From Certainty to Uncertainty, Pt. 2

From Certainty to Uncertainty, Pt. 2

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Peat, F. David. From Certainty to Uncertainty: The Story of Science and Ideas in the Twentieth Century. Washington D.C.: Joseph Henry Press, 2002.

Chapter Six:  From Clockwork to Chaos

We have already seen a number of ways in which that pivotal year 1900 stood as the watershed between certainty and uncertainty.  This chapter introduces yet another of these revolutions-the introduction of chaos into the heart of science.  Today chaos theory, along with its associated notions of fractals, strange attractors, and self-organizing systems, has been applied to everything from sociology to psychology, from business consulting to the neurosciences.  As a metaphor it has found its way into contemporary novels.  As a technique it is responsible for the special effects of so many movies.

Chaos theory has become ubiquitous, but to discover its origins we must go back to 1900 and a study made by the mathematician and philosopher Henri Poincare.  Poincare was investigating another of those certainties, one that the human race had lived with since the beginning of time: “The sun will always rise in the morning and set in the evening.”  In questioning the inevitability of things he was challenging our certainty that the earth’s orbit around the sun will continue to repeat itself.  In his research, Poincare was touching something very deep, no less than civilizations’ entire way of understanding time and what it means to live within a cyclical nature.  In doing so he was touching the seeds of chaos, and maybe this is the reason that the term “chaos” and the notion of a chaos theory has proved to be so disturbing to a mind that seeks order, regularity, and predictability.

The Womb of Time

Early human societies were embraced within the rhythms of nature.  They lived with the rising and setting of the sun, the heat of midday and the cool of the evening breeze, the long days of summer and frosty nights of midwinter. Nature’s rhythms were so ubiquitous that humans bent to their demands.

Then at the end of the thirteenth century the first mechanical clocks appeared on public buildings and, in towns at least, people became aware of a new quality of time.  It was a time measured mechanically, a time divided and subdivided into equal proportions.  No longer did it matter if it was winter or summer, if there was plowing or harvesting of grain to be done, for the mechanical hours each lasted the same duration.  Irrespective of work to be completed, or the amount of daylight remaining, clocks ticked away the hours and minutes equally throughout the seasons. (Before the advent of clocks the “hour” was probably of a more flexible nature.)

Where previously time’s qualities had been measured by cycles, seasons, the waxing and waning of the moon, the canonical periods of prayer, and the chanting of the offices of the day, now time was quantified and reduced to numbers.  But numbers can be easily arranged on a line-which mathematicians refer to as “the number line.”  So it was quite natural that, in place of cycles within cycles, time should also be strung out on a line and counted off in so many hours and minutes.  Now instead of time cycling and returning it would stretch out indefinitely from past to future.

Time in other cultures was the god Chronos, the rotations of the gods of the Mayan calendar, the old man with his scythe, the figure calling us toward the grave.  Now time was number, and number was time.

This new sense of time, based on the mechanical clock, became the standard against which other aspects of life could be measured.  Events took place “as regular as clockwork.”  Even human beings could be clocklike.  The philosopher Immanuel Kant took his daily walk with such regularity that his neighbors set their clocks by him.  By the start of the nineteenth century particularly accurate clocks were called “regulators,” a name that had previously been applied to certain judges and commissions.  The rule of clockwork had become a metaphor for law and the good order of society.  Within a clockwork universe there could be no surprises and no ambiguities, only a series of certainties strung out along the line of time.

This metaphor of the clock also applied to the heavens, as in the phrase “Newtonian clockwork.”  Isaac Newton had demonstrated that all motion, from the fall of an apple to the orbit of the moon around the earth, could be explained on the basis of three simple laws.  With their aid it is possible to predict eclipses of the sun and moon for centuries to come.  Because of this regularity, the solar system was compared to a clock, a mechanism that is stable, predictable, and understandable and which holds no irregularities or surprises.

The philosopher Wilhelm Leibniz satirized Newton’s God as someone who wound up his watch at the moment of Creation and then allowed the universe to tick away by itself.  Yet Newton’s vision was magnificent.  By stripping away the qualities of things, their taste, feel, and color, it became possible to arrive at an essence of movement-the mathematical principles of structure and transformation that underlie the material world.

Just as in the previous chapter we saw how Renaissance painters discovered the trick of linear perspective by which they could express the space and depth of the world, so too Newtonproduced a faithful representation of the movements of the universe in terms of number.  The French mathematician Pierre Simon de Laplace claimed that if he had stood beside God at the moment of Creation he could have used Newton’s laws to predict the entire future of the universe.

Laplace’s fantasy exposes another aspect to this metaphor of Newtonian clockwork.  Laplace imagined himself standing beside God.  This meant that he was no longer a part of the universe.  Instead of being a participator within a living cosmos, he stood outside and observed its inner working in a dispassionate manner.  This is also an image of Newtonian science itself.  While it was possible to describe the motions of the heavens using mathematics, this “universe” turned out to be less a home in which to live than an object standing before us to be understood, described, predicted, and controlled.  The values and qualities, the tastes and smells of the universe become less important, or essentially irrelevant, when compared to its mathematical description in terms of mass, position, and speed.

Newtonian clockwork also had its applications here on earth.  As the moon orbits around the earth it pulls on the oceans and so produces the alternation of high and low tides.  Such events, the time and height of tides, are entirely predictable, except for the minor perturbations caused by irregularities in coastline, the meeting of tidal streams in estuaries, and so on.  But knowing the exact time and height of a tide is important; it even proved to be a key element in the plot of John Buchan’s famous spy novel The Thirty-nine Steps.  [1]

Newtonian clockwork appeared, at first sight, to be a perfect mechanism.  There was however, a tiny grain of sand hidden deep within its wheels and cogs.  When it comes to the moon’s motion around the earth, or the earth’s orbit around the sun, Newton’s laws can be solved exactly and the appropriate numbers calculated to any accuracy desired.  But what about the small additional pull of the moon on the earth as the earth orbits the sun?  And what is the precise effect of the gravitational pull of the asteroid belt on the orbit of Jupiter?  These tiny effects are analogous to the perturbations that coastline irregularities have on tide predictions.  Scientists call this astronomical problem the three-body problem.  It asks: How do three or more bodies move under their mutual attractions of gravity?

While the two-body problem can be solved exactly, there is no simple solution to the three-body problem.  No single equation can be written down directly and used to calculate numerical answers to any degree of accuracy.  This does not mean that Newton’s laws are incorrect or approximate.  Rather, the corresponding mathematical equations present insurmountable difficulties that make it impossible for a general solution to be written down in a direct way.  In the case of the simpler two-body problem, it is just a matter of inserting the numerical values for the position, speed, and mass of the earth and sun into the relevant equation, and the answer pops out.  But when the mutual pulls of earth, sun, and moon act together on each other this simple approach no longer works.

Astronomers found a way around this problem using an approach called “perturbation theory.”  In perturbation theory, you begin with the reasonable assumption that the moon’s effect on the earth’s orbit around the sun is very small.  Start with the simple two-body problem, the earth’s orbit around the sun (neglecting the moon), and then apply a small correction (called the “perturbation”) to take into account the much smaller pull of the moon.  To this first correction apply another, even smaller, correction.  And then a third correction, and so on ad infinitum.  In practice, scientists don’t need to add too many of these corrections because, after the first, the size of successive corrections becomes so tiny as to make no practical difference to the value of earth’s orbit of the sun.

It is a case of “wheels within wheels.”  [2] Using perturbation theory astronomers made tiny corrections to the orbits of the planets to account for the gravitational pulls of smaller third and fourth bodies.

The results satisfied astronomers but left mathematicians feeling uneasy.  Astronomers were adding together a number of tiny corrections-admittedly each one was much smaller than the other.  It is not unreasonable to assume that a few very small things add up to another small thing.  But what about adding up an infinite number of very small things?  How do we know that these won’t sum up to something large?

Mathematicians love to play with patterns of numbers and devise ways for summing up infinite series of ever-smaller numbers.  Take, for example, the series known as 1/n exp 2.(3) The first member of the series is (1/2)exp 2, that is, 1/4.  So imagine our unperturbed answer is 1.0000.  Adding this first member of the series, which we can also think of as a “correction” to 1.0000, gives us 1 + 1/4, or 1.25.  The next member of the series of “correction,” is smaller 1/3) exp 2, that is  or 0.1111.  This is now added to the first correction.  The new “corrected” answer is 1.3611.  The third correction is (1/4)exp 2, or , which equals 0.0625 and brings the answer to 1.4236.  Additional corrections are even smaller 1/25, 1/36, and 1/49, but there are an infinite number of them.

