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We promised we'd follow up on our previous article on complexity theory, so here goes. Please see that article to understand some of the lexicon in this follow-up.

Indeterministic, or complex systems, are those systems where the hand of time does matter, and these systems can also be explained quite easily with a few examples.

Everyone knows that weather patterns cannot be predicted with any confidence. The reason for this is that weather systems represent the collision of an infinite number of factors interacting in random and unpredictable ways. The most famous saying with respect to weather patterns is that a butterfly flapping its wings in Africa can create a hurricane on the East coast. Given this complexity, weather patterns cannot be reverse engineered. Though we know that Florida has experienced several hurricanes last Fall, we cannot determine what specific conditions necessarily created these storm systems. Maybe it was a butterfly, maybe it was a collision of two minor low pressure systems. We can never know. In these types of chaotic, highly complex systems, time is irreversible and the product of these infinitely complex interactions cannot be reverse engineered. We cannot "undo" the course of time and determine what set of initial conditions created this storm or that tornado. Time marches relentlessly on. In this respect, time reversibility is a key difference between deterministic systems that adhere to Newton's laws, and those indeterministic systems that exhibit randomness and non-linearity.

Why all this talk about time reversible vs. time irreversible processes? Why talk about deterministic, inviolable laws vs. complex, non-linear behaviors? For this reason: the acceptance of Newton's laws of motion, and their power in changing the very environment that man found himself in, caused a tectonic shift in the understanding of science and nature in the 18th century. Where the Greeks and ancient philosophers could not come to a conclusion about whether nature was chaotic or deterministic, Newton's laws seemingly answered that question: the world was deterministic, governed by a set of known laws that, if perfected, would allow man to largely harness the resources around him. For the next several hundred years, this view washed over every academic discipline known to man: economics, physics, chemistry, sociology, politics, biology, and many more. The goal of discovery became perfecting our understanding of those laws and our application of them to build taller buildings, manage economies, and enact new forms of government. Understood in this light then, Newton's laws had a massive impact on the world.

One of the first scientists to question this view was the biologist Charles Darwin in the mid 1840s. By looking not at the individual animals themselves but rather populations of them, he was able to make some logical generalizations about the manner in which these species evolved. Darwin was then able to provide a framework - a model - for how biology "worked" through time - the theory of natural selection. Individual variability, subject to natural selection pressures, could produce a large, unpredictable and irreversible change in an entire class of animals. The deterministic world view held in biological circles up to that time could provide absolutely no answer for why different animal species even existed. Darwin's theory of natural selection, a uniquely indeterministic model that appreciated the chaotic, non-linear forces at work in natural systems, did. And thus Darwin was one of the first scientists to break the stranglehold that determinism and linearity had on science.

Later in the early 1900s, the physicist Henri Poincaré started his important work on instability by showing that planetary movements were inherently unstable, particularly where three or more planets interacted. Poincaré mathematically proved that tiny uncertainties in the measurement of initial conditions (position, speed, direction) of three body planetary systems could result in enormous uncertainties in the final predictions. Most importantly, these uncertainties remained even if the initial uncertainties were shrunk to the smallest imaginable size. Thus he proved that instability - non-linearity - is inherent in natural systems. Since that time, other researchers have gone on to show that this instability is actually an essential characteristic of such natural systems; that its presence gives rise to the macroscopic equilibrium that we see in, say, the manner in which planets move around the sun.

A general appreciation for the time irreversible, non-linear, complex forces at work in natural systems was a slow endeavor. Darwin's theory only slowly came to be appreciated for how radical it was. It wasn't until the exploration of chaos theory in the early 1960s that scientists even started to appreciate the fact that there were forces at work in natural systems that could not be precisely modeled or reverse engineered, could not be understood using then existing laws of mathematics and physics, and exhibited behavior that was deemed complex. It was meteorologist Edward Lorentz in 1963 that coined the phrase 'butterfly effect' to describe the possibility that entire weather systems could be generated by something as insignificant as a butterfly flapping its wings thousands of miles away. Gradually it came to be known that even the smallest imaginable discrepancy between two sets of initial conditions would always result in a massive discrepancy at later times as that initial "aberration" was magnified many times over by the non-linearity inherent in the system. This has become a hallmark of chaotic systems.

Though physicists like Poincaré were exploring non-linearity in the early 1900s, there was simply no mathematics to model or solve non-linear problems until the 1960s. The more scientists looked, the more non-linear systems they found all around them. Robert May, Benoit Mandelbrot, Mitchell Feigenbaum, David Ruelle, Floris Takens, and others worked on chaos theory from the 1960s onward, trying to model it and determine its important characteristics. This work on chaos theory, including the importance of fractals, is not necessary to detail here. The important points however are these: chaos theory had the effect of moving science forward in its appreciation for non-linear, time irreversible processes and of forcing scientists to realize that the existence of chaotic, non-linear processes within natural systems is actually, and ironically, essential for their long run stability. Slowly physicists came to appreciate that, in nature, chaos was the norm rather than the exception.

The widespread acceptance of chaos theory allowed the scientific community, and all the disciplines that previously embraced determinism (economics, biology, physics, politics, etc.) to start to accept the fact that the world is in fact a mix of deterministic and indeterministic systems, a mix of chaotic and linear dynamics, of time reversible processes and time irreversible processes. This insight, though it may seem trifling, is almost as revolutionary as the development of Newtonian mechanics itself. And academics are just now starting to understand the manner in which this insight can be put to use in all disciplines.

We just hope we've got a slight head start as that idea applies to negotiated financial markets.

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