Applied Mathematics: A Very Short Introduction

I am not now, and never have been, an applied mathematician. In fact, in my youth, I took a G.H. Hardy-like approach to the whole subject and turned up my nose at it. Unlike Hardy, however, my attitude softened as I grew older and more experienced, and I came to realize that applied mathematics could be quite interesting and beautiful in its own right. Even if I had not reached this conclusion on my own, however, this lovely little book would have convinced me.

This is an entry in the Very Short Introduction series published by Oxford University Press, a series that has been extant for more than 20 years and now numbers just over 600 books, only about 20 of which, unfortunately, are in mathematics. They are generally written by experts in the field (including, for example, Fields medalist Timothy Gowers, Ian Stewart, Robin Wilson, and Jacqueline Stedall) and are characterized by the fact that they are physically very small and quite short. Each book has roughly the same length and width of the paperback books that I used to buy in the 1960s (I believe paperbacks are larger now), but are much thinner, usually only a fraction of an inch thick, with typically about 150 pages or so per book.

Tiny books like these have several advantages: they are inexpensive ($11.95 per book seems to be the common rate on Oxford’s website), easy to shelve, and, above all, you can actually read them. I read this book from cover to cover on a lazy Saturday morning. It took a bit more than four hours, and was time well spent.

It didn’t take more than the very first paragraph of the preface to convince me that I was going to enjoy this book. The author describes as follows the inevitable conversations that he has at parties after explaining that he is an applied mathematician: “After the awkward pause that sums up most of my human interactions, I look for the closest exit, convinced that further contact would inevitably deepen my existential crisis.” Clearly, Goriely knows how to turn a phrase, a conclusion reinforced a few pages later when he describes applied mathematics as “mathematics’ oxymoronic sibling”.

This book is intended as Goriely’s response to the question “What is applied mathematics?” He begins by attempting a characterization of the subject, though he ultimately concludes, in the spirit of Justice Potter Stewart’s famous comment about pornography (“I know it when I see it”), that no really precise definition is possible. However, in the process of attempting to characterize the subject, he focuses on three constituent parts of it: modeling, theory and methods. He does this in the context of a physical phenomenon that is not as simple as it might first appear (a burning candle) and insightfully observes that

For the applied mathematician, the candle is not the object of study, but a paradigm for a large class of autocatalytic phenomena described by various physical principles leading to the same mathematical description. Once we understand the candle, we understand a multitude of other systems. On the one hand, understanding these equations from a mathematical perspective provides insights on the phenomena. On the other hand, studying the details of these different systems motivates us to better understand the underlying mathematical structure that they share.

Subsequent chapters look at particular examples of how applied mathematics deals with the real world, and the tools used in this endeavor. In chapter 2, Goriely considers the principle of dimensional analysis and gives several examples. One particularly fascinating one concerns Geoffrey Taylor, who used dimensional analysis to estimate (quite accurately), using publicly available photographs from Life magazine of the Trinity atomic bomb explosion, the amount of energy released by the explosion (something that the government did not want to be publicly available). “The story goes that Taylor was properly admonished by the US Army for publishing his deductions from their (unclassified) photographs.”

Chapter 3 returns to the subject of creating mathematical models of physical phenomena. Starting with the relatively simple example of a thrown ball, Goriely demonstrates that multiple models can be created, depending on the complexity of one’s assumptions. He points out that simple models can be more easily solved but have less predictive power: “It is in that Goldilocks zone (not too big, not too small!) that modern mathematical modeling thrives….”. Then, after a discussion of the physics paradigm for modeling, the author closes the chapter with a brief discussion of the modern practice of modeling.

The common theme of the next two chapters is on differential equations: ordinary ones in chapter 4, partial ones in chapter 5. In chapter 4, the author begins by discussing equations in general and how they might not have solutions, looks at alternatives to finding explicit solutions to equations, and then briefly runs through a history of ordinary differential equations and their uses, culminating in a discussion of the prey-predator problem, first involving rabbits and foxes and then adding wolves, resulting in chaotic orbits. In chapter 5, partial differential equations are discussed with particular emphasis on linear waves and nonlinear solitons.

The general theme of the next chapter is data analysis. Starting with medical imaging and proceeding through analysis of DNA, the author eventually arrives at compressed sensing and a brief look at the use of linear algebra in applied mathematics. Then, in the next chapter, to illustrate the idea that one never knows what subjects in pure mathematics might prove to have useful applications, the author discusses first quaternions and then the theory of knots. No prior background in either of these areas is assumed. In fact, in the case of quaternions, Goriely spends more time sketching their history and properties than in describing their applications, though some indication of their use is provided. In the case of knots, Goriely addresses their applications to problems in cellular biology, such as DNA analysis.

The final chapter of the book looks at some modern issues in applied mathematics. The author begins with a discussion of networks and then discusses current issues in the analysis of the human brain, an activity that involves not only networks but other areas of science and mathematics. Anybody teaching or taking a course in graph theory could certainly do worse than glance at this chapter.

Throughout, the book is written with style and humor. It has some mathematical prerequisites: as is clear from the chapters on differential equations, for example, it assumes at least a basic understanding of calculus on the part of the reader. Some prior knowledge of physics would also be useful to the reader as well. So is a willingness to deal with mathematical equations and arguments, presented at an accessible but honest level. For people with these prerequisites, this book provides an informative, and entertaining, reading experience. I thoroughly enjoyed it.

Mark Hunacek (mhunacek@iastate.edu) teaches mathematics at Iowa State University.