DETERMINISTIC CHAOS AND THE SCIENCES OF COMPLEXITY: PSYCHOANALYSIS IN THE MIDST OF A GENERAL SCIENTIFIC REVOLUTION

Vann Spruiell, M.D.

This was a Plenary Address presented at the Annual Meeting of the American Psychoanalytic Association, May 10, 1991. It was published in the Journal of the American Psychoanalytic Association in 1993, in Volume 41: 3-44. I am indebted to the following colleagues for helpful criticisms and suggestions: Paul Ecker, M.D., Gerald Fogel, M.D., Robert Galatzer-Levy, M.D., Stanley Goodman, M.D., Michael Moran, M.D., Paul Mosher, M.D., and Alan Pollack, Ph.D., M.D. I also gratefully acknowledge help with the mathematical concepts by Slawomir Kwasik, Ph.D., a Professor of Topology at Tulane University.

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A combination of new concepts and the enormous powers of computation now available have created the beginnings of a major new scientific revolution. It does not have to do with psychoanalysis per se, rather it has to do with basic assumptions among the sciences generally. The result is already showing in new visions of nature variously called "deterministic chaos," "nonlinear dynamics," or "sciences of complexity." For the first time it is possible to study complex systems in process, over time. This paper, especially attending to issues of separation and integration, focuses on two components of these new understandings: fractal geometry (part of the mathematics of topology), and deterministic chaos (the tendency of nonlinear systems to oscillate toward and away from absolute chaotic disorganization). In a diverse group of intellectual disciplines, it is now possible to describe systems in operation in detail, in terms of nonlinear differential equations. From these, computer models of multiple variables in interaction can be produced. In turn, these possibilities will allow experimentation on the models by altering the live variables in the subject being investigated. At the present time, however, psychoanalysis can only use deterministic chaos and fractals metaphorically. In the future, especially if psychoanalysis is seen in terms of process or organismic theory, it is likely that such models will be preduced the origins of major changes among those disciplines having to do with mind, brain, and the reorganization of concepts of their interactions.

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PART 1.


INTRODUCTION

A MAJOR SCIENTIFIC REVOLUTION is underway -- not of psychoanalysis itself, but of some basic assumptions among the sciences generally. This revolution will add to, rather than replace, most traditional theories and practices that were once considered to be the pars pro toto of science. Most of those altered theories and practices will be put in their places as parts of the science of the next century. The older versions of science will be joined by new opportunities to study forms and processes in nature, some of which could hardly be imagined previously. The purview of the sciences will be expanded enormously during these coming decades -- and integration of the various disciplines will be furthered.

The sciences of complexity have come about because technological and theoretical advances in mathematics and physics have allowed new integrations. That makes it possible to study the interactions of parts within systems in process. The sciences of complexity include new concepts in physics, computer science, and mathematics (in particular, number theory and topology). They have made it possible to study living and some non-living systems in process. They make it possible to reveal remarkable, previously unknown forms that nature, metaphorically, "prefers" over other conceivable forms. The existence of oscillations, in living and nonliving complex systems, toward and away from total disorder -- deterministic chaos -- can now be clearly shown.

The radical opening of science is less than two-and-a-half decades old. It has become widely known among scientists and mathematicians only during the last decade. Psychoanalysts have been in its midst, although few recognized that fact. As a subversive discipline in the form of its theory of mind, psychoanalysis played a part in preparing the intellectual ground for a new subdivision of science to catch fire. The revolution will have enormous effects on our long-range aims and expectations, both for development as a special discipline and its integration with other intellectual disciplines.

I need to "place" myself for the reader. For brevity's sake, I shall identify myself as a full-time analyst and a Freudian -- in the sense that most evolutionary biologists consider themselves Darwinian (Freudians and Darwinians today are not "true believers," or should not be). I am competent as a psychoanalyst—not as a mathematician, physicist, or general biologist—certainly not as a computer scientist. To the extent that I appreciate philosophy, particularly the philosophy of science, I am a regretful but mostly shameless autodidact. I have had experiences on two large-scale, multidisciplinary research teams, the first (Greenblatt and Solomon, 1953) only on the periphery, as a medical student, the second (Lief et al., 1960), a five-year study of a large number of medical students, individually seen in panel interviews, each for three or four years, as a full member. I have carried out several small-scale research projects and pilot studies. I have been particularly interested in studies on information, especially archival projects that will support our collective psychoanalytic knowledge. One, which has engrossed me for several years involves the archiving of published papers and reviews and making the material easily available to all people desiring them -- at cost. Another was the attempt to produce on disks all of the psychoanalytic literature in the major journals in English. (Beginning in 1997,) I have been assembling a regional web site, connecting all the psychoanalytic organizations in the South. But primarily I have been interested in research from the point of view of a full-time psychoanalytic practitioner immersed in work with patients.

These experiences have left me fascinated with the problems of making sense out of the enormous number of variables that confront the psychoanalyst. We do make sense, at least to other analysts. But how do we make sense to each other? How do we recognize one another? And how can we make scientific sense to open-minded "outsiders"? In wondering about these questions, it has been exciting to try to catch up with older ideas, and learn what I could of new ones. With help, I could follow some of the language of simple mathematical notations. With still more help, I could comprehend others through analogies in words. I understand that all translation incurs distortion. But what else can diverse human beings do but try to translate?

This essay also reflects a long-standing fascination with the world of ideas that used to be called natural philosophy and moral philosophy. I am interested in integration and separations, within bodies and minds and among people, which if they occur prematurely and too sharply lead not to healthy differentiation and division of labor, but to pathology, physical and mental. Metaphorically, these same ideas may apply to larger groups of people, whole cultures, perhaps even theories.

I shall make some raw assertions without the opportunity to argue or support them. The discoveries of the past ten years are too complex and too diverse to bear condensation in one paper. I shall also add some speculations. My aim is not necessarily to convince, but to interest. In one way it would have been better, had it been feasible, to include some of the beautiful graphic images of features of complexity theory that have been generated on computers. Many of the concepts can be understood more easily that way. They lend themselves to the human capacities of pattern recognition. There are special reasons to put it this way, and they will be discussed below. However, such images might distract the viewer from more central issues of what complexity theory has to do with psychoanalysis and vice versa.

In what follows, I shall first discuss the collapse of some previously accepted scientific "certainties" during the 20th century, then sketch the basic features of the new sciences of complexity. For those psychoanalysts already familiar with the ideas, this part should present no problems. For those who are not, some of the concepts may seem very strange. For the latter, I suggest that they "read through" parts that seem obscure. The whole argument does not depend on understanding each mathematical concept completely.

I shall then turn to psychoanalysis as a theory of a complex system -- an extraordinarily complex system. Then, I take up the opportunities the new scientific concepts offer for future scientific research in psychoanalysis itself. If I succeed in interesting readers, they will find much more thorough and accurate entry level descriptions and examples of deterministic chaos and fractal geometry in other places. I especially recommend for persons unacquainted with such ideas Gleick's ( 1987) Chaos: Making a New Science, Stewart's ( 1990) Does God Play Dice' The Mathematics of Chaos and Casti's (1990) Searching for Certainty.

Uncertainties

Before sketching the nature of the revolution and setting forth ideas about its meanings for psychoanalysis, I shall discuss some intellectual idealizations of science and mathematics, along with subsequent disillusionments with them. They are: (1) the existence of misleading myths about "science" and the parts of mathematics most of us were taught; (2) faulty myths -- mostly held by non-mathematicians -- about the supposedly ideal certainties that might be obtained through mathematical operations; (3) the challenges of these myths offered by complexity theory, which will be described; (4) opportunities presented for the unifications of knowledge by the demise of the old and simple "clockwork universe" of certainties promised, especially in the western world, by positivistic philosophies.

Misleading Myths About Traditional Science and Mathematics

The putative "facts" most of us were taught about science and mathematics were very distant from the actualities faced by working scientists and mathematicians in the present. Of course we could see the tangible evidence of stupendous progress made during the past 200 years. Old and frivolous gods had been partially replaced by rational understandings. But in that event, a new mythology was introduced, and most of us took a long time to recognize it. As Toulmin (1982) put it, "the myths of the twentieth century . . . are not so much anthropomorphic as mechanomorphic . . . in the main it is because our contemporary myths are scientific ones that we fail to acknowledge them at all" (p. 24).

Leaving to one side relativity and quantum theory, normal science as it has existed even well into the present century, among most scientists, powerful as it is when applied to phenomena of relatively short duration and on human scales of perception, is in many of its fundamental assumptions too flawed or too limited to serve contemporary purposes.

These assumptions include: the belief that the rational capacity to predict events is theoretically unlimited; the pretense that scientific studies are public in nature; that there is a necessity, if the intention is to achieve scientific veridicality, to reduce variables and study large numbers statistically; that certain assumptions about what constitutes valid experimentation, verification, and the nature of "truths," can be codified as the "canons of science," which supposedly define the "scientific method"; that scientific progress occurs in small increments; and that it is justified to have increasing confidence in the near omniscience of measuring devices.

