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Chaos: The Geometrization Of Thought

Talk given to the Chaos in Psychology Association


As a result of the popular books and magazine articles that have appeared over the last few years the topic of chaos theory has become familiar to many people. While some psychologists may not be comfortable with the mathematical details of the theory they are probably acquainted with its broad outlines and general concepts. Thus, for example, the image of “butterfly effect” is often applied to systems so extraordinary sensitive that a perturbation as small as the flapping of a butterfly’s wings produces a large scale change of behavior. While chaos theory holds that such systems remain strictly deterministic they are, nevertheless, so enormously complex that the exact details of their behavior are, in practice, unpredictable even with the aid of the largest computers.

On the other hand, since such systems remain within the grip of their strange attractor while the details of their fluctuations appear to be random, nevertheless, their chaos is contained within a particular range of all possible behaviors. Their dynamics may, for example, exhibit a fractal structure in which similar patterns are repeated at smaller and smaller scales of space and intervals of time. As an example, while it is impossible to predict the exact value of a particular share on the stock market at an arbitrary date in the future one may be able to say something about its general pattern of fluctuation over a month, day or even an hour.

In a sense, therefore, chaos theory is something of a misnomer for it is not so much the study of systems in which all order has broken down in favour of pure chance but rather of those which exhibit extremely high degrees of order involving very subtle and sensitive behavior. The full description of such systems would require an enormous, potentially an infinite, amount of information. On the other hand, highly complex behavior can sometimes be simulated in very simple ways through the constant repetition of an iterative processes such as Prigogine’s baker’s transformation or the non-linear feedback associated with the changing size of insect populations.

While chaos theory and fractal descriptions are capable of simulating a wide variety of natural processes it remains an open question as to the extent to which such theories actually offer a full account of the inner workings of nature and society. For example, while repeated iterations can generate complex results this does not necessarily mean that such iterations are part of the actual generative processes of nature itself. Another pertinent question is to what extend dues absolute randomness and chaos occurs within the universe. While chaos theory is purely deterministic may there exist certain natural processes that are essentially chaotic, indeterministic and random? Quantum theory would be an obvious choice, for the time at which a radioactive nucleus disintegrates is, according to the theory, absolutely indeterministic – it is a matter of pure chance. David Bohm, however, has produced a deterministic version of quantum theory which perfectly accounts for all the empirical findings and predictions of the theory without invoking the assumption of absolute chance.

Another area in which intrinsic randomness occurs is in the sequence of digits of an irrational number. But what is the ontological basis of such numbers in nature? Are they a manifestation of intrinsic randomness in the universe or do they represent the abstract limits of processes that involve an infinite amount of information? At present there seems to be no way of deciding whether pure chance and randomness plays a role in the cosmos or if all systems are essentially deterministic in nature.

Chaos, Non-linearity and Geometry

Chaos theory is itself a branch of more general fields of study that include non-linear dynamics and general systems theory. From these come such important concepts as, for example, bifurcation points – a region of phase space (or behavior space) in which a tiny change in external conditions can produce an overall qualitative change in a system’s behavior. Non-linear dynamics also deals with limit cycles and quasi-periodic behavior; that is, with systems that settle down into repetitive behavior highly resistance to external perturbations. Solitons may also be produced in non-linear systems, these are localized entities that move like particles and are apparently independent yet have their origin in the dynamics of the system as a whole.

A significant characteristic of non-linear and chaotic dynamics is that although the underlying theory can be treated in a purely formal way, using abstract equations, it lends itself to an easily visualizable expression in terms of the movement of a system through a landscape of valleys, hills, mountain ridges, saddles and so on. Although the spaces involved may in fact be multidimensional phase-, configurational- or behavior-spaces, nevertheless, there is much to be gained from understanding nature in a purely visual way and geometrical way. Thus, for example, the mathematical description of a limit cycle can be easily conceptualized in terms of a system trapped to move within the confines of a deep circular valley. One can also imagine a system perched on a mountain ridge and about to plunge down one side or the other depending on the slightest wind. Thanks to computer simulations it is also possible to explore the nature of chaotic motion through the various scales of its strange attractor.

