Read This!

The MAA Online book review column

Briefly Noted

March 2006

At this point, we are adding some 30 reviews a month to MAA Reviews. This column represents an idiosyncratic selection of some of the reviews that have appeared there (fairly) recently.

Paul Halmos's "automathography" is a classic of the genre. First published in 1985, it contains Halmos's memories of his (then) 50-year career as a mathematician, from the 1930s to the 1980s. This is an essentially unchanged reprint of the original Springer edition. The changes are cosmetic: the cover is prettier, it's a paperback, and a few of the photographs have been replaced or retouched.

Halmos's basic approach is made clear in the "overture":

Sure, I had parents (two) and wives (two, one at a time, the present one for forty years), and cats… I like Haydn, long walks, Nero Wolfe, and dark beer, and for a few years I tried TM. All that is true, but it's none of your business — that's not what this book is about.

Instead, the book is about his life as a mathematician among mathematicians. It shows us a little about how Halmos thinks about mathematics, about what interested and motivated him, and about how he interacted with others. It includes a lot of what might, somewhat uncharitably, be described as "gossip": stories and anecdotes about mathematics, mathematics departments, and mathematicians.

In my experience, mathematicians love this sort of thing. Those of my colleagues who have read this book have enjoyed it. My students have liked it much less, partly because they aren't that interested in the world of mathematics, partly because they feel "turned off" by what they describe as Halmos's "arrogance." I think what bothers them is Halmos's bluntness about what counts and what does not count as significant in mathematics. That Halmos's harshness is mostly directed at his own work doesn't change my students' assessment. If all this famous guy can say, after trying for fifty years, is "I want to be a mathematician," they argue, then we students have no chance at all.

Well, there's some truth to that impression. Very few of us can aspire to being as good as Paul Halmos. And the mathematics community does tend to have very high standards for "what counts," sometimes overly high standards. Outsiders find this strange. Robert Preston, writing in the New Yorker in 1992, said that "a certain impression I had of mathematicians was… that they spent immoderate amounts of time declaring each other's work trivial. " There isn't too much of this kind of thing in I Want to Be a Mathematician, but there certainly is some.

But Halmos's book was never really intended for outsiders. For us members of his community, who aspire to be mathematicians ourselves, it gives us a glimpse of what one particularly successful mathematical life was like, shows us a little about what our community was like in the past, and yes, shares some juicy gossip.

[Fernando Q. Gouvêa; posted to MAA Reviews 01/27/2006]

For over 50 years, the MAA has sponsored the American High School Mathematics Exam (AHSME) and its successor, the American Mathematics Competitions (AMC 10/12.) These contests involve thousands of students throughout the country. Students who do well at this level can move on to higher level competition in the American Invitational Mathematics Exam (AIME) and the USA and International Mathematical Olympiads. In order to simplify the grading process, the exam is given in a multiple choice format, with penalties for incorrect answers.

This volume is the seventh in a series of books containing problems and solutions from the AHSME and AMC 10/12 contests. The book contains 275 multiple choice problems and complete solutions with explanations. The problems are taken from the 1995 through 1999 AHSME exams and the 2000 AMC 10/12 exams. In addition to the AHSME and AMC problems, the book includes a collection of 23 additional open ended problems that are not in the multiple choice format. The editor has also included a detailed classification that makes it easy to find problems on particular topics.

This book is an important reference for high school students and their coaches preparing for the AMC. Although the problems are generally much less challenging than problems in other mathematics competitions, this book may also be of some use to students preparing for other mathematics competitions.

[Brian Borchers; posted to MAA Reviews 2/27/2006]

Statistics is all around us. Just by going to work in the morning we can observe numerous examples of the use of statistics and immediately think of methods/models to analyze the observed data. Freedman'sgreat qualityas an author is the ability to provide you with this statistical vision, if you don't posses it already.

This book is truly an eye opener. It provides essential rigorous insight into statistical modeling. It differs from others in many aspects. Most statistics books, especially the more technical ones, are filled with theorems, proofs and examples you will never encounter in practice, either because of they are too simple or because they are extremely complex and are there to serve only as counterexamples. By contrast, this book provides real examples taken from real studies. The theorems and the corresponding proofs are presented in an elegant and intelligent manner. The author answers the questions the reader/researcher should ask. Among modeling books, this one is a gem.

