth rate, and (1 – x) the limits to growth inherent in the ecological niche occupied by the fish. The function is recursive, and when it is iterated the population first goes through rather wild fluctuations, after which it may settle down to a stable equilibrium state – i.e. its attractor is a single point. By increasing the value of r the final population level (the attractor) is heightened. If this value (xlast) is mapped against growth rate (r) on a two-dimensional graph, a simple, rising curve is thus produced (Figure Three – left half).
However this is only true if the value of the parameter r is kept below a certain level. If not, strange developments take place. First the single, final population level splits. The fish now exhibit stable biannual fluctuation between two states instead of remaining always at one. On the graph the curve divides in two. If r is further increased the curve bifurcates again, then again, and again, and again: As the growth rate increases the population stabilizes on two, four, eight, sixteen, thirty-two etc. different levels – undergoing increasingly complex cycles. But as growth rates increase, the bifurcation rate also increases – the curve splits faster and faster, until all semblance of order disappears into a haze and annual fluctuations seem completely random – population never stabilizes at all – its attractor is no longer a stable shape, but "strange", chaotic, as Hénon's graph.
For in fact, these violent and unpredictable fluctuations are not random, but chaotic. No linear or statistical analysis would show it, but there is concealed order in this system. Bits and pieces of cycles flicker into being and disperse forever, semblances of order crop up and dissolve before our eyes. But there is order here. This becomes apparent if we continue increasing the growth-rate by very small increments. Suddenly, for certain values of r, stable attractors reappear. The population fluctuates with perfect regularity in five-, six- or eight-year cycles. But as the parameter is further increased, the process of accelerating bifurcation repeats itself and we return to chaos. As we vary the value of r such islands of order appear again and again, without warning and with no predictable sequence. (Figure Three)
May's graph is a slightly more complex metaphor of choice than the experiment with magnets. In certain situations it has one or a limited number of equilibrium states – i.e. the attractors are points, circles, or more complex, but still repetitive shapes – in which case choice may be described as stable and scalar. At other times the attractors are "strange", as the boundary zone between magnets. Then there is no predictability or repetition in the dynamics of choice – it is non-scalar. To complete the metaphor, the parameter r must be conceived of as connoting the "kind of choice" we make – how well we know what we are looking for, how much pressure we are under, how fast we must choose etc. By varying these factors we are precipitated into fundamentally different processes of choice, determined by the different kinds of attractors affecting them.
This discussion may be made somewhat less abstract by linking it to an analysis of ambiguity by Dan Sperber (1977). Sperber tries to explain why certain impulses (e.g. smells) have the power to awaken vivid memories of scenes, atmospheres and places, like those of childhood – while others (like most words) evoke only a neutral and limited response. He asserts that the mind functions on two "levels". On the first and most accessible level, knowledge is organized in clearly delimited and labeled categories – e.g. concepts. On the second level knowledge exists as continuous sequences of experience, memories, images. An unambiguous statement reaches the first level of the mind, attaches itself to a concept, and we then know how to relate to it. An ambiguous impulse will find no category on the first level. It is therefore "put in quotes" and allowed to "sink" to the level below. There it drifts till it attaches itself to some part of an image or memory, which it "evokes" to consciousness unedited.
I like Sperber's imagery but find it misleading on some points. First of all his model gives the "explicit" mind undue precedence. If we must conceive of the mind as "leveled", I tend to believe that the primary receptor is not the explicit, conscious and prosaic level, but that of ambiguity, emotion and poetry. We feel first, think later. As Gregory Bateson puts it:
"Poetry is not a sort of distorted and decorated prose, but rather prose is poetry which has been stripped down and nailed to a Procrustean bed of logic." (1967, p.136)
But beyond that, I wonder if the metaphor of levels – common as it is – may not itself obscure the issue. The idea that the mind is organized in "levels" derives from logic and gives associations to a defined hierarchy in which some statements are more abstract than others. Sperber uses the word in a looser and more metaphorical sense, but the central idea of degrees of "abstractness" still carries over into his metaphor. This seems to me to narrow our understanding of the mind unnecessarily – for is not the essence of ambiguity that it fails to distinguish abstract from concrete? Since the concept is only approximate anyway, might not a less rigid metaphor be preferable?
