The ethics of the sand pile. History stands poised on the brink of catastrophe. The very existence of the human race is precarious. Edward Skidelsky is awed by the implications of a radical new physics

Ubiquity: the science of history . . . or why the world is simpler than we think

Mark Buchanan <e

What a pity that this remarkable book has been so misleadingly presented. According to the blurb, it uncovers "a new law of nature" that applies to subjects as diverse as the "behaviour of forest fires, the extinction of species, the pattern of earthquakes, the rise and fall of financial markets, the flow of traffic, the growth of cities, the outbreak of wars and even trends in fashion, music and art". The new law is "unifying science", and will "make it easier for us to manage and control the future".

This kind of language may bolster sales, but it will lead thinking people to dismiss Mark Buchanan as a charlatan. It recalls the worst pretensions of Marxism, such as Engels's attempt to construct a "dialectic of nature" to complement the dialectic of history. Attempts to unify the whole of human knowledge under a single rubric are now terminally unfashionable, and any endeavour to "make it easier for us to manage and control the future" is regarded with peculiar suspicion.

It would be a great shame if Ubiquity were lumped together with Marxism and other discredited grand narratives. As a matter of fact, Buchanan's purpose is diametrically opposed to that of previous system-builders. The "new law of nature" described in Ubiquity is not a law of nature in the traditional sense at all. It is not an attempt to explain facts in the manner of Newton, by relating them back to deep underlying causes. Rather, it is the admission that, for a wide variety of phenomena, there are no deep underlying causes, just an accumulation of tiny accidents. This is a radically new kind of physics, and it suggests a new approach to the study of society. The example of Newtonian physics inspired such thinkers as Marx to seek out the underlying law of historical change; the new science suggests that such a law does not exist. A postmodern physics has inspired a postmodern history.

Ubiquity is not an original work of science: it is, rather, an attempt to summarise and bring together the work of scientists in many different fields. Buchanan's gift is for synthesis and lucid exposition. His background is in science journalism, and he has a journalist's feel for the intellectual limitations of his readership. The maths is applied with restraint, and is supplemented with concrete illustrations wherever possible. My own career in science terminated ingloriously with a single GCSE in chemistry, yet I found Ubiquity fell (just) within my grasp. Nor did I suspect - as I often do when reading popular science - that this feeling of understanding was merely a gratifying illusion spun by artful prose. The core ideas in Ubiquity really are very simple - that is their beauty. It is curious that the progress of science is not always in the direction of ever greater complexity; simplicity is often the mark of an intellectual breakthrough.

All the phenomena discussed here obey a certain mathematical formula known as a power law. As they increase in scale, so they decrease in frequency. When the size of an earthquake doubles, it becomes four times as rare. When the size of a stock-market fluctuation doubles, it becomes 16 times as rare. The exact fraction does not matter; it is the general law that counts. What this law indicates, translated into English, is that there is no such thing as an average size for an earthquake or a stock-market fluctuation. There is no median point around which they all cluster.

Certain phenomena - human height, intelligence - do cluster around a central point. Plotted on a graph, they form the notorious bell curve. But imagine for a moment, Buchanan says, that human height obeyed a "power law" instead. What this would mean, in practice, is that there would be no possible way of predicting the height of the next person you bumped into. You might crush her under your foot or, alternatively, she might crush you under her foot. And whatever size you happened to be, the situation would be the same. Whether you were as tall as a mountain or as small as an ant, the human landscape would look roughly the same. It would be, in the jargon, "scale invariant" - identical on every possible scale of magnification. This is precisely how the landscape of earthquakes, stock-market crashes, forest fires and wars looks. There is, in the strictest sense, no such thing as a typical earthquake or a typical war.