In this case, mathematicians know the precise answer when all these terms are summed.  Starting with the number 1 and adding an infinite number of corrections we arrive at the answer 1.6449.  The initial answer of 1.0000 has been somewhat perturbed, but even with an infinite number of corrections it remains finite.

There are many such series where an infinite number of corrections add up to a finite answer.  But what about the series: 1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 and so on?  Again each correction becomes smaller and smaller.  However, in this case mathematicians know that when an infinite number of these corrections are added together the answer blows up in our face.  It is infinite.  This was what worried mathematicians when they used perturbation theory to solve the three-body problem.  How did astronomers know that in every planetary case the effect of an infinite number of small corrections would always result in a finite correction to an orbit?  What happens if these corrections blow up?  What does this mean for the orbit of a planet or an asteroid?

Toward the end of the nineteenth century Henri Poincare attempted to lay this problem to rest.  While he was still unable to solve the three-body problem specifically, he found a way to say something general about the overall shape and behavior of its solutions.

Poincare showed that, in most cases, things turn out as everyone expected-small influences result in small effects, and the more accurate solutions are close to the simpler two-body solutions.  Nevertheless, this need not always be the case.  In some very exceptional instances, the perturbed solutions “blow up.”  The addition of a large number of very small effects accumulates rapidly, and instead of planets being “regular as clockwork,” for certain critical arrangements the system becomes unstable.  In other words, Poincare had discovered chaos hidden within the heart of the Newtonian universe.  Newtonian clockwork was only regular under certain conditions.  Outside this boundary, physics was faced with uncertainty.

How does this happen?  The rotation of earth round the sun presents a relatively simple problem for Newtonian science.  The sun pulls the earth toward it; likewise the earth exerts a pull on the sun.  Now add in the effect of the moon.  As the moon rotates around the earth it exerts a slight pull, which has the effect of slowing down and speeding up the earth’s motion.  It is alternately pushing the earth toward the sun and pulling it away again.

In the case of the earth-sun system, the moon’s effect is not very large.  But for certain critical arrangements of other planets, “resonance” can take place.  To understand resonance, think of a heavy man on a swing and a small child who gives the swing a nudge from time to time.  (These nudges can be thought of as perturbations to the motion of the swing.)  Over all, these nudges have little effect on the man’s regular swinging back and forth.  But suppose the child nudges the man each time he reaches the highest point of his swing.  If each tiny nudge is timed exactly the nudges will begin to add up.  In swing after swing, the man goes higher and higher.  This effect of a very small perturbation that accumulates from oscillation to oscillation is called resonance.

In the case of two planets in orbit around the sun, the second may be nudging the first in such a way that these nudges resonate exactly with that planet’s “year.”  In turn, the first planet also nudges the second.  In this way, tiny effects accumulate until the entire system acts wildly.  Effects from one planet are feeding back into the orbit of the other.  [4]

The same thing can happen with critical arrangements of the orbits of two planets around the sun.  The very tiny perturbing effect of one on the other feeds back with each orbit of the planet, amplifying until the whole system becomes unstable.  In this way, Poincare pointed out that within one of the most basic of all certainties-that the sun will rise each morning-was hidden the potentiality for instability, surprise, uncertainty, and even chaos.

Poincare published his result in 1900.  It was the same year as Planck’s hypothesis about the quantum nature of energy.  Five years later Einstein’s special theory of relativity appeared and then the flurry of contributions from Bohr, Sommerfeld, Heisenberg, Schroedinger, Pauli, Fermi, and Dirac that established modern quantum theory.  No wonder Poincare’s remarkable result was marginalized and did not remain within the center of the scientific limelight.  Physicists and mathematicians were also discouraged by the difficulties they would have to face if progress were to be made beyond Poincare’s initial result.  After all, many scientists prefer to work on problems that will yield results for publication, since publication often leads to promotion!

It was not until halfway through the twentieth century that breakthroughs occurred that gave birth to the present science of chaos theory.  Three Russian mathematicians, A. K. Kolmogorov, Vladimir Arnold, and J. Moser, came up with general ways to picture the sort of problems on which Poincare had been working.  Another advance was the development of computers that could solve highly complicated equations numerically and display the solutions on a screen.  Today scientists and mathematicians can visualize complicated systems and “see” what the solutions look like.

Kolmogorov, Arnold, and Moser (KAM) confirmed that chaos and stability could exist within the solar system.  In most situations the orbits of the planets remain stable for tens of millions of years, but for certain critical arrangements of orbits, the tiny pull of one planet on another planet accumulates and “feeds back” to the first planet.  This may be the explanation for the gaps in Saturn’s rings.  Calculations show that if a rock were placed in one of these gaps its orbit would become so erratic that it would fly off into outer space or collide with material in the other rings.  A similar explanation may also account for the absence of a planet lying between Mars and Jupiter.  Matter that attempted to coalesce in this region may then have been subject to erratic forces.  As a result, instead of forming a planet, it gave rise to the asteroid belt, a collection of rocks and mini planets.

KAM’s approach, along with high-speed computers, applied not only to the solar system but also to a host of other situations, including weather, water waves, the stock market, fluctuation in the size of insect populations, the spreading of cracks and faults in metals, traffic patterns, brain activity, heart beats, prison riots, the mutual interaction of certain chemicals, and turbulence in a pan of heated water.  Today mathematicians, engineers, physicists, chemists, scientists, biologists, environmentalists, economists, sociologists, and even psychotherapists use ideas from chaos theory and work on ever more complex systems to the point where they find themselves joined by artists, designers, animators, filmmakers, composers, and computer hackers.

A variety of names is associated with this new science: nonlinear systems theory, catastrophe theory, chaos theory, complexity theory, self-organizing systems, open systems, general systems theory, fractals, strange attractors, far-from-equilibrium systems, autopoiesis, and so on.  In popular accounts, they all tend to be lumped together under the general rubric of chaos theory.  As with quantum theory, chaos theory places strict limits on certainty.  It indicates that we must always be willing to accept some degree of “missing information.”

But what exactly is chaos?  Take something as simple as a pan of water on the stove.  Water at the bottom of the pan begins to heat and, being warmer, it is less dense and therefore tends to rise.  Water at the top of the pan, at room temperature, is heavier and begins to fall.  Warm rising water therefore fights for space against cooler descending water.  Inevitably the result is chaos-a complex series of competing flows within the pan to the point where it seems impossible to predict how the water will behave from region to region.  Similar forms of turbulence occur in a host of different systems: when winds encounter city skyscrapers, as speedboats rush across lakes, or when commuters entering a subway station must fight an exiting crowd.

Machines that vibrate out of control, static produced in electronic devices, rivers in flood, atmospheric storms, fluctuations in the stock market, and fibrillation of the heart are all examples of systems that appear unpredictable and out of control, systems in which what happens from moment to moment appears to be a matter of pure chance rather than of scientific law.

Until KAM and high-speed computers came along such chaotic systems were regarded as too messy to be within the province of science.  Theoretical physicists and engineers preferred not to think about them.  If you push a steam engine or an automobile too fast it begins to shudder and quake to the point where it may self-destruct.  Such behavior is to be avoided rather than made the subject of research.  And if you adjust the settings on an amplifier and the room is filled with static then clearly you have a badly designed amplifier.

Today all such systems are open for study using the approach known as chaos theory.  And if scientists have given up hope of ever fully describing a chaotic system, at least they have come a long way toward understanding them.

Chaotic Systems

Scientists no longer throw up their hands in horror at chaotic systems for they know that such systems conceal many interesting secrets.  Chaos itself is one form of a wide range of behavior that extends from simple regular order to systems of incredible complexity.  And just as a smoothly operating machine can become chaotic when pushed too hard (chaos out of order), it also turns out that chaotic systems can give birth to regular, ordered behavior (order out of chaos).  Chaos theory explains the ways in which natural and social systems organize themselves into stable entities that have the ability to resist small disturbances and perturbations.  It also shows that when you push such a system too far it becomes balanced on a metaphoric knife-edge.  Step back and it remains stable; give it the slightest nudge and it will move into a radically new form of behavior such as chaos.