May's diagram offers such an alternative. Different impulses do in fact have different effects on us – but not because they have some kind of inherent quality in themselves ("concrete" ambiguity or "abstract" explicitness). The variation implied by the parameter r in May's function says nothing about the impulse impinging on us at all, but instead distinguishes between different kinds of choice. If an impulse enters our mind, we must choose what it means – this is Sperber's problem. But the question is then not what kind of impulse we are dealing with but how we go about our "search" for its meaning. As the value of r changes, we "search" in different ways. A smell encountered in one context will not engender the same kind of search as the same smell in another context. Some searches lead to unambiguous answers – the pendulum of attention comes to rest or describes a definite pattern between fixed points (Search Pattern One). Other searches never stabilize at all. They end in strange attractors, "fields" throughout which our search continues forever in erratic patterns, which never repeat themselves (Search Pattern Two).
Such chaotic fields are not "levels" but rather "wells" in the mind, within which all scalar categories – such as "levels" of consciousness – dissolve. Since they are non-scalar, microscopic variations in the brain's processing of ideas can cascade through these "wells" up to the "macroscopic" level of conscious attention. Distorted remnants of pattern may be rediscovered, new patterns spontaneously form. Slight changes in the parameters of the search (r in May's function) may transport us from chaos to stable equilibrium states or vice versa. The two search patterns in Figure Four were generated from a function, which behaves much like May's. To transform chaos into stable eleven-point equilibrium it was enough to change the value of the parameter c by a single point in the sixteenth decimal place – a third of a tenth of a quadrillionth of its total meaningful variability range!
These phenomena may actually be seen at work in an experiment performed by James Crutchfield (1968) in which a face on a video screen is stretched and folded by a recursive function somewhat like Hénon's. After a few iterations the screen is reduced to a uniform gray. Iteration continues however, and suddenly the face reappears – an island of order in chaos. This happens again and again, at times the face is inverted, split, distorted or occurs in many copies. There is no way of calculating when or how it will reemerge. In a mind or culture past or subliminal experiences may be concealed in this way – and reappear as unexpectedly and in as many permutations as the face when our attention searches in non-scalar wells.
The metaphor casts consciousness and culture in a dramatic and precarious light. A slight difference in how we choose to see may evoke vivid images out of the past, like those described by Sperber. But the model affords insight into other mental processes as well. This may be how inspiration works: After aimless meandering the mind arrives at a sudden point of clarity. We also sense the threat and potential of non-scalar wells, from which the new may be dredged, or where we may loose our grip on reality irretrievably.
The metaphor of Chaos changes our thinking about culture in many ways. Theories of boundary zones are an interesting example, which (in various guises) play a central role in anthropology. Both Victor Turner (1964) and Mary Douglas (1975) have pointed out that such "liminal" states often are seen as potent threats to the social order. This accords well with the idea that areas of ambiguity often are non-scalar and open for "cascading" from below – sudden and unpredictable change. Turner's ideas are of particular interest. He notes that periods of transition – the elaborate rites-de-passage of traditional groups, the empty "waiting" so typical of our own society, the violent transformations of revolution and war – all threaten society in this way. Their impact must therefore be limited, but since they are at the same time vital sources of change and renewal one cannot simply ignore them. The solution is to encase them in elaborate ritual and symbolism, thus focusing attention on them without permitting them to spread. The symbolic "framework" built around such zones may be seen as an example of a non-scalar structure built to utilize and contain the spontaneous non-scalar processes of transition. This metaphor gives Turner's analysis increased weight and urgency. On the one hand we see how statistically insignificant influences may be amplified in liminal fields and have fundamental effect on society as a whole. On the other hand we realize that the problem confronting a structure, which contains and utilizes such change is one of aesthetics. Any attempt to understand (let alone design) such structures in functionalist or utilitarian terms is thus not only futile but seriously misleading. The complexity surrounding rites-de-passage is – like the fractal – witness to the fundamental meaning of beauty.
