| | The Princeton Companion to Mathematics |
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 2008-12-24 - Frolicking with Bourbaki
Even more than the program for the 2009 meeting of the American Mathematical Society, this book exhibits the Great Divide in today's mathematics.
On one side of the divide we have mathematicians playing a parlor game called Bourbaki with other mathematicians. The game goes like this. Think up definitions for a handful of cute nouns, verbs, adjectives and adverbs: oblate corkscrewed doubly-banded sub-farkleoid. Now write a stream of papers classifying the nouns with their adjectives and composing the verbs with their adverbs. Tenure is only another 10 turns of the crank away. This mathematics 100% content-free; synthetic problems worked in synthetic settings.
On the other side of the Great Divide we have mathematicians talking to students. Here we have politically correct (or at least not politically incorrect) problems cast in severely dumbed-down non-Bourbaki mathematics. Have the student push the numbers around for an hour or two and mix in a couple x's and y's until they start to feel really good about saving the world from something.
As an aside, it is obvious why we aren't interesting students in becoming mathematicians. The dull students don't think they'll ever be able to play Bourbaki. The smart ones don't ever want to.
Except for Part VII, The Influence of Mathematics, the Companion is all on the Bourbaki side of the Great Divide. This is not to say that there aren't some execellent sections. When an author really knows the subject you become convinced as you read that you understand it too. Barry Mazur on Algebraic Numbers is a wonderful example as is Computational Complexity by Oded Goldreich and Avi Wigderson. And the biographies in Part VI are by and large written with a light yet informative and insightful touch.
But there is so much drivel, dross and heavy breathing. The Companion is almost entirely unedited and unsupervised meanderings of second and third rate mathematicians trying desperately to convince their peers that they can play Bourbaki with the best of them. Who knows? They probably can. And that is the failing of this book.
Read Ball's History of Mathematics, Whittaker and Watson's Modern Analysis, and Aigner and Ziegler's Proofs from the BOOK and you'll know more about what mathematics is than this book will ever tell you.
 2008-12-16 - math
This book has a lot of information concerning mathematics, both of a historical and a practical nature. Everyone can learn something from reading the book.
2008-11-08 - A major event in mathematical publishing
The Princeton Companion to Mathematics is such an extraordinary book that I am still amazed that the chief editor, Timothy Gowers, managed to pull it off. The renowned mathematician Doron Zeilberger announced that if he could take only one book with him to a desert island, it would be the Princeton Companion to Mathematics.
Why such high praise? Simply put, the PCM gives a single-volume overview of all of pure mathematics, with a clarity and coherence that cannot be found anywhere else. To be sure, there do exist several good books on the history of mathematics that give a good overview of elementary mathematics and introduce the reader to some of the great mathematicians of the past. There also exist excellent "popular science" books by writers such as Martin Gardner and Ian Stewart, that explain selected topics in advanced mathematics to the lay reader in an engaging and clear manner. And there are also encyclopedias (including Wikipedia) that delineate the main branches of mathematics and give succinct definitions of all the main concepts. But only the PCM does all of these things at once, in only a thousand pages.
The PCM is all things to all people. If your mathematical background is limited, you can still learn a great deal from the more elementary sections of the book, as well as from the biographical sketches of nearly a hundred famous mathematicians of the past. At the other end of the scale, even professional mathematicians will learn something from the articles on branches of mathematics other than their own specialty. Gowers made a systematic effort to find contributors who are not only world experts in their subject, but who write extremely well. He also forced the contributors to write in as accessible and elementary a manner as possible. The result is that even highly abstruse areas of mathematics are explained here with a clarity that is difficult to find anywhere else in the mathematical literature. The PCM is thus especially valuable to mathematics majors and graduate students.
Despite the ambitious scope of the book, it retains a strong sense of unity and coherence, by consistently emphasizing the forest rather than the trees. It also gives the reader a holistic view of mathematics by devoting different sections of the book to different perspectives on the subject. For example, one section organizes mathematics by sub-discipline, while another section highlights the main results and open problems of mathematics, while yet another section picks out the most important concepts. By putting all these aspects together in one volume, the PCM gives the reader a bird's-eye view of the whole subject that is not available from Wikipedia or from a shelf full of popular books on disparate topics.
The PCM is so well-written that it can be read either cover-to-cover, or browsed at random, or consulted as a reference when needed.
One word of warning: As Gowers himself notes, the book would be more accurately titled, "The Princeton Companion to Pure Mathematics." While applications of mathematics to other fields are touched on briefly, Gowers consciously limited the book primarily to pure mathematics, in order to keep the scope of the book manageable.
Should you still have doubts about the book, you can browse parts of the book for free: Selections from the book may be found at the book's official website, and many of the contributing mathematicians have posted their own sections on their own websites (you can find these easily using Google). And for more reviews of the book, see Gowers's blog.
 2008-10-07 - Level please?
I am not rating this book, in any strong sense, having never read it. My rating goes more to the book descriptions. The descriptions are intriguing, but lacking in a way most books on various subjects within mathematics are.
I want to know what level it is written at. I am currently in my first semester of calculus, and I intend to major in mathematics education. But is this a book I could understand most of now? Some of? Better wait a few years before getting?
2008-10-05 - very useful but not perfect
This is a wonderful book trying to offer a spherical view of mathematics.
In general it is quite successful in that. There are however, a number
of deficiencies:
(a) there is no special chapter on mathematical physics, which is quite
strange (to say the least) for such an enormous in importance branch of mathematics. Physics is found under "mirror symmetry" and "vertex
operator algebras" which is strangely inadequate; I could not find
a discussion of mathematical physics as a mathematics discipline with its own distinguishing character and concepts at an abstract enough level to
see its deep connections with geometry;
(b) similarly, there is no clear, deep and unifying
concept and description of mathematical informatics in this book; this
topic is described in separate non-connected chapters with computational
informatics (called "theoretical computer science" in this book) that deals with information processing discussed quite separately from information communication (in the end of the book), as for the topic of
storage/retrieval informatics, this is completely absent (see the "geometry of information retrieval" by Rijsbergen for a nice intro to the relevant mathematics there);
(c) lots of space spent on mathematicians and other topics towards the end of the book could have been used in order to cover more essential topics that inform the reader about what he wants to know
when he buys this book, i.e. the concepts and methods of mathematics.
Certainly not perfect, but enjoyable and highly recommended.
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