All these systems exhibit what is called nonlinear behavior.  Nonlinear systems behave in rich and varied ways.  In a linear system a tiny push produces a small effect, so that cause and effect are always proportional to each other.  If one plotted on a graph the cause against the effect, the result would be a straight line.  In nonlinear systems, however, a small push may produce a small effect, a slightly larger push produces a proportionately larger effect, but increase that push by a hair’s breadth and suddenly the system does something radically different.  Put gentle pressure on the accelerator pedal of your car and the speed increases.  The greater the pressure, the faster the car goes.  This is linear behavior.  But when the accelerator pedal is pressed to its limit, the passing gear kicks in and the car jumps forward in a nonlinear way.  In the case of three astronomical bodies, all those tiny pushes and pulls on the orbits can feed back into each other and, when resonance occurs, accumulate into a much larger overall effect.

Over a limited and fixed range of behavior, external influences can have a predictable effect on a nonlinear system.  But when the system reaches a critical point, a knife-edge called a “bifurcation point,” it will jump in one of several different directions, often in an unpredictable way.  Put a ball bearing at the bottom of a bowl and a small push will send it a little way up the side until it falls back again.  But balance it on the lip of the bowl and a single breath of wind will cause it to fall back into the bowl or alternatively fall onto the floor and roll away into the corner of the room.

A system at a bifurcation point, when pushed slightly, may begin to oscillate.  Or the system may flutter around for a time and then revert to its normal, stable behavior.  Or, alternatively it may move into chaos.  Knowing a system within one range of circumstances may offer no clue as to how it will react in others.  Nonlinear systems always hold surprises.

What is of particular relevance to the argument of this book is that such systems are also discovered in human organizations, the stock market, traffic patterns, spread of diseases, fluctuations in population size, and so on.  In all these cases, and many more, a tension exists between what can be known and determined for sure, and what lies beyond our predictive capacity.

Chaotic Populations

Planetary chaos was introduced through the metaphor of grit in the Newtonian clock.  There are other examples where regular, cyclical behavior conceals the seeds of chaos.  Take as an example of regular behavior a reedy lake containing trout and pike.  If the pike are too rapacious they will consume their source of food and start to die out.  But there are always a few trout hiding in the reeds and, freed from the threat of so many predator pike, their numbers increase.  Soon the lake is well stocked with trout and the few remaining pike discover they can have a field day.  Pretty soon the pike population increases and many of the trout get eaten.  Now the cycle begins again.  Hungry pike discover that their prey cannot be easily found and so they begin to die of starvation.  Year after year, and generation after generation, the number of trout and the number of pike oscillate up and down in a stable and predictable way.

Cyclical oscillations of predator and prey look very much like the ticking of a clock.  But in this case the origin of the pendulum swing is not mechanical; rather the results of one cycle feed back into the next in a repetitive way.  Mathematicians call this form of repetition “iteration.”  Iteration means that the output of one calculation, or of one cycle, is the input for the next.  Some iterations lead to stable situations, such as the population of pike and trout, while others produce chaos.

To see how chaos can emerge out of regular population cycles, let us change the example slightly.  Instead of pike and trout we’ll take rabbits.  Release a pair of rabbits on a virgin continent and they will breed until they have spread over the entire land.  But suppose these rabbits arrive on a small desert island.  At first they breed and multiply, but soon they are eating the vegetation as fast as it can grow.  Like the pike that eat too many trout, the rabbits begin to die out. 

There are two competing factors at work on the population, one causing it to expand by breeding and the other causing it to die off because of limited space in which to live and limited food to eat.  Like the example of the pike and trout, population size is determined by an iterative situation because young rabbits of one season become the breeding pairs of the next.  It turns out that the mathematical equation that models this behavior is quite simple and, provided you put in a value for the birth rate, the population can be predicted for years and years to come. 

To explore this example even further we must now forget about real rabbits and deal with hypothetical computer rabbits whose birthrate can be adjusted as we choose.  Real rabbits don’t work in this way, unless they are given hormones, but the example itself applies to a host of other real-life situations, from the spread of rumors and the distribution of genes in a population to certain chemical reactions and insect damage to crops. 

With a low birthrate the initial pair of “rabbits” breeds and the population increases until it reaches a stable level that remains static from generation to generation.  This population is exactly in balance with the resources of the island.  It is the stable sustainable population for that particular environment. 

With a higher birthrate the population increases more rapidly, temporarily overpopulating, then falling again, and after a time rising again.  The result is a stable oscillation in population size, a predictable sequence of fat and lean years that exactly mirrors the behavior of pike and trout in a lake. 

But suppose the birthrate is higher still.  The result of a mathematical analysis shows that within the first oscillation of fat and lean years can be found a second oscillation, a subcycle, a cycle within a cycle.  It now takes four turns of the cycle to come back to the starting point. 

Increasing the birthrate even further means that new oscillations are added.  Now it is a case of wheels within wheels within wheels, or oscillations within oscillations within oscillations.  The situation becomes increasingly complex; population scientists would have to gather data from many years to work out the complex pattern of oscillations and so be able to predict the population for the following years. 

While the cycles within cycles are complex, the underlying mathematical equation is quite simple, and with the help of a computer, scientists can watch the way cycles increase in complexity each time the birthrate is increased very slightly.  It would be natural to assume that the end result would be an infinite number of cycles, a vast machine of incredible proportions containing a limitless number of cogs within cogs.  But, infinitely complex as this may be, this is still a regular form of behavior, since if one were willing to wait for an infinitely long period of time the same behavior would cycle round again. 

This, in fact, is not what happens.  The system reaches a critical point at which the very slightest increase in birthrate no longer generates an additional cycle but, rather, chaotic behavior.  The population now jumps at random from moment to moment.  No amount of data collection can be used to predict the population at the next instant.  It appears to be entirely without order.  It is truly random and chaotic. 

With the aid of this example we encounter one of the paradoxes that lies within the heart of chaos theory: What does it mean to say that something is random or that it has no order?  Toss a quarter in the air and you can’t predict if it will come down heads or tails.  Throw a ball into a roulette wheel and you don’t know if it will end on black or red.  The result is random.  Knowing the results of a long sequence of coin tosses is no help in predicting the next result.  If someone has tossed six heads in a row the chance of the next throw coming down heads remains exactly 50:50.  The sequence of heads and tails is random.  But this does not mean that the process by which the coin lands head or tails is itself without any order.  Each time you flick a coin you use a slightly different amount of force and so the coin spins in the air for a slightly different amount of time.  During this same period it is buffeted by chance air currents and when it lands on a table it bounces and spins before settling down heads or tails.  In coin tossing, the coin is subject to a large number of perturbations and disturbances that are beyond our control.  Moreover these contingencies are so complex as to be beyond any normal sort of calculation.  Nevertheless, at every instant of the coin’s flight everything is completely deterministic. 

The same thing happens in a pinball machine or a roulette wheel.  In both cases the ball is buffeted in a complex series of collisions.  Rather than there being no order there is an order so complex as to be beyond prediction. 

Likewise, while chaos theory deals in regions of randomness and chance, its equations are entirely deterministic.  Plug in the relevant numbers and out comes the answer.  In principle at least, dealing with a chaotic system is no different from predicting the fall of an apple or sending a rocket to the moon.  In each case, deterministic laws govern the system.  This is where the chance of chaos differs from the chance that is inherent in quantum theory.  When a rock rolls down a mountainside it is knocked here and there by the contours of the slope.  The end result, where it finally lands, is random.  Yet each individual bump is totally deterministic and obeys Newton’s laws of motion.  Its extreme complexity arises out of the huge number of external perturbations acting on the rock.  Chaos and chance don’t mean the absence of law and order, but rather the presence of order so complex that it lies beyond our abilities to grasp and describe it. 

Chance isn’t always caused by external perturbations.  In a fast-flowing river an individual, tiny region of water is being pushed this way and that by the river itself, like a rock rolling down a mountainside.  The entire system acts as a complex series of internal perturbations pushing and pulling on each aspect of itself.  Feedback and iteration act within the system to create ever greater complexity.  The result is a turbulent river. 

