The Best Writing in Mathematics 2016

This is the seventh volume in one of my favorite series of books, an annual compilation of high-quality mathematical writing combined with a valuable list of additional resources prepared by the editor. This year’s edition contains 30 moderately short pieces (only a handful exceed 20 pages in length, and quite a few are under 10 pages) covering, as is typical, lots of different aspects of mathematics.

Many of these articles are not about actual substantive mathematical results or theories but are instead about various aspects of mathematics as a discipline. These address pedagogical, historical and philosophical issues. For example, among the pedagogically-themed articles, there are two (one by Schoenfeld and the other by Beals and Garelick) that discuss the Common Core controversy; an article by Hyman Bass about how to teach mathematics; and one by Ian Stewart offering suggestions on how to write a popular math book. One article that is really four articles in one (it consists of four pieces by David Acheson, Rachel Levy, Gilbert Strang, and Peter Turner) discusses aspects of teaching applied mathematics.

Historical articles include one by Victor Blasjo, who argues that the purpose of Leibniz’s famous 1693 paper was not to prove the Fundamental Theorem of Calculus, but actually to describe a construction for evaluating integrals when no anti-derivative existed; a piece by Serrano and Suceava on the early history of the concept of curvature and the contributions of Nicole Oreseme to this endeavor; and an article by Richeson that traces the history of the formula \(C=\pi d\). In addition, there is an article by Daniel Silver about G.H. Hardy’s A Mathematician’s Apology and public reaction to it. Hardy is also the subject of another article, by Christenson and Garcia, which discusses his work in genetics (the Hardy-Weinberg law) and the tension between that and his professed disdain for applied mathematics.

On the philosophical end of the spectrum, there is an article by John Stillwell about what constitutes “depth” in mathematics; he subsequently expanded these ideas into a book, Elements of Mathematics: From Euclid to Godel, that was reviewed in this column about nine months ago. In another article raising philosophical issues, Abbott addresses the question of whether mathematics really is as “unreasonably effective” as Wigner famously said it was. Interestingly, this issue was just recently raised in a review in this column of McDonell’s The Pythagorean World; the 1980 Monthly article by Hamming that is referenced in that review is an important part of Abbott’s article as well.

Stillwell’s article, mentioned above, also discusses substantive mathematics in the sense that it describes a number of mathematical results that he considers “deep”. Other articles also discuss mathematical theorems or theories. On the easy end of the spectrum, Steven Strogatz has an article — first published in The New Yorker magazine, and therefore accessible to a general audience — that discusses a proof of the Pythagorean theorem that may have been discovered by a young Einstein. On a more sophisticated level, Erica Klarreich writes about astonishing connections, illuminated by string theory, between the Monster group and the j-function that comes up in the theory of modular forms.

String theory is also the subject of an article by Brian Greene, who asks whether this theory is “revealing reality’s deep laws” or is a “mathematical mirage that has sidetracked a generation of physicists.” Discussing some of the experimental problems that have arisen and the new assumptions that have been made, he concludes that while he has only “modest hope that the theory will confront data during my lifetime”, the pull of string theory “remains strong”.

Other substantive articles include one by Joshua Bowman discussing dynamical systems in the context of billiard ball paths on tables of various shapes: the author proves some results and states more difficult ones without proof. In addition, Burkard Polster is the author of two articles in this volume, one giving an elementary proof of a result on stacking circles in a rectangle, and the other discussing the card game Spot It!, an endeavor which results in connections with projective planes.

One article, by Davide Castelvecchi, uses substantive mathematics — specifically, the abc conjecture in number theory — as a springboard for a personality-driven discussion. The conjecture itself is briefly and informally described, but the bulk of the article concerns a purported proof of it that has been released by Shinichi Mochizuki. The proof, which is 500 pages long and invents an entire new branch of mathematics that Mochizuki calls “inter-universal geometry”, is apparently so complex that even experts in the field cannot understand it. The current status of the proof, and conjecture itself, are, thus, in limbo.

A good many of the articles contain full-color illustrations; this is particularly valuable, for example, in the article by Dauben and Senechal on mathematics at the Metropolitan Museum of Art: the authors tour the museum and reproduce photos of artwork there that illustrate mathematical ideas. (The authors indicate in their article that another article, still in the planning stages, will focusing on symmetry in works of art at the Met. For a book’s worth of examples of such art, see Symmetry by Emil Makovicky.)

There is more, of course; space considerations do not allow a description of all the articles in this volume. As I have said before in previous reviews, the omission of an article from this discussion does not reflect my opinion of its quality.

As mentioned above, in addition to the articles, there is a lengthy (15 pages) list of various articles, interviews and reviews that did not make the cut for this volume. In addition, there is a shorter piece listing various books published in this period.

Based on a random selection, I think that most, if not all, of the articles in this book can be found online. Of course, this doesn’t mean that you, or your university library, shouldn’t purchase the book: although I had previously run across one or two of the articles on my own, most of them were new to me, and I would likely have never found them without this book. And of course there is always value in having the articles assembled in one easy-to-find place.

One thing about these volumes has always puzzled me. Although this book contains the word 2016 in the title, it was in fact published in early 2017, and the articles that appear in it were originally published in 2015, not 2016. This pattern seems to be true in general, making me wonder what the significance of the year in the title is. (I suppose one answer is that this is the year in which the editor did all the work.) This minor quirk, however, is more amusing than anything else, and certainly does not detract from the value of this very nice series of books. It seems to me that anybody who enjoys reading this column will find something — more likely lots of things — that are interesting in this book, and also in the others in the series.

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