The topics covered are the usual ones, such as the MLE, logit/probit modeling, path modeling and bootstrap, among the others. All of these are more or less familiar to every statistician and statistics student. Statistical Models is intended as a second course, so it builds onthe standard introductory material, but adding something special: of the theory and complexity of the subject. The author is very careful in presenting the theoretical ideas. He strives to explain almost all the bits and pieces. Rather than just presentinga theorem and its proof he gives the reasoning behind it. This is what should be highly appreciated. The more you read on, the morethis bookslooks like a step-by-step guide to statistical modeling. The writing is so clear and attractive that it doesn't allow you to get confused or lost so that you would stop reading.

I think that the most important part of the book (and generally where the most understanding will potentially come from) are the exercises. These are truly teaching exercises. If you take this book on as means for a second course in statistics by just simply reading the book without doing the exercises you will not get far. It is hard to describe the form of the exercises. Some concentrate on the basic understanding of the subject with questions (very good for class discussions) like "is MLE biased on unbiased?", some are in the form of a study, some analyze the output of the model, some are proofs, etc. There are exercises for everyone's taste, so to speak. And, even better, there are solutions at the end of the book!

Who is this book for? It should definitely find its place on the graduate student's bookshelf as well as on the bookshelf of a serious statistical researcher. Having completed a serious first course in statistics and some linear models the book could be easily used for self-study.

It is definitely not enough to know just how to plug one model into the software and get its output. We also need the "insider information," and this is exactly what this book offers. In any case, it will definitely raise you to the next level.

[Ita Cirovic Donev; posted to MAA Reviews 1/26/2006]

Roger Baker and his co-translators have put us all in their debt. Everyone knows that Bernhard Riemann was one of the most important mathematicians of the 19th century, and that his influence has been considerable. Reading his papers, however, has been an option only for those who could read German. There were exceptions, of course, such as the papers on the zeta function (translated in H. M. Edwards' The Riemann Zeta Function) and on the foundations of geometry (translated in various different sources). But, as far as I know, this is the first translation of the whole (well, almost the whole) set of papers included in Weber's edition of Riemann's Gesammelte Mathematische Werke.

Baker, Orde, and Christenson have translated all the mathematical and physical papers in Weber's volume. Only three items have been omitted: an article on "The Mechanism of the Ear" and two fragments on philosophy. The numbering of the papers in Weber's edition is preserved. Weber's notes have been translated, and also Dedekind's essay on Riemann's life. Finally, notes on each of the papers have been added. These provide brief comments and pointers to the literature (both historical and mathematical).

The book is nicely, if simply, produced. My only wishes would have been a heavier paper stock for the cover and a non-computer modern font (but I know most of my colleagues don't much worry about the latter). The contents are what matters most, and those are very welcome indeed. This is an indispensable book. No library should be without it.

[Fernando Q. Gouvêa; posted to MAA Reviews 10/13/2005]

Publication Data

The Contest Problem Book: American Mathematics Competitions 1995-2000 Contests, Book VII, compiled and augmented by Harold B. Reiter. Mathematical Association of America, 2006. Paperback, 181 pp., $43.95. ISBN 0-88385-821-5.

Statistical Models: Theory and Practice, by David Freedman. Cambridge University Press, 2005. Paperback, 414 pp., $34.95. ISBN 0-521-67105-1.

Bernhard Riemann, Collected Papers, translated by Roger Baker, Charles Christenson and Henry Orde. Kendrick Press, 2004. Paperback, 555 pp., $60.00. ISBN 0-9740427-2-2.

Fernando Q. Gouvêa teaches at Colby College in Waterville, ME. He often asks his students to read books about mathematics and mathematicians. He describes himself as a "historian wannabe" and he thinks volumes of collected mathematical works are really cool.

Brian Borchers is a professor of Mathematics at the New Mexico Institute of Mining and Technology. His interests are in optimization and applications of optimization in parameter estimation and inverse problems.

Ita Cirovic Donev is a PhD candidate at the University of Zagreb. She hold a Masters degree in statistics from Rice University. Her main research areas are in mathematical finance; more precisely, statistical mehods of credit and market risk. Apart from the academic work she does consulting work for financial institutions.