Of course, scientists -- at least competent ones -- never believed that their measuring devices could ever be absolutely accurate. And they were well aware that when they studied complex phenomena in nature they selected only some of many variables for study. But they believed that their measurements were exact enough, and tiny quantitative variations of initial conditions could be safely ignored. Thoughtful scientists understood that the work -- as opposed to the reporting -- of actual investigations cannot really be observed closely and directly by "outsiders." At least that holds for the present and the foreseeable future, even if the "outsiders" are members of other scientific disciplines, much less if they are laymen. It is not possible to adequately understand the work of scientists in other disciplines in detail without extensive training and practice. Furthermore, broadly educated scientists eschew myths of certainty, or try at least to qualify any such claims within their own disciplines.

The notion that there is a "scientific world" with close communications among its parts flies in the face of the reality that most scientific disciplines have become almost completely isolated from each other. The idealization of incremental progress—that steady advances are made in regular, small increments—is demonstrably untrue (Kuhn, 1962, 1977). The idea that presently the sine qua non of normal science is the capacity to express its laws mathematically often rings gorgeously true to outsiders, but hardly at all to some members of other scientific disciplines. The fatuous idea that to be a science, the "canons" of science must be strictly applied as they have been expressed from Bacon's and Mills' times, sets irrational injunctions and constraints on investigation, even on conjecture, especially in the biological sciences.

Actually, there exists a motley collection of scientific fields, tenuously held together by more or less individualized, systematic methods of investigation and verifications of truths accessible by a variety of methods. The practitioners of individual sciences are typically experts only within their own areas, and their wisdom is not automatically transportable to other areas. The insular lack of knowledge by most scientists of scientific disciplines other than their own is emphasized in a popular book by Hazen and Trefil (1990).

Yet, idealizations and myths and their "freezing into belief systems" (Spruiell, 1987) have been characteristic of too many working scientists trapped in the cells of their own disciplines, too many psychoanalysts, and far too many nonscientists. Traditional science was (and is) mostly valid about what it can address -- a small part of nature. It is simply misleading if taken to mean it can reach grand scale of "truth." It generalizes (to students) from its demonstrable successes and tends to foster idealized pretensions. It is not "exact." It is not "hard" (in the sense of providing certainties, especially certainties about causes or predictions over any long period of time [Casti, 1990]). It cannot be applied accurately to a seemingly simple operation, such as an adult playing with a child on a swing -- not even to a pendulum swinging in a vacuum or even the timing of the best quartz crystals. Nor is science "tough-minded" except within the confines of its private temples. And it is merely evangelistic or brutal (depending on who has power over whom) to proclaim dogmatically that the only reliable truths are those determinable by particular scientific methods.

Traditional science cannot account for the patterns of branching of limbs on trees. It cannot give more than an arbitrary account of the length of a nation's coastline, the alterations between the drips and small steady flows of a leaky faucet, the delicate traceries of a sea dollar, or the "noise" ("aberrations") in its most carefully designed experiments. It cannot come close to accounting for the complexities of the brain or mind. And its methods of simplification of variables and their interactions have too often led to mistaken long-range predictions about the results of processes in even the most seemingly precisely known complex systems.

The science of the future will be less restrictive, range further, and be more modest in its claims to approach truth. At the same time it will benefit from the cross-currents of information created by interdisciplinary studies. The idealizations of what was thought to constitute science have been almost totally cast aside by a growing body of the best educated contemporary scientists, especially physicists. Very few non-scientists understand these developments. For psychoanalysis the problem has not been that our field is "unscientific." The problem has had to do with the limitations of science as we knew it—or thought we knew it.

Misleading Myths About Certainties in Mathematics

Traditional mathematics reached an interesting point in the first decade of this century. It might amount to a caricature to draw sharp distinctions between the two more or less non-communicating groups of mathematicians that emerged. Taking that risk, on one side were mathematicians and positivists who believed that a conjoining of an ideal mathematical and an ideal logical "world" was possible. Within those seemingly perfect worlds of artificial logical systems, all deductions that obeyed the requirements of the fundamental postulates would generate provably true statements within themselves. Russell and Whitehead (1910-1913), in Principia Mathematica, seemed to secure the positive base for at least the ideal construction of a completely reliable logical system. (However, Whitehead soon began to doubt some of its "certainties" by the first revision of the book a few years later, and began the development of his own philosophy of process [see Monk, 1990, p. 219; Whitehead, 1929])¹. In contrast to this form of mathematics, a very different form was developed by Henri Poincairé and his followers -- but the fertile ideas about the dynamics of complex systems and topology had to wait for several decades to be appreciated by physicists. The stage was set for the current scientific revolution largely from major intellectual revolutions in physics, the theories of relativity and quantum mechanics, and the work of a collection of mathematicians, especially Poincairé, Gödel, and Turing. In 1931, Gödel published a theorem that soon shocked the world of mathematics and later the world of philosophy. It continues to interest mathematicians, and it has had important influences on epistemology.

Gödel's most important theorem is expressed by his fellow mathematician, Casti (1990), in casual words:

Gödel began the demolition of Russell's certainties by using Principia Mathematica as the example! We became bereft of yet another false ideal. Earlier, we could at least imagine perfectly logical systems. But we are better off with the disillusionment.

The Development of Sciences of Complexity

The new sciences of complexity are often identified by a part, deterministic chaos. As a number of authors have pointed out, the simple term "chaos theory" is awkward inasmuch as what is intended is not absolute randomness. Furthermore, it is no longer a theory—the actuality of oscillations of nonlinear systems toward and away between near-periodicity and near-chaos are experimentally reproducible characteristics found both in nature and in pure mathematical systems. However, theories to explain the different forms of chaos are only just beginning to emerge, become documentable mathematically, and become reproducible. The term "deterministic chaos" is preferable to "chaos." The Royal Society, meeting in 1986, came up with an even more awkward definition of near-chaos, "Stochastic [random] behavior occurring in a deterministic system." Still another term, nonlinear dynamics, is often used (and the rationale for that will be explained below). In what follows, I shall usually use the broader "sciences of complexity" to mean the application of the related tools of computation, number theory, topology, and deterministic chaos to previously existing disciplines.

These new tools came about through the development of ideas concerning possibilities and limitations of mechanical computation (Turing, 1937). For those who are not mathematicians or computer scientists, the works of Penrose (1989) and Stewart (1990) are good sources for comprehension. When computers that could cope with the monumental complexities of nonlinear differential equations became available, it became practical to solve equations describing the interactions of multiple variables within a system. Earlier, they had been almost impossible to study scientifically. The result has been a spread of the tools to practically every serious intellectual center of the world. A partial list of the fields using them includes: mathematics (especially topology and number theory), physics (including quantum physics), fluid mechanics, biology (including ecology, physiology, and evolutionary theory), medicine (including neurology, neuropsychology, the brain sciences, psychiatry, cardiology, circulatory physiology, epidemiology), meteorology, and astronomy. Chaos has been applied in fields not ordinarily considered scientific economics, international relations, highway engineering even the stock market! As one might expect, the results of the latter experiments have varied, but among the more organized research groups, early results are exciting.

The Demise of the "Clockwork" Theory of Causation

The new ideas and discoveries have joined earlier revolutions, relativity and quantum mechanics, to create a major new scientific revolution -- I use Kuhn's (1962, 1977) definitions. But there are other definitions. For example, Toulmin, a philosopher of science, is quoted by Pool (1989) as saying, "People assumed for more than 200 years that the Newtonian world was predictable . . . Chaos shows that this was always a mistaken assumption" (pp. 26 -28). Toulmin believes that chaos is immensely important as a tool, but not a revolution by his own definition. The definitions to one side, the meanings are comparable. All intellectual disciplines will be influenced, if not by a revolution at least by the jettison of old certainties.

The Newtonian theories of causality, best enunciated by LaPlace (see Toulmin, 1982), were that if a supreme intelligence knew all the causes existing at one instant it could predict every subsequent event in the universe. This fantasy has been destroyed. The notion of a "clockwork" universe has lost credibility because of demonstrable properties of complex systems. But this loss of limitations in the past so eagerly adopted by positivists has opened the way to new, previously unimaginable opportunities to extend science.

Edward Lorenz (1963), a professor of meteorology at the Massachusetts Institute of Technology (M.I.T.), began to work in a way that most of his peers thought odd. He had a Royal McBee LGB-300 computer, a primitive thing that worked at a tiny fraction of the speed that home computers now have. Most of his colleagues at that time were suspicious of computers, and his interest in weather prediction was anathema to them. They did believe the weather could be accurately predicted months ahead—some day—providing that thousands or millions of sensors were placed about the world and coordinated with a computer with powers then almost unimaginable. But until then, they thought, weather forecasting was for television.

The Discoverer

Lorenz had been interested in mathematics since boyhood, and would have become a mathematician but for the interruption of World War II. Caught in an eddy of its maelstrom, he was swept into meteorology. Essentially, he assigned three variables and an equation borrowed from another field, hydrodynamics, and applied them to three major determinants of the weather, from howling storms to dead calm, from deluges to droughts, in process over time. He assigned special values to each and equations with which to relate them. The computer, slow and unreliable as it was, generated a little model of the weather for him to observe in his safe M.I.T. office. He could reconstruct from the interactions of these variables all kinds of changes—the storms, windy times, calm, rain, etc. One day, he decided to repeat a sequence and add to it. Rather than repeat the old sequence in its entirety—it took hours to run—he decided to restart it in the middle and halve the time. He entered the different variables as they had been recorded for that time, and set the thing going.