This introduction of geometrical and pictorial images is particularly important when it comes to psychology and sociology for it opens up new ways of thinking that are not always possible when one deals with nature in purely abstract and numerical ways. Indeed, spatial imagery seems particularly appropriate; after all, we tend to use spatial metaphors when talking about our inner life; we are “up in the air”, “in a strange space”, “loosing direction”, “following a path” and “becoming disoriented”. That great summing up of the medieval world, Dante’s Divine Comedy, drew upon a sacred geometry in which everything and everyone had their place. The poem begins with one who has lost his way in a dark wood. By making a journey though a highly structured landscape the traveler moves towards harmony and balance. Dante’s landscape is at one and the same time theological, cosmological, social and individual; it is an image of the integration of the psyche and of the dynamics of the solar system, for both individual and cosmos are subject to that same love that moves the sun and stars.

This image of a sacred landscape, and of a path that must be taken towards wholeness, is found in many other cultures. Several of my Native American friends speak of having “a map in the head”, and appear to perceive the landscape around them as richly structured and enfolded in space and time. Thus, for example, the land of the Blackfoot was created by Napi as he moved on his journey north. In his other metamorphoses he created the lands of the Ojibwaj, Cree and Naskapi. Likewise, during the “Dream Time” the Australian landscape was created by the ancestors and within certain rock formation the ancestor, rock, present time and Dream Time all coexist. Related accounts of creation and the landscape can be found in Indian where places of pilgrimage are associated with various parts of the body. In all cases, therefore, a sacred geometry exists both internally and externally, it is part geography, part cosmological relationship, part history, part social order and partly the evolution and nature of human consciousness.

Some of the deepest as well as the earliest ways of understanding ourselves and the cosmos are expressed in geometrical patterns such as the mandala, sacred hoop, four directions, world tree, the snake that consumes its own tail, etc. In all cases these spatial arrangements have a great numinous power of their own. In our Western society Carl Jung referred to them as archetypes of the collective unconscious. My Native American friends caution me that these are not mere symbols, representations or metaphors but are powers in themselves that contain great energy.

Clearly we are touching something very deep in these universal forms and images that are at one and the same time manifestations of the internal dynamics of the cosmos and the structuring of human consciousness. One wonders to what extend the marriage of chaos theory and psychology is beginning to touch upon similar ground. Could it be that the images and concepts we are now investigating are not mere abstract conceptual representations or short hand ways of thinking but possess a numinous power of their own?

Physics has traditionally relied upon quantitative measurement, numerical descriptions and algebraic manipulation for the construction of its theory but I wonder to what extent this reliance upon the quantitative and abstract may be less appropriate within psychology.

Jungian theory suggests that one comes to know the world and oneself by means of the Four Functions. Jung classified Thinking and Feeling as concerned with evaluation and thus termed them the “rational functions”. By contrast Intuition and Sensation are “irrational functions” for they do not measure or evaluate but work directly through direct perception. Jung warned of the danger of allowing one of these functions to dominate the others. In this sense the geometrical or pictorial descriptions that are emerging out of chaos theory could be said to be more perceptive and sensate and thus provide an important counterbalance to our prevailing quantitative, evaluative approaches to nature.

And there are even good arguments to suggest that this geometrical, pictorial approach may be an inherently important way of describing the natural world, and our own perceptions of it. In their search for deeper theories at the most fundamental levels of physics some theoreticians have be looking at new geometries, cohomologies and topologies as ways of exploring the most fundamental level of physics. The study of loops and knots, for example, has recently been shown to make connections to theories of the subquantum world, such as Edward Witten’s axiomatic field theory.

Relationships such as “inside/ outside”, “contained in”, “next to”, “enfolded in”, “connected to”, “excluded from”, etc can all be expressed in purely topological and geometrical terms without reference to number and measurement. Not only do such relationships appear to be more appropriate at the quantum level of the world they also lie very close to the ways in which our brains gather and process information about the world. Thus topological and geometrical relationships may be a more fundamental way of understanding both matter and consciousness. 

Some Burning Questions

Thus it appears that notions of form, pattern, geometry and structure can be found at the deepest levels of matter and the psyche. It is at this point that a number of questions pose themselves:

  • How are we develop this approach within psychology and the social science?
  • How can we enrich our current pictures and geometries?
  • How will we include such notions as quality and value within the geometrical realm?
  • How are we to deal with the dualities of subjectivity and objectivity?
  • Can we develop new geometries and structures that will both enrich our theoretical understanding of the psyche and serve as integrative function for those who seek an inner development?