Back Issues:

• April 1999: Reprints of "classics", books on applied mathematics.
• May 1999: Books on math and physics, both popular and technical.
• July 1999: Various kinds of geometry, and classics in applied mathematics.
• August 1999: What's happening, Dirichlet, and Cryptonomicon.
• October 1999: Graph theory sources, John von Neumann, and Noncommutative Geometry.
• November 1999: Number theory in nine chapters, mathematics at work, and Jacques Hadamard.
• February 2000: Fallacies, Flaws, and Flimflam, measuring the universe, and primary sources at USMA.
• March 2000: Cardano and Manifold: Time.
• May 2000: Grassmann, Smarandache, and a really big prime.
• June 2000: A problem book, a history book, a collection of biographies, and a historical survey of reciprocity laws.
• August 2000: acrostics, field theory, categories, and partial differential equations.
• September 2000: excursions, history, and some high-level expository writing.
• November 2000: a festival of math videos and the Fourier-analytic proof of quadratic reciprocity.
• December 2000: Analysis problems, mathematical mysteries, and industrial mathematics.
• January 2001: Euler, Kolmogorov, and Honsberger.
• March 2001: history of mathematics, in three flavors: Harriot, prime numbers, and geometrical exactness.
• May 2001: three very different "applied mathematics" books.
• August 2001: introductions to algebraic number theory and to topology, and problem books.
• November 2001: Wilbur Knorr, John H. Conway, and vector analysis.
• February 2002: biological time, celestial mechanics, and things rarely talked about.
• March 2002: history of recent mathematics and differential geometry.
• June 2002: mathematics in other cultures, Archimedes, probabilistic thinking, and Dover's Phoenix.
• August 2002: Euclid's Elements, Hurewicz's Lectures, and Olympiads 2001.
• October 2002: Algebra from A to Z, Problems in Probability, and a really old trigonometry textbook.
• December, 2002: group representation theory, Irish mathematicians, and integration.
• January, 2003: careers, logic, dueling idiots, and differential geometry.
• March, 2003: the Putnam, the Liber Abaci, and the Millennial Conference.
• April, 2003: analysis, reprints, and Rithmomachia.
• May, 2003: conversation starters, dissections, and Fibonacci numbers.
• July, 2003: mathematical circles, the Scottish Café, and the zeta function.
• September, 2003: history — an encyclopedia, historiography, and algebra.
• October, 2003: selected papers and second editions.
• December, 2003: calculus, relativity, and biology.
• February, 2004: cryptography, generating functions, Wallis, and Stirling.
• March, 2004: differential equations, fundamental groups, reprints, and spherical buildings.
• April, 2004: Ramanujan graphs, time, and Medieval Islam.
• May, 2004: History: of Greek mathematics, of mechanics, of tables, of Reverend Bayes.
• June, 2004: The Grothendieck-Serre correspondence, mathematical biology, olympiad problem books, and units.
• July, 2004: Groups, Probability, Olympiads, and Isaac Newton.
• October, 2004: Klein, dogs, rivers, physics, geometry.
• November, 2004: Karl Pearson, chessboard problems, John Wallis, and contemporary art.
• January, 2005: Abel's legacy, reductive groups, curriculum foundations, and irresistible integrals.
• February, 2005: Differential equations, graphs, optimization, algorithms... and arithmetic on function fields of characteristic p.
• March, 2005: William Rowan Hamilton, π, Adolph Zeising, and Jesuits.
• April, 2005: differential equations, abstract algebra, and two kinds of number theory.
• May, 2005: algebra, chance, variations, and Chinese.
• Late May, 2005: motives, historical sources, Lebesgue, Cauchy.
• July, 2005: calculus, control theory, instabilities, biostatistics.
• August, 2005: difference equations, Galois theory, algebra, martingales in finance.
• October, 2005: 4-manifolds, geometry, Turing, and statistics in sports.
• December, 2005: lattices, irrational numbers, topology, and linear models.
• January, 2006: algebra, geometry, bubbles, jaws, Greek mathematics, and quotations.
• February, 2006: classics in statistics and in mathematics education, a math league, and transcendence.

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Read This! is the MAA Online book review column. Contributions are welcome; contact the editor if you'd like to be one of our reviewers. Books for review should be sent to the editor: Fernando Gouv&ecric;a, Dept. of Mathematics, Colby College, Waterville, ME 04901. Publishers, please check our reviews information page.