Deep in my mind, somewhere, there is a list of "can't miss" authors, whose books I buy without question whenever they come out. Science fiction writer Gene Wolfe is on that list, as is novelist Frederick Buechner. And now, so is T. W. Körner. Having already written what's probably the best book on Fourier Analysis, Körner here turns his hand to writing a book directed at the "general public", and it's a winner.
Körner says, in his preface, that this book is "meant, first of all, for able school children of 14 and over and first-year undergraduates who are interested in mathematics and would like to learn something of what it looks like at a higher level." I don't know about English school children; there won't be many American 14-year-olds who will read this book. But that doesn't matter: for those who do read it (and I hope many potential math majors do read it), it offers a unique look at what mathematics, especially applied mathematics, is like.
Rather than attempt a book on "mathematics for poets", Körner explains that he decided to talk as if he were speaking to another mathematician. Almost all of his topics involve only elementary mathematics (though here and there an occasional remark or exercise goes quite a bit deeper than that), but the attitude is quite sophisticated. As he says in the preface, this means that most of the book's intended readers will find at least a few points where they are in over their heads. Körner urges them to skip over such parts or, even better, to find some professor willing to discuss them. Read this way, Körner's The Pleasures of Counting is really a pleasure, and may well attract many students to mathematics. It strikes me, in many ways, as the ideal book for independent reading or for a first-year seminar.
So what's in the book? Basically, it is a series of stories about using mathematics to understand the world around us. Körner first explains how statistics was used to understand cholera epidemics, allowing Dr John Snow to figure out how cholera was transmitted without having any idea about the role of bacteria in causing disease. Next, he goes on to talk about submarine warfare during both World Wars, about doing science "in a darkened room" (i.e., with "pure thought", without direct reference to experiment), about algorithms and the halting problem, about code-breaking during World War II, and so on. Along the way, Körner discusses quite a bit of (mostly elementary) mathematics and gives an insider's view of how mathematics is done and how it is applied to the world. This is particularly apparent in the many charming footnotes that tell stories, point to references, and even include some of the comments made by the mathematicians who read the book to evaluate it for publication.
For example, let's consider chapter 5, "Biology in a Darkened Room" (the first chapter in Section 2, "Meditations on Measurement"). It begins with a section reproducing a portion of Galileo's Dialogues Concerning Two New Sciences, in which the characters discuss falling bodies. The main point is a "thought experiment" that leads to the conclusion that Aristotle was wrong about how fast bodies fall. Körner comments on Galileo's text, explaining the reasoning and telling us that Galileo went on to do actual experiments that confirmed the hypothesis that (if we neglect the effects of air resistance) all bodies fall at the same speed. He then goes on, in section 5.2, "The long and the short and the tall", to apply Galileo's method to think about the size of animals. He begins, again, with Galileo, this time on the strength of columns, then quotes from J. B. S. Haldane's famous essay "On Being the Right Size", and goes on to apply the mathematics of scaling to study how a number of animal characteristics (e.g., metabolic rates) vary with size, to discuss "variations in design" in animal species, and even to consider how large diving mammals can be and how high animals in general can jump. He then compares his back-of-the-envelope results with real data. This is all quite elementary, and at the same time quite fascinating.
And throughout, there are those footnotes! Is Galileo's publisher Elzivir related to today's Elsevier? (No, unfortunately.) Did Galileo actually do the experiment of dropping two weights? ("It used to be fashionable to doubt whether such experiments were, in fact, performed, but historical research reveals so many instances that anyone passing under a tall building must have seemed in constant danger from falling weights dropped by enquiring philosophers.") Is Galileo always right? (No, but "pioneers are judged by what they get right, not what they get wrong".) Plus lots more: references to books (including a science fiction novel), side notes about interesting related facts (central towers in Medieval churches, soldiers' habits of using sub-machine-guns to open bottles), and so on.
One should note that an appendix contains sources, listed by page, for all citations in the book. For example, he lists S. Drake's Galileo at Work as the reference for the "historical research reveals" statement from the footnote quoted above. This list of sources, and the large bibliography, are very useful to readers who wish to find out more about one of Körner's topics.
The book concludes with two sample mathematics lessons, one extracted from Plato's Meno and one based on Körner's experience at Cambridge. The latter is quite well done, including some nice mathematics (basically, a discussion of the basic axioms about the integers) and a discussion of why and how mathematicians prove things. This makes a very nice complement to a book which focuses mostly on applied mathematics.
Besides the listing of sources, there is an appendix on notation and a very useful appendix listing books for further reading. The choices are idiosyncratic, but that comes as no surprise. The appendix concludes with a delightful extended quote from Lebesgue about Camille Jordan's Cours d'Analyse. Lebesgue describes what makes the book so good in terms that will resonate with the experience of many students... and that many authors of mathematics books ought to take to heart!
One of the notable qualities of the book is the way it humanizes mathematics by emphasizing the people who engage in it, discussing their personalities and their mathematics together. This is a particularly welcome feature in a book for the "general reader", who has often acquired a feeling that mathematics is an inhumanly precise, cold and austere subject. Körner shows that it doesn't feel that way to those who engage in it, and therefore that it needn't feel that way at all.
Of course, every book has its flaws. There are perhaps a few too many examples about war in this book. Though this does include Richardson's admirable attempt to apply mathematics to understand the causes of war (and thereby to find ways to avoid it), some people might feel that the author is a bit too enthusiastic about the contributions of mathematics to fighting and winning wars. A second minor flaw, from the point of view of a teacher who would love to assign this book for independent reading, is the lack of an appendix with solutions or hints for the many exercises (some of them quite hard) that are found along the way. But the book is already very long, and such an appendix would have made it significantly longer, which may account for the omission.
All in all, however, this is a wonderful book, a welcome addition to the rather sparse bibliography of mathematics books that are (mostly) non-technical but nevertheless have real substance. If you know a teenager who is interested in mathematics, this might be just the right book for her. Though I haven't yet tried to use it like this, it should work very well for independent reading at the undergraduate level. It also might be just the thing for a course or seminar designed to attract students, especially those with an "applied" bent, to mathematics. I'm looking forward to trying it out!
Fernando Q. Gouvêa (firstname.lastname@example.org is Associate Professor and Chair of the Department of Mathematics and Computer Science at Colby College. His research area is in number theory (especially modular forms and Galois representations), with side interests in the history of mathematics and in expository writing in mathematics. He is also the editor of MAA Online.
Preface; Part I. The Uses of Abstraction: 1. Unfeeling statistics; 2. Prelude to a battle; 3. Blackett; 4. Aircraft versus submarine; Part II. Meditations On Measurement: 5. Biology in a darkened room; 6. Physics in a darkened room; 7. Subtle is the Lord; 8. A Quaker mathematician; 9. Richardson on war; Part III. The Pleasures Of Computation: 10. Some classic algorithms; 11. Some modern algorithms; 12. Deeper matters; 13. Enigma; 14. The Poles; 15. Bletchley; 16. Echoes; Part IV. The Pleasures Of Thought: 17. Time and chance; 18. Two mathematics lessons; 19. Last thoughts; Appendix A. Further reading; Appendix B. Sources; Appendix C. Some notations; Bibliography; Index.