Journal of the American Psychoanalytic Association http://apa.sagepub.com
Finding Your Way Through Chaos, Fractals, and Other Exotic Mathematical Objects: a Guide for the Perplexed Robert M. Galatzer-Levy J Am Psychoanal Assoc 2009; 57; 1227 DOI: 10.1177/0003065109347905 The online version of this article can be found at: http://apa.sagepub.com
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Robert M. Galatzer-Levy
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FINDING YOUR WAY THROUGH CHAOS, FRACTALS, AND OTHER EXOTIC MATHEMATICAL OBJECTS: A GUIDE FOR THE PERPLEXED
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ome psychoanalysts and neuroscientists assert that mathematical ways of thinking, variously called nonlinear dynamics, complexity theory, chaos theory, catastrophe theory, small-world theory, and artificial life (to name a few), hold keys that integrate and enrich the study of human minds, feeling, and brain function. They refer to ideas like nonlinearity, complexity, chaos, the fractal nature of reality, and emergence, suggesting that puzzling aspects of mental life can be comprehended using them. Is there content behind these obscure words? Can mathematics contribute to psychoanalysis? Does this project continue the intellectual imperialism of the “hard sciences” over psychology? How can an analyst come to a position of understanding and rationally evaluate these claims or even learn enough to appropriately use these ideas? In this review I will try to provide a road map for analysts to begin an intellectually credible exploration of nonlinear dynamics. I have deliberately omitted any discussion of specific applications of such dynamics to psychoanalysis.1 My purpose is to help you find your way into them from a sound base in the underlying theories. 1 The applications of these new ideas to psychoanalysis, including gender (Harris 2005), analytic process (Palumbo 1999; Coburn 2002; Galatzer-Levy in press; Levinson 1994; Lyons-Ruth 1999; Stern et al. 1998; Pincus, Freeman, and Modell 2007; Procci 2002; Stolorow 1997; Taerk 2002), child analysis (Tyson 2005), development (Galatzer-Levy 2004; Hershberg 2006), mind-brain theory (Davis 2002; Freeman 2007: Levin 2003), therapeutic impasse (Harris 2009); postmodern theory (Leffert 2008), metaphor (Antal 2008; Stein 1999), relational psychology (Seligman 2002), group therapy (Pincus and Guastello 2005; Rubenfeld 2001), character (Piers 2000), psychoanalytic education (Levin 2006), transference (Pincus, Freeman, and Modell 2007), and other psychoanalytic issues have led some of us (Coburn 2000, 2007; Moran 1991; Palumbo 2007; Priel and Schreiber 1994; Galatzer-Levy 1995; Seligman 2005; Spruiell 1993) to claim that a whole new paradigm for thinking about psychoanalysis is emerging.
DOI: 10.1177/0003065109347905 Downloaded from http://apa.sagepub.com at UNIV OF CHICAGO on January 1, 2010
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HISTORICAL CONTEXT
To understand the nonlinear revolution requires a brief excursion into the intellectual world from which it emerged. Before the sixteenth century the physical world was seen as the work of a masterful human-like intelligence, explainable in terms of His intentions and their meanings. Amazing advances in astronomy and technology led to the realization that many phenomena that had seemed comprehensible only as the product of a human-like will could be explained as the product of simple laws of nature. This “mechanization” of the scientific worldview—the idea that the world is best explained as one would the operation of a machine (Dijksterhuis 1950)—has dominated intellectual life ever since. In the hands of Newton, Leibniz, and their followers, using the incredibly powerful mathematical tools they invented, the new worldview resulted in a new type of human understanding, whose impact is matched only by that of religion. By the mid-nineteenth century, biology adopted this worldview, and by the beginning of the twentieth century Freud, following Fechner, helped make it a centerpiece of the study of human psychology (Boring 1950). Like all intellectual activity, scientific study is dominated by its methods. This dominance is most profound when the impact of method is implicit and unrecognized so that features of science’s picture of the world that arise from its methods are misconstrued as intrinsic features of its objects of study. The mechanistic worldview’s most potent method is the study of differential equations, mathematical expressions relating quantities and their rates of change. In elementary physics, for example, we learn that force equals mass times acceleration, F = ma, or that acceleration equals force divided by mass, a = F/m. Acceleration is the rate of change of velocity, which is itself the rate of change of position. This is a differential equation—the rate of change of the rate of change of position equals the force on the body divided by its mass. The study of differential equations tells us that if we know the initial position and velocity of an object and the forces acting on it, this differential equation predicts the body’s future position. Combined with a knowledge of physical forces, this simple equation made the physics and technology that reshaped the material and intellectual world possible. Differential equations put a man on the moon, provided energy previously unimaginable, and led to our understanding the intimate workings of the neurons that compose our brains. No wonder differential equations came to be thought of as the natural way to comprehend anything subject to rational contemplation. 1228 Downloaded from http://apa.sagepub.com at UNIV OF CHICAGO on January 1, 2010
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However, to think about differential equations critically, in a manner that allows one to understand precisely what is going on, requires substantial mathematical sophistication and education. Indeed, differential equations are commonly used mechanically without much understanding and, so, without appreciation of the assumptions built into them. The combination of great power and intellectual difficulty gave the worldview derived from differential equations enormous authority and more or less uncritical acceptance. The way in which the language of differential equations shaped thought was obscured. The obscurity increased when ways of thinking derived from differential equations were applied without using the equations themselves. For example, when Freud (1914) asked “what happens to the libido that has been withdrawn from objects” (p. 74), he posited a U-tube model relating narcissistic and object libido (Kohut 1971)—since the total amount of libido remains the same when one form of libido is increased, another form must be decreased. Freud was drawing on concepts of physical energy conservation detached from their grounding in mathematical physics. The result was that it remained unclear for many decades that psychic energy is not conserved in the same way physical energy is. The U-tube analogy does not apply. Only a small subclass of differential equations could be used for scientific computation before the development of the digital computer. These equations shared the property that they predict that small changes in initial conditions result in small changes in the evolution of a system. (If you throw a rock a little harder, it goes a little further. It doesn’t suddenly go into an exotic orbit in space.) These equations usually share the quality that their elements, the rates of change of their variables, appear in the equation as sums of those elements multiplied by constants. They are called “linear” because they have the same form as the algebraic expression for straight lines. All other differential equations are “nonlinear.” This is a bit like referring to most mammals as “non-platypus mammals.” Linear equations were thought of as normal because they are tractable; nonlinear equations were treated as strange because there were few ways to successfully explore them. But in fact if one looked dispassionately at differential equations, only the tiniest portion of them would be linear. Norbert Wiener, a mathematical genius who benefited greatly from his own psychoanalysis (Wiener 1956, 1961), observed that the distinctive feature of biological and psychological organisms is their use of feedback to shape future action. Feedback occurs when a system’s previous output changes what it does next—that is, the results of actions become part of 1229 Downloaded from http://apa.sagepub.com at UNIV OF CHICAGO on January 1, 2010