The Psychological Dimension

Anyone who has dealt with chaos theory will be aware of the associations it evokes in others. The very term brings with it unresolved feelings associated with the break down of order; social chaos; fears at loss of control; anxiety at sudden, disruptive changes; concerns about the transformation of values; and apprehension at the disintegration of our and the loss of all we hold dear.

To what extent are such profound and disturbing feelings being captured and contained within such concepts as fractals, chaos, bifurcation points and strange attractors? To what extent are these very forms and patterns becoming charged with a numinous power that can aid in both understanding and healing?

Concepts and Clues from Physics

In many ways as we seek to enrich psychology from the perspective of chaos theory we are all groping in the dark. Since my own background lies in the physical sciences it is natural that I should look to it for clues and suggestions. Indeed, this approach has tended to be the pattern of the last decades for other disciplines look to physics for their metaphors. However, I am well aware of the caution needed in this wholescale importation of ideas. I can remember a discussion I had with Stanislav Grof who dreamed of a time when psychology would produce its own deep insights about the worlds of consciousness and matter and that these, in turn, would flow in the other direction and enrich physics and aid in the understanding of fundamental matter. In light of this qualification, the ideas and suggestions outlined below should be taken with a generous grain of salt. While they do not all apply directly to chaos theory or the geometrization of thought but I think that each one contains a fruitful germ of enquiry.

a. Fundamental Law

Often psychology and the social sciences seek fundamental principles and laws, as well as the elementary structures on which behavior and consciousness is based. This approach is clearly taken from physics which has always sought out the most fundamental laws and most elementary levels of matter structures in its effort to answer such questions as:

  • Why is there something rather than nothing?
  • Why is the universe the way it is?

Such laws have a curious ontological existence for they appear to stand outside matter and space and have an existence prior to time. Thus, for example, fundamental law is invoked to bring the universe of time, space, matter and energy into existence within a primordial Big Bang. Now while physicists like Stephen Weinberg and Stephen Hawking are firm believers in such an ultimate law – and possible even in a most fundamental level of matter – not everyone agrees with this approach.

Another way of looking at things is that rather than governing natural processes these laws are, in fact, generated out of the processes of nature themselves. In this way of thinking the laws are always provisional and context-dependent; they operate at a certain limited level and break down in more extended contexts. Likewise, “fundamental particles” and ultimate material levels are always subsistent on something beyond them. In this fashion, rather than seeking for fundamental laws and equations one begins to ask how systems generate their own regularities and patterns of behavior.

Take, as an example, the gas laws of Charles and Boyle. At one time they were thought to express the fundamental properties of matter. Later they were discovered to be the result of statistical averages over an enormous number of molecular processes that make up the gas. Later still these molecular processes were discovered to originate in collisions that had to be expressed quantum mechanically.

Today physics seeks to explain one law in terms of behavior at a smaller and presumably more fundamental level. But it could also turn out that processes at the sub-quantum level are conditioned by the universe in the large, and by large scale non-local processes. Thus it could be that the regularities or laws that are uncovered at one context, or level, are the result of processes operating at many other levels.

To sum up, physics has traditionally pictured the world as created out of elementary “building blocks” whose behavior is governed by simple, fundamental laws. An alternative approach would be to consider a universe of process and flux out of which unfolds, always within certain limited contexts, a variety of patterns, regularities and invariances which we take to be “The Laws of Nature”.

One wonders if psychology and sociology will ever be based on fundamental laws and levels or if one should rather seek the patterns and forms that emerge out of complex processes and the interpenetration of levels.

b. Inertia and Form

It has always struck me that the fundamental behavior of matter and of thought are in many ways similar. Thus, for example, thought tends to cling to forms and patterns as does matter cling to its motion.

According to Newton the motion of matter is always similar to itself. A free body moves in a straight line or remains at rest – at each instant its motion is exactly similar to itself. When, however, this body is acted on by an external force its motion is not disturbed in an irregular way, rather the change of motion is always exactly similar to itself – uniform acceleration.

