For more than two thousand years some familiarity with mathematics has been regarded as an indispensable part of the intellectual equipment of every cultured person. Today the traditional place of mathematics in education is in grave danger. Unfortunately, professional representatives of mathematics share in the responsibility. The teaching of mathematics has sometimes degenerated into empty drill in problem solving, which may develop formal ability but does not lead to real understanding or to greater intellectual independence. Mathematical research has shown a tendency toward overspecialization and overemphasis on abstraction. Applications and connections with other fields have been neglected. However, such conditions do not in the least justify a policy of retrenchment. On the contrary, the opposite reaction must and does arise from those who are aware of the value of intellectual discipline. Teachers, students, and the educated public demand constructive reform, not resignation along the line of least resistance. The goal is genuine comprehension of mathematics as an organic whole and as a basis for scientific thinking and acting.

Some splendid books on biography and history and some provocative popular writings have stimulated the latent general interest. But knowledge cannot be attained by indirect means alone. Understanding of mathematics cannot be transmitted by painless entertainment any more than education in music can be brought by the most brilliant journalism to those who never have listened intensively. Actual contact with the content of living mathematics is necessary. Nevertheless technicalities and detours should be avoided, and the presentation of mathematics should be just as free from emphasis on routine as from forbidding dogmatism which refuses to disclose motive or goal and which is an unfair obstacle to honest effort. It is possible to proceed on a straight road from the very elements to vantage points from which the substance and driving forces of modem mathematics can be surveyed.

The present book is an attempt in this direction. Inasmuch as it presupposes only knowledge that a good high school course could impart, it may be regarded as popular. But it is not a concession to the dangerous tendency toward dodging all exertion. It requires a certain degree of intellectual maturity and a willingness to do some thinking on one's own. The book is written for beginners and scholars, for students and teachers, for philosophers and engineers, for class rooms and libraries. Perhaps this is too ambitious an intention. Under the pressure of other work some compromise had to be made in publishing the book after many years of preparation, yet before it was really finished. Criticism and suggestions will be welcomed.

At any rate, it is hoped that the book may serve a useful purpose as a contribution to American higher education by one who is profoundly grateful for the opportunity offered him in this country. While responsibility for the plan and philosophy of this publication rests with the undersigned, any credit for merits it may have must be shared with Herbert Robbins. Ever since he became associated with the task, he has unselfishly made it his own cause, and his collaboration has played a decisive part in completing the work in its present form.

R. Courant

New Rochelle, N. Y.
August 22, 1941


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