Later, he came back to the machine and found that the second half of the second sequence indeed began in the same place as the first. But to his surprise the lines of the second run quickly began to diverge; eventually they diverged totally. Like most of us would, he first blamed the computer (it had gone crazy), then himself (he had got the figures wrong). Finally he understood. The computer kept values to six decimal places, but the printouts showed only three places. Instead of using ".506127" he used ".506," naturally assuming that such minute differences as those from the tenthousandth to less could make no difference. Lorenz was able to test this with his computer, and discovered the way to some very surprising facts, including the proof that tiny differences in initial conditions of a nonlinear dynamic system can lead to major consequences relatively soon.

Sensitivity to Initial Conditions

From this, came the famous "butterfly" analogy: a butterfly flapping its wings in Japan at a certain time can create a tiny atmospheric disturbance that eventually could influence the other variables in major ways, say, a month later, in distant parts of the world. I do not know who used the butterfly analogy first, but subsequent writers have chosen different examples. That gives leave to suggest that such a creature might influence whether a hurricane devastated New Orleans or wandered off into Texas -- or over to Florida.

It is hard to imagine that we shall ever have instruments sensitive enough to predict the weather specifically more than a few days in advance—or make them about other complex systems. Time spans, for one thing, for systems to oscillate chaotically extend from moments for molecular systems to eons for astronomical systems. For another thing, the unpredictability applies to specifics, not general trends. Two flecks of foam above a waterfall have unpredictable relations below it—but the waterfall's tendencies can be reasonably well anticipated.

Topology, a branch of mathematics first developed about the turn of the century by Poincaire, had demonstrated some of the same principles many years earlier. And some had also been demonstrated by physicists. But the two sets of findings were never related. The Lorenz paper was published in 1963 by the Journal of Atmospheric Sciences. The meteorologists did not take it seriously, and nobody from the other sciences read it. Hardly anybody in any other discipline even knows of that journal's existence, and few scientists of whatever stripes would have been able to understand it without help.

Differential equations express the process of a system—its "movement" or its "becoming"—from one configuration to another. This holds whether the system is purely mathematical, without known correlates in nature, or it does happen to describe events in nature. In either case, if successive states can be graphed on a computer, characteristic forms will emerge. Thus, the "exact" sciences are capable of utilizing mathematical equations to study some systems in process by using differential equations. But before advanced computers existed these had to be "linear" in nature equations that "behave well" in computations. If they are graphed in analytic geometric forms they create orderly lines—not necessarily straight ones, but, for example, regular curved ones, as in sine waves.

Complex systems, after all, the most interesting parts of nature, cannot be described (except in separated parts) by the relatively easily solvable linear equations. When scientists tackled the problems of conditions in nature that, if described in mathematical terms, would demand nonlinear differential equations, they added "corrective" constants. The Ptolemaic system had to have epicycles added in order to make planetary movements somewhat predictable. It was assumed incorrectly that their orbits were circular rather than elliptical.

What are nonlinear differential equations? They are those in which the variables of the equation interact with each other in complex rather than additive ways. These latter differential equations are typical of all living and other complex systems. To repeat, these living systems could not be studied, except empirically or by the laborious assemblage of separately derived statistics, by the science that existed two decades ago. Poincaré long ago illustrated the extreme complexity when only three bodies interacted; usually the equations describing the interactions could not be "solved"—graphed in a way that would demonstrate the system in process. But with the development of powerful computers even these can be parsed. Imagine the increase of complexity when there are more than three variables in mutual interaction—as there usually are in living systems! Psychoanalysts, of course, do imagine it somehow, but it seemed in the past unlikely that these multiple variables could be quantified, much less modeled in action. Not many analysts thought mathematics had any applicability at all! Matte Blanco (1975), Thom (1975), and Sashin (1985) were exceptions.

I can illustrate nonlinear dynamics in words. The behavior of an ice hockey puck after a player has slammed it depends on multiple factors, including the mass of the puck, its inertia (or momentum and vector if it were in motion when hit), the mass and propulsive strength of the stick governed by the player's muscles (and the vector of the player's motion), the distance over the ice to the goal, the conditions of the ice (its temperature, amount of liquid on it, how much it has been cut up previously by the skates, etc.), the "hook" or "slice" of the stick that puts a spin on the puck, etc. These represent only some of the variables, but I need to focus on only two: the changing speed of the puck and the friction of the ice. The variables interact: the faster the puck moves, the less the friction. Furthermore, there is not a smooth, "linear" relation between the two (because friction and speed are themselves determined complexly). These variables (and the others) interact with each other, and any set of rules to construct a computer model of these events would have to be expressed in nonlinear differential equations. Again, within a system, every functioning part influences every other part.

Yet, somehow hockey players do quite often score goals without thinking the matter out. They do it by non-conscious "computations" far, far more complex than any currently imaginable digital or analogue computer's "artificial intelligence." They do it with an almost inconceivably complex system (that is why it seems so simple) of pattern recognitions that take into account all the variables mentioned above and more—roughly, in a "fuzzy" way, at non-conscious levels. These patterns, or Gestalten, are learned and improved over time in marvelous ways as "automatic" habits and techniques. But just how, we are far from understanding. Very likely these complex events are instances of relatively simple equations that are capable of generating extraordinary levels of complexity. But they are by no means consistent, and seemingly inexplicable cycles of breakdowns of play oscillate with standard play and occasional superb play—and here we introduce, in addition, the state of mind of the player.

Strange Attractors

Lorenz constructed an equation for only three key variables to create a simplified model of a weather system. At each successive moment in the activity of such a system, it is possible to make a point on a special graph that expresses the interactions of all the variables. If the points are connected, they describe an "orbit" around a central point that exists in "phase space"—not "actual space." Phase space is shown if two interacting variables are graphed in a space that includes both positive and negative numbers and "real" and "imaginary" numbers. We are familiar with "real" numbers, but "imaginary" numbers are not imaginary at all—they are numbers generated if "real" numbers are multiplied by "-1." They can thus provide space for the plotting of the square roots of negative numbers. Each variable in an equation amounts to one dimension. Three dimensions would create a three-dimensional graph. There can be as many dimensions as mathematicians might like. More than three or four cannot be imagined by ordinary people, but many biological systems have far more variables operative than three or four. Nine dimensions would choke the most elaborate main frame computer!

Reduced to three, Lorenz's feeble computer ground out the solutions to nonlinear differential equations that traced a process. If they had been relatively tractable to solution, they could have been solved by a calculator, or even by hand—but that would have taken years.

The point that is "orbited" in phase space is called an "attractor." If a pendulum's movements, an apparently simple system, are plotted on the graph, a spiral is created as the pendulum is slowed by friction. The stopping of the pendulum is indicated on the graph by a point. It is as if the stilled pendulum "aims" at a point, and if we imagined it existed in actual space, we might conclude that its target was the center of the earth—ignoring that the center of this roiling earth is constantly changing. It would be tempting to conclude that there is an actual "attractor" somewhere pulling the pendulum toward it. But an "attractor" exists as an abstraction, just as the connected successive points, each representing the interactions of all the variables of the system, and collectively representing the abstract "orbits" are abstractions. They describe the process of the system. These abstractions can exist in purely mathematical terms, but many of them also describe models of actual events in nature.

In more complex systems, attractors "move" (in phase space), split, combine and recombine with others; they can also be graphed and comprehended as images or forms. It is even possible on the computer to make cross-sections of their patterns of movements. In the case of linear equations, the "orbits" described over time are regular—periodic—like circles, ellipses, parabolas, in two dimensions (or like smooth spheres in three). But nonlinear differential equations behave "strangely." If plotted on a computer, a strange attractor not only "moves" with the changes in its system in process, the "orbits" do not overlap but may come infinitesimally close to each other.

To return to the seemingly simple example of the pendulum, we are accustomed to think that a pendulum is slowed only by friction. We know "intuitively" that unless it is given small, regular, external boosts, it will eventually stop. We assume that an ideal pendulum, swinging free of friction, would not stop. Yet, most of these assumptions are incorrect if fine enough measurements can be made. The most exquisitely designed pendulum mechanisms, even if run in a vacuum, would not operate with perfect periodicity. If observed carefully and long enough, irregular oscillations toward and away from deterministic chaos, disorder, would be reflected in our phase space graph. There is deterministic chaos even in the two-dimensional phase space of simple pendulums! Deterministic chaos can be demonstrated in the best quartz timers.

The orbits demonstrated in simple, two-dimensional phase space, as in a pendulum, seem strange, and in three-dimensional space, as in Lorenz's three-variable weather system, the image generated becomes even more strange. The totality of its orbits, none exactly overlapping another, but coming infinitesimally close, as if laminated, also moves in three dimensions. The product is an abstract structure that seems to have not only length and breadth, but "thickness"—a sort of sculpture rather than a sort of picture. The image after a sufficient number of orbits looks vaguely like a butterfly (of all things), with two beautiful wings and something like a body projecting at right angles in the third dimension. What it would look like after a thousand times that number of orbits, I do not know. Remember that the form being described is three-dimensional because Lorenz used three variables.

