Shifting Sands

Peter Lipton

  • How Nature Works: The Science of Self-Organised Criticality by Per Bak
    Oxford, 212 pp, £18.99, June 1997, ISBN 0 19 850164 1

Taken alone, the basic laws of physics suggest a bleak universe, a thin, cold soup of atoms in motion. From this point of view, the complex dynamic structures we actually find, animate and inanimate, seem a miracle, so prompting the famous argument for the existence of God, that such manifest design requires a Designer. The argument from design has serious weaknesses, but one may still wonder what to put in its place. Per Bak, a physicist at the Niels Bohr Institute in Copenhagen, thinks he has the answer: self-organised criticality.

Bak applies this idea to a stunning range of disciplines, from geology, economics, biology and neurology to the science of traffic jams. The flagship example of self-organised criticality, however, is the humble sandpile. Imagine that grains of sand are dropped one by one onto a flat surface. The situation is not exciting at first, with each grain staying pretty much where it lands, but in time a pile will form, and eventually new grains will trigger avalanches of various sizes.

What impresses Bak about the sandpile is that the grains of sand build up to their own ‘critical’ state, without external guidance. In the beginning, the system is simple, since the grains do not interact, but as they fall they build their own structure, which eventually behaves in complex ways. The resulting pattern of avalanches is also important. Unsurprisingly, there will be fewer large ones than small ones, but Bak claims that the distribution will obey a theoretically significant ‘power law’, the simplest form of which would have the number of avalanches inversely proportional to their size.

Bak discerns this pattern of behaviour emerging from effectively random processes all over the place. It is supposed to characterise earthquakes, the ‘punctuated equilibrium’ that Stephen Jay Gould and Niles Eldridge claim for biological evolution, stock prices, neural activity, and the movement of cars on the M25.

In 1987, along with Chao Tang and Kurt Wiesenfeld, Bak published a technical account of self-organised criticality which, he reports, has been cited over two thousand times, making it the most cited paper in physics of the past ten years. His attempt at popularisation is only partially successful, however. His prose is down-to-earth, but his tone is grandiose: not everyone will join him in wondering whether historians of science will come to rank self-organised criticality with Newton’s laws. He can also be difficult to follow; and the effect of the book’s many graphs, diagrams and pictures is primarily rhetorical and aesthetic.

Nevertheless, there are things we can learn from Bak’s sandpile. The point about the avalanche is not just that a single grain causes a large slide, but that if one grain had not done the job, another one would have done. The avalanche is overdetermined. Normally, when one thing causes another, if the cause had not occurred, the effect would not have occurred either. Indeed, according to some philosophers, this counterfactual describes the essence of causation. In cases of overdetermination, however, there is causation without the counterfactual. If the grain that actually caused the avalanche had not fallen, there would still have been an avalanche, set off by another grain.

Although overdetermination is not usual, it is not rare either, and, as Bak recognises, it has interesting consequences for the explanation of the behaviour of systems that exhibit it. One of these is that the appropriate level of explanation moves from the microscopic to the macroscopic level. Overdetermined systems exhibit a form of emergence: the whole has features not present in the parts. A good example of this phenomenon is given by Alan Garfinkel in his Forms of Explanation (1981), where he discusses predator-prey relations between foxes and rabbits. When the fox population is high and a fox kills a rabbit, there is overdetermination. The fox caused the death of the rabbit, but had that fox not done it, another one nearby would have. Suppose we are asked why the rabbit died. What is wanted is a cause of the event, but one satisfying the counterfactual that most causes satisfy but which overdetermined causes do not. We can find it by moving up from the micro level of the guilty fox to the macro level of the fox population. The fact net that this was so high is a cause that makes a difference, since the rabbit would probably have survived had there not been so many foxes about.

The situation is the same for the sandpile and for most of the other systems Bak considers. The particular grain that triggers the avalanche is a cause but not an explanatory one. To find an explanation, we have to move from the motion of the particular grain to the overall structure of the pile. This shows how a reductionist approach to the understanding of complex systems may fail.