This affects another type of "boundary-theory" as well. As I have mentioned in earlier chapters, anthropologists often assume that the function of ethnicity is to express contrast between groups – articulate the boundary between them. Thus, in Harald Eidheim's analysis of ethnic mobilization among Norwegian Saamis (1971) this group is said to emphasize its divergence from majority culture in order to promote solidarity and unified political action. Elaborating cultural uniqueness is functional. But this may be misleading. Perhaps the boundary itself engenders ornamentation. The Saamis may be less concerned with attaining a goal than with the ambiguity of the situation in which they find themselves. Their aesthetics reflect the moral fatality of the choice they face, rather than its political expediency.
In a wider sense however, boundaries are a more pervasive phenomenon. Any social aggregate is built up of separate parts with innumerable zones of ambiguity between them. If many of these are non-scalar, the consequences might be momentous. Roger M. Keesing (1970) has proposed a role-theory according to which a social role ("father", "doctor", "queen") is not, as generally accepted, a given reference for action, but itself an aggregate. Each conscious role is assembled out of "building-blocks" – bits of expectation, duty, emotion, symbolism – which have no conscious meaning in themselves. In routine situations the standard assemblages are sufficient, but if change is necessary new roles may be put together quickly from these "bits". This process of assembly involves multiple ambiguities of choice, and therefore a complete role must in fact be a non-scalar aggregate rather than a set of prescriptions or rules. Its pattern would then be pure ornament and have potential for spontaneous, non-functional change.
The same "ambiguity of assemblage" may have important repercussions for society as a whole. We might think of the classical idea of "social equilibrium", which is explicitly or implicitly evoked in almost all analyses of social stability. Equilibrium is commonly thought of as a state of balance between two "principles", "groups" or "forces". In fact however, the systems involved consist of thousands or even millions of idiosyncratic human beings – playing ambiguous roles interrelated by ambiguities of boundary. All in all May's graph seems a better model of this kind of reality. "Stable states" may then be slippery islands of order in chaos, on which the system may loose its footing as a result of seemingly trivial changes. Or else "stability" may be only one of several possible equilibrium states, between which the system fluctuates spontaneously (without parameter change) in the long run. This has been advanced as an explanation of very long-term climatic change. Ice ages may not need causal explanation at all – they are just another "natural" equilibrium state for the Earth's weather. If the same is true for society the impact on theories of social change will be profound.
But as we have repeatedly seen, social structure cannot be understood in non-scalar terms exclusively. There must be a balance between openness and enclosure; "wells" of non-scalar pattern are contained by overriding, scalar rules. We shall now look somewhat closer at this distinction.
First of all it is clear that the metaphor of Chaos is not applicable to all kinds of meaning or social organization. Sperber points out the importance of ambiguity in perception – but it is obvious that not all perception is ambiguous. We might think of the research of Vilayanur S. Ramachandran (1988) on depth perception and shading. One of his experiments shows that the mind assumes that a rounded shape shaded in its lower half is convex, while shading in the upper half is interpreted as concave. Since the mind "knows" that light usually falls from above it gives the shading factor priority in perception of depth. Ramachandran's illustrations are striking. Two disks with opposite shading are seen clearly as a bulge and a hollow. But when the page is rotated both are inverted, even though we follow them with our eyes throughout the movement. This abrupt transformation, and the priority given to a single cue, is evidence of scalar perception. There is no fractal boundary zone here, but a clear-cut distinction, which overrides many other qualities of the picture. When our perspective changes, our world changes.
