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The Princeton Companion to Mathematics
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 musi—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. Accessible in 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 include:
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
Unparalleled in its depth of coverage, The Princeton Companion to Mathematics surveys the most active and exciting branches of pure mathematics. Accessible in 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 include:
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
1056 pages, Hardcover
First published October 5, 2007
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Displaying 1 - 30 of 42 reviews
December 17, 2011
So, I will probably never finish this; as i step through my college math classes it works as a handy reference. I can gleam an intuition on concepts that have me muddled. It covers what I would imagine to be everything we know about math. I can also use it to hammer nails and subdue burglars.
June 10, 2014
Just five words: "General Map of Mathematical World".
No, you're not supposed to read it from beginning to the end. No, you're not supposed to know it all by heart. But use it as reference, check which area you want to know, and you will be happy (-ier).
(ignoring that advice may lead to psychological hazard)
No, you're not supposed to read it from beginning to the end. No, you're not supposed to know it all by heart. But use it as reference, check which area you want to know, and you will be happy (-ier).
(ignoring that advice may lead to psychological hazard)
May 8, 2015
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.
November 28, 2009
finally spent some serious quality time with this, reading it through aside from those sections (only about 1/4; not too bad, not too bad!) of which i could make neither heads nor tails, lacking too many years' preparatory material (primarily the geometry. analysis and algebra were both just wonderful, especially since they got the magnificent Terrence Tao to write most of the analysis). i think i actually finally learned some calculus of variations from this exercise -- hurrah (lord knows it's been long enough; i first heard of functional analysis a decade ago or so, and have just never had the time)! everyone should own this book.
----
Amazon 2008-10-24. It finally got published (on my 28th birthday, none the less!), after a long year of eager waiting! Ooooooooh, I'm excited! I forget where I first heard about this, but here's a recent review from Not Even Wrong:
I just recently got my hands on a copy of the new Princeton Companion to Mathematics, and I fear that this is likely to seriously impact my ability to get things done for a while, as I devote too much time to happily reading many of its more than 1000 pages.
The book is an amazing document (and physically, a beautiful, if weighty object), unlike anything else I know of. Its coverage of mathematics and mathematical culture is very wide and sometimes deep, but it makes no attempt to be comprehensive. Thus, the accurate title “Companion to” rather than “Encyclopedia of”. The most remarkable aspect of the book is the extremely high quality of the contributions from a large number of different authors. It includes many wonderful long expository articles, mostly at a level that a good undergraduate math student could hope to appreciate, with much of the book accessible to an even wider audience. The articles are often written by some of the best researchers and expositors around. For example, one can find Barry Mazur writing on Algebraic Numbers, Janos Kollar on Algebraic Geometry, Cliff Taubes on Differential Topology, Ingrid Daubechies on Wavelets, Persi Diaconis on Mathematical Statistics, and many, many others of similar quality. The table of contents is available here.
The book also includes extensive articles on historical topics in mathematics and short biographies of a large number of mathematicians, as well as coverage of applications and a section largely devoted to describing the art of problem-solving and how mathematics really gets created. This section includes a beautiful set of five essays called “Advice to a Young Mathematician”, which give five different equally fascinating perspectives from some of the best in the subject about how they achieved what they did, as well as what they have learned from years of helping students become researchers. The authors of these pieces are Michael Atiyah, Bela Bollobas, Alain Connes, Dusa McDuff, and Peter Sarnak. Luckily for all young (and old) mathematicians, this chapter is freely available here.
The person most responsible for this is clearly the editor (and author of some of the pieces), Fields Medalist Timothy Gowers, who had help from many others, including fellow Fields Medalist Terry Tao. Gowers has a weblog, and he has written about the book in these entries (and there’s a podcast interviewing him on the book web-site at PUP). Terry Tao has a posting about the book here.
If you’re looking for a gift for someone with a serious interest in mathematics, no matter what their background, you won’t do any better than this.
----
Amazon 2008-10-24. It finally got published (on my 28th birthday, none the less!), after a long year of eager waiting! Ooooooooh, I'm excited! I forget where I first heard about this, but here's a recent review from Not Even Wrong:
I just recently got my hands on a copy of the new Princeton Companion to Mathematics, and I fear that this is likely to seriously impact my ability to get things done for a while, as I devote too much time to happily reading many of its more than 1000 pages.
