The Princeton Companion to Mathematics
Encyclopedia
The Princeton Companion to Mathematics is a book, edited by Timothy Gowers with associate editors June Barrow-Green and Imre Leader
, and published in 2008 by Princeton University Press
(ISBN 978-0-691-11880-2). It provides an extensive overview of mathematics
, and is noted for the high caliber of the contributors. The book was a 2011 winner of the Euler Book Prize
of the Mathematical Association of America
, given annually to "an outstanding book about mathematics".
rather than applied mathematics
, although it does also cover both applications of mathematics and the mathematics that relates to those applications;
it provides a broad overview of the significant ideas and developments in research mathematics. It is organized into eight parts:
Despite its length, the range of topics included is selective rather than comprehensive: some important established topics such as diophantine approximation
are omitted, transcendence theory
, differential geometry, and cohomology
get short shrift, and the most recent frontiers of research are also generally not included.
and MathWorld
, the articles in the Princeton Companion are intended to be more reflective and discursive, and to convey the beauty and depth of modern mathematics. Quoting a passage from Bertrand Russell
that "Pure Mathematics is the class of all propositions of the form p implies q", the editor of the Companion states that it "is about everything that Russell’s definition leaves out."
The core sections of the Companion are aimed primarily at readers who are already familiar with mathematics at the undergraduate level. Much of the rest of the book, such as its collection of biographies, would be accessible to a mathematically inclined high school student, and there is enough depth of coverage in the book to interest even professional research mathematicians. Reviewer Jonathan Borwein
summarizes the audience for this book broadly:
, considered to be the top honor in mathematics. Other contributors include Fields medalists Michael Atiyah
, Alain Connes
, Charles Fefferman
, and Terence Tao
, and well-known mathematicians Noga Alon
, George Andrews, Béla Bollobás
, John P. Burgess
, Clifford Cocks
, Ingrid Daubechies
, Persi Diaconis
, Jordan Ellenberg
, Oded Goldreich
, Andrew Granville
, Jeremy Gray
, Frank Kelly, Sergiu Klainerman
, Jon Kleinberg
, János Kollár
, Peter Lax
, Dusa McDuff
, Barry Mazur
, Carl Pomerance
, Eleanor Robson
, Peter Sarnak
, Madhu Sudan
, Clifford Taubes
, and Avi Wigderson
.
of the Mathematical Association of America
, given annually to "an outstanding book about mathematics".
The Princeton Companion was also listed as an outstanding title by Choice Magazine, a publication of the American Library Association
, in 2009.
Imre Leader
Imre Bennett Leader is a British mathematician and Professor of Pure Mathematics, specifically combinatorics, at the University of Cambridge....
, and published in 2008 by Princeton University Press
Princeton University Press
-Further reading:* "". Artforum International, 2005.-External links:* * * * *...
(ISBN 978-0-691-11880-2). It provides an extensive overview of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, and is noted for the high caliber of the contributors. The book was a 2011 winner of the Euler Book Prize
Euler Book Prize
The Euler Book Prize is an award named after Leonhard Euler and given annually at the Joint Mathematics Meetings by the Mathematical Association of America to an outstanding book in mathematics that is likely to improve the public view of the field....
of the Mathematical Association of America
Mathematical Association of America
The Mathematical Association of America is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university, college, and high school teachers; graduate and undergraduate students; pure and applied mathematicians; computer scientists;...
, given annually to "an outstanding book about mathematics".
Topics and organization
The book concentrates primarily on modern pure mathematicsPure mathematics
Broadly speaking, pure mathematics is mathematics which studies entirely abstract concepts. From the eighteenth century onwards, this was a recognized category of mathematical activity, sometimes characterized as speculative mathematics, and at variance with the trend towards meeting the needs of...
rather than applied mathematics
Applied mathematics
Applied mathematics is a branch of mathematics that concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge...
, although it does also cover both applications of mathematics and the mathematics that relates to those applications;
it provides a broad overview of the significant ideas and developments in research mathematics. It is organized into eight parts:
- An introduction to mathematics, outlining the major areas of study, key definitions, and the goals and purposes of mathematical research.
