The Princeton Companion to Mathematics

First an advice: please read the Editorial reviews, for no review from a single reader is likely to do better than the former taken collectively. Having said that, I feel that I might have more freedom to confine myself to a totally personal and partial viewpoint in what follows. Moreover, my account here is mainly intended towards those contemplating a career in Mathematics, although it might be also of some use to others. K.J.Devlin once said in a review that when T.Jech's "Set Theory" first came out in 1978, the graduate logic students went without food in order to buy it. I didn't know whether Devlin's statement was justified, but I did follow his advice to buy it in my graduate years - fortunately still with something to eat after the spending. In the case of the Princeton Companion, I would have no hesitation to buy it even if it meant that I had to starve. And I recommend a budding mathematician to do the same, if necessary. Why is the Companion so highly recommended? It is mainly because of the increasingly extreme specialization taking place within today's Mathematics (and other sciences, perhaps to a lesser extent). People often complain that they don't know what the mathematicians are doing. Yet it will be more embarrassing if the mathematicians themselves also admit that they don't know much about Mathematics either. For it seems fair to say that today an average PhD candidate in Math will be familiar with less than 1% of the topics under investigation by their colleagues. To make the word "familiar" more definite in this context, I will adopt the following rough, working definition: Suppose you are able to get access to any graduate course or seminar in any university in the world. Now randomly go to any such course/seminar. If you become able to follow and participate in their discussions after one month's study and struggle, then I will count you as "familiar" with that course/seminar topic. And my claim is that the probability for an average PhD candidate to get lost in the math topics currently under study will be more than 99%. Here I will give no discussion on how my claim is to be justified or whether - if it is true - any mathematician should worry about it at all - if all that is desired is to stay in one's chosen niches of specialization and continue producing specialized articles and books to survive the fierce academic competition. To some extent the over-specialization is indeed inevitable, due to the vast explosion of human knowledge during the last 100 years. But if you are unhappy with your own unfamiliarity with Math and want to do something about it, then as far as I know this Companion will be your best aid. As I have said, I heartily agree with most of the Editorial reviews and they will already give you a fair assessment of the content of the Companion. There is no point to repeat their remarks. As for my own perceptions, I am most surprised to discover that the Companion provides so many surprises. First of all, I am surprised by its readability and accessibility. I bet that even an undergraduate student can have a fair share of the gems contained therein. So far I have joyfully read about one-tenth of this tome, in spite of my previous ignorance of 99% of its content. I am eager to learn more from it when I have more time. But this accessibility is not done by making its content shallow or superficial or confining itself to pre-20-century mathematics. E.g. I'm surprised to be enlightened by many insights even from those topics where my knowledge is better, therefore not expecting much from such supposedly "introductory" accounts beforehand. How the editors and authors have managed to achieve this combination of readability and depth at the same time still seems somewhat mysterious to me. But there is no doubt that they have thrown in huge efforts for that purpose. Another surprise is to see the willingness of many first-rate mathematicians to speak their mind. Mathematicians are always passionate about their researches, but this passion is seldom manifest in their articles or books. When they start reporting their discoveries to others, they often behave ice-cold and give little clues about how the hell they had discovered or arrived at their results in the first place. This is partly because the actual process of discovery is usually very long, devious and full of false starts. It will be both less dignifying for the revered mathematicians to exhibit their human weaknesses to the readers and usually there will not be enough space in the articles anyway. Moreover, mathematical arguments must be highly logical in structure, which forces their presentation to be more analytical rather than synthetical, although the discovery process will usually be more synthetical in nature. So it is quite easy for a reader to know all the leaves while still not seeing the tree itself when reading a piece of math, let alone participating in the actual creative process spanning across diverse mental states of the authors during their investigation. It is therefore unusual that the Companion offers so many insights on the more psychological and human side of mathematical research. Some such examples are in the sections "Advice to a Young Mathematician", "The Art of Problem Solving" and also sprinkled elsewhere throughout the book. I especially wish that in my student years I could have read something like the 10-page "Advice to a Young Mathematician" by five fine mathematicians. But actually, even if I had done so, I might be too narrow-minded or cocky or ignorant to appreciate their counsel at that stage. Alas, one has to learn from one's own mistakes. Nevertheless, if a budding mathematician buys the Companion, reads those 10 pages and carefully reflects on them, then in my opinion it is already worth the money spent - even if nothing else in the book is made use of.

The Princeton Companion manages to be so much more than your typical popular mathematics book. While obviously a thousand pages could never hope to include everything that has been written on this vast subject, where this text shines is its uncanny ability, relative to its page limit, to paint a compelling picture of the modern mathematics landscape (emphasis on modern; if you don't know what I'm talking about read the preface) that is both thorough and also motivating. More precisely, this book gives an overview of essentially all of the most important areas of active research mathematics, while striking a balance between being too glib versus overly dry and verbose. If you're looking for the former see pop math books galore. The latter being something like a graduate mathematics textbook or monograph assuming all sorts of advanced prerequisites that might take semesters or even years to understand. Clearly it would be impracticable to attempt to include that level of detail here. 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.