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the input to future steps. Feedback is central in biology, from the broad form of evolution to the motion of a hand in lifting an object. Engineers attempted to build machines using similar adaptive feedback. This turned out to be far more difficult than anticipated because the underlying mathe matics of feedback mechanisms seemed to the best mathematicians of the day to be devilishly hard (something the physicist A. J. Lotka had discovered independently in studying ecology). Wiener simultaneously struggled with the mathematics of designing machines that use feedback and applying the resulting insights to biological and psychological systems. Working when the possibilities for digital computers were emerging, Wiener empha sized the comparison between computers and the mind-brain. In doing so he confronted the differences between linear and nonlinear equations and the very different pictures of the world to which they lead. Wiener attempted to solve some of these problems using the mathematics then available. These attempts produced pages of fearsome-looking equations, best passed over by today’s readers except as illustrations of how intractable these problems were to the best mathematicians of the mid-twentieth century. Wiener’s masterwork, Cybernetics (1961), is a prelude to the application of nonlinear dynamics to psychological problems. Despite their seeming intractability, nonlinear differential equations have attracted the attention of some of the greatest mathematicians. At the beginning of the twentieth century Henri Poincaré, considered by many the greatest mathematician of his time, realized that what is most important about a differential equation is not its numerical solution, even though such solutions are crucial to technological applications, but the qualitative features of the solution—does the predicted motion repeat so that periodic motion results, does it gradually damp down so that the system becomes static, do the oscillations predicted grow ever larger, and so on? This qualitative theory of differential equations led to the insight that linear equations produce a small number of possible qualitatively different results, but that the results yielded by nonlinear equations are complicated, varied, and deeply surprising. In the 1960s, the French mathematician René Thom demonstrated that a group of nonlinear differential equations predict types of motion observed in the real world but not encompassed in the field of linear differential equations. These motions involve abrupt changes and motions that when reversed do not return to their starting points. Thom’s enormously influential book, Structural Dynamics and Catastrophe Theory (1975), whose impeccable mathematics and narrow scope put it beyond most psychoanalytic readers, nonetheless rewards 1230 Downloaded from http://apa.sagepub.com at UNIV OF CHICAGO on January 1, 2010
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close attention to the pages he devotes to the implications of his work for the process of gathering empirical knowledge. Eugene Wigner (1960) summarized two millennia of amazement with the usefulness of mathematics in the observation that mathematics is “unreasonably effective” in the sciences. How is it that again and again mathematical formulations have provided deep and quantitatively impressive understandings of natural phenomena? Thom provides a psychological answer to this question. He observes that absent a theory that encompasses observations, researchers ignore observations themselves. Lacking a theory and, hence, a language with which to talk about abrupt change, investigators either paid no attention to them or treated them as deviant variations from a picture of an orderly, smoothly transitioning world. Thom’s observation is particularly important for psychoanalysts because the central heuristic Freud employed to understand apparent discontinuity is the idea that discontinuities are merely apparent and that all psychological actions are linked in a continuous manner, though the links are often disguised and hidden. Analysis fills in the steps hidden from awareness. Freud’s foundational works on dreams, slips, symptoms, and jokes are largely detailed demonstrations of the power of this heuristic. Every analyst shares some of this experience. But our commitment to the heuristic is so powerful and automatic that it blinds us to the genuine discontinuities that Thom showed are part of the physical and mathematical world. The further development of catastrophe theory has a sad and telling dimension. It was immediately obvious that the theory was enormously suggestive for research in the human sciences and might explain dramatic social phenomena such as prison riots. Led by the mathematician Christopher Zeeman (1977), researchers claimed that catastrophe theory could answer a wide range of questions and presented these claims with impassioned enthusiasm that seemed like “hype” to many scholars. As a result, catastrophe theory faded from the mathematics of the social sciences. During this same period, computer-based researches produced investigators like Benoit Mandelbrot (1982) and Edward Lorenz (1993). Lorenz’s retelling of the story in The Essence of Chaos is at the same time an engaging read for nonmathematicians and an accurate description of the theory’s basic ideas. Lorenz, an MIT meteorologist, tackled the enormously difficult and important problem of long-range weather forecasting using computational muscle. The differential equations for weather are clearly correct, but there is no way to write down their solution as an equation; starting with the explicit rules about how weather changes over time, 1231 Downloaded from http://apa.sagepub.com at UNIV OF CHICAGO on January 1, 2010
