What is The Princeton Companion to Mathematics about?

The Princeton Companion to Mathematics is a reference work unlike most reference works: it is meant to be read, not just consulted. Edited by Fields Medalist Timothy Gowers with contributions from dozens of leading mathematicians, it covers the breadth of modern mathematics in a way that assumes a serious reader — someone comfortable with university-level calculus and proof — but does not require expertise in any specific area. It is a book for mathematicians who want to understand the parts of the field they don't work in, and for educated non-specialists who want to understand what modern mathematics actually is.

The book is organized in sections. The early parts cover the language and foundations of mathematics: what proofs are, how mathematical objects like numbers, sets, and functions are defined, and what it means for something to be true in mathematics. These sections are particularly valuable for readers whose training emphasized computation over reasoning — they explain why mathematicians care about rigor and what the alternative would look like. The historical essays that trace how mathematical ideas developed over centuries are among the most readable pieces in the volume.

The core of the book is a series of essays on mathematical concepts and their connections — covering branches like algebraic geometry, number theory, topology, analysis, combinatorics, and probability — written by specialists trying to convey the essential flavor of their area to intelligent outsiders. The quality is uneven, as it must be in a multi-author work, but the best essays are genuinely illuminating. They show not just what a field studies but why its questions matter and what makes its methods interesting.

This is not a book to read cover to cover, though some readers do. It is a book to dip into, to read around a topic you've encountered elsewhere, and to return to when a concept in another book or paper demands more context. The companion functions best as a guide to the landscape of mathematics rather than a complete map of any territory. For anyone serious about understanding what mathematics has become in the last two centuries, there is no comparable single volume.