Now let us take another glance into the mirror of chaos and discover that what we take to be a random system without any visible order can also be seen as an order of infinite richness and complexity.  Each aspect of a chaotic system is deterministic and governed by internal feedback, constant iteration, or complex external perturbations.  Similarly a computer calculation must simulate these physical effects.  Working with a turbulent river or a population with a varying birthrate means that the result of one stage of the calculation (its output) becomes the input for the next round of calculations.  Like the system itself the computer calculation iterates itself again and again, with each output being the input of the next cycle. 

This brings us to yet another paradox of chaos, for, although the equations of a system are totally deterministic, the final results can never be calculated exactly.  Even the fastest and largest computers are finite.  They may carry out a calculation to ten decimal places, which is good enough for most purposes.  But this means that there is always uncertainty in the final decimal place-one part in ten billion.  This seems unimportant until one realizes that this tiny uncertainty is being iterated around and around in the calculation.  Under critical conditions the cycling of even an almost vanishingly small uncertainty begins to grow until it can dominate the entire result.  

The meteorologist Edward Lorenz discovered this in 1960 when he was iterating some simple equations used in weather prediction.  To speed up the calculation he dropped some decimal places and, when the calculation was finished, to his great surprise he discovered that the resulting weather prediction was vastly different from his initial, more accurate calculation.  A small uncertainty in the initial data of the weather system had swamped the final calculation. 

By analogy to the computer calculation, when a real weather system is in an unstable condition a small perturbation can produce a radically different change of weather.  With a system balanced on a knife-edge or at a bifurcation point, even the flapping of a butterfly’s wing can send it in a totally different direction.  The ancient Chinese drew attention to the interconnectedness of all things by saying that the flapping of a butterfly’s wings can change events on the other side of the world.  In chaos theory this “butterfly effect” highlights the extreme sensitivity of nonlinear systems at their bifurcation points.  There the slightest perturbation can push them into chaos, or into some quite different form of ordered behavior.  Because we can never have total information or work to an infinite number of decimal places, there will always be a tiny level of uncertainty that can magnify to the point where it begins to dominate the system.  It is for this reason that chaos theory reminds us that uncertainty can always subvert our attempts to encompass the cosmos with our schemes and mathematical reasoning. 

There is yet another reason why a system, deterministic in principle, can be unpredictable in practice.  Leaving the limitations of computers aside, it is impossible to collect all the data needed to characterize a system exhaustively; that is, without any degree of error or uncertainty creeping in.  In turn, that uncertainty rapidly blows up when systems iterate within themselves.  Take the world’s weather again.  A mathematical argument (based on the properties of fractals) demonstrates that there can never be a sufficient number of weather stations to collect all the information needed to describe the fine details of the weather at any one time.  (The fractal dimension of weather is larger than the fractal dimension of any network of weather stations.)  While it is possible to predict weather trends for days in advance we can predict exactly what the weather will be at any precise instant.  It may look like heavy rain tomorrow but we cannot know just how many millimeters of rain will fall at a particular spot, or the exact time that rain will begin. 

In addition, just as in quantum theory, the very act of observation of a system disturbs the properties of the system: the effect of introducing a probe or making a measurement when a system is at a bifurcation point or in a chaotic state can also cause that system to respond in an unpredictable way.  Although it is always possible to adjust and fine-tune a linear system, things are entirely different when it comes to nonlinearity.  In certain regions of behavior the system may respond to a corrective manipulation; in other regions a small correction may push the system in an unexpected direction. 

Chaos, Chaos Everywhere 

The previous chapter argued that the way we represent the world has a deep influence on what we see.  Chaos theory provides an excellent contemporary example of this phenomenon.  Today we tend to “see” the world, ourselves, and our organizations in terms of attractors, chaos, self-organization, and the butterfly effect.  Economists and financial analysts look for patterns of self-similarity within the daily and hourly fluctuations of the stock market.  Therapists speak of strange attractors governing the repetitive behavior of their clients.  Community leaders and business consultants are concerned with the dynamics of self-organizing systems.  Moviemakers create planetary geographies using fractal generators.  Suddenly chaos, complexity, and self-organization surround us to the point where the general public is using terms more generally associated with mathematicians and theoretical physicists, whereas just half a century ago, no one had ever heard of such terms. 

Only a few decades ago the fluctuations of the stock market were seen as purely random.  Organizations and businesses were studied in terms of rules and hierarchies and good and bad managers.  And “chaos” itself?  It was simply a word used to mean a pattern without any order, an aberration, something not worth studying or taking seriously.  Chaos was the garbage can into which everything was thrown that could not be represented by means of simple rules and behaviors.  And what we now know as fractal orders were once called by mathematicians “a gallery of monsters.” 

How did such a striking change in attitude come about?  Why did people begin to take an interest in chaos and notice strange, new, complex patterns of order in what they had previously taken as random events?  Again, the short answer is that we mainly see what we already know.  Or to put it another way, we could only begin to “see” the inner world of chaos once we had discovered ways of representing it.  Once we are given a mental map of the world of chaos we can begin to discern its features. 

The development of high-speed computers and new mathematical approaches made it possible to describe the general nature of chaotic systems, apparently random fluctuations, and highly complex patterns.  These features of nature had always been present, but until the means to represent them had been discovered they were essentially invisible to us.  These very important aspects of the world had been ignored because we had no real way of looking at them.  In 1900, we saw a world of law, order, and certainty in which chance and randomness were unwanted exceptions.  Today uncertainty and chaos are seen as essential to the hidden order of the cosmos. 


For the past few hundred years Western science, and the Western mind, have been preoccupied with notions of certainty, predictive power, and the exercise of control.  Other societies are willing to accept flux and uncertainty.  They live in the Tao, within the flow of things, and tolerate the fact that they will never know all there is to know about the universe.  By contrast, the Western mind has been seeking a story with a definite ending.  Science wants theories that are finite and rounded off.  A good theory should not leave gaps, areas of ambiguity, or uncertainty.  Moreover, as in some Freudian death wish, physics seeks to bring about its own end.  It desires the ultimate answer, the “theory of everything” that will bring closure to its activities.  With the ultimate equation, theory will be finished, all questions will be answered.  We will know once and for all the story of the universe.  In fact that term, “the universe story,” has been used by Thomas Berry and Brian Swimm as the title of a book and a project to provide a contemporary scientific account of the universe of similar mythic proportions to that of Dante in the Middle Ages. 

Most societies have their origin stories, ways of linking their present world and society to the creation figures of the past.  Some stories concern the creation of the world.  But often the world is already present as given and the stories are about the naming of things, the origin of medicine, language, cooking, and writing.  Berry and Swimm intend something similar with their Universe Story.  Yet traditional origin stories have an open quality to them or involve the role of clowns and tricksters such as Coyote, Raven, or Brer Rabbit who turn things upside down and subvert the order of the world. 

Until the twentieth century forced us to face the basic uncertainty of the universe, we asked science to present us with comfortable bedtime stories, ones in which “everything comes out all right at the end.”  Science believed in the parsimony of the universe and applied Occam’s razor.  [5] There could only be one right theory and every choice should be judged as being good or bad.  Now chaos theory is telling us that if we desire total certainty, if we want to hold the universe in the palm of our hands, we have to leave the human race behind and become godlike beings who can observe and measure a system without in any way disturbing it.  As in Laplace’s fantasy of being present at the creation of the universe, such beings are omniscient to the point where they can gather complete and total information about a system.  They possess computers of infinite power, computers the size of the universe itself, that enable them to understand the inner workings of that same universe. 

But we are finite creatures.  Total knowledge and predictive power will always be beyond us.  We have to accept that we can never know the universe fully and totally.  We must learn to live with a measure of uncertainty, paradox, and ambiguity.  We must acknowledge that vital pieces of information may always be missing.  That is the price we pay for entering fully into the life of the cosmos, for becoming participators in nature instead of mere observers.  Living in the universe gives us obligations and responsibilities.  Each of our acts of observation will in some way disturb the universe and we must accept full responsibility for the consequences of these actions. 

Feeling Out Trends 

This does not mean that we must wash our hands of chaotic systems.  While their fine details remain forever beyond us, we may still be able to detect patterns within their behavior that are not totally random.  Stable systems, such as predator and prey (see the earlier example of pike and trout), are in the grip of what scientists call an attractor.  Just as a magnet attracts iron filings into a fixed pattern, so the attractor of a complex system pulls its dynamics, or behavior, into characteristic repetitive directions.  Perturb the system and its attractor pulls it back on track.  Attractors are a little like Jungian archetypes, always acting in the background.  If a person is in the grip of a particular archetype-the hero, the puer aeternis (eternal golden youth), the devouring mother-this will influence the pattern of behavior within relationships, work, and so on.  Likewise knowing the shape of an underlying attractor helps us to predict what a system’s behavior will be. 

