Structure and Change
Since the early days of biology, philosophers and scientists have noticed that living forms, in many seemingly mysterious ways, combine the stability of structure with the fluidity of change. Like whirlpools, they depend on a constant flow of matter through them; like flames, they transform the materials on which they feed to maintain their activities and to grow; but unlike whirlpools or flames, living structures also develop, reproduce, and evolve.
In the 1940s Ludwig von Bertalanffy called such living structures open systems” to emphasize their dependence on continual flows of energy and resources. He coined the term Fliess- gleichgewicht (“flowing balance”) to express the coexistence of balance and flow, of structure and change, in all forms of life. 1 Subsequently ecologists began to picture ecosystems in terms of flow diagrams, mapping out the pathways of energy and matter in various food webs. These studies established recycling as a key principle of ecology. Being open systems, all organisms in an ecosystem produce wastes, but what is waste for one species is food for another, so that wastes are continually recycled and the ecosystem as a whole generally remains without waste.
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Green plants play a vital role in the flow of energy through all ecological cycles. Their roots take in water and mineral salts from the earth, and the resulting juices rise up to the leaves, where they combine with carbon dioxide (C0 2 ) from the air to form sugars and other organic compounds. (These include cellulose, the main structural element of cell walls.) In this marvelous process, known as photosynthesis, solar energy is converted into chemical energy and bound in the organic substances, while oxygen is released into the air to be taken up again by other plants, and by animals, in the process of respiration.
By blending water and minerals from below with sunlight and C0 2 from above, green plants link the earth and the sky. We tend to believe that plants grow out of the soil, but in fact most of their substance comes from the air. The bulk of the cellulose and the other organic compounds produced through photosynthesis consists of heavy carbon and oxygen atoms, which plants take directly from the air in the form of C0 2 . Thus the weight of a wooden log comes almost entirely from the air. When we burn a log in a fireplace, oxygen and carbon combine once more into C0 2 , and in the light and heat of the fire we recover part of the solar energy that went into making the wood.
Figure 8-1 shows a picture of a typical food cycle. As plants are eaten by animals, which in turn are eaten by other animals, the plants’ nutrients are passed on through the food web, while energy is dissipated as heat through respiration and as waste through excretion. The wastes, as well as dead animals and plants, are decomposed by so-called decomposer organisms (insects and bacteria), which break them down into basic nutrients, to be taken up once more by green plants. In this way nutrients and other basic elements continually cycle through the ecosystem, while energy is dissipated at each stage. Thus Eugene Odum’s dictum “Matter circulates, energy dissipates.” 2 The only waste generated by the ecosystem as a whole is the heat energy of respiration, which is radiated into the atmosphere and is replenished continually by the sun through photosynthesis.
Our illustration is, of course, greatly simplified. The actual food cycles can be understood only within the context of much more
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complex food webs in which the basic nutrient elements appear in a variety of chemical compounds. In recent years our knowledge of those food webs has been expanded and refined considerably by the Gaia theory, which shows the complex interweaving of living and nonliving systems throughout the biosphere—plants and rocks, animals and atmospheric gases, microorganisms and oceans.
The flow of nutrients through an ecosystem’s organisms, moreover, is not always smooth and even, but often proceeds in pulses, jolts, and floods. In the words of Prigogine and Stengers, “The energy flow that crosses [an organism] somewhat resembles the flow of a river that generally moves smoothly but from time to time tumbles down a waterfall, which liberates part of the energy it contains.” 3
The understanding of living structures as open systems pro-
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vided an important new perspective, but it did not solve the puzzle of the coexistence of structure and change, of order and dissipation, until Ilya Prigogine formulated his theory of dissipative structures. 4 As Bertalanffy had combined the concepts of flow and balance to describe open systems, so Prigogine combined “dissipative and structure” to express the two seemingly contradictory tendencies that coexist in all living systems. However, Prigogine’s concept of a dissipative structure goes much further than that of an open system, as it also includes the idea of points of instability at which new structures and forms of order can emerge.