What if such an overriding perspective is imposed on a non-scalar system? Think of Hénon's attractor. The points out of which it is constituted appear at random throughout the figure as the recursive function is iterated. Let us now approach this graph as a metaphor of some social phenomenon – e.g. kinship. A kinship system is usually considered a list of terms labeling persons by what the native defines as relevant ties of parentage, siblinghood and marriage. Much effort has gone into finding the principles underlying the complex patterning of kinship. Functionalism applied a juridical metaphor – defining a few abstract "laws" which could describe the entire pattern. As attention turned to the pragmatic utility of kinship this approach was supplanted by more process-oriented analyses. Maurice Bloch (1971) thus proposed that a kinship term is not a label for a specific type of person ("father", "aunt") but for a range of moral expectations pertaining, most typically, to that type. "Father" means "fatherly". The native can thus transfer the label to anyone from which he expects or demands "fatherly" behavior.
By using Hénon's attractor as a metaphor of kinship we gain new understanding of this discussion. The kinship system may now be seen from several points of view. As a participant you are "labeled" and see your label – in Bloch's terms – as a moral rule governing your interaction. The closest parallel to this "rule" is the function producing Hénon's graph. You know and follow it, but the pattern it produces – the graph as a whole – is probably inaccessible to you. Or you may be an outsider. In this case, you collect data on the system as a whole. If you are a functionalist you look for its underlying pattern – the regularities of the Hénon graph. But this pattern is not even vaguely similar to the "rule" followed by a participant. The system's patterning cannot be deduced from the morality producing it – it can only be seen from without. This does not make one view or the other more "true". It's a matter of choosing your perspective. This point becomes obvious while you are in the process of discovering the pattern of the whole. Only a very specific "point of view" makes the pattern intelligible. This is clearly the case with Hénon's function. If you analyze its output as a linear or statistical sequence no pattern emerges at all. It can only be seen if its graph is plotted, and only a few graphical perspectives will do – you won't even see the pattern without a computer. This is the reason why the often oblique reasoning indulged in by functionalists does not render their results invalid – this may be the only way the pattern can be seen at all.
Such choices of perspective are reminiscent of the abrupt inversions demonstrated by Ramachandran. Two views of the same scene reveal utterly different objects. The objects themselves may be non-scalar – as the kinship system or the graph – but the switch that brings the pattern out always implies a scalar choice – an inversion of perspective. This intimate relationship between scalarity and non-scalarity is obvious in visual art. Take one of Cézanne's paintings of trees and water. Its woven lines and shapes has the beauty of fractal patterns. But the choice of perspective, the "place" from which the pattern becomes visible, is a different matter:
"The desire to render the image of what we see, without any falsity due to emotion or intellect, any sentimental exaggeration or romantic 'interpretation'; indeed, without any of the accidental properties due to atmosphere or even light – Cézanne declared more than once that light does not exist for the painter." (Herbert Read...)
Cézanne's "objectivity" is a scalar, non-chaotic choice, a realization that the "image of what we see" – the true pattern – only emerges when the correct perspective is chosen.
The non-scalar systems we have investigated all have such limits, even Hénon's graph. It varies endlessly but never goes outside a clear-cut – though infinitely convoluted – boundary. The chaotic fields of May's function have similar limitations. Moreover, the meaningful range of variation for the parameter r is very narrow. If it drops below 0 or exceeds 4.0 neither chaos nor equilibrium is produced, but an asymptotic curve towards negative infinity. You must know "which r's" to use, or you cannot produce pattern. Boundaries are thus implicit even in purely mathematical systems. Their variation must remain "within limits" or else dissipate into the void. What is chaotic is the freedom of motion within these limits – the fractal interweavings of Cézanne's picture are framed by his choice of perspective.
