D SPERRY FINLAYSON

• Ancient origin myths: the formation of Order out of Chaos. • Apollo and Dionysos, and Greek science. Aristotle and Ptolemy and the Roman Church. • Medieval Alchemy. Galileo and Descartes. The Enlightenment and the Industrial revolution. • Natural systems almost impermeable to scientific investigation. • The “fudge factor” and “noise.” • Dark Matter and Dark Energy: 95% of universe. • Newton’s equations linear; thus valid in only a narrow set of circumstances. • Nature is predominantly nonlinear. • Henri Poincaré’s “bizarre” mathematical discoveries. • Computers and the study of chaos and change. • “Irrational” numbers like pi and phi indicate relationships ubiquitous in Nature. • Chaos is essential to the orderly cosmos. • Second Law of Thermodynamics. • Orderly systems made by human beings: treasures. • Necessity of both order and disorder in the body. • Two kinds of chaos: the “entropic,” and the “creative.” • A fashion for disorder in the arts. • Chaos and order in snowflakes: a symbiosis. • Dynamic harmony; our love for it and dependence upon it.

Almost all the questions of most interest to speculative minds are such as science cannot answer.…Science tells us what we can know, but what we can know is little, and if we forget how much we cannot know we become insensitive to many things of great importance.

[Bertrand Russell, quoted in Mary Midgely, Wisdom, Information and Wonder, p 115]

[I]f you take regular physical systems which have been analyzed to death in classical physics, but you take one little step away in parameter space, you end up with something to which all of this huge body of analysis does not apply.

[Physicists Doyne Farmer and Norman Packard, in James Gleick, Chaos: Making a New Science, pp 250-51]

In human attempts to “make sense of things,” certain grand conceptual divisions have been found useful since earliest times—between the seen and the unseen, the physical and the spiritual, the orderly and the chaotic. Ancient stories and myths of the creation of the world told of the birth of Order out of a pre-existing primal Chaos. This was described and symbolized in a variety of ways, but chaos was generally seen as underlying existent order and integral to it; it was something both vast and generative, the fertile ground out of which all forms came into being—not in a one-time event, but as an ongoing process.[1] The Egyptian Sun god Ra was born out of a formless abyss; in ancient Babylonian myth a multitude of new forms developed out of chaos and became the foundation of the structure of the universe; in China, the giant who brought order into the world at its beginning broke at birth out of the egg of chaos.

The early Greeks personified order and chaos in the gods Apollo and Dionysos—Apollo who drove the chariot of the sun in its dependable arc across the sky, the god of light, of music and the other arts, of order and reason; Dionysos, the god of drunkenness, of disorder and randomness—a contrasting duality underlying the unity of All That Is.[2] For Greek science, the cosmos was entirely orderly; the obvious presence of the unpredictable and chaotic in everyday experience could be explained by stories of the dangerously whimsical and fickle Olympic gods, of the Fates and Furies. [3] Though the Greeks sought to know the nature of the whole physical world, they made no attempt to prove their explanations by experiment. Their theories, founded in verbal logic, reached ultimate expression in the works of Aristotle. These were eventually woven into the dogma of the Roman Catholic Church, and with the Alexandrian Ptolemy’s geocentric astronomy, remained the official basis of scientific thinking in Europe until the time of Galileo, that is for nearly two thousand years.[4]

Medieval alchemists, whose ideas descended from ancient Babylonian beliefs, nevertheless gave chaos a place in their explanations of reality by assigning to it the function of a creative matrix from which all forms arise.[5] The experimental sciences evolved out of alchemy, and by the 1600s their findings had begun to conflict with the dogma of the Church. In 1616 the Church forbade the teaching of Copernicus’s hypothesis that the earth moves round the sun—rather than the sun round the earth as in Ptolemy’s system—on the grounds that the Copernican claim contradicted the Scriptures. The great Italian astronomer, physicist, and mathematician Galileo—who employed mathematical logic and knowledge based on observation as against Aristotle’s logico-verbal approach to scientific discovery—was found guilty of supporting the Copernican view. In 1634 he was forced to recant, and afterwards confined to his estate near Florence for the last eight years of his long life.

Galileo’s near-contemporary René Descartes was a French mathematician, scientist, and philosopher who lived and worked in the relative safety and intellectual freedom of Protestant Holland. He avoided Galileo’s difficulties with the Church to some degree by declaring a separation between the physical and the spiritual, matter and mind,[6] between those things that are of the world and the things that are of God. Descartes claimed that the only absolute certainty, in a world where the senses can be fooled and logic mistaken, is certainty of the existence of the thinker’s own thinking mind. Spinoza rephrased Descartes’ famous statement, “I think, therefore I am” more clearly: “I, in being conscious, am existent.”[7]

In Descartes’ understanding, the only assurance of the reality of everything outside the thinker’s consciousness of his or her own being in any given moment, is the reality of a perfect, benevolent, and non-deceiving God—since a real effect must necessarily be the product of a real cause. Only God, he concluded, exists eternally. All that exists in time, depending on God for its continuance, is either a body (having three-dimensional extensity and potential mobility), or a self (having awareness, feeling and volition).[8]

On this conceptual foundation Descartes built a consistent and coherent scientific method of discovery and demonstration for the exploration of the physical world (written, significantly, in accessible French rather than in the usual scholarly Latin).[9] His method could lead someone seeking certain knowledge from a few very general principles to conclusions which anyone could verify. His first understanding of this “wonderful discovery” and “marvelous science” came to him in a dream: he saw that physics could be reduced to geometry, and all the sciences be interconnected “as by a chain.” To be considered true knowledge, the results in every science must be as clear, as controllable, and as certain as are those in mathematics. Before Descartes, the lovely certainties of mathematics were thought to be beyond the reach of those sciences based on empirical data; it was assumed that each science’s method must vary with the materials it studied.

