Customer Reviews

The Princeton Companion to Mathematics

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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.

This is a wonderful book. I have the Kindle version, but disappointingly found that the
purely mathematical portions, i.e., equations, etc. has not been incorporated into the text.
Equations, etc. appear to be low resolution images that are barely readable and need to be
"double tapped" and then appear independent of the text and are nearly pix elated.
This is obviously an example of a great book that was converted to the e-book version
in haste and has proven an obstacle to reading it in this format.
Too bad because this practice will set back adoption of the e-book revolution.
My advice: Do not buy it in the Kindle format.

Take Gowers's delightful little book, "Mathematics: a very short introduction", make it about twenty times as long, bring in a host of excellent contributors to write specialized articles, put the whole thing together very nicely, and you have the present book.

This book is not an encyclopedia, but it does offer a sweeping panorama of mathematics, written at an accessible level. It includes introductory articles on what mathematics is and basic concepts, more advanced (but still accessible) articles introducing various key concepts and areas of mathematics, articles on history of mathematics and biographies of mathematicians, descriptions of key theorems and problems, essays on the applications of mathematics, and more. There is something in here for everyone with an interest in mathematics.

As a professional mathematician, I am familiar with most of the introductory material, but I still like seeing it so nicely expressed and might use it as a teaching resource. Among the more advanced articles, there is lots of material which I feel like I "should" know, but actually don't.

The editors did an amazing job of finding really top-level people to write the specialized articles, who are both renowned experts in their areas and excellent expositors. The quality of the writing is infinitely superior to most articles in wikipedia or other online math encylopedias.

As I said, this not a comprehensive reference. The articles are introductory and designed for "bedtime reading". (Although if you read this book in bed you will probably have to sit up and put it on your lap because it is as big as a phone book.)

Anyway, I was very pleasantly surprised when I received this book. I expect to spend lots of time in the next few months browsing through it to brush up on my basic mathematical literacy. I think it will be even more useful for undergraduate mathematics students who want a good overview of what mathematics is about.

UPDATE: There is a useful page of errata, and discussion thereof, on Gowers's weblog.

Got my copy a week ago. What an exceptional book! Any of the random samples I read so far provides a informative, yet pleasant read. Gowers (Rouse Ball Professor of Mathematics in Cambridge) did a fantastic job in editing the many articles into a coherent and surprisingly accessible overview of modern mathematics. From inception to publication of this book took Gowers and his associate editors some 6 years. The amount of editorial attention given to this publication clearly shows and translated into a book that is - unlike any other math book I know of - easy to read and of high quality.

This book provides lots of material that is of interest to non-mathematicians. As is mentioned in one of the other reviews here, this heavy volume does not contain a separate chapter on mathematical physics, yet as a physicist I found lots of material directly relevant to physics. There is a very interesting chapter on the general theory of relativity, and lots of material on quantum mechanics. Also fundamental concepts highly relevant in physics such as spherical harmonics, dynamical systems, deterministic chaotic behavior, phase transitions, Lie groups, etc. are covered in inviting shorter sections. Each of the subjects is introduced in such a way that the reader first gains an intuitive understanding of the concept, that subsequently gets deepened via a more rigorous approach.

If only there was a similar 'companion' to modern physics! (The book of Oxford's Emeritus Rouse Ball professor Roger Penrose, The Road to Reality: A Complete Guide to the Laws of the Universe comes close, but falls short of being truly PCM's equivalent in physics.)

If you're interested in math, don't hesitate and buy this book. (And be quick: I bought it here at Amazon for just over US$71. In the meantime, the price has increased already by more than US$5... ;-)

I have both the hard copy and the Kindle copy. I love the hard copy. However, I don't see well, so I bought the Kindle copy, thinking that the high Kindle price meant that the many equations involving mathemathical symbols would re-size the symbol fonts along with the regular text. Wrong. These equations still appear as nearly microsopic smudges on the screen, regardless of how large one makes the regular text font size. The same defect remains on the PC screen, so that Kindle-PC views are just as unreadable. For a book with this cost? Whether on the Kindle DX screen or my 27" PC monitor screen? From Princeton? Unacceptable.

This is a review of the Kindle version of the book. I own the physical version as well, and heartily agree with reviews of the overall excellent content, clear explications and wonderful cross-references in the original volume.

I do not give the Kindle edition five stars for two reasons: The frequent and highly relevant cross-references, which are ripe for an electronic hypertext cross-linking capability, are not at all exploited on the Kindle. This is, I am certain, far more the fault of the publisher than the Kindle device itself. One of the things I was looking forward to with the ebook version was being able to skip around dynamically, guided by the cross-linked articles. Now, that is very difficult. The only way to accomplish it is via manual searches.

Also, many of the more esoteric symbols, such as matricies or entities with super/sub-scripts seem to be "washed out". This isn't a terrible issue, as they are still legible.

Overall, the lack of hyperlinked cross-references is the most significant shortcoming.

For such an expensive ebook, one would have expected more attention to detail.

I learned about this book while I was reading a blog post on Timothy Gower's first blog post.

