What Is Mathematics? An Elementary Approach to Ideas and Methods, 2nd Edition (1996)

In the summer of 1937, when I was a young college student, I was studying calculus by going through my father’s book Differential and Integral Calculus with him. I believe that is when he first conceived of writing an elementary book on the ideas and methods of mathematics and of the possibility that I might help with such a project.

The book, What is Mathematics?, evolved in the following years. I recall participating in intensive editing sessions, assisting Herbert Robbins and my father, especially in the summers of 1940 and 1941.

When the book was published, a few copies had a special title page: Mathematics for Lori, for my youngest sister (then thirteen years old). A few years later, when I was about to be married, my father challenged my wife-to-be to read What Is Mathematics. She did not get far, but she was accepted into the family nonetheless.

For years the attic of the Courant house in New Rochelle was filled with the wire frames used in the soap film demonstrations described in Chapter VII, §11. These were a source of endless fascination for the grandchildren. Although my father never repeated these demonstrations for them, several of his grandchildren have since gone into mathematics and related pursuits.

No really new edition was ever prepared since the original publication. The revised editions referred to in the preface were essentially unchanged from the original except for a few corrections of minor errors and misprints; all subsequent printings have been identical to the third revised edition. In his last years, my father sometimes talked of the possibility of a major modernization, but he no longer had the energy for such a task.

Therefore I was delighted when Professor Ian Stewart proposed the present revision. He has added commentaries and extensions to several of the chapters in the light of recent progress. We learn that Fermat’s Last Theorem and the four-color problem have been solved, and that infinitesimal and infinite quantities, formerly frowned upon as flawed concepts, have regained respectability in the context of “nonstandard analysis.” (Once, during my undergraduate years, I used the word “infinity,” and my mathematics professor said, “I won’t have bad language in my class!”) The bibliography has been extended to the present. We hope that this new edition of What Is Mathematics? will again stimulate interest among readers across a broad range of backgrounds.

Ernest D. Courant

Bayport, N. Y.
September 1995