Descartes proposed that since reason is the same in all men, there then can be one universal method which may be applied in the same way to all kinds of problems and data, producing a single and universal science rather than a “collection of curiosities.” He began by analyzing reason as it is employed in mathematics: first its methods, and then the subjects to which it is applicable, like physics and chemistry, these being “sciences of number, weight and measure.”[10]

“The whole of philosophy is a tree whose roots are metaphysics, whose trunk is physics, and whose branches are the other sciences,” Descartes wrote. However this reasoning—by which the physical and spiritual aspects of reality, though distinct, were encompassed and interdependent within the same system of thought—was not carried forward by his most influential followers. Those followers generally ignored his metaphysical views, fundamental though he considered them to be; they expanded only on the mechanistic and materialistic elements of his thinking.[11] Thus the phenomenon of “modern science” came into being; Descartes’ metaphysical “roots” withered away.

Over time, the technological triumphs built on Descartes’ theoretical foundations, and achieved by newly organized scientific thought and experiment, produced the Industrial Revolution. By the late nineteenth century in some circles, and generally by mid-twentieth century, it was believed that science would eventually be able to explain all mysteries and solve all human problems by patient and rigorously impersonal exploration. Thus science seemed capable of satisfying the human desire identified by the great twentieth century theologian Paul Tillich as fundamental (though foolish because hopeless)—the desire for security, certitude and perfection. Science gradually came to be venerated as a kind of god.

Enlightenment philosopher David Hume (1711-1776) had expressed the determination to find such certainty and security in his famous adage, known as “Hume’s Fork”: “If we take in our hand any volume—of divinity or school metaphysics, for instance—let us ask, Does it contain any abstract reasoning concerning quantity or number? No. Does it contain any experimental reasoning concerning matter of fact and existence? No. Commit it then to the flames, for it can contain nothing but sophistry and illusion.”[12] If a statement contains no abstract meaning having to do with quantity or number—i.e. mathematics—or cannot be meaningfully tested against the world of facts, then it is nonsense; it follows of course that theology, metaphysics and so on are nonsense. To follow such advice allows one to live in a mental world that appears reassuringly certain, secure, and perfect, though vast areas of human experience must be ignored.

Roughly two hundred years later beliefs such as Hume’s were still alive, and were explicitly stated in the goals of the “logical positivists” of the Vienna Circle, as we have seen.[13] By the early twentieth century, “sophistry and illusion” in the form of religious beliefs having been more or less cleared out of the way for many, it was time to get rid of philosophical ideas held since at least the time of the ancient Greeks. The late professor of philosophy and art critic Arthur Danto has described the “revolution” in American academic philosophy brought about by these ideas when they arrived from Europe during and after the Second World War, brought by refugee philosophers fleeing the Nazis.[14] Professor Danto writes: “It was their view that in a certain sense philosophy as it had been known down the centuries had come to an end. In time it was replaced by something intellectually responsible, namely, science….[I]f a proposition happens to have no verifiable consequences, it is meaningless, or as they bluntly said, was nonsense.” This came to be known as “the verifiability criterion,” a longed-for standard by which to judge which among human ideas were valid. It was, and is, irresistibly seductive to many people, but it turned out that in the real world it cannot be applied in any satisfactory way. As Professor Danto points out,

A number of exceedingly sharp formulations of [this] seemingly lethal logical weapon demonstrated that as soon as one makes the principle tight enough to exclude as nonsense the philosophy the positivists sought to demolish, the principle forthwith excluded a lot of the science they were anxious to put forward as the very paradigm of meaningfulness [for instance all scientific generalizations from the particular or from a finite group]. And when it was loosened up enough to admit the latter, nonsense kept gushing in. It became a challenge to fix the criterion up to withstand these linked pressures, but in the end no one found out how…. The positivists continued to insist upon it as if it were true and fatal, but finally, except as a stratagem for intimidation, it stopped being interesting. Still, philosophy proceeded as if it were true.

[Arthur Danto, After the End of Art, p 142]

In my experience, almost everybody proceeded, and proceeds, as if it were true; very few people are aware that it isn’t. And its efficacy as a stratagem for intimidation remains powerful, which at least partly explains the rush to establish every academic discipline as a kind of science. Our craving for security, certainty and perfection—and its complement, the fear and rejection of the mysterious—lead us to an abject acceptance of putative scientific “facts,” founded on an ill-informed faith in the reductionist and ultimately inapplicable “verifiability criterion.”

At the same time, the general optimistic hubris was made possible by a necessarily limited vision. That vision focused on the great achievements of the previous 300 years or so and ignored the fact that in order to arrive at its successes, science had thus far been obliged to leave aside, to dismiss as “noise,” or to manage by mathematical adjustments, attributes of reality that were in fact absolutely fundamental. Chaos was dismissed as being merely complexity so great that at present scientists couldn't follow it; they would be able to do so in time, no doubt.[15] The general confidence was not diminished by the fact that natural systems—the migrations of butterflies, the rhythms of the human heartbeat, the workings of the brain—had so far proven virtually impervious to reductive scientific inquiry. Science, in presenting its theories and discoveries to the press and public, rarely mentions the mathematical maneuverings sometimes necessary to make the equations come out right, or indeed those vast areas which remain beyond our understanding.

