I was very pleased today to obtain a courtesy copy of the Princeton Companion to Mathematics, which is now in print. I have discussed several of the individual articles (including my own) in this book elsewhere in this blog, and Tim Gowers, the main editor of the Companion, has of course also discussed it on his blog. Browsing through it, though, I do get the sense that the whole is greater than the sum of its parts. One particularly striking example of this is the final section on advice to younger mathematicians, with contributions by Sir Michael Atiyah, Béla Bollobás, Alain Connes, Dusa McDuff, and Peter Sarnak; the individual contributions are already very insightful (and almost linearly independent of each other!), but collectively they give a remarkably comprehensive and accurate portrait of how mathematical progress is made these days.
The other immediate impression I got from the book was the sheer weight (physical and otherwise – the book comprises 1034 pages) of mathematics that is out there, much of which I still only have a very partial grasp of at best (see also Einstein’s famous quote on the subject). But the book also demonstrates that mathematics, while large, is at least connected (and reasonably bounded in diameter, modulo a small exceptional set). I myself certainly plan to use this book as a first reference the next time I need to look up some mathematical theory or concept that I haven’t had occasion to really use much before.
Given that I have been heavily involved in certain parts of this project, I will not review the book fully here – I am sure that will be done more objectively elsewhere – but comments on the book by other readers are more than welcome here.
[Update, Sep 29: link to advice chapter added.]