Martin H. Krieger's Doing Mathematics aims to "provide a description of some of the work that mathematicians do, employing modern and sophisticated examples." As a mathematician, I found the book interesting at times, but mostly disappointing. The author says he is aiming at lay readers as well, but I cannot imagine such readers finding themselves efficiently informed by this book.
The body of the book consists of a preface, six chapters, and an epilog, some 260 pages in all. Chapter 1 is an introduction, and the remaining five chapters are intended to be read in any order. Chapters 2-5 are case studies, corresponding in order to the four words of the subtitle. Chapter 6 has a much more speculative feel. The rest of the book consists of reprints of mathematical physics papers of Yang and Fefferman-Seco, in support mainly of Chapter 4, new translations from French of writings of Leray and Weil, in support of Chapters 3 and 5 respectively, extensive notes, a bibliography, and an index, some 190 pages.
Chapter 2, "Convention: How Means and Variances are Entrenched as Statistics," would be the best model for a better version of this book. Mathematics, like all disciplines, has conventions, some appropriate, some arbitrary to the point of inappropriateness. Krieger looks at one. Suppose one has a list of numbers x1, ..., xn. Why is it that the mean of these numbers is generally viewed as more important than the median, or the midrange (xmax + xmin)/2? Krieger defines and uses the idea of Lp. He identifies mean as an L2 notion, median as an L1 notion, and midrange as an Linfinity notion. He similarly asks why the L2 notion of variance is generally viewed as more important than its L1 or Linfinity counterparts. A theme is that informed people know perfectly well about L1 and Linfinity but more casual users of statistics are bound by convention into thinking that means and variances are always the right thing. But even in this best chapter, the focus is weak and there is no real conclusion.
Chapter 3, "Subject: The Fields of Topology," is about "the development of a field and its subject matter." Krieger describes the story of this chapter as "sibling sub-fields in tension," the siblings being algebraic topology and point set topology. Already, there is something of a problem, as another reasonable take on this situation would be "unrelated fields coincidentally having the same last name." The problem is felt throughout, as there is just too much presented, from a diagram informing readers that strongly paracompact spaces are collectionwise normal to a statement of the Künneth formula. There is no letting up: entirely misplaced in this chapter is a six-page "excursus into mathematical physics," even though it is conceded that "the subject of topology appears only briefly."
Chapter 4, "Calculation: Strategy, Structure, and Tactics in Applying Classical Analysis" is mainly about the interplay between strategy and tactics in mathematical research. It begins nicely: "Strategy is about the larger meaning of what you are doing, the relationship of means to ends... Tactics are the line-by-line computations you perform, and the tricks you use that do the work of the moment even if you are not sure why they work well." How strategy and tactics interact is an interesting aspect of doing mathematics, and could be well illustrated by modest examples. But Krieger protests against "highly simplified toy problems" and instead illustrates his points with some sixty pages on decades of work on two topics, the Ising model and the stability of matter. The text becomes a barrage of paper titles, schematic summaries, lists of equations, and citation after citation. Even Krieger writes, "The level of detail in this chapter is perhaps deadening, and the knowledge required to follow some of the technical arguments is that of an expert."
Chapter 5, "Analogy: A Syzygy Between a Research Program in Mathematics and a Research Program in Physics" is even broader in scope than the previous chapter. The research program in mathematics is the Langlands program, which is already a huge program in and of itself. I know enough about the Langlands program to see that Krieger's presentation of it is so vague that a mathematician without prior knowledge would have a very hard time extracting its main points. I am that befuddled reader with respect to the other program, something Krieger calls the Onsager program in statistical physics. A few choice specific mathematical examples would have gone a long way. Instead we get various versions of analogies within the programs and analogies between the analogies, versions Z, Y, A, B, C, D, E, F, G, H, I, J, in Krieger's organization. Being told that the story of this chapter is "The Witch is the Stepmother" does not enhance understanding either.
Chapter 6, "In Concreto: The City of Mathematics" is speculation about "suggestive linkages" and "concrete resonances" between mathematics and the world. It begins, "In effect we have been examining the legacies of Gauss and Faraday, of Riemann and Maxwell." It goes on to speculate about connections with "the city, the body, and God," which Krieger hopes are "each worthy of a monograph." Having found the vagueness of previous chapter frustrating, I was not won over by sentences like "A multilayered surface, whether it be a sheaf of stalks, a Riemann surface, or an urban palimpsest, allows for the dream of a unity in multiplicity."
Krieger has made some effort to accommodate different levels of readers, for example structuring his text so that lay readers are alerted to sections that can be safely skipped and paragraphs that provide non-technical summaries. However even the non-technical part of the text is forbidding. Very little is done to motivate non-mathematician readers and less is done to truly inform them on a mathematical level. In summary, I'd say that mathematicians may want to browse through this book, but lay people should definitely look elsewhere to find out about doing mathematics.
David Roberts is an associate professor of mathematics at the University of Minnesota, Morris.