It is often said that the human brain itself is the most complex physical structure in the universe. Scientists have estimated that the number of possible permutations and combinations of brain activity is greater than the number of elementary particles in the known universe. And the activities of the frontal lobes of the brain, which are involved in such aspects of our being as our moral sense and wisdom, are “the most mysterious of all.”[16]

Currently the “Big Bang” theory of the origin of the universe is widely accepted and promoted, and yet it is dependent on the proposition that ordinary matter—made up of electrons and atomic nuclei—constitutes only about 4.5 percent of the total energy of the universe[17] About 23 percent is made up of the particles of “dark matter,” which do not interact with ordinary matter or radiation, and are only known in that they appear to exert gravitational influence on those known realities. Most of the energy of the universe, about 72 percent, has been labelled “dark energy,” which “resides” in space itself, not in the masses of any sort of particle.[18] Thus that which forms 95 percent of the energy of the universe appears at present to be entirely unknowable. Certainly “dark” matter and energy manifest nothing whatever of the “number, weight, and measure” demanded by Descartes. The continuing search for them, at staggering expense of both money and effort, requires what could be called a “leap of faith”—in mathematics. The particle physicist Stephen Weinberg has written that “[t]he explanation of dark energy is now the deepest problem facing elementary particle physics.”[19] Even to a layperson, the explanation of 95 percent of the universe does indeed appear to be a deep problem.

The evolving method of what is called “classical” western science has been reductive: systems and entities are broken down into smaller and smaller parts, and those parts analyzed and studied more and more exhaustively, in the hope that when the pieces are put back together the orderly whole will be understood. Reductive scientists have found plenty of interesting things to investigate without venturing into such realms as volatility and turbulence, where chaos rules. The dream is that eventually all will be understood, and everything will thus be seen to play its part in some kind of universal order, explained by one great, all-encompassing "Final Theory."

In his surprise best-seller A Brief History of Time (1988), the British physicist Stephen Hawking wrote:

It turns out to be very difficult to devise a theory to describe the universe all in one go. Instead, we break the problem up into bits and invent a number of partial theories. Each of these partial theories describes and predicts a certain limited class of observations, neglecting the effects of other quantities, or representing them by simple sets of numbers. It may be that this approach is completely wrong. If everything in the universe depends on everything else in a fundamental way, it might be impossible to get close to a full solution by investigating parts of the problem in isolation. Nevertheless, it is certainly the way that we have made progress in the past.

[Stephen W. Hawking, A Brief History of Time, p 11]

That in fact everything in the universe does depend on everything else in a fundamental way is increasingly seen as likely, even by scientists. However classical science studies the universe as if it were a vast Cartesian machine made of independent parts which, from microcosm to macrocosm, can be taken apart in the mind, examined, and put back together again—looking for and identifying the orders inherent in it, and when possible making conceptual connections among those orders. As far as we can tell, if it is a machine, it is a machine with no beginning or end, no boundaries or any other limits.

For centuries, aberrations which could not be incorporated into the rational mathematical description of a phenomenon were seen as being side effects of the multitude of systems operating at the same time, bound to interfere with one another to some small degree. The advances of reductionist science have been made possible by ignoring such “noise,” or by modifying data slightly to account for it. This is perfectly respectable—it was necessary if investigation were to go forward, and the stunning advances in understanding and in technological development that it allowed have doubtless justified it. But one obvious trouble with the faith that reductive science will one day explain Everything is that living systems, all the creatures and plants that live on Earth, don't fit into its equations, and neither do the patterns of bird flight, or turbulence—whether in the ocean or a laboratory beaker. Such systems are both chaotic and orderly, entirely interdependent with other systems, and infinitely subtly nuanced. In fact phenomena which do fit perfectly into the reductionist cause-and-effect equations are very much the exception rather than the rule.

Complex systems [which include all living systems, and non-organic systems like the weather and the stock market]—both chaotic and orderly ones—are ultimately unmanageable, irreducible into parts, because the parts are constantly being folded into each other by iterations and feedback…and the system as a whole is constantly changing, bifurcating, iterating….

Time thus becomes an expression of the system's holistic interaction, and this interaction extends outward. Every complex system is a changing part of a greater whole, a nesting of larger and larger wholes leading eventually to the most complex dynamical system of all, the system that ultimately encompasses whatever we mean by order and chaos—the universe itself.

[John Briggs and David F. Peat, Turbulent Mirror, p 147]

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Sir Isaac Newton's equations describing and predicting the movement of our planet around the sun, or the moon around the earth, are accurate only when no more than two such bodies are considered. When a third celestial body is symbolically introduced into the calculations, the equations can produce no single answer. And of course there are never only two elements involved in any situation whatever. This fact is seldom pointed out to students, since the dominant story of science is that its march is triumphantly forward, from inviolable base camp to new outpost eventually to be made inviolable. (This claimed certainty can make high school science and mathematics dreary and dull to some students, attractive to others.)

Newton's equations were “linear,” that is they produced only one possible answer. They describe a world of simple cause and effect. But there is another class of equations, which are nonlinear, and produce two or many more possible answers. There are infinitely many linear equations—but there is an infinitely greater number of possible nonlinear equations, and they are very strange.

Nonlinear equations are like a mathematical version of the twilight zone. Solvers making their way through an apparently normal mathematical landscape can suddenly find themselves in an alternate reality. In a nonlinear equation a small change in one variable can have a disproportional, even catastrophic impact on other variables[[20]].…In linear equations, the solution allows the solver to generalize to other solutions; this isn't the case with nonlinear equations. Though they share certain universal qualities, nonlinear solutions tend to be stubbornly individual and peculiar. Unlike the smooth curves made by students plotting linear equations in high school math classes, plots of nonlinear equations show breaks, loops, recursions—all kinds of turbulence.…In the nonlinear world—which includes most of our real world—exact prediction is both practically and theoretically impossible.

