What are the five most influential mathematical theories of the 20th century? Indeed, how should such a selection be made, and on what merits should any particular development be held in higher esteem than another?
This was the task that John L. Casti set himself, and in the first book he wrote on the subject, Five Golden Rules , he settled on the Minimax theorem, the Brouwer fixed point theorem, Morse's theorem, the Halting theorem and the Simplex method. All of these he explained in a language accessible to anyone who has some basic mathematical background. However, despite an eloquent and thorough justification for his choices, such a small snapshot of the past century's advances in mathematics cannot do justice to the huge collective efforts in all areas of the subject.
It is therefore hardly a surprise that Casti has continued his journey through the vast expanse of modern mathematics, in the guise of Five More Golden Rules . This time he settles on topics from such a diverse choice of areas as knot theory, dynamical systems, control theory, functional analysis and information theory. Again, Casti tries to appeal to the mathematical novice, but on the other hand has added a little more meat to his exposition than previously, which will give more knowledgeable readers something to get their teeth into.
In many ways this book carries on precisely where Five Golden Rules left off, and on several occasions makes the assumption that the reader is familiar with its predecessor, but this is not a prerequisite. In addition, Casti never claims to be providing an exhaustive description of each problem and provides extensive lists for further reading.
Part of the beauty of Five Golden Rules was the frequent references to "real-world" applications. This sequel continues in the same vein through, among others, the application of knot theory to the way in which DNA strands get tangled (and untangled), and an idea of how control theory helped the Viking landers to negotiate their way onto the Martian surface. These touches make it, on the whole, an enjoyable read, but perseverance is needed in places, none more so than the description of the Kalman filter in chapter three. This theory is in essence a method for minimising errors in navigation, whether by a Boeing 777 or by the Apollo 11 lunar module, and is part of the much wider area of control theory. Unfortunately, Casti's enthusiasm for this particular subject (for which he has written a graduate-level text) will, I suspect, make the extensive background he gives rather heavy going for the lay reader.
The only other anomaly in the book is his treatment of the Hopf bifurcation theorem, a very powerful tool when used in the correct context (this theorem in essence guarantees, under certain hypotheses, periodic phenomena). The corresponding chapter as a whole is a succinct survey of the world of dynamical systems theory, from simple population models to fractals, but the main result due to Hopf is almost swallowed up by the wealth of material presented.
There are, of course, many questions left unanswered on completing this book, all of which centre on those topics left out rather than those included, but that is part of the appeal. None of these criticisms really detracts much from the book as a whole, and the later chapters more than compensate for any lapses earlier on. I would encourage anyone who would like to discover, or rediscover, the many delights of modern mathematics to read both this book and its predecessor.
David Wood is lecturer in mathematics, University of Warwick.
Five More Golden Rules: Knots, Codes, Chaos and Other Great Theories of 20th-Century Mathematics
Author - John L. Casti
ISBN - 0 471 32233 4
Publisher - Wiley
Price - £18.50
Pages - 268