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A Century of Advancing Mathematics

Stephen F. Kennedy, Donald J. Albers, Gerald L. Alexanderson, Della Dumbaugh, Frank A. Farris, Deanna B. Haunsperger, and Paul Zorn, Editors
MAA Press
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The Basic Library List Committee strongly recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Jeff Ibbotson
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Hooray for the MAA! The year 2015 marks the 100th anniversary of the founding of that society. The volume at hand is a celebration of that long history and contains a unique selection of the best in expository writing. This should come as no surprise, since the American Mathematical Monthly was one of the prime elements in the early formation of the society. As one can learn from reading David Zitarelli’s article in this book, Herbert Slaught’s vision for the Monthly in 1915 (the journal had been started some years prior by Benjamin Franklin Finkel) was that the object of the journal was “to stimulate activity on the part of college teachers…that may lead to production.” I think it is easy to see the continuing success of the Monthly in this regard. Even more, though, the MAA has been a major force in coordinating curriculum across America. The early CUP and CUPM reports are a testament to that. The full story is covered in several of the articles in this book. This volume covers nearly every aspect of the broad mission of the MAA — exposition, remembrances of leaders, forecasting of the future of curriculum, technology and the discipline itself. I found myself dipping into it again and again — just when I had decided I was done reading I found myself drawn into yet another of the articles.

Representing mathematical developments over the course of the century is a choice selection of expository articles written by mathematicians integrally involved in their development. The developments mentioned are Thurston’s work on three-dimensional geometry and topology, complex dynamics, the Four Color Theorem, optimal results in geometry, the Stanley-Wilf conjecture of combinatorics, the Prime Number Theorem, and Elliptic Curves. It’s hard to argue that any of these are not truly important developments over the last century. Francis Bonahon’s article on “The Hyperbolic Revolution” is a delight and easy to read: from torus knots to Thurston’s Geometrization theorem in 10 pages or less. Likewise, Alexander and Devaney’s article on complex dynamics delivers a strikingly beautiful and simple explanation of progress in that field (and an interesting view of how the ties between the German mathematical community and the nascent community in America came about).

The expository articles are definitely pitched at differing levels of rigor: Andrew Granville’s digression on the Prime Number Theorem(s) is full of classical analysis (both real and complex) and probably not considered “poolside reading”. Joe Silverman’s article on elliptic curves is thorough and accessible to the casual reader. In particular, a motivated undergraduate could learn much about the use of zeta functions, algebraic cryptology and modular functions from it without too much difficulty. Frank Morgan’s piece on milestones in Geometry addresses the exciting work of Perelman on the Poincaré Conjecture, Honeycombs, the Kelvin Conjecture and the Double Bubble Theorem. Eric Egge’s article on the Stanley-Wilf Conjecture is a charming story of the limits of computational complexity and asymptotic growth of k-stack sortable sequences. In several of these articles one notices the recurring roles played by research by undergraduates and intelligent use of technology (alongside some very classical and concrete mathematics — no sheaves or ramified blue transcendent categorical jimjams in sight!).

The MAA’s mission to improve college-level teaching still remains at the forefront of its agenda. To that end this volume features Michael Starbird writing on Inquiry-Based Learning, the experience of being in a Math Circle from the Kaplans, Joe Gallian on undergraduate research in mathematics, and Paul Zorn on the Calculus Reform Movement.

In a splendid example of an enticing lecture, we have Gilbert Strang’s thoughtful article on several delightful ways to introduce the exponential function. This function is likely the first transcendental function encountered by students of calculus and Strang weaves a multicolored skein of the ways in which various starting points (as a solution to a differential equation, as the limit of a series of polynomials, as a full-fledged power series) can be used. The paths connecting these points are then exhibited in ways which will enlarge anyone’s understanding of mathematics.

Next up, Tom Banchoff writes of the early days of visualization tools for geometry and Jon Borwein exhibits his own powers of prognostication in forecasting the future of mathematics. Bonnie Gold leads us into the post-Gödelian universe and reviews the newer philosophies of mathematical knowledge and praxis. Gerald Alexanderson write of some of the famous expositions of old (E.T. Bell, Constance Reid’s books on Hilbert and Courant, G. H. Hardy, Pólya and more). I was especially pleased to see him mention Imre Lakatos’s book Proofs and Refutations. This book was a model for my own learning about conducting conversation-driven classes and should be read by every person who teaches mathematics. It deserves to be better known.

Perhaps my favorite chapters of the book were those which peeked into the lives of “giants” of MAA’s past. Ivan Niven, Ralph Boas, Leonard Gillman, Paul Halmos, George Pólya, Lida Barrett and Henry Alder are all given their own space. To see the human sides of such figures of yore is to be reassured about one’s own path through the discipline. Too many humanists choose to believe that all mathematicians must fall into some version of the wild-eyed professor who only has eyes for a Platonic ideal far removed from reality. Read these chapters and you will learn otherwise. Our discipline is a human and humane one. Lest anyone forget this, the Mathematical Association of America is there to remind us otherwise. What was begun all those years ago at AMS meetings in December 1915 has yielded much fruit indeed.

Jeff Ibbotson holds the Smith Teaching Chair in Mathematics at Phillips Exeter Academy.