[Ibid. p 24]

James Gleick points out that it is the solvable systems that are presented in textbooks:

They behave. Confronted with a nonlinear system, scientists would have to substitute linear approximations or find some other uncertain backdoor approach. Textbooks showed students only the rare non-linear systems that would give way to such techniques. The[se] did not display sensitive dependence on initial conditions. Nonlinear systems with real chaos were rarely taught and rarely learned. When people stumbled across such things—and people did—all their training argued for dismissing them as aberrations. Only a few were able to remember that the solvable, orderly, linear systems were the aberrations. Only a few, that is, understood how nonlinear nature is in its soul.

[James Gleick, Chaos: Making a New Science, p 68]

In classical reductive science, any randomness and chaos disturbing a system—such as a pendulum swinging in a vacuum or the revolving planets of our solar system—could only come from chance events outside such systems. Undisturbed, they would continue in their courses forever, unvarying.[21]

Late in the nineteenth century, however, it occurred to Henri Poincaré, yet another brilliant French mathematician, physicist and philosopher, that reductionism might be an illusion—that Newton's equations, essential as they were, might be fundamentally flawed as predictors of events in the natural world. Poincaré questioned Newton's conclusions regarding the stability of the solar system, since, as noted earlier, for more than two celestial bodies the Newtonian equations cannot be solved: a series of approximations is required to “close in” on an answer.[22] Poincaré added a third body to the two-body equations, and found that most predicted orbits would be affected only a very little by the presence of the third body, but that in some cases even a very small third gravitational pull might cause one of the planets to behave in an erratic, even chaotic way, to weave or wobble—or to fly out of its orbit, out of the solar system altogether.

To Poincaré, these discoveries were so disturbing that he put them aside, finding them “so bizarre that I cannot bear to contemplate them.”[23] And they were in any case entirely overshadowed by the new theories of relativity and quantum mechanics which appeared soon afterward. Poincaré had shown mathematically that chaos and turbulence, which had been thought to be a kind of infection coming from outside a given system, could arise within the system itself at any moment, given time, even if the system were sealed away from the influence of all other systems or events. This was a direct challenge to the Newtonian view of the world which had been the basis of classical science for nearly 200 years. It meant that chaos, disorder, and unpredictability were inherent in orderly systems.

In Science and Method, Poincaré wrote:

A very small cause which escapes our notice determines a considerable effect that we cannot fail to see, and then we say that the effect is due to chance. If we knew exactly the laws of nature and the situation of the universe at the initial moment, we could predict exactly the situation of that same universe at a succeeding moment. But even if it were the case that the natural laws had no longer any secret for us, we could still know the situation [only] approximately. If that enabled us to predict the succeeding situation with the same approximation, that is all we require, and we should say that the phenomenon had been predicted, that it is governed by the laws. But it is not always so; it may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible.

[Henri Poincaré, Science and Method, p 671]

Poincaré's equations lay untouched for half a century, until they were taken up again in the 1950's. By this time, early analogue computers were available and could be used to perform the mind-numbingly repetitive and seemingly endless (far beyond human capacity) computations necessary for exploring the multiple possible solutions to non-linear equations. At last a systematic study of chaos and change was possible, and it has continued, using computers to model complex systems.

Chaos theory—along with cybernetics, catastrophe theory, and information theory— was listed in a Scientific American article in 1995 as one of the failures of the 20th century attempt to find a theory of “well, almost everything.” Chaos, according to one of its former enthusiasts, “has had a declining output of interesting discoveries.”[24] If this is the case, it is at least partly for the reason suggested in another article in the same magazine; that many of the relationships in natural systems are far too complex to be modelled by the highest-powered computers available. Nevertheless, the continuing relevance and importance of the investigation of chaos was recognized in 2002 by the award of the prestigious Japan Prize to Professor James York, a founding theorist of Chaos in the 1960s.[25]

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Some of the most fundamental and obvious relationships in the physical world, such as that between the circumference of a circle and its diameter, must be represented by irrational numbers, as we may have learned in school. These are numbers that cannot be expressed precisely by either a final or a repeating decimal; they appear capable of going on forever, each digit appearing randomly.[26] Which is to say that their exact quantity cannot be expressed in the language of numbers (a language hugely more precise than the language of words, though limited in its application to those things which can be counted and measured). For practical purposes we can produce these relationships geometrically, using stakes and string or ruler and compass, or they can be approximated by numbers with a finite number of decimal points. The sharper the pencil, the more finely calibrated the ruler, or the greater the number of decimal points, the closer the approximation. Pi, which expresses the ratio between a circle’s circumference and its diameter, is an irrational number; the root of two—the relationship between the sides of a square and its diagonal—is another. The golden mean, section, or division is another; it is ubiquitous in both nature and art, and we will look at it later, in chapter eleven.

The parts in these instances have a precise relationship to each other and to the whole, but that relationship can’t be exactly quantified, which seems wonderful to the lover of mysteries, and merely a manageable nuisance to the reductive mind.[27] Like the sides of a square and its diagonal, the circumference of a circle and its diameter are said to be incommensurate, that is, they cannot be measured to a finite number on the same scale. If one dimension measures a rational number of units, the measure of the other will be an irrational number. At the same time, the circle, the square, and the golden rectangle are all geometrically inter-related.