Just as an attractor governs a stable system, so a chaotic system is governed by what is called a strange attractor.  This means that, although behavior may on the surface appear totally chaotic and infinitely complex, it nevertheless originates from an underlying pattern, for the strange attractor itself has an underlying fractal structure.  Fractals are complex patterns in which a particular element of the pattern is repeated at ever decreasing scales ad infinitum.  Likewise, while the behavior of a system in the grip of a strange attractor is chaotic, varying unpredictably from moment to moment, these jumps in behavior mirror each other at ever decreasing scale and take place within a certain zone, or range, of possibilities. 

Economists have compared the behavior of the stock market to a system in the grip of a strange attractor.  While there are overall trends that indicate which stocks are going to rise over the next weeks and which will fall, within these trends can be found fluctuations that, at first glance, appear random.  Yet the “random” fluctuations that occur over say, one hour, mimic similar random fluctuations over a day, and over a week.  Mathematicians call this self-similar behavior.  A fractal displays similar patterns at ever decreasing scales, likewise small fluctuations within the stock market have a fractal structure, and while remaining unpredictable in their fine details, the overall patterns are imitated at smaller and smaller time intervals. 

Although the detailed moment-to-moment behavior of a chaotic system cannot be predicted, the overall pattern of its “random” fluctuations may be similar from scale to scale.  Likewise, while the fine details of a chaotic system cannot be predicted one can know a little bit about the range of its “random” fluctuation. 


Up to now we have looked at systems in which simple order breaks down, or disappears, into that highly complex swirl of behavior called chaos.  Yet the theory of nonlinear systems presents us with a paradox, for behind the door marked “chaos” lies a world of order, and behind that door marked “order” can be discovered chaos. 

Let us return to the sudden burst of noise from an electronic apparatus-an amplifier connected to a loudspeaker perhaps.  Electronic engineers know of a problem called intermittency.  This occurs when the regular, ordered output of an amplifier is suddenly swamped by random “noise.”  These periods of random noise can also cease suddenly and give way to periods of regular behavior.  When intermittency is occurring, we have the alternation of randomness with simple order. 

It would be easy to say that a defect in the design of the amplifier (in fact a nonlinear amplifier) results in the occasional breakdown of regular behavior to produce chaos.  On the other hand, it is equally true to say that periods of chaos (highly complex behavior) break down to leave regular behavior.  In one case, chaos emerges out of simple order, in the other order emerges out of chaos. 

Human societies have their periods of chaos-Carnival, Mardi Gras, Oktoberfest-in which normal social rules are abandoned.  Men dress in women’s clothing, married people indulge in sexual license, there are orgies of eating and drinking, night is turned into day, authority is mocked, and the Fool rules the day.  This can be seen as a temporary breakdown of the stable order of society and the lapse of rule.  On the other hand it could be that within the apparent chaos of the carnival can be found the source of a society’s order over the rest of the year.


In some cases chaos rules when order is relaxed, in others order has its seeds in the realm of chaos.  Go back to that example of a heated pan of water.  Competition between hot water rising from the bottom and cooler water descending from the surface produces haphazard, chaotic behavior.  But with the right degree of heating these apparently random flows and counterflows suddenly settle down and organize themselves into a regular pattern of hexagonal cells of rising and falling water.  This pattern remains stable, provided that there is a constant flow of energy, as heat, through the system.  Similar patterns are found in deserts, where the competing flow of hot air rising from the sand meets cooler air falling from above.  The result is that regular patterns of rising and falling air move grains of sand until hexagonal patterns form on the desert floor, just like the cells in a bee hive. 

The cells in a heated pan of water, or the movement of sand in a desert, are examples of order arising out of chaos.  They all occur in what scientists call open systems.  When energy flows through a system, such as heat in a pan of water, the system can order itself into a stable structure. 

A river provides another example of what is termed self-organization.  During the summer it flows slowly with hardly a ripple to disturb its surface.  Where there is a rock in the river the water divides and flows gently past the disturbance.  But once the spring rains arrive, the river flows faster.  In many ways, the movement of particular regions of water appears chaotic and turbulent, but notice what happens as fast-flowing water encounters a rock.  Now a vortex appears downstream from the rock.  It is a stable form that has emerged out of the chaotic order.  These vortices are remarkably stable.  Throw in a stone and the vortex may be disturbed for only a moment, but then continues as before. 

A vortex is an example of the way an open system organizes itself to produce a stable structure.  Unlike the pan of water, in which an energy flow produced stable patterns, this time it is matter-water-that is flowing through the vortex.  As long as the river is in spate, this structure is remarkably stable.  As soon as the flow subsides, the vortex disappears. 

Natural and social open systems exhibit many examples of self-organization, systems in which regular behavior and stable structures emerge out of chaos.  These are found in everything from traffic flows, economic systems, the movements of goods and services, to certain types of waves in canals and rivers and even Jupiter’s giant Red Spot.  Some, like the vortex, are open to a flow of matter, others to a flow of energy, or even information. 

A city can be thought of as a self-organized system that has structured itself over a historical period.  It maintains its form by virtue of a complex network of flows-money, food, energy, people, and information.  Provided that these flows are maintained at a certain level, the city will sustain itself, garbage will be moved, people will have enough to eat, taxes will be paid, and social services will function.  But if any one of these flows should be interrupted for a long enough period, the city would collapse and chaos would reign. 

Again one of the powerful lessons of this book is being repeated for us.  That is, our acceptance of a degree of uncertainty is the very essence of being alive in the universe.  Many systems in nature and human society have evolved through processes of self-organization.  They were not put together in a mechanical way, by bringing various parts together and arranging them according to some hierarchical scheme and overarching law.  Rather they emerged through the interlocking of feedback loops and out of flows to and from the external environment.  In this sense, the stabilities of our lives, of our organizations and our social structures, do not arise out of fundamental certainties but from out of the womb of chance, chaos, and openness.  Patterns in a pan of heated water and the vortex in a river are particularly simple examples of order emerging out of chaos.  Likewise human society itself, with its cities, international governments, and global economics can only exist through this dynamical dance between chaos and order. 

The open systems that fall under the umbrella of chaos theory have a large number of components that interact together and engage in mutual feedback.  Traditionally, physicists preferred to study isolated systems where all conditions could be carefully controlled.  Such systems behave in regular ways and contain no surprises, so that carefully controlled experiments always match the predictions of theory.  But today we realize that nature’s open systems are far richer and more interesting.  Their behavior is a product of their ability to organize themselves and respond in varied ways to changing environments.  It is only relatively recently, because of the long-standing theoretical difficulties involved, that such systems have begun to be studied in a systematic way. 

This contrast between the versatility and flexibility of self-organization and the behavior of mechanical systems can be illustrated by comparing life in a village to that within a traditional army.  To function effectively during war, an army must have a predetermined and well-understood hierarchy of soldiers, noncommissioned officers, officers, and so on up to the general staff of generals and field marshals.  Recruits are put through a rigorous training and drill that teaches them to obey orders without question and to carry out tasks in a repetitive way.  As soldiers they can be slotted in, like cogs in a wheel, so that, as with any smooth-running machine, they function efficiently as replaceable units.  Officers are trained both to obey and to give orders and, in certain situations, to show initiative. 

Each person entering the army fits neatly into a particular slot, so that during battles and campaigns the army machine continues to function despite a turnover in personnel.  This also means that, with the exception of the highest ranks or individual acts of heroism, the skills and personalities of any particular individual have little significance.  Soldiers fit into the army, rather than the army accommodating them. 

Compare this with the village of Pari, Italy, where I now live.  Here, while everyone has skills in common, idiosyncrasies of personality are important.  Although there is a village association, often plans and decisions are made in the evening as people sit together chatting in the square, or as they stop and gossip while walking around the village.  Sometimes a village meeting is called and resolutions are voted on.  In other situations, people simply turn up to help when assistance is needed. Over hundreds of years, and out of necessity, the village has learned to organize itself in a way that maintains its traditions and respects people for the particular skills they bring. 

