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The Best Writing on Mathematics 2013

Mircea Pitici, editor
Princeton University Press
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Sandra Keith
, on

Mircea Pitici is a PhD candidate in mathematics education with a specialization in writing. By scanning journals and books from 2012 and passing essays through a fine sieve of high profile reviewers (who frequently disagreed among themselves), he has created this fourth in an annual series collecting writings on mathematics. With a foreword by Roger Penrose, the book is a surprisingly integrated collection of essays for the most part understandable to an undergraduate. Revolving around the theme of “what is mathematics in the cyber-age,” these essays scrutinize history, the arts, philosophy, and the sciences. Perhaps more concerned with the best accessible mathematics writing than simply with the best writing in mathematics, the editor has cast a wide net to loop in young as well as distinguished authors in diverse professions.

Taking this book as an anthology of popular mathematics, it is a gentle, likable collection. On the other hand, if an academic publisher such as Princeton University Press decides to take on an annual project which collects the “most memorable” writings of the year, one might want to sit up and take notice. There is a fair chance that this series could become part of the curriculum for a science writing or mathematics appreciation course. And then it becomes potentially formative, not only of style but of shifts in the discipline. Penrose notes that the selection unfortunately pays no attention to physics applications, although physics and mathematics form one of the most productive and significant mergings of our age. But even so, this is a book with palpable influence.

But why writing? In his introduction, Pitici views mathematical thinking as integral to critical thinking and discusses the need for a broader audience. I couldn’t agree more. Not long ago, one was expected to rise up to the level of the professional literature. This had the effect of narrowing the audience and erecting barriers between mathematics and other disciplines. Since then, two historical changes in the world of mathematics have ameliorated this problem: mathematical fields have tended to become more overlapping, and mathematicians have had to assume a broader perspective. As a result, just as mathematics has evolved over the last generation, so has the importance of writing about it. Currently there is no shortage of popularizing mathematics books; the editor lists more than sixty such books published in 2012 alone.

Such observations, however, don’t address the question of what characterizes good writing. It is unfortunate that the book provides no guidelines. In his (too brief) introduction, Pitici finds the usefulness of writing in plainness and transparency, which isn’t really saying much, as these are characteristics of mathematical and scientific style that date to Francis Bacon in the 1600s. In fact, most of the essays here do follow this premise. Particularly enjoyable to me in the more mathematical category are a piece on Bézier curves and their link to parabolas (by Renan Gross, undergraduate). Nevertheless, some of the articles seem to fall short on the goal of openness and transparency, while some other articles, selected presumably for simplicity or theme, fail to develop an idea as fully as one might want. One piece on fashion, aimed at students, offers no clarity about the claim that a particular designer was influenced by geometry, especially since that designer seems to minimize the idea. A piece on luck and randomness in video games involves no mathematics or statistics. The explanation of Durer’s fascination with a parabola would be much better with a little more math. And similarly, an article written for cognitive psychologists seems to claim little more than a need for more frontal cortex exploration of math anxiety.

Since this review is about writing, perhaps a brief side-track into the history of the theory of scientific writing would be useful. Aristotle, that great categorizer, divided discourse into specialist and public (or rhetorical) kinds. The latter he divided into (1) Political: deliberation about the future, (2) Forensic: accusation and defense relating to the past, and (3) Epideictic: ceremonial expression regarding the present status of a person or work. But, as Aristotle was clearly aware, this division between specialist and public discourse was difficult to sustain: think Nobel prizes! In the classic Renaissance debate about the appropriate style for scientific explanation, Bacon’s demand for plainness and transparency was in conflict with the traditional expectation that matters of scientific importance demanded a Ciceronian oratorical and philosophical point of view, to distinguish the specialists from the common folk. Although Baconians roundly “won” that debate in important ways, it seems to be with us in slightly different terms. The issue of tone, of pedagogical friendliness and looseness of style, seem to be largely what Pitici is after.

Established writers can move beyond any sense of contested ego. In his discussion of infinite machines, Pavlus’ easy range of reference and example across current science and technology makes him a confident explainer, as is Ian Stewart in his discussion of pattern and symmetry. And reading Philip Davis is like collapsing into a comfortable sofa, he is so completely at home with his subject and humanistically warm. In contrast, some of the authors who are not coffee room names may have been included for their persona, which may come as a surprise to the reader. Anna Sfard does not say much that is new, but has a style of charm recognizable to every woman on this planet and thus is a voice not often heard in math journals. David Lloyd adopts a “forensic” style to debunk the idea that Paleolithic man must have known his Platonic solids. And the manner of a “political” blog serves Frank Quinn well in his discussion of the math revolution. (On the other hand one could wonder if the editor has bailed on his own criterion of plainness and transparency when the author leaves the “how to” discussion to his references.)

But these would probably not be Pitici’s terms for describing effectiveness in writing, and my frustration lies in the lack of any alternative explanatory framework for what makes writing good in the items he has selected. The risk in adopting “plainness” as your evaluative guideline is that it is basically no guideline at all.

To bring us back to practicalities, let me suggest that most of my problems with this otherwise enjoyable book would be solved quite simply with: (1) An additional introduction that summarizes commentary from the involved reviewers, with reference to the topics included or not, (2) An introduction by the editor that speaks less superficially about writing, and that offers up a rationale for the selection process, (3) Individual prefaces accompanying each essay that set a critical thinking model for the reader rather than simply a marketing one, and (4) The placement of original publishing venue and biography of the author after each article instead of the far back.

Lastly, here are some questions that this suggestive book raised for me. What is the status of popular mathematics writing internationally, since so much of this writing uses English? Would it be possible, or even advisable, to provide an online forum for commentary on books? I tend to value exploratory and discussion writing as an important form of mathematical discourse for both discovery and learning. The scientific philosopher and mathematician C.S. Peirce placed “abductive” discourse as an important form of scientific discussion basic to scientific literacy. In this dialectic model, writing encourages more writing. This is another way of keeping the field open. There should be no last word.

Sandra Keith is professor emerita at St. Cloud State University, MN. She has long been interested in mathematics writing. 

Foreword Roger Penrose ix
Introduction Mircea Pitici xv
The Prospects for Mathematics in a Multimedia Civilization Philip J. Davis 1
Fearful Symmetry Ian Stewart 23
E pluribus unum: From Complexity, Universality Terence Tao 32
Degrees of Separation Gregory Goth 47
Randomness Charles Seife 52
Randomness in Music Donald E. Knuth 56
Playing the Odds Soren Johnson 62
Machines of the Infinite John Pavlus 67
Bridges, String Art, and Bézier Curves Renan Gross 77
Slicing a Cone for Art and Science Daniel S. Silver 90
High Fashion Meets Higher Mathematics Kelly Delp 109
The Jordan Curve Theorem Is Nontrivial Fiona Ross and William T. Ross 120
Why Mathematics? What Mathematics? Anna Sfard 130
Math Anxiety: Who Has It, Why It Develops, and How to Guard against It Erin A. Maloney and Sian L. Beilock 143
How Old Are the Platonic Solids? David R. Lloyd 149
Early Modern Mathematical Instruments Jim Bennett 163
A Revolution in Mathematics? What Really Happened a Century Ago and Why It Matters Today Frank Quinn 175
Errors of Probability in Historical Context Prakash Gorroochurn 191
The End of Probability Elie Ayache 213
An abc Proof Too Tough Even for Mathematicians Kevin Hartnett 225
Contributors 231
Notable Texts 237
Acknowledgments 241
Credits 243