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it is not possible to write an equation for what the weather will be at a particular time. Enter the computer, with its capacity for untiring rapid calculation. The computers available to Lorenz were primitive by today’s standards. An accidental feature of these computers was that while they carried out computations to ten decimal places, they printed results to only eight. Because the computations were so time-consuming, Lorenz sometimes had to turn off the computer overnight and resume the computations using numbers from the printout (which lacked the ninth and tenth decimal places). Intuition suggested that this should not yield results significantly different from those achieved by simply letting the computer continue with the computation. After all, the difference between the two corresponded to about one part in ten million. This is much smaller than the roughly one-part-in-a-hundred error in the measurement of meteorological variables like air pressure and temperature, so if it was possible to predict long-term weather from meteorological data at all, this computational error should make no difference. Then came the shocker. The weather predicted by the computers that were restarted missing the last two decimals quickly became entirely different from that predicted when the computation was allowed to continue uninterrupted. Something in the equations implied that minimal changes in the initial conditions, far smaller than could actually be measured, quickly caused profound changes in the results. Rather than treating these findings as an aberration or throwing up his hands in despair, Lorenz explored the underlying mathematics. These rock-solid equations of classical physics predicted that arbitrarily small changes in initial conditions predict large changes in the weather. (The idea was popularized as the “butterfly effect”—a butterfly flapping its wings in Sumatra could determine whether there was a snowstorm in Chicago a week later.) Years earlier mathematicians had developed the concept of an attractor, a pattern of motion toward which movement governed by a differential equation tends. For example, pendulums, even if disturbed, tend to return to swinging in a predictable periodic fashion. The attractors that were commonly studied had simple structures. Lorenz found that the attractor of the equations he was studying were far more complex; tiny changes in initial conditions yielded dramatic changes in the resulting motion. It was as if by starting a pendulum swinging a millimeter away from its usual starting point, it was made to suddenly fly off in an entirely different direction, doing curlicues as it went. What looked like a computational error turned out to be a window into a new mathematical 1232 Downloaded from http://apa.sagepub.com at UNIV OF CHICAGO on January 1, 2010
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world. It was quickly recognized that there are a host of “strange attractors”—strange in the sense that that slight changes in conditions lead to radically different outcomes. Approximately a decade later, another advance in technology, the development of computer graphics, led Mandelbrot to discover fractals. In a lecture available on the internet (Mandelbrot 2001), he describes how his long-standing interest in rough surfaces led him to search for simpler descriptions of them and to the discovery that extremely simple equations can generate surfaces of literally infinite complexity. The resulting graphics are not only stunningly beautiful but strikingly resemble configurations appearing in nature—coastlines, ferns, trees, clouds. Several books and websites are devoted to fractals from an essentially aesthetic point of view (Sprott 2008; Mandelbrot 1982; Devaney and Keen 1989; Peitgen, Jürgens, and Saupe 2006; Peitgen and Richter 1986). They are well worth the analyst’s attention not only because of their beauty, but also because they clearly demonstrate how enormous complexity can arise from simple rules. Of these works, Peitgen and Richter’s The Beauty of Fractals: Images of Complex Dynamical Systems (1986) is at the same time the most aesthetically pleasing and the most mathematically informative. The math will be too challenging for most analysts, but the pictures will wow you and give some intuitive sense of what this is all about. Sprott’s Fractal Gallery website (1992) includes hundreds of fractals, including “the fractal of the day” randomly generated each morning, and is a wonderful place to wander around and observe the variety of the nonlinear dynamical world. It offers little by way of explanation, however. From a conceptual viewpoint, the most amazing feature of fractals is that the rules that determine them can be extremely simple. This quality of richness arising from simplicity and the configurations arising from elements that one would not imagine would lead to an end result is called emergence. It is a quality not limited to computer simulations, but one that may be manifest in any system with even a modest degree of complexity. It is centrally important in the study of biology, ecology, evolution, and economics. For example, each of us is born with a small number of genes that specify the early development of white blood cells, a small menu of antibodies and antibody precursors, and a brief set of rules for their transformation and combination. Set in motion and confronted with the ordinary vicissitudes of life, each of us evolves a unique immune system that differentiates normal body tissue from invaders and mounts effective 1233 Downloaded from http://apa.sagepub.com at UNIV OF CHICAGO on January 1, 2010
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defenses against them. This system is believed to be able to recognize and respond to every possible organic molecule. Yet the rules on which it operates can be summarized in a page. The Nobel laureate Gerald Edelman, who was among the discoverers of this process, argues in Neural Darwinism (1987) that the brain operates along similar lines.2 The recognition that complex phenomena can emerge from simple rules is profoundly useful to the psychoanalyst in search of explanations for the labyrinths created by the human mind. For example, Freud saw that the enormous number of delusions constructed by Daniel Schreber could all be described as the product of the application of a simple set of rules (projection) to anxiety-arousing mental content. Emergence also refers to the coming together of elements to form new configurations with new functions. A beautiful example of such emergence, described in Thelen and Smith’s A Dynamic Systems Approach to the Development of Cognition and Action (1994), is demonstrated by children learning to walk. This work is particularly rewarding for psychoanalysts because it clearly illustrates how the concept of emergence can be studied in relation to realworld phenomena and because it emphasizes the centrally useful feature of emergent systems—there are many routes to good functional development. This could serve as a major corrective to traditional psychoanalytic views that there is but a single normal developmental line, deviation from which risks psychopathology. At the turn of the twentieth century, long before the development of powerful computer graphics, Poincaré recognized that the most potent approach to understanding all types of motion is through topology, the study of the fundamental form of a motion, such as its being repeated or bounded. Since Descartes, mathematicians have understood that anything that can be represented as equations can also be represented in an equivalent 2 In brain research we see how researchers continue to focus on linear systems because of their promise of clear results. Some of the most exciting work in neuro science concerns the worm C. elegans, chosen because its extremely simple nervous system is entirely determined by its genetics. This organism cannot possibly model major features of our nervous system because our entire genome is, by several orders of magnitude, inadequate to fully specify our brains. Similarly, by their very design most neuroimaging investigations of mental function, focused on the localization of brain functions, do not include the idiosyncratic and dynamic features of brains, while generating an ever larger volume of data suggesting static configurations. See Freeman (2000) for a brilliant discussion of why the brain must be conceptualized as a nonlinear dynamic system. Thom’s warning that data tend to be collected that support an accepted worldview at the expense of exploring alternatives, no matter how obvious they may be, applies well to the situation in current neuroscience.