Prigogine’s theory interlinks the main characteristics of living forms in a coherent conceptual and mathematical framework that implies a radical reconceptualization of many fundamental ideas associated with structure—a shift of perception from stability to instability, from order to disorder, from equilibrium to non- equilibrium, from being to becoming. At the center of Prigogine’s vision lies the coexistence of structure and change, of “stillness and motion,” as he eloquently explains with a reference to ancient sculpture:
Each great period of science has led to some model of nature. For classical science it was the clock; for nineteenth-century science, the period of the Industrial Revolution, it was an engine running down. What will be the symbol for usP What we have in mind may perhaps be expressed by a reference to sculpture, from Indian or pre-Columbian art to our time. In some of the most beautiful manifestations of sculpture, be it the dancing Shiva or in the miniature temples of Guerrero, there appears very clearly the search for a junction between stillness and motion, time arrested and time passing. We believe that this confrontation will give our period its uniqueness. 5
Nonequilibrium and Nonlinearity
The key to understanding dissipative structures is to realize that they maintain themselves in a stable state far from equilibrium. This situation is so different from the phenomena described by
classical science that we run into difficulties with conventional language. Dictionary definitions of the word “stable” include
fixed,” “not fluctuating,” and “unvarying,” all of which are inaccurate to describe dissipative structures. A living organism is characterized by continual flow and change in its metabolism, involving thousands of chemical reactions. Chemical and thermal equilibrium exists when all these processes come to a halt. In other words, an organism in equilibrium is a dead organism. Living organisms continually maintain themselves in a state far from equilibrium, which is the state of life. Although very different from equilibrium, this state is nevertheless stable over long periods of time, which means that, as in a whirlpool, the same overall structure is maintained in spite of the ongoing flow and change of components.
Prigogine realized that classical thermodynamics, the first science of complexity, is inappropriate to describe systems far from equilibrium because of the linear nature of its mathematical structure. Close to equilibrium—in the range of classical thermodynamics—there are flow processes, called “fluxes,” but they are weak. The system will always evolve toward a stationary state in which the generation of entropy (or disorder) is as small as possible. In other words, the system will minimize its fluxes, staying as close as possible to the equilibrium state. In this range the flow processes can be described by linear equations.
Farther away from equilibrium, the fluxes are stronger, entropy production increases, and the system no longer tends toward equilibrium. On the contrary, it may encounter instabilities leading to new forms of order that move the system farther and farther away from the equilibrium state. In other words, far from equilibrium, dissipative structures may develop into forms of ever-increasing complexity.
Prigogine emphasizes that the characteristics of a dissipative structure cannot be derived from the properties of its parts but are consequences of “supramolecular organization.” 6 Long-range correlations appear at the precise point of transition from equilibrium to nonequilibrium, and from that point on the system behaves as a whole.
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Far from equilibrium, the system’s flow processes are interlinked through multiple feedback loops, and the corresponding mathematical equations are nonlinear. The farther a dissipative structure is from equilibrium, the greater is its complexity and the higher is the degree of nonlinearity in the mathematical equations describing it.
Recognizing the crucial link between nonequilibrium and nonlinearity, Prigogine and his collaborators developed a nonlinear thermodynamics for systems far from equilibrium, using the techniques of dynamical systems theory, the new mathematics of complexity, which was just being developed. 7 The linear equations of classical thermodynamics, Prigogine noted, can be analyzed in terms of point attractors. Whatever the system’s initial conditions, it will be “attracted” toward a stationary state of minimum entropy, as close to equilibrium as possible, and its behavior will be completely predictable. As Prigogine puts it, systems in the linear range tend to “forget their initial conditions.” 8
Outside the linear region the situation is dramatically different. Nonlinear equations usually have more than one solution; the higher the nonlinearity, the greater the number of solutions. This means that new situations may emerge at any moment. Mathematically speaking, the system encounters a bifurcation point in such a case, at which it may branch off into an entirely new state. We shall see below that the behavior of the system at the bifurcation point (in other words, which one of several available new branches it will take) depends on the previous history of the system. In the nonlinear range initial conditions are no longer “forgotten.”
Moreover, Prigogine’s theory shows that the behavior of a dissipative structure far from equilibrium no longer follows any universal law but is unique to the system. Near equilibrium we find repetitive phenomena and universal laws. As we move away from equilibrium, we move from the universal to the unique, toward richness and variety. This, of course, is a well-known characteristic of life.
