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The science of "complexity" has reached the attention of the chattering classes, or at least of the writing ones, and the subject is experiencing the predictable backlash following the honeymoon. The well-written but frankly adulatory Complexity by Mitchell Waldrop, and the heavily publicised The Quark and the Jaguar by Murray Gell-Mann, were among the books which rode the crest of the wave, but now there is a spate of books which seem destined through no fault of their own to hit the trough, of which Frontiers of Complexity by Peter Coveney and Roger Highfield is one. The earlier books emphasised the role of a small institute in Santa Fe to which Gell-Mann belongs (as well as a remarkably high percentage of other authors of complexity books), somewhat unfairly to what is a very broad field of endeavour, pursued by groups of scientists worldwide. As a result, the Santa Fe Institute has become identified as the defining institution of the field, in this book as elsewhere.

If I had to give my own definition of "complexity science", I would say that it is the search for general concepts, principles and methods for dealing with systems which are so large and intricate that they show autonomous behaviour which is not just reducible to the properties of the parts of which they are made. It is the search for the operative principles and concepts, as opposed to the reductionist study of detail, which distinguishes the "complexologist", and as such he is subject to such calumnies as that he is searching for a Theory of Everything. What is true is that he is searching for a theory, not a recipe: to take the example of a successful complexologist, natural selection, not the family tree of the horse.

Complexity is an enormous, rapidly growing and diversifying field. Coveney and Highfield is arguably the best general book so far on this highly "complex" subject, and would for instance make a useful text for an undergraduate seminar. It covers, in some descriptive detail, most of the subjects which comprise this emerging science, referring to detailed interviews with many of the major players and citing passages from a wide selection of the important books and articles. There is little or no factually incorrect material, and most controversial material is clearly labelled so. I felt, however, that with more effort and a wider coverage, it could have been better.

First let us sketch what Frontiers of Complexity is about. Complexity is a subject which has an annoying propensity to define itself, almost in spite of the efforts of many workers (including myself) to avoid the word as a vague, undefinable generalisation, welding together a congeries of different ideas. In fact, I was at first confused by finding "complexity" here defined twice, or perhaps many times, in several incompatible ways. On page seven it is defined in terms of emergent phenomena in macroscopic systems, a concept originating in evolutionary biology and introduced into the physical sciences (perhaps by your reviewer in 1967) without reference to the supposedly essential role of the modern computer, which is added in on pages nine and ten. I am bemused to find that Darwin and his successors in evolutionary biology and ecology, in the absence of computers, cannot have been dealing with "complexity". I guess we can all go along with "nonlinearity", another stated essential piece, and perhaps with "irreversibility", which the authors also toss into the pot. I am unhappy, nonetheless, with definitions which exclude the essentially equilibrium phenomena of broken symmetry and broken ergodicity, the few-dimensional aspect of (true, technically defined) chaos, and the marvellous analytic treatments of neural networks and of complex optimisation by such as Gerard Toulouse, Giorgio Parisi, Marc Mezard, David Thouless, Sompolinsky and many others, as well as leaving out most evolutionary biology.

For my money "complexity" is a state of mind, embracing any study of a realistic system which negates the strong reductionist (what I once called the constructionist) point of view which assumes everything follows from the fundamental laws, and emphasises the appearance of emergent phenomena of all kinds, at least intellectually independent of the microscopic substrate in which they appear. This attitude is often mistaken for holism or for a rejection of scientific ("weak") reductionism but is neither. Many of its ideas appeared either as parts of computer science or as results of computer investigations, but many did not, and in this reviewer's opinion the greatest problem the field faces is maintaining its sometimes tenuous connection with the nonvirtual world, especially considering the seductive nature of computer work and the fascination of the lay public with the images the computer produces. I am not sure that this book will be very helpful in cementing this connection.

Complexity science does in the main result from the creative tension between two intellectual traditions: the creative side of computer science, and the natural science of complex systems; and the book alternates its attentions between these.

Starting with an affecting prologue on the twin founding geniuses of complexity theory and computer science, Alan Turing and John von Neumann, we encounter a couple of introductory chapters on complexity itself and on discrete mathematics and its various overlaps with complexity. Next we are introduced to the computer via a very brief history, from Babbage to the quantum computer. The natural science strain comes in via chapters on computing schemes based vaguely on natural analogies: neural networks, cellular automata, simulated annealing, the genetic algorithms. Nature here appears primarily as a model, not as an intrinsically complex object of study. A final chapter in this sequence juxtaposes three quite incompatible bedfellows: the Brussels "dissipative structure" school, chaos and the concomitant relation of chaos to fractal structures, and self-organised criticality. The major links between these seem to be that they all have a relationship to complex phenomena observed in nature, and they lead to striking pictorial illustrations.

Next come two long chapters on life, first "Life as we know it", and then "Life as it could be", ie natural and artificial life. "Natural" covers studies on mechanisms of the origin of life, on pattern formation and morphogenesis, on economics and other forms of behaviour, and on ecology; "artificial" covers the whole gamut of artificially constructed and artificially evolved creatures, as well as a number of artificial worlds in which creatures or agents compete, ie artificial ecologies. A final substantive chapter treats both aspects of the mind: computational attempts to mimic brain and neurological function, and scientific attempts to understand the mind as an emergent phenomenon.

Perhaps my uneasiness about the coverage can be best justified by discussing their treatment of the field in complexity theory I have dealt with from its inception, "spin glass" and its ramifications. The book's scenario rather oddly places the paper of David Sherrington and Scott Kirkpatrick as the central event in the field. This paper was an obvious next step from Sam Edwards's and my earlier work, and it seemed to all of us that it could not be wrong, but it was - that was the main way in which it was important, because the search for this error led to a still vital part of theoretical physics, as well as to insights into the nature of complex systems and complex problems which still have to be appreciated by most computer scientists. These results follow from the Thouless-Toulouse-Parisi concept of replica symmetry breaking, which stands out in many minds as a truly spectacular breakthrough. Also missing is the tie-in to the ongoing effort to understand complex behaviour in glasses, gels, polymers, vortex lattices and the like. This is, to be sure, a very tough subject to popularise, but a perhaps prejudiced view is that its importance as a balance to the computer-oriented slant of much of the rest of the book would have made it worthwhile.

The above criticism could perhaps result from my personal involvement with this area, but there are a few other false notes at several other points. In discussing the Brussels school, the "principle of minimum entropy production" is seriously quoted, despite the fact that it is essentially never obeyed (entropy production is a maximum in the linear regime, irrelevant everywhere else). In a similar vein, reaction-diffusion systems like the B-Z reaction are extensively used in morphogenesis, but they are never allowed to form their own patterns: that is under strict genetic control. The significance of all this work to the real study of complex systems' behaviour is still questionable.

I was disturbed by the space given (though, admittedly, with carefully footnoted caveats) to problems of non-computability and discrete mathematics a la Godel and Roger Penrose, and to microtubules as the new pineal gland. And surely giving the dubious philosopher Nancy Cartwright space at all is carrying the overview of the field beyond its boundaries.

I believe firmly, with Coveney and Highfield, that complexity, however defined, is the scientific frontier. It takes a lot of flak from its critics in the more conventional sciences, some of which is brought on by the more extravagantly stated claims of its proponents. These two authors have been accused of wide-eyed acceptance of these claims, but I felt the sceptical side was often well-represented. This book will make clearer what both sides in this debate are on about, even if it does not provide the most critical possible assessment of where the real "meat" in the field is to be found. Perhaps that would be expecting too much, at this point in history.

P. W. Anderson is Joseph Henry professor of physics, Princeton University.