By Jane Stewart Adams
In 1952, a paper appeared in the Bulletin of Mathematical Biology titled The Chemical Basis of Morphogenesis. The paper’s aim was to provide an analysis of how the genes of a zygote realize the anatomical structure of the organism. The paper was written by Alan Turing, and he was determined to explain the phenomenon of morphogenesis—the capacity of all life-forms to develop increasingly complex bodies out of impossibly simple beginnings—in already well-understood physical and mathematical terms. Instead of positing new forces, substances, properties, or laws, Turing presented a mathematical model wherein a system containing simple entities following simple rules could evolve into a dynamic structure of incredible depth and complexity.*
The key to this phenomenon is the instability of equilibrium, a seemingly counter-intuitive notion as it is generally given to be true that equilibrium is a stable state. Consider: a marble perched atop a sphere or a pencil standing on its point are both states of unstable equilibrium. Though the marble or the pencil may be placed at the precise point where the net forces cancel out (and thus the system will not change), the smallest deviation from that point causes the entire system to come crashing down. Random disturbances among the simple entities in such systems cause the symmetry of the system to break down, giving rise to unforeseeable change and growth. “Such a system, although it may originally be quite homogenous, may later develop a pattern or structure due to an instability of the homogeneous equilibrium, which is triggered off by random disturbances.”** This is the birth of the principle of sensitive dependence on initial conditions—how small changes give rise to drastic changes in a system—and it is central to my own work on emergence in complex systems.
It is appropriate that Turing’s paper pays special attention to how the repeated patterns of flowers could be encoded in their seeds, for the seeds of Turing’s revelation had been sown long before 1952, had in fact been germinating in the rich soil of scientific thought since the birth of atomism, if not before. Like so much of his work, Turing’s Morphogenesis paper was a flash of insight that launched a scientific revolution, becoming the cornerstone of the new, cross- disciplinary sciences of chaos and emergence, but that flash did not strike out of the blue. Turing’s revelation is rooted deeply in the work of Leucippus, Democritus, Epicurus, and Lucretius; the spirit of modern chaos bears an uncanny resemblance to the spirit of ancient atomism.
Early ancient atomists proposed a powerful and consistent materialistic picture of the natural world in the simple terms of atoms and void. Their task was to provide an account of the origin of every phenomenon in the natural world—every force, every property, every complicated, complex structure of matter, living and non-living—in terms of the interactions of atoms—tiny, indivisible building blocks with only the humblest of properties, such as size and shape.*** Importantly, as it is presented by Aristotle in On Generation and Corruption, the major motivation for atomism was to explain change without claiming that something must at some point come from nothing. Atomists attempted to explain change as the re-combination of pre-existing entities, namely atoms. For the early atomists to succeed, it appeared that they would need to add new atomic properties, or new properties of atomic interactions, or new forces acting upon the atoms—just as it appeared that in order for mathematical (and otherwise) biologists to succeed in explaining morphogenesis, they would need to add new properties, laws, etc. None of these additions would have been easy bullets for the early atomists to bite as they were intent on maintaining their strict materialism.
Their task was not unlike Turing’s task: to explain impossible complexity in terms of impossible simplicity. Presented with an analogous problem, it is unsurprising (but nonetheless extraordinary) that the early atomists came up with an analogous solution.**** Like Turing, the early atomists were determined not to posit new fundamental forces, properties, and the like. And, like Turing, they did this by proposing a solution in dynamical (as opposed to static) terms, meaning terms that do not just describe a system by its state at t1, t2,…, tn, but rather in terms of how a system evolves through those states over time. Like modern chaos, the hypotheses presented by the early atomists were concerned with change, and importantly not just with change of state. The nature of dynamical systems is the very nature of change.
Ilya Prigogine and Isabelle Strengers provide an excellent description of the set of equations required to describe such dynamical systems:
At every instant, a set of forces derived from a function of the global state…modifies the state of the system. Therefore this function as well is modified: from it, a moment later, a new set of forces will be derived. To resolve a dynamic problem is, ideally, to integrate these differential equations and to obtain the set of trajectories taken by the points of the system…It is evident that the complexity of the equations to be integrated varies according to the more or less judicious choice of the canonic variables that describe the system. *****
The differential equations described here ideally model the nature of change, but Prigogine and Strengers are quick to point out that very few dynamical systems are such that their equations are easily integrated. Rather, most dynamical systems are unpredictable and include irreversible processes in which interactions must themselves be taken into account, meaning that among the “canonic variables” required to describe the system is the very modification of the set of forces used to describe the state of the system. Put in very simple terms by James Gleick, in such non-linear dynamical systems (as they are called) “the act of playing the game has a way of changing the game.”