Both irrational numbers and nonlinear equations describe fundamental features of the real world. And many of the conclusions arrived at by the scientists of chaos are fascinating, not only as ideas about the natural world, but because of what they may suggest about the relationships between us, our art-making, and the universe around and far beyond us. I was surprised to find these three passages in a book on Chaos Theory:

According to the theory of Paul LaViolette and William Gray, nuances circulate all the time from the emotional and perceptual centers of our brains only to become rapidly simplified by our cortex into thoughts that are categorical and “organizationally closed.” Everything we regard as our knowledge of the world is organizationally closed.[[28]] But our wondering, uncertainty, and questioning are full of nuance. In experiencing nuance we enter the borderline between order and chaos, and in nuance lies our sense of the wholeness and inseparability of experience

[John Briggs and David F. Peat, Turbulent Mirror p 195]

The tension between similarities and differences [within the poem given as example]…creates for us a sense of unpredictability and randomness in the creative work, a sense that what we’re experiencing is organic, is both familiar and unknown.…[E]verything not only affects everything else in the poem, it is everything else.

[Ibid. p 197]

[T]he right algorithm [in contemporary music] means one that creates nuances by balancing the randomness with self-similar features [order through repetition]. The resulting piece forces the listener to constantly interact with the music by recognizing it as a cloud of sounds which are obviously ordered and similar to each other but are also constantly unexpected and different.

[Ibid. p 199]

Similarly, in a New York Times review of a production of Verdi's opera Forza del Destino, which was first performed in 1862:

Put the right kind of voices in “Forza,” and audiences can always find their way through its sometimes bewildering expanse. Amid the apparent clutter, the music sweeps the listener up in the destinies of these three tortured souls as they intersect and diverge, and then, finally, catastrophically converge.

Step back a bit, though, and look and listen, and you begin to realize that the clutter and sprawl are in fact a large part of the point of “Forza.” No one will argue [but] that it is a taut piece of dramatic construction [i.e.: a highly resolved orderly structure], but it aims for the grand scale and fine texture, the combination of logic and randomness, the detail and pulse of life.

[The New York Times, 1995]

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Order is fundamental to beauty, and to life itself. At the same time, chaos is as essential to the universal reality which includes living organisms like us as order is, and chaos can be a generating source of order, just as the ancients proposed in their mythology.

Artists have probably always sensed the necessity for a measure of chaos, of “the contingent” in every situation, and that may be why they tend to be situated at the outside edge of any orderly social system.[29]

Some years ago, at a party, a young scientist suggested to me that the findings of the study of chaos and complexity are pleasing to lay people because they are so “intuitive;” they seem to support our own experience of the world. A woman standing nearby said, “Isn’t it great? We used to have to think of chaos as bad. Now we know it’s good!” Well, yes, but only to a degree. The understanding that chaos is not outside the orderly cosmos, but an integral part of it, is now scientifically supported. But that doesn’t mean that unrestricted chaos is ever “good” for living creatures.

The much-quoted Second Law of Thermodynamics tells us that entropy, which is one kind of chaos, is “the ultimate stage reached in the degradation of matter and energy of the universe.”[30] The universe, and all systems within it, tend to slide toward such entropy—toward a condition of increasing disorder, mixing, and randomness—this movement being as irreversible as time itself. However, and marvelously, all living creatures operate against this general movement toward disorder, dissolution and decay. In fact they produce new patterns of order, both physical and mental, as they evolve as entire species, and, subtly, even within a single lifetime.

Human beings make orders, making art, making laws, developing theories. An achieved order—such as the grammar and syntax of a language, or the way of being in the world proposed by a religion with its attendant metaphors and supportive rituals—is a significant communal accomplishment, a treasure, not something to be thrown away lightly, unless in favor of a more evolved order.[31] As Iris Murdoch says, chaos should be used judiciously, even within a single work of art: “Art which is nothing but such reminders [of the apparently disorderly world] ceases to be art.” In making order we make coherence, and even sometimes something wonderful, a kind of life. Yet a too-strict order is likely to collapse into chaos and entropy. This turns out to be the case even within our own living bodies:

The schizophrenia victim is suffering from too much order—trapped order—which paradoxically appears, in the epileptic seizure, as a massive attack of chaos…for the brain, chaos is entirely normal, but the chaos induced by too much order is devastating. One is reminded of Wallace Stevens' line, "a violent order is disorder."

[John Briggs and David F. Peat, Turbulent Mirror, p 167]

[A] heart that beats with the steadiness of a Swiss watch may actually be showing signs of debility or disease…a [healthy]young person with an average heart rate of 60 beats per minute may experience variations of up to 20 beats per minute every few heartbeats….

[“Arias from the Heart,” Harvard Alumni Magazine March/April 1996]

A researcher studying the patterns of the beating heart made an intriguing discovery:

[In]an unexpected dividend…[w]hile looking at a mathematical analysis of a heart-rate time series, [researcher Ary Goldberger] realized that it closely resembled a chart of the changes in pitch in classical music…colleague Chung-Kang Peng suggested they make a game of putting the heartbeats to music. Goldberger thought they would end up with “junk.” [He was] surprised by the tune that emerged instead….[H]is son…translated the heartbeat rhythms into piano melodies….

“Maybe music is therapeutic or restorative because it retunes our inner rhythms”[says Goldberger].

[Ibid.]