Currently mathematicians are using probabilistic tools to approximate solutions with "fuzzy logic," just as minds approximate "solutions," as our hockey player did with the stupendous numbers of variables entailed by any shot. Reductionism is unavoidable; the selection of key variables from the total presumably will always be necessary even if computer models of complex systems are to be made. A strange attractor only "seems" to be organizing the system—that is, acting upon it in remarkable ways. It is actually the product of the organization. Mathematically and physically, it is the point that completely describes the state of the system at that particular moment. At intervals, even in the direct graphing of nonlinear equations, as in Lorenz's weather system, the direct graphing of the variables shows branching, and these are periods when the characteristic forms of near-chaos develop. There are periodic movements toward disorder and away from it, toward order and regularity.

These oscillations toward and away from disorder are characteristic of all complex systems, and their variations, their natures, even possibly their functions, are beginning to spawn research in very different intellectual disciplines.

Recursive Equations, Fractal Geometry, and Nature's Preferred Shapes

In some recursive equations, those in which the "output" of the equation is fed back as "input" in the succeeding equation, one finds regular approaches to near-chaotic forms, and at specific, regular levels the graphic form bifurcates. Again, this is true whether it is purely mathematical in form or reflects some material system in nature. One of the pioneers in this area, Mitchell Feigenbaum (for the best descriptions, see Gleick, 1987; Stewart, 1990 -- Feigenbaum's actual papers are not for nonmathematicians!) used a hand-held computer and a simple recursive equation that happens to describe a parabola when graphed. After enormous effort, he discovered a constant of nature that provided an understanding of the periodic branching or bifurcation of any nonlinear system at periods approaching chaos. These changes are demonstrated even more dramatically when materials are strongly perturbed, or undergo state changes, e.g., from ice to water, or from water to gas. Feigenbaum is one of the few men to discover a constant in nature, like Pi. In fact, he discovered two (and may be the only man to have done that, as far as I know). These constants for splitting hold mathematically, using numbers alone, and hold true in nature as in Lorenz's artificial weather. This chaotic "doubling" accounts for the impossibility of predicting a system after several such episodes.

Although the equation itself may be relatively simple, the results may be wondrously complex, some like those seen in nature, sometimes not like images we have not seen but which appear strangely familiar. What is characteristic of these images, which are the result of multiple iterations (repetitions) of the recursive equation plotted with complex numbers, is the development of scaling. The same fundamental form is duplicated with variations at every scale of magnifications. Complex numbers are "real numbers" between minus infinity and plus infinity added to the so-called "imaginary numbers" mentioned above. The most famous of the recursive equations is z - - > z² = C, the equation used in Mandelbrot's "Julia Set." The mathematics can be easily followed in either Geleick's (1987) or Stewart's (1990) discussions.

Mandelbrot (1986), a resident genius of IBM., is a sort of "superstar" of this field, which he named fractal geometry. This geometry deals with forms that are fractions between the familiar dimensions (1, a line; 2, a plane, 3, an object with height width and length). Fractals are easier to understand when diagrams can be used. The characteristic results are sometimes remarkable. But Mandelbrot is by no means the only investigator of fractals. Others have been able to find multitudes of equations which also produce fractal images -- and they are also abstractions, but look amazingly like ferns, trees, branching blood vessels, or bronchi -- or dust. It is possible to calculate the approximate fractal number for characteristic forms in nature.

Even more fascinating is that if one zooms down on the computer's monitor, mimicking a microscope, the same basic forms can be seen, with seemingly endless variations. These figures will retain "self-sameness." With an ideal computer, these scalar effects should go to infinity. The reasons these computerized "forgeries" appear so "realistic" is that nature is in fact "rough," that is, characteristically "fractal" rather than "smooth and straight." Anyone can buy programs with the algorithms to create such figures, supply the figures for the variables -- and discover forms on a home computer that perhaps have never been seen by humans before. At scales from the microscopic to the terrestrial to the planetary, probably to the galactic, the same nonlinear dynamic principles appear to be at work; the same patterns, always unique, but always similar in given classes. Structures like "Russian dolls" exist in nature—figures within figures within figures. Originally, fractal geometry seemed very far from chaos theory, but as Stewart (1990) points out:

Edward Lorenz (1963), a professor of meteorology at the Massachusetts Institute of Technology (M.I.T.), began to work in a way that most of his peers thought odd. He had a Royal McBee LGB-300 computer, a primitive thing that worked at a tiny fraction of the speed that home computers now have. Most of his colleagues at that time were suspicious of computers, and his interest in weather prediction was anathema to them. They did believe the weather could be accurately predicted months ahead—some day—providing that thousands or millions of sensors were placed about the world and coordinated with a computer with powers then almost unimaginable. But until then, they thought, weather forecasting was for television.

The Discoverer

Lorenz had been interested in mathematics since boyhood, and would have become a mathematician but for the interruption of World War II. Caught in an eddy of its maelstrom, he was swept into meteorology. Essentially, he assigned three variables and an equation borrowed from another field, hydrodynamics, and applied them to three major determinants of the weather, from howling storms to dead calm, from deluges to droughts, in process over time. He assigned special values to each and equations with which to relate them. The computer, slow and unreliable as it was, generated a little model of the weather for him to observe in his safe M.I.T. office. He could reconstruct from the interactions of these variables all kinds of changes—the storms, windy times, calm, rain, etc. One day, he decided to repeat a sequence and add to it. Rather than repeat the old sequence in its entirety—it took hours to run—he decided to restart it in the middle and halve the time. He entered the different variables as they had been recorded for that time, and set the thing going.

Later, he came back to the machine and found that the second half of the second sequence indeed began in the same place as the first. But to his surprise the lines of the second run quickly began to diverge from those of the first eventually to diverge totally. Like most of us would, he first blamed the computer (it had gone crazy), then himself (he had got the figures wrong). Finally he understood. The computer kept values to six decimal places, but the printouts showed only three places. Instead of using ".506127" he used ".506," naturally assuming that such minute differences as those from the tenthousandth to less could make no difference. Lorenz was able to test this with his computer, and discovered the way to some very surprising facts, including the proof that tiny differences in initial conditions of a nonlinear dynamic system can lead to major consequences relatively soon.

Sensitivity to Initial Conditions

From this, came the famous "butterfly" analogy: a butterfly flapping its wings in Japan at a certain time can create a tiny atmospheric disturbance that eventually could influence the other variables in major ways, say, a month later, in distant parts of the world. I do not now who used the butterfly analogy first, but subsequent writers have chosen different examples. That gives leave to suggest that such a butterfly might influence whether a hurricane devastated New Orleans or wandered off into Texas.

It is hard to imagine that we shall ever have instruments sensitive enough to predict the weather specifically more than a few days in advance—or make them about other complex systems. Time spans, for one thing, for systems to oscillate chaotically extend from moments for molecular systems to eons for astronomical systems. For another thing, the unpredictability applies to specifics, not general trends. Two flecks of foam above a waterfall have unpredictable relations below it—but the waterfall's tendencies can be reasonably well anticipated.

Topology, a branch of mathematics first developed about the turn of the century by Poincaire, had demonstrated some of the same principles many years earlier. And some had also been demonstrated by physicists. But the two sets of findings were never related. The Lorenz paper was published in 1963 by the Journal of Atmospheric Sciences. The meteorologists did not take it seriously, and nobody from the other sciences read it. Hardly anybody in any other discipline even knows of that journal's existence, and few scientists of whatever stripes would have been able to understand it without help.

Differential equations express the process of a system—its "movement" or its "becoming"—from one configuration to another. This holds whether the system is purely mathematical, without known correlates in nature, or it does happen to describe events in nature. In either case, if successive states can be graphed on a computer, characteristic forms will emerge. Thus, the "exact" sciences are capable of utilizing mathematical equations to study some systems in process by using differential equations. But before advanced computers existed these had to be "linear" in nature equations that "behave well" in computations. If they are graphed in analytic geometric forms they create orderly lines—not necessarily straight ones, but, for example, regular curved ones, as in sine waves.

Complex systems, after all, the most interesting parts of nature, usually cannot be described by the relatively easily solvable linear equations. When scientists tackled the problems of conditions in nature that, if described in mathematical terms, would demand nonlinear differential equations, they added "corrective" constants. The Ptolemaic system had to have epicycles added in order to make planetary movements somewhat predictable. It was assumed incorrectly that their orbits were circular rather than elliptical.

But let us go back: What are nonlinear differential equations? They are those in which the variables of the equation interact with each other in complex rather than additive ways. These latter differential equations are typical of all living and other complex systems. To repeat, living systems could not be studied, at least while the creature was alive, except empirically or by the laborious assemblage of separately derived statistics, by the science that existed two decades ago. Poincare long ago illustrated the extreme complexity when only three bodies interacted; usually the equations describing the interactions could not be "solved"—graphed in a way that would demonstrate the system in process. But with the development of powerful computers even these can be parsed. Imagine the increase of complexity when there are more than three variables in mutual interaction—as there usually are in living systems! Psychoanalysts, of course, do imagine it somehow, but it seemed in the past unlikely that these multiple variables could be quantified, much less modeled in action. Not many analysts thought mathematics had any applicability at all! Matte Blanco (1975), Thom (1975), and Sashin (1985) were exceptions.