The existence of bifurcations at which the system may take several different paths implies that indeterminacy is another char-
acteristic of Prigogine’s theory. At the bifurcation point the system can choose —the term is used metaphorically—from among several possible paths, or states. Which path it will take will depend on the system’s history and on various external conditions and can never be predicted. There is an irreducible random element at each bifurcation point.
This indeterminacy at bifurcation points is one of two kinds of unpredictability in the theory of dissipative structures. The other kind, which is also present in chaos theory, is due to the highly nonlinear nature of the equations and exists even when there are no bifurcations. Because of repeated feedback loops—or, mathematically, repeated iterations—the tiniest error in the calculations, caused by the practical need to round off figures at some decimal point, will inevitably add up to sufficient uncertainty to make predictions impossible. 9
The indeterminacy at the bifurcation points and the “chaos- type” unpredictability due to repeated iterations both imply that the behavior of a dissipative structure can be predicted only over a short time span. After that, the system’s trajectory eludes us. Thus P r ig°gi ne ’ s theory, like quantum theory and chaos theory, reminds us once more that scientific knowledge offers but “a limited window on the universe.” 10
The Arrow of Time
According to Prigogine, the recognition of indeterminacy as a key characteristic of natural phenomena is part of a profound recon- ceptualization of science. A closely related aspect of this conceptual shift concerns the scientific notions of irreversibility and time.
In the mechanistic paradigm of Newtonian science, the world was seen as completely causal and determinate. All that happened had a definite cause and gave rise to a definite effect. The future of any part of the system, as well as its past, could in principle be calculated with absolute certainty if its state at any given time was known in all details. This rigorous determinism found its clearest expression in the celebrated words of Pierre Simon Laplace:
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An intellect which at a given instant knew all the forces acting in nature, and the position of all things of which the world consists— supposing the said intellect were vast enough to subject these data to analysis—would embrace in the same formula the motions of the greatest bodies in the universe and those of the slightest atoms; nothing would be uncertain for it, and the future, like the past, would be present to its eyes. 11
In this Laplacian determinism, there is no difference between the past and the future. Both are implicit in the present state of the world and in the Newtonian equations of motion. All processes are strictly reversible. Both future and past are interchangeable; there is no room for history, novelty, or creativity.
Irreversible effects (such as friction) were noticed in classical Newtonian physics, but they were always neglected. In the nineteenth century this situation changed dramatically. With the invention of thermal engines, the irreversibility of energy dissipation in friction, viscosity (the resistance of a fluid to flow), and heat losses became the central focus of the new science of thermodynamics, which introduced the idea of an “arrow of time.” Concurrently, geologists, biologists, philosophers, and poets all began to think about change, growth, development, and evolution. Nineteenth-century thought was deeply concerned with the nature of becoming.
In classical thermodynamics irreversibility, although an important feature, is always associated with energy losses and waste. Prigogine introduced a fundamental change of this view in his theory of dissipative structures by showing that in living systems, which operate far from equilibrium, irreversible processes play a constructive and indispensable role.
Chemical reactions, the basic processes of life, are the prototype of irreversible processes. In a Newtonian world there would be no chemistry and no life. Prigogine’s theory shows how a particular type of chemical processes, the catalytic loops that are essential to living organisms, 12 lead to instabilities through repeated self-amplifying feedback, and how new structures of ever-increasing complexity emerge at successive bifurcation points. “Irreversibility,”
Prig°gi ne concluded, “is the mechanism that brings order out of chaos.” 13
Thus the conceptual shift in science advocated by Prigogine is one from deterministic reversible processes to indeterminate and irreversible ones. Since the irreversible processes are essential to chemistry and to life, while the interchangeability of the future and the past is an integral part of physics, it seems that Prigogine’s reconceptualization must be seen in the larger context discussed at the beginning of this book in connection with deep ecology, as part of the paradigm shift from physics to the life sciences. 14
Order and Disorder
The arrow of time introduced in classical thermodynamics did not point toward increasing order; it pointed away from it. According to the second law of thermodynamics, there is a trend in physical phenomena from order to disorder, toward ever-increasing entropy. 15 One of Prigogine’s greatest achievements has been to resolve the paradox of the two contradictory views of evolution in physics and biology—one of an engine running down, the other of a living world unfolding toward increasing order and complexity. In Prigogine s own words, “There is [a] question, which has plagued us for more than a century: What significance does the evolution of a living being have in the world described by thermodynamics, a world of ever-increasing disorder?” 16
In Prigogine’s theory the second law of thermodynamics is still valid, but the relationship between entropy and disorder is seen in a new light. To understand this new perception it is helpful to review the classical definitions of entropy and order. The concept of entropy was introduced in the nineteenth century by Rudolf Clausius, a German physicist and mathematician, to measure the dissipation of energy into heat and friction. Clausius defined the entropy generated in a thermal process as the dissipated energy divided by the temperature at which the process takes place. According to the second law, that entropy keeps increasing as the thermal process continues; the dissipated energy can never be re-
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covered; and this direction toward ever-increasing entropy defines the arrow of time.