But music is more than simply rhythm, it manifests patterns of regular and irregular rhythm, and of all the other elements which constitute it. The woman happy to find that “chaos is good” had forgotten that there are kinds and degrees of chaos, and that “unfettered randomness” is as much a death-dealer as ever. Nevertheless, for many the Second Law’s grim prognosis has appeared to account for much of the disintegration and chaos of the modern world, and some artists and art theorists have used it as a sort of scientific justification for chaotic work. They are not alone in this, as James Gleick points out:

However expressed, the Second Law seems a rule from which there is no appeal. In thermodynamics that is true. But the Second Law has had a life of its own in intellectual realms far removed from science, taking the blame for disintegration of societies, economic decay, the breakdown of manners, and many other variations on the decadent theme.…[T]hese secondary metaphorical incarnations of the Second Law now seem especially misguided.…

Somehow, after all, as the universe ebbs toward its final equilibrium in the featureless heat bath of maximum entropy, it manages to create interesting structures.…[T]he slippery notion of entropy [is] reasonably well-defined for thermo-dynamic systems in terms of heat and temperature, but devilishly hard to pin down as a measure of disorder…thermodynamic entropy fails miserably as a measure of the changing degree of form and formlessness in the creation of amino acids, of microorganisms, of self-reproducing plants and animals, of complex information systems like the brain….The important laws, the creative laws, lie elsewhere.

[James Gleick, Chaos: Making a New Science p 308]

One thing that makes the notion of entropy so “slippery” is that at least two kinds of chaos are recognized: one is “entropic,” as described by the Second Law, and leads toward death and total disorder; the other is so-called “creative” chaos. However the words “chaos” and “entropy” are often used as if they were fully synonymous, which produces unfortunate confusion. Despite such warnings as that quoted above (published in 1987), the concept of entropy has not lost its popularity as an explanation for many aspects of contemporary life and art. An article in Harvard Design magazine by a theorist in landscape architecture can serve as an example:

Those conversant with the language of contemporary art know that entropy was a particular preoccupation of Robert Smithson. Several of his earthworks can be interpreted as pedagogic exercises in entropy. Smithson dumped asphalt into a quarry and let it run randomly down a slope; he piled dirt on the roof of a woodshed until the supporting beams cracked. His Nonsites—sculptures created by collecting materials from a place, sorting them into bins, and exhibiting them along with maps and photographs of their sites—might be described as efforts to reverse the effects of entropy, if only temporarily.…“The ‘pastoral,’ it seems, is outmoded. The gardens of history are being replaced by sites of time”—by sites, that is, that manifest the transformative effect of human action or of natural processes like erosion and decomposition.…[Smithson] wrote: “A bleached and fractured world surrounds the artist. To organize this mess of corrosion into patterns, grids, and subdivisions is an esthetic process that has scarcely been touched.”

[Harvard Design Magazine, Fall 2000, p 62]

How curious the results of such play with scientific concepts! Entropy is exemplified and then “organized” or “reversed, if only temporarily;” the pastoral gardens of the past are being replaced by “sites of time.” These are wastelands devastated by human activity, the detritus occasionally organized into boxes and piles in the name of art or counter-entropy. No landscape, unfashionably pastoral or not, can avoid being a “site of time,” and a testament to the motivations of its human inhabitants; this is more true than ever now, when bulldozers and explosives make almost any sort of ruination possible. Such landscapes are destroyed not by time, but by us.

The truth is that art, like all forms of life, moves against entropy: great art is not disordered, not random. It is likely to incorporate enough disorder, undetectable or overt, to make it live, but its parts are more or less clearly ordered.[32] “Things in a state of arrested disruption” are not art, unless that disruption is made part of some kind of overall order. The article goes on to say that in the contemporary study of complexity scientists attempt to understand the patterns found in natural systems which are dynamic and mutable, but at the same time self-organizing:

…for instance in the emergence of life from inert matter, in the evolution of more elaborate life forms from simpler ones, and in the increasingly intricate interdependencies within complex ecosystems like coral reefs and rain forests.…Landscape architecture is today exhibiting, in its own way, the tendency toward greater organization and complexity described by theorists and scientists, and in so doing is attempting to keep at bay randomness and disorder. And it is this tension—between order and disorder, between organization and entropy—that provides much of the narrative power of contemporary landscape architecture.”

[Ibid. p 63]

Thus opinion on art, however breathlessly, tries to keep up with, to emulate, science—as do so many academic disciplines. At the same time the writer of the piece, in his un-self-conscious humanity, experiences the power of the tension between organization and disorder when both are manifest in the same work. Art of which landscape is the medium, like all art, has always exhibited a “tendency toward organization.”[33] As to the degree of obvious complexity in art, this is now, and has always been, a choice open to the artist, made in response to constraints of various kinds, and influenced by motivation, personal taste, and technical capacity.

An increasing tolerance of disorder has been characteristic of the most “advanced” art for a hundred years and more. Perhaps, as adaptive animals, we have become progressively more inured to the ever-greater chaos we experience in our everyday environment. Some even claim to have developed a preference for disorder, as being “honest,” and “of our time.” But complete entropic chaos, “a confused mass or agglomerate of matters or heterogeneous items that are hard to distinguish, isolate, or interpret”[34] is the opposite of art, and is anti-life. Alex Ross, music critic of the New Yorker, has written,

“Noise” is a tricky word that quickly slides into the pejorative. Often, it’s the word we use to describe music that we don’t understand. Variations on the put-down “That’s just noise” were heard at the premiere of Stravinsky’s “Rite of Spring,” during Bob Dylan’s first tours with a band, and on street corners when kids started blasting rap. But “noise” can also accurately describe an acoustical phenomenon and it needn’t be negative. Human ears are attracted to certain euphonious chords based on the overtone series; when musicians pile on too many extraneous tones, the ear “maxes out.” This is the reason that free jazz, experimental rock, and experimental classical music seem to be speaking the same language: from the perspective of the panicking ear, they are. It’s a question not of volume but of density. There is, however, pleasure to be had in the kind of harmonic density that shatters into noise. The pleasure comes in the control of chaos, in the movement back and forth across the border of what is comprehensible.

