Encyclopedic Dictionary of Mathematics: Second Edition

Rating:
Summary: Complete and accurate.
Review: I have found this survey of mathematics very useful both as research and teaching tool. It includes almost every topic you can imagine treated in a complete and concise way. It also gives a lot of valuable and difficult-to-find original references, of crucial importance when doing research.

I saw this dictionary for the first time in my Polythecnic Institute's library (Mexico City). I think every math library should own its.

Please read my other reviews in my member page (just click on my name above).

Rating:
Summary: best all-round math book for the mathematician's bookshelf
Review: I've been using this book in my work as a mathematician since I bought the first english-language edition in 1984. The second english-language edition is not enormously different to the first, but it is an improvement. Both have been by far the most useful reference on my bookshelf for 18 years. I have always found that the coverage is in-depth and yet comprehensible (with a bit of pen-on-paper work). It's especially useful for accessing results from areas other than my own speciality. I've found the differential geometry coverage literally better than the dozen texts on DG which I have bought. It must be worth more than 100 books on the shelf. Indexing and cross-referencing are both excellent. Historical context is very good. I use this encyclopedia at least 10 times a week. Virtually every definition I need is here, and every important theorem is summarised.

Rating:
Summary: Indispensable. How did I ever get on without it?
Review: If my house were on fire and I had only sufficient time to rescue four books, I would likely grab my four-volume Encyclopedic Dictionary of Mathematics, Second Edition (EDM2). Truly, this is one of the most useful books I own. As testimony to this fact one need only observe that there are more bookmarks protruding from my copy of EDM2 than there are pages (well, almost).

If you are a mathematician, or if mathematics is central to what you do, you will likely appreciate this collection as it contains wonderfully concise yet informative and authoritative entries on nearly every branch of modern mathematics. Need to refresh your memory on Radon-Nikodym derivatives and their properties? No problem. Are you up on Grassman algebras? If not, you can look it up in EDM2. Interested in game theory? It's in there. What about semi groups, elliptic integrals, perturbation theory, lattice theory, Hilbert spaces, projective geometry, integral geometry, measure theory, geometrical optics, and non-standard analysis? All there!

But simply listing the topics covered in EDM2 will not give you an adequate picture of its utility. What is amazing about the book is how much information it can pack into very few pages, yet manage to keep the discussion quite readable. Don't get me wrong; it doesn't read like a Stephen King novel (nor would you want it to). But the entries are self-contained and cogent enough that you can actually learn a good bit about topics that are totally new to you. Of course, you will want to avail yourself of the many cited references to gain a more complete understanding of any given topic, but you will be well on your way to getting acquainted with fundamental definitions and techniques of a hitherto unfamiliar branch of mathematics.

Here are several examples: If you look up "polynomial approximation" you will find a succinct discussion that rigorously defines such terms Bernstein polynomials, Chebyshev system, Haar's condition, degree of approximation, moduli of continuity, approximation by Fourier expansions, trigonometric interpolation, Lagrange interpolation, and orthogonal polynomials, and all in FOUR terse but readable pages, with plenty of references at the end. The entry on "geometric optics" covers Fermat's principle, Gauss mappings, Malus's theorem, and aberration, all in TWO pages. The succinct one-page biography of David Hilbert is followed by a one-page synopsis of Hilbert spaces. In a mere eight pages on function spaces it provides what amounts to a condensed survey of functional analysis, covering norms, dual spaces, Besov spaces, the Sobolev-Besov embedding theorem, Kothe spaces, etc.

Of course, what you will not find in this book is a single proof. Nor will you find up-to-the-minute esoteric theorems. But then I cannot imagine how such a reference could encompass such things; mathematics is far too vast. Nonetheless, EDM2 has amazing breadth and depth for a meager four-volume collection. And it is written with mathematicians in mind, so the discussions are crisp and rigorous. It's exceedingly well done.

Rating:
Summary: Indispensable. How did I ever get on without it?
Review: If my house were on fire and I had only sufficient time to rescue four books, I would likely grab my four-volume Encyclopedic Dictionary of Mathematics, Second Edition (EDM2). Truly, this is one of the most useful books I own. As testimony to this fact one need only observe that there are more bookmarks protruding from my copy of EDM2 than there are pages (well, almost).

If you are a mathematician, or if mathematics is central to what you do, you will likely appreciate this collection as it contains wonderfully concise yet informative and authoritative entries on nearly every branch of modern mathematics. Need to refresh your memory on Radon-Nikodym derivatives and their properties? No problem. Are you up on Grassman algebras? If not, you can look it up in EDM2. Interested in game theory? It's in there. What about semi groups, elliptic integrals, perturbation theory, lattice theory, Hilbert spaces, projective geometry, integral geometry, measure theory, geometrical optics, and non-standard analysis? All there!