I can illustrate nonlinear dynamics in words. The behavior of an ice hockey puck after a player has slammed it depends on multiple factors, including the mass of the puck, its inertia (or momentum and vector if it were in motion when hit), the mass and propulsive strength of the stick governed by the player's muscles (and the vector of the player's motion), the distance over the ice to the goal, the conditions of the ice (its temperature, amount of liquid on it, how much it has been cut up previously by the skates, etc.), the "hook" or "slice" of the stick that puts a spin on the puck, etc. These represent only some of the variables, but I need to focus on only two: the changing speed of the puck and the friction of the ice. The variables interact: the faster the puck moves, the less the friction. Furthermore, there is not a smooth, "linear" relation between the two (because friction and speed are themselves determined complexly). These variables (and the others) interact with each other, and any set of rules to construct a computer model of these events would have to be expressed in nonlinear differential equations. Again, within a system, every functioning part influences every other part.

Yet, somehow hockey players do quite often score goals without thinking the matter out. They do it by nonconscious "computations" far, far more complex than any currently imaginable digital or analogue computer's "artificial intelligence." They do it with an almost inconceivably complex system (that is why it seems so simple) of pattern recognitions that take into account all the variables mentioned above and more—roughly, in a "fuzzy" way, at nonconscious levels. These patterns, or Gestalten, are learned and improved over time in marvelous ways as "automatic" habits and techniques. But just how, we are far from understanding. Very likely these complex events are instances of relatively simple equations that are capable of generating extraordinary levels of complexity. But they are by no means consistent, and seemingly inexplicable cycles of breakdowns of play oscillate with standard play and occasional superb play—and here we introduce, in addition, the state of mind of the player.

Strange Attractors

Lorenz constructed an equation for only three key variables to create a simplified model of a weather system. At each successive moment in the activity of a system like the one Lorenz constructed, it is possible to make a point on a special graph that expresses the interactions of all the variables. If the points are connected, they describe an "orbit" around a central point that exists in "phase space"—not "actual space." Phase space is shown if two interacting variables are graphed in a space that includes both positive and negative numbers and "real" and "imaginary" numbers. We are familiar with "real" numbers, but "imaginary" numbers are not imaginary at all—they are numbers generated if "real" numbers are multiplied by "-1." They can thus provide space for the plotting of the square roots of negative numbers. Each variable in an equation amounts to one dimension. Three dimensions would create a three-dimensional graph. There can be as many dimensions as mathematicians might like. More than three or four cannot be imagined by ordinary people, but many biological systems have far more variables operative than three or four. Nine dimensions would choke the most elaborate main frame computer!

Reduced to three, Lorenz's feeble computer ground out the solutions to nonlinear differential equations that traced a process. If they had been relatively tractable to solution, they could have been solved by a calculator, or even by hand—but that would have taken years.

The point that is "orbited" in phase space is called an "attractor." If a pendulum's movements, an apparently simple system, are plotted on the graph, a spiral is created as the pendulum is slowed by friction. The stopping of the pendulum is indicated on the graph by a point. It is as if the stilled pendulum "aims" at a point, and if we imagined it existed in actual space, we might conclude that its target was the center of the earth—ignoring that the center of this roiling earth is constantly changing. It would be tempting to conclude that there is an actual "attractor" somewhere pulling the pendulum toward it. But an "attractor" exists as an abstraction, just as the connected successive points, each representing the interactions of all the variables of the system, and collectively representing the abstract "orbits" are abstractions. They describe the process of the system. These abstractions can exist in purely mathematical terms, but many of them also describe models of actual events in nature.

In more complex systems, attractors "move" (in phase space), split, combine and recombine with others; they can also be graphed and comprehended as images or forms. It is even possible on the computer to make cross-sections of their patterns of movements. In the case of linear equations, the "orbits" described over time are regular—periodic—like circles, ellipses, parabolas, in two dimensions (or like smooth spheres in three). But nonlinear differential equations behave "strangely." If plotted on a computer, a strange attractor not only "moves" with the changes in its system in process, the "orbits" do not overlap but may come infinitesimally close to each other.

To return to the seemingly simple example of the pendulum, we are accustomed to think that a pendulum is slowed only by friction. We know "intuitively" that unless it is given small, regular, external boosts, it will eventually stop. We assume that an ideal pendulum, swinging free of friction, would not stop. Yet, most of these assumptions are incorrect if fine enough measurements can be made. The most exquisitely designed pendulum mechanisms, even if run in a vacuum, would not operate with perfect periodicity. If observed carefully and long enough, irregular oscillations toward and away from deterministic chaos, disorder, would be reflected in our phase space graph. There is deterministic chaos even in the two-dimensional phase space of simple pendulums! Deterministic chaos can be demonstrated in the best quartz timers.

The orbits demonstrated in simple, two-dimensional phase space, as in a pendulum, seem strange, and in three-dimensional space, as in Lorenz's three-variable weather system, the image generated becomes even more strange. The totality of its orbits, none exactly overlapping another, but coming infinitesimally close, as if laminated, also moves in three dimensions. The product is an abstract structure that seems to have not only length and breadth, but "thickness"—a sort of sculpture rather than a sort of picture. The image after a sufficient number of orbits looks vaguely like a butterfly (of all things), with two beautiful wings and something like a body projecting at right angles in the third dimension. What it would look like after a thousand times that number of orbits, I do not know. Remember that the form being described is three-dimensional because Lorenz used three variables.

Currently mathematicians are using probabilistic tools to approximate solutions with "fuzzy logic," just as minds approximate "solutions," as our hockey player did with the stupendous numbers of variables entailed by any shot. Reductionism is unavoidable; the selection of key variables from the total presumably will always be necessary even if computer models of complex systems are to be made. A strange attractor only "seems" to be organizing the system—that is, acting upon it in remarkable ways. It is actually the product of the organization. Mathematically and physically, it is the point that completely describes the state of the system at that particular moment. At intervals, even in the direct graphing of nonlinear equations, as in Lorenz's weather system, the direct graphing of the variables shows branching, and these are periods when the characteristic forms of near-chaos develop. There are periodic movements toward disorder and away from it, toward order and regularity.

These oscillations toward and away from disorder are characteristic of all complex systems, and their variations, their natures, even possibly their functions, are beginning to spawn research in very different intellectual disciplines.

Recursive Equations, Fractal Geometry, and Nature's Preferred Shapes

In some recursive equations, those in which the "output" of the equation is fed back as "input" in the succeeding equation, one finds regular approaches to near-chaotic forms, and at specific, regular levels the graphic form bifurcates. Again, this is true whether it is purely mathematical in form or reflects some material system in nature. One of the pioneers in this area, Mitchell Feigenbaum (for the best descriptions, see Gleick, 1987; Stewart, 1990 -- Feigenbaum's actual papers are not for nonmathematicians!) used a hand-held computer and a simple recursive equation that happens to describe a parabola when graphed. After enormous effort, he discovered a constant of nature that provided an understanding of the periodic branching or bifurcation of any nonlinear system at periods approaching chaos. These changes are demonstrated even more dramatically when materials are strongly perturbed, or undergo state changes, e.g., from ice to water, or from water to gas. Feigenbaum is one of the few men to discover a constant in nature, like Pi. In fact, he discovered two (and may be the only man to have done that, as far as I know). These constants for splitting hold mathematically, using numbers alone, and hold true in nature as in Lorenz's artificial weather. This chaotic "doubling" accounts for the impossibility of predicting a system after several such episodes. Although the equation itself may be relatively simple, the results may be wondrously complex, some like those seen in nature, sometimes not like images we have not seen but which appear strangely familiar. What is characteristic of these images, which are the result of multiple iterations (repetitions) of the recursive equation plotted with complex numbers, is the development of scaling. The same fundamental form is duplicated with variations at every scale of magnifications. Complex numbers are "real numbers" between minus infinity and plus infinity added to the so-called "imaginary numbers" mentioned above. The most famous of the recursive equations is z - - > z² + C, the equation used in Mandelbrot's "Julia Set." The mathematics can be easily followed in either Geleick's (1987) or Stewart's (1990) discussions.

Mandelbrot (1986), a resident genius of IBM., is a sort of "superstar" of this field, which he named fractal geometry. This geometry deals with forms that are fractions between the familiar dimensions (1, a line; 2, a plane, 3, an object with height width and length). Fractals are easier to understand when diagrams can be used. The characteristic results are sometimes remarkable. But Mandelbrot is by no means the only investigator of fractals. Others have been able to find multitudes of equations which also produce fractal images -- and they are also abstractions, but look amazingly like ferns, trees, branching blood vessels, or bronchi -- or dust. It is possible to calculate the approximate fractal number for characteristic forms in nature.

Even more fascinating is that if one zooms down on the computer's monitor, mimicking a microscope, the same basic forms can be seen, with seemingly endless variations. These figures will retain "self-sameness." With an ideal computer, these scalar effects should go to infinity. The reasons these computerized "forgeries" appear so "realistic" is that nature is in fact "rough," that is, characteristically "fractal" rather than "smooth and straight." Anyone can buy programs with the algorithms to create such figures, supply the figures for the variables -- and discover forms on a home computer that perhaps have never been seen by humans before. At scales from the microscopic to the terrestrial to the planetary, probably to the galactic, the same nonlinear dynamic principles appear to be at work; the same patterns, always unique, but always similar in given classes. Structures like "Russian dolls" exist in nature—figures within figures within figures. Originally, fractal geometry seemed very far from chaos theory, but as Stewart (1990) points out:

Topology, Catastrophe Theory, and "Self-sameness"

The concepts of topology introduced by Poincaire have infiltrated and made possible many of the developments listed above—and many more than either I could understand or had space to describe. Topological theory deals with shapes of any size, but makes no distinction between a square and a circle. It does, however, make distinctions among surfaces that are folded, stretched, twisted, knotted—and constrained within a defined space. It is often called "stretch and fold" theory.