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graphic form and that exploring the geometric forms associated with equations leads to enriched understanding of both form and equation. Topology is an extension of geometry that studies the properties forms maintain when they are stretched, or compressed, or undergo similar deformation without being cut or pasted together. For example, a teacup and a doughnut are topologically equivalent because a teacup can be transformed into the shape of a doughnut simply by stretching and compressing parts of it, but a doughnut and a ball are not because transforming a ball into a doughnut requires cutting. Let me briefly interrupt my narrative, because this is the point at which many analysts’ eyes start to glaze over. Most of us were attracted to psychoanalysis because it provides an excuse to study people’s lives in depth, lingering on the details and particulars that make life interesting. Abstractions like topology, while they may be interesting in their own right, seem to move in precisely the opposite direction from this core pull of psychoanalysis. Struggling with this problem, Bion (1977) observed that the power of abstraction to call attention to fundamental configurations that can easily be lost among a swarm of details, and to allow precise statements of otherwise inarticulable matters, complements rather than detracts from the detailed study of lives. The analyst who contemplates the topological configurations in an analysand’s associations is not pulled away from their particulars (unless the contemplation is used defensively) but instead is invited to appreciate elements of the associations that might otherwise remain invisible. Topology applied to differential equations addresses the form of the motion predicted by the equation. Does it repeat? Does it move back and forth between a few distinct types of motion? Does it fly off into space? Just as graphs can represent any number of things (the Dow-Jones average vs. the year, the acceleration of a rocket vs. its height), so too the “spaces” we are talking about can represent anything we choose (the viscosity of a substance in relation to temperature and pressure, the intensity of negative affect in relation to the extent the analyst has been empathic and to the duration of the analysis). But because the topological approach gave too little information for the physical and technological problems for which differential equations were primarily used, it was not widely adopted. Instead mathematicians either approximated equations that could not be solved directly with ones that could be, or did numerical computations to find approximate solutions, even though they knew these methods to be fundamentally flawed. 1235 Downloaded from http://apa.sagepub.com at UNIV OF CHICAGO on January 1, 2010
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In the 1960s, Stephen Smale revived Poincaré’s work in order to study nonlinear differential equations. The term nonlinear commonly confuses psychoanalysts and other nonmathematical readers, who equate it with complicated or semi-mystical. Remember high school algebra. The equation for a line (y = ax + b, where a and b are constants) involves only multiplying variables by constants. The equation for a parabola (y = ax2 + b) or hyperbola ([x-a])y = c) involves multiplying or dividing variables by one another. Equations having the form of the equation of a line are called linear; those that do not are called nonlinear. This is all the term nonlinear means. The term does not entail anything more mysterious, vague, or outside of ordinary experience. However, it does turn out that linear differential equations (differential equations in which variables and rates of change of those variables appear multiplied only by constants and never by each other) have much simpler properties than nonlinear differential equations (equations that do not meet these requirements). Smale showed that the topology associated with differential equations, in particular the nature of the connectedness of those forms and the possibility of moving from one place in the form to another, provides rich insights into the properties of the dynamical system in question. For the psychoanalytic reader who is attracted to geometry and topology (not to mention quite funny cartoons), Abraham and Shaw’s wonderful Dynamical Systems (1992) provides an introduction to the field that is at once accurate, deep, and accessible to anyone with a minimal knowledge of geometry and a willingness to stretch his or her mind. By the mid 1980s, a small group of researchers became profoundly enthusiastic about nonlinear dynamics. This enthusiasm was more than captured by the journalist James Gleick (1987), who in one of the bestselling popular science books of recent decades described this exotic area of mathematics and its almost as exotic creators, as well as imprinting the field with the humorously intended name chaos theory. The term had been introduced, with fine irony, by the mathematicians T. Y. Li and J. Yorke (1975) to underline the idea that deep order exists in apparent chaos. Shortly after Li and Yorke’s paper, the mathematical biologist Robert May (1976) showed that many long-standing problems in fields ranging from ecology to sociology are usefully approached using the concepts of chaos theory. Gleick hyped chaos theory in a manner that made the hyping of catastrophe theory look mild, suggesting that problems ranging from neuron function to world peace would yield to its methods. However, the book fails to describe the specifics of nonlinear dynamics. It is one of those 1236 Downloaded from http://apa.sagepub.com at UNIV OF CHICAGO on January 1, 2010
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unfortunate books that create the illusion but not the reality of understanding. It has contributed as much as anything to a confusion of terms in which chaos theory and “nonlinear” have come to be used in reference to any situation where a simple sequential explanation appears impossible. Such usage introduces pseudoexplanations into discussions and obscures the actual power of nonlinear dynamics in the areas in which it is applicable. When the psychoanalytic literature refers to nonlinear dynamics, readers should ask themselves, Is this merely a way of asserting that the described phenomena are real, though not readily explained, or is it a reference to specific mathematical findings that help explain the situation? Shortly after the publication of Gleick’s book, he collaborated in publishing a suite of computer programs designed to illustrate the concepts of chaos theory (Gordon, Rucker, and Walker 1990). Unlike the book, these programs, which are available free on the internet, make no claim beyond what they actually do. They beautifully and simply illustrate some of the main themes of nonlinear dynamics (http://www.cs.sjsu.edu/faculty/rucker/ chaos.htm). A few hours of playful engagement with them, in an interactive way that requires no knowledge beyond the ability click a mouse, will