Although the dissipation of energy into heat and friction is a common experience, a puzzling question arose as soon as the second law was formulated: What exactly causes this irreversibility? In Newtonian physics the effects of friction had usually been neglected because they were not considered very important. However, these effects can be taken into account within the Newtonian framework. In principle, scientists argued, one should be able to use Newton’s laws of motion to describe the dissipation of energy at the level of molecules in terms of cascades of collisions. Each of these collisions is a reversible event, so it should be perfectly possible to run the whole process backward. The dissipation of energy, which is irreversible at the macroscopic level, according to the second law and to common experience, seems to be composed of completely reversible events at the microscopic level. So where does irreversibility creep in?
This mystery was solved at the turn of the century by the Austrian physicist Ludwig Boltzmann, one of the great theorists of classical thermodynamics, who gave a new meaning to the concept of entropy and established the link between entropy and order. Following a line of reasoning developed originally by James Clerk Maxwell, the founder of statistical mechanics, 17 Boltzmann devised an ingenious thought experiment to examine the concept of entropy at the molecular level. 18
Suppose we have a box, Boltzmann reasoned, divided into two equal compartments by an imaginary partition at the center, and eight distinguishable molecules, numbered from one to eight like billiard balls. How many ways are there to distribute these particles in the box in such a way that a certain number of them are on the left side of the partition and the rest on the right?
First, let us put all eight particles on the left side. There is only one way of doing that. However, if we put seven particles on the left and one on the right, there are eight different possibilities, because the single particle on the right side of the box may be each of the eight particles in turn. Since the molecules are distinguishable, these eight possibilities all count as different arrangements.
Similarly, there are twenty-eight different arrangements for six particles on the left and two on the right.
A general formula for all these permutations can easily be de-
Figure 8-2
Boltzmann’s thought experiment.
rived. 19 It shows that the number of possibilities increases as the difference between the numbers of particles on the left and right becomes smaller, reaching a maximum of seventy different arrangements when there is an equal distribution of molecules, four on each side (see figure 8-2).
Boltzmann called the different arrangements “complexions” and associated them with the concept of order—the lower the number of complexions, the higher the order. Thus, in our example, the first state with all eight particles on one side displays the highest order, while the equal distribution with four particles on each side represents the maximum disorder.
It is important to emphasize that the concept of order introduced by Boltzmann is a thermo -dynamic concept, where the molecules are in constant motion. In our example the partition of the box is purely imaginary, and molecules in random motion will keep going across it. Over time the gas will be in different states—
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that is, with different numbers of molecules on the two sides of the box—and the number of complexions for each of these states is related to its degree of order. This definition of order in thermodynamics is quite different from the rigid notions of order and equilibrium in Newtonian mechanics.
Let us look at another example of Boltzmann’s concept of order, which is closer to everyday experience. Suppose we fill a bag with two kinds of sand, the bottom half with black sand and the top half with white sand. This is a state of high order; there is only one possible complexion. Then we shake the bag to mix up the grains of sand. As the white and the black sand get mixed more and more, the number of possible complexions increases, and with it the degree of disorder, until we arrive at an equal mixture in which the sand is of a uniform gray and there is maximum disorder.
With the help of his definition of order, Boltzmann could now analyze the behavior of molecules in a gas. Using the statistical methods pioneered by Maxwell to describe the molecules’ random motion, Boltzmann noted that the number of possible complexions of any state measures the probability of the gas being in that state. This is how probability is defined. The more complexions there are for a certain arrangement, the more likely will that state occur in a gas with molecules in random motion.
