All I Want Are Some (Math) Books for the Holidays...

by Fernando Q. Gouvêa

"Mathematicians love books," said Paul Halmos in his autobiography, I Want to be a Mathematician. And it's true; we do. We like to read them, and we also like to press them on others, in the hopes that they'll come to love our subject as much as we do.

But which books should we give? There are lots of math books out there (our Read This! column has reviewed more than 400 books over the last few years), so it's hard to choose. Well, perhaps some "specialists" can help. I contacted several veteran MAA Online reviewers and the members of the FOCUS/MAA Online Editorial Board, and asked them for suggestions of two kinds. First, I asked them to suggest a book that might serve as a gift for their favorite mathematician. Second, I asked for a book that might be given to an interested non-mathematician. This article collects the results of this thoroughly unscientific survey, with commentary. When possible, we have provided a link to MAA Online review of the book; if one was not available, we linked to the publishers' web site (if we could locate it). So as to not play favorites with online book sellers, we've left it up to our readers to choose one.

For Your Favorite Mathematician

What should you give your favorite mathematician? The responses ranged all the way from quite technical to more popular books. David Roberts told us that this Christmas one of his presents will be The Grothendieck-Serre Correspondence, which has just been published in a dual-language edition, the original French and an English translation appearing on facing pages. (I own a copy of the original French edition, so I agree that this is a book worth having, both for the mathematical content of the letters and for its value as a historical document.) David acknowledges that this is a rather specialized choice, but points out that "the giver might want to ferret out the preferences of the mathematician recipient before making a choice."

Michael Berg asked himself, "what would I want?" The answer, he says, is the Collected Works of Jean-Pierre Serre. He adds, "I hope this doesn't suggest that I am my own favorite mathematician..." Serre's works were originally published in four very expensive hardcover volumes, three of which seem to have recently been reprinted in paperback, so getting Michael the first three volumes wouldn't put too big a hole in one's budget.

Darren Glass also went for something technical: Number Fields, by Daniel Marcus. He said that

Not a month goes by that I don't recommend this book to someone. If you don't know any algebraic number theory, then this book is a very friendly introduction to the field, which doesn't get bogged down in trying to explain things in full generality and only deals with special cases, but which gives the flavor of the field. If you do know algebraic number theory, then this book serves as a nice compact reference for the statements and proofs of many theorems in the number field case. The author assumes little more than a good undergraduate algebra course, and as such I have seen this book be useful to advanced undergraduates, to masters students, to Ph.D. students, and to mathematicians in other areas. The exercises in the book are particularly nice.

Peter Ruane took us quite literally:

Dear Hermann Grassmann,

Since you were so passionate about all you did, and since your achievements were so remarkable, I send you my warm regards and best wishes for a Happy Christmas . Would contemporary mathematics be the same without your contributions? We may never know.

However, reading your work is so difficult for me — almost like trying to read the Rig Veda without the aid of the translation that you were the first to provide. Nonetheless, it's unfair to level the charge of impenetrability against your work because you seemingly constructed universal algebra with no mathematical predecessors upon whose work you could build.

Since many Christmasses have elapsed since your demise, a long overdue token of my appreciation is enclosed. It is a copy of H. G. Forder's (out-of-print) book that sought to develop your ideas and make them accessible to a wider audience. I didn't find this easy going either, but I'd appreciate your thoughts about the extent to which you feel that it encapsulates your work.

Here it is then, The Calculus of Extension by H. G. Forder (Cambridge University Press, 1941). Enjoy! (As they say in Wendy's breakfast-bars.)

An infinity of Happy New Years to my favourite mathematician, who now resides I know not where.

Peter Ruane

Several expository and/or "popular" mathematics books got the nod from our reviewers. Staying true to form, Ed Sandifer (MAA Online columnist and Euler fan extraordinaire) recommended William Dunham's Euler: The Master of Us All. He says it "has good mathematics, good history, wonderfully explained. Euler evaluates zeta(2), discovers generating functions to find partition numbers, factors huge numbers previously thought to be prime, lays the foundations of complex variables, and Dunham shows us how he did it."

Sharon Cutler-Ross suggested The Riemann Hypothesis, by Karl Sabbagh. She described it as "a nice, albeit somewhat quirky, presentation of the hypothesis' history and current status." Two other recent books also discuss the Riemann Hypothesis: Prime Obsession, by John Derbyshire, and The Music of the Primes, by Marcus du Sautoy. I think I'd choose Derbyshire for a more mathematical person, but all three books have had good reviews. An interesting recent article by Jordan Ellenberg discussing all three of these books and more appeared in the November 2003 issue of the literary journal The Believer.

Several other books of this type were mentioned. Herb Kasube suggested The Honors Class: Hilbert's Problems and their Solvers, by Benjamin Yandell, which he reviewed for MAA Online some time back. Herb also mentioned Four Colors Suffice: How the Map Problem was Solved, by Robin Wilson, about which he says, "I actually had trouble deciding in which category to put this wonderful book. It is written in a style that makes it extremely readable, even by the non-mathematician. Robin includes enough mathematics to give it credibility, but not so much as to turn away the non-mathematician. At the same time, every mathematician should read and appreciate this book."

Ioana Mihaila suggested that both mathematicians and non-mathematicians would enjoy Raymond Smullyan's puzzle books, particularly Satan, Cantor and Infinity, The Riddle of Scheherazade, and The Lady or the Tiger? and Other Logic Puzzles. Maria Fung recommended 100 Great Problems of Elementary Mathematics: Their History and Solution, by Heinrich Dörrie. She describes the book as "an enticing compendium of elegant and timeless mathematics."

