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On daily basis I need to check (and re-check and re-check...) some definitions and main theorems that are not in my research area. Usually I accomplish this by a Google-search and/or a visit to our library. Unfortunately this doesn't work too well as the local library is a small one and internet seems to be a contradictive entity on its own.

Are there any must-to-have mathematical encyclopedia that one should invest to when starting to work in a math-oriented research field? I'm mainly interested in discrete mathematics and logic, but it definitely wouldn't hurt to have a wider scope in the book (say, for example, optimization, calculus and some probability theory).

I'm not interested in any study material, but a (probably very heavy) book with short listing / explanation of the basic definitions and the useful theories from different areas.

Any suggestions?

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This perhaps isn't quite what you're looking for, but the Princeton Companion to Mathematics (see is a great reference for the big picture of many fields of mathematics. – Peter Humphries Apr 7 '11 at 7:41
@Peter, that is at least close to what I'm looking for. Also, I just realized that mentioning discrete mathematics was perhaps a mistake, because it seems my bookshelf collects books from the areas that are close to my work, thus the encyclopedia should actually contain information from other areas. – user10891 Apr 7 '11 at 12:35
I agree, wholeheartedly. And let me add that the primary reason it is so good is, in my opinion, that Timothy Gowers (who is active here at MO) did a fantastic job as editor. He asked me to write a section, and I felt I did a quite good job with my first draft, but even though it was on a subject not in his field, he read it carefully and made excellent suggestions, and wasn't satisfied until many drafts later. That said, I think for many purposes Wikipedia is an excellent source for what you ask for. It is not 100% dependable but it is remarkably good and always getting better. – Dick Palais Apr 7 '11 at 22:39
up vote 5 down vote accepted

The suggesstion by Peter Humphries in the comments is good. This book contains a nice overview of many fields in mathematics, although you might find that it does not contain the level of detail you're looking for.

There is also Springer's online Encyclopaedia of Mathematics at that might be what you're looking for. Personally though, I usually do a google search and end up at Wikipedia, or look into a reference book of a specific field, for example Blackadar's "Operator Algebras".

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The Encyclopedic Dictionary of Mathematics ( sounds closest, at the level of MO. It is two volumes, and is of Japanese origin. The Springer Encyclopaedia of Mathematics, all online now at, is an updated version of a Soviet work. Neither of these, however, is going to be a good reference for discrete mathematics. I'm not aware of a book that does what you ask.

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Since you mention Discrete Mathematics the 'Handbook of Combinatorics' (in two volumes, total of 2000+ pages) could be interesting to you. Roughly, it is an organized collection of survey articles covering major fields and techniques of combinatorics (few or no proofs, but lots of definitions, results, and references), each written by an expert in the respective subfield.

I think it is a very nice book; yet, at least a couple of years ago, it was (in my opinion) a bit expensive.

There is another book that seems similar 'Handbook of discrete and combinatorial mathematics' (1000+ pages), which however I do not know.

The more precsise bibliographic informations are:

Handbook of combinatorics. Vol. 1, 2. Edited by R. L. Graham, M. Grötschel and L. Lovász. Elsevier.

Handbook of discrete and combinatorial mathematics. Edited by Kenneth H. Rosen, John G. Michaels, Jonathan L. Gross, Jerrold W. Grossman and Douglas R. Shier. CRC Press.

There are similar handbooks for other fields, too.

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I was looking for a similar book myself, but my requirements are different. I need a book which covers most of the undergraduate level mathematics in one book! I am told no such book exsists, so I've been using "The handbook of mathematics" from Springer - for a while. Here is the Table of contents :

It is a nice book but the problem is that I found it too wide for my needs, and there are no problems to work on! I need a similar but detailed book with examples and problems. However, it might fit your requirements perfectly as it has a wide range of topics covered, so please check it out. I have not checked The Princeton Companion book, but I think it will not be that useful for me.

Sorry for hijacking your thread, but if someone have a recommendation for me please comment.

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I suggest you ask at or one of the other sites listed in the FAQ. – S. Carnahan Apr 14 '11 at 3:59

Some of the best references are people. Reference librarians can help in ways most mathematicians can't. If you have the opportunity, see what Interlibrary Loan programs your local library has. The people at that library may be able to find online sources or services that will deliver whole books or sections to you electronically. Also, building up your list of favorites (and saving it somewhere in the computing cloud) will be useful.

If your research interests are specific enough, you might find an Internet mailing list or discussion group which has their list of favorite references. However, MathOverflow will probably not give you that list, and suggest you try searching other forums for such.

Gerhard "Ask Me About System Design" Paseman, 2011.04.10

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