Advances in topological theory led in many directions. One of them made possible Thom's (1975) "catastrophe theory," a relative of deterministic chaos, that attended to discontinuities in nature (related to the folding). It pointed to important possibilities in the study of the development of form in biological systems, particularly in embryology. Possible applications for psychoanalysis were explored by Sashin (1985) and Galatzer-Levy (1978). Unfortunately, catastrophe theory was initially too enthusiastically accepted, and failed to show the range of applications hoped. It lost ist "reputation," despite the correctness of its mathematics. But Stewart (1990) points to its continued worth as a way to understand the remarkable movements and alterations in developing embryos.

SUMMARY

Topology demonstrates remarkable phenomena. Take, for example, a pizza-maker who pounds his dough to a certain thinness, folds it, pounds it again, over and over. The fate of two drops of ink on the original shape is indeterminable -- they may wind up far apart on the dough, or close, But a curious thing happens and can be demonstrated startlingly if a picture of Poincaire himself (or anybody else) is painted on the "pizza" of a computer screen. The picture is artificially "stretched" repeatedly and yet constrained within the same space, thus folded. It soon become unidentifiable, lines form, then lines that show unmistakable signs of chaotic behavior, then, oddly enough, several small but identifiable likenesses of the original picture reappear! Following that, they also disappear, and what eventually takes their place is a slightly degraded version of the original image.

A repeatedly applied transformation of a mathematical system constrained by a bounded space will return to states near its original state -- infinitely (Stewart, 1990, p. 58).

Theories of complexity include a collection of concepts from various scientific disciplines: relativity, quantum mechanics, number theory, coding, possibly aspects of "artificial intelligence," topology (fractal geometry), and deterministic chaos. I have discussed only the latter two. I refer the reader to several excellent books available.¹ Deterministic chaos is related to features of fractal geometry (and to other applications of topology, e.g., catastrophe theory). It is possible to use fractal geometry to mimic some systems in nature. I have alluded to the central features: the characteristics of nonlinear differential equations; the sensitivity to initial conditions; the impossibility of detailed predictability of complex systems; the existence of recursiveness and scaling—the same processes and patterns occurring on submicroscopic and astronomical levels, their "self-sameness" on all these scales, the "Russian doll" effect.

I draw the conclusion that while it was necessary for the sciences to specialize in order to develop to present levels, the specialization increased to the point that there were almost no cross-communications. Nowhere was that more obvious than in the deterioration of relations, formerly so close, between modern physicists and mathematicians. According to Ralph Abraham, a mathematician,

Ordinary Observations Without Special Equipment

To turn from technical terms and abstractions, it is important to simply look about us. It is not necessary to bother with a computer: simply stare for a long time at a waterfall, or a movie of an oil well out of control. Take a look at a snowflake, seemingly infinite in its diversity, and then remember some other time when you are skin diving near banks of coral: see how they compare. Take a look from an airplane at the waves of sand in the desert below, and at some other time stand by a clear stream and look at the pattern of sand in its bottom. Peer through a big telescope and see the red spot of Jupiter, or examine the evidence for the chaotic orbit of Pluto or Saturn's moon, Hyperion. For the evening news, look at a satellite picture of an approaching hurricane. Watch a dust-devil spinning in a field. Listen to one recording of one session of an ordinary psychoanalysis in process. Listen to the transitions that take place in a well-functioning psychoanalytic case conference. Once one is aware of deterministic chaos, one sees it everywhere.

Chaos is part of all living systems and all complex nonliving systems—and even, perhaps, some abstract systems hitherto assumed to be "perfectly regular." The "strange" behavior is not strange; deterministic chaos applies to ecological systems and the movement of the blood in the heart. Is it surprising that it applies to brains and that it applies almost certainly to minds? And to systems in interaction with each other, like two stars or two people? It is awesome that certain mathematical principles exist not as theories we invented, but laws that can be documented. They apply to everything.

Self-analysis, Personal Analyses, and Scientific Fantasies

Grossman (1982, 1989; Grossman and Simon, 1969) has been especially interested in the relation between scientific theories and inner conscious and unconscious fantasies. His scholarly paper, "Hierarchies, Boundaries, and Representation in the Freudian Model of Mental Organization" (1989), presents a convincing argument that Freud derived the basic features of his most complex model of the mind from his self-analysis. He constructed the scientific theory largely from his own endopsychic perceptions and fantasies, and then compared its parts to rational external experiences. The theoretical structure applied to both the neurological and the psychological theories. The point is important, because one of the characteristics of the theories of complexity is that they operate according to mathematical principles that are relevant to families of nonlinear dynamic systems.

The same nonlinear differential equations may apply to systems that seem on the surface to be totally different from each other and are studied by entirely different disciplines.

Although Freud used different models at different times, the model created in On Aphasia in 1891 (see A.-M Rizzuto, 1989, 1991; and Valerie D. Greenberg²) and repeatedly used in one way or another throughout his career, followed almost the same fundamental organizing principles that are now assumed in the model here called the process model (Weinshel, 1990; Freud, 1913, p. 130), or organismic model (Loewald, 1970). The structural model has been used since 1923. All of the other terms clearly suggest, in English, a dynamic system, in constant flux (See Tyson and Tyson (1990). These terms mean the same things if they imply the same ideas Freud intended by the 1923 "structural theory," when he clearly compared the structural theory of the mind to the processes that take place within a living cell in relations with its external world. In keeping with ideas of Freud, more recently stressed by Friedman (1988) and separately by Abrams (1987, 1988) and Weinshel (1990), I emphasize (1983, 1990) that with this model in mind, the crucial features were the "free-floating attention" of the analyst, and the similar but not identical "free association" of the analysand (used here in the sense it was used by Kris [1982]). The form of relatively unconstrained associative processes sought by analyst and patient are contrasted to processes characteristic of ordinary "rational," or "logical," or "objective" thought.

The two forms are often related to secondary and primary processes, and it is tempting to term them "linear" and "nonlinear" thought. The temptation should be resisted. The obvious analogy to linear and nonlinear mathematical equations is a faulty one. The mathematical definitions are sharply distinguishable—differential equations are either linear or nonlinear. The phenomena of the two forms of thought are not dualistic because the terms linear thought and nonlinear thought refer to scanning symbols in consecutive order and global pattern recognitions, respectively (A. Pollack, 1991, personal communication). It is preferable to think of a continuum between relatively directive/directing conscious thought and relatively nondirective/directing conscious (associational) thought. When functioning is optimal, there is oscillation along this continuum, with at least some degree of volitional control, more frequently and consciously deliberately invoked by the analyst, but also to a growing degree consciously and deliberately invoked by the patient.

Others have focused upon these oscillating patterns of thought (especially, Lewin, 1955). When the oscillations are most efficient, they favor the perceptions of inner states. In setting aside, as much as is possible, rational and moral constraints of fantasy, they provide both access to endopsychic perceptions—fantasies and feeling states that are "closer" to unconscious processes—and their evaluation by way of secondary process thought. Freud ( 1 91 5b) ascribed this method of investigation to scientific creativity. In a letter to Ferenczi on April 8, 1915, he applied this mechanism as the "succession of daringly playful fantasy and relentlessly realistic criticism." In the recently recovered draft of the "lost" Chapter 12 of his magnum opus on metapsychology, Freud (1915b) revealed the astonishing scope of his interest in "endopsychic perception." Much earlier (1901), he had revealed something of radically revolutionary nature, in Freud's words:

On Aphasia contested Meynert's view of point-by-point localization of the representations of external perceptions in the brain. To the contrary, Freud said, the periphery is contained as representations in the cortex "much as a poem contains the alphabet" (Freud, 1891, p. 53). Following Hughlings Jackson, Freud believed that no point-for-point projection of peripheral information in the cortex exists; the different elements are combined in new ways in different relations to each other. Further, there can be displacements from representations to other representations. Thus, a specific transference, a form of displacement, is never exactly like, for example, a lost infantile object, and relations with another person based largely on transferences are doomed to disappointments (Grossman, 1989).

Freud (1900, 1923) added two important elements to his "organismic" model. The first, apparently germinating in his self-analysis during the same period in which the basic scientific fantasy grew, was the fundamental insight that the mind of the doctor and the mind of the patient are qualitatively similar. They utilize almost identical processes. They have a quality of self-sameness. The second has to do with internalization processes that take place with the environment, especially after loss (Schafer, 1968).

Freud applied this theoretical template to a whole series of related living systems: periphery to cortex, cortex to psyche, and within the psyche a system of word presentations partially corresponding to representations of things or patterns. Each system exists in hierarchical relation to the next. Grossman (1989) mentions that this epigenetic theory of development can be compared to "Russian dolls," in which similar but not identical dolls are nested within each other. It also is the referent to Freud's "archaeological" fantasy of the mind. The influence of this model can be traced through the various developments of Freud's theory, not only in terms of theories about the neural system and the system of mind, but also in terms of the very techniques used to devise the theory, i.e., Freud's use of selfanalysis and his use of analysis with patients.