While its title may lead one to presume that it is perhaps a collection of contest problems, it is in fact a survey of problems that have challenged the great mathematicians of history, including Archimedes, Newton, Gauss, Omar Khayyam, and Euler, among others. Its breadth and the self-contained, concise nature of its essays lend it to pleasant browsing and quick reference for essential ideas underlying famous results that many mathematicians have heard about but may not necessarily have studied in detail. It is particularly strong in algebra and number theory, as well as classical geometry. ... It would be a fine resource for math club seminar topics, and in usual Dover fashion, it is a true bargain.

For an Interested Non-Mathematician

Ed Sandifer made two different suggestions. For a mathematics student, he suggested William Dunham's Journey through Genius. "I've had my students read it several times," he says. "Several of them proclaimed it the best book they've ever read, not just the best mathematics book, but the best book ever." For someone who is not a mathematics student, Ed suggested E. T. Bell's Men of Mathematics. "Though he was blind to the fact that women do mathematics too, and though goes overboard in his enthusiasm and sometimes bends the facts a little bit, his love of mathematics is infectious."

Michael Berg also suggested something biographical (autobiographical, in fact): André Weil's Apprenticeship of a Mathematician, "because it's such a wonderful tale of what an actual (great) mathematician is like!" As an alternative in a similar spirit, he also mentioned Constance Reid's Hilbert.

As he did when asked about a book for a mathematician, David Roberts told us of an actual Christmas present. "I'm giving one of my teenage nephews [Timothy] Gowers' Mathematics: A Very Short Introduction," he says, pointing out that I didn't need to be told much more because I wrote the review for MAA Online. I'll add that this very nice little book also worked wonderfully when I asked my Abstract Algebra students to read it. Surprisingly, students seem to appreciate Gowers' philosophy that "it doesn't matter what it is, it only matters what it does."

We've already mentioned Robin Wilson's Four Colors Suffice: How the Map Problem was Solved, and noted that Herb Kasube felt it'd work for almost anyone, mathematician or not, interested in mathematics. Hortensia Soto-Johnson mentioned Life by the Numbers by Keith Devlin, adding that "it is an excellent book for anyone interested in mathematics. I also recommend it for high school mathematics teachers who want to portray the importance of mathematics." Darren Glass suggested Simon Singh's The Code Book:

Of all of the recent "pop-math" books, I think that this book is the one that is the best written. This book is a history of cryptography, which according to the subtitle ranges from ancient Egypt to quantum cryptography. Cryptography gives an easy entrance point to much of contemporary (and not-so-contemporary) mathematics, and so a reader who does not know mathematics is not left with the same questions of "why do we care?" which several of my non-mathematician friends have asked after reading the books on Fermat's Last Theorem or the Riemann Hypothesis. Singh has done an excellent job of covering a large number and variety of topics, and kept the book quite enjoyable.

Peter Ruane addressed his recommendation to world leaders:

To all leaders of involved in the World's centres of conflict,

I realise that you are taken up with political or religious imperatives and that these may require you to take up arms; but YOU, and WE, deserve some Yuletide respite from present day horrors.

Poetry, music or meditation may provide such relief, but what about a bit of mathematics? You may well be rusty or fearful of our subject, yet this need not be the case. For example, I invite you to have a look at this wonderful book by David Wells. It is called The Penguin Dictionary of Curious and Interesting Geometry and its title bears strict relationship to its contents.

Enticing diagrams, very clear commentary, many fascinating anecdotes and mini-biographies. If you don't enjoy this I'll be... well, I'll be absolutely amazed. The only pre-requisite for accessing this book is some capacity to visualize and some desire to enlarge your understanding of the nature of the world that seems to be at your command.

A peaceful Xmas to you all.

Peter Ruane

P.S.: Many previous political leaders have benefited from their excursions into mathematics. Just ask Napoleon or Thomas Jefferson etc.

Finally, two of our contributors suggested works of fiction. Sharon Cutler-Ross said that David Guedj's The Parrot's Theorem is "a fairly complete math history wrapped in a mystery that touches on what it means to do mathematics." Maria Fung recommended The Curious Incident of the Dog in the Night-time, by Mark Haddon, pointing to her MAA Online review for why. (Tidbit: the book "succeeds amazingly well at interweaving mathematics and mathematical modes of thought into a gripping story line that is surprisingly passionate in its relentlessly analytical tone.") Haddon's book has appeared in several lists of best books of the year.

And me?

And what would I recommend? I'll add two recommendations in each category. Using the "what would I like to get" criterion, I'd suggest the new MAA edition of G. Waldo Dunnington's Gauss: Titan of Science, particularly because of the valuable additions by Jeremy Gray. The Green Lion Press edition of Euclid's Elements would also make a handsome gift. Of course, if price were no object, what I'd really like would be the complete Series Prima of the Euler Opera Omnia!

For a non-mathematician, I can't help but recommend two books that had a crucial role in my own discovery of the glories and joys of mathematics. When my Uncle Clovis gave me a copy of Edward Kasner and James Newman's Mathematics and the Imagination, he probably didn't expect it to have the impact it had on me. It describes some fairly standard bits of mathematics (large numbers, transfinite numbers, an introduction to calculus, special constants like e and π) that show up in lots of popular mathematics books, but it is done with a verve and passion that I have never forgotten. (I still remember trying to explain to my friends about different sizes of infinite numbers. They must have thought I was off my rocker!) He was also the one who gave me the original edition (in Portuguese) of Malba Tahan's The Man Who Counted, a mock Arabian Nights adventure (written by a Brazilian high school mathematics teacher using an Arabic-sounding pseudonym). It's great fun, easy to read, and it gives the reader a good sense of the fascination of mathematics.