give you a feel for many of the central ideas of nonlinear dynamics. A much more extensive suite of software demonstrating the concepts of nonlinear dynamics, Chaos Demonstrations (Sprott and Rowlands 2009), is not quite so nicely packaged but allows the user to go much further. In the early 1990s a shift occurred in the focus of nonlinear dynamics that is particularly important to psychoanalysts. Whereas earlier applications of nonlinear dynamics might be summarized as, “Isn’t it amazing that the apparently disorderly nonlinear world has a deep and beautiful underlying order!” the newer viewpoint suggested, “Most of the really interesting order we see in all but the simplest systems arises from the nonlinear dynamics inherent in such systems.” This approach, which has come to be called “complexity theory,” emerged in Stuart Kauffman’s magisterial The Origins of Order: Self-organization and Selection in Evolution (1993), which offers a rich mosaic of speculations about how biological systems are ordered and in the process demonstrates how a general theory of complex systems might operate. Central ideas of the field, such as how various levels of organization interact, how the connections between elements of a system affect its evolution, as well as the ever present observation that simple processes can in combination produce extraordinarily rich configurations, are clearly developed for the first time in this volume. Unfortunately, in addition to being quite long, the book 1237 Downloaded from http://apa.sagepub.com at UNIV OF CHICAGO on January 1, 2010
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will challenge most analytic readers, as Kauffman’s polymathic understanding of many fields with leave many readers in the dust. More contemporary expositions of complexity theory (Érdi 2007; Gros 2008) are far easier to follow and more systematic, but lack much of the thrill of discovery evidenced in the Kauffman volume.3 The study of how order and richness can evolve within systems seems to invite grand claims and equivalently large volumes. Stephen Wolfram’s monumental A New Kind of Science (2002)—it runs twelve hundred pages—extends earlier investigations of computation by making the broad claim that the phenomena are best understood not in terms of differential equations but as the result of underlying computational rules. Any rule that can be expressed as a differential equation can be expressed also as a computational rule, but the reverse is not true. For example, the information encoded in DNA is in discrete bits that specify ongoing computational activity within the cell. The new kind of science Wolfram is suggesting is a formulation in which the central concept is “cellular automata.” In the 1930s Alan Turing demonstrated that any possible computation can in theory be performed by a device with very elementary operations and procedures. Cellular automata are similar devices, except that their processes and outcomes have a natural graphic representation that can be very appealing. Wolfram exhaustively develops the power of cellular automata and is particularly clear about how emergent phenomena, surprising new configuration that are inherent in a system but not predictable from simple scrutiny of that system, arise in cellular automata. Wolfram argues that computational science can and should replace differential equations as a means of describing phenomena. Wolfram’s approach is likely to be highly influential in covering the same ground as nonlinear dynamics, but the sheer heft of this volume will discourage many readers. TEXTBOOKS
As nonlinear dynamics, chaos theory, and complexity theory became part of the language of any well-educated scientist, textbooks on many levels 3 A current point of view in complexity theory that may be particularly interesting to psychoanalysts is termed “frustration”; corresponding rather well to the psychoanalytic idea of conflict, it holds that conflicting tendencies in various parts of a system may result in entirely new configurations (dreams, neurotic symptoms, etc.). Complexity theory adds a language for discussing several forms of conflict that have not been systematically separated out in psychoanalytic theory (Binder 2008).
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began to appear. These require varying degrees of “mathematical sophistication,” a concept not easily defined. On the simplest level, it involves a familiarity with the language of mathematics. Although some of the texts discussed assume virtually no knowledge of mathematical language, most assume that the reader has learned the vocabulary of elementary calculus if not a facility in solving the problems that are the bane of college students.4 Mathematical sophistication also refers to the ability to conceptualize matters abstractly, to understand that situations that may seem very different may share important underlying qualities and patterns. For some people the idea that the same mathematical concepts can be used to explain stock market prices and the concentrations of substances and chemical reactions is easy to accept, while for others such an idea seems all wrong. (Quick personality test: Do you find the phrase “Consider the spherical cow” [a] funny and intriguing, [b] bizarre and repulsive? If your answer is “a” you are probably mathematically sophisticated; if it is “b” you have a way to go.) Understanding nonlinear dynamics requires this kind of mathematical sophistication, and few of the authors I have cited (Stewart being an exception) take readers’ lack of this kind of sophistication into account. I advise the analyst who bridles at sentences like “consider love as a 5-dimensional vector (something having both quantity and direction) . . .” to take a moment (before discarding the book in disgust) to make sure the idea is clearly understood and to then engage in some “willing suspension of disbelief ” in order to see where the author is going. Often there are rewarding surprises ahead. Finally, mathematical sophistication refers to the ability to appreciate rigorous mathematical arguments, to understand the difference between an illustration that makes a mathematical idea plausible and a proof that demonstrates its truth in the sense of explicitly showing the assumptions underlying it and proving it from those assumptions. The advantage of a reasonable level of rigor is not so much that it demonstrates the truth of what is described (this is seldom in doubt) but that it often leads to a level of clarity about an idea that can be achieved in no other way. Most of the texts mentioned here do not demand rigorous reading, and, where they do, these portions of the text can often be skipped without loss of continuity. However, the reader who really wants to understand what is going on in nonlinear dynamics is likely to find close reading of this part of the text particularly rewarding. The 4 Sawyer’s What Is Calculus About? (1961) is a brief, lucid introduction to the ideas and vocabulary of calculus adequate to read most of the texts discussed here.
