Read This!

The MAA Online book review column

Briefly Noted

January 2003

Well, it's many years later, and here is the second edition of 101 Careers in Mathematics. On pages 66 and 67, you will find a profile of Yawa, who is now a senior associate at KKO and Associates, working on mathematical modeling related to transportation planning.

Of course, I'm bursting with pride to see her there. (There's one other Colby student in the book, too!) But perhaps her story can serve as a testament to the value of this book. By giving many profiles of people who used their mathematics degrees in many different ways, it will open interesting and valuable possibilities for any student who reads it.

One minor complaint: I don't much like the title. First of all, there are more than 101 profiles here. Second, there is other stuff: several appendices with tips and hints on finding jobs. So why 101? I have no idea. Surely not because of the dalmatians?

In one sentence: every mathematics department should have one of these lying around in its common room for the students to browse. [Fernando Q. Gouvêa]

In the Preface of their First Course in Mathematical Logic, the authors state the following. "The scope of the present volume comprises the sentential theory of inference, inference with universal quantifiers, and applications of the theory of inference developed to the elementary theory of commutative groups, or, as it is put in the text, to the theory of addition. Because of the complexities introduced by existential quantifiers, their consideration has been delayed and will be given in the subsequent volume, A Second Course in Mathematical Logic." Checking Suppes' website, where his complete list of publications is given, I was unable to detect the indicated sequel: a mathematical tradition holds! But this book certainly stands on its own as a very solid introduction to mathematical logic, as distinguished from "logic for mathematicians."

There is no doubt that the authors have much more in mind than simply to expose the reader to the basic logic tool-kit every mathematician needs: it's very much the case that the reader should be, if not an aspiring mathematical logician, then certainly keenly interested in formal logic for its own sake. After all, how many of us would know when properly to use the descriptive phrase "modus ponendo ponens" (hint: we do this almost all the time), or "modus tollendo tollens" for that matter (it's contraposition, essentially). So there is a substantial amount of logician's arcana to be had in the book --- and one wouldn't expect anything else from Suppes.

The book of course succeeds beautifully in what it sets out to do: it has stood the test of time very well indeed (having first appeared in 1964). It has a lot of good exercises and is very thorough. It is fine for self-study as well as for use in a (somewhat reactionary) beginning course on mathematical logic proper, both for mathematicians and philosophers of the right disposition. It's a very nice book! [Michael Berg]

Another belated notice, but hey, if the publishers don't send 'em, we can't review 'em. Sometimes we aren't even aware they exist!

OK. Paul Nahin is the author of An Imaginary Tale: the story of the square root of -1, reviewed here several years ago. Ed Sandifer found it fascinating but demanding: "Like complex numbers themselves, this book has two parts. The first half of Nahin's book is a pleasant and anecdotal introduction to complex numbers, full of ideas and stories that are seldom seen in the popular literature. The second half requires a good deal more concentration, and, appropriately, leads us into some greater complexities."

Nahin's Dueling Idiots and Other Probability Puzzlers is also a blend of the pleasant and anecdotal and the more technical, but the mix is more uniform. The book is a collection of puzzles and problems in probability. There are 21 problems, from "How to Ask an Embarassing Question" (how to get honest answers to a question that people may be embarassed to answer truthfully) to "When Theory Fails, There is Always the Computer" (simulate away!). Each problem is stated in a short essay. The solutions of the problems follow, also embedded in short essays. In all, stating and explaining the problems takes 80 pages, and the solutions use up 94 pages. At the back of the book, there is a longer essay on "random" number generators. There are also many pages of program listings for the many MATLAB programs the author uses.

Nahin is an engineer, and it shows. The approach and the problems are practical and down-to-earth. There are lots of formulas and equations, so this is not really a book for the mythical "general" reader, but it is a book that people who aren't afraid of mathematics and who like problems will enjoy. [Fernando Q. Gouvêa]

Though I am definitely not an expert on the subject, I find elementary differential geometry fascinating and I love to teach it. Whenever I do, however, I find I have to make up my mind on a very basic question. Modern differential geometry is built upon a very elaborate theoretical framework (from differential forms all the way to connections and cohomology) and a correspondingly elaborate notation. It seems necessary to avoid most of this theoretical baggage in an undergraduate course. At the same time, one should try to teach a course that will prepare students for future courses, that includes some points of contact that can help students deal with the heavier notation they may meet in future courses.

When I taught the course this Fall, I used Andrew Pressley's newish book, Elementary Differential Geometry. Pressley takes the simplest route with respect to all the technical setup: avoid all of it. Instead of covariant derivatives, use derivatives with respect to local coordinates. Use moving frames without mentioning connections. Mention the Christoffel symbols very quickly, but don't do very much with them. For the most part, it works.

One place where this approach runs into problems is with respect to the Weingarten operator (aka the shape operator). There's no simple way to define this in Pressley's setup, so it ends up appearing only after quite a lot of buildup, and basically as a product of two matrices that just happens to include a lot of information. I felt it would have been better to actually introduce the covariant derivative and define the Weingarten operator properly. (And that's what I did in class.) The other problem with this section is the strangely uneven use of linear algebra. Pressley uses matrices and ends up appealing to a big theorem (self-adjoint operators have real eigenvalues), but he seems to avoid using linear transformations directly. Since my students did know what linear transformations were, I used that language; on the other hand, since they had never seen the big theorem, I presented an easy proof.

The section on geodesics has essentially nothing on parallel transport, which is a pity. On the other hand, the book does include several versions of the Gauss-Bonnett theorem, allowing the professor to end the course with a bang. (This may require some judicious skipping of earlier sections.) All in all, I was quite happy with the book. [Fernando Q. Gouvêa]

Publication Data

A First Course in Mathematical Logic, by Patrick Suppes and Shirley A. Hill. Dover Publications, 2002. Softcover, 288pp., $12.95. ISBN: 0486422593.

Dueling Idiots and Other Probability Puzzlers, by Paul Nahin. Princeton University Press, 2000 (paperback with new preface, 2002). Paperback, 269 pp., $18.95. ISBN 0-691-10286-4.

Elementary Differential Geometry, by Andrew Pressley. Springer Undergraduate Mathematics Series, Springer-Verlag, 2002. Paperback, 336 pp., $39.95. ISBN 1-85233-152-6.

Michael Berg (mberg@lmu.edu) is professor of mathematics at Loyola Marymount University in Los Angeles, CA. His research interests are algebraic number theory and non-archimedian Fourier analysis.

Fernando Q. Gouvêa (fqgouvea@colby.edu) is the author of several books, including, most recently, Math through the Ages, written in collaboration with William Berlinghoff.

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.

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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 Math&CS, Colby College, Waterville, ME 04901. Publishers, please check our reviews information page.