I have read quite a bit and I think is a wonderful book. While it does lack a lot of detail. It does excel at giving a general view at what mathematics is all about.

If you ever wonder

What is mathematics?

or what do mathematicians do. This book is a good start. The book is a compilation of essays in different topics in mathematics many written by first class mathematicians including Timothy Gowers and Terence Tao both recipient of the Fields Medal in mathematics the equivalent to the Nobel prize and many others.

I believe the book should be part of any mathematicians or aspiring mathematician library.

The book covers mathematical history and also mathematics itself. Great parts of the book could be read by high school students but for some other parts is necesary to have at least and undergrad degree to be able to understand it. I wonder if a new Ramanujan found this book if he will be able to reivent the whole of mathematics from this book?

Someone asked in Prof. Gower's Blog What knowlege prerequisites are required to be able to profit from this book and what follows is part of the answer that professor Gower's gives and I believe it could be also very useful to others considering buying this book.

"That's a good question and one that doesn't have a straightforward answer. When we started out on the book, we hoped to have a more or less uniform level of difficulty throughout. But it fairly soon became clear that that was not practical, since some parts of mathematics are much harder to explain than others, so we modified our goal to one of trying to explain everything as accessibly as possible (and ideally more accessibly than one could easily find elsewhere), even if that level of accessibility varied from article to article.

The result? I would say that if you have done high-school mathematics and were good at it, then you will understand at least some of the book, enough to make it worth reading if you have a genuine interest in the subject, a wish to learn more, and a willingness to think quite hard as you read. If you are taking university courses in the subject, then the proportion you will understand will be much higher: some parts will be heavy going, but other parts will give you very useful insights into the concepts that are being thrown at you all the time. And if you are a graduate student or professional mathematician, then the book will be a very helpful resource if you are interested in getting at least some understanding of parts of mathematics that are not your own speciality."

My simplest advice is

If you love mathematics or like mathematics this is a book to get and treasure!

I first heard about this book on Gowers' blog, where he linked to a collection of sample articles (to read some of them, just google the title and explore the first few links). When I began reading the articles, I had no intention of buying the book. But from the first few articles, I was awed by the clarity of exposition, and by the frequency of deep and illuminating insights. Very soon, I ordered Princeton Companion to Mathematics (PCM) from Amazon. This book has far exceeded my expectations.

The place where PCM really shines is mathematical exposition using non-technical language. Don't mistakenly think the minimum of equations means the book shies away from very deep math. On the contrary. I think non-technical language emphasizes the big picture and the connections between seemingly disparate areas. Just for context, I am a math professor at a research school. I got my bachelor's and master's in math, and PhD in theoretical computer science (don't worry, though; this book offers a lot for an undergrad, or even a motivated high school student). Although grad school taught me a lot, that experience was more about depth. Reading PCM has greatly increased my breadth. I can't say enough good things about this book; it's perhaps my favorite book ever written about mathematics (compare with, "Proofs from the book"; but PCM is accessible to a wider audience). I wish I could rate it 6 stars out of 5.

When I run across a mathematical term or topic that I don't understand, my first stop typically is Wikipedia, sometimes Wolfram Mathworld. But the articles there don't always contain the best writing for my interest. For example, I may be left wondering what a "sheaf" is really all about. What was the motivation to create it? What is the true essence of it? I can follow the hyperlinks in Wikipedia and learn that the "elephant has a trunk" from one article and that it has a tail from another article and yet another article may talk about the legs. But I may still not really get it: What is the "elephant" (or sheaf or Teichmüller space or...) really all about? Wikipedia can sometimes be too fragmented and, ironically, too technical. It tries to be accurate and detailed and founded on authoritative references but it isn't always the best source to get the intuition about something, at least for mathematical topics.

Some might say you should make the investment and get a book on the topic you want to know more deeply. Yes, I could buy a book about algebraic geometry. Indeed, I did. But, typically, a book requires a pretty big investment of time and focus to work through. And, without the intuition up front, it isn't clear that the payoff at the end will be worth it.

The Princeton Companion fills the gap. The focus is on providing a "feel" for topics ranging from elementary to very advanced: motivations, simple examples, intuitive exposition. There is not an emphasis on proofs or completeness. One aspect I really like is that for any particular topic at random, it typically contains the full range of the topic (from simple to advanced), so that no matter what your current understanding is, you can find a place within the narrative to connect with the material and then go from there. The authors seem to have taken the view of extreme editing, leaving out as much as they can, so that what they leave in is a kind of distillation of the essence of each topic. What is left is fairly well integrated. As a result, reading it is like being on a grand tour of mathematics: You get to sample the best restaurants, see the most beautiful art, wander the nicest shops, without having to commit to living there full-time.

A superb and fascinating collection of articles on mathematics topics, somehow retaining clarity for the interested layman, to the student, to the professional. An absolute necessity for anyone studying math as well.

Everything about this book is great. The writing quality is remarkably high. The production quality is excellent. There is a sampling of interesting stuff in the articles for everyone.

I really, really wish I had had a book like this in college. One can get a clearer picture of some fields just by reading its three-page summaries than by taking a whole course in many cases.

Not sure what else to say, just that the book is wonderful.