The book is an amazing document (and physically, a beautiful, if weighty object), unlike anything else I know of. Its coverage of mathematics and mathematical culture is very wide and sometimes deep, but it makes no attempt to be comprehensive. Thus, the accurate title “Companion to” rather than “Encyclopedia of”. The most remarkable aspect of the book is the extremely high quality of the contributions from a large number of different authors. It includes many wonderful long expository articles, mostly at a level that a good undergraduate math student could hope to appreciate, with much of the book accessible to an even wider audience. The articles are often written by some of the best researchers and expositors around. For example, one can find Barry Mazur writing on Algebraic Numbers, Janos Kollar on Algebraic Geometry, Cliff Taubes on Differential Topology, Ingrid Daubechies on Wavelets, Persi Diaconis on Mathematical Statistics, and many, many others of similar quality. The table of contents is available here.
The book also includes extensive articles on historical topics in mathematics and short biographies of a large number of mathematicians, as well as coverage of applications and a section largely devoted to describing the art of problem-solving and how mathematics really gets created. This section includes a beautiful set of five essays called “Advice to a Young Mathematician”, which give five different equally fascinating perspectives from some of the best in the subject about how they achieved what they did, as well as what they have learned from years of helping students become researchers. The authors of these pieces are Michael Atiyah, Bela Bollobas, Alain Connes, Dusa McDuff, and Peter Sarnak. Luckily for all young (and old) mathematicians, this chapter is freely available here.
The person most responsible for this is clearly the editor (and author of some of the pieces), Fields Medalist Timothy Gowers, who had help from many others, including fellow Fields Medalist Terry Tao. Gowers has a weblog, and he has written about the book in these entries (and there’s a podcast interviewing him on the book web-site at PUP). Terry Tao has a posting about the book here.
If you’re looking for a gift for someone with a serious interest in mathematics, no matter what their background, you won’t do any better than this.
July 21, 2021
Definitely haven't actually read the whole thing, but she lives next to my bed and I worship the ground she sits on, so I didn't want her wasting away forever on my currently reading shelf. Extremely well-rounded, readable, and with a focus on intuition over rigour, it does everything a companion to mathematics should and is also the best 3am insomnia reading to boot.
September 13, 2016
This is really a big book mainly oriented to be a consulting sort of enciclopedic book,is as a USA mirror image of well known russian Mathematics,Its Contents Meaning and Significance by Aleksandrov and others,yet at a higher level.
The book vith a lot of contributors has 8 parts:Introduction(what is mathematics about ,the general goals of mathematical resarch),The Origins of Modern Mathematics,Mathematical Concepts,Branches of Mathematics(this to last are the bulk of the book),Theorems and Problems(key problems in modern mathematics as "the independence of the continuum hypotesis" and many others),Mathematicians(a brief biography of many mathematicians and his most important works),The Influence of Mathematics(its aplications to sciences and tecnology),Final Perspectives(for example"adwice to young mathematicians"or "the ubiquity of mathematics).
A book that gives a universal panoramic of the state of to day mathematics,relatively easy to read for those with a level at leat of first college couse level and by that requires more mathematical maturity that the russian book by Aleksandrov.
A much recomended book for those with a serious interst in mathematics
The book vith a lot of contributors has 8 parts:Introduction(what is mathematics about ,the general goals of mathematical resarch),The Origins of Modern Mathematics,Mathematical Concepts,Branches of Mathematics(this to last are the bulk of the book),Theorems and Problems(key problems in modern mathematics as "the independence of the continuum hypotesis" and many others),Mathematicians(a brief biography of many mathematicians and his most important works),The Influence of Mathematics(its aplications to sciences and tecnology),Final Perspectives(for example"adwice to young mathematicians"or "the ubiquity of mathematics).
A book that gives a universal panoramic of the state of to day mathematics,relatively easy to read for those with a level at leat of first college couse level and by that requires more mathematical maturity that the russian book by Aleksandrov.
A much recomended book for those with a serious interst in mathematics
April 5, 2021
Макар и да се считам за далечен на материята, прочетените няколко глави се оказаха ясно написани, при все че примерите, използвани от авторите за илюстрация на концепциите, бяха преднамерено прости. Хареса ми акцентът във II част върху историята на модерната математика, оформянето на съвременната представа за математическо доказателство от древността и средновековието насам и т.н.
