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THERE is no mathematical notation in Frontiers of Complexity. There are almost no equations. It is a gesture of abnegation, a reassurance to entice that shy beast, the General Reader, Ignoramus Legens, represented in this case by me, who has read perhaps two books of popular science in the past year, one sloppy and sensational and the other very poetic and translated from the French. Ignoramus ignores equations anyway, just as he will skip anything that looks like poetry. But the gesture is appreciated. Ignoramus toddles in.

Peter Coveney is a senior research scientist in the Schlumberger Cambridge Research Laboratory, and Roger Highfield is science editor of The Daily Telegraph. It sounds like a good team, one scientist, one science journalist. The question must be, has their book properties that could not have been predicted from either author alone? Or can we predict anything from their previous combination, which produced The Arrow of Time? Has Ignoramus time to have a look at that? He has not.

The flight of time鈥檚 arrow is nowhere more swift than in this pocket universe, the field of informational topology that is journalism. Beyond the event horizon of the deadline, no light shines. Which is a shame, truly, because this is not a book to sum up in a week. This is a book to tackle, to come to terms with, to mull over. Its 鈥減hase scape鈥�, its 鈥渟earch space鈥�, is bloody enormous. Maths, biology, physics, chemistry, pharmaceuticals, economics, neurophysiology, psychology, metallurgy. To list them is not the point, as the authors explain, for complexity is the study of systems, how they behave; and in particular the study of the sort of behaviour that cannot be predicted from the system鈥檚 individual components. Complexity is about the whole that is always greater than the sum of its parts.

After a foreword by Baruch Blumberg, at whose college Roger Highfield enjoyed a sabbatical fellowship, after acknowledgments and introductions to the argument and to the prophets and progenitors Alan Turing and John von Neumann, we are led briskly across the foothills of the intertwined histories of mathematics and logic, right to the Edge: to the theoretical limit, the place where mathematics and logic fall away into uncertainty, incompleteness and undecidability. It鈥檚 a severe sort of route, a sheer assault on an exponential slope of evolution where all but the fittest ignoramuses must fall away panting, and go back to the telly. No, no. Up, up, and on. The title means what it says, and we have great distances to travel.

Frontiers of Complexity is a serious, scrupulous book, and remains sober at intellectually intoxicating altitudes. The prevailing wind blows straight from Darwin. Turing and von Neumann reappear at every bend, it seems, having theorised the whole route beforehand, but the authors are keen to ensure that we are not distracted by 鈥渃olorful accounts of personalities鈥�. They have too much else to do, guiding us up each salient, roping us up for the perilous bits. We get five-minute breathers while they tell us about fuzzy logic, parallel processing, logic landscapes and simulated annealing. Since we will not appreciate simulated annealing properly without an explanation of the peculiar ferromagnetic properties of spin glasses, then an explanation we shall have. Just to be on the safe side, they remind us what ultrasonic means, what Pythagoras theorised, what Pavlov did to dogs. If I was that bad, thinks Ignoramus, puffing grumpily along, I should never have come.

One prospect is paramount. The study of complexity arises now because it can. Charles Babbage predicted as much: 鈥淎s soon as an Analytical Engine exists, it will necessarily guide the future course of the science.鈥� So the history of maths gives way to the history of computing, a rapt account of the technological ascent. Complexity is a function of the logarithmic expansion of the power to handle large bodies of organised data: to move them around, to orchestrate them, to bring them into collision, to breed them. Drawn from and applied to the modelling and analysis of everything from tornadoes to purchasing preferences, it is the science of experimentation in silico 鈥� a new phrase to Ignoramus, and one which actually says it all.

Coveney and Highfield then direct our gaze ahead, to the Future Computer, which they fancy may be optical: information woven on looms of light. Meanwhile we have the 鈥渦niverse synthesisers鈥�: cellular automata, inspired by John Conway鈥檚 Game of Life. Elements arranged in grids react to their neighbours, or the lack of them. Modelled in virtual space, they can simulate the formation of snowflakes, the percolation of oil through rock, or help to explain and predict the hitherto intractable mystery of what happens to cement when it sets. Coveney and Highfield show how problems in economics or turbine flow can be elucidated by genetic algorithms, which evolve to fit the problem, benefiting from material interchanges inspired by sexual reproduction. 鈥淢uch of what we discuss in this chapter is almost certainly wrong in detail,鈥� they warn, embarking on the topic 鈥淟ife As We Know It鈥�.

By now Ignoramus is desperately grateful there are no equations. He marvels at the interdisciplinary athletics. He is astonished by the view across these heady peaks of innovation and science and metascience. Fattened in the lush, lazy, oxygen-rich valleys of fiction, he finds himself gasping for an anecdote. Well, there are some, among the theories, facts and developments. There are good stories, like Denis Noble鈥檚 virtual heart cells, a mass of which pulsate in a counter-intuitive way that proves to be exactly what happens in the real thing. You will have to forgive me if I claim the reviewer鈥檚 perquisite of nicking the best joke, which I think must be philosopher Daniel Dennett鈥檚 theory that the human mind consists of 鈥渓ots of sub-agencies and coalitions and competitive functionaries working simultaneously to create the illusion that one Boss Agent is actually in control 鈥� rather like,鈥� he supposes, 鈥渢he Reagan presidency鈥�.

There is much good stuff in the notes, culled from the literature, interviews, correspondence, e-mail. The tangential comments and useful parentheses are almost swallowed by the profuse undergrowth of technical apparatus, abbreviated titles and page references, and when you start to add the addresses of World Wide Web sites, you have a clear argument for the superiority of hypertext. As it is, Ignoramus has to read every note to see if it might be for him.

However cautious and responsible, no book can fail as a source of images to delight a science fiction writer when it reveals that in Minnesota you can now watch the imagination at work, as lower-voltage activity in the relevant regions of the brain. There are the virtual ants helping to design new call-routeing programs for BT 鈥� they scavenge, lay down pheromone pathways between food and hive, and when one crashes, any other can take its place. There are the digital wildlife that may come to roam the Internet, if Tom Ray and Kurt Thearling get their way: shy, nocturnal programs that flee ever westwards, to graze in off-line machines. There is also the electronic aquarium stocked by the University of Toronto with virtual fish that have to learn how to swim, how to prey and avoid being preyed upon.

Coveney and Highfield approve of complexity 鈥渢hrowing light on questions that once lay exclusively in the province of philosophy and mysticism鈥�, and disposing of what, less patiently, they call 鈥渁 lot of somewhat nebulous theorising鈥� about concepts such as life, intelligence and consciousness. Complexity is already quantifying the sort of correspondences previously reserved for poets and painters, defining a sense in which mountain avalanches and stock market crashes may even be the same phenomenon. Coveney and Highfield readily reproduce Richard Dawkins鈥檚 perceptions that chain letters are a virus, a coded instruction saying 鈥淐opy Me鈥�, as are the appeals of televangelists for money.

It is at this point that even Ignoramus has to begin to wonder how much of this glorious new language will prove, in time, to be metaphors for mistaken or misleading assumptions. Or is it rather that science is always metaphorical, a cultural epoch talking to itself about the Universe? As a piece of postmodern thought, the digitising of poststructuralism, the theory of complexity comes near to touching the void.