At the very heart of dynamical systems, then, we find fundamental unpredictability side by side with unstable equilibria and sensitive dependence on initial conditions. Turing and the early atomists each proposed something like this fundamental unpredictability in their own unique, yet clearly analogous, terms. Turing proposed random disturbances; Leucippus and Democritus proposed the atomic whirl; and Lucretius and Epicurus proposed the atomic swerve. Each of these descriptions captures something about the nature of change itself that descriptions in terms of states simply cannot. By introducing fundamental unpredictability, not a new force or a new law or a new property, the early atomists and many after them provided an account of how impossible complexity can emerge from impossible simplicity, how a world of the richness and depth that we experience can emerge from the interactions of humble atoms. The result of Turing’s random disturbances—and of the atomic whirl, and of the atomic swerve—is all the same: “the disorderly behavior of simple systems [acts] as a creative process. It [generates] complexity: richly organized patterns, sometimes stable and sometimes unstable, sometimes finite and sometimes infinite, but always with the fascination of living things.”******
The astounding thing is that the spirit of the atomic whirl and of the atomic swerve is so evident in the spirit of modern chaos that they are barely, if at all, distinguishable. That the rejection of what even today are considered givens—that equilibrium is a stable state, and that the act of describing the state of a system does not alter the state of that system (though this latter given is being steadily upheaved by quantum mechanics)—was the fuel for the fire of these world- changing ideas inspires me to propose an embedded urban project that explores how the principles of morphogenesis are operating in everyday urban activity and life, with references to the history of these principles leading back to the ancient atomists. This project will comprise installations of data art throughout the city using turnstile data published by the MTA, juxtaposed with the language of ancient atomists.
Ideally, this project will capture and reflect back to the city’s residents how we, while minding our own simple, individual business (entering the subway to go to work, leaving the subway to go home, etc.) are participating in the complex and rich evolution of the city as a whole. Engaging with these installations will introduce subtle changes to the behavior of residents (for example, a resident may stop to watch a simulation of the number of people entering and exiting the subways across the city just before entering the subway herself), capturing the very spirit of unpredictability in the moment.
The piece will also be explorable on the Internet. Here, individuals will be able to interact with the piece to zoom in and out of different time and space scales, e.g, to a particular week or to a particular station. In doing so, they will hopefully see how the pattern of the whole subway system for an entire year is simply the pattern of one station, or one week, or one day, or one turnstile, even, writ large.
The opacity of the piece, how visible it is, will change dynamically with the number of individuals viewing it at any given time. If no one is viewing it, it will be invisible, and it will be more visible the more people are viewing it. In this way, I hope it will inspire an even simpler understanding of how the systems in which we participate in our daily lives make sense only through our participation. They do not control us or ordain what will be for us, they are us.
*“Instead of positing new forces, substances, properties, or laws…” Phenomena that resist explanation in already well- understood terms sometimes lead scientists (and, more frequently, philosophers) to propose new fundamental forces.
**At first blush, the addition of a “randomness” component could be seen as just as much of a cop-out as the addition of new fundamentals. Turing’s paper appeared in the very early years of a new mathematical and scientific approach; at around the same time, the founders of the Copenhagen interpretation of quantum mechanics—most notably, Werner Heisenberg—were proposing that randomness is, in fact, fundamental. The early work of his time ultimately led to the legitimization of randomness in terms of non-linearity, sensitive dependence on initial conditions, and chaos.
***See David Furley Two Studies in the Greek Atomists
****I should perhaps say that Turing came up with an analogous solution to that of the early atomists when presented with an analogous problem, as opposed to the other way around. However, it is not evident that there is a time relation, or even a causal relation, between analogs. If there is, it is a topic for another time.
*****See Ilya Prigogine, Isabelle Strengers quoted in John Rahn The Flow and the Swerve: Music’s Relation to Mathematics
******See James Gleick Chaos: Making a New Science