The Princeton Companion To Mathematics - Topics and Organization

Topics and Organization

The book concentrates primarily on modern pure mathematics rather than applied mathematics, although it does also cover both applications of mathematics and the mathematics that relates to those applications; it provides a broad overview of the significant ideas and developments in research mathematics. It is organized into eight parts:

• An introduction to mathematics, outlining the major areas of study, key definitions, and the goals and purposes of mathematical research.
• An overview of the history of mathematics, in seven chapters including the development of important concepts such as number, geometry, mathematical proof, and the axiomatic approach to the foundations of mathematics. A chronology of significant events in mathematical history is also provided later in the book.
• Three core sections, totalling approximately 600 pages. The first of these sections provides an alphabetized set of articles on 99 specific mathematical concepts such as the axiom of choice, expander graphs, and Hilbert space. The second core section includes long surveys of 26 branches of research mathematics such as algebraic geometry and combinatorial group theory. The third describes 38 important mathematical problems and theorems such as the four color theorem, the Birch and Swinnerton-Dyer conjecture, and the Halting problem.
• A collection of biographies of nearly 100 famous deceased mathematicians, arranged chronologically, also including a history of Nicolas Bourbaki's pseudonymous collaboration.
• Essays describing the influences and applications of mathematics in the sciences, technology, business, medicine, and the fine arts.
• A section of perspectives on the future of mathematics, problem solving techniques, the ubiquity of mathematics, and advice to young mathematicians.

Despite its length, the range of topics included is selective rather than comprehensive: some important established topics such as diophantine approximation are omitted, transcendence theory, differential geometry, and cohomology get short shrift, and the most recent frontiers of research are also generally not included.