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This is a one-of-a-kind reference for anyone with a serious interest in mathematics. Edited by Timothy Gowers, a recipient of the Fields Medal, it presents nearly two hundred entries, written especially for this book by some of the world's leading mathematicians, that introduce basic mathematical tools and vocabulary; trace the development of modern mathematics; explain essential terms and concepts; examine core ideas in major areas of mathematics; describe the achievements of scores of famous mathematicians; explore the impact of mathematics on other disciplines such as biology, finance, and music--and much, much more.
Unparalleled in its depth of coverage, The Princeton Companion to Mathematics surveys the most active and exciting branches of pure mathematics, providing the context and broad perspective that are vital at a time of increasing specialization in the field. Packed with information and presented in an accessible style, this is an indispensable resource for undergraduate and graduate students in mathematics as well as for researchers and scholars seeking to understand areas outside their specialties.
- Features nearly 200 entries, organized thematically and written by an international team of distinguished contributors
- Presents major ideas and branches of pure mathematics in a clear, accessible style
- Defines and explains important mathematical concepts, methods, theorems, and open problems
- Introduces the language of mathematics and the goals of mathematical research
- Covers number theory, algebra, analysis, geometry, logic, probability, and more
- Traces the history and development of modern mathematics
- Profiles more than ninety-five mathematicians who influenced those working today
- Explores the influence of mathematics on other disciplines
- Includes bibliographies, cross-references, and a comprehensive index
Contributors incude:
Graham Allan, Noga Alon, George Andrews, Tom Archibald, Sir Michael Atiyah, David Aubin, Joan Bagaria, Keith Ball, June Barrow-Green, Alan Beardon, David D. Ben-Zvi, Vitaly Bergelson, Nicholas Bingham, Béla Bollobás, Henk Bos, Bodil Branner, Martin R. Bridson, John P. Burgess, Kevin Buzzard, Peter J. Cameron, Jean-Luc Chabert, Eugenia Cheng, Clifford C. Cocks, Alain Connes, Leo Corry, Wolfgang Coy, Tony Crilly, Serafina Cuomo, Mihalis Dafermos, Partha Dasgupta, Ingrid Daubechies, Joseph W. Dauben, John W. Dawson Jr., Francois de Gandt, Persi Diaconis, Jordan S. Ellenberg, Lawrence C. Evans, Florence Fasanelli, Anita Burdman Feferman, Solomon Feferman, Charles Fefferman, Della Fenster, José Ferreirós, David Fisher, Terry Gannon, A. Gardiner, Charles C. Gillispie, Oded Goldreich, Catherine Goldstein, Fernando Q. Gouvêa, Timothy Gowers, Andrew Granville, Ivor Grattan-Guinness, Jeremy Gray, Ben Green, Ian Grojnowski, Niccolò Guicciardini, Michael Harris, Ulf Hashagen, Nigel Higson, Andrew Hodges, F. E. A. Johnson, Mark Joshi, Kiran S. Kedlaya, Frank Kelly, Sergiu Klainerman, Jon Kleinberg, Israel Kleiner, Jacek Klinowski, Eberhard Knobloch, János Kollár, T. W. Körner, Michael Krivelevich, Peter D. Lax, Imre Leader, Jean-François Le Gall, W. B. R. Lickorish, Martin W. Liebeck, Jesper Lützen, Des MacHale, Alan L. Mackay, Shahn Majid, Lech Maligranda, David Marker, Jean Mawhin, Barry Mazur, Dusa McDuff, Colin McLarty, Bojan Mohar, Peter M. Neumann, Catherine Nolan, James Norris, Brian Osserman, Richard S. Palais, Marco Panza, Karen Hunger Parshall, Gabriel P. Paternain, Jeanne Peiffer, Carl Pomerance, Helmut Pulte, Bruce Reed, Michael C. Reed, Adrian Rice, Eleanor Robson, Igor Rodnianski, John Roe, Mark Ronan, Edward Sandifer, Tilman Sauer, Norbert Schappacher, Andrzej Schinzel, Erhard Scholz, Reinhard Siegmund-Schultze, Gordon Slade, David J. Spiegelhalter, Jacqueline Stedall, Arild Stubhaug, Madhu Sudan, Terence Tao, Jamie Tappenden, C. H. Taubes, Rüdiger Thiele, Burt Totaro, Lloyd N. Trefethen, Dirk van Dalen, Richard Weber, Dominic Welsh, Avi Wigderson, Herbert Wilf, David Wilkins, B. Yandell, Eric Zaslow, Doron Zeilberger
- Sales Rank: #49939 in Books
- Published on: 2008-09-28
- Original language: English
- Number of items: 1
- Dimensions: 2.50" h x 8.10" w x 10.00" l, 5.70 pounds
- Binding: Hardcover
- 1034 pages
Review
Winner of the 2011 Euler Book Prize, Mathematical Association of America
Honorable Mention for the 2008 PROSE Award for Single Volume Reference/Science, Association of American Publishers
One of Choice's Outstanding Academic Titles for 2009
"The Princeton Companion to Mathematics makes a heroic attempt to keep [abstract concepts] to a minimum . . . and conveys the breadth, depth and diversity of mathematics. It is impressive and well written and it's good value for [the] money."--Ian Stewart, The Times
"This is a panoramic view of modern mathematics. It is tough going in some places, but much of it is surprisingly accessible. A must for budding number-crunchers."--The Economist (Best Books of 2008)