The same duality is essential to social structure on a more mundane level. We might consider Laura and Paul Bohannan's analyses of economy and politics among the Tiv of Nigeria (1952, 1959; and Sahlins 1961). In this tribe of semi-mobile agriculturalists, economic circulation was contained by three hierarchic and exclusive spheres, reserved for women and bride-wealth, cattle and prestige "money", and certain foods-stuffs respectively. This was a scalar organization established to hinder uncontrolled and destabilizing circulation of items vital for the survival of the group. The latter function is most obvious in a fourth sphere – which the Bohannans do not mention though it is clearly there – that of land-rights. Since land was the basis of Tiv existence it was essential that it could not be bartered for other goods. Clear-cut scalar boundaries isolated its sphere of circulation and provided security. But within the sphere other rules pertained. Land was acquired by inheritance along non-scalar and fractal lines: As a whole Tiv territory was inherited by the group from its apical ancestor, Tiv himself. Descendants of his sons split this area between them, their sons' descendants re-split it, and so on – in self-similar pyramids within pyramids – down to the individual household. In theory the settlement pattern mirrored this segmentary hierarchy directly – close relations lived close, distant relatives far off. But despite its seeming rigidity, the system was eminently adaptable and flexible, as all non-scalar systems. Politically it provided a convenient basis for mobilization. In case of conflict between two groups each commanded the support of their "neighbors" up to the generation below their first common ancestor. Neighborhood quarrels thus never escalated – nevertheless the whole tribe mustered instantly against outsiders. Economic flexibility was no less striking. The segmentary lineage system was constantly reinterpreted. If a group moved due to land-pressure, it would in the course of few generations acquire a new place in the genealogical charter consonant with its territorial position. It was simply "inserted" as a new branch on the already overwhelmingly complex fractal tree. Thus scalar containment of the economy in spheres secured society's basic needs. But non-scalar "wells" were enclosed by at least one sphere to ensure the system's flexibility.
Two opposite principles thus underlie social organization, reflecting and demanding radically different patterns of choice. In a non-scalar system choice is a matter of balance and skill. The redefinition of Tiv genealogies exemplifies this. It presupposes an equilibrium between individual and group, an ability to make others "forgive and forget" by proving that you are indeed necessary and deserve the identity you seek. It is reminiscent of Simone Weil's image of the boat on the sea, maintaining balance between the forces of nature and a single man. It is also close to Bateson's definition of skill – the erasure of levels to create a non-scalar "well" (see below). But the boat must be constructed, the "well" must have walls, the balance must rest on a fulcrum. This implies choice of another kind, closer to that exhibited by Tiv in their separation of economic spheres. Here also manipulation is possible. Goods could be "converted" between spheres, and on the surface this may seem much like redefining genealogy. But conversion has quite different consequences. If you "lift" some item from a lower to a higher sphere, you simultaneously transform it into an object of higher prestige. You redefine its "placement", the perspective from which it shall be seen – "invert" it, as one of Ramachandran's disks.
All non-scalar wells must have such limits. In the body, blood vessels split and re-split on many scales. But a smallest capillary and a largest artery do exist. Clouds are non-scalar – they have the same kind of structure no matter how far off or close you stand. But surely not if you stand on Mars or in front of an electron microscope? Even natural chaos has limits, and social chaos all the more so: A similar non-scalar syntax repeats itself in language on the level of narrative, anecdote, paragraph, sentence and phoneme in words. But some statements are too small or large for words – they are no longer linguistic. So also in esthetics. The lack of scale in clouds, waves or gothic cathedrals is beautiful – but so is the strict, scalar geometry of the pyramids.
