On the edge of anarchy

Nonlinearity, Chaos and Complexity is a messianic account of the evolution of quantitative science.

The first of three themes to be expounded is that of "linear"

modelling, which is later roughly equated to "reductionism" and which seems to be taken to mean the "linear superimposability of individual behaviours without interaction" rather than the more usual "just keep the essentials".

The second theme, and the book's most attractive section, is nonlinearity and chaos. The account of the Lorenz attractor will be widely accessible.

The climactic theme is the caliginous theory of "complexity". This concept is said to account for turbulence in fluids, the weather, the evolution of life, the formation of organised structures in all types of society and "phase changes" in science and society, including revolutions.

Thermodynamically oriented readers will identify it with Ilya Prigogine's "far from equilibrium" vision.

The book repeatedly signposts complexity as lying between linear theory and nonlinear/chaos theories, the dividing line between nonlinearity and chaos never being detailed. Even though complexity is stated more precisely to be at the "edge" of chaos, many readers will struggle to grasp its significance. One possibly helpful illustration is to think of the Earth's crust, which has evolved into such a delicate state that construction of even quite small dams can initiate earthquakes.

Readers are encouraged to grasp the holistic nettle of complexity even though it means discarding long-cherished quantitative science and redefining mathematics. The resulting vision of mathematics ignores the gigantic gaps in our knowledge of what the authors call "linear science"; for example, there is no hint of the unpredictable behaviour of solutions of linear differential equations with delay. Meanwhile, the statement that "in the domain of nonlinearity, it appears that the behaviour of systems cannot be described by comprehensible rules" denigrates centuries of intellectual gains. And there is no mention of "homogenisation" or "multiscaling", or even of the enormously successful theory of statistical mechanics.

The authors give the layman a good introduction to some of the different types of traditional mathematical modelling and its success and failures.

The book may appeal to those who lean towards grand unified theories, but down-to-earth readers will dislike the undisciplined exposition, the mathematical inaccuracies and the woeful index. It is most likely to resonate with scientists who feel that mathematical models are inadequate to describe most problems in the social sciences and many in the physical sciences.

Damien Challet is research associate and John Ockendon is lecturer, Oxford Centre for Industrial and Applied Analysis, Oxford University.