- An overview of the history of mathematics, in seven chapters including the development of important concepts such as number, geometry, mathematical proof, and the axiomatic approach to the foundations of mathematics. A chronology of significant events in mathematical history is also provided later in the book.
- Three core sections, totalling approximately 600 pages. The first of these sections provides an alphabetized set of articles on 99 specific mathematical concepts such as the axiom of choice, expander graphExpander graphIn combinatorics, an expander graph is a sparse graph that has strong connectivity properties, quantified using vertex, edge or spectral expansion as described below...
s, and Hilbert spaceHilbert spaceThe mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
. The second core section includes long surveys of 26 branches of research mathematics such as algebraic geometryAlgebraic geometryAlgebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
and combinatorial group theoryCombinatorial group theoryIn mathematics, combinatorial group theory is the theory of free groups, and the concept of a presentation of a group by generators and relations...
. The third describes 38 important mathematical problems and theorems such as the four color theoremFour color theoremIn mathematics, the four color theorem, or the four color map theorem states that, given any separation of a plane into contiguous regions, producing a figure called a map, no more than four colors are required to color the regions of the map so that no two adjacent regions have the same color...
, the Birch and Swinnerton-Dyer conjectureBirch and Swinnerton-Dyer conjectureIn mathematics, the Birch and Swinnerton-Dyer conjecture is an open problem in the field of number theory. Its status as one of the most challenging mathematical questions has become widely recognized; the conjecture was chosen as one of the seven Millennium Prize Problems listed by the Clay...
, and the Halting problemHalting problemIn computability theory, the halting problem can be stated as follows: Given a description of a computer program, decide whether the program finishes running or continues to run forever...
. - A collection of biographies of nearly 100 famous deceased mathematicians, arranged chronologically, also including a history of Nicolas BourbakiNicolas BourbakiNicolas Bourbaki is the collective pseudonym under which a group of 20th-century mathematicians wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. With the goal of founding all of mathematics on set theory, the group strove for rigour and generality...
's pseudonymous collaboration. - Essays describing the influences and applications of mathematics in the sciences, technology, business, medicine, and the fine arts.
- A section of perspectives on the future of mathematics, problem solving techniques, the ubiquity of mathematics, and advice to young mathematicians.
Despite its length, the range of topics included is selective rather than comprehensive: some important established topics such as diophantine approximation
Diophantine approximation
In number theory, the field of Diophantine approximation, named after Diophantus of Alexandria, deals with the approximation of real numbers by rational numbers....
are omitted, transcendence theory
Transcendence theory
Transcendence theory is a branch of number theory that investigates transcendental numbers, in both qualitative and quantitative ways.-Transcendence:...
, differential geometry, and cohomology
Cohomology
In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex. That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries...
get short shrift, and the most recent frontiers of research are also generally not included.
Target audience
The book's authors have attempted to keep their work accessible by forgoing abstraction and technical nomenclature as much as possible and by making heavy use of concrete examples and illustrations. Compared to the concise and factual coverage of mathematics in sources such as WikipediaWikipedia
Wikipedia is a free, web-based, collaborative, multilingual encyclopedia project supported by the non-profit Wikimedia Foundation. Its 20 million articles have been written collaboratively by volunteers around the world. Almost all of its articles can be edited by anyone with access to the site,...
and MathWorld
MathWorld
MathWorld is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Digital Library grant to the University of Illinois at...
, the articles in the Princeton Companion are intended to be more reflective and discursive, and to convey the beauty and depth of modern mathematics. Quoting a passage from Bertrand Russell
Bertrand Russell
Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, and social critic. At various points in his life he considered himself a liberal, a socialist, and a pacifist, but he also admitted that he had never been any of these things...
that "Pure Mathematics is the class of all propositions of the form p implies q", the editor of the Companion states that it "is about everything that Russell’s definition leaves out."
