This is a translation of a book first published under the title Das Spiel in 1975. It is an ambitious book whose aim is to convey to the reader what it is to have a well-furnished scientific mind. Some years back, C.P. Snow persuaded us that the diagnostic characteristic of such a mind is familiarity with the second law of thermodynamics. His particular choice of a scientific law was unfortunate, because it is easier to talk nonsense about the second law than almost anything else, but in principle he was on the right track. A knowledge of theories is more relevant than a knowledge of facts. Biologists have to know a lot of facts, while physicists seem to know almost nothing. But although it is true that a well-educated scientist will be familiar with a number of theories, from Newton’s laws to the central dogma of molecular biology, I do not think that this is the critical distinction between understanding science and not understanding it. I suggest, instead, that it is a familiarity with the ways in which systems with different structures and relationships are likely to behave. It is this familiarity that Eigen and Winkler try to convey.
I can best explain this by giving an example. One of the things I learnt when I was an aircraft engineer during the war was that if a control system has a time delay in the feedback loop – that is, if some time elapses between a control action and its effects on the object to be controlled – then the system is likely to oscillate. Consequently, whenever I come across a system which is oscillating, whether it be the menstrual cycle or the numbers of hares and lynxes in Canada, I look for delayed feedback. In doing so, I am assuming that structure determines behaviour. That is, if the components of a system are related to one another in particular ways, then the system will behave accordingly. The behaviour is determined by the structure, and not by whether the components are electrical circuits, hormones or animals.
What a scientist has to acquire, then, is not merely a knowledge of specific theories relevant to his particular field, but an understanding of how the behaviour of a system depends on the relationship between its parts. Typically, the behaviour is described mathematically. One learns that simple harmonic motion is described by a particular differential equation, whether the variable that is behaving harmonically is the current in a tuning circuit or the velocity of a weight suspended from a spring. The major difficulty in communicating scientific ideas to a non-professional audience, therefore, is the lack of a common mathematical language.
Eigen and Winkler attempt to bridge this language gap. The concepts they aim to explain are not the relatively simple engineering examples of delayed feedback and harmonic motion. Instead, they discuss many of the most fundamental and difficult ideas in contemporary science. To give a list, which may be meaningful only to scientists, they discuss the concepts of equilibrium, competition, natural selection, entropy and information, symmetry and group theory, pattern formation, and conservative and dissipative structures. At the same time, they apply these concepts to different levels of organisation, from chemistry through development, ecology and evolution to economics and music.
Despite this vast range of topics, the authors avoid the use of mathematics. I can imagine two reasons why they made this choice. The first, and obvious, reason is that they want to communicate with non-mathematical readers. But I suspect that there may have been a second and more interesting reason: they wanted to convey something which cannot be communicated by mathematics alone. If I claim to understand the behaviour of some system, I mean rather more than that I understand the mathematical description of it. I mean that I can in some way analyse it and play with it in my head, imagining how it would behave in various circumstances. For want of an alternative, this ability can be described as having a ‘physical intuition’ about the system. This should not be taken to mean that one can have such intuitions only about physical systems, since they are just as relevant in biology. More nearly, what is meant is that one can make use of one’s everyday experience of physical objects to visualise how imaginary objects might behave. Such an intuition is normally a complement to mathematical description, rather than an alternative to it. If the mathematical analysis of some system predicts that it will behave in a particular way, one usually tries to gain some insight into why it should do so. Personally, if I cannot gain such an insight, I check the algebra, or the computer programme, very carefully, and expect to find a mistake. Scientists seem to differ a good deal in the extent to which they rely on physical intuition as opposed to algebra, but it would be fatal to try to do without either. One reason Eigen and Winkler avoid mathematics may be that they want to encourage their readers to develop their physical intuition.
The method they adopt, in the place of mathematical description, is to draw analogies between physical processes and games. A game has rules, but the course of any particular game is also influenced by chance events which, depending on the game, can include variations in the initial position, the throwing of dice or shuffling of cards, and the unpredictable decisions of the players. Real sequences of events resemble games in that they are constrained by laws (e.g. Newton’s laws, the laws of genetics), but the actual sequence is influenced by chance. Thus one cannot predict the course of a particular game of chess, even if the opponents have played before, and still less could one predict the course of a hand at bridge before the deal was made, yet in both cases one can predict that certain rules will be obeyed. Similarly, one cannot predict the future history of any given species, but one knows that these processes will be governed by laws. Of course, we have a complete knowledge of the laws of chess, and only a partial knowledge of the laws of physics or genetics, but that does not invalidate the analogy.
