Chaos
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"Chaos breeds life, when order breeds habit."
- Henry Adams
In the late 1700s, the eminent work initiated by the English mathematician and physicist Sir Isaac Newton was carried forward by the French mathematician and physicist Pierre-Simon Laplace, also known as the “French Newton.” Laplace firmly believed in determinism, the idea that the past completely determines the future, that given enough information about a system, it is possible to make completely accurate predictions about the future state of that system.
In 1814, Laplace published the first formal articulation of determinism, called Laplace’s Demon (1):
“We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes.”
By the 1800s, determinism was all the rage in mathematics and physics. There were certain “special cases” that did not fit with the idea of determinism, but these misfits were pushed aside.
Over one century later, on a winter day in 1961, the mathematician and meteorologist Edward Lorenz studied one such special case - the weather. Lorenz used a model based on 12 equations to run simulations that predicted weather patterns on his rather archaic digital computer. To save time, he entered data from the mid-point of his earlier simulations so that he could examine the later weather predictions of his model without having to wait through the entire simulation. When Lorenz entered the data, he rounded it off to three digits whereas the data from the earlier simulations contained six digits, a difference he assumed to be inconsequential since temperature is rarely measured to within one part in a thousand. However, when Lorenz compared the new simulations with the earlier ones, he saw this:
The two simulations matched each other closely at the start, but after several model “months” they wildly diverged, not even remotely resembling each other in the longer term. Upon completing an extensive analysis, Lorenz realized that his assumed inconsequential round-off error was in fact highly consequential, that given enough simulation time even the tiniest round-off errors produced massive changes in the entire system.
Fascinated, Lorenz pursued these observations with an even simpler model, consisting of only three basic equations, that described rolling fluid convection in a box full of gas heated from below. These were the three equations:
Dx/dt = P(y-x)
Dy/dt = Rx-y-xz
Dz/dt = xy-By
When Lorenz plotted these equations separately in one dimension, the traces were mundane:
Yet when he plotted the traces together in three dimensions, something remarkable appeared:
This is now known as the Lorenz attractor, a model of rolling fluid convection so simple that it can be described by three basic equations, yet even given all the information about the system it is not possible to make accurate predictions about its future state - the rolling fluid convection system never enters the same state twice, and as such the plotted line that gives shape to the Lorenz attractor never intersects itself. It is not possible to make accurate predictions about the future state of such a system. Yet it is clear to anyone with eyes that the Lorenz attractor contains an orderly pattern.
The Lorenz attractor shows us that even given all the information about the state of a system, its future state cannot be accurately predicted. Yet despite the unpredictability, there is clearly some kind of order. The Lorenz attractor would have been unwelcome in Laplace’s deterministic world, if Laplace's deterministic world actually existed.
The Beautiful Order In Chaos
The observations of Lorenz and his description of the Lorenz attractor embody the phenomenon of chaos, a phenomenon that is difficult to pin down with words. That said, a chaotic system contains at least two core features:
(1) Extreme sensitivity to initial conditions.
Even the tiniest change in the system will have a massive impact on its future state (shown in Lorenz's weather system).
(2) Non-periodicity.
The system never, ever enters the same state twice (shown in Lorenz's rolling fluid convention system).
Chaos teaches us that even the simplest systems may exhibit completely unpredictable behaviour, yet while that behaviour is unpredictable and may even seem random, there lies within that behaviour a mysterious, beautiful order.
The Lorenz attractor is just one of many examples of a strange attractor, a set of numerical values that a chaotic system tends to evolve towards. A chaotic system is described by its strange attractor. Although the simplest chaotic systems exhibit completely unpredictable behaviour that may even seem random, the mysterious, beautiful order is apparent in their corresponding strange attractor. Here are more examples of strange attractors:
Duffing attractor. |
Clifford attractor. |
The Beautiful Order In Life
Many people today prefer predictable things over unpredictable things; they want to know what they will be doing at work before they actually go to work, or where they will be going before heading out for the evening, or where they will be staying before going on a vacation. For a variety of reasons, there exists a strong urge to predict future events before those future events are actually experienced.
This urge to plan and predict everything has led us to a unique point in history, an era in which an inordinate amount of power has been granted to policy-makers who chase prediction and order through excessive laws, academic theories, and naive ideologies. However, since excessive laws, academic theories, and naive ideologies promise predictability and order, they cannot endure in a world defined by chaos, a world where accurate prediction is an illusion, where true order arises from chaos. The result is a false sense of order, a kind of pseudo-order that is, to put it lightly, unsatisfying.
History and the world we live in today have not been shaped by policy-makers but by risk-takers, people who embrace the unpredictability of chaos and make discoveries based on action in the real world. Through the looking glass of history, it is clear that entrepreneurs such as Henry Ford and Bill Gates have had more impact than any bureaucrat, that inventors such as Thomas Edison and Leonardo da Vinci have had more impact than any academic, and that heroes such as Saladin and Martin Luther King Jr. have had more impact than any politician.
The predictable, orderly life marketed by policy-makers doesn't really exist. It is an illusion. Risk-takers know that we live in a reality of chaos, and it is in chaos that we find the only true order in existence - unpredictable, mysterious, beautiful.
The only order worth seeking.
Solace.
References
(1) Laplace, PS. 1814. Essai philosophique sur les probabilités.
