The Princeton Companion to Mathematics

The Princeton Companion to Mathematics

Packed with nearly 200 articles by some of the world’s leading mathematicians, The Princeton Companion to Mathematics provides an unparalleled look at the field. Organized thematically, this indispensable resource is a vital tool for mathematics students as well as researchers seeking to understand areas outside their specialties.

From basic principles to specialized fields

The Companion is divided into eight sections. The first provides an overview of mathematics and explains its basic concepts. For example, one article describes the principal number systems (natural, integer, rational, real and complex), before describing important algebraic structures and the basics of mathematical analysis. Another article summarizes the goals of mathematical research; other pieces describe groups, vector spaces and more.

The book’s second part is a collection of essays on the historical development of mathematics. What ideas drove the growth of geometry from Euclid’s day to our own? How was mathematical analysis put on a rigorous footing? How did the idea of proof evolve? You’ll find the answers here.

The next section explains more than 75 mathematical objects, concepts and buzzwords, from axiom of choice to zeta function. Want to know what a symplectic form or the incompressible Euler equation is? What about a Sobolev space or the ideal class group? The answers await you.

This is followed by a collection of 24 articles—the heart of the entire book—about different mathematical areas. These articles, considerably longer than those of the previous section, tackle everything from algebra and algebraic geometry to analysis and combinatorics. “Succinctly put, algebraic geometry is the study of geometry using polynomials and the investigation of polynomials using geometry.” So begins János Kollár’s 11-page article on algebraic geometry, which manages to touch on points such as the classification of curves, moduli spaces and singularities and their resolutions. Additional articles in this section address computation, geometry and logic and set theory, as well as number theory and probability.

Profiles of mathematicians, notable theorems and more

We continue with a set of 70 biographical profiles of influential mathematicians, from Apollonius to Hermann Weyl. We read, for example, about Kurt Gödel, whose incompleteness theorem of 1931 destroyed Hilbert’s dream that all mathematical statements should be derivable from a fixed set of axioms.

The following section focuses on notable theorems and open problems in mathematics, from the four-color theorem to the Riemann hypothesis. Here, for example, you’ll find a description of the fundamental theorem of arithmetic, which asserts that every positive integer can be expressed in exactly one way as a product of prime numbers. There are also deft descriptions of Gödel’s theorem and the insolubility of the quintic [equation].

Moving on, we’re treated to a section that covers the intellectual and practical influences of mathematics on other disciplines, such as philosophy, art, biology, chemistry, economics and music. Interested in the mathematics of traffic in networks? If so, an article provides an overview of the mathematical models used to describe everything from cars on the highway to the path of information across the Internet. Another article examines the mathematics of algorithm design, and a third critiques the field of mathematical biology.

The final section includes an assortment of writings. For example, the brilliant geometer Michael Atiyah contributes “Advice to a Young Mathematician,” in which he shares his personal views from a long and distinguished career. Herbert Wilf’s article, “Mathematics: An Experimental Science,” highlights the growing use of computing in mathematics, and Anthony Gardiner’s “The Art of Problem Solving” offers valuable philosophical insights.

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Hardcover: 1008 pages

Publisher: Princeton University Press ( March 01, 2008 )

Item #: 13-0094

ISBN: 0691118802

Product Dimensions: 8.0 x 10.0 x 0.0 inches

Product Weight: 92.0 ounces