Thus the number of possible complexions for a certain arrangement of molecules measures both the degree of order of that state and the probability of its occurrence. The higher the number of complexions, the greater will the disorder be, and the more likely the gas will be in that state. Boltzmann therefore concluded that the movement from order to disorder is a movement from an unlikely state to a likely state. By identifying entropy and disorder with the number of complexions, he introduced a definition of entropy in terms of probabilities.
According to Boltzmann, there is no law of physics that forbids a movement from disorder to order, but with a random motion of molecules such a direction is very unlikely. The larger the number of molecules, the higher the probability of movement from order to disorder, and with the enormous number of particles in a gas
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that probability, for all practical purposes, becomes certainty. When you shake a bag with white and black sand, you may observe the two kinds of grains drift apart, seemingly miraculously, to create the highly ordered state of complete separation. But you
are likely to have to shake the bag for a few million years for that event to happen.
In Boltzmann s language the second law of thermodynamics means that any closed system will tend toward the state of maximum probability, which is a state of maximum disorder. Mathematically this state can be defined as the attractor state of thermal equilibrium. Once equilibrium has been reached, the system is not likely to move away from it. At times the molecules’ random motion will result in different states, but these will be close to equilibrium and will exist only for short periods of time. In other words, the system will merely fluctuate around the state of thermal equilibrium.
Classical thermodynamics, then, is appropriate to describe phenomena at equilibrium or close to equilibrium. Prigogine’s theory of dissipative structures, by contrast, applies to thermodynamic phenomena far from equilibrium, where molecules are not in random motion but are interlinked through multiple feedback loops, described by nonlinear equations. These equations are no longer dominated by point attractors, which means that the system no longer tends toward equilibrium. A dissipative structure maintains itself far from equilibrium and may even move farther and farther away from it through a series of bifurcations.
At the bifurcation points, states of higher order (in Boltzmann’s sense) may emerge spontaneously. However, this does not contradict the second law of thermodynamics. The total entropy of the system keeps increasing, but this increase in entropy is not a uniform increase in disorder. In the living world order and disorder are always created simultaneously.
According to Prigogine, dissipative structures are islands of order in a sea of disorder, maintaining and even increasing their order at the expense of greater disorder in their environment. For example, living organisms take in ordered structures (food) from their environment, use them as resources for their metabolism,
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and dissipate structures of lower order (waste). In this way order “floats in disorder,” as Prigogine puts it, while the overall entropy keeps increasing in accordance with the second law. 20
This new perception of order and disorder represents an inversion of traditional scientific views. According to the classical view, for which physics was the principal source of concepts and metaphors, order is associated with equilibrium, as, for example, in crystals and other static structures, and disorder with nonequilibrium situations, such as turbulence. In the new science of complexity, which takes its inspiration from the web of life, we learn that nonequilibrium is a source of order. The turbulent flows of water and air, while appearing chaotic, are really highly organized, exhibiting complex patterns of vortices dividing and subdividing again and again at smaller and smaller scales. In living systems the order arising from nonequilibrium is far more evident, being manifest in the richness, diversity, and beauty of life all around us. Throughout the living world chaos is transformed into order.
Points of Instability
The points of instability at which dramatic and unpredictable events take place, where order emerges spontaneously and complexity unfolds, are perhaps the most intriguing and fascinating aspect of the theory of dissipative structures. Before Prigogine, the only type of instability studied in some detail was that of turbulence, caused by the internal friction of a flowing liquid or gas. 21 Leonardo da Vinci made many careful studies of turbulent flows of water, and in the nineteenth century a series of experiments was undertaken that showed that any flow of water or air will become turbulent at sufficiently high velocity—in other words, at sufficiently large “distance” from equilibrium (the motionless state).
