COMPLEXITY

a scientific theory which asserts that some systems display behavioral phenomena that are completely inexplicable by any conventional analysis of the systems' constituent parts. These phenomena, commonly referred to as emergent behaviour, seem to occur in many complex systems involving living organisms, such as a stock market or the human brain. For instance, complexity theorists see a stock market crash as an emergent response of a complex monetary system to the actions of myriad individual investors; human consciousness is seen as an emergent property of a complex network of neurons in the brain. Precisely how to model such emergencethat is, to devise mathematical laws that will allow emergent behaviour to be explained and even predictedis a major problem that has yet to be solved by complexity theorists. The effort to establish a solid theoretical foundation has attracted mathematicians, physicists, biologists, economists, and others, making the study of complexity an exciting and evolving new scientific theory. This article surveys the basic properties that are common to all complex systems and summarizes some of the most prominent attempts that have been made to model emergent behaviour. The text is adapted from Would-be Worlds (1997), by the American mathematician John L. Casti, and is published here by permission of the author. John L. Casti Additional reading General works John L. Casti, Complexification (1994), is a wide-ranging historical introduction to the interdisciplinary study of complexity, and Would-be Worlds (1997), contains a nontechnical overview of the growing role of computers in modeling complex dynamical systems and an examination of the fundamental principles of complexity.Ian Stewart, Does God Play Dice?, 2nd ed. (1997), is an introduction to the mathematical properties of chaos. Specialized works Benoit B. Mandelbrot, The Fractal Geometry of Nature, updated and augmented (1983; originally published in French, 1975), is a lavishly illustrated introduction to mathematical fractals and fractal forms in nature. John L. Casti, Reality Rules: Picturing the World in Mathematics, 2 vol. (1992, reissued 1997), is an introduction to creating mathematical models of complex dynamical systems from the social and natural sciences. E. Atlee Jackson, Perspectives of Nonlinear Dynamics, 2 vol. (198990, reissued 1992), offers a fairly rigorous mathematical introduction to nonlinear dynamics.John S. Nicolis, Chaos and Information Processing (1991), is a technical presentation of the author's model of human cognition, based upon ideas from complex dynamical systems and information theory. John L. Casti