A short "review" for the The Princeton Companion to Mathematics.
Saturday I found a bookshop that I like. It has four floors: basement (bargains), ground (fiction), first (humanistic subjects), second (scientific subjects). This is contrast to the usual fiction and fiction bargains, plus some nude pictures counting as "art".
I stayed on the second floor for five minutes before deciding to buy The Companion. It is remarkable that it is expensive: 80 euro. It is also remarkable that within those five minutes two other people decided to buy it.
Compressed preface. The book leans towards examples rather than formalism. The core is a set of chapters on branches of mathematics that are being developed actively by today's researchers, such as, arithmetic geometry, numerical analysis, and theoretical computer science. Other parts worth mentioning are the concept index (useful for when you are too ashamed to admit you are completely lost in someone's argument) and the set of biographies of mathematicians. I'll let the other goodies surprise you.
My opinion after one hour. Contributors were chosen by expertise and expository skill so I was surprised to not see Knuth. I guess he's getting old and needs to finish his book. The typesetting uses Palatino fonts and is really well done. (Even the annoyingly small parentheses that Terence Tao uses have been fixed and now have the proper size. :p) OK, now let's get to less trivial things! The book is awesome! It's something I would have absolutely loved to have when I was in high-school and it's something that I will use as a bed-time reading for a while. It is very different in flavour from Knuth's books, though. It covers a lot of breadth and intuition but does not go into depth. That is intended, since in-depth study is fairly easy to carry on now that most scientific articles are available online. The big picture, however, can be acquired only by attending conferences, and few are lucky enough to be able to do it.
PS: It's a "review" and not a review because a proper review is done after you read the whole book. That won't happen any time soon.