The Princeton Companion to Mathematics Illustrated Edition
Timothy Gowers
(Editor)
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Editorial Reviews
Review
"One of Choice's Outstanding Academic Titles for 2009"
"Honorable Mention for the 2008 PROSE Award for Single Volume Reference/Science, Association of American Publishers"
"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 numbercrunchers." ― The Economist
"Although the editors' original goal of text that could be understood by anyone with a good background in high school mathematics provided shortlived, this wideranging 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 BarrowGreen 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 1000page 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 oneofakind 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, farreaching effort to survey the current landscape of (pure) mathematics. Chief editor Gowers and associate editors BarrowGreen 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 mathematicianprofessional 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 morewhich have led to a wellintegrated 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, MathBlog.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, bitplayer.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
Review
"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
From the Inside Flap
"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 neededand will become a much usedreference work. In fact, it will stand alone as the reference work in mathematics."John J. Watkins, Colorado College
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 neededand will become a much usedreference work. In fact, it will stand alone as the reference work in mathematics."John J. Watkins, Colorado College
About the Author
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Product details
 Publisher : Princeton University Press; Illustrated edition (September 28, 2008)
 Language : English
 Hardcover : 1034 pages
 ISBN10 : 0691118809
 ISBN13 : 9780691118802
 Item Weight : 5.69 pounds
 Dimensions : 8.38 x 2.63 x 10.31 inches

Best Sellers Rank:
#86,016 in Books (See Top 100 in Books)
 #19 in Encyclopedias (Books)
 #22 in Mathematics Reference (Books)
 #98 in Mathematics Study & Teaching (Books)
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Customer reviews
Top reviews from the United States
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This is not so with The Companion. To give a concrete example, consider this definition of a scheme given by the book in its chapter on Algebraic Geometry: "Roughly speaking, a scheme is an algebraic set where we also keep track of the multiplicities and of the directions they occur in". On the one hand this lacks the formalism that would be necessary for an Algebraic Geometer. But it is also about as good as one could expect in a book this size, and indeed the concepts leading up to this, algebraic sets and multiplicities, are adequately explained without handwaving.
And this is the real virtue of the book: it provides an intuitive understanding of concepts, similar to an introductory textbook on a particular mathematics topic like say linear algebra that might forgo the abstract definition of a vector space over a field for the sake of efficiently providing very concrete examples over R or C. This can be done without talking about bases or dimension or everything that you would learn in a graduate level course. The book does something similar in its chapter on Algebraic Numbers, focusing on quadratic number fields specifically for most of the chapter until the very end when it becomes more appropriate to generalize the concept afterwards.
Overall a profound and inspiring mathematics book. I haven't seen anything else quite like this book before and I've been a passionate reader of mathematics for over a decade. If you have any interest in math do yourself a favor and purchase a copy of this book for yourself. And then you can purchase the books in the "Further Reading" sections once you're ready to learn even more about these topics. This book is a gateway drug to math you've been warned.
 Encyclopedia Of Mathematics (Science Encyclopedia)
 The Princeton Companion to Mathematics
 Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables (Dover Books on Mathematics)
 NIST Handbook of Mathematical Functions (Being of course the 2010 update of the Abramowitz classic above).
Given these four, there is hardly a topic from among the current 495 math fields of study that isn't at least explained in enough detail to save LOTS of time on link expeditions. At minimum, these give head starts on alphabetized keywords that will quickly fill holes in any research project, class, or syllabus.
 Reviewing areas of math you already know
 Intros to areas of math you want to know more about
 Discovering areas of math you may want to learn more about
Of course the articles are mostly Big Picture overviews, but for the purposes above, that is exactly what is needed. Great book. You'll find yourself perusing it often.
I'm an exmath major picking his way through Galois Theory thirty years after graduation (and I wander into a lot of other topics, too). That's what I use this for. It would be interesting to hear what some research mathematicians think of it.
Top reviews from other countries
If you are looking for a reference book or encyclopedia then do not buy this book for that reason, as the editor explains it will be too complicated for some and not complicated enough for others. The review on the back page by John J. Watkins is complete nonsense regarding this being 'The stand alone reference in mathematics'. It is a good book though for what it is. A companion.
It is not a textbook, and you won't earn a college degree by reading it, but anyone reading part 1 will understand fundamentals which will make textbooks much easier to read. Part 2 is also a good path to understanding the history of mathematics, and both why and how it was developed to solve real problems that man has faced down the years.
Further sections explain key concepts in language as close to laymans's terms as possible, and give more detailed guides to major subtopics, significant contemporary questions, prominent figures, applications to other area of science and life as a mathematician.
The book is not 'light' reading in any sense. Maths can be inherently difficult, and the contents of the Companion are essays without illustrations. The hard copy is also physically daunting, which is why I would recommend the digital kindle version for each of access and use.
Nonetheless, if you have an interest in mathematics and have some level of maths education then, whether you use it as a guide to the subject or as a companion to deeper study, this is a book you should have.