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world of nonlinear dynamics is exciting precisely because it contains so many elements that are contrary to “common sense” (i.e., widely held unexamined beliefs). Texts on nonlinear dynamics all assume that the reader will be able to abandon common sense in favor of clearly demonstrable mathematics, but some authors are much gentler in that they appreciate these difficulties. Analysts who want to contribute to the use of nonlinear ideas in psychoanalysis or to think critically about such applications must familiarize themselves with the field in reasonable depth. As with any other kind of significant knowledge, this familiarity should include some active experience in working with the ideas in a concrete way. Popular, easily read books with rare exception leave the reader with an insufficiently precise picture of nonlinear dynamics to allow a critical engagement with its problems as they emerge in psychoanalysis. The situation is rather like that of someone whose knowledge of tennis comes from the pages of Sports Illustrated, but who has never picked up a racket. For the reader willing to learn how to play the game from a master coach, Sprott’s Chaos and Time-Series Analysis (2003) can serve as a wonderful guide. Ideas are set forward in a clear, organized way and profusely illustrated with realworld applications. Each chapter is accompanied by numerous first-rate problems that give readers a chance to immerse themselves in the ideas and to see them in action. (Many of the problems involve writing small computer programs—a skill that can be learned at the level required in an afternoon. One of the few places Sprott leads the reader astray is in recommending GW Basic as a first programming language. Although GW Basic is in many ways superior to Microsoft’s Visual Basic, the latter has far more supports for the novice programmer, who will gladly forgo looking at a blank, incomprehensible screen at the beginning of his or her education.) As his title suggests, Sprott’s text is particularly strong on the study of time-series, the description of a system’s evolution over time. Time-series analysis is a natural discipline for psychoanalytic investigation. We are always asking in what sense the patient’s associations or life history grows from a meaningful pattern of change across time. With some important exceptions, investigations of analyses are limited to unsystematic verbal descriptions. They could, however, be quantified with relative ease and be subjected to time-series analysis. A second excellent text, Steven Strogatz’s Nonlinear Dynamics and Chaos (1994) is at a higher degree of mathematical sophistication in that terms are precisely defined and theorems rigorously proven. Its discussions 1240 Downloaded from http://apa.sagepub.com at UNIV OF CHICAGO on January 1, 2010
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of applications largely from the physical sciences and engineering are clear and to the point, but some of them are now outdated. The book is delightfully written and rock-solid in its content. Strogatz has a facility for explaining complex ideas simply yet accurately. The analyst who wants a real sense of how topology can provide deep insight into differential equations and to get a sense of what serious mathematical thought in this area is like would do well to look into Abraham and Shaw’s brilliant Dynamics: The Geometry of Behavior (1992). Through a series of cartoon-like illustrations, the authors show how questions about change over time can be translated into questions about the properties of geometric figures. Though it contains almost no technical language and no problems or exercises, analysts will find it requires careful reading; it stretches the imagination in ways that those unfamiliar with mathematics will find challenging. The reward of careful reading, however, is a real understanding of one of the major developments of twentieth-century mathematics and an expansion of the imagination useful in comprehending any system, such as the human psyche, that evolves over time. F O R T H E A N A LY S T W H O WA N T S A S O U N D O V E R V I E W
Analysts who do not want to devote the time and energy needed to develop a technical knowledge of nonlinear dynamics but who want to be able to follow discussions of nonlinear dynamics applied to psychoanalysis have several routes open to them. Since Gleick’s popular but ill-conceived Chaos was published, several authors have tried to describe aspects of nonlinear dynamics in a way that engages and informs the nontechnical reader. Often they fall into the same trap as Gleick, substituting emphatic hand waving and the evocation of exotic personalities for the ultimately more interesting and dramatic mathematical findings. There are two outstanding exceptions to this, each of whom has the virtue of underlining centrally important concepts from nonlinear dynamics and bringing the reader more up to date with these developments. Strogatz’s Sync (2003) is a masterful example of excellent popular science writing by a working mathematician. Strogatz weaves solid ideas about nonlinear dynamics into a narrative of the field’s development, often presenting vivid pictures of the players. The book’s title comes from the fascinating and psychoanalytically pertinent question of how it is that two or more organisms come to synchronize their activities. How is it that 1241 Downloaded from http://apa.sagepub.com at UNIV OF CHICAGO on January 1, 2010
Book Essay