Real systems always combine scalar and non-scalar properties. But sometimes non-scalarity will predominate. Gleick, in his eminently readable introduction to the "new science", writes:
"’Our feeling for beauty is inspired by the harmonious arrangement of order and disorder as it occurs in natural objects – in clouds, trees, mountain ranges, or snow crystals...' [A]rt that satisfies lacks scale, in the sense that it contains important elements at all sizes." (1987, p.117)
This is certainly true of Bach's fugues or Dostoevsky's novels. Chartres is beautiful from ten miles or ten inches off, a high-rise can be appreciated only from a helicopter. Non-scalarity in art acts like a "well" – by negating scale it allows vital impulses to cascade into consciousness. It permits the new and unexpected to emerge by dissolving the "compartments" and "levels" we imprison it in. This is what Bateson (1967) speaks of in the following quote:
"[T]he problem of grace is fundamentally a problem of integration and... what is to be integrated is the diverse parts of the mind – especially those multiple levels of which one extreme is called 'consciousness' and the other the 'unconscious'. For the attainment of grace, the reasons of the heart must be integrated with the reasons of the reason." (p.129)
But beauty has another aspect as well, that of perspective, geometry, proportion. The size of an object is never insignificant. Chartres magnified to skyscraper proportions would be as monstrous as Kunst der Fuge with a 300-piece orchestra. Many structures are only possible on certain levels of scale. A mouse cannot be magnified to the size of a horse, all its proportions must change or else its body will collapse under its own weight. In our own age of increasingly sophisticated transformative technology this should not be forgotten. Mikhail Bulgakov (1925) has demonstrated the fatality of misjudged proportion in a sinister and moving story about a dog who received a man's brain. The dog – a lively, independent mongrel – became a coarse, violent and suffering human being. In 1925, when the story was written, it raised so many questions of scale that it was only published in the Soviet Union 65 years later: Can a "scientifically" conceived revolution transform a teeming proletariat into a cultural elite? What is the cost of leaping from one state to another? In the fractal chaos of revolution it may seem possible, the cost bearable. But what about later, when scale reasserts itself?
For even fractal beauty is not brought out before a scalar choice of perspective is made. The flat facade of a high-rise is ugly. But New York is not. New York – seen from the right perspective – is as fractal and non-scalar as the Grand Canyon. As Claude Lévi-Strauss points out:
"The beauty of New York has to do not with its being a town, but with the fact, obvious as soon as we abandon our preconceptions, that it transposes the town to the level of an artificial landscape in which the principles of urbanism cease to operate, the only significant values being the rich velvety quality of the light, the sharpness of distant outlines, the awe-inspiring precipices between the skyscrapers and the sombre valleys, dotted with multicoloured cars looking like flowers." (1955 , p.96)
Lévi-Strauss' scalar "transposition to another level" makes the non-scalarity of the city visible. His view of art, as opposed to Bateson's, is therefore intimately related to scale. All art, he states (1962, p.23), rests on reduction of perceptible dimensions – a painting is flat, enclosed by frames – hence much of reality is erased in it. This reduction is the overriding factor that focuses the image. Like the walls containing a gas under pressure, it gives it force and effect. For Lévi-Strauss, the archetypical work of art is the miniature. It is the point of view, the "framing", the "selection" – not the deep "well" it contains – which is beauty. The Egyptian pyramids are – in their absolute and terrifying denial of all non-scalarity – perhaps the most perfect statement of this theme.
This brings us to the last aspect of our discussion. In spite of their differences, both pyramids and gothic cathedrals are bounded – but in opposite ways. The boundary of the pyramid resembles the frame of a painting; it is a scalar choice of perspective. The cathedral's boundary is itself fractal – meandering within endless meanderings. But this difference conceals a deeper similarity, without which neither building would attain beauty. Both forms of architecture combine scalar and non-scalar traits. The cathedral clusters around a core, the altar, a symbolic point of focus and perspective, without which all the variation would be pointless. The pyramid has no core. The burial chamber is hidden and perhaps even designed to be empty. Instead, the pyramid contains an infinite number of pyramids – it is a massive block of self-similarity. It is a stone frame around stone within stone.
The tightly woven interdependence of the scalar and non-scalar becomes still clearer when we consider that the centerpiece of the cathedral is itself a pyramid, and each fragment of the pyramid is a cathedral of non-scalarity. An endless – self-similar – regression may be traced in each structure. What makes them different is the final choice.
Every frame must contain something unframeable, and within every non-frameable growth must reside a core of fixedness. Any tool – even the most utilitarian – is a way of "utilizing" aesthetics (a well-used hammer is beautiful because of its detail). And any object of art – even the most useless – must have a functional center. Scale and chaos interact to build up the objects, values and acts our lives consist of. Just as the size and placement of the cathedral at Chartres reflect a scalar choice, the mirroring of the pyramid in a single grain of desert sand it is non-scalar.