Likewise, in relativity the form of the laws of nature are always similar to themselves no matter in which particular frame of motion those laws are expressed. One could even think of the persistence of material bodies as arising in a similar way, for the form of matter is always exactly similar to itself from moment to moment.

Thus, if I were to think of a basic principle of nature it would be this notion of a clinging to form, a principle which seems to occur at all levels of behavior of matter. One wonders to what extent this is also present within consciousness.

c. Algebras of Thought

Attempts have been made to create algebras of thought and curiously enough these same formal structures have currently been shown to have relevance within fundamental physics. The 19th century mathematician Herman Grassman created the algebra that bears his name in an effort to show how one thought unfolds out of another. Grassman believed that algebra was not about the physical world but about the movement of thought itself. He observed that each thought contains the trace of that which has gone before and anticipate that which is to come. His algebra reflects that dynamical movement of unfolding and enfolding.

Unfortunately later mathematicians focussed on the static aspects of this work and neglected the more radical nature of his algebra. Grassman’s work was, however, revisited by William Kingdon Clifford and William Roland Hamilton who appeared to understand the radical nature of what he was attempting. Today Grassman and Clifford algebras play an important role in fundamental physics. But since most mathematicians do not appear interested in fundamental questions of thought there is little written on this aspect of their work. A colleague of mind, Basil Hiley, who was working with the late David Bohm on the application of these algebras to matter and consciousness, has suggested that the best approach for those interested would be Grassman’s “Gasammette Mathematische und Physikalische Werke” Leipzig 1894 – some of which has been translated into English – also W. R. Hamilton “The Mathematical Papers” Vol #, Algebras pp15-16. Ed H. Halberstarn and R.E. Ingram, Cambridge U Press 1967.

d. Collective variables

The relationship between an underlying flux and the appearance of order was investigated by the physicists David Bohm and David Pines in the context of a plasma in a metal. In essence, Bohm showed how within the apparent random thermal motions of a vast number of electrons are enfolded regular, collective oscillations of the Plasma. Likewise, the random motions of individual electrons are conditioned by the overall motion of the plasma. (Technically speaking the long range electrical interactions between electrons are shielded by the plasma so that individual electrons move randomly under short range forces.)

Bohm and Pines work seems to me to be to be full of valuable metaphors for psychology and the social science. Take for example, the way in which large scale behavior unfolds out of the random motion of an enormous number of tiny elements and, at the same time, how each individual random motion is conditioned by the collective. Thus, while society emerges out of a group of individuals, individual behaviors unfold out of the collective. One can also speculate on the relevance of this image for the brain – for large scale behavior across the brain develops out of the action of an enormous number of neurological elements. In turn, this large scale activity conditions the behavior and response of the individual elements and changes their interactions.

I do not know of any non-technical account of this work. It is described in the original papers, such as David Bohm and David Pines, Physical Review 92, 609-625 (1953), but these papers are quite mathematical for the non-technically minded.

e. Global behavior

The work referred to in the previous sections demonstrated the way in which a dynamical system can separate itself into two modes of behavior – one dealing with the apparently chaotic motion of individual elements and the other dealing with the large scale and more regular behavior – yet each dynamics interpenetrates the other and coexists at the same time.

Indeed this raises important questions of how microscopic and macroscopic descriptions can exist side by side and how one and the same system can separate into fast and slow variables. It seems to be of great importance that similar investigations should be made into the functioning of the brain and consciousness for here both local and global behavior appear to be performing different yet interpenetrating functions. Thus, it is often said that conscious awareness arises at the global level – yet the global arises out of the local and, in turn, acts to condition the local.

This duality between local and global structures is also connected with the sorts of geometrical pictures that are present in chaos theory and non-linear theories. In these approaches a system is treated as a point moving through a richly structured landscape of planes, valleys, saddles, peaks and fractal dimensioned strange attractors. Yet the system point itself is not structureless for it has its own internal dynamics, the fact that is responds to the vagaries of the external landscape means that its internal structure is coupled to that of its outer environment.