The sandpile thus illustrates a sense in which complex systems may exhibit emergence that is neither spooky nor trivial. It also exhibits a second type of emergence, which we can see if we shift our focus from the situation where the pile is already in the ‘critical’ state, poised for avalanche, to the early stages, where it forms itself. This is a shift from asking why complex systems behave as they do to asking how they developed in the first place, from the features of ‘criticality’ to the features of ‘self-organisation’. Here the main point is the lack of design: the structure itself emerges from the random behaviour of the falling grains that eventually make it up. We can thus explain the emergence of design without positing an intelligent designer.

One natural question is how the mechanism of self-organised criticality relates to the other, better-known mechanism for generating design without a designer, evolution by natural selection. Bak devotes considerable attention to the question of the pace of evolutionary change, arguing that self-organised criticality may show why Gould and Eldridge are right to argue for a picture of punctuated equilibrium, in which the pace is highly uneven, with short periods of dramatic change separated by long periods of relative stasis.

According to Bak, the periods of dramatic change are the avalanches of evolution, and the pace of change is yet another example of self-organised criticality. This, however, does not appear to answer the question of the relationship between the two mechanisms. On the race of it, natural selection is fundamentally different from self-organised criticality, because, while genetic mutation may be random, selection is anything but. Selection mechanisms explain why organisms have a particular feature by positing a filter for that feature in the environment, a filter that lets creatures possessing it through to thrive and reproduce, while creatures without it die off.

The filters of natural selection appear not to have a parallel in the mechanism of criticality. Moreover, the two mechanisms appear to explain different things. Where self-organised criticality may explain why a system develops into one that generates spurts of avalanche-like behaviour, selection mechanisms explain why organisms have the persisting traits they do, and why they are well-suited to their environments. Both mechanisms begin with randomness, but where one leads to catastrophes, the other leads to equilibrium.

There are nevertheless interesting parallels between the mechanisms; I will mention two. The first is that, like self-organised criticality, selection mechanisms support an explanatory shift from the micro to the macro level, although in the case of selection the switch occurs in the phenomenon that is explained, rather that in the explanation itself. Rather surprisingly, a selection mechanism may explain why every organism of a particular type has a certain trait without explaining why each of them has it. The fact that only people with red hair are admitted to a club will explain why everyone in the club is red-haired, but it does not explain why you, who are a member of the club, have red hair.

The second parallel concerns prediction, or the lack of it. The overdetermination of the sandpile makes avalanches difficult and effectively impossible to predict, and there is a similar difficulty in predicting evolutionary change, because of the random aspect of genetic variation and the contingencies of environmental change. This shows, among other things, a fundamental difference between explanation and prediction, since there can be one without the other, and goes against the influential tradition according to which explanation and prediction are fundamentally the same, since both consist in the deduction of a conclusion describing the phenomenon predicted or explained from premises consisting of laws and initial conditions. According to this tradition, prediction moves from premises to conclusion whereas explanation moves in the other direction; but the structure is the same.

In life, situations where we can explain but not predict are commonplace. Seeing the brick lying on the living room floor, I have no difficulty explaining why the window is shattered, though this is not something I would have predicted. Often, both in life and in science, the reason we can give an explanation only after the fact is that the phenomenon to be explained itself gives us essential evidence for its explanation. Astronomers can explain why the light from a star is shifted towards the red end of the spectrum in terms of the speed at which the star is receding and the Doppler law that links the two. The faster the recession, the lower the frequency, just as the sound of a siren is reduced in pitch as the ambulance passes. The red shift is thus explained even if the only way the astronomer can determine how fast the star is moving is by observing the red shift. Recession explains red shift, as red shift makes it possible to determine recession. This rules out prediction, since we have to know the red-shift first, but it is a virtuous circle so far as explanation is concerned.

