Every once in a while, I am struck by
how often mathematics textbooks sound just like each other. Glance at the
table of contents of your typical analysis or algebra textbook, and you can
be 90% sure of seing exactly the same sequence of topics each time. There
are honorable exceptions of course, and I'm always glad to see one, because
they indicate that (at least some) mathematicians are still actively
thinking about what should be taught, in what order, and how.So here is the first volume in the Princeton Lectures on Analysis, entitled Fourier Analysis: an Introduction and written by Elias M. Stein and Rami Shakarchi. The series wants to serve as an integrated introduction to "the core areas in analysis." The following volumes will treat complex analysis (volume 2), measure theory, integration, and hilbert spaces (volume 3), and selected other topics (volume 4). The basic pre-requisite for the series seems to be a standard undergraduate introduction to analysis covering the basic theory of convergence, derivatives, and the Riemann integral. Some basic familiarity with the complex numbers and elementary functions (e.g., complex exponentials) is assumed. So the book is aimed at graduate students and maybe advanced undergraduates.
The new series begins with Fourier analysis because the authors feel that this subject plays a central role in modern analysis and because it has played an important historical role. It is also much more concrete than abstract measure theory or functional analysis. Finally, the authors plan to use results from volume one in the following volumes, emphasizing that analysis is a coherent whole rather than a collection of disjointed topics.
The first book covers the basic theory of Fourier series, Fourier
transforms in one and more dimensions, and finite Fourier analysis. The
last topic allows the authors to present, as an application, the proof of
Dirichlet's theorem on primes in arithmetic progressions. The result would
make a great book for independent study courses with advanced
undergraduates, and, I think, would also be useful for graduate
courses. It's definitely worth a look.
[Fernando Q. Gouvêa]
Dover's new series of
Phoenix Editions is dedicated to providing
high-quality hardcover editions of books for which there has been a small
but steady demand. So far, Dover has shown excellent taste in its choice of
volumes, and I hope the mathematics community will support their effort to
keep these useful books in print.
One of the highlights, for me, of the latest batch of Phoenixes is Felix Klein's Lectures on the Icosahedron. Despite the old-fashioned language and notation, this remains an important book. It connects the icosahedron to group theory (via its symmetry group), to function theory (via elliptic and modular functions), and to solving the quintic. The book is historically important (anything written by Klein would be!), but it's also interesting because there are very few modern works dealing with this material (Jerry Shurman's Geometry of the Quintic is the only one I know of).
I'm also delighted to see Stillman Drake's
Galileo at Work: His Scientific Biography back in print. A historian
once told me that no one knew more about Galileo than Stillman Drake. Drake
wrote extensively about Galileo. Reacting to Koyré's depiction of
Galileo as a Platonist whose experiments were almost all thought
experiments, Drake dug up solid evidence of Galileo the
experimentalist. This book fits into that project. Starting from a very
careful investigation of Galileo's writings, letters, and (most
importantly) notes, Drake puts together a picture of Galileo "at work",
actually doing science.
Drake's scholarship is impressive and respected, so this is a book that is definitely worth reading. Of course, Drake's view of Galileo has its dubious points. At times he seems to protest too much, trying too hard to make Galileo be correct. Sometimes this is a valuable correction to scholars who are too quick to conclude that Galileo was wrong, but Drake comes out so consistently on the great man's side that one starts to be suspicious.
The other contentious aspect of Drake's picture of Galileo is his
insistence on the dramatic opposition between Galileo and the philosophers
of his time. In Drake's view, Galileo wanted to do a non-metaphysical kind
of science that makes no philosophical claims at all, which put him in
sharp opposition to the natural philosophers, who were interested in far
more than a merely descriptive account of how nature behaves. Whether or
not one agrees, Drake's account of Galileo's scientific life is essential
reading. [Fernando Q. Gouvêa]
Worthy reprints have been frequently
mentioned in this column, in part because I'm convinced that certain books
have permanent value and that publishers should be encouraged to keep them
in print. Here are two more examples.
Most algebraic number theorists have had occasion to refer to Helmut Koch's Galloissche Theorie der p-Erweiterungen, first published some thirty years ago and still an essential reference on the subject of p-extensions (that is, Galois extensions whose Galois groups are p-groups) of algebraic number fields. But readers like myself have found ourselves hobbled by (a) the difficulty of finding a copy and (b) the inadequacy of our German reading skills.
No more. Thanks to Franz Lemmermeyer, here is Koch's Galois Theory of p-Extensions in English, supplemented by a "Postscript" that summarizes in a few pages what has happened since 1969. This new edition will keep Koch's work available for many years yet, and I'm sure many readers will be grateful that Springer has sprung for it.