David Bohm and Gideon Carmi looked at the whole question of the way in which a dynamical system divides itself into an internal part and an external environment – Physical Review A133, 319-331 &332-350 (1964). The mathematician Roger Penrose has also explored the way in which a series of elements can generate a collective, global order which in turn acts back to define the system. For example, Penrose examined the mathematical relationships of a network of quantum mechanical spinors – the most elementary quantum elements possible. He discovered that when these networks become sufficiently large they begin to define the first elements of a Euclidian space, namely angles in a three dimensional space. (Roger Penrose “Angular Momentum: An approach to combinatorial space-time” in “Quantum Theory and Beyond” ed Ted Bastin, Cambridge University Press, 1971)

Likewise, when groups of harmonic oscillators are coupled together they begin to define structures that mimic those of space and time. The same thing can be done in Hamilton-Jacobi theory in which waves are can be brought together and interfere. Out of their “beats” one can begin to define relationships in space and time.

There are many other examples of the ways in which a collection of elements spontaneously define a global order. Indeed, the orders of space and of time can be generated in this way. In turn, these generating elements themselves need not be a priori primitive but may be the manifestation of some deeper underlying order. In is intriguing to speculate if large scale structures within consciousness and the physical brain could be created in similar ways. Indeed, both human brains and human societies may spontaneously generate global orders which then act back on their elements to order them in new ways.

f. Coherence

Another approach to this question is that of “coherence”. Plasmas, superconductors and superfluids all exhibit long range, coherent order in which their individual elements work together in response to an overall form. Electrical resistance in a normal metal is caused by the scattering of electrons. An electron in a superconductor, however, will not scatter because its motion is guided by the overall form of the global wave function. Again the notion of pattern and form plays a key role in the physics of coherent systems.

The theoretical physicist H. Frohlich has speculated that examples of coherent systems go beyond that of the laser, superconductor and superfluid, indeed that they are ubiquitous and a fundamental characteristic of life. Living systems are always open systems, ones that can be pumped to higher energy levels by an influx of matter and energy. In this way, Frohlich argued, a living system is able to maintain a collective and coherent state, one in which a global order prevails and acts back to condition the individual elements of the system and coordinate their behavior.

One would suppose that coherence, and global coordination would also be present in the brain, consciousness and society. It acts as an ordering or integrating principle and arises out of the openness of a system to its environment. Isolate such a system from its environment and all its internal coherence will decay.

Coherent systems exhibit long range order and could be said to respond to global forms, or patterns of information. I have always been interested to speculate on how such ideas apply to such things as brains and the immune system. When global information is available to the entire system it becomes possible for distant parts to be coordinated so precisely that tiny disturbances within a system could propagate without dissipation or destructive interference. Ripples at the edge of a pond will normally interfere destructively so that they effects are soon smoothed out. But if these ripples are able to respond to a global pool of information then it would be possible for them to become coordinated exactly in phase. In such a case they would interfere constructively, grow in amplitude and converge towards the center of the pond. By analogy a brain, or for that matter a neural net, in which global, long range order prevails – or in which non-local correlations are present – could direct disturbances from all over its surface into particular regions. This highly localized activity could then fold back into the entire brain only to appear in a localized form within some other region.

The operation of such a system would require a global field of active and subtle information. But as we know, systems that are termed chaotic, are highly sensitive and could be said to carry a large amount of information. By contrast, systems described by limit cycles tend to loose the information of their initial conditions. Hence a high degree of “sensitive chaos” would appear to be a prerequisite for brains, immune systems and societies.

g. Chaos or Madness

In regard to questions of coherence, chaos and global order one could ask to what extend should the world chaos be applied in health and sickness? In informal discussions during the conference the ideas of chaos and mental illness were at times used in an interchangeable way which, I believe, is confusing and unfortunate.

Healthy individuals exhibit integrated purposeful behavior, there is a center to their existence, they can direct attention in meaningful ways and have a rich sense of inner life. At the same time they are flexible and adjustable, and can be sensitive to what is occurring outside without being swamped or overwhelmed by a sense of mere contingency.

There are also times when that sense of identity diminish or vanish although – such as when falling in love, deeply engaged in creative work, or out in nature. Some have had been granted mystical experiences, others though training and discipline have been able to enter transcendental or extended states of consciousness in which there is a total loss of the “I” and a deep identification with some transcendent reality; yet others have acted as what are sometimes called “Shamans”, prepared to take journeys into other worlds for the sake of the society. Yet, even when there may appear to be a total dissolution of boundaries, what could be called the interior sense of integration retains its watchful care over the organism. Thus, there are stories of “shamans” and medicine people who, at the high of some ceremony involving drumming or the ingestion of drugs, are aware in a very practical sense of the proximity of the central fire and of the welfare of those around them.