Is the seventh chapter of The Interpretation of Dreams (1900) merely a translation of the neurological theory? Or was the original neurological theory in On Aphasia a translation of Freud's already existing psychological insights? The two versions do have the characteristic of self-sameness. Freud warned, however, that psychological processes should not be confused with physical processes. "The psychic," he said, "is a process parallel to the physiological, 'a dependent concomitant'" (Freud, 1891, p. 55). The words, "a dependent concomitant" were in English in the original, a quotation from Hughlings Jackson. Freud usually adhered to a unitary conception of brain-mind. But at other times he seemed to imply a dualistic, interactive theory. Although Freud's model was first presented in print in On Aphasia, we assume that this work must have been a crystallization of processes extending to his roots in the past. Its intellectual precursors must have been there long before, at least as far back as adolescence. A huge literature has grown up about his development during late adolescence and young adulthood. Freud was not only innately gifted, he had the advantage of a thorough, classical education. During that time there was a rich assortment of heroes for the boy and young man, including contemporary heroes. Freud developed an extraordinarily original model for psychoanalysis out of these intellectual conflicts. It was (and is) a model of a complex system.

These were times of great diversity of thought ( Practically speaking for most serious scientists of the era—and publicly but not necessarily privately for Freud—the "respectable" view was all on the side of materialism and positivism, and tended to include only the familiar methods of reduction of variables and operations according to the "canons" of Bacon and Mills. This was the science that relied solely on linear equations. It also attempted to exclude the subjectivity of the observer. Nonetheless, the early psychoanalysts did learn to accept in themselves—as we accept in ourselves—the fundamental insights Freud had learned from his self-analysis. It was a completely different way of thinking compared to that supposed to be proper for scientists.

Published accounts regularly make it appear that the creations resulted from inexorable logic, "objectivity," and proper experimentation. Yet there is abundant evidence that most creative scientists, perhaps all, recognize the role of nondirective thought in their creations—and call it intuition, artistic "feel," or undefined "inspiration." Just the kind of thought that was supposed to be avoided by scientists is one key to originality. It is a way of thinking that deals with complex variables in a completely different way than the most logical forms of thought.

Nondirective thought, if it does not encounter internal resistances, is, on a conscious level, effortless. It "comes from below." As Freud (1915b) put it, "I maintain that one should not make theories—they must fall into one's house as uninvited guests while one is occupied with the investigation of details" (p. 83). As Paul Ecker said (1990, personal communication), "All along psychoanalysts have been the primary practitioners of complexity theory. The problem of psychoanalytic research on these matters is something else again—a very difficult problem."

I do not mean to imply that the older version of scientific methods will be replaced; they will not. Logic is logic, and that means rational, "linear" thought, organized thought, critical thought, cleansed as much as possible of dreamy absurdities and external inconsistencies. Nondirective/directed thought takes the form of reveries—patterns, images, mixed with varying quantities of verbal and other symbolic forms—unconstrained by conscious agendas. Both are necessary in the practice of psychoanalysis.

With this Janus-faced approach, psychoanalysis has contributed greatly to knowledge. In another place (Spruiell, 1989), I have discussed the surprising amount of research that continues to be conducted within psychoanalysis, despite the severe contemporary problems of finding financial and institutional support. And the interests in building bridges between what is known about the mind and what is known about the brain continues, with reciprocation from the neural sciences.

It is impossible, in this paper, to do justice to the scientific work that has been done and is continuing along more or less traditional scientific lines over the past decade. Examples include work by teams, reported by Hoffman and Gill (1988), Luborsky et al. (1980), Koenigsberg et al. (1985). A recent collection edited by Dahl et al. (1988) can provide an entry for the reader. For examples of research done by analysts and nonanalysts, see Masling (1983, 1986, 1991). For considerations of future possibilities and limitations in relation to the neural sciences, see Reiser (1984), Stent (1975, 1978), Changeaux (1985). For overviews of the present scientific status of psychoanalysis, see Wallerstein (1983, 1986, 1988).

New Ways to Think About Psychoanalysis

Because the new sciences of complexity can utilize computers, it is possible to demonstrate forms of dynamic systems graphically, in process. It is possible to see in the patterns of systems to demonstrate differing oscillations toward and away from chaos at differing levels of complex organization, and to begin to understand them better, perhaps to come to understandings of the developmental and hierarchical levels of communicating systems, the capacities to generate truly new and nonobvious innovations, and to pose new questions about human realities, our own natures, and the nature of our universe.

While we cannot predict where two adjacent bits of foam at the top of a waterfall will wind up in relation to each other at the bottom, we may be able to predict possibilities of relatively free or relatively constrained interactive communications or influences within the whole of a system and within definite ranges (according to the limits of proscriptive rules rather than the dictates of prescriptive rules [Spruiell, 1983]). Thus, we can begin to understand how a system under threat from impingements or perturbations from the external world may alter itself by cutting off or distorting communications among its parts either damping or intensifying its "optimal oscillating levels" of chaotic behavior, but also restricting certain functional interactions.

That is what psychoanalysis aims to do in reverse, and when successful, does do—is it not? Help catalyze the analysis of unconsciously initiated separations (repression, isolation, screening, splitting, etc.) and thus restore previously lost connections? If psychoanalysts pay attention to what they actually do within the analytic frame, they will regain hope and a measure of freedom from the older constraints of obsolete and narrow conceptions of nature, e.g., the "clockwork universe." And it will be pleasant to no longer need to worry about whether psychoanalysis is a science. Instead we shall ask ourselves, and be asked: In what ways does psychoanalysis fit in with other nonlinear dynamic systems? In what ways can our discipline become more scientific?

Unfortunately, our particular scientific burdens are stupendous. We use nonlinear processes constantly, in attending to associations and feeling states, to fantasies and images, in ourselves and in our patients—we take in and "process" spectacular numbers of variables. But we must "compute" large parts of this material endopsychically, often unconsciously. And we must not only scan external perception, we must scan the patterns we perceive inwardly and bring them into conjunction with the more "rational" self-criticisms of linear thought.

We can translate these meanings for others, but we know how difficult that is, even in the most favorable circumstances. We have not been able to delineate our variables clearly enough, much less make them quantifiable enough to be monitored, even less to find ways to monitor them except within the confines of the psychoanalytic consultation room—and our own minds. We do not even keep good records, and only recently have begun to establish our bibliography and begin to archive transcripts, audio tapes, and video tapes of actual analytic work in progress. Our best proofs at present are demonstrated in the slow workings of the therapeutic and intellectual marketplaces. Our clearest consensuses are reached only within the safety of small group discussions of peers. Our best safeguards are to continue to do the work of analysis and improve that work. Our hopes still reside in expansion of the research already going on in traditional science.

We are nowhere close to having an ability to set up plausible computer models for mental activities.

Separations

The theme of separation has recurred throughout this paper. I have used the term to encompass literal interferences or alterations of communications among subsystems of the body or mind—not only among people—that go beyond the necessary separations and individuations that culminate, if all goes well, in various levels of maturation called adulthood. Physically and mentally, maturity implies the furtherance of integration of functions. Developmentally, new functions, as well as parallel, redundant systems, are conceivable in terms of the mind as well

35

as the body, and make [unctional "self-repair" possible, or at least potentially possible within the psychoanalytic situation.

Physical symptoms are the experiential responses to perceptions of actual chemical or tissue damage. Psychological symptoms, of course, may mimic organic disorders, and may be particular responses to perceptions, valid or invalid (from the point of view of the observer) of impending danger. Most psychological symptoms, whether recognized as seemingly alien phenomena or not recognized and mistaken to be part of character, are also experiential responses to self-inflicted, functional interferences to free communication among the systems of the mind, whether they were originally protective in nature or attempts to maintain infantile gratifications that were dangerous. But we also know that they are, as are any other complex psychic actions, the results of particular compromise formations (Brenner, 1982). Their only formal difference from other compromise formations is the "high price" in human life such patients must pay. But quantitative differences at some point can change into qualitative differences (Galatzer-Levy, 1978).

Chronically fixed compromise formations are, from the individual's perspective, inescapable, necessary, the best result possible. It is very difficult for life to teach other lessons, although one way to these other lessons is available for at least some people, and that is self-analysis. But even self-analysis in depth is usually available only to people who are able to find the protection of an analytic situation set up by a professional who is competent to maintain it. It is the rare person who can go on this quest very far alone.

Whether we speak of the body or of the mind, such separations require alterations in all other systems that integrate particular functions, for example, the transportation of oxygen to individual cells, or the much more complex sets of functions having to do with genuinely interactive social adaptations (Spruiell, 1983).

What do these considerations have to do with the future influence on psychoanalysis of complexity theory? Among other things, they have to do with the functional significance of the "ordinary" oscillations toward and away from chaos. Unusual perturbations of the systems ultimately result in great divergences because nonlinear dynamic systems are extraordinarily sensitive to (some) very small alterations of variables (as Lorenz's classical experiment first demonstrated). There seem to be "normal" oscillations in individual subsystems, some subsystems "go into action" if perturbed by stimuli that are not overwhelming; other subsystems damp their tendencies to chaotic conditions when they do "go into action."