The core sections of the Companion are aimed primarily at readers who are already familiar with mathematics at the undergraduate level. Much of the rest of the book, such as its collection of biographies, would be accessible to a mathematically inclined high school student, and there is enough depth of coverage in the book to interest even professional research mathematicians. Reviewer Jonathan Borwein
Jonathan Borwein
Jonathan Michael Borwein is a Scottish mathematician who holds an appointment as Laureate Professor of mathematics at the University of Newcastle, Australia. Noted for his prolific and creative work throughout the international mathematical community, he is a close associate of David H...
summarizes the audience for this book broadly:
Contributors
The contributors to The Princeton Companion to Mathematics consist of 133 of the world's best mathematicians. Timothy Gowers, its editor, is the recipient of the Fields MedalFields Medal
The Fields Medal, officially known as International Medal for Outstanding Discoveries in Mathematics, is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union , a meeting that takes place every four...
, considered to be the top honor in mathematics. Other contributors include Fields medalists Michael Atiyah
Michael Atiyah
Sir Michael Francis Atiyah, OM, FRS, FRSE is a British mathematician working in geometry.Atiyah grew up in Sudan and Egypt but spent most of his academic life in the United Kingdom at Oxford and Cambridge, and in the United States at the Institute for Advanced Study...
, Alain Connes
Alain Connes
Alain Connes is a French mathematician, currently Professor at the Collège de France, IHÉS, The Ohio State University and Vanderbilt University.-Work:...
, Charles Fefferman
Charles Fefferman
Charles Louis Fefferman is an American mathematician at Princeton University. His primary field of research is mathematical analysis....
, and Terence Tao
Terence Tao
Terence Chi-Shen Tao FRS is an Australian mathematician working primarily on harmonic analysis, partial differential equations, combinatorics, analytic number theory and representation theory...
, and well-known mathematicians Noga Alon
Noga Alon
Noga Alon is an Israeli mathematician noted for his contributions to combinatorics and theoretical computer science, having authored hundreds of papers.- Academic background :...
, George Andrews, Béla Bollobás
Béla Bollobás
Béla Bollobás FRS is a Hungarian-born British mathematician who has worked in various areas of mathematics, including functional analysis, combinatorics, graph theory and percolation. As a student, he took part in the first three International Mathematical Olympiads, winning two gold medals...
, John P. Burgess
John P. Burgess
John Burgess is a John N. Woodhull Professor of Philosophy at Princeton University. He received his Ph.D. from UC Berkeley's Group in Logic and Methodology of Science. His interests include logic, philosophy of mathematics and metaethics...
, Clifford Cocks
Clifford Cocks
Clifford Christopher Cocks, CB, is a British mathematician and cryptographer at GCHQ.He invented the widely-used encryption algorithm now commonly known as RSA, about three years before it was independently developed by Rivest, Shamir, and Adleman at MIT...
, Ingrid Daubechies
Ingrid Daubechies
Ingrid Daubechies is a Belgian physicist and mathematician. She was between 2004 and 2011 the William R. Kenan Jr. Professor in the mathematics and applied mathematics departments at Princeton University. In January 2011 she moved to Duke University as a Professor in mathematics. She is the first...
, Persi Diaconis
Persi Diaconis
Persi Warren Diaconis is an American mathematician and former professional magician. He is the Mary V. Sunseri Professor of Statistics and Mathematics at Stanford University....
, Jordan Ellenberg
Jordan Ellenberg
Jordan S. Ellenberg is a mathematician working as a Professor of Mathematics at the University of Wisconsin–Madison. His research covers a wide variety of topics within arithmetic geometry. He received both the A.B. and Ph.D...
, Oded Goldreich
Oded Goldreich
Oded Goldreich is a professor of Computer Science at the Faculty of Mathematics and Computer Science of Weizmann Institute of Science, Israel. His research interests lie within the theory of computation...
, Andrew Granville
Andrew Granville
Andrew James Granville is a British mathematician, working in the field of number theory.He has been a faculty member at the Université de Montréal since 2002. Before moving to Montreal he was a mathematics professor at University of Georgia from 1991 until 2002...
, Jeremy Gray
Jeremy Gray
Jeremy Gray is an English mathematician primarily interested in the history of mathematics.He studied mathematics at Oxford University from 1966 to 1969, and then at Warwick University, obtaining his Ph.D. in 1980 under the supervision of Ian Stewart and David Fowler. He has worked at the Open...