В книгата се подчертава ценността на различните подходи към разрешаването на даден проблем, различните "гещалти" тъй да кажем (конкретни примери: техниките от алгебрата спрямо анализа при доказването, че някакво нелинейно уравнение е положително за всяко реално Х; или че различните критерии за "измерение", описани в III част, може да си противоречат помежду си и измерността на множествата да варира според критерия).
В този смисъл, изглежда, че дори в математиката се открива пространство за многообразие и творчество.
В книгата се подчертава ценността на различните подходи към разрешаването на даден проблем, различните "гещалти" тъй да кажем (конкретни примери: техниките от алгебрата спрямо анализа при доказването, че някакво нелинейно уравнение е положително за всяко реално Х; или че различните критерии за "измерение", описани в III част, може да си противоречат помежду си и измерността на множествата да варира според критерия).
В този смисъл, изглежда, че дори в математиката се открива пространство за многообразие и творчество.
Want to read
February 8, 2009
Want to read
November 15, 2008The editor is Timothy Gowers, not Imre Leader, who is listed as the second of two associate editors, the first of whom is June Barrow-Green. A recent purchase from Amazon, this is a beautiful book, to be read in bits here and there and savored for the exquisite beauty of the mathematics it surveys. I am one of those people who seek out mathematical texts to read for pleasure.
Want to read
November 12, 2009a buffet of maths
December 29, 2022
Mathematics, one says, is a young man’s game. Is there any hope left for those of us past the age? Perhaps so, if one consider why. Unlike what is the case in the humanities, where scholarly excellence rests upon seasoned judgment to be acquired only through sustained reading and many years of reflection, in mathematics, once equipped with a minimum of technique, all a neophyte needs is to internalize a handful of axioms and he can be off and running. For – let us suppose one’s research in a field not subject to overgrazing – there are plenty of implications to explore which nobody has determinedly pursued before, and, ordinarily, everyone hearkens to a familiar genius who prompts him towards an angle of his own for which his abilities uniquely suit him. This being so, the many astonishing discoveries in pure mathematics and theoretical physics won by youthful brilliance should not come as a surprise. In more applied disciplines, say condensed-matter theory or organic chemistry, it can take decades to assimilate and master a large mass of empirical facts and theoretical techniques before becoming privy to any very great insight, of a kind no one will have thought of before.
Thus, the lament over declining capacity often to be met with in celebrated mathematicians past their prime really reflects an immature wish to stay forever young rather than to engage in a type of intellectual work appropriate to their age (case in point, G.H. Hardy in A Mathematician’s Apology from the Cambridge University Press, first edition 1940). Indeed, anyone who has perused the collected works of luminaries such as Georg Cantor, David Hilbert, Henri Poincaré, Hermann Weyl – who enjoyed the good fortune not to have their lives cut short as sadly happened to the precocious Niels Henrik Abel, Évariste Galois, Bernhard Riemann, Sofia Kovalevskaia, Ramanujan, Alan Turing, John von Neumann and so forth – will see that their mature and lasting contributions to the edifice of mathematics tended to be made well into middle age, if not later (the immortal Gauss, on the other hand, maintains a prodigious level of productivity all through his long life).
What is the distinguishing feature of these so-called universals? They were widely learned in many disciplines and profited from the catholicity of their vision. The narrow specialization that has become the rule in our day and that trends in faculty hiring and grant support incentivize is therefore most regrettable. Is there anything to be done about this situation? Recall that self-education [Bildung] is a work, not of a day, but of a lifetime. Every little bit helps and knowledge patiently accumulated over year after year builds up to a broad perspective that might, in due course, rival that of our forebears; even if one cannot hope ever to equal them, emulation of their supreme achievement is to be enjoined upon us.
Therefore, any resources that might assist such an endeavor of self-education will come as welcome. It can be quite a challenge to break into a mature branch of mathematics, since the popular literature, if any there be, tends to the superficial while the professional literature will have become highly technical and nobody bothers to explain elementary things, for everyone who counts is presumed already to know them. That is why the present Princeton Companion to Mathematics, edited by the eminent British mathematician and Fields medalist Timothy Gowers and published by the Princeton University Press in 2008, turns out to be so handy.
