Ian Stewart commences the proceedings in The Mathematics of Life by claiming that five great “revolutions” have altered the way in which scientists think about life: the “invention of the microscope, the systematic classification of the planet’s living creatures, the theory of evolution, the discovery of the gene, and the discovery of the structure of DNA” (p. 1). This salubrious state of affairs notwithstanding, the author contends that a sixth revolution is now upon us, and it concerns the almost routine use of mathematics in contemporary biology. In this regard, what is particularly dramatic is the extent to which mathematical methods “are starting to set the agenda in some areas of biology” (p. 319). This book contains a detailed and exciting account of the many facets of this sixth revolution.

The author’s discussion of revolution number three - i.e., the theory of evolution - begins by rightly noting that, in contemporary times, there can be no meaningful discussion of evolution without also saying something sensible about the role of God in this process. Two points in this discussion are worth emphasizing. First, in Victorian England and in most of Europe, people gradually came around to the idea that the creation story in Genesis is a metaphor, and they accepted the “discoveries of science as insights into God’s creation” (p. 76). In contrast, those who believed in the literal truth of the Bible painted themselves into an intellectual corner by unconvincingly attempting to deny a vast body of scientific evidence. Second, we are told that the entire corpus of scientific evidence notwithstanding, if one is able to convince people that the Earth is only 10,000 years old then the battle against what some have called “evilution” is won and the idea of evolution can be shown to be a nonsensical concept. This is why “Creationists now routinely dispute the scientifically established figure for the age of the Earth, which is about 4.6 billion years” (p. 66).

Can geometry helps us shed light on the behavior of viruses? In discussing this tantalizing question, Stewart points out that mathematicians have developed a general theory of the geometry of viruses based on symmetry principles that are closely related to the group theory of the regular solid known as an icosahedron. He explains that the present knowledge of mathematicians about viruses fully “justifies the view that abstract geometry in higher dimensions can tell us a lot of useful things about real viruses in three dimensions” (p. 157).

In a chapter cryptically titled “Hidden Wiring,” the author discusses a variety of issues concerning brain size and intelligence. He begins by debunking the popular notion that there is a necessary connection between brain size or weight and intelligence. He then notes that because of significant advances in the development of a whole host of experimental techniques, it is now possible to image the activity of the brain in real time. Even so, because the brain is so complex, it often makes sense to study particular features of the nervous system rather than trying to comprehend the entire brain in one go. In the author’s colorful words, our “brains are so complicated that, ironically, they may be inadequate to understand...our brains” (p. 180).

John Maynard Smith is generally recognized to be the foremost expositor of game theory in evolutionary biology. This fact is acknowledged by the author in his unfortunately rather terse commentary on the ways in which game theory can explain the mating behavior of lizards. More generally, we learn that game theory can help explain the evolution of new species when alterations in the environment render a single species strategy evolutionarily unstable. When this happens, a mutant can successfully invade and, given sufficient time, a random mutation can arise. The author is careful to point out that while this does not explain the phenomenon known as “speciation,” it does “determine circumstances under which it might or might not be possible” (p. 220).

The penultimate chapter in this book contains a fascinating discussion of the potential existence of life outside our own planet. This discussion results in three noteworthy points. First, we are told that it is imprudent to dismiss reasonable possibilities of life outside our planet simply because “there are no observations to support them” (p. 303). Second, although it can be useful to create lists of the special features of the Earth, one cannot infer from these features alone that they are necessary for life. In other words, there is a fundamental difference between conditions that are necessary for life and those that are sufficient to support it. Finally, if one is interested in comprehending how probable alien life might be, then one ought not to “focus on conditions that are virtually identical to those found on this world, and then argue—typically confusing sufficiency with necessity—that only those conditions are suitable for life” (p. 315).

In the last four decades, biomathematicians such as Colin Clark (see the 1976 book Mathematical Bioeconomics) and Marc Mangel (see the 1985 book Decision and Control in Uncertain Resource Systems) have done much to show how mathematical analysis can substantially improve our understanding of natural resource management in particular, and the dynamic and stochastic behavior of jointly determined ecological-economic systems such as fisheries, forests, and rangelands. The only shortcoming of The Mathematics of Life is that it unfortunately pays no attention to this important line of research. Subject to this caveat, Stewart's book shows, in an engaging literary manner, the many ways in which mathematics has and will continue to revolutionize scientific thinking in biology.