Eigen and Winkler do more than point to the general analogy between games and real events. They invent a number of games which mimic particular biological and physical processes. By playing the games, one should acquire an intuition about the processes of which the games are analogues. The approach can best be illustrated by describing the simplest game in the book. This game shows how a system can approach a stable equilibrium by a series of chance events. The game can be played on a board with 36 squares (six by six). Each square is occupied by a bead, which can be black or white. To make a move, two dice are thrown. The numbers on the dice then specify one of the squares on the board – for example, three up and four across. The bead on that square is then replaced by one of the opposite colour. This completes a move. Suppose, for example, that the game is started with all the beads black. After a number of moves, it will be found that the numbers of black and white beads are approximately equal, the numbers fluctuating either side of equality. In this simple form, it is easy to predict what will happen without actually playing the game, although there are questions whose answers are not immediately obvious. For example, what is the distribution of frequencies with which various numbers of black beads are present in a long sequence of throws? How big a board must one have to be reasonably certain that, after reaching the equilibrium, the board will not again come to have beads of only one colour?
This game is an analogue of the way in which equilibria are reached in physical systems. Even relatively simple changes in the rules can lead to quite rich behaviour, which it would be difficult to predict without playing the game. For example, suppose that, once a square has been chosen by throwing the dice, the bead on it is replaced only if some specified minimum number of the beads on neighbouring squares are also of the opposite colour. Such ‘co-operative’ games can give rise to spatial patterns, in a way analogous to the appearance of patterns in the development of living organisms.
How far do the authors succeed? This is a hard question to answer. When writing about science for non-scientists, it is as well to have an imaginary reader in mind. I have two imaginary readers. One is an intelligent but ignorant 16-year-old: myself when young. The other is an intelligent but even more ignorant British civil servant, bent on improving his mind. How would these two fare with The Laws of the Game? The civil servant would, I suspect, fare very badly, if only because it might not occur to him that it is necessary actually to play through the various games in order to gain an insight into the behaviour which follows from particular rules. If this was obvious, there would be no need to acquire physical intuition: we would all be born with it, and Eigen and Winkler are wasting their time. In any case, for all I know they do not allow dicing at the Reform Club.
I fear that my 16-year-old might also fare badly, although for different reasons. He might enjoy playing the games, and if so would gain some insight into physics and biology. The trouble is that there are a great many things in the book he would not understand. There are a fair number of things I found difficult, and I have been thinking about these problems for some time. It is hard to tell how a relatively ignorant reader would react. The right policy would be to press on regardless, accepting that some things will remain obscure, and enjoying the bits that make sense. Some readers, however, may merely get discouraged.
In effect, my fear is that Eigen and Winkler may have attempted the impossible. Eigen himself did brilliant work in chemical kinetics, and then turned to biology. Unlike some other physical scientists who have made this switch, he has grasped the fundamental ideas of biology. In particular, he understands the principle of natural selection, which usually seems to defeat physicists. He has spent some extremely fruitful years working on the origin of life: that is, on how chemical processes gave rise to biological ones. As a consequence of this history, Eigen has thought hard and long about the problems discussed in this book. It may be that the understanding which he and Dr Winkler are trying to convey can be acquired only by years of work with science. Perhaps the only way to acquire a well-furnished mind is to spend a lifetime, or at least a long apprenticeship, in furnishing it.
For those who would write popular science, this is a counsel of despair. The authors might offer two replies. The first is that it is better to convey a vague idea than no idea at all. They quote with justified disapproval Wittgenstein’s dictum: ‘Whatever can be said at all can be said clearly, and whatever cannot be said clearly should not be said at all.’ If accepted, this would rule out all poetry and much of science, although it would leave us with most of mathematics and the London telephone directory.
The second reply they might make is that I have misidentified their imaginary reader. Perhaps they were writing for readers with more knowledge of science than I have supposed: not for me when young but me now. Indeed, I did find their games illuminating, partly because, being originally physical chemists, they tend to draw their analogies in the opposite direction to me. They understand predator-prey systems by seeing that they are just like the law of chemical mass action, whereas I understand chemistry by imagining animals running about. The ideal readers for this book may be teachers of science. There is a rich source of ideas here for anyone trying to teach the structure of scientific thinking. There seems little doubt that young mammals play games because in that way they acquire skills they will need later. Why, then, should we not play games to learn science?
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