[Alex Ross, The New Yorker, 7/13/99]

Again, science has identified two fundamental kinds of chaos. One is “a state of utter confusion, completely wanting in order, sequence, organization or predictable operation.” This is disorder when order has completely and irrevocably broken down in the process of time, as the Second Law of Thermodynamics predicts —the disorder of death. It is entropy, “the ultimate stage reached in the degradation of the matter and energy of the universe, a state of inert uniformity of component elements: absence of form, pattern, hierarchy, or differentiation.”[35] It is known as a “near-to-equilibrium” situation, that is, near to perfect balance.

The second kind of chaos is “a state of things in which chance is supreme: nature that is subject to no law…especially the confused unorganized state of matter before the creation of distinct and orderly form;” this is a “far-from-equilibrium” situation, that is, disorder before, and potentially generative of, order. [36]

Ilya Prigogine, winner of the Nobel prize in chemistry in 1977, was the first scientist to discover that new orderly systems may arise in “far-from-equilibrium” states. He is said to have spoken of such potentially creative chaos in “almost mystical” terms, and to have refused to define it.[37] Other scientists describe chaos in different ways, as, for instance, “the irregular, unpredictable behavior of deterministic, nonlinear dynamical systems” or “apparently random recurrent behavior in a simple deterministic (clockwork-like) system.”[38]

Since creative disorder, however subtle its expression, is a necessary characteristic of living systems such as human beings, it is unsurprising that it is also intrinsic to the works of art we make. “Nothing’s perfect” is not merely a resigned or disapproving evaluation of the reality that confronts us, it is an observation of the character of the great Logos in which we are at home. And yet to attain, in our actions, the living relationships which are < >´ takes skill, inspiration, dedication, and/or luck.

In art-making we transpose aspects of, and add to, the world around us, by imitating not only the forms and movements and sounds of that world, but the relationships among them. If the relationships within the work are “right” we are satisfied, which is why entirely abstract art can be convincing. Such relationships are to be found not only in the outside world but within ourselves. Chaos theorists speak of “optimistic time,” in which evolution and growth occur, and of the “pessimistic time” of decay and dissolution. It is not surprising that as healthy living creatures we prefer to experience the former in what we make, in what we see and hear and taste and touch, and in what we do. Pouring asphalt into a perfectly good quarry and calling it a “site of time” seems to me like making a mess and giving a fancy “scientific” explanation for it. It could conceivably be done so that < >´ occurs and art is made, though asphalt is a rather nasty substance when used for anything other than the surface of a road.

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As our appreciation for the nuances of chaotic behavior has matured by exposure to natural examples, novelties have emerged. Chaos and order have been found to co-exist in a curious symbiosis. Imagine a very large egg-timer in which sand is falling, grain by grain, to create a growing pile of sand in equilibrium, just on the verge of collapse. The pile amasses in an erratic manner. Sand-falls of all sizes occur, and their effect is to maintain the overall gradient of the pile of sand in equilibrium, just on the verge of collapse. This self-sustaining process has been dubbed “self-organizing criticality” by its discoverer Per Bak.

[John D. Barrow, The Artful Universe, p 244]

It may be that the condition I am calling “< >´ ” is some kind of “self-organizing criticality, an equilibrium just on the verge of collapse.” In a lovely example, we have the snowflake, an ever-different, yet always the same, entity which has long excited human delight and curiosity.

Ice crystals form in the turbulent air with a famous blending of symmetry and chance, the special beauty of six-fold indeterminacy….

Sensitive dependence on initial conditions serves not to destroy but to create. As a growing snowflake falls to earth, typically floating in the wind for an hour or more, the choices made by the branching tips at any instant depend sensitively on such things as the temperature, the humidity, and the presence of impurities in the atmosphere. The six tips of a single snowflake, spreading within a millimeter space, feel the same temperatures, and because the laws of growth are purely deterministic, they maintain a near-perfect symmetry. But the nature of turbulent air is such that any pair of snowflakes will experience very different paths. The final flake records the history of all the changing weather conditions it has experienced, and the combinations may well be infinite.

[James Gleick, Chaos: Making a New Science, pp 309, 311]

+

In this chapter I may have fallen into the very trap I warned against, that is of using only partly understood scientific “facts” as a base for flights of fancy in support of my own thesis. But the thesis came out of experience; I discovered the science by chance, and it is presented simply as a possibly persuasive support. The chapter is a layperson’s attempt to write coherently and succinctly at the edges of a vast subject. I have not ventured to describe phenomena such as fractals, strange attractors, solitons, and all the other marvelous discoveries of the science of chaos, although since they are inherent in the world around us they may well be relevant to my subject here.

The artist’s response to nature, and our own responses to both art and nature, are much deeper and more subtle than the usual understanding of Plato’s claim that the arts imitate nature would allow. Artists do often imitate the surface aspects of things, but it is the fundamental lively harmony of nature, its “naturalness” or “rightness” that we all crave and feed on, and try to imitate or find within ourselves. Some of the insights of chaos theory—however clumsily presented here—may help us to understand that dynamic harmony, and therefore to better perceive the powerful bonds connecting art, nature, and our own spirit. We are part of nature, of course, and so, at least theoretically, is what we make. Our best actions of every kind are each an instance of that “blending of symmetry and chance” that we so love because it enlivens us. It makes our life worth living, and indeed is probably essential to that life.