As a Princeton alum, I can safely state that the University can teach you anything you want to know. The place is awash with resources, professors, technicians, and visiting scholars to whom you can approach and ask them for explanation or guidance. P U Press is the publishing component of that, and this book is emblematic of its catalog. I am a physical scientist who constantly needs to check a mathematical subject to support my work. So much of everything I need from the basic to the esoteric is right here. The book rarely leaves my desk. This does not mean it's superficial. While not comprehensive in every subject, it is solid enough for provide a proper grounding or refresher. And at this price it's a bargain

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.

I bought this book along with the Princeton Companion to Applied Mathematics and have no regrets whatsoever. It has brought me nothing but joy and fascination so far, after reading several pages and skimming all across the book. Just perfect for a layman with a math undergrad degree who wants to sample diverse topics without diving into the sea of badly-written or poorly-curated articles that is Wikipedia or StackOverflow or Reddit. The writing has so far (in my admittedly cursory reading) been nothing but superb. Timothy Gowers and his collaborators seem to have a knack for making things “as simple as possible and no simpler”, which typically reflects mastery.

A great book with many topics covered. But still, despite its size, it skips on a some subjects, ignoring some fundamental structures. May be it could have been reduced on history that does not directly relate to a topic, and on mathematicians as the selection criteria are debatable and add unnecessary volume. I suggest also less probability-statistics and more arithmetic and group theory (mainly permutation groups). I find it very readable, but still leaving me with some frustration as I often wanted more. For the editor, certainly a challenge in the selection and the degree of depth! In spite of all the weaknesses a great book in my library!

When I run across a mathematical term or topic that I don't understand, my first stop typically is Wikipedia, sometimes Wolfram Mathworld. But the articles there don't always contain the best writing for my interest. For example, I may be left wondering what a "sheaf" is really all about. What was the motivation to create it? What is the true essence of it? I can follow the hyperlinks in Wikipedia and learn that the "elephant has a trunk" from one article and that it has a tail from another article and yet another article may talk about the legs. But I may still not really get it: What is the "elephant" (or sheaf or Teichmüller space or...) really all about? Wikipedia can sometimes be too fragmented and, ironically, too technical. It tries to be accurate and detailed and founded on authoritative references but it isn't always the best source to get the intuition about something, at least for mathematical topics. Some might say you should make the investment and get a book on the topic you want to know more deeply. Yes, I could buy a book about algebraic geometry. Indeed, I did. But, typically, a book requires a pretty big investment of time and focus to work through. And, without the intuition up front, it isn't clear that the payoff at the end will be worth it. The Princeton Companion fills the gap. The focus is on providing a "feel" for topics ranging from elementary to very advanced: motivations, simple examples, intuitive exposition. There is not an emphasis on proofs or completeness. One aspect I really like is that for any particular topic at random, it typically contains the full range of the topic (from simple to advanced), so that no matter what your current understanding is, you can find a place within the narrative to connect with the material and then go from there. The authors seem to have taken the view of extreme editing, leaving out as much as they can, so that what they leave in is a kind of distillation of the essence of each topic. What is left is fairly well integrated. As a result, reading it is like being on a grand tour of mathematics: You get to sample the best restaurants, see the most beautiful art, wander the nicest shops, without having to commit to living there full-time.

It's a gem. The articles are compact but by no means lacking in scope and depth. The authors are excellent. Every genuine mathematician, i.e. a someone who seeks to understand theorems and prove them, and who loves the structure of this vast structure, will cherish it.

This epic single volume spans all major areas of modern mathematical research. Each section was written by some of the most eminent mathematicians of our time, and covers topics in Algebra, Statistics, Topology, Computation, Combinatorics and so forth, and contains the basics of topics written in a relatively accessible but still rigorous manner. The explanations are concise, yet precise, and are designed for readers who are new to the topic. Make no mistake, however, "relatively accessible" still means that this book requires some background in university-level mathematics. Though the book does contain an introductory chapter that goes over all prerequisite material, readers who are not used to thinking about advanced mathematics will likely find many sections to be quite a slog to read through, especially once the notation and the abstraction starts stacking up. But it would be unfair to fault the authors and editors for this -- writing rigourously yet concisely about Modular Forms, Orbifields, Differential Topology, etc. is an impossible task for people with no background at all. Indeed an illustrative example would be the overview of trigonometric functions, which would likely be a chapter in a high school textbook but is covered in the Companion in a single page. To readers with a bit of mathematical background, however (perhaps a few proof-based courses in university), I reckon the Companion to be an invaluable reference for all areas of mathematics.

Amazingly coherent to understand. Although print is too small. Takes time to read.

Una buena lectura. Te hará pasar buenas e interesantes horas de lectura. No es libro de texto, sin embargo, encontrarás ramas, hechos, problemas, teoremas, conjeturas de las matemáticas que sin duda te interesarán.

An encyclopedia of maths, worth the money.

Reading this book to understand the various fields in mathematics is so revealing. The chapters re written by subject experts, which I am not. I am verse in topics in applied mathematics, especially mathematical physics, but not pure math. This is a reavealing landscape painting of the broad field of mathematics.