There is a tremendous amount of wonderful expository work being done on mathematics, but I doubt that I am alone in finding insufficient time to explore it all. I must be deliberate about finding ways to get myself to read the things I want (writing book reviews helps), but I know that I am doomed to be forever overwhelmed. The articles outside of my immediate interests, those that explore the diversity and the broader discipline of mathematics, are the ones that most often lose out to other priorities.
A collection like editor Mircea Pitici’s Best Mathematical Writing on Mathematics can ease this frustration. All of the material is accessible to a broad mathematical audience and includes essays on the history, philosophy, education, and practice of mathematics. The book includes articles, columns, books, and presentations of various sorts, including the popular press and journals of mathematics and mathematics education. All of the material was published in 2009. Many sources are sufficiently obscure that some of these papers could have easily been overlooked by even the most voracious mathematical consumer.
There is considerable variety here. Among the thirty-five pieces (not counting a thorough introduction by Pitici and a nice forward by William Thurston), we get personal reflections like Freeman Dyson’s discussion of types of contributors to mathematics and physics in his AMS Einstein Lecture “Birds and Frogs,” educational commentary such as a criticism of the role of calculus texts in education from Ann Kajander and Miroslov Lovric, historical exposition such as Judith Grabiner’s exploration of Lagrange’s interest in proving Euclid’s parallel postulate, and some popular goodies like Steven Strogatz’s New York Times column on modeling romantic relationships with differential equations. One can always quibble about what should or should not have been chosen for inclusion (Pitici points out that many decisions were guided by reprinting restrictions), but the material is thoughtfully selected overall.
What the reader will not find is much deep mathematical content. Indeed, there is almost no mathematical notation to be found in the collection. The editor emphasizes in the introduction that this is the best writing on mathematics, not in mathematics. Advanced topics such as information-based complexity and circle packing do appear, but they are usually broad invitations or narratives of mathematicians’ journeys through these topics and not detailed expositions. Everything in the book is meant to be accessible to an undergraduate.
The articles are grouped into vaguely titled sections (Mathematics Alive, Mathematicians and the Practice of Mathematics, Mathematics and Its Applications, Mathematics Education, History and Philosophy of Mathematics, and Mathematics in the Media), but most do not fit squarely under any one of these categories and so the assignment of essay to section is largely arbitrary. That’s not really a complaint so much as a concession to the nature of the compilation. Original citations are found in the table of contents; short biographies of the authors can be found at the end of the book. (I’ll nitpick briefly and suggest these data be presented right along with the articles.)
This is the kind of book with which mathematics students would do well to have a chance encounter — something that would do well lying around a math lounge. It is not the book one recommends for a student to learn how to mimic good writing for an impending thesis nor to learn any real new mathematics, but rather it offers a glimpse into how we mathematicians think and talk about our discipline — and why many of us are so passionate about it. The book is well-suited for college libraries; even though it compiles material published elsewhere, the sources are uncommon.
This is the first in what is intended as an annual publication and I am grateful for it, since I almost certainly would have missed most of these articles. I found the content variously engaging, informative, inspiring, silly, and annoying (identifying which is which is left as an exercise for the reader), but I feel enriched for having read all of them. It is a recommended collection for anyone with an interest in the practice of mathematics at pretty much any level.
Bill Wood is a mathematician, board game enthusiast, lousy disc golf player, and impending faculty member at the University of Northern Iowa.
Foreword by William P. Thurston xi
Introduction by Mircea Pitici xv
The Role of the Untrue in Mathematics by Chandler Davis 3
Desperately Seeking Mathematical Proof by Melvyn B. Nathanson 13
An Enduring Error by Branko Grunbaum 18
What Is Experimental Mathematics? By Keith Devlin 32
What Is Information-Based Complexity? By Henryk Woz´niakowski 37
What Is Financial Mathematics? By Tim Johnson 43
If Mathematics Is a Language, How Do You Swear in It? By David Wagner 47
Mathematicians and the Practice of Mathematics
Birds and Frogs by Freeman Dyson 57
Mathematics Is Not a Game But… by Robert Thomas 79
Massively Collaborative Mathematics by Timothy Gowers and Michael Nielsen 89
Bridging the Two Cultures: Paul Valery by Philip J. Davis 94
A Hidden Praise of Mathematics by Alicia Dickenstein 99
Mathematics and Its Applications
Mathematics and the Internet: A Source of Enormous Confusion and Great Potential by Walter Willinger, David L. Alderson, and John C. Doyle 109
The Higher Arithmetic: How to Count to a Zillion without Falling Off the End of the Number Line by Brian Hayes 134
Knowing When to Stop: How to Gamble If You Must—The Mathematics of Optimal Stopping by Theodore P. Hill 145
Homology: An Idea Whose Time Has Come by Barry A. Cipra 158
Adolescent Learning and Secondary Mathematics by Anne Watson 163
Accommodations of Learning Disabilities in Mathematics Courses by Kathleen Ambruso Acker, Mary W. Gray, and Behzad Jalali 175
Audience,Style and Criticism by David Pimm and Nathalie Sinclair 194
Aesthetics as a Liberating Force in Mathematics Education? By Nathalie Sinclair 206
Mathematics Textbooks and Their Potential Role in Supporting Misconceptions by Ann Kajander and Miroslav Lovric 236
Exploring Curvature with Paper Models by Howard T. Iseri 247
Intuitive vs Analytical Thinking: Four Perspectives by Uri Leron and Orit Hazzan 260
History and Philosophy of Mathematics
Why Did Lagrange "Prove" the Parallel Postulate? By Judith V. Grabiner 283
Kronecker's Algorithmic Mathematics by Harold M. Edwards 303
Indiscrete Variations on Gian-Carlo Rota's Themes by Carlo Cellucci 311
Circle Packing: A Personal Reminiscence by Philip L. Bowers 330
Applying Inconsistent Mathematics by Mark Colyvan 346
Why Do We Believe Theorems? By Andrzej Pelc 358
Mathematics in the Media
Mathematicians Solve 45-Year-Old Kervaire Invariant Puzzle by Erica Klarreich 373
Darwin: The Reluctant Mathematician by Julie Rehmeyer 377
Loves Me, Loves Me Not (Do the Math) by Steven Strogatz 380
The Mysterious Equilibrium of Zombies and Other Things Mathematicians See at the Movies by Samuel Arbesman 383
Strength in Numbers: On Mathematics and Musical Rhythm by Vijay Iyer 387
Math-hattan by Nick Paumgarten 391