The five golden rules of this book's title are five mathematical theories selected by John L. Casti to form the basis of the five chapters of his book. Each chapter concerns itself with a central theorem or algorithm, and the problems to which this theorem has been applied. Casti's aim is to show how these theorems arise from real-life problems, and that they have theoretical and practical applications. Casti writes for a nonspecialist audience but assumes readers will not be frightened by a few simple equations.

The first chapter centres on John von Neumann's minimax theorem. This gives a complete answer to the problem of finding the optimal strategy in two-player zero sum games - simple models of conflict situations where the interests of the two players are diametrically opposed. Casti starts with some well-chosen examples vividly illustrating von Neumann's insight that often the best strategy is to play randomly, but within a precisely computed framework of relative odds. After this Casti moves briskly through multi-player zero sum games to two-player non-zero sum games. Casti's main example here is the paradoxical "prisoner's dilemma", where each player is tempted to betray the other, leading to a result that is bad for both parties. There seems to be no way to define a "best" strategy for such games, but Casti describes how the computer simulations of Robert Axelrod and Anatol Rapoport suggest that, in an evolutionary context, cooperative and altruistic behaviour can develop when the prisoner's dilemma is repeatedly played in a large population.

The next chapter is on L. E. J. Brouwer's fixed-point theorem. This topological theorem on continuous maps on compact convex sets may appear arcane at first, but Casti motivates it well by considering the problem of ranking American college football teams: one wants to rank teams based not just on the number of their victories but also on the strength of the teams they have beaten. After ranking, each team's position can be adjusted to take account of their opponents' rating, and then the process can be repeated continually. Brouwer's theorem guarantees that there is a "fixed-point" of this process: some ranking of the teams stays stable even after adjusting for the strength of the opposition.

The third chapter deals with the theorems of Marston Morse and Rene Thom's theorem on critical values of functions. This work led to the development in the 1960s and 1970s of "catastrophe theory" that aimed to provide a mathematical account of processes involving sudden and discontinuous change. The controversies that accompanied the birth of catastrophe theory are briefly mentioned, but Casti wisely attends more to the mathematical background. While many other authors of popular accounts give diagrams of the elementary catastrophes and examples of their use, Casti does all this, and explains the origins of these catastrophes in unfolding of singularities of functions, thus giving a deeper understanding of the underlying mathematics.

The penultimate chapter gives a very swift ride through the theory of computation, starting with Alan Turing's axiomatisation of the notion of computation, and proceeding via Kurt Godel's incompleteness theorem and its philosophical implications, Gregory Chaitin's work on algorithmic information theory, to the theory of computational complexity and NP-completeness.

The final chapter centres on an algorithm - the simplex method of linear programming due to George Dantzig - rather than on a theorem. This has been one of the most widely applied quantitative techniques since its development in the late 1940s. After illustrating the method, Casti applies it to network flow problems and then discusses the much harder problem of nonlinear optimisation.

This book is a fine piece of mathematical exposition, in which even experienced mathematicians will find novelties.

Robin Chapman is lecturer in mathematics, University of Exeter.

## Five Golden Rules: Great Theories of 20th-Century Mathematics and Why They Matter

Author - John L. Casti

ISBN - 0 471 00261 5

Publisher - Wiley

Price - £16.99

Pages - 235

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