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The Princeton Companion to Mathematics Format Kindle
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It is very well organized in order to lead you step by step from basic concepts to more complicated mathematics with a middle section that tells you everything you need to know about the people who made mathematics what it is today. In short, it is a must-have for anybody who's mathcurious : from novice to expert.
Beware though, it is not an introduction or a student module. It is not conceived to actually teach mathematics. It is a companion, a guide-through, not a textbook with keys at the end.
But on top of everything, the book is a wonder to the eye and fingers. It is marvelously edited and a piece of jewelry to stand on any bookshelf. A perfect gift for enthusiasts.
The physical book is also high quality material.
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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 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... ;-)
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.
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.
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