Prigogine’s studies showed that this is not true for chemical reactions. Chemical instabilities will not automatically appear far from equilibrium. They require the presence of catalytic loops, which bring the system to the point of instability through repeated self-amplifying feedback. 22 These processes combine two different
phenomena: chemical reactions and diffusion (the physical flow of molecules due to differences in concentration). Accordingly, the nonlinear equations describing them are called “reaction-diffusion equations. They form the mathematical core of Pngogine’s the- ory, allowing for an astonishing range of behaviors. 25
The British biologist Brian Goodwin has applied Pngogine’s mathematical techniques in a most ingenious way to model the stages of development of a very special single-celled alga. 24 By setting up differential equations that interrelate patterns of calcium concentration in the alga s cell fluid with the mechanical properties of the cell walls, Goodwin and his colleagues were able to identify feedback loops in a self-organizing process, in which
structures of increasing order emerge at successive bifurcation points.
A bifurcation point is a threshold of stability at which the dissipative structure may either break down or break through to one of several new states of order. What exactly happens at this critical point depends on the system’s previous history. Depending on which path it has taken to reach the point of instability, it will
follow one or another of the available branches after the bifurcation.
This important role of the history of a dissipative structure at critical points of its further development, which Prigogine has observed even in simple chemical oscillations, seems to be the physical origin of the connection between structure and history that is characteristic of all living systems. Living structure, as we shall see, is always a record of previous development. 25
At the bifurcation point, the dissipative structure also shows an extraordinary sensitivity to small fluctuations in its environment. A tiny random fluctuation, often called “noise,” can induce the choice of path. Since all living systems exist in continually fluctuating environments, and since we can never know which fluctuation will occur at the bifurcation point just at the “right” moment, we can never predict the future path of the system.
Thus all deterministic description breaks down when a dissipative structure crosses the bifurcation point. Minute fluctuations in the environment will lead to the choice of the branch it will fol-
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low. And since, in a sense, it is those random fluctuations that lead to the emergence of new forms of order, Prigogine has coined the phrase “order through fluctuations” to describe the situation.
The equations of Prigogine’s theory are deterministic equations. They govern the system’s behavior between bifurcation points, while random fluctuations are decisive at the points of instability. Thus “self-organization processes in far-from-equilibrium conditions correspond to a delicate interplay between chance and necessity, between fluctuations and deterministic laws.” 26
A New Dialogue with Nature
The conceptual shift implied in Prigogine’s theory involves several closely interrelated ideas. The description of dissipative structures that exist far from equilibrium requires a nonlinear mathematical formalism, capable of modeling multiple interlinked feedback loops. In living organisms these are catalytic loops (that is, nonlinear, irreversible chemical processes), which lead to instabilities through repeated self-amplifying feedback. When a dissipative structure reaches such a point of instability, called a bifurcation point, an element of indeterminacy enters into the theory. At the bifurcation point the system’s behavior is inherently unpredictable. In particular, new structures of higher order and complexity may emerge spontaneously. Thus self-organization, the spontaneous emergence of order, results from the combined effects of nonequilibrium, irreversibility, feedback loops, and instability.
The radical nature of Prigogine’s vision is apparent from the fact that these fundamental ideas were rarely addressed in traditional science and were often given negative connotations. This is evident in the very language used to express them. Non- equilibrium, nonlinearity, instability, indeterminacy, and so on, are all negative formulations. Prigogine believes that the conceptual shift implied by his theory of dissipative structures is not only crucial for scientists to understand the nature of life, but will also help us to integrate ourselves more fully into nature.
Many of the key characteristics of dissipative structures—the sensitivity to small changes in the environment, the relevance of
DISSIPATIVE STRUCTURES
previous history at critical points of choice, the uncertainty and unpredictability of the future—are revolutionary new concepts from the point of view of classical science but are an integral part of human experience. Since dissipative structures are the basic structures of all living systems, including human beings, this should perhaps not come as a great surprise.
Instead of being a machine, nature at large turns out to be more like human nature unpredictable, sensitive to the surrounding world, influenced by small fluctuations. Accordingly, the appropriate way of approaching nature to learn about her complexity and beauty is not through domination and control, but through respect, cooperation, and dialogue. Indeed, Ilya Prigogine and Isabelle Stengers gave their popular book, Order out of Chaos, the subtitle “Man’s New Dialogue with Nature.”
In the deterministic world of Newton there is no history and no creativity. In the living world of dissipative structures history plays an important role, the future is uncertain, and this uncertainty is at the heart of creativity. “Today,” Prigogine reflects, “the world we see outside and the world we see within are converging. This convergence of two worlds is perhaps one of the important cultural events of our age.” 27
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