We do not know yet what meanings these findings, which are just beginning to emerge, will have for psychoanalysis. For example, Freeman and his colleagues (Skarda and Freeman, 1987) have been able to demonstrate in living animal brains the waxing of levels of chaotic behavior from the stimulation of nasal sensory nerves, with specific projections to the olfactory bulb. Such stimulation ignites increases of chaotic behavior, and from there to the multiple projections in various parts of the cerebrum. In these experiments, stimulation intensifies levels of chaos. On the other hand, recent studies of the expectable levels in which brief periods of chaotic activity in the normal pulse of a healthy heart are damped when the person is near death as a result of some specific heart diseases. The pulse becomes much more regular, and periodic in nature. The understanding of the function of chaotic fluctuations is only beginning. It is safe to presume that there are at least analogies between the systems that make up the brain and the psychological systems that make up the mind. But whether that is true, or it can be plausibly demonstrated not to be true, psychoanalysis has a scientific future that will inevitably amaze us. And, inasmuch as the principles suggested above presumably apply to all living systems at whatever scale, they may well apply to the integration of individuals in groups, organizations, even ideas and belief systems (Spruiell, 1987, 1989).

Surely there are phenomena analysts observe that call for observation in the study of systems in other disciplines—as Freud implicitly suggested so long ago. Psychoanalysis has learned extremely important things about patterns of hierarchical arrangements of systems; related to the development of subsystems is the evolution of boundary functions—as biological membranes have complex functions. Other examples from psychoanalysis have to do with the redundancy of most subsystems; related, are capacities for self-repair. Still other insights concern interactions between systems of lower and higher levels of complexity within the individual and between individuals of higher and lower organizations (Loewald, 1971), and, finally, interactions within groups made of similar components.

Not only do similar mathematical principles govern all systems, but the same equations apply to seemingly totally different systems. That leads us to wonder what they have in common on a deep level of structure. For example, it has already been shown that measles epidemics over the years in Baltimore and New York follow the same patterns, the same movements toward and away from chaos, and operate according to the same mathematical equations. Further, similar activities and fluctuations can be demonstrated to have occurred in the cotton exchange over many years!

The Current Relevance of Complexity to Psychoanalysis

Galatzer-Levy (1978, 1986, 1988) has produced several contributions on the impact of computer science on psychoanalysis. Among them is a discussion of the relation of catastrophe theory to psychoanalysis (specifically to a discussion of the ways qualitative changes become converted into qualitative changes, as they can be shown to do in the worlds of physics and mathematics through the "stretching and folding" involved in topology). Another has to do with the mathematical problems posed by an attempt to cope with many variables at once, as in calculating the shortest route from one point to a distant point in a large city—the solutions have to do with the use of "fuzzy logic," probabilistic answers—in the same way our minds cope with such problems. A third has to do with the concept of working through, using a model from artificial intelligence. It was mentioned above that catastrophe theory was developed by Thom (1975); it used topological principles to study discontinuities in nature (and so is a close "relative" of chaos theory). Sashin (1985) also used catastrophe theory to consider phenomena of affects. Galatzer-Levy's and Sashin's papers have not received the attention deserved.

The first specific published paper in an established psychoanalytic journal on the relationship of complexity theory and the newest mathematics was by Moran (1991). It is an excellent introduction, soundly based on its mathematics, and modest and challenging in its speculations about possible research applications. Fogel (1990) published an exciting review, packed with details, of Gleick's book on Chaos, with an important addendum on our changed expectations of scientific predictions by Mosher (1990).

I shall not attempt to summarize the rich ideas advanced, but only emphasize several points. Moran mentions that "there is no clearly elaborated thing or even (single) conceptual analogue for which the ('strange') attractor stands" (pp. 17-18). But there may be such referents in psychoanalysis, among them the more or less chronic unconscious fantasies of the patient. 6'Through the mode of the strange attractor, the unconscious fantasies, though themselves very simple underlying structures, become manifest in complex, multidimensional behavior" (p. 18). Elaborating this, he points to the impact of small, repeated, well-timed perturbations in the form of interpretations—keeping in mind the extreme sensitivity to initial conditions of systems approaching chaotic states. Study of the impacts of the analytic relationship itself may provide new insights into the processes of intrapsychic change. So will awareness of the phenomena of scaling help, as Teller and Dahl (1986) have shown in their study of the microstructure of free association, based on the recorded words of one patient. A person's personality "signature" comes through all the variations of all the "magnifications" of behavior, the "Russian dolls within Russian dolls," all in interaction—from a slip of the tongue, through a dream, through the most complex and long-lasting mental phenomena and actions. Most of all, Moran stresses the "opening up" of research possibilities for the future. Fogel's review makes similar speculations, particularly concerning scaling, dreams, and transference.

Moran's and Fogel's speculations represent the use of chaos theory as a "way of thinking about psychoanalysis," like other perspectives, e.g., the "ways of thinking" provided by differing versions of frame theory, or rule theory (Spruiell, 1983). They can be used as organizing templates for novel views of psychoanalysis. The picking up of similarities between psychoanalysis and deterministic chaos depends on our human capacities for complex pattern recognition. We understand the similarities through the use of endopsychic perceptions—in conjunction with the "rigorous criticism" to which Freud alluded. The surprises of psychoanalysts come from such inner conjunctions. The research problem is whether we can find ways to demonstrate them externally, as representations of realities in nature.

At present it would be wishful thinking to believe psychoanalysts can exploit these ideas in the form of external research productions, demonstrable to nonanalysts. Ultimately we need to find ways to identify "clean variables" to which reliable values can be set. It is known that certain linked variables seem to be identical or at least closely bound with linked variables in seemingly very different systems. We usually cannot understand the similarities on the level of "deep structure." In the past it was practically impossible to study such correlations. But now, these may provide alternative ways to create plausible, even if simplified, computer models of mental systems in operation. It has been known for a long time that large multidisciplinary studies of human beings generate statistically significant correlations among variables derived from disciplines as superficially distant as physiology, the study of recorded psychiatric interviews, sociological ratings of behavior in groups, and psychological tests (Greenblatt and Solomon, 1953). It already is possible to crudely infer, from external physiological sensors, the existence of certain affective states, although nothing reliable can be perceived about their actual content, from external physiological sensors. It is likely that much can be learned that is relevant to psychoanalysis from experiments in other disciplines that utilize variables closely correlated with those of psychoanalysis.

Some day it may be possible to understand certain mental functions by way of better mapping of brain functions. The use of imaging techniques such as PET, MRI, are only in their infancy. When it becomes possible to create our own computer models—and I believe but do not know that it will—then variables can be altered, or removed, or added—for the observer to see what happens that might have relevance to the understanding human minds. When and if these possibilities come about, psychoanalysis will have made a quantum leap as a science. Then we shall begin to learn presently unimaginable things rather than discover correspondences to the private knowledge of our own hearts, or metaphorical similarities with the things other disciplines have learned.

But we cannot forget that a revolution is only beginning. Nobody in the thick of battle and disorganization can see clearly, and even if it were possible to see clearly, accurate predictions would still be out of the question. We bear, with all the great privileges of experiencing revolutionary ideas, the burdens of uncertainty, the possibility of grandiose expectations coupled with pathetic disillusionment, the ever-present needs to simplify and cheapen the complex and the awe-inspiring.

Alan Pollack (as far as I know the only analyst with a Ph.D. in mathematics who taught the subject at M.I.T.) states that deterministic chaos "does have applicability to the systems that regulate affect, attention, mood, action, etc. It may even help build better neurobiological models of these and other aspects of brain functioning." But currently, he cautions, "I don't think it has much applicability to clinical psychoanalysis—except inspirationally. The inspiration can be very valuable, in the way that grand orienting metaphors are usually valuable—and simultaneously limiting, of course" (1990, personal communication). And Ian Stewart (1990) ends his book, "At this stage in the development of chaotic dynamics, it's important not to be too carried away by rampant speculation. The subject is exciting, it's fashionable, it cuts across numerous disciplines, it's moving very fast. In such circumstances, misunderstandings are all too easy" (p. 327).

It will be a great professional privilege to assume such scientific burdens in the future, with all their pitfalls and difficult methodological problems. The mathematics of deterministic chaos is the "glue" that holds the sciences together (and possibly some fields now considered nonscientific), to bridge pathological separations and alienation—analogous to physical and psychological pathologies. If so, psychoanalysts will be integrationists, at the crossroads.

FOOTNOTES

1. For an appreciation of the beauty of fractal geometry, see Peitgen and Richter's The Beauty of Fractals ( 1986). A six-part series on chaos appeared in Science during 1989 about general conceptions, and many articles on specific pieces of research have appeared. The same is true of Nature. Several important articles have also appeared in Scientific American. Books by Pagels (1983,1988) and Penrose (1989) help provide an understanding of number theory as it applies to computations and codes in complexity theory. I have found Casti's (1990) Searching for Certainty especially instructive.

2. (1998) Valerie D. Greenberg has this year published, Freud and His Aphasia Book. It is a beautifully researched study of Freud's background, motivations, times, and "influences." This book will vastly expand our appreciation for these subjects, and is a fine example of the power of multi-disciplinary studies.

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