Princeton Companion to Mathematics

I finally got a copy of The Princeton Companion to Mathematics. PCM is over 1000 pages of extremely well written, self-contained essays on a variety of topics in pure and applied mathematics. The book aims to be comprehensive in its coverage, an amazing ambition that seems to have been achieved, at least at the level of overview for non-specialists. Modern mathematics is such a broad and deep subject that PCM will be of use to students and experienced researchers alike. Every essay I have looked at is a pleasure to read.

Because the essays can be read independently, I think the ideal form for this book would be as an online document. Perhaps the publishers could work out a system for granting online access to people who buy the book (copy control issues notwithstanding)?

For hard core researchers, the alternative is the Japanese Encyclopedic Dictionary of Mathematics (EDM), which was prepared by the Mathematical Society of Japan. This book is highly compressed (even at 2000 pages) and is not the place to look for a cursory overview or readable introduction. One reviewer wrote:

EDM 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.

...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.

This lack of heuristic background and examples has made the EDM very difficult for me to use. It's OK for looking up results, but not for getting a feel for a completely new area. Perhaps there are people who are sufficiently strong that they can just pick up and read the EDM the way I can read the PCM, but probably not very many!

PCM was edited by Fields Medalist Timothy Gowers, who recently posted the latest errata on his blog.

Below are the contents:

I. What is Mathematics?
II. Ideas of Mathematics
III. Mathematical Objects
IV. Branches of Mathematics
V. Mathematicians
VI. Theorems and Problems
VII. The Influence of Mathematics
VIII. Miscellaneous

The following sample articles can be accessed as described below. Enter Username “Guest” and Password “PCM” at this site, then click “Resources” in the sidebar, then “Sample articles” in the sidebar.

Section Title

I Some Fundamental Mathematical Definitions
I The Language and Grammar of Mathematics
II Geometry
III Braid Groups
III Designs
III Determinants
III Distributions
III The Exponential and Logarithmic Functions
III Function Spaces
III Hilbert Spaces
III Knot Polynomials
III Metric Spaces
III Normed Spaces and Banach Spaces
III Permutation Groups
III Riemannian Metric
III Quaternions
III The Exponential and Logarithmic Functions
III The Euclidean Algorithm and Continued Fractions
III The Simplex Algorithm
III The Spectrum
IV Algebraic Geometry
IV Algebraic Numbers
IV Arithmetic Geometry
IV Differential Topology
IV Dynamics
IV Enumerative and Algebraic Combinatorics
IV High-Dimensional Geometry and Its Probablistic Analogues
IV Moduli Spaces
IV Operator Algebras
IV Probabilistic Models of Critical Phenomena
IV The Fourier Transform
V George Birkhoff
V János Bolyai
V Arthur Cayley
V Pierre Fermat
V Kurt Gödel
V Jacques Hadamard
V David Hilbert
V Sonya Kovalevskaya
V Nicolai Ivanovich Lobachevskii
V Pierre-Simon Laplace
V Isaac Newton
V Emmy Noether
V Jules Henri Poincaré
V Karl Weierstrass
VI Dvoretzky's Theorem
VI Gödel's Theorem
VI Liouville's Theorem and Roth's Theorem
VI The Atiyah–Singer Index Theorem
VI The Banach–Tarski Paradox
VI The Classification of Finite Simple Groups
VI The Fundamental Theorem of Algebra
VI The Fundamental Theorem of Arithmetic
VI The Insolubility of the Quintic
VII Analysis, Mathematical and Philosophical
VII Mathematical Biology
VII Mathematics and Chemistry
VII Mathematics and Economic Reasoning
VII Reliable Transmission of Information
VII Routing in Networks
VII The Mathematics of Algorithm Design
VII The Mathematics of Money
VIII Advice to a Young Mathematician
VIII Mathematics: An Experimental Science
VIII The Art of Problem Solving