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 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.

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.

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.