# The order in chaos / Review of 'Does God Play Dice?' by Ian Stewart

Does God Play Dice? by Ian Stewart, Basil Blackwell, pp 310, Pounds

sterling 15

FEW books have reached the advertisements on the sides of the escalators

in London’s Underground, but James Gleick’s book on chaos is one of them.

If the passengers on the underground can see chaos all around them, without

the help of scientists, and there is already one popular book on chaos,

what need is there for another one? Ian Stewart’s Does God Play Dice? gives

the answer. Gleick’s book is essentially a personal account of the American

contribution to the science of chaos. Stewart set himself the more daunting

task of putting the science into its historical context, and explaining

what it is about.

The title of the book recalls a letter from Einstein to Max Born ‘You

believe in a God who plays dice, and I in complete law and order.’ When

Einstein refused to believe that God played dice, it was quantum mechanics

that he had in mind; but his philosophy captures the attitude of an entire

age to classical mechanics. The metaphor of dice for chance applies across

the board. Does determinism leave room for chance? We now know that classical

mechanics is more mysterious than even Einstein imagined. The very distinction

he was trying to emphasise, between the randomness of chance and the determinism

of law, is called into question. Deterministic laws can produce behaviour

that appears random. Order can breed its own kind of chaos. The question

is not so much whether God plays dice, but how God plays dice. It is this

discovery that has given us the science of chaos and its implications have

yet to make their full impact on our scientific thinking.

Deterministic chaos has its own laws, and inspires new experimental

techniques. There is no shortage of irregularities in nature, and some of

them may be manifestations of chaos. Turbulent flow of fluids, reversals

of the Earth’s magnetic field, irregularities of the heartbeat, the convection

patterns of liquid helium, the tumbling of celestial bodies, gaps in the

asteroid belt, the growth of insect populations, the dripping of a tap,

the progress of a chemical reaction, the metabolism of cells, changes in

the weather, the propagation of nerve impulses, oscillations of electronic

circuits, the motion of a ship moored to a buoy, the bouncing of a billiard

ball, the collisions of atoms in a gas, the underlying uncertainty in quantum

mechanics – these are quoted in the book as a few of the problems to which

the mathematics of chaos has been applied.

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How can the applications be so wide? Because chaos is a consequence

of nonlinearity, and most phenomena are nonlinear. The very word nonlinearity

is deceptive: it is as if most of biology were called the study of non-elephant

life.

There are few mathematicians with Stewart’s sense of history, and the

ability to put the ideas of mathematics into words with the help of analogy

and the apt quote. He starts with the Greeks and the clockwork model of

the Universe that developed from Newton’s success in applying his laws of

motion to the Solar System. He contrasts it with probability theory which

provided the laws of error, and points out how the dichotomy of determinism

and probability diverted attention from a whole range of problems which

could not be treated using the traditional methods.

The historical introduction finishes with Poincare, as the last universalist,

and the first of the moderns, who paved the way for the science of chaos

in the last decades of the 19th century. Then the going gets tougher: linearity

and nonlinearity, phase space of many dimensions, attractors strange and

otherwise get their explanations and their pictures. Some of the explanations

succeed, but a few do not, as they are unlikely to be followed by anyone

who has not met the subject before. For example, Smale’s work has had a

profound influence on the mathematics of dynamics, but it is very difficult

to put his ideas across at this level. But then the examples follow, and

the clarity returns.

Sometimes the literary allusions are overdone: for example, Japanese

haiku poems are very elegant, but it looked to me as if they were dragged

into the book by the scruff of the neck. Perhaps the author was aiming for

the Japanese market? Stewart’s book can be read at more than one level.

For an elementary introduction to the science of chaos, skip the difficult

bits and read the introductions to the chapters and the conclusions before

tackling the meat in the middle, which may be too difficult to digest anyway.

At a more advanced level, if you have been brought up in the tradition of

linear mathematics, and you want to learn about the wonders of nonlinearity,

read the historical introduction, and go through the mathematical sections

in order of difficulty, rather than the order in which they are written.

In any case, this is a book well worth reading, and a valuable contribution

to the literature on chaos.

Ian Percival is Professor of Mathematics at Queen Mary College, London.