Read This!

The MAA Online book review column

Briefly Noted

January 2006

Alan Beardon's Algebra and Geometry introduces the concepts of algebra, primarily group theory and linear algebra, by focusing on classical geometric maps: symmetries, isometries, linear transformations, and Möbius transformations. Beardon's goal is to present some beautiful material in a way that emphasizes the unity of mathematics over the compartmentalization often found in mathematics curricula, and at a level that is accessible to a junior or senior mathematics major. It is a vision for a beautiful book but, unfortunately, he only partially succeeds in fulfilling it.

Naturally, the book does not cover all of the topics typically found in an undergraduate algebra or geometry text. Beardon focuses on groups and vector spaces. The basics of these subjects are introduced and then given meaning through geometry. For example, eigenvalues and orthogonal matrices are introduced so that the isometries of Euclidean space can be described; the group of Möbius transformations is shown to be isomorphic to the quotient of the special linear group (with complex entries) and the subgroup {±I}. Spherical geometry and Euler's formula for a sphere are introduced so that the Platonic solids can be identified.

The algebraic ideas are tied together nicely by the geometric theme. The algebraic concepts are all introduced for the purpose of describing or studying geometric maps: symmetries, isometries, etc. However, I see the book in this light only in hindsight. This focus did not become clear to me until near the end of the book and I do not believe that most students will see this common thread.

The material is accessible to upper-level undergraduate mathematics majors. The pace of the book is good: low dimensional cases are presented before the general n-dimensional case is presented; groups are introduced long before their properties are discussed. The exercises are a nice mix of concrete calculations and abstract proofs. The book, however, could have been more accessible to students. For a book with geometry in the title, there are very few pictures. The exposition would be greatly enhanced if there were more diagrams to go along with the written descriptions.

Furthermore, in some sections, particularly early in the book, there are decidedly few examples. For example, in the section in which groups are introduced, the only example is the group of linear functions on the reals under the operation of composition. Moreover, when examples are given, they tend to come after the theorems and their proofs. The example of the group of linear functions, for example, comes at the end of the section. The exercises do make up somewhat for the lack of examples, as they provide a number of concrete examples and calculations but, this is not the order for best learning the material. The examples should come first and then the theorems.

So I long for what might have been. Beardon has written a fine introductory text. It could have been great. Despite my longing for what might have been, let me encourage you to consider using Algebra and Geometry, if you do not want to teach a standard abstract algebra course. Beardon outlines a beautiful course.

[Stephen T. Ahearn; posted to MAA Reviews 01/04/2006]

This video production by Springer VideoMath is probably best understood as being a collection of nine short video productions on the general topic of numerical simulations. An added introduction and introductory skits performed by a robot actor bring these separate productions together to give the illusion of being a single production, but some lack of structural continuity is apparent. Although the video begins with an introduction to numerical simulation and a brief discussion of grids and quadrature, there is a sudden transition to advanced applications with no looking back to any basic tools of the trade.

Often, discussion of a complex application includes a quick overview of the simulation methods, but the emphasis here is on quick. There may be a brief review of some partial differential equations and a statement that the PDE's are solved numerically, but the viewer should not expect to gain any more than a cursory understanding of the methods or background of any particular simulation.

The viewer will, however, be introduced to the great power of numerical simulation and will, in a short time, be introduced to a dazzling array of significant successes in this field. From crystal formation to fluid flow through biofilms, from the control of robot arms and the modeling of adaptive automobile suspensions to the design of surgical procedures the viewer will be treated to a survey of a wide variety of applications.

The visual graphics make the production a pleasure to watch, but we may pause to reflect on what the educational value of this video might be. As stated before, no one will learn how to do numerical simulations by watching this video, but it is possible to learn to be interested in this important field of technology. The video could serve to motivate a young student to pursue a career in mathematical modeling and numerical simulations, but the file may be of more use for the crusty old-timer.

If there are any traditional scientists left who think that numerical simulation does not apply to their particular field then this video may help them to see that there could be some unexpected possibilities.

[Paul E. Cohen; posted to MAA Reviews 12/28/2005]

The twentieth century was a time of big changes in our understanding of Greek mathematics. Many parts of what David Fowler has called "the standard story" have been challenged, and the scholarly consensus has certainly changed as a result. Three examples will have to suffice:

1. The standard story suggested that the discovery of incommensurability represented a kind of "foundational crisis" for Greek mathematics, requiring a rethinking of, for example, the concept of ratio. This consensus has been challenged, and many scholars have come to doubt its correctness.
2. In the early 20th century, the consensus was that texts such as Book II of Euclid's Elements were best understood as "geometric algebra". This idea was sharply criticized by Sabetai Unguru in the 1960s and has largely been abandoned by historians (who are now more interested in understanding how the Greek approach differs from what later became known as "algebra").
3. The standard story paid little attention to social context. Historians today are interested in finding out how Greek mathematicians lived, what they did from day to day, and how the very sophisticated pure mathematics of the great texts got transmitted (or not) to scientists and practitioners.

