The Princeton Companion to Mathematics

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.

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.

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 behaveice-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.