I'm also happy to see the second volume of the A K
Peters reprint of Winning Ways, by Elwyn R. Berlekamp, John
H. Conway, and Richard K. Guy. The original edition was in two volumes
(and, if memory serves me correctly, used a lot of color). It was quite
expensive and is now essentially impossible to get. (Searching for it on
abebooks.com, I found four copies of
the new edition but none at all of the original.) The new edition splits
each of the original volumes into two, and uses much less color. At almost
$40 a pop for each of the four volumes, it doesn't come out to be much less
expensive, but the new format at least allows one to buy it piecemeal.
The book itself is so well known that one hardly needs to comment. This is the book that created the modern theory of combinatorial games, a lively subject that continues to generate interest. (See for example Games of No Chance, reviewed here several years ago, and its sequel More Games of No Chance, reviewed here this month.) Volume one dealt with the basic theory of such games. This volume, subtitled "Change of Heart!", explores what happens when some of the rules established in that first book are broken. So we get discussions of loopy games and other infinite games, misère play, and so on. So: interesting mathematics, fun games, and a whole new area to explore make this an indispensable book. [Fernando Q. Gouvêa]
Rithmomachia. Say it with me, boys and
girls: Rith-mo-mach-i-a. It's a game, sort of like chess but involving
arithmetic. According to Thomas More, that's what they play in
Utopia instead of "gambling with dice or other such ruinous games."
That and a game about the contest between vices and virtues.
The game is played on an elongated chessboard. The pieces have numbers on them, chosen so that their ratios will exemplify the various categories of ratio discussed in Boethius' Arithmetic. Pieces are captured by setting up various arithmetical identities, so to play the game one has to be able to do mental calculation. The details of the rules varied, but the basic idea was always the same.
Sound boring? Well, I guess it does. But rithmomachia seems to have been quite popular with — or at least was played by — educated men in Northern Europe for 500 years, between the 11th and the 16th century. Whether they actually enjoyed it or just used it as a way to sharpen their arithmetic skills is anyone's guess.
Ann Moyer's book The Philosophers' Game is an account of rithmomachia and its connection to the teaching of arithmetic in the tradition of Boethius' book (which is in turn based on Nichomachus' Introduction to Arithmetic, so is a descendent of Pythagoreanism). It also includes an edition of one of the various books describing the game and its rules. (So if any of you out there is thinking of using the game to torture students, here's your chance.) Moyer thinks that the link between the game and Boethius is very important, and that the game started to lose its appeal exactly when the newer approaches to arithmetic and algebra began to become prevalent in the 16th and 17th centuries. The book has the potential to be the source of various interesting student activities or projects; consider asking your library to get a copy. [Fernando Q. Gouvêa]
Fourier Analysis: An Introduction, by Elias M. Stein and Rami Shakarchi. (Princeton Lectures in Analysis, volume one.) Princeton University Press, 2003. Hardcover, 320 pp., $49.95. ISBN 0-691-11384-X.
Lectures on the Icosahedron, by Felix Klein. Dover Phoenix, 2003. Hardcover, 304 pp., $42.50. ISBN 0-486-49528-0.
Galileo at Work: His Scientific Biography, by Stillman Drake. Dover Phoenix, 2003. Hardcover, 560 pp., $65.00. ISBN 0-486-49542-6.
Galois Theory of p-Extensions, by Helmut Koch. Springer Verlag, 2002. Hardcover, 190 pp., $65.00. ISBN 3-540-43629-4.
Winning Ways for Your Mathematical Plays, Volume 2, by Elwyn R. Berlekamp. A K Peters, 2003. Paperback, 212 pp., $39.00. ISBN 1-56881-142-X.
The Philosophers' Game: Rithmomachia in Medieval and Renaissance Europe, with an edition of Ralph Lever and William Fulke, "The Most Noble, Auncient, and Learned Playe" (1563), by Ann E. Moyer. University of Michigan Press, 2002. Hardcover, 232 pp., $60.00. ISBN 0-471-11228-7.
Fernando Q. Gouvêa (fqgouvea@colby.edu) reads science fiction, teaches at Colby College, edits FOCUS and MAA Online, and writes books, including, most recently, Math through the Ages, with William Berlinghoff.
|
Go to...
|
Find out...
|
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.
MAA Online is edited by Fernando Q. Gouvêa (fqgouvea@colby.edu). Last modified: Wed Apr 23 13:15:52 EDT 2003