Whereas, in the army, soldiers are forced to sacrifice a measure of their personal freedom so as to fit in and obey, within the village a wide range of behavior, even verging on the eccentric, can be tolerated.  The former type of structure is a metaphor for mechanical, hierarchical organization; the latter stands for the self-organization seen in many natural and social systems. 

It is even possible to see such behavior in physics, as in the case of plasma vibrations in a metal.  Back in the 1940s, the physicist David Bohm was working on the theories of the plasmas, that curious “fourth state of matter” as distinct from a solid, liquid, or gas.  Plasmas occur in the upper atmosphere and the corona of the sun as well as within metals.  They are composed of electrically charged particles-positively charged nuclei and negatively charged electrons-and their mutual attraction and repulsion give the plasma its special properties. 

When he was working on plasmas, Bohm was struck by the way they formed an electrical shield almost as if to protect themselves when an electrical probe was introduced.  It was as if they were living organisms, he thought.  At the same time that he was puzzling over their behavior, he was thinking about the future of American society.  He knew that America was founded on a strong sense of individualism and personal freedom, but he was also concerned about how the good of the collective could be maintained.  Did people have to sacrifice their individual freedom for the good of the whole?  How was it possible to have free individuals and at the same time put the good of society first? 

David Bohm realized that the two systems-the plasma and human society-illuminated each other.  In physics, he could treat the plasma in two mathematical ways.  In one, he dealt with an undisciplined crowd of individual electrons.  In the other, he treated the plasma as a single entity, a sort of vibrating cloud.  As Bohm studied the problem he discovered that, mathematically speaking, each of these descriptions is enfolded within the other.  The collective behavior of the vibrating cloud unfolds out of the individual motion of the free electrons.  Likewise, individual motion unfolds out of collective vibrations.  But this mutual unfolding introduces a subtly different slant on the nature of individuality.  The electrical charges on electrons cause them to affect each other at long distances.  But the collective aspect-the vibrating plasma cloud-modifies or shields out the long-range electrical forces that operate between individual electrons.  The result is that, within the ambience of the plasma, individual electrons act as if they only experience electrical forces when other electrons are very close to them.  Because each individual electron contributes to the whole plasma these individual electrons are ever freer. 

Bohm concluded that hidden within the apparently chaotic motion of individual electrons could be found the collective vibrations of the whole plasma.  Conversely, concealed within the vibrations of the plasma are the motions of free electrons.  Likewise, within a human society each individual makes free choices that in some small way may change the course of that society.  Conversely, the choices we make are influenced by the meanings we find in life, and very often these meanings are the product of the society in which we live.  Thus the freedom of individual choice is enfolded within the whole of society, and the meaning of that society can be discovered within each individual.  While chaos theory is, in the last analysis, no more than a metaphor for human society, it can be a valuable metaphor.  It makes us sensitive to the types of organizations we create and the way we deal with the situations that surround us. 

Organizing Self-Organization

It is a major leap from the simple examples of the vortex in a river and patterns in a heated pan of water to the New York Stock Exchange and the growth of the Internet.  While the latter examples do have features in common with the former they are vastly more complex in their internal structure and range of behavior.  Indeed, when it comes to socially based systems we reach the limits of the more simplistic metaphors of chaos theory.  Such systems involve a delicate balance of dynamical structures that involve feedback loops at many levels.  Their internal complexity allows them to remain open to the contingencies of the external world while maintaining internal stability. 

Take, as a starting point in increasing complexity, a single living cell.  To preserve their internal chemistry, cells have evolved a semipermeable membrane.  This membrane allows nutrients to enter and metabolic waste products to leave.  At the level of these exchanges, the cell is open to its environment, yet at the same time the stability of its internal chemistry is also being isolated from the outside world.  To survive and divide, a single cell must be sufficiently open to a two-way traffic with its environment, yet at the same time it must shield its internal structure from undesired fluctuations in that same external environment. 

The human body is even more complex.  Collections of cells have gathered to form organs and, in turn, organs make up the body itself.  The body displays a rich hierarchical structure that is maintained through the interaction of its many feedback loops involving the blood stream, nervous system, immune system, and flows of hormones and other chemicals. 

The human body must be open to its environment.  It scans the horizon for food.  It seeks a mate.  It avoids danger and eliminates waste.  And while it is looking outward it must also preserve its internal environment.  From day to night, winter to summer, the body must maintain a stable core temperature.  It must monitor and control levels of sodium, potassium, oxygen, and carbon dioxide in the blood.  In this way a complex web of interactions maintains the activities of the brain, circulatory systems, waste elimination, and so on.  Clearly with this level of organizational complexity, the human body and its functioning are a far cry from patterns in a pan of heated water. 

Through the long processes of evolution, the human body developed highly sophisticated control mechanisms to maintain a high degree of internal stability (homeostasis) within a contingent world.  Shift core temperature by a few degrees and the result is coma or death by hypothermia.  While the message of chaos theory is that natural and social systems can self-organize out of underlying chaos, the more sophisticated the resulting system, the more a balance must be maintained between chaos and order and the more complex (and robust) must be its internal structures and control mechanisms.  At one level the body appears to function in a hierarchical fashion, with its particular functions designated to semiautonomous players such as the immune system, brain, and circulatory system.  At the same time, all these players are richly interconnected through a wide variety of feedback loops. 

Yet despite, or indeed because of, its stability, chaos also plays a role within the body’s structures and processes.  Take the human heartbeat, for example.  When it is totally regular this indicates disease or even the onset of a heart attack.  If it demonstrates too much chaos then it has entered a state of fibrillation, and death may ensue.  Instead the healthy heart maintains small (“chaotic”) fluctuations around its steady beat.  Good health therefore depends on allowing a small amount of chaos into the system.  Something similar applies to brain patterns.  When they are totally regular and free from fluctuations, this indicates that a person is in coma or under a deep anesthetic.  A beating heart and a functioning brain are complex systems resulting from the cooperative behavior of many smaller subsystems.  In this sense, the brain and heart are self-organized systems that, for their continued health, must combine an overall goal (a regular beating heart, for example) with a measure of individuality (fluctuations within the regular beats). 

The same applies to human behavior itself.  We often think of “madness” as being irrational and without any order.  But generally the opposite is true.  Those who suffer from severe mental illness, psychosis, and so on often have restricted and repetitive behavior.  An obsessive compulsive, for example, cannot tolerate the least degree of uncertainty in the environment and so such people engage in elaborate repetitive rituals such as arranging the objects in their room and touching each one in turn.  By contrast, those who are mentally healthy are capable of a wide range of responses and forms of behavior.  They can adjust to changes in their environment, tolerate ambiguity and uncertainty, take intuitive leaps, and make plans even when they do not have full information as to a particular situation. 

Organizing Organizations 

The individual who knows many things is more likely to survive and prosper in today’s rapidly changing world than the highly specialized expert who has restricted his or her knowledge to one skill alone.  It is possible, for example, to design a system in a rigid way so as to protect its inner functioning.  Provided the environment is stable, such systems can survive indefinitely.  Evolution is full of such examples of insects, plants, and animals that have evolved to fill a particular ecological niche within an unchanging environment.  For them, fluctuation, chaos, and change would present a real danger.  Others, including the rat and the human being, are able to exploit change and uncertainty to their own advantage. 

The same is true of human organizations.  Some businesses have evolved to do one thing extremely well and to go on doing it in response to a relatively constant demand.  It would make no sense to introduce sweeping changes or a new range of products in some circumstances.  Yet, when the environment in which these businesses operate undergoes a major change, they will die out and be replaced by something entirely different.  Dynamically changing environments, which include many of the social and economic environments of our present world, demand social organizations that are sufficiently flexible to adjust to unforeseen fluctuations, to adapt to the unknown and be willing to exploit new pathways and strategies as circumstances change. 

Chaos theory cautions us that complete knowledge and control will always elude us.  Nevertheless, just as the human body must retain a measure of homeostasis when all around is changing, so too a business cannot operate through total unpredictability, chance, and contingency.  While it may be open to change, a business must also make a profit, or at least avoid heavy losses, even when the market is unstable.  Economists need to know the effects of changes in the bank rate.  Governments have to make policies for years ahead.  How then can organizations function effectively while at the same time tolerating a measure of ambiguity and uncertainty in the world around them?  The answer is that a measure of flexibility and what perhaps could be called biodiversity is required. 