In this way the healthy individual and society is characterized by a strong sense of meaning and direction, by extreme sensitivity and internal order, by an openness to exteriors, by richness of response, by creativity in action and, at the same time, a deep sense of interior integration.

By contrast, the sick person is characterized by a paucity and rigidity of response, by repetitive behavior and inappropriate action. Their inner experience may range from a sense of loss of connection, to that of being overwhelmed by external events and uncontrolable inner promptings. They may experience a general lack of meaning, an interior blankness and a loss of sensation. Or conversely the external world may appear violent, overwhelming and meaningless. In both cases the normal sense of integration, of openness and appropriate response to the interior and exterior worlds has been compromised.

The question is, how should this be discussed in terms of “Chaos theory” and with non-linear dynamics? Is chaos to do with the mad or with the healthy?

Healthy, organic systems all tolerate a degree of chaos, for they are open and rich in their response. By contrast, systems which are less sensitive and far from chaos tend to settle into rigid, repetitive behavior such as limit cycles. For example, a healthy heart exhibits a certain degree of fluctuation in its activity, whereas a strictly repetitive cycle presages a heat attack. Likewise a individual who is psychically trapped in a Basin attractor will undergo repetitive, mechanical responses. Their behavior will be obsessional, neurotic and somewhat predictable.

From an informational point of view limit cycles are particularly simple, for the information inherent in a system’s initial conditions is rapidly lost. No matter where you enter the limit cycle you end up being locked into the same repetitive cycle. Hence, the more you are trapped in a repetitive response the more your personal story becomes impoverished and meaningless. Likewise your ability to process information in a creative way is diminished. Indeed, there may well appear, to an individual within such a state of mind, to be no way of escaping from this predicament. Indeed it is only by increasing the “dimension” of the system, as it where, will it ever be possible to enter into a new range of behavior.

The more mechanical the behavior, the less creative and organic is the response and, as a consequence, coherence becomes lost. Coherence is the rich integration of the whole organism that leads to subtle behavior and appropriate response. Coherent systems exhibit a global form in which meanings and perceptions, even those lying far apart, are integrated together. By contrast, the person trapped in a limit cycle becomes impoverished. Speech and thought, for example, may become increasingly mechanical. In extreme situations the lack of internal coherence may suggest what is popularly thought of as “chaos” – that is, randomness in thought, speech and behavior. But it is important to remember that what we take for behavioral “chaos” is in fact the result of a strictly limited, mechanical order in which no inner structure or subtlety is present. True chaos, by contrast, is rich in information and highly sensitive to contexts and changes in the environment.

Thus mechanical, repetitive behavior, and even “chaotic” and “random” responses are the result of an impoverishment of the organism in respond to its environment, an inability to process both interior and exterior information in appropriate ways. The result is a lack of interior coherence results in a breakdown of inner order and outwardly directed behavior

In other circumstances the boundaries between inner and outer may be become too loose, and interior integration so weak, that a person becomes overwhelmed by external contingencies. Similarly the interior can overwhelmed by fear and oppression with a consequent sense of lack of control. Again all these are examples of the inability to process coherent information in appropriate ways.

h. Enriching the Geometry

The analogies between physical and mental systems are of course strictly limited for human beings live with values, feelings, emotions and other qualities. Bearing with these reservations and limitations in mind is it possible to enrich the current images of chaos theory and non-liner dynamics to include other qualities?

Similar questions have been raised in elementary particle theories where physicists desire to enrich their current geometrical pictures by referring to “internal symmetries” of the elementary particles. One way in which this can be done is through Gauge theories and Fibre Bundle theories. Here a rich structure is superposed upon each point in space, the theory then explores the nature of the relationships between these structures at different space-time points. One could speculate that something analogous could be constructed on a “behavior” space of consciousness or some sort of “social space.”

