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.1
They key to this phenomenon is the instability of equilibrium, a seemingly counter-intuitive notion as equilibrium is generally considered to be 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, Turing claimed, 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.”2
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 the topic at hand.
Turing’s paper was not widely recognized at first, and he died shortly after its publication. Within a decade, however, Morphogenesis had become the cornerstone of a “new science”, and scientists and mathematicians across disciplines had latched onto the central themes of Turing’s paper to develop the legitimate sciences of chaos and emergence.
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 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, 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.3 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.4 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.4 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.
One approach, as John Rahn points out in The Flow and the Swerve: Music’s Relationship to Mathematics, is “to describe change in terms of states, at least two”. But, he asks, “do two such static descriptions capture the feeling of change itself?” Descriptions of atoms and their interactions in purely static terms could not explain how complexity emerges from a simple system of atoms and void. Thus, early atomists breathed dynamical life into their world picture, breaking with “a larger thread…in pre-Socratic Greek philosophy” that focused on examining the structure of “flash-frozen slices of the universe.” The problem with this thread was that it neither accommodated nor captured the nature of change itself.
The second approach is more difficult than the approach in terms of states. Rahn brings such words as “unpredictability”, “turbulence”, and even “free will” to bear on the nature of change itself, features of change that almost by definition resist explanation in static terms. Where in the description of an atom’s size, shape, position, and velocity is there room for a variable of unpredictability or randomness? Such dynamical properties exist between states of a system; they are not properties of states, but properties of processes. To accommodate these dynamical properties, the ancient atomists proposed systems not altogether unlike the system proposed by Turing.
The point of instability in the Leucippian and Democritan picture appears to be the point at which there are “enough atoms” present such that the atomic whirl is instantiated. It is unclear if motion is an intrinsic dispositional property of atoms, or if atoms move as a result of the forces created by the whirl. Both accounts present significant difficulties for Leucippus, but these are beyond the scope of this paper. The point is that once the atomic whirl is established, atoms begin to arrange themselves into structures according to their properties. These structures in turn affect how the whirl evolves, and so on and so forth. On this view, very small changes in the local arrangements of atoms give rise to large global changes in the system as a whole.5
“Lucretius, following Epicurus, modeled the universe as a frame of atoms … falling (naturally, according to their weight) forever in parallel lines, with this important tweak: occasionally, for no reason, an atom will swerve in its fall.”6 The atomic swerve is nothing short of delightful. If we may get a little carried away for a moment, the atoms in the Lucretian and Epicurean picture have almost charming personalities. Now, echoing Leucippus and Democritus, Epicurus and Lucretius proposed that “as the velocity of the laminar flow increases, vortices arise spontaneously,…irreversible processes of turbulence deriving catastrophically from infinitesimal changes in initial conditions.” Laminar flow or streamline flow is the flow of a fluid in parallel layers, with no disruption between the layers. Without the emergence of vortices at high velocities, and without the atomic swerve, atoms would never deviate from their laminar atomic flow, as it is called, and complex structures and patterns would never emerge.
Ilya Prigogine and Isabelle Strengers provide an excellent description of the set of equations required to describe such dynamical system:
At every instant, a set of forces derived from a function of the global state (such as the Hamiltonian, the sum of kinetic and potential energies) 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.7
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.”8 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.
“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 C W W Taylor The Atomists: Leucippus and Democritus ↩
See John Rahn The Flow and the Swerve: Music’s Relation to Mathematics ↩
See Ilya Prigogine, Isabelle Strengers quoted in Rahn ↩
See James Gleick Chaos: Making a New Science ↩