[In the following excerpt, Campbell offers a positive assessment of Chaos.]

. … James Gleick's Chaos tells an exhilarating tale. It starts a quarter of a century ago with work on weather forecasting by Edward Lorenz and finishes with an account of the penetration of ‘chaos’ research into sciences as different as epidemiology and astronomy. Science has traditionally turned a blind eye when a graph fluttered unmanageably and thus the future could not be plotted on a straight line or a smooth curve. It is such non-linear phenomena which chaos research investigates.

‘Chaos’, as it is used in this context, is confusing. It is not, for instance, a synonym for ‘random’. Weather is chaotic: you never know exactly when the next cyclone will come across the Atlantic—but it is not random. The behaviour of warm damp air, on any scale from a cupful to a cyclone, follows the general laws of expansion, contraction and movement which apply in more predictable systems. In terms of averages and general patterns, it is stable and describable. Science up to now has looked to deal with phenomena which settle to regular patterns. It has assumed that in chaotic systems like the weather the problem was a scarcity of data, a confusion of superimposed patterns and rhythms, all interacting with each other. If only you could get all the detail right, the rest would look after itself. How can a process be both part of a world which obeys physical laws and also impossible to predict in principle?

Lorenz's discovery of the meaning which can be given to this proposition occurred when he was trying to model world weather. He had set up a highly unrealistic, very simple computer program in which variables (temperature, air pressure and so on) produced a circulation pattern. The picture the program gave of swirling air masses bore a decent resemblance to real maps of world weather. But the interaction of the equations was complex. Lorenz could not predict what was coming up next. At one point he wanted to examine one part of a computer run at greater length. As a shortcut he keyed in the values as they stood at the starting-point of the section he was interested in, rather than re-run the whole thing from the beginning. At first, the output chugged out of the printer in the same pattern as it has on the first run, but it quickly began to diverge, and soon bore no resemblance to the original run. Eventually Lorenz worked out what he had done: the computer worked to six decimal places, the print-out only showed three and in keying in the values he had assumed that the difference—one part in a thousand—was inconsequential. It was not. The evidence that equations which gave a good approximation of weather were so sensitive to small changes in initial conditions was bad news for the future of long-range weather forecasting. Even if your sampling grid was six feet (and not sixty miles, which is the scale used at present), the effect of small sampling errors would spread through the system very fast; and no matter how fine you made your grid, the amount of information needed for perfect predictions would be a set further away. As you moved towards the infinitely small, the impossibility of knowing the effect of an even finer dimension would pursue you. Small errors, Lorenz showed, can have large effects. It is not surprising that the huge computing power put into weather forecasting has had only fair results.

Inherent in this discovery was rather shocking news. The assumption that a good approximation will always produce results in due proportion had been proved wrong. The figures had only to be a little bit out for some predictions to be way out, and this limit applied even in systems like damped pendulums which seemed, on the face of it, unlikely to behave chaotically. Chaotic effects can be found in very simple systems. A waterwheel fed by a regular stream will behave chaotically at some rates of flow, and not at others. Regions of instability may be surrounded by stable regions. At some rates of increase animal populations fluctuate randomly from year to year, at others they stabilise, or follow regular cycles of increase and decrease.

Computers were the tool which made the mapping of chaos possible. When millions of computations are made, patterns begin to appear. A few hundred dots may make no pattern, but as they build up (apparently randomly), shapes begin to appear. For many researchers, the experience of watching patterns arise on a screen as unpredictable results arrive seems to have been crucial. The topological transformations which drew the patterns out of the data opened new windows. When these phenomena were investigated, Mandelbrot's fractal geometry began to come into the picture. His discoveries dealt with patterns which repeated at smaller and larger scales. A coastline has gulfs which have bays which have smaller bays which have ragged rocks which have jagged edges, and so on. The mathematics of patterns which repeat themselves in this way at different scales was relevant to chaotic systems. One example is provided by the theoretical orbits of stars in galaxies which, when examined at higher and higher energies, turn out to behave with patterned unpredictability. The ability to reveal an infinity of detail, to be complex but patterned, suggests a better analogy for brain function and biological morphogenesis than models based on linear descriptions.

To the layman the ideas of chaos are wonderfully attractive. They are difficult but not opaque, they seem to correspond to experience. To some (Kuhn amongst them) they constitute a paradigm shift, a new view which the mis-match between the evidence of experience and an existing theoretical structure forces on a sceptical scientific community.

Gleick's book is enthralling. It is hard for the innumerate, like me, to hold back the thought that the world is chaotic in this precise way. The illustrations he gives of images generated on computer screens are just such combinations of the mechanical and the unpredictable as instinct would suggest underlie the natural world. The astonishing thing is that it is possible to simulate them. The exploration of the universe of numbers has become possible. It turns out to be more diverse than anyone guessed and that this diversity seems to have a strong connection with diversity in the physical universe. Perhaps that is why God gets seven mentions in the index. Chaos theory seems to resolve at least the more mechanistic anxieties about determinism, and among the answers to Einstein's remark about God not playing dice is one from Joseph Ford: ‘he does, but they are loaded.’

Gleick and [Ed] Regis [in Who Got Einstein's Office?] both tell their stories using the common science writer's mix of biographical sketch, exposition and quotation. They are lucky to have such stories, and readers are lucky that they tell them so well. …