The Princeton Companion to Mathematics was extensively reviewed, and often praised, all over the mathematical and scientific blogosphere, see e.g. here, here, here and here. Most of this praise is probably well deserved. But where should an interested student (or even a professional mathematician who wants to extend her or his professional range, for that matter) go in order to deepen the knowledge acquired from PCM without getting bogged down into the details of the proofs and other such subtleties that abound in the specialized literature?
Of course, there is plenty of possible answers to this one, and you are welcome to share yours in the comments. However as far as “classical” (basically more or less up to the early XXth century level) mathematics goes, the Oxford User’s Guide to Mathematics appears to provide, at least for me, a reasonable, if not quite perfect, enhancement for PCM.
OUGM has many omissions of its own and certainly could use more editing and proofreading — in particular, in order to make it somewhat more self-contained, but nevertheless this book provides a fairly broad and reasonably deep (for the beginner) panorama of the “classical” mathematics as defined above. For instance, it does not cover category theory and related stuff. However, by and large, OUGM does a quite decent job in helping the beginner to advance her/his understanding of a great number of mathematical disciplines from abstract algebra to probability theory, and I certainly recommend to have a serious look into this book if you really want to deepen your knowledge of the “classical” subjects beyond the PCM level.
P.S. I just cannot miss this opportunity to wish merry Christmas and happy New year to the readers of this blog 🙂
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Alternatively, the ‘interested student’ could try
http://en.wikipedia.org/wiki/Portal:Mathematics
(Another possible use for such sources is to brush up on the material that you’re supposed to know, but don’t.)
Sure! I didn’t say OUGM is the only option but rather an option 🙂 although perhaps it is, as a whole, a bit less uneven than (the totality of) math Wikipedia entries. But, as I have already said in the post, the main problem is OUGM just doesn’t cover many subjects at all.
If breadth of coverage is required, then the student could try the two volume ‘Encyclopedic Dictionary of Mathematics’ (MIT Press), an English translation of a two volume work originally published by the Mathematical Society of Japan. Concise entries, though; more a ‘dictionary’ than an ‘encyclopedia’.
Similarly, Springer provides online access to their Encylopedia of Mathematics and its Applications at
http://eom.springer.de/
It is a translation of ‘Matematicheskaya entsiklopediya’ originally published in the Soviet Union, and subsequently extended with supplements up to 2002. Again, broad coverage, but concise articles, less verbose than PCM.
We’re in agreement about the idea of multiple options.
EoM is nice but also pretty concise. The point I’m trying to make all along is that, for a number of “classical” subjects, OUGM provides longer narratives that fit nicely in between the PCM (where many articles are fairly concise but typically longer than, say, in EoM) and the (lighter) textbooks for students. By the way, if you know of other books that fit roughly the same niche as OUGM (essentially, the books for mathematicians who need to grasp the basics of some subfield which is new to them), please share the info.
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