To take an example, an electron can have either a positive or negative charge. At first sight this idea of charge seems to have little to do with the underlying spatial structure but suppose that at each point in space we associate a sort of dual internal structure that corresponds to two internal degrees of freedom – the two possible choices for the sign of its electrical charge. In the absence of an external electrical field it is quite arbitrary which of the two possible charges is called positive and which negative. There is, therefore, a symmetry associated with electrical change at each point in space. However, once we make a convention of, for example, calling the electron negative at one point in space then this convention must be upheld at all other points.

In other words, we begin with an arbitrary binary structure at each point in space but now we must introduce a convention or set of transformations or connections between each point in space, to ensure that out arbitrary choice of positive or negative is consistent at all points in space. It turns out that the transformations or connections required, called gage equations, are identical in their form with Maxwell’s equations for the electromagnetic field. This is quite a remarkable result for it means that one of the most important fields in physics – the electromagnetic field – can be derived in a purely geometrical way by considering structures and transformations between space-time points.

It is persuasive to speculate that the behavior spaces associated with psychology and sociology could be enriched by adding a variety of structures, or values, at each point in behavior space. At first sight this may seem like a tall order but I am encouraged every time I look at a painting by Cezanne. Three dimensional space can of course be portrayed in a more or less mechanical way by means of Perspective. Cezanne, however, tried to do something deeper, for by means of color; boundaries; structures which orient at different angles, axes of rotation, edges and fields he was able to associate each region of the canvas with a very rich set of values. In turn, these regions assume new values in relationship to the rest of the canvas. Thus the whole painting is built up in a very complex way, each region being context-dependent on the whole, and the whole being built up out of local and non-local relationships between regions. Cezanne was therefore able to create a very rich order on a two dimensional surface, one that employed such relationships as locality and non-locality, context dependent value and the enfolded relationship between the parts and the whole. I think that we could take Cezanne as an example of how rich and satisfying structures can be built up in a formal way.

The Great Work

Chaos theory has struck such a deep chord in so many different disciplines that it is difficult to account for its attraction in terms of technological advances alone. One reason may be that chaos theory has taken us away from algebraic and numerical abstraction into the geometry of pattern and form. But another, and even more compelling reason may be that chaos has an archetypical power of its own.

Chaos theory enables us to touch the irrational elements within our lifes, to dialogue with the breakdown of internal order and unforeseen change. Chaos theory suggests that it is still possible to hold on to a certain sort of rationality even within the breakdown of all order. It suggests that when we are forced to give up control over our lives we may be giving ourselves to a deeper form of wisdom and guiding principle. It implies that within the heart of chaos lie new forms of subtle order.

In this respect I am struck by the remark made by the physicist Wolfgang Pauli that physics must learn to accept the irrational in matter. For Pauli this irrational element was the absolute chance inherent in quantum processes – a chance that transcends all causal accounts or rational explanations. Seeking the irrational can also be seen in Jungian language as an attempt to balance the rational functions of evaluation by those of perception.

One can also enter chaos theory through the alchemical door and picture chaos in terms of the Spirit Mercury – the Trickster of the Cosmos. Thus one experiences the cosmos as a dance between law and order, on the one hand, and chaos and the transformations of the trickster, on the other.

Indeed, it is as if the practitioners of Chaos Theory were participating in what was once called The Great Work, the Ars Magna or the Royal Road. That is, in the search for spirit within matter. The Great Work seeks to loosen the material bonds of the spirt through the processes of purification, and the spirit that is liberated is not free to return to matter in an act of reanimation and renewal.

Those who practiced the Ars Magna in the Middle Ages did not simply belive, as Carl Jung has proposed, that they were concerned with their own personal processes of Individuation. Rather, they were participating within the aboriginal act of creation and renewal of the universe. It may seem farfetched to suggest that Chaos Theory has such elevated goals. Nevertheless, in attempting to touch chaos within matter and psyche we come in contact with powerful forces and attempt to contain them through a variety of symbols. Chaos is concerned with the breakdown of order, with sudden transitions, with the appearance of the Trickster in people’s lifes, with that last throw of the dice when there is nowhere left to fall. Under such circumstances one moves from the world of strict causality into that shadow realm of sycnhronicity where matter and psyche mirror each other. Work on chaos theory may well have begun to touch this new universe.