Book Review: Chaos Theory Tamed
Fortunately, there is a (partial) solution to this problem. Chaos Theory Tamed, by Garnett P. Williams, clearly explains much (but not all) of the mathematics invoked in The Black Swan: the mathematics of complexity theory. (Note: There are epistemological distinctions between complexity theory, chaos theory, fractals, etc. However, for the purposes of this book review, I will mostly ignore those distinctions. At any rate, Garnett Williams' book covers math that are cross-disciplinary and would be useful for any of those aforementioned categories.) Uniquely, Garnett Williams' book manages to explain the mathematics of complexity, chaos, and (to some extent) fractals in a way that is both accessible yet sophisticated.
Most books on chaos theory, complexity theory, fractals, etc., fall into two categories. The first category are books that are of a 'pop science' / 'pop math' variety; relatively easy to understand but whatever knowledge one can glean from these books are given with a lot of hand-waving and not a lot of ways for more sophisticated readers to get beyond generalities and the 'gee whiz' factor. The second category of books are for the more technically minded. The more technical books are (hopefully) good for the initiated but are over the heads of the uninitiated. Frankly, even for those comfortable with the mathematics and scientific jargon invoked by these technical works, the more formal papers and books are usually unpleasant to read and may deny people the sense of epiphany that one should get from good science.
There are a handful of books that attempt to bridge this gap between pop sci/math and more formal literature, but most of these books, frankly, fail. Garnett William's book is the one bright exception ... a positive Black Swan.
The mathematical prerequisites for reading Chaos Theory Tamed are light. The reader only needs to have the equivalent of a good American high school math education (basic algebra, and hopefully, some exposure to pre-calculus), and the patience to go through the logic (and equations) presented throughout the book. Unlike the few other gap bridging books that are in the marketplace, Chaos Theory Tamed's mathematical prerequisites are very minimal.
This low level of mathematical pre-knowledge should not be mistaken with a low level of mathematical sophistication. On the contrary, Garnett Williams' book does a great job of covering math that is actually invoked and used by professional researchers in the fields of chaos and complexity theories (as opposed to hand-waving toy models presented in other pop sci/math books and, even, in the gap bridging books on chaos theory). Without assuming an active knolwedge of calculus, Garnett Williams explains the workings of difference equations and differential equations of the type that Edward Lorenz and Mitchell Feigenbaum used to 'discover' chaos theory. Like other books on chaos theory, Chaos Theory Tamed discusses topics like 'strange attractors','bifurcation,' etc.; unlike those other books, Garnett Williams actually explains what those terms mean and the logic (math) behind them.
More relevant to Nassim Taleb's The Black Swan, Chaos Theory Tamed does an excellent job of explaining probability theory and the mathematics of the power law.
Garnett Williams does an excellent job of explaining the nuts and bolts of probability theory and mathematical statistics to the uninitiated. In fact, if that is all he did in his book (and he did much more than that), it would be a worthy book in it of itself since he provides the sort of explanations that would be useful to anyone wanting to learn probability and statistics beyond an elementary level. He also explains information theory along with the idea of entropy (in both the thermodynamics and Shannon information theory senses) and ties those ideas in with probability theory. Again, Garnett Williams' explanation of these complex topics -- topics that are so riddled with difficulties that more technical books try to avoid it -- are both so ambitious and helpful to the uninitiated that I could recommend the book on these grounds alone.
But Chaos Theory Tamed doesn't stop with explanations of 'strange attractors,' 'entropy,' and 'autocorrelation in time series.' Garnett Williams' book gives the best accessible explanation of power laws that I've encountered. Chaos Theory Tamed explains power law and scalability (scale free, scale invariant, etc.) in terms of 'dimensions.' Dimensions are essentially the kind of dimensions that we are all familiar with ... three dimensions of space (four dimensions if you include time), two dimensions of a sheet of paper or a fictional 'flatland,' and the single dimension of a Platonic straight line. With power laws, the dimension of what is being measured is the exponent and is relatively invariant (within some range). When graphed on log-log axes, a power law is a straight line and the power law exponent (represented as 'alpha' in The Black Swan) is the slope (usually, negative) of this line.
Garnett Williams uses the concepts of dimensions and scales in order to give a very clear-headed explanation of fractals. Chaos Theory Tamed doesn't dwell on fractals as much as other books that are more specifically focused on fractals, but, when the book does deal with fractals, the explanations of what fractals are and how they relate to chaos theory are brilliant for its clarity. Basically, fractals are defined in this book as being "fractional dimensions" -- i.e., instead of 1, 2, or 3 dimensions (integer dimensions), fractals are fractional (non-integer) dimensions (1.2, 1.4, 2.8, etc.). I actually find this sort of definition to be much more useful and interesting then the thousands of pretty pictures of fractals I have seen in other books, magazines, and on the web.
One can tie this fractal dimension idea in with power laws in the following way: The power law exponent (the dimension) is usually not a whole number. For example, 1.1 is the exponent (as provided by Nassim Taleb in The Black Swan, p. 264) for the net worth of Americans; that power law exponent translates to the top 20% of the wealthiest Americans having 86% of the wealth (from p. 265 of The Black Swan). Again, this way of thinking about fractals is at least as valuable as seeing a pretty picture of fractals.
Another distinguishing feature of Chaos Theory Tamed is that, unlike other books on chaos theory, it goes into how one might empirically detect and measure chaos and/or complexity; i.e., it goes beyond mere scientific speculation or even theorizing. The concept of dimensions (i.e., the power law exponent; the 'alpha' in The Black Swan) is important to the empirical study of chaos and complexity. There are different standards/measures of dimensionality including Hausdorff dimension, information dimension, and correlation dimension; they are distinct but similar ways of measuring dimensions. Information theory, in the form of Kolmogorov-Sinai entropy and mutual information, is also important to the quantitative study of chaos, and Garnett Williams does a commendable job of explicating these topics.
Garnett Williams is also quite frank about the limitations and difficulties inherent in trying to empirically detect and measure complexity. This note of caution fits in well with Nassim Taleb's warnings about not placing too much weight on 'precise' (frankly, there aren't any) measures of 'alpha' when thinking about randomness from a Mandelbrotian perspective.
One final point to praise in Chaos Theory Tamed is its glossary. Its glossary -- clearly defining terms in chaos theory, probability, mathematical statistics, information theory, etc. -- alone is worth the price of the book!
Garnett Williams' professional background is worth noting. He was a geologist for the U.S. Geological Survey. A geologist might not be the first person to come to mind in writing a book about the mathematical backbone of chaos theory, but in many ways he is the ideal person to write the book. The distribution of earthquakes and other natural geological phenomena follow power laws and have been studied by complexity theorists as being examples of complexity in nature. In fact, many natural -- as opposed to social -- phenomena seem to be consistent with Black Swan theory. Thus, Nassim Taleb's ideas should not be thought of as being confined to social sciences only but as being applicable to natural sciences as well.
Bottom-line: If you're looking for a clear-headed explanation, that is both accessible and sophisticated, of the mathematics behind The Black Swan (and complexity/chaos/fractal theory, in general), then Chaos Theory Tamed is a great place to start ... in fact, it's the best place to start. Frankly, I wish there were more books like Chaos Theory Tamed and more authors like Garnett Williams; books that aren't afraid to cater to the intellectually ambitious and science/math writers who aren't afraid to lay bear the equations that tend to mystify science and math to the uninitiated.