Review of contents: as a reference work designed to appeal to those who already possess some grounding in the field, the emphasis is on conveying a mature perspective on general concepts, many of which one will have seen before but here set forth in a systematic context with all the insight of the genuine expert. Thus, Part I takes up the question of what mathematics is, its language and grammar and the goals of research, exemplified with several concrete examples. In Part II on origins, the development of number systems, algebra, the idea of proof and the crisis in foundations that shook the mathematical world around 1900. The real meat comes in Parts III and IV, the former a series of concise encyclopedia-like entries on all manner of things: from set-theoretic axioms to current topics such as braid groups, Calabi-Yau manifolds, moduli spaces, Ricci flow, von Neumann algebras and so forth. Selection a mix of traditional material and present-day hot topics. For a less piecemeal approach, refer to Part IV, a series of miniature introductions to various branches of mathematics from the point of view of a practitioner pitching to those potentially interested in engaging in advanced research. Mostly good on motivating why one ought to find the current work of interest, whether one feels compelled to go into the respective field or not. The expository part of this companion is rounded out by Part V on theorems and problems, a conspectus of classic results or problem areas one might like to know about (say, Hilbert’s Nullstellensatz or the status of the Mordell conjecture). A reasonable sampling to afford a view of the kinds of things people think about these days.
Articles are written by competent authorities and assume enough background that one can reach a discussion of the exciting ideas at the forefront of current research. The author is identified by name when sufficiently famous (for instance, Terence Tao for articles on topics in analysis). Although the book is a thousand pages long most of the articles remain at a fairly high level so it reads quickly.
A couple highlights: a very good article on set theory in Part IV [pp. 615-634] and another on logic and model theory [pp. 635-646]. Worth mentioning for this reason: these caught this reviewer’s attention because, unlike an article on partial differential equations with which he is already somewhat conversant, as a trained theoretical physicist, they taught him something new in an area to which he could direct his research in the future. That is why reading the present book could be recommended to almost anyone, at whatever stage of professional development from beginning graduate student onwards. For anyone who has pursued the life of the scholar and researcher will have many suggestions circulating inchoately in his mind for years, and with the right stimulus could be persuaded to follow up on some of them – one doesn’t necessarily know which it will prove be, in advance!
As one might expect, editorial decisions as to how much space to allot to various topics can occasionally be questionable. The sixteen pages devoted to mirror symmetry, for instance, are absurdly disproportionate to the intrinsic importance of the subject; similarly for the eleven pages on vertex operator algebras. In this reviewer’s opinion, there is moreover too much on combinatorics and computational complexity overall.
The capsule biographies of ninety-five major mathematicians in Part VI [pp. 733-825] end up being a little dissatisfying: highlights of their respective careers are duly listed, but the summaries are too concise to set the context in a convincing enough way that one could really appreciate the import of each’s great accomplishment, somewhat like the difference between reading a review of a book and consulting the original, or satisfying oneself with the caricature of an old result in a modern textbook versus following the full derivation in the original monograph.
Parts I-V aim at a comprehensive coverage of the entire field of well established mathematics. Since the editorial intent seems to be not just to serve as a reference work but also to encourage young people to enter the field, Parts VII and VIII have been appended to bring the reader up to date by considering a few non-trivial applications, to chemistry, biology, wavelets, algorithm design, communications and cryptography, economics and finance, statistics and probability and finally music and art (physics skipped over here, perhaps because it is so integral to all of mathematics and has received adequate treatment in earlier sections). The articles are, of course, readable and informative but probably unnecessary for anyone who has already ‘caught the bug’ of pure mathematics, so to speak. Lastly, Part VIII concludes with advice to young mathematicians from Michael Atiyah, Béla Bollobás, Alain Connes, Dusa McDuff and Peter Sarnak – an arsenal of tips from some who have succeeded quite well, could help root out a bad habit or confirm a good habit.
Four stars. Succeeds admirably in its goal of offering an up-to-date reference work for the serious student of mathematics. Not for the rank beginner, but ideally suited to those who have attained to a certain degree of mathematical maturity and are looking to branch out into a new field, or two!