What does all this prove? So far, it looks like nothing more than a curious but idle observation. It is all very well to gesture to these mysterious correspondences; the important thing is to explain them. This is what Buchanan goes on to do. Scale invariance is, for him, no more than a symptom pointing to a common underlying pattern of organisation. Only this common underlying pattern does not correspond to any "law of nature". This marks a crucial departure from traditional physics. The timeless equations of Newton or Einstein are of no use in trying to understand phenomena such as earthquakes or forest fires, in which accident plays a central role. The best way to understand them is to construct a "game of chance", combining elements of randomness with elements of regularity. It functions as a stylised model of the real situation, bringing out its central features and explaining its behaviour. Such games are, needless to say, played on computers. Indeed, this whole area of science is unthinkable without the computer.

The game that Buchanan comes back to again and again - which functions as an organising metaphor for his thought - is the "sand-pile game". Devised by the physicists Bak, Tang and Weisenfeld in 1987, the game involves sprinkling grains of sand, one at a time, on to a table top. The grains soon form piles, which grow steeper and steeper until an errant grain triggers an avalanche and the pile flattens out again. They then asked a simple question: what is the average size of an avalanche? But however many times they ran the game through the computer, they could arrive at no answer. A falling grain might dislodge anything from one to a million other grains. The distribution of avalanches obeyed our old friend the power law: double the number of grains involved, and the avalanche becomes slightly more than twice as unlikely.

What is it about grains of sand on a table that produces this curious statistical result? The precise physical properties of sand are irrelevant. The computer simulation captures what one might call the logic of the situation, while discarding all that is accidental to it. This logic turns out to be identical in all the phenomena discussed by Buchanan. The composition of the individual items does not matter at all; they might be grains of sand, trees in a forest, atoms in a magnet or traders on the stock-market floor. All that matters, for the purpose of the game, is that they are organised into what is known as the "critical state". This is what makes the application of the new physics to human society more than merely metaphorical. There is a sense in which grains of sand and human beings really are behaving according to similar laws. The historic division between the social and the natural sciences may at last be coming to an end.

One of the most important conclusions of Ubiquity is that, for systems organised into the critical state, there is no difference in principle between the small and the large. There is no special class of "great events" that requires special explanation. "Large events are just magnified copies of smaller ones, and arise from the same kinds of causes." This is important, because it runs against our natural inclination to believe that great events must have great causes. Geologists have sought out the causes of great earthquakes, and economists have sought out the causes of great crashes, as though these could somehow be set apart from the tiny tremors that daily afflict the earth's crust and the financial markets. Buchanan suggests that they are, in fact, nothing more than larger versions of these tiny tremors, and require no special analysis.

Applied to history, this theory suggests that great wars and revolutions demand no explanation beyond a narration of the precise chain of events that compose them. In the sand pile, it is impossible to specify the cause of a huge avalanche other than by tracing its exact progress right back to the original grain that triggered it all off. There are no "laws of avalanches" distinct from the laws governing the movement of the individual grains. And any grain - the unfortunate Gavrilo Princip in Sarajevo springs to mind here - can, if it falls at the right time and place, start an avalanche. The only way to understand the history of the sand pile is to recount it; old-fashioned narrative history turns out to be the most scientific of all.

The vision of history that emerges from Ubiquity is tragic. It is the vision of the Iliad. History stands permanently poised on the brink of catastrophe; the abduction of one woman can lead to the destruction of cities. Instability is an inalienable feature of human life. We flatter ourselves that we have overcome it through the development of rules and institutions, not realising that those very rules and institutions are equally subject to its depredations. The very existence of the human race is precarious; the tiniest fluctuation at some random point in the ecosystem could unleash on us an avalanche of extinctions.

The predicament of the individual is equally tragic. Human society is configured in such a way that each individual action may have consequences that are vast but totally unforeseeable. Each of us stands, potentially, at the pivot of world history; anyone can be the grain that brings the pile tumbling down. We are thus burdened with an awesome responsibility, yet at the same time denied any means of knowing how to discharge it. What practical consequences follow from this? Is there an "ethics of the sand pile"? This question awaits another book.