Chaos theory invites us to reflect upon the structures and organizations that surround us, from our workplace to the community in which we live, our golf club, religious organization, school, and even the national government, the United Nations, and multinational corporations.  How do these organizations function?  Do they appear rigid and hierarchical?  Can they tolerate a degree of uncertainty?  Are they able to respond to the needs of individuals?  How easy is communication within and between the different levels of the organization?  Are suggestions appreciated and acted on?  Is the image the organization presents to the outside world different from that seen by its employees?  How rich are its feedback loops?  How complex is its internal structure?  How flexible is it to degrees of unpredictability? 

The structures of organizations are always present in both explicit and implicit ways.  When a corporation occupies a high-rise, its structure is quite obvious.  Directors and managers occupy the upper floors.  They have their own individual offices, washrooms, and dining room.  Those on the floors below work in open-plan offices and use a cafeteria.  They are clearly lower on the pecking order.  Sometimes a physical building expresses the essence of an organization, the face it wishes to present to the world.  In other cases it is something left over from an earlier period of the organization’s history that no one has taken the trouble to change.  But in all cases, physical surroundings have a subtle effect on those who work in the building.  For example, what role is played by that oil painting behind the vast mahogany desk?  Is it there to impress clients?  or to bolster the ego of the director?  Mussolini positioned his desk at the end of an extremely long room so that each person summoned by the dictator became diminished as he or she was forced to walk that long distance under the scrutiny of Il Duce’s eyes.  By contrast the highly respected Canadian politician Mitchell Sharp, when he was minister of External Affairs, chose to queue up and eat in the staff cafeteria along with everyone else.  He not only represented a democracy but practiced the spirit of this democracy in his daily life. 

And how is creativity encouraged and used within an organization?  How rich are the lines of communication and feedback between individuals and the various sections of the organization?  What level of initiative do people have?  Are the rooms and open-plan offices anonymous?  Or do they express the personality of each occupant?  Do the employees feel that they are only carrying out the tasks that have been assigned to them?  Or are they contributing something essential of themselves?  Are they bringing their own particular skills and life experience to the organization?  Are they being respected both as persons and as skilled workers?  In short, are they engaged creatively so that they feel a deep satisfaction by the end of each week’s work? 

I was once walking around a research organization with a scientist from the Massachusetts Institute of Technology (MIT).  One of the researchers at the bench asked why his own organization had not produced scientific work of comparable status to that of the Boston research institute.  The MIT scientist replied, “That’s easy to understand.  I drove past your labs at seven o’clock last night and all the lights were out.  At MIT the lights are still blazing after ten o’clock!”  One organization was offering challenges and personal engagement; the other was presenting routine.  [6] 

Organizations and Attractors 

Organizations can be similar to human personalities.  They have their family histories and personal stories; decades later they may still be playing out the consequences of past traumas.  I remember an old lady, quite comfortably off, who lived very parsimoniously, even chewing crusts of stale bread rather than throwing them away.  She had lived through the Great Depression and World War II in England, one a time of crushing poverty, the other of rationing and deprivation.  The traumatic memory of those events had never left her.  In this area of her life, she had become closed to change and trapped in the strange attractor of her past. 

The French psychoanalyst Jacques Lacan noticed that people can even become trapped by a name.  Suppose, as is sometimes the case, a baby is given the name of a dead relation.  That child grows within a certain matrix of stories told about the dead relation: memories, anecdotes, and amusing or tragic stories.  It is as if, when the child looks in the mirror, he sees not so much himself and his own face but a vague image of the person he is supposed to represent.  Rather than identity being an interior matter, Lacan observed, in such cases the patient identifies with that exterior image-a dead relation or some sort of ideal that parents have projected onto their child.  In other words, he did not feel himself as inhabiting his own body but as being elsewhere.  He felt bound by certain dimly understood drives to fulfill a role and to become that person who can only lie outside himself.  Again, what applies to an individual can apply to a business, an organization, or even a government. 

It is often the case that a company calls in efficiency experts or business consultants to observe its operation and offer advice.  Just how effective this proves to be depends on factors that are also seen in the relationship between psychotherapist and patient.  Many therapists set great store by the initial interview-that first meeting between therapist and patient in which the patient attempts to position herself in relation to the therapist.  Clearly this can be a very tense time.  The patient is admitting that she is having problems in her life.  She is asking for help and anticipates having to go into painful and embarrassing details.  It is out of this tension that the therapist notices many of the patterns that have been underpinning the patient’s past life.  Does she relate to the therapist as an authoritative parental figure?  Or as someone who can be seduced into giving in, making deals about fees, cutting corners, and arriving at compromises?  Is she afraid that the therapist may not always be there for her at the same day and same hour?  Does she feel that in some way she is being cheated out of the 50-minute therapeutic hour she has paid for?  Will she adopt strategies to win a few minutes more?  Or attempt to invade the therapist’s private life by discovering details of home, family, and background?  Will the therapist end up colluding with the patient?  Or take a vicarious enjoyment in her shady sexual and business exploits? 

Within that first therapeutic encounter the future course of therapy may be made or broken.  It is as if the therapist is always in danger of being contaminated by what could perhaps be called the patient’s attractor, that history of relationships and repetitive pattern of behavior.  If the therapist is strong, firmly centered, experienced, and alert, the therapy will go well.  But sometimes a therapist becomes sucked into playing along with a lifelong survival strategy established by the patient.  The patient may win over the therapist to her side, or lean on the therapist for months to come, or use the therapist to gain approval of her behavior with a partner or business figure. 

The same thing applies to organizations.  To the extent that they are gripped within their own history they are incapable of engaging fully in a creative act of growth and of maintaining flexibility in the face of change and uncertainty.  Individuals and organizations that behave in repetitive ways are always following some limited set of goals and repeating their mistakes.  They are similar to self-organized systems in the grip of an attractor.  No matter if employees and directors come and go, no matter if computers are exchanged for typewriters, or even if the company moves from a Victorian building to a modern high-rise, a hidden magnetic attraction will still be present. 

Some consultants refer to the “story” of an organization and the way this continues to play itself out decades later.  A large organization operated with two corporate executive officers (CEOs) rather than the more common single CEO.  Naturally this gave rise to all manner of tensions and conflicts within the organization that compromised its efficiency.  It was no surprise to learn that, well over a century earlier, the company had been founded by two brothers at a time when their country was involved in civil war.  It is as if some sort of memory was operating within the organization, a type of attractor that created dualism and division. 

In such cases, feedback loops have become fixed and do not re-adjust to new circumstances.  Likewise, iterations continue to flow throughout the system to support a set of fixed responses.  Like a human heart that exhibits too much order in its rhythms, these systems have become overly rigid and no longer embrace the creative side of chaos.  Maybe at some time in the past when the economic, business, or political environment changed, that organization closed itself off from the full potential of the outside world to the point where it now only engages the marketplace in a limited number of strategies. 

On the other hand, as with any living organism, an organization may have a natural lifetime.  Some wither and die.  Others occupy a sort of fossilized position in the marketplace, like one of those curious animals found in odd ecological niches of the world.  They may still be making money, yet generate little satisfaction for those who work within their walls.  The organization simply “isn’t going anywhere,” and so workers become indifferent to its goals. 

It is also true that an organization can undergo a radical form of renewal.  It can grow creatively.  It can accept the challenge of a changing world and employ the creativity of its employees to the full.  But if such an organization wishes to adjust, learn, grow, and renew itself, it must be willing to go through a period of reorganization.  This may mean opening up the feedback loops, changing the pathways whereby information and meaning circulate around the organization, maybe even changing the way computer systems, rooms, corridors, work hours, meeting rooms, and the like are structured.  To carry out this renewal, the organization will have to face an initial period of chaos.  Many people fear chaos because for them it means lack of control.  Familiar routines may become disrupted.  New relationships will have to be made.  People may be required to learn new tasks, and a variety of formal and informal groups may have to be reorganized. 

Lessons from chaos theory show that energy is always needed for reorganization.  And for a new order to appear an organization must be willing to allow a measure of chaos to occur; chaos being that which no one can totally control.  It means entering a zone where no one can predict the final outcome or be truly confident as to what will happen. 

