An old New Yorker cartoon—Two scientists at a blackboard. On the left and right of the blackboard are complex equations. The phrase, “Then the miracle occurs.” connects them. One scientist says to the other, “I think you should be more explicit about step two.”
Today that “miracle” is often made explicit using the mathematics of nonlinear systems. We psychoanalysts have lots of miracles for which nonlinear dynamics can help.
Pick up a psychoanalytic publication and it is hard not to come across the term “nonlinear.” The popularity of the word reflects a sense that something is going on that is not captured well by our common sense vision of orderly interactions, something complicated and perhaps mysterious happens in the complex world of mental function. The term “nonlinear” in mathematics indeed refers to amazing features of the world that were very hard to understand until recent years.
Nonlinearity is a quality of processes seen in virtually all complex systems, a quality that means surprising and unexpected things will happen even though the process follows the laws of cause and effect. This often occurs because in nonlinear systems tiny changes in causes can make huge differences in outcomes. A good way to think about it is to imagine a road dividing: Go a few feet to the right, you end up in New Hampshire, a few feet to the left and your destination is Southern California.
The study of nonlinear systems began in the early 20th century, but serious progress had to wait for developments in mathematics and other theoretical disciplines in the 1960s. A small group of psychoanalysts began to explore their application to our field in the 1970s.
The basic idea is this: When more than two objects interact, even when their interactions are governed by simple rules, the result is not just the sum of their individual interactions. Something new and different occurs.
Let’s start with Newton who calculated how individual planets moved around the sun. His solution was exact. He could write an equation for planetary motion and plug in numbers that gave very good but imperfect results. But he knew that, in addition to the gravity between the sun and each planet, the planets exerted a gravitational pull on one another. He could write equations that included these forces but he could not solve them. Nor could anyone else in the succeeding 350 years. The “obvious” solution of making a correction to the equations for the interactions of the individual planets doesn’t work as well as one might hope. It improves the solutions but leads to physical impossibilities
after a while. The equations were impossible to solve. Even the seemingly simple general problem of predicting the motion of three bodies under the force of gravity remained intractable and yielded predictions that were simply wrong. Such questions as whether the planets would eventually crash into the sun or spin off into space could not be answered clearly. A new kind of math was needed.
That new math, sometimes called chaos theory or nonlinear dynamics systems theory or the theory of nonlinear differential equations, described a new world, a world that expands our notions of what is possible, a world that should reshape our “common sense” and, with that, enlarge our picture of psychological configurations.
Imagine, for example, that a patient comes to a session with a novel ambition for which you cannot find a convincing antecedent. Using a linear model, you say to yourself, “This must come from somewhere. Maybe some repression was lifted freeing the patient to think a new thought.” But the resulting interpretation feels forced. However, in a linearly based model of mental function it or something like it, such flawed interpretations are the only possible ones. Perhaps, you noticed an antecedent for the development, but it seemed too minor to have a profound effect.
In nonlinear systems, however, qualitative changes do emerge from quantitative ones, even small ones. Just as when water cools below 32° Fahrenheit and turns to ice, a substance qualitatively different from liquid water, so too, at the right point tiny quantitative changes in psychological systems can precipitate novel mental configurations, even ones not there in latent or unconscious form. This idea expands our capacities for conceptualizing change and is reason for optimism about the possibilities that can emerge from the psychoanalytic process.
Our picture of development is transformed by a nonlinear worldview. If, as is true of nonlinear systems, small changes can yield big consequences, Robert Frost’s image from “The Road Not Taken,” where a minimal change “made all the difference,” becomes a good metaphor for the reality that tiny changes in the course of development can be life altering. Recognizing this as a possibility in a complex system enlarges the range of our understanding. For example, development may be transformed by seemingly small events. This profound effect of small events might suggest that the latent meaning of an event is greater than is obvious (the standard analytic understanding). Or, it may be that occurring at the right moment, minor events open new pathways leading to the emergence of something new and surprising.
Analysts love narratives and find them useful explaining psychological events. Convincing narratives often come from mythologies or plausible stories of development. We can think of narratives in terms of the ways characters and elements move through time and space. Unknown to either of them, Oedipus meets his father on the road where the father will not yield right of way. The usefulness of the myth in psychoanalysis is that it describes a path close to those that emerge in many analyses. Mathematicians call such pathways “attractors.” For a long time, the study of motion was limited to systems that corresponded to the unfolding of a single story line. But, today, a far larger range of attractors is recognized with properties such as sudden jumps from one pattern of motion to another. These strange attractors suggest new types of narratives, for example narratives that suddenly jump between distinct patterns of change, closer to James Joyce’s Ulysses than Sophocles’ Oedipus Rex.
Analysts have long known that a closely examined bit of analytic process may reveal the structure of the entire analysis. This quality of a tiny portion representing the structure of the whole is typical of “fractals,” the beautiful structures seen everywhere in nature from coastlines to snowflakes to plants and the lung. Fractals provide a model for the analytic claim that the study of material at different levels of magnification from an appropriate 15 seconds of interchange to an analysis may yield basically the same information. They provide a rationale for this connection. Since one of the major impediments to researching analysis is the overabundance of process material, this fractal property in which a small part very well represents a whole is a road to simplifying this difficult work.
Analytic sessions sometimes feel like a dance. The back and forth between analyst and patient builds something new and curative. A similar process between infant and caretaker is part of normal development. For hundreds of years, physicists have known that when vibrating systems come into contact, new configurations of vibration are likely to appear in each of the systems and in the newly formed dual system. The nonlinear mathematics that describes these
processes has only come into being in recent decades. Applying it to the psychoanalytic process suggests how two people moving through their own repetitions again and again may create something new when they come together.
The New Yorker cartoon’s “Then the miracle occurs” refers to “emergence,” the appearance of new properties of complex nonlinear systems seemingly out of nowhere. A few examples: the complex hexagonal form of snowflakes, the emergence of new species in Darwin’s theory of evolution, the stable configuration of the internet out of millions of individual connections or of elaborate colonies made up of individual ants each of which follows simple rules. So far the main use of this concept in psychoanalysis is to open our eyes to the reality that such things happen. There are emergent properties of analytic processes and psychological function generally. When we see something that seems new in an analysis or in development we neither need to assume that it can be reduced to something that was already there nor do we have to declare it a “miracle,” albeit by some other name. It is likely an emergent property of the complex system we call the human mind.
The field of nonlinear mathematics is enormous. It has implications for psychoanalysis ranging from questions about boundaries to the statistical methods appropriate for exploring psychoanalytic data to the way practicing analysts listen to our patients.
Want to know more?
Ian Stewart’s Does God Play Dice? remains the best introduction to chaos theory while Melanie Mitchell’s Complexity: A Guided Tour is a lively introduction to the broader field of complexity. Both are accessible without an extensive mathematical background to those who do not instantly become anxious when mathematics is mentioned. Applications of nonlinear dynamics to psychoanalysis can be found using PEP with the search terms “nonlinear” and/or “chaos theory.” My own book on the subject, Nonlinear Psychoanalysis: Notes from 40 Years of Chaos will be published by Routledge in the coming year.