In Classics in the History of Greek Mathematics, Christianidis has provided a very useful collection of papers that encompass many of these changes. Here one can find Unguru's original papers (and the angry responses they elicited), discussions of the supposed "crisis of foundations", papers on the origin of Greek axiomatics, on geometry and "algebra". The papers, alas, are in their original languages (German, French, and English), but anglophone readers can be reassured that most of the articles are in English, especially the more recent ones. In any case, this is an excellent place to go to learn what the historians have been up to, and the give-and-take of debate makes it fascinating. The book is probably too expensive for non-fanatic individual readers, but it is definitely a "must buy" for libraries.

[Fernando Q. Gouvêa; posted to MAA Reviews 1/11/2006]

I love quotes. For years, I have collected them, used them in my email "signature" file, included them in assignments, and generally annoyed people with them. My collection includes many quotes about mathematics, which range for the insightful to the spectacularly wrong-headed. (After all, as John Kenneth Galbraith once pointed out, "If all else fails, immortality can always be assured by spectacular error.") I'm happy to welcome Mathematically Speaking to my shelves.

The authors have put together a "dictionary of quotations" about mathematics in fairly standard format, which includes (thank goodness) a proper set of indices. Those indices alone mean that this book surpasses by far the competition. Memorabilia Mathematica and Out of the Mouths of Mathematicians are both useful collections, and 777 Mathematical Conversation Starters is fun, but when you're looking for a partly-remembered quote none of them is as easy to use as this book. In addition, the authors of Mathematically Speaking have tried to find precise sources for all their quotes. A bit too often, they give up and simply say "quoted by X in Y", but even then they still give Y as precisely as they can. Still, in these days of the internet and its free-ranging quotes, it's good to have even such "quoted by" sources indicated.

A lot depends, of course, on the authors' taste: they have to decide what quotes are worth quoting. Gaither and Cavazos-Gaither are pretty good, though the inevitable clinkers do get in. I'm not sure, for example, what Alan Turing meant by "Science is a differential equation. Religion is a boundary condition." Or what "The Cartesian criterion of truth" (that's the whole quote), by Unknown, source unknown, is doing here.

The authors describe their choices saying "Some of the quotations are profound, others are wise, some are witty, but none are frivolous." Well, too bad, I say. The frivolous ones are sometimes the best. The book does have a certain earnestness to it, though it includes Mae West's remark that "A figure with curves always offers a lot of interesting angles." (That's not frivolous?) Overall, the quotes are well chosen, and several are new to me. I'm glad to have the book. I think most of the readers of Read This! would enjoy it too.

[Fernando Q. Gouvêa; posted to MAA Reviews 1/11/2006]

Publication Data

Bubbles, Jaws, Moose Tests, and More: The Wonderful World of Numerical Simulation, by Hans-Joachim Bungartz, Ralf-Peter Mundani, and Anton Christian Frank. Springer VideoMATH Series. Springer Verlag, 2005. VHS/NTSC, $39.95. ISBN 3-540-21168-3. DVD Version, $39.95, ISBN 3-540-21167-5. VHS/PAL Version, $59.95, ISBN 3-540-21169-1.

Classics in the History of Greek Mathematics, ed. by Jean Christianidis. Boston Studies in The Philosophy of Science 240. Kluwer Academic Publishers, 2004. Hardcover, 461 pp., $187.00. ISBN 1-4020-0081-2.

Mathematically Speaking: A Dictionary of Quotations, ed. by C. C. Gaither and A. E. Cavazos-Gaither. Institute of Physics, 1998 (Distributed by CRC/Chapman&Hall). Paperback, 484 pp., $39.95. ISBN 0-7503-0503-7.

Stephen T. Ahearn (ahearn@macalester.edu) teaches mathematics at Macalester College in St. Paul, MN. His primary research interests are in algebraic topology and computational topology/geometry but allows himself to be distracted by other interesting topics as in his article "Tolstoy's Integration Metaphor from War and Peace." He also enjoys hiking, swimming, baking bread, and reading.

Paul Cohen received his Ph.D. from the University of Illinois, was appointed as a Member of the Institute for Advanced Study by Kurt Gödel, and has taught at the University of Tennessee and at Lehigh University. He currently lives in Maine and is teaching at Colby College.

Fernando Q. Gouvêa is Professor of Mathematics at Colby College and the co-author, with William P. Berlinghoff, of Math through the Ages. He somehow finds time to also be the editor of FOCUS Online and of MAA Reviews.

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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.