, Frank Kelly, Sergiu Klainerman
Sergiu Klainerman
Sergiu Klainerman is a mathematician known for his contributions to the study of hyperbolic differential equations and general relativity. He is a Professor in the Department of Mathematics at Princeton University, a position he held since 1987. From 1980 to 1987 he was a faculty member at New...
, Jon Kleinberg
Jon Kleinberg
-External links:**** Stephen Ibaraki*Yury Lifshits,...
, János Kollár
János Kollár
János Kollár is a Hungarian mathematician, specializing in algebraic geometry. He is a member of the National Academy of Sciences since 2005 and received the Cole Prize in 2006....
, Peter Lax
Peter Lax
Peter David Lax is a mathematician working in the areas of pure and applied mathematics. He has made important contributions to integrable systems, fluid dynamics and shock waves, solitonic physics, hyperbolic conservation laws, and mathematical and scientific computing, among other fields...
, Dusa McDuff
Dusa McDuff
Dusa McDuff is an English mathematician. She was born in London, England as the daughter of the noted biologist Conrad Hal Waddington. Her mother, Justin, born Justin Blanco White, was an architect, while her maternal grandmother was the feminist Amber Reeves, a lover of H.G. Wells and an author...
, Barry Mazur
Barry Mazur
-Life:Born in New York City, Mazur attended the Bronx High School of Science and MIT, although he did not graduate from the latter on account of failing a then-present ROTC requirement. Regardless, he was accepted for graduate school and received his Ph.D. from Princeton University in 1959,...
, Carl Pomerance
Carl Pomerance
Carl Bernard Pomerance is a well-known number theorist. He attended college at Brown University and later received his Ph.D. from Harvard University in 1972 with a dissertation proving that any odd perfect number has at least 7 distinct prime factors. He immediately joined the faculty at the...
, Eleanor Robson
Eleanor Robson
Eleanor Robson is a Reader in History and Philosophy of Science at the Department of History and Philosophy of Science, Cambridge University, vice-chair of the British School of Archaeology in Iraq and a Quondam Fellow of All Souls College, Oxford....
, Peter Sarnak
Peter Sarnak
Peter Clive Sarnak is a South African-born mathematician. He has been Eugene Higgins Professor of Mathematics at Princeton University since 2002, succeeding Andrew Wiles, and is an editor of the Annals of Mathematics...
, Madhu Sudan
Madhu Sudan
Madhu Sudan is an Indian computer scientist, professor of computer science at the Massachusetts Institute of Technology and a member of MIT Computer Science and Artificial Intelligence Laboratory.-Career:...
, Clifford Taubes
Clifford Taubes
Clifford Henry Taubes is the William Petschek Professor of Mathematics at Harvard University and works in gauge field theory, differential geometry, and low-dimensional topology.-Early career:Taubes received his Ph.D...
, and Avi Wigderson
Avi Wigderson
Avi Wigderson is an Israeli mathematician and computer scientist, a professor of mathematics at the Institute for Advanced Study in Princeton. His research interests include complexity theory, parallel algorithms, graph theory, cryptography, distributed computing, and neural...
.
Awards
Gowers and the Princeton Companion were the 2011 winners of the Euler Book PrizeEuler Book Prize
The Euler Book Prize is an award named after Leonhard Euler and given annually at the Joint Mathematics Meetings by the Mathematical Association of America to an outstanding book in mathematics that is likely to improve the public view of the field....
of the Mathematical Association of America
Mathematical Association of America
The Mathematical Association of America is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university, college, and high school teachers; graduate and undergraduate students; pure and applied mathematicians; computer scientists;...
, given annually to "an outstanding book about mathematics".
The Princeton Companion was also listed as an outstanding title by Choice Magazine, a publication of the American Library Association
American Library Association
The American Library Association is a non-profit organization based in the United States that promotes libraries and library education internationally. It is the oldest and largest library association in the world, with more than 62,000 members....
, in 2009.
External links
- Book homepage at Princeton University Press; contains several sample chapters
- Princeton Companion To Mathematics category in Gowers's blog