But simply listing the topics covered in EDM2 will not give you an adequate picture of its utility. What is amazing about the book is how much information it can pack into very few pages, yet manage to keep the discussion quite readable. Don't get me wrong; it doesn't read like a Stephen King novel (nor would you want it to). But the entries are self-contained and cogent enough that you can actually learn a good bit about topics that are totally new to you. Of course, you will want to avail yourself of the many cited references to gain a more complete understanding of any given topic, but you will be well on your way to getting acquainted with fundamental definitions and techniques of a hitherto unfamiliar branch of mathematics.

Here are several examples: If you look up "polynomial approximation" you will find a succinct discussion that rigorously defines such terms Bernstein polynomials, Chebyshev system, Haar's condition, degree of approximation, moduli of continuity, approximation by Fourier expansions, trigonometric interpolation, Lagrange interpolation, and orthogonal polynomials, and all in FOUR terse but readable pages, with plenty of references at the end. The entry on "geometric optics" covers Fermat's principle, Gauss mappings, Malus's theorem, and aberration, all in TWO pages. The succinct one-page biography of David Hilbert is followed by a one-page synopsis of Hilbert spaces. In a mere eight pages on function spaces it provides what amounts to a condensed survey of functional analysis, covering norms, dual spaces, Besov spaces, the Sobolev-Besov embedding theorem, Kothe spaces, etc.

Of course, what you will not find in this book is a single proof. Nor will you find up-to-the-minute esoteric theorems. But then I cannot imagine how such a reference could encompass such things; mathematics is far too vast. Nonetheless, EDM2 has amazing breadth and depth for a meager four-volume collection. And it is written with mathematicians in mind, so the discussions are crisp and rigorous. It's exceedingly well done.

Rating:
Summary: The Consumate Personal Mathematics Reference
Review: Prepared by the Mathematical Society of Japan, this two-volume set provides an outstanding reference of mathematics. It is considered by many to be the best available work that is both definitive and encompassing. Treatment is in depth, and presentations assume a solid mathematical background of the reader. This reference is excellent for the researcher working at the doctoral level. Cost of the paperback edition is very reasonable.

Rating:
Summary: The Consumate Personal Mathematics Reference
Review: Prepared by the Mathematical Society of Japan, this two-volume set provides an outstanding reference of mathematics. It is considered by many to be the best available work that is both definitive and encompassing. Treatment is in depth, and presentations assume a solid mathematical background of the reader. This reference is excellent for the researcher working at the doctoral level. Cost of the paperback edition is very reasonable.

Rating:
Summary: Incomparably Great
Review: The Encyclopedic Dictionary of Mathematics is an astonishing achievement. The result of an extraordinary, decades-long collaboration among literally hundreds of celebrated Japanese mathematicians, it will not only never be equalled but in all probability will never be challenged. In two massive volumes, the EDM surveys the whole of the mathematical sciences, both pure and applied, through a series of pithy articles containing the key definitions, methods, and results of every mathematical subdiscipline sufficiently coherent to have a name. It also tabulates vast amounts of information -- homotopy groups of spheres, symmetries of ordinary differential equations, characters of finite groups, class numbers of algebraic number fields, and so forth, seemingly, ad infinitum -- available, as far as I know, in no other single reference work.

Equipped with a detailed and extensive system of indexes, the EDM makes its myriad resources readily available even to the befuddled; the vaguest, most dimly remembered hint is generally enough to track down a topic or result quickly and easily. Each entry, too, offers its own references to the mathematical literature -- and these invariably include the seminal contributions to the particular area under discussion. But it is important to note that the EDM was written by the Japanese, for the Japanese: many of its references direct the reader to (untranslated, Japanese-language) works in (often inaccessible) Japanese mathematics journals.

It is likewise only fair to point out that the EDM is a tool for serious research mathematicians. To keep its component articles brief, it makes full, unapologetic use of a wide variety of notational and expositional economies. The EDM seldom if ever provides a heuristic explanation of anything; although it often gives a bare outline of the historical development of a subject area, it resolutely eschews Toeplitz's "genetic" exposition, in which the crucial problems and examples that engendered a field are placed in the foreground. Only those persons comfortable with a very considerable level of compactness and abstraction in the exposition of mathematical ideas will find the EDM easy reading.

A further cautionary note: although the EDM does offer the same tables of integrals and lists of the zeroes of Bessel functions that make up the bulk of a book like Abramowitz and Stegun's, it makes no attempt to be a handy desk reference for the harried engineer who imagines he may someday need a tidbit of mathematical legerdemain to cope with the odd ODE. First and foremost, the EDM is a sophisticated survey of all extant mathematical knowledge. Those who come to it seeking only the solution to some thorny integral will find, besides the solution they seek, a staggeringly vast, undreamt-of universe of profound mathematical ideas. The experience may well leave them stunned and bewildered for days.

Rating:
Summary: best all-round math book for the mathematician's bookshelf
Review: This dictionary is great. No other work is as comprehensive and detailed. This is a must buy if you are interested in modern mathematics.