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Complexity in Mathematics

By Ian Stewart

In complexity science, the phrase ‘complex system’ has a specific technical meaning. It is not merely something very complicated. In fact, there is a sense in which complex systems are not complicated at all – even if that is how they seem. The key feature of a complex system is that it is composed of large numbers of entities or agents, interacting according to specific rules. Usually the number of distinct types of entity is small, and the rules for interaction are fairly simple. The surprise is that these simple ingredients can generate astonishingly complicated ‘emergent’ behaviour, which often appears to transcend the limitations of the individual entities and rules.

The brain is an example. Here the entities are nerve cells, which interact by transmitting signals. Consciousness is an emergent property. In a crowd, the entities are individual people, which interact by avoiding occupying the same location; emergent properties are things like stampedes. An ecosystem is a third example: the entities are living creatures, the interactions are things like predation or reproduction, and the emergent properties are most of the things we encounter in daily life. Stockmarkets, national economies, living cells, and storm clouds can also be viewed as complex systems.

Classically, science and mathematics have modelled systems of this type by aggregating the individual entities into some kind of continuum, and measuring the state of the system by large-scale averages or other ‘smoothed’ quantities. A crowd, for example, becomes a kind of fluid, where what matters is the local density of people, not the people themselves. An economy is also a fluid, and what flows is money. But it is becoming ever more apparent that this classical approach can miss important kinds of behaviour, and is inadequate for many purposes. A crowd, for instance, can flow in opposite directions along the same corridor. People naturally play follow-my-leader through gaps in the approaching crowd, so the flow can change direction completely from one person to the next. Fluids are not like that.

During the last quarter of the 20th century, a number of research centres started to develop alternative approaches, in which the entities were modelled as individuals. Perhaps the best known is the Santa Fe Institute, which pioneered this point of view before it became fashionable. Much of the early work relied on computer simulations, which revealed many interesting new phenomena, but at a price. Simulations are seldom ‘realistic’, in the sense that they include all relevant features of the real world. On the contrary, they deliberately simplify or ignore many aspects of reality, hoping to focus on those aspects that are most important for the phenomena under investigation.

When modelling a crowd, for example, many aspects of human psychology are ignored. What matters is that at each moment, each individual has a target direction, where they want to go, and a small range of options – move into a nearby space, stay put, bump into someone. Real people are less limited. Nevertheless, models of this kind are sufficiently accurate to be used commercially in the design of large public buildings, like railway stations. They predict places where a crowd may become dangerously dense, and they make it possible to investigate potential methods of crowd control when the building is still just an architect’s plan in a computer.

There is a big opportunity here for mathematics. We need to develop an improved understanding of the link between the small-scale rules governing individual entities, and the large-scale features of the entire system that become apparent in simulations. Which rules give rise to which features? In some areas of the physical sciences, such questions have led to important discoveries, and computer simulations alone may not offer sufficient insight. A wealth of new and important mathematics awaits discovery.


‘Complexity Science is one of the most fascinating and unorthodox areas at the frontiers of science and mathematics. To those with imagination, it offers new hope of understanding apparently intractable problems of huge importance to humanity. To its critics, its aspirations exceed its achievements. However, when classical approaches to such problems fail – as they are doing, spectacularly, in global economics, to name one topical example – it only seems wise to consider alternatives.’

Ian Stewart FRS, is Emeritus Professor of Mathematics and Digital Media Fellow. His research interests include dynamical systems, bifurcation theory, pattern formation, and biomathematics. He is also a writer of popular science and of science fiction. He was awarded the Royal Society’s Michael Faraday Medal for furthering the public understanding of science, and has also delivered the Royal Institution’s Christmas Lectures.


Professor Ian Stewart FRS

Mathematics Institute

Ian Stewart's website