It is fitting that chaos theory got its start in the humble but frustrating field of meteorology. Why does it seem impossible for all our hot-shot meteorologists, armed as they are with ever more efficient computers and ever greater masses of data, to predict the weather?

Two decades ago, Edward Lorenz, a meteorologist at MIT stumbled onto chaos theory by making the discovery that ever so tiny changes in climate could bring about enormous and volatile changes in weather. Calling it the Butterfly Effect, he pointed out that if a butterfly flapped its wings in Brazil, it could well produce a tornado in Texas.

Since then, the discovery that small, unpredictable causes could have dramatic and turbulent effects has been expanded into other, seemingly unconnected, realms of science.

The conclusion, for the weather and for many other aspects of the world, is that the weather, in principle, cannot be predicted successfully, no matter how much data is accumulated for our computers. This is not really “chaos” since the Butterfly Effect does have its own causal patterns, albeit very complex. (Many of these causal patterns follow what is known as “Feigenbaum’s Number.”)

But even if these patterns become known, who in the world can predict the arrival of a flapping butterfly?

The stock markets are said to be nonlinear, dynamic systems. Chaos theory is the mathematics of studying such nonlinear, dynamic systems. Does this mean that chaoticians can predict when stocks will rise and fall? Not quite; however, chaoticians have determined that the market prices are highly random, but with a trend. The stock market is accepted as a self-similar system in the sense that the individual parts are related to the whole. Another self-similar system in the area of mathematics are fractals. Could the stock market be associated with a fractal? Why not? In the market price action, if one looks at the market monthly, weekly, daily, and intra day bar charts, the structure has a similar appearance. However, just like a fractal, the stock market has sensitive dependence on initial conditions. This factor is what makes dynamic market systems so difficult to predict. Because we cannot accurately describe the current situation with the detail necessary, we cannot accurately predict the state of the system at a future time. Stock market success can be predicted by chaoticians.

Manus J. Donahue III
An Introduction to Chaos Theory and Fractal Geometry

The upshot of chaos theory is not that the real world is chaotic or in principle unpredictable or undetermined, but that in practice much of it is unpredictable. And in particular that mathematical tools such as the calculus, which assumes smooth surfaces and infinitesimally small steps, is deeply flawed in dealing with much of the real world. (Thus, Benoit Mandelbroit’s “fractals” indicate that smooth curves are inappropriate and misleading for modeling coastlines or geographic surfaces.)

Chaos theory is even more challenging when applied to human events such as the workings of the stock market. Here the chaos theorists have directly challenged orthodox neoclassical theory of the stock market, which assumes that the expectations of the market are “rational,” that is, are omniscient about the future. If all stock or commodity market prices perfectly discount and incorporate perfect knowledge of the future, then the patterns of stock market prices must be purely accidental, meaningless, and random (“random walk”), since all the underlying basic knowledge is already known and incorporated into the price.

The absurdity of believing that the market is omniscient about the future, or that it has perfect knowledge of all “probability distributions” of the future, is matched by the equal folly of assuming that all happenings on the real stock market are “random,” that is, that no one stock price is related to any other price, past or future. And yet a crucial fact of human history is that all historical events are interconnected, that cause and effect patterns permeate human events, that very little is homogeneous, and that nothing is random.

With their enormous prestige, the chaos theorists have done important work in denouncing these assumptions, and in rebuking any attempt to abstract statistically from the actual concrete events of the real world. Thus, the chaos theorists are opposed to the common statistical technique of “smoothing out” the data by taking twelve-month moving averages of monthly data-whether of prices, production, or employment. In attempting to eliminate jagged “random elements” and separate them out from alleged underlying patterns, orthodox statisticians have been unwittingly getting rid of the very real-world data that need to be examined.

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