Thus, the lament over declining capacity often to be met with in celebrated mathematicians past their prime really reflects an immature wish to stay forever young rather than to engage in a type of intellectual work appropriate to their age (case in point, G.H. Hardy in A Mathematician’s Apology from the Cambridge University Press, first edition 1940). Indeed, anyone who has perused the collected works of luminaries such as Georg Cantor, David Hilbert, Henri Poincaré, Hermann Weyl – who enjoyed the good fortune not to have their lives cut short as sadly happened to the precocious Niels Henrik Abel, Évariste Galois, Bernhard Riemann, Sofia Kovalevskaia, Ramanujan, Alan Turing, John von Neumann and so forth – will see that their mature and lasting contributions to the edifice of mathematics tended to be made well into middle age, if not later (the immortal Gauss, on the other hand, maintains a prodigious level of productivity all through his long life).
What is the distinguishing feature of these so-called universals? They were widely learned in many disciplines and profited from the catholicity of their vision. The narrow specialization that has become the rule in our day and that trends in faculty hiring and grant support incentivize is therefore most regrettable. Is there anything to be done about this situation? Recall that self-education [Bildung] is a work, not of a day, but of a lifetime. Every little bit helps and knowledge patiently accumulated over year after year builds up to a broad perspective that might, in due course, rival that of our forebears; even if one cannot hope ever to equal them, emulation of their supreme achievement is to be enjoined upon us.
Therefore, any resources that might assist such an endeavor of self-education will come as welcome. It can be quite a challenge to break into a mature branch of mathematics, since the popular literature, if any there be, tends to the superficial while the professional literature will have become highly technical and nobody bothers to explain elementary things, for everyone who counts is presumed already to know them. That is why the present Princeton Companion to Mathematics, edited by the eminent British mathematician and Fields medalist Timothy Gowers and published by the Princeton University Press in 2008, turns out to be so handy.
Review of contents: as a reference work designed to appeal to those who already possess some grounding in the field, the emphasis is on conveying a mature perspective on general concepts, many of which one will have seen before but here set forth in a systematic context with all the insight of the genuine expert. Thus, Part I takes up the question of what mathematics is, its language and grammar and the goals of research, exemplified with several concrete examples. In Part II on origins, the development of number systems, algebra, the idea of proof and the crisis in foundations that shook the mathematical world around 1900. The real meat comes in Parts III and IV, the former a series of concise encyclopedia-like entries on all manner of things: from set-theoretic axioms to current topics such as braid groups, Calabi-Yau manifolds, moduli spaces, Ricci flow, von Neumann algebras and so forth. Selection a mix of traditional material and present-day hot topics. For a less piecemeal approach, refer to Part IV, a series of miniature introductions to various branches of mathematics from the point of view of a practitioner pitching to those potentially interested in engaging in advanced research. Mostly good on motivating why one ought to find the current work of interest, whether one feels compelled to go into the respective field or not. The expository part of this companion is rounded out by Part V on theorems and problems, a conspectus of classic results or problem areas one might like to know about (say, Hilbert’s Nullstellensatz or the status of the Mordell conjecture). A reasonable sampling to afford a view of the kinds of things people think about these days.
Articles are written by competent authorities and assume enough background that one can reach a discussion of the exciting ideas at the forefront of current research. The author is identified by name when sufficiently famous (for instance, Terence Tao for articles on topics in analysis). Although the book is a thousand pages long most of the articles remain at a fairly high level so it reads quickly.
A couple highlights: a very good article on set theory in Part IV [pp. 615-634] and another on logic and model theory [pp. 635-646]. Worth mentioning for this reason: these caught this reviewer’s attention because, unlike an article on partial differential equations with which he is already somewhat conversant, as a trained theoretical physicist, they taught him something new in an area to which he could direct his research in the future. That is why reading the present book could be recommended to almost anyone, at whatever stage of professional development from beginning graduate student onwards. For anyone who has pursued the life of the scholar and researcher will have many suggestions circulating inchoately in his mind for years, and with the right stimulus could be persuaded to follow up on some of them – one doesn’t necessarily know which it will prove be, in advance!
As one might expect, editorial decisions as to how much space to allot to various topics can occasionally be questionable. The sixteen pages devoted to mirror symmetry, for instance, are absurdly disproportionate to the intrinsic importance of the subject; similarly for the eleven pages on vertex operator algebras. In this reviewer’s opinion, there is moreover too much on combinatorics and computational complexity overall.
The capsule biographies of ninety-five major mathematicians in Part VI [pp. 733-825] end up being a little dissatisfying: highlights of their respective careers are duly listed, but the summaries are too concise to set the context in a convincing enough way that one could really appreciate the import of each’s great accomplishment, somewhat like the difference between reading a review of a book and consulting the original, or satisfying oneself with the caricature of an old result in a modern textbook versus following the full derivation in the original monograph.