"Although the editors' original goal of text that could be understood by anyone with a good background in high school mathematics provided short-lived, this wide-ranging account should reward undergraduate and graduate students and anyone curious about math as well as help research mathematicians understand the work of their colleagues in other specialties. The editors note some advantages a carefully organized printed reference may enjoy over a collection of Web pages, and this impressive volume supports their claim."--Science
"This impressive book represents an extremely ambitious and, I might add, highly successful attempt by Timothy Gowers and his coeditors, June Barrow-Green and Imre Leader, to give a current account of the subject of mathematics. It has something for nearly everyone, from beginning students of mathematics who would like to get some sense of what the subject is all about, all the way to professional mathematicians who would like to get a better idea of what their colleagues are doing. . . . If I had to choose just one book in the world to give an interested reader some idea of the scope, goals and achievements of modern mathematics, without a doubt this would be the one. So try it. I guarantee you'll like it!"--American Scientist
"Accessible, technically precise and thorough account of all math's major aspects. Students of math will find this book a helpful reference for understanding their classes; students of everything else will find helpful guides to understanding how math describes it all."--Tom Siegfried, Science News
"Once in a while a book comes along that should be on every mathematician's bookshelf. This is such a book. Described as a 'companion', this 1000-page tome is an authoritative and informative reference work that is also highly pleasurable to dip into. Much of it can be read with benefit by undergraduate mathematicians, while there is a great deal to engage professional mathematicians of all persuasions."--Robin Wilson, London Mathematical Society
"Imagine taking an overview of elementary and advanced mathematics, a history of mathematics and mathematicians, and a mathematical encyclopedia and combining them all into one comprehensive reference book. That is what Timothy Gowers, the 1998 Fields Medal laureate, has successfully accomplished in compiling and editing The Princeton Companion to Mathematics. At more than 1,000 pages and with nearly 200 entries written by some of the leading mathematicians of our time and specialists in their fields, this book is a one-of-a-kind reference for all things mathematics."--Mathematics Teacher
"Overall [The Princeton Companion to Mathematics] is an enormous achievement for which the authors deserve to be thanked. It contains a wealth of material, much of a kind one would not find elsewhere, and can be enjoyed by readers with man different backgrounds."--Simon Donaldson, Notices of the American Mathematical Society
"This is an enormously ambitious book, full of beautiful things; I would wish to keep it on my bedside table, but that could only be possible relays, since of course it is far too large. . . . To sum up, [The Princeton Companion to Mathematics] is really excellent. I know of no book that will give a young student a better idea of what mathematics is about. I am certain that this is the only single book that is likely to tell me what my colleagues are doing."--Bryan Birch, Notices of the American Mathematical Society
"The book is so rich and yet it is well done. A rare achievement indeed!"--Gil Kalai, Notices of the American Mathematical Society
"My advice to you, reader is to buy the book, open it to a random page, read, enjoy, and be enlightened."--Richard Kenyon, Notices of the American Mathematical Society
"Massive . . . endlessly fascinating."--Gregory McNamee, Bloomsbury Review
"This volume is an enormous, far-reaching effort to survey the current landscape of (pure) mathematics. Chief editor Gowers and associate editors Barrow-Green and Leader have enlisted scores of leading mathematicians worldwide to produce a gorgeous volume of longer essays and short, specific articles that convey some of the dense fabric of ideas and techniques of modern mathematics. . . . This volume should be on the shelf of every university and public library, and of every mathematician--professional and amateur alike."--S.J. Colley, Choice
"The Princeton Companion to Mathematics is a friendly, informative reference book that attempts to explain what mathematics is about and what mathematicians do. Over 200 entries by a panel of experts span such topics as: the origins of modern mathematics; mathematical concepts; branches of mathematics; mathematicians that contributed to the present state of the discipline; theorems and problems; the influences of mathematics and some perspectives. Its presentations are selective, satisfying, and complete within themselves but not overbearingly comprehensive. Any reader from a curious high school student to an experienced mathematician seeking information on a particular mathematical subject outside his or her field will find this book useful. The writing is clear and the examples and illustrations beneficial."--Frank Swetz, Convergence
