What Is Mathematics?: An Elementary Approach to Ideas and Methods

Richard Courant

“Mathematical knowledge has been regarded for more than two thousand years as an essential component of every cultivated person’s intellectual arsenal. Sadly, the traditional role of mathematics in education is now seriously threatened. Mathematical instruction and learning have devolved into rote memory exercises that produce a passable level of formal proficiency but no true comprehension or increased intellectual independence. This issue is addressed in the new version of the classic work by Herbert Robbins and Richard Courant. Its objective is to give mathematics a new purpose.

What is Mathematics? Second Edition is a brilliant collection of mathematical jewels that gives an engaging and approachable image of the mathematical world. It was written for novices and scholars, students and teachers, philosophers and engineers. This fascinating survey enables readers to explore mathematics as an organic whole rather than just a dry practice in problem-solving. It covers everything from natural numbers and the number system to geometrical constructions and projective geometry, from topology and calculus to matters of principle and the Continuum Hypothesis. Readers can simply pick and select topics of particular interest without hurting their grasp of succeeding portions, thanks to chapters that are essentially independent of one another and sections that lead upward from basic to more sophisticated talks.

What is Mathematics? Second Edition offers new insights into recent mathematical advancements and describes proofs of the Four-Color Theorem and Fermat’s Last Theorem, problems that were open when Courant and Robbins wrote this masterpiece but ones that have since been solved. It has been updated with a new chapter by Ian Stewart.

Formal mathematics is similar to spelling and grammar in that it depends on using regional rules correctly. Like journalism, meaningful mathematics provides a compelling tale. The tale must be true, though, unlike some forms of journalism. The best mathematics is similar to literature in that it engages you both intellectually and emotionally by bringing a story to life before your eyes. Like a superb piece of literature, What is Mathematics offers anyone interested a doorway into the realm of mathematics.”