[1] John Briggs and David F. Peat, Turbulent Mirror: An Illustrated Guide to Chas Theory and the Science of Wholeness, p 19.

[2] For a rich discussion of this, see Friedrich Nietzsche, The Birth of Tragedy, 1872.

[3] It is believed that Pythagoras and his followers discovered the existence of irrational numbers, which were frighteningly inconsistent with this great Cosmic Order. But they were sworn never to reveal the existence of such numbers; it is said that to do so was punishable by death or banishment.

[4] Encyclopedia Britannica Vol 1, p 534-5, and Simon Blackburn, The Oxford Dictionary of Philosophy p 310.

[5] A contemporary version of this story: “No cosmological concept is as widely known as the big bang: from a state without physical order, lacking even space and time, matter appeared.” [Philip and Phyllis Morrison in Scientific American, Feb 2001, p 93] And of course in Genesis, “And the earth was without form, and void….” Time, and comprehensible existence of any kind, are still seen to begin with the establishment of order. In entropic chaos there is no meaning.

[6] Simon Blackburn, The Oxford Dictionary of Philosophy.

[7] Encyclopedia Britannica Volume 7, p 283 b.

[8] Ibid. p 284.

[9] In his Discourse on the Method of properly Guiding the Reason in the Search for Truth in the Sciences.

[10] Ibid. pp 282 and 283

[11] Encyclopedia Britannica Vol 4, p 974-5.

[12] David Hume, Inquiry Concerning Human Understanding, (XII(iii)) quoted in Simon Flew, A Dictionary of Philosophy p 156

[13] Chapter 7, “Arts and Sciences,” p 80.

[14] Arthur Danto, After the End of Art, p 141. In the pages following this, Professor Danto tells a fascinating story of the mid-twentieth century intellectual ferment in the United States and Europe which shaped some of the attitudes to art that we now take for granted.

[15] John Briggs and David F. Peat, Turbulent Mirror, p 22

[16] Berkeley neuroscientist Vilayanur S. Ramachandran, in the first of his BBC Radio “Reith Lectures,” 2003.

[17] Energy and matter are equivalent, at least under some conditions, as Einstein theorized and has now been demonstrated.

[18] Stephen Weinberg, “The Missions of Astronomy,” The New York Review of Books, 10/22/2009.

[19] Ibid.

[20] This is regularly experienced when painting a portrait, in the alternating discovery and loss of the mysterious and elusive “likeness.” At any point in the making of the painting, a tiny brushstroke on the painted eyelid, for instance, can destroy the work’s “likeness” to the individual living being that is the sitter. The painter has lost that singular and indescribable pattern of being, and cannot be sure of finding it again. A good portrait walks the edge of caricature, since it can be no more than a recognizable simplification of the full pattern of relationships—the logos—of the person.

[21] In this we can see reductive science as an ingenious human construction: a structure of concepts and measurements and equations to which many parts of reality appear to conform, and others can be made to conform. Useful, indeed. But we are ourselves entangled and caught in such nets of words, measurements, and concepts, so that it is only by conscious and practiced effort or by accident or luck that we experience the verbally mute suchness, the reality of the world and of the arts.

[22] John Briggs and David F. Peat, Turbulent Mirror, p 27.

[23] Ibid. p 29

[24] John Horgan, “From Complexity to Perplexity,” Scientific American, June 1995.

[25] National Public Radio, WBUR Boston, The Connection, 1/8/03

[26] John Briggs and David F. Peat, Turbulent Mirror, p 70.

[27] Pi, the number used to calculate what many consider the most perfect and ordered object of our imagination—the circle—can never be stated exactly. Even in the Euclidean world, order and chaos go hand in hand. [Ibid.]

[28] Re the idea that the brain’s function is in fact to filter and organize the information brought to us by our senses so that we can hope to rationalize it: see Aldous Huxley, The Doors of Perception, p 21

[29] In the 1960s, when signing a contract for a car rental in England, I was told by the agent not to write “artist” as my occupation, since the company didn’t rent cars to tinkers, gypsies, and artists.

[30] Webster’s Third New International Dictionary

[31] Heraclitus wrote: “The people must fight on behalf of the law as though for the city wall”. (Fragment 44, Diogenes Laertius IX}

[32] In a formal landscape, tree branches and their shadows move, fountains blow, and clouds come and go in unpredictable patterns moving within and above the easily apparent, often bilaterally symmetrical, order.

[33] It seems likely that over the aeons of our existence, our species has learned that chaotic situations in which some degree of orderliness is experienced are potentially interesting, capable of evolving into a new kind of order and for that reason exciting, even promising. Nowadays a pattern of some sort suddenly detected in a seemingly chaotic musical or theatrical performance can revive the interest of a despairing listener or observer. In the same way an astronomer scanning a random view of the universe through a telescope is intrigued by what appears to be an area of orderliness in the night sky. The tolerance of, and even preference for one or another degree of disorder varies from person to person and from situation to situation, as in the advice to the artist offered by the meticulous literary stylist Flaubert: to live like a bourgeois so that one can be wild in one’s work.

[34] Webster’s Third New International Dictionary

[35] Ibid.

[36] John Briggs and David F. Peat, Turbulent Mirror p 136

[37] Ibid.

[38] James Gleick, Chaos, p 306