"Every research mathematician, every university student of mathematics, and every serious amateur of mathematical science should own a least one copy of The Companion. Indeed, the sheer weight of the volume suggests that it is advisable to own two: one for work and one at home. . . . Even an academic sourpuss should be pleased with the attention to detail of The Companion's publishers, editors, and authors and with many judicious decisions about the level of exposition, level of detail, what to include and what to omit, and much more--which have led to a well-integrated and highly readable volume."--Jonathan M. Borwein, SIAM Review
"Edited by Gowers, a recipient of the Fields Medal, this volume contains almost 200 entries, commissioned especially for this book from the world's leading mathematicians. It introduces basic mathematical tools and vocabulary, traces the development of modern mathematics, defines essential terms and concepts, and puts them in context. . . . Packed with information presented in an accessible style, this is an indispensable resource for undergraduate and graduate students in mathematics as well as for researchers and scholars seeking to understand areas outside their specialties."--Library Journal
"The book I'm talking about is The Princeton Companion to Mathematics. If you are in an absolute rush, the short version of my post today is, buy this book. You don't have to click on the link with my referral if you don't want to, seriously just pick up a copy of this book, I can guarantee you that it will be love at first sight. . . . The Princeton Companion to Mathematics is not only a beautiful book from an aesthetic standpoint, with its heavy, high quality pages and sturdy binding, but above all it's a monumental piece of work. I have never seen a book like this before. . . . [T]he bible of mathematics. . . . I believe this is the kind of book that will still be in use a hundred years from now."--Antonio Cangiano, Math-Blog.com
"I'm completely charmed. This is one of those books that makes you wish you had a desert island to be marooned on."--Brian Hayes, bit-player.org
"This has been a long time coming, but the wait was worth it! After many years of slogging through textbooks that presented too many proofs and demonstrations that were left to the student or lacking numerous intermediate steps, after encountering numerous 'introductions' that were obtuse and highly theoretical and after digesting far too many explanations with maximal equations and minimal verbiage, we arrive at the happy medium. This book is a companion in every sense of the word and a very friendly one at that. . . . For a comprehensive overview of many areas of mathematics in a readable format, there has never been anything quite like this. I would urge a trip to the local library to have a look."--John A. Wass, Scientific Computing
"This book is supremely accessible. Many in the sugar industry with a fairly good grasp of mathematics will probably not struggle with it, and will invariably marvel at its richness and diversity. [A] great companion."--International Sugar Journal
"The book contains some valuable surveys of the main branches of mathematics that are written in an accessible style. Hence, it is recommended both to students of mathematics and researchers seeking to understand areas outside their specialties."--European Mathematical Society Newsletter
From the Back Cover
"This is a wonderful book. The content is overwhelming. Every practicing mathematician, everyone who uses mathematics, and everyone who is interested in mathematics must have a copy of their own."--Simon A. Levin, Princeton University
"The Princeton Companion to Mathematics fills a vital need. It is the only book of its kind."--Victor J. Katz, professor emeritus, University of the District of Columbia
"I think that this is a wonderful book, completely different from anything that has been written before about mathematics and mathematicians."--Endre Süli, University of Oxford
"The Princeton Companion to Mathematics is a much needed--and will become a much used--reference work. In fact, it will stand alone as the reference work in mathematics."--John J. Watkins, Colorado College
About the Author
Timothy Gowers is Rouse Ball Professor of Mathematics at Cambridge University and was a recipient of the Fields Medal for Mathematics, awarded for 'the most daring, profound and stimulating research done by young mathhematicians'.
Most helpful customer reviews
236 of 241 people found the following review helpful.
A major event in mathematical publishing
By Timothy Chow
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.
63 of 64 people found the following review helpful.
A physicist's perspective
By J. Koelman
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... ;-)
97 of 104 people found the following review helpful.
mathematics: a not-so-short introduction
By Nim Sudo
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.
See all 76 customer reviews...
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