Parts I-V aim at a comprehensive coverage of the entire field of well established mathematics. Since the editorial intent seems to be not just to serve as a reference work but also to encourage young people to enter the field, Parts VII and VIII have been appended to bring the reader up to date by considering a few non-trivial applications, to chemistry, biology, wavelets, algorithm design, communications and cryptography, economics and finance, statistics and probability and finally music and art (physics skipped over here, perhaps because it is so integral to all of mathematics and has received adequate treatment in earlier sections). The articles are, of course, readable and informative but probably unnecessary for anyone who has already ‘caught the bug’ of pure mathematics, so to speak. Lastly, Part VIII concludes with advice to young mathematicians from Michael Atiyah, Béla Bollobás, Alain Connes, Dusa McDuff and Peter Sarnak – an arsenal of tips from some who have succeeded quite well, could help root out a bad habit or confirm a good habit.
Four stars. Succeeds admirably in its goal of offering an up-to-date reference work for the serious student of mathematics. Not for the rank beginner, but ideally suited to those who have attained to a certain degree of mathematical maturity and are looking to branch out into a new field, or two!
Currently reading
February 27, 2021A goal to finish this book by year-end. It might seem a huge window but given what the book covers and is about, I plan to use it as a reference guide while also diving deep down to some of the concepts if I need to. Just reading this one out of curiosity though and because I have always enjoyed mathematics but never really got the opportunity to link its various parts together.
February 22, 2013
Amazing book. The most amazing, encyclopaedic overview of mathematics ever written. This is the book a mathematics lover wants to take with him to the desert (along with Mathematics: From the Birth of Numbers by Jan Gullberg, another take-me-with-you-when-you-get-to-a-deserted-island book).
May 16, 2022
not really a book you finish. more like a shrine you intermittently visit.
December 18, 2020
A Luxury not a Necessity
This is solid, hefty encyclopedic/ reference book. The print quality is outstanding. The breadth of the book is enormous and the depth is somewhat lacking, but given the breadth that's to be expected. I have not read the whole thing and I don't even think it's meant to be read cover to cover. I received this as a gift early in my undergrad and consulted it many times over the years to get a quick overview of various problems, theorems and mathematicians.
There first two sections and the last two sections read nicely. There are also several nice articles in the final section about mathematics as a whole. The sections on Concepts and Theorems and Problems are not organized pedagogically and therefore serve better as a reference. The section on Mathematicians contains a section on most major Mathematicians from Pythagoras to Bourbaki and is ordered chronologically by birth. Each section is short but interesting and there are interesting facts in each one.
I think every mathematician or science enthusiast would enjoy having this on their bookshelf, but it's definitely a luxury not a necessity.
This is solid, hefty encyclopedic/ reference book. The print quality is outstanding. The breadth of the book is enormous and the depth is somewhat lacking, but given the breadth that's to be expected. I have not read the whole thing and I don't even think it's meant to be read cover to cover. I received this as a gift early in my undergrad and consulted it many times over the years to get a quick overview of various problems, theorems and mathematicians.
There first two sections and the last two sections read nicely. There are also several nice articles in the final section about mathematics as a whole. The sections on Concepts and Theorems and Problems are not organized pedagogically and therefore serve better as a reference. The section on Mathematicians contains a section on most major Mathematicians from Pythagoras to Bourbaki and is ordered chronologically by birth. Each section is short but interesting and there are interesting facts in each one.
I think every mathematician or science enthusiast would enjoy having this on their bookshelf, but it's definitely a luxury not a necessity.
June 16, 2022
I can't rate this whole textbook 5/5 truthyfully because there a huge sections of the book that I skipped over and others that I don't fully understand. However, the first chapters that are about the history of mathematics are amazing. I have never read math related like it, not only is it super digestible and easy to understand, this particular section basically talks about Elementary level all the way to post graduate studies level math and connects it all together. I honestly wish everyone could read this part because it connects the "dots" on what you have learned in school and university so well.
July 3, 2018
I read a tiny part of the book at a time. Every time reading it my excitement never stops running high. Speechless. Masterpiece. It gives a big picture of all famous modern math concepts so far.
