What is mathematics?

About the book

What Is Mathematics? is an exposition of mathematical ideas by Richard Courant, a mathematician, and Herbert Robbins, written for readers who want more than a formula sheet. It treats mathematics as a connected field rather than a set of separate techniques. The book ranges across number theory, geometry, topology, calculus and the foundations of analysis, building each topic from elementary questions toward substantial results.

The authors aim to convey what mathematicians do: construct arguments, test conjectures and see how ideas link. They favor explanation over formal machinery, so a reader meets the reasoning behind a theorem as well as its statement. The result is a survey with real intellectual weight. It assumes curiosity and some persistence, and it rewards readers who work through the examples instead of reading passively.

How to read it

Approach the book as a course of study. Its arguments unfold in sequence, and later sections lean on ideas introduced earlier. Keep paper nearby and attempt the problems; the text often teaches by inviting the reader to reconstruct a step before it is supplied.

The prose is clear but dense in places, especially where topology and analysis turn abstract. Expect to reread passages. The book does not hide difficulty behind friendly graphics, and its examples carry conceptual weight. Readers without a background in calculus will find the later material demanding, though the early chapters on numbers and geometry are more accessible.

Who it suits

This suits adults who studied some mathematics and want to understand its structure, teachers seeking richer explanations of familiar topics, and scientifically minded readers who prefer argument to summary. It also serves students preparing for more advanced work, provided they are ready to engage actively. Anyone wanting a quick overview of formulas will find it slow going.

Reading considerations

The book reflects mathematics as presented in the mid-twentieth century, and some notation and emphasis will feel dated. Later editions include supplementary material by other hands, so check which edition you hold before relying on pagination or specific passages. Treat its claims about foundations as a starting point for further reading rather than a final word.