In cases such as the star, the barrier to prediction is broadly technological: we may have no independent way of determining the rate of recession. In the case of critical systems, however, the difficulty seems to lie deeper. One reason for this we have already seen: avalanches, unlike red shifts, are overdetermined. Another source of unpredictability may be the ‘power laws’ that characterise critical systems. In practice, the ability to predict often depends on being able to ignore small effects, but this requires that there not be too many of them. This condition will be satisfied if the number of effects is independent of their size. Thus, if there are roughly as many large avalanches as small ones, we may ignore the small ones: their net contribution will be negligible. As I understand it, however, the power laws show that critical systems are not like this. There are in fact many more small avalanches than large ones, so that, in the simplest case, the avalanches of each size make the same net contribution. This means that no effect is small enough to be ignored: the predictor’s nightmare.

Much of Bak’s book can be seen as an extended game of spot-the-criticality, with the suggestion that it applies to more or less everything. That suggestion is almost certainly false, but the game is beguiling. As a philosopher of science, I ask, for example, whether the development of science itself exhibits the tell-tale emergences of self-organised criticality. Scientific research may seem a particularly unpromising place to look, since the theories that emerge are complex abstract objects which, unlike sandpiles and geological faults, do have intelligent designers. Nevertheless, explanations that appeal to intelligent designers have only limited power where what is at issue is the development of fundamentally new ideas. One wants to explain the emergence of new theories without tacitly assuming that they are not really new or that the theorists who develop them enjoy perfect foresight or perfect understanding of the complex web of social interactions characterising any scientific community.

As it happens, the sandpile idea appears to fit surprisingly well with Thomas Kuhn’s provocative picture of scientific research. According to Kuhn, research in a particular specialty is normally structured by finding exemplary solutions to concrete problems that themselves guide new work. Eventually, these exemplars stop working, suggesting new problems without supporting new solutions. This drives the science into a crisis state, one possible outcome of which is a scientific revolution, where new exemplars are adopted and the scientists’ work is refashioned.

In sandpile terms, the exemplars are the grains, the crisis the critical state, the revolutions the avalanches. There are minor differences between scientific research and sandpiles, but a Kuhnian picture of science may display the two types of emergence that characterise self-organised criticality. Consider first the question of overdetermination. For a research tradition in crisis, the failure of an exemplar is a cause of revolution. There is almost certainly overdetermination here, since if one exemplar had not failed, another one eventually would have done, normal science being a process where exemplars are pushed to breaking point. Hence, the particular anomaly that provokes a crisis will not explain why there was a revolution. To explain this, we need to move to the macro level, and appeal to the structure of normal research which, says Kuhn, guarantees its own eventual destruction.

What about the other kind of emergence, that of self-organisation? Here, too, the Kuhnian picture is promising, since exemplars serve not just as guides to the application of an independently articulated theory, but also as sources of theory itself. Exemplars give content to the theory: they determine what it says and where it applies. They also determine the structure of research during the long periods between scientific revolutions, since scientists will choose new problems that are similar to the exemplary solutions, attempt to solve them with techniques similar to those that worked with the exemplar, and judge their success by the standards that the exemplar exemplifies. From the exemplars emerges rulelike behaviour in the absence of prior rules, and the changes of exemplar that trigger scientific revolutions restructure the system of research, just as the grains that trigger an avalanche reshape the pile of sand.

In the case of the sandpile, the critical behaviour is a consequence of the way the grains form patterns of interaction. To articulate the Kuhnian parallel, one would want to find analogous patterns of interaction within the research community. Such a detailed comparison would surely also bring out some important dissimilarities. For example, Kuhn’s dramatic contrast between normal and revolutionary science does not appear to fit the moral of the power law of criticality, that the frequent small changes make a net contribution comparable to the rare large ones. The source of this dissimilarity, however, may be Kuhn’s picture of scientific research, rather than the real thing.