Does God Play Dice?: The New Mathematics of Chaos (Penguin Mathematics) [Paperback]

Beautiful fractals, the butterfly effect and unpredictable systems were the images that chaos conjured up in my imagination before I sat down and read this book. Within its pages the incredible diversity of chaotic systems; and the diversity is remarkable; is presented and explained. It is staggering to see the picture unfold, the gradual realisation that 'the' scientific statement of the eighteenth century; that the universe runs according to a set of immutable laws; is unable to explain much of the behaviour in even the simplest of classical systems. The discovery of a whole new world, and one that has been in existence since the beginning of the universe: chaos. This book is merely an introduction to a comparatively new and exciting area of mathematics; but using the word merely is doing it an injustice, since it encapsulates the topic superbly and leaves the reader with a desire to study the mathematics of chaos in more detail.

Fittingly the opening chapter commences with the backdrop to this word 'chaos'. Three hundred years ago, Newton published, 'The Mathematical principles of Natural Philosophy'. This work is unrivalled in the field of mathematics; its basic message has been absorbed into our culture: "Nature has laws and we can find them." Unfortunately, although mathematics allows us to calculate the solutions to many difficult problems, we are still left in an unordered world, where apparently simple motions, on closer inspection, become unpredictable and hence unexplainable in the language of mathematics. It is appropriate at this point to introduce the nature of chaos. Stewart is quick to point out that since this branch of mathematics is still in its formative stages, giving it a precise definition is not possible or wise. However to get us off the mark he gives the definition reluctantly reached by the Royal Society in 1986: "Stochastic behaviour occurring in a deterministic system." More roughly speaking, random behaviour in a system governed by laws.

Where is the dividing line between order and chaos? The chapter 'The Laws of Error,' introduces another field of mathematics, Probability theory - the mathematics of chance. Mathematicians had found that analysing the detailed workings of large systems was too involved and complex. Probability theory grew out of a need to simulate detail without actually having to examine it. As Stewart states: "Mathematicians could calculate the motion of a satellite of Jupiter, but not that of a snowflake in a blizzard."

The book continues with a look at one of the prime examples of chaotic systems in our World, weather systems. This century has seen many attempts to write equations that will linearize weather and use them to predict exactly how weather systems will behave. As we are well aware short-term predictions are accurate a large percentage of the time, but long term predictions are much harder to make. What we learned in the 'Strange Attractors' chapter can be applied here. The initial conditions that we feed into any model we have will have finite accuracy. Even if we obtain data exact to many decimal places, it will not take many iterations before it digresses from the path that the described weather system follows. Lorenz stumbled upon this when computing weather systems. After examining results from two separate calculations involving the same set of data, albeit with different rounding accuracies, he discovered that his results were very similar for a short period of time, but then diverged extremely rapidly and followed distinct paths. This breakthrough was later to be named the 'Butterfly effect', illustrating the manner in which a trivial dynamic can upset a disproportionably large system. .

At this stage in the book, Stewart leads us into a chapter entitled 'Recipe for chaos'. It firstly attempts to describe the workings of chaos as analogous to a recipe, presumably in an attempt to simplify the concepts and avoid any complex mathematics. It does not really achieve the desired effect. This chapter was the hardest to grasp, which is a shame since it contains many of the fundamental facts about chaos and its axioms if one can use such a word.

Fractals are important part of chaos that joins the discussion at this point. They present us with a language to describe what we see happening with chaos. A fractal, generally speaking, is a geometric object, which continues to exhibit detailed structure over a wide range of scales. Self-similarity exhibited again. Interestingly the method of describing the detail level of a fractal is by allocating it a dimension, known as its Hausdorff dimension. These dimensions tend to be fractional (hence fractal).

The final three chapters are new to this, the second edition of the book, and they describe some of the advances in the subject since 1989, namely the prediction and control of chaotic systems, which are both perfectly possible.

In its totality, this book gives any discerning reader an opportunity to delve into the World of chaos and come away with a greater understanding of the topic as a whole and a glance at the variety of areas and applications it covers. The mathematics of chaos is involved; this is not surprising since the initial discovery of the topic was due to the inability of conventional mathematics to describe certain behaviours. Nevertheless, on the whole Stewart does a good job of explaining concepts and then illustrating them with simplified examples, avoiding the need for much of the mathematics. However there were one or two places where his desire to seek analogies for his models overlooks the aim of aim of the analogy in the first place that is to aid the readers understanding of the topic. Helpful too, was the inclusion of a fair number of diagrams and schematics that in several cases proved invaluable to my understanding of the book. This is a great introduction to a subject that is becoming increasingly important and perhaps indispensable in mathematics.