Sumatran fireflies spread over miles of a riverbank bring their flashing into unison, or how is it that the back-and-forth of conversation, first visible in the infant’s interaction with its mother, comes into being and is sustained throughout life in the normal turn-taking of conversational interchange. Explicating the ideas of a former graduate student of his, developed under the rubric “small world theory,” Strogatz shows how the synchrony so central to the psychoanalytic interchange can be mathematically modeled. Ian Stewart (2002), the mathematician author of several superb expository texts, brings all his skills to making nonlinear dynamics, its history, and its implications clear in Does God Play Dice? The title refers to Einstein’s assertion against the probabilistic interpretation of quantum mechanics, “God does not play dice with the universe.” One of the most dramatic findings of nonlinear dynamics, which I have argued is central to a revision of psychoanalytic theory, is the concept of emergence (Galatzer-Levy 2002). As noted earlier, a central heuristic of Freudian investigation is that apparently abrupt and meaningless changes reflect a hole in the description of the situation and that the analyst’s clinical and theoretical work consists largely in discovering how this apparent gap results from the blocking of part of the story from awareness. This approach rests on the assumption that things cannot appear from nowhere, that apparently novel configurations must reflect an underlying continuity. This assumption is at odds with physical phenomena we can observe every day. Liquids like water are abruptly transformed into qualitatively different substances, like ice, by simply changing the temperature. Water left to boil on a stove can be observed to circulate, its center rising to the surface, its edges falling again to the bottom, though nothing other than heat has been added to create this structure. On a much grander scale, Darwin convincingly argued that the forces of natural selection operating over time have resulted, without the benefit of a overarching plan, in the hugely complex biological world in which we live. Indeed, later studies have shown that the biological world that has emerged under natural selection is in large measure the result of small accidents (Gould 1977). It seems very likely that mental life includes many emergent phenomena whose apparent discontinuity with a reasonable narrative unfolding is not the result of repression of part of the narrative.5 So the analyst will want to be familiar with the concept of emergence as a possible alternative explanation to the source of the apparent gap in observed processes, particularly developmental processes. Emergence, 5
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From the “classical” analysis of a snowflake: Analyst: From what you have told me, you are remarkably like your many siblings. You all have sixfold symmetry and delicate branching structures. Yet each of you
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by John Holland (1998), a pioneer of artificial intelligence, provides a gentle yet accurate description of this concept, a presentation focused primarily on its specific instances in various real-world phenomena. OTHER SOURCES
For those who enjoyed Mr. Wizard as children, and for those too young or old to have seen the show, the simple hands-on demonstration of scientific principles in experiments carries a delight. In what must be the ultimate nerd flick, Strogatz visits several colleagues who demonstrate the operation of nonlinear systems in mechanical, chemical, and even verbal contexts. The film, approximately forty minutes long (available online at http://hdl.handle.net/1813/97 and also, in segments, on YouTube) conveys the excitement of translating math into experiment and showing that it actually works. Strogatz’s fine lecture on synchrony is also available on YouTube. They both more than repay watching. Another online resource of particular charm is a lecture that Benoit Mandelbrot gave at MIT describing the history of his interest in rough surfaces (2001). The analytic listener will not only feel informed by this terrific lecture, but will also wonder about the psychological wellsprings of a lifelong commitment to a deep intellectual pursuit. A Nova program on fractals (Nova 2008) is accurate and beautiful but somehow does not capture the excitement of investigation of the three films I have just mentioned. There are thousands of online resources for the study of nonlinear dynamics. Sprott also maintains a website with valuable features including tutorials, links to other websites, and, perhaps most charming, a “fractal of the day,” an ongoing illustration of the infinite and lovely possibilities of this mathematical form (Sprott 2009).
is different from the others. This doubtless reflects a wish to be your own individual snowflake. Snowflake: You are right that we are all different, but try hard as I can, I cannot recall, despite having recollections going all the way back to life inside mother cloud, when or why I became different. It just feels like I was always that way. Analyst: In the highly conformist culture from which you come, being different must feel absolutely necessary and, at the same time shameful, so you have repressed the events that lead to your particular structure. Snowflake: (as defiantly as it can get) Perhaps your assumption that my mind operates like your conscious mind, by way of coherent narratives, is a function of your unanalyzed narcissism. Perhaps a few more years on the couch would prepare you to analyze a snowflake!
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Book Essay
There are many professional journals concerning nonlinear dynamics. Like most professional journals, they assume a good working knowledge of the field of study and are likely to frustrate nonspecialists. For psychoanalysts, the journal Nonlinear Dynamics, Psychology and Life Sciences, published by the Society for Chaos Theory in Psychology and Life Sciences, can provide a look at what is being done today in related disciplines. The society’s newsletter tracks publications, conferences, and ideas in this rapidly expanding field. Its conferences are particularly friendly and informative and usually include a day of courses taught by outstanding scholars, all at prices friendly to struggling academics. (Information about the society is available at www.societyforchaostheory.org.) Direct applications of nonlinear dynamics to the social sciences are at their beginning. Although some of these applications can be directly extended to psychoanalysis, most will be of interest primarily as illustration of how nonlinear dynamics can be integrated with psychology. Their most extensive use has been in the study of organizations. Guastello (2002) has summarized many of these efforts, and additional reviews are available in an excellent book he coedited (Guastello, Koopmans, and Pincus 2009), but readers without a working knowledge of nonlinear dynamics may find both books frustrating. Applications of nonlinear dynamics to small groups (Arrow, Mcgrath, and Berdahl 2000) and marital dynamics (Gottman et al. 2002) illustrate how researchers have succeeded in joining the two fields. CONCLUSION