Yet, in the last analysis, all organizations and groups deal in human relationships.  And this means they deal in fears, disappointments, and aspirations.  It means taking into account those who, for several years, may have felt slighted, snubbed, not given proper respect, or not listened to.  Change may offend vested interests and threaten those who simply want to keep doing the same old job.  This is where the metaphor of chaos theory has its limits, for organizations are composed of human beings and not abstract sets of feedback loops.  Human beings need to feel respected.  Most of them like being part of a group.  Most of them wish to feel challenged and their creativity fully used.  What’s more, human beings need to feel that there is a meaning and purpose to their lives.  Part of this meaning comes from the warmth of their family and friends and part from their work.  It is not money alone that attracts an employee or manager but the challenge of work.  It is the possibility of learning new skills, of extending oneself, and of feeling that one is doing something useful and meaningful in the world.  Highly creative people are willing to take a drop in salary to move into an organization or field in which they feel truly creative, or one that is ethically satisfying in that it does something positive for society or for the environment. 

Many of the social and political movements that arose in the past decades spoke to people who felt themselves marginalized and disenfranchised-people of particular races or sexual orientations, women, or those who have particular mental or physical disabilities that prevent them functioning in the same way as the majority of the population.  People may feel themselves discriminated against, often through the subtle and largely unconscious attitudes of others.  It is only when we are open to change and renewal that we realize that we belong inside society, that a healthy and creative group, society, or organization is not something external to us but, in the last analysis, it becomes the expression of each one of us, and each of us shares in its meaning. 


If we can never have total certainty, and if our abilities to predict and control the world around us are inherently limited, then the metaphor of chaos theory will lead us to rethink what it means to take corrective action.  What does it mean to make plans, execute policies, and aim at goals in a world that always contains a measure of uncertainty and ambiguity?  In short, what guidance can chaos theory give us when we feel the need to take action? 

Newspapers write of fighting crime, the war on drugs, the war on want, and now the war on terrorism.  Doctors speak of taking aggressive measures in fighting a disease.  Issues are to be challenged and confronted.  The rhetoric of combat, conflict, and aggression is all around us and seems unnecessarily violent, considering that these are issues regarding social and medical matters.  It suggests a mindset desperate to retain control over each and every situation, so that deviation from a preconceived plan, goal, or ideal is seen as involving something akin to a moral lapse that requires correction and punishment.  Action of this nature cannot tolerate uncertainty. It uses the language of confrontation, a language in which problems are to be dominated and overcome.  Such rhetoric is also used to whip up support at elections by suggesting that a wrong or inherent evil has been pinpointed and, like an enemy, is going to be beaten to its knees.  This same rhetoric places issues and problems as lying outside us.  It seeks to apportion blame to extraneous factors and is tailor-made for the creation of the “other”-ethnic, social, economic, or religious-group that can then be blamed for all of society’s ills.  Scapegoating has been going on for millennia.  It is easier and more convenient to lump people together under a flag, skin color, or religion than it is to take into account the wide range of human individuality and diversity. 

Once again we encounter a central issue of this book.  It is that of objectifying the world and attempting to stand outside a system as a supposed omniscient observer.  It is the action of distancing oneself and seeing the world in terms of “problems” and “solutions,” instead of realizing that societies, cities, nations, and economic systems are immersed in complex webs of meaning that give them their cohesion and from which they take their values.  People may be good or bad, stupid or creative, ignorant, uneducated, traumatized, or in some cases simply evil.  We can never place ourselves outside the system as observers; our behavior, goals, and values are always set within that matrix of meaning that emerges out of the multilayering of family, group, society, global economics, and so on.  Any policy or plan, any action taken, unfolds out of this matrix and its accompanying values and meanings.  In turn it acts back upon it.  Going to “the heart of the problem” may be important, yet it can also mean ignoring all the factors that gave rise to that situation in the first place, or to those factors that are ameliorating the situation at the present moment and causing it to persist.  When we look at the world as object, or “problem,” we forget that we too are an essential part of the pattern we see around us. 

If we have an overly rigid approach to life we treat the world in a mechanical way.  If a clock, or any other mechanical system, malfunctions we take it apart and look for the cause.  Such a system is composed of parts connected together.  When it doesn’t work we suspect that one of those parts has failed or come loose.  And so we take the mechanism apart and look for the bits that don’t function. 

This approach works perfectly with clocks, toys, car engines, and other mechanisms.  But how well does it apply to a city, a society, a human being, a polluted lake, or the stock market?  When we view the world as a machine, we think of it and act toward it in a mechanical way.  When we deal with a machine, we believe that every malfunction can be analyzed and reduced to a problem associated with some defect in a component.  Such problems always have easy solutions because components can be repaired or replaced.  And so we end up responding to the world in mechanical ways because we see it as no more than a particularly elaborate machine. 

To build a clock or a car you take parts off the shelf and assemble them together.  But in the case of self-organized and open systems the “parts” are expressions of the entire system.  A river isn’t composed of smooth water and vortices glued together.  Rather, the vortex, while remaining stable and identifiable, is an aspect of the entire river.  Likewise, the volunteer groups in a community are expressions of the cohesion and meaning of that town or city. 

When a social or natural system malfunctions, this can sometimes be traced to a fault in a particular aspect.  More often it is a deficit of the entire system.  Take, for example, the human body.  Falling and breaking an arm or leg appears at first sight analogous to a defect within a machine-we can no longer walk, or lift things, because of a defective component.  On the other hand this failure may be an expression of a long-term defect in the entire system.  The leg may have broken because of osteoporosis-a lack of calcium in the bones.  This could be the result of a faulty diet, but is most probably a calcium deficiency resulting from the body’s metabolic changes caused by aging.  Or a person could have fallen because he was preoccupied and did not notice where he was going, or because he had been drinking in an attempt to relieve a high level of stress.  In turn, people’s jobs and the need to make more money to support a particular lifestyle produce such stress.  And so the failure of a particular component ends up being connected to many other factors and meanings. 

When a group of people is exposed to the same virus some become quite ill, some experience a couple of days of tiredness and slight fever, and others notice nothing untoward.  Why is this?  Why do some people become ill and others remain well?  Issues of the effectiveness of the immune system touch on a wide variety of factors: on the negative side, stress, lifestyle, and exposure to low levels of contaminants in the environment; on the positive, the ability to laugh, a life full of meaning, a deep interest in friends and relations, and a feeling of something positive to aim for in life.  In a wider sense the health of the immune system becomes embedded in our work, family, and the values and structure of our entire society. 

In turn, what applies to the human body and the life of the individual also applies to a society and an environment.  When problems surface, the causes may be complex and interlocking.  In so many cases, they depend upon levels of meaning and contexts. 

Chaos theory tells us that we may not always be able to “control” or “fix” a particular situation.  We know that some systems are highly resistant to change.  Others may be oversensitive so that a small interaction may flip the system in unpredictable ways.  Rather than seeing such systems in mechanical terms it would be more effective to feel out and understand the ways such systems function at an organic level.  We need to sense them as living, functioning systems, to see how they depend upon complex levels of meaning so that any action we take flows from an understanding of this underlying meaning. 

Action need no longer be violent and confrontational.  We don’t need to think in terms of problems to be tackled, or of making war on defects.  Rather we must work at many levels simultaneously-at both the practical level and the level of meaning, dealing with both particulars and generalities, looking at both a specific defect and the overall context in which this defect occurs.  We must remember that whenever we look at some system outside ourselves we are also looking inward at ourselves, at our projections and prejudices and our fantasies of how things should be. 

Psychotherapists know this when they say that the patient is there to cure the doctor!  Our drive to correct and improve things must always be open to question.  We must ask why we make such an effort to deal with the world.  Are we reacting to environmental disaster out of fear and anger or out of a deep love and empathy with the natural world?  Do we want to heal because we don’t feel whole inside?  Do we wish to improve the world around us because we feel inadequate?  Do we engage in endless activity because our own lives are empty?  Every action flows out of who we are and the meanings we value.  We are constantly bringing ourselves to the world, and who we are and the values we hold are aspects of that world of which we are an essential part. 

The move from certainty to uncertainty that characterized the twentieth century has brought with it a great responsibility.  Each of us today realizes our connection to the society in which we live through countless feedback loops.  Each of us helps to generate and sustain the meaning by which that society functions.  What’s more, chaos is no longer something to be afraid of; it is an expression of the deep richness that lies within the order of the cosmos and our very lives.