It’d be great for those who would like to survey the whole math world before making a decision to settle with one Ph.D. dissertation topic. I should’ve found and learned from it myself before starting grad school.
It’d be great for those who would like to survey the whole math world before making a decision to settle with one Ph.D. dissertation topic. I should’ve found and learned from it myself before starting grad school.
Currently reading
April 5, 2020I am reading this book now. But although it aims readers from a wide scope,some parts of it are really dense and sometimes it can be difficult to understand but it is natural in mathematics though. I was thinking of read 20 pages everyday with a real comprehension when I started but it decreased to 10 after that and now it is 7-10 in a time about 2 hours.
October 16, 2023
Interesting and well written, but also kinda not sure why it exists or what it's for. Technical enough to scare away non-academics, but there's no real reason to use it it as a reference over subject textbooks. Open it up and pick a subject for the hell of it.
I keep it on my coffee table to scare non-math people.
I keep it on my coffee table to scare non-math people.
July 7, 2018
A fabulous compilation of brief articles on a wide range of mathematical topics.
September 16, 2020
My perpetual companion.
December 18, 2021
a must read for potential phd mathematics students
Currently reading
December 27, 2022This book serves as an improved offline version of wikipedia for mathematicians.
June 18, 2021
This is probably one of the longest (if not the longest) books I have read cover-to-cover. Having said that, it was quite enjoyable because the book is so easy to read chapter-by-chapter (or section-by-section). The breadth and depth of the book is stunning. It can be read by those with just a high school level of mathematical sophistication (while I think this goal is admirably accomplished for most of the book, there are definitely large stretches of the book where one only gets a thorough appreciation if you have a great deal more math training). Covering a vast array of mathematical subjects is a difficult thing to do, and I was constantly impressed by the writing, the independence of sections with useful cross-references, and the logical, building-up structure (even with basically independent sections!).
The book starts with an introduction to reading math; the history of mathematics; then various branches, problems and theories; brief biographies of mathematicians; math applied in non-physics fields; and, finally, advice for new mathematicians.
This is an excellent volume for anyone interested in mathematics. While it can be read cover-to-cover, it can also serve as an encyclopedia of mathematics for select topics, but where whatever topics it does cover, it covers well. I have enjoyed learning the importance of equivalence classes in their enormous varieties and gained an appreciation for some of the more pure number theory aspects of mathematics, among the many interesting ideas in this book. It really is a wonderful resource and despite being published in 2008, very little of the material has aged.
The book starts with an introduction to reading math; the history of mathematics; then various branches, problems and theories; brief biographies of mathematicians; math applied in non-physics fields; and, finally, advice for new mathematicians.
This is an excellent volume for anyone interested in mathematics. While it can be read cover-to-cover, it can also serve as an encyclopedia of mathematics for select topics, but where whatever topics it does cover, it covers well. I have enjoyed learning the importance of equivalence classes in their enormous varieties and gained an appreciation for some of the more pure number theory aspects of mathematics, among the many interesting ideas in this book. It really is a wonderful resource and despite being published in 2008, very little of the material has aged.
June 16, 2009
This is a thorough and outstanding reference to the more complex ideas in mathematics. It also has a lengthy section on biographies of noteworthy mathematicians and the history of mathematics. Other sections include concepts, branches of mathematics, theorems and problems, influences,, and perspectives. Not one for the bookbag though, at 1034 pages. I do refer to it often, however!
December 20, 2015
This hefty tome is an extremely comprehensive reference work. If you want to find out the general basics of a field of mathematics, chances are it's in here. The sheer breadth means it's not nearly as complete on depth, but that's not its purpose. I think it gives a good, overall picture of the state of mathematics today.
July 5, 2016
An incredible overview of what all the major branches of mathematics. It can serve as a reference book or just as an introduction into mathematics for the curious people. It gives historic references to the development of every branch of modern mathematics, there're some articles about the philosophical issues. Definitely a must have for undergraduate students in math.
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December 12, 2012Both teaching and rational inquiry, at their creative and inspired best, thus lead us to the very threshold of ultimate mystery and induce in us a sense of profound humility and awe.
~~~ Theodore Meyer Greene
~~~ Theodore Meyer Greene
March 26, 2015
you can't actually finish this book like you can't finish a dictionary...but of the articles in this book, it gives a very good reference to any kind of math and shows you more. I wish there was more emphasis on applications and real world examples.
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