Nonlinear dynamics and complexity theory hold the key to new ways of thinking that are likely to reshape our worldview as deeply as the mathematical view of nature reshaped Western thought at the beginning of the modern era. Just as was the case for that revolution, much of the influence of nonlinear thinking is likely to be implicit and carried forward without knowledge or appreciation of its underlying sources. But just as those willing and able to appreciate the actual work of the scientific revolution were better able to see and develop its implications for the new worldview, so those who understand nonlinear dynamics will be able to understand, appreciate, and critique its role in reshaping analytic thinking. This review is an attempt to provide interested analysts a road map for exploring this new territory. I hope you will use it—and enjoy the journey. 1244 Downloaded from http://apa.sagepub.com at UNIV OF CHICAGO on January 1, 2010
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REFERENCES
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Galatzer-Levy, R. (2004). Chaotic possibitilies: Toward a new model of development. International Journal of Psychoanalysis 85:419–441. Galatzer-Levy, R. (in press). Good vibrations: Analytic process as coupled oscillators. International Journal of Psychoanalysis 90. Gleick, J. (1987). Chaos: Making a New Science. New York: Viking. Gordon, J., Rucker, R., & Walker, J. (1990). Chaos: The software. Sun Rafael, CA: Autodesk. Gottman, J., Murray, J., Swanson, C., Tyson, R., & Swanson, K. (2002). The Mathematics of Marriage: Dynamic Nonlinear Models. Cambridge: MIT Press. Gould, S.J. (1977). Ontongeny and Phylogeny. Cambridge: Harvard University Press. Gros, C. (2008). Complex and Adaptive Dynamical Systems: A Primer. New York: Springer. Guastello, S.J. (2002). Managing Emergent Phenomena: Nonlinear Dynamics in Work Organizations. Mahwah, NJ: Erlbaum. Guastello, S.J., Koopmans, M., & Pincus, D., eds. (2009). Chaos and Complexity in Psychology: Theory of Nonlinear Dynamics. New York: Cambridge University Press. Harris, A. (2005). Gender as Soft Assembly. Hillsdale, NJ: Analytic Press. Harris, A. (2009). “You must remember this.” Psychoanalytic Dialogues 19:2–21. Hershberg, S. (2006). Pathways of growth in the mother–daughter relationship. Psychoanalytic Inquiry 26:56–69. Holland, J. (1998). Emergence: From Chaos to Order. New York: Perseus. Kauffman, S. (1993). The Origins of Order: Self-Organization and Selection in Evolution. New York: Oxford University Press. Kohut, H. (1971). The Analysis of the Self. New York: International Universities Press. Leffert, M. (2008). Complexity and postmodernism in contemporary theory of psychoanalytic change. Journal of the American Academy of Psychoanalysis & Dynamic Psychiatry 36:517–542. Levin, C. (2006). “That’s not analytic”: Theory pressure and “chaotic possibilities” in analytic training. Psychoanalytic Inquiry 26:767–783. Levin, F. (2003). Psyche and Brain: The Biology of Talking Cures. Madison, CT: International Universities Press. Levinson, E. (1994). The uses of disorder: Chaos theory and psychoanalysis. Contemporary Psychoanalysis 30:5–24. Li, T.Y., & Yorke, J.A. (1975). Period three implies chaos. American Mathematical Monthly 82:985–992. Lorenz, E. (1993). The Essence of Chaos. Seattle: University of Washington Press.
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Seligman, S. (2005). Dynamic systems theories as a metaframework for psychoanalysis. Psychoanalytic Dialogues 15:285–319. Sprott, J.C. (1992). Sprott’s fractal gallery. Available online: http://sprott .physics.wisc.edu/fractals.htm Sprott, J.C. (2003). Chaos and Times-Series Analysis. New York: Oxford University Press. Sprott, J.C. (2009). Sprott’s gateway. Available online: http://sprott.physics .wisc.edu/ Sprott, J.C., & Rowlands, G. (2009). Chaos Demonstration, Version 3 (computer software). Raleigh, NC: Physics Academic Software. Spruiell, V. (1993). Deterministic chaos and the sciences of complexity: Psychoanalysis in the midst of a general scientific revolution. Journal of the American Psychoanalytic Association 41:3–44. Stein, A. (1999). Whose thoughts are they, anyway? Dimensionally exploding Bion’s “double-headed arrow” into coadapting, transitional space. Nonlinear Dynamics, Psychology, & Life Sciences 3:65–92. Stern, D.N., Sander, L.W., Nahum, J.P., Harrison, A.M., Lyons-Ruth, K., Morgan, A.C., Bruschweiler-Stern, N., & Tronick, E.Z. (1998) Noninterpretive mechanisms in psychoanalytic therapy: The ‘something more’ than interpretation. International Journal of Psychoanalysis 79:903–921. Stewart, I. (2002). Does God Play Dice? 2nd ed. New York: Blackwell. Stolorow, R. (1997). Dynamic, dyadic, intersubjective systems: An evolving paradigm for psychoanalysis. Psychoanalytic Psychology 14:337–346. Strogatz, S. (1994). Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering. Cambridge, MA: Westview. Strogatz, S. (2003). Sync: The Emerging Science of Spontaneous Order. New York: Theia. Taerk, G. (2002). Moments of spontaneity and surprise: The nonlinear road to something more. Psychoanalytic Inquiry 22:728–739. Thelen, E., & Smith, L. (1994). A Dynamic Systems Approach to the Development of Cognition and Action. Cambridge: MIT Press. Thom, R. (1975). Structural Stability and Morphogenesis: An Outline of a General Theory of Models. Reading, MA: Benjamin. Tyson, P. (2005). Affects, agency, and self-regulation: Complexity theory in the treatment of children with anxiety and disruptive behavior disorders. Journal of the American Psychoanalytic Association 53:159–187. Wiener, N. (1956). I am a mathematician, the later life of a prodigy; an autobiographical account of the mature years and career of Norbert Wiener and a continuation of the account of his childhood. In Ex-prodigy. Garden City, NY: Doubleday.
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Wiener, N. (1961). Cybernetics: Or Control and Communication in the Animal and the Machine. 2nd ed. Cambridge: MIT press. Wigner, E. (1960). The unreasonable effectiveness of mathematics in the natural sciences. Communications in Pure and Applied Mathematics 13:1–14. Wolfram, S. (2002). A New Kind of Science. Champaign IL: Wolfram Media. Zeeman, C. (1977). Catastrophe Theory: Selected Papers 1972–1977. Reading, MA: Addison-Wesley. 22 South Michigan Avenue Suite 1407 Chicago, IL 60603 E-mail:
[email protected]
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