Anything but simple

October 6, 1995

Peter Coveney says there is a growing breed of scientists using computers to formulate the universal features of"complexity" defined as the behaviour of large collections of single units with the potential to evolve.

Fads sweep through science at regular intervals. Some areas are more susceptible: physics, particularly theoretical physics, suffers from the vicissitudes of fashion possibly more than any other discipline. One of the reasons may be the grandiose nature of its aims, including the most ambitious goal of all, to formulate the theory of everything; another reason is that trends are set by a relatively small number of people, with everyone else jumping on the bandwagon. Trendy subjects which have attracted widespread attention in recent times include chaos, catastrophe theory, cellular automata, neural networks, self-organised criticality and the edge of chaos. Now there is "complexity", a field just as prone to hyperbole, but one whose existence does herald an important change in the way we think about science. Its emphasis is on emergent phenomena where the whole is invariably greater than the sum of the parts.

The problem with fads is that they lead to extravagant claims which, when they eventually fail to live up to overblown expectations, are followed by a backlash, one that may be so heightened in its intensity that even such merits as there may have been in the trendy ideas are snuffed out.

A lot of emphasis has been placed on certain personalities and work done at the Santa Fe Institute for Studies in the Sciences of Complexity, New Mexico, and one is left with the impression that many United States scientists are claiming credit for the discovery of concepts such as self-organisation and dissipative structures for which Ilya Prigogine of the Free University of Brussels was awarded a Nobel Prize in 1977. But the vagueness of the terms, coupled with the boldness with which they have been asserted, is once again threatening a backlash.

In spite of the exaggerated claims, there is indeed a rich field of scientific endeavour which can be generally referred to as the study of complex systems and which is here to stay. I define complexity as the study of the behaviour of large collections of simple units which have the potential to evolve.

A complex system of this kind will be non-linear: the interactions between the myriad units then lead to coherent collective phenomena, sometimes referred to as emergent properties, which can only be properly described at higher levels than those of the individual units for a complex system, "the whole is more than the sum of the parts". This is as true for a human society as it is for a raging sea or the electrochemical firing patterns of neurons in a human brain. Remarkably, the long-term behaviour of just three balls on a frictionless billiard table is unpredictable, even though the equations of motion describing the system are precisely known.

Conventional science is frequently blind to the connections that can be drawn between such apparently disparate things as frustration in antiferromagnets, the workings of the brain, the rise and fall of stock markets, and a host of other phenomena. Today, most scientists restrict themselves to the detailed study of one aspect of a single sub discipline within one branch of the tree of science, be it the large-scale structure of the universe or the molecular structure of a protein from the Human Immunodeficiency Virus. There is a growing community of scientists swimming against this tide. Motivated by a desire to establish connections across conventionally separate disciplines, they want to show that there is an economy of concepts necessary for understanding the way the world works. The ultimate goal is to come to grips not only with the complexity of any single phenomenon but also with the universal features of complexity itself, whether manifested by evolution in a rain forest or within the core of a computer, by the drip of a leaky tap, the spirals of colour that can form in a chemical reaction, or the workings of a conscious brain. The quest is to find unity in diversity, to explain how order can emerge from a mass of evolving agents, whether atoms, cells or organisms.

Just as we cannot understand any human language without reference to its grammar, we can only fully grasp and manipulate complexity by appealing to its own grammatical structure expressed in the language of mathematics. The term complexity is often used in an infuriatingly vague sense by scientists, who may mean different things by it. But within the realm of mathematics, the definition of complexity is unambiguous. There the complexity of a problem is defined in terms of the number of mathematical operations needed to solve it. Sizing up the degree of complexity of a given problem is the mission of mathematical complexity theory. It tells us whether the problem will be tractable that is, whether it will be practical to attempt to solve it by systematic means.

Many important real-world problems, such as that of the peripatetic salesman who has to devise the most economical way to visit a set of cities, can be formulated simply enough, but attempts to find their solutions by systematic means rapidly become impractical as the problem's size (for the salesman, the number of cities) increases beyond a small number. Other examples of mathematically complex problems include descriptions of how brains learn from their interactions with the external world and how evolution led to organs as complicated as the brain in the first place. Such problems lie beyond the scope of pen, paper and analytical mathematics; computers provide the only means of solving them.

The essential role played by the computer in the modern study of complex systems explains in large part why such a rich field of investigation was overlooked for so long. Before the advent of the digital computer, it was impractical for any person to sit down and feed thousands, even millions, of numbers into a set of equations describing a given complex problem a whole lifetime would have elapsed without the likelihood of obtaining a single useful result. Thus the science of complexity is intricately entwined with and crucially dependent on computer technology. The dizzying increase in computer power over the past 50 years has enabled scientists to model progressively more complex phenomena.

These approaches to complexity are so successful that life itself is now gaining a new meaning. Neither actual nor possible life is determined by the matter of which it is composed. Rather, life is a process, and it is the form of this process, not the matter, that is the essence of life. One can therefore ignore the physical medium and concentrate on the logic governing the process. In principle, one can thus achieve the same logic in another material clothing, totally distinct from the carbon-based form of life we know.

The implications of separating living complexity from its medium are stunning. Imagine creating an artificial world and planting in it the logical seeds of life. Given enough time, you could watch evolution in action, as a primitive organism proliferates and mutates into a vast diversity of offspring.

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