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

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

PREFACE

PREFACE TO THE SECOND EDITION

What Is Mathematics? is one of the great classics, a sparkling collection of mathematical gems, one of whose aims was to counter the idea that “mathematics is nothing but a system of conclusions drawn from definitions and postulates that must be consistent but otherwise may be created by the free will of the mathematician.” In short, it wanted to put the meaning back into mathematics. But it was meaning of a very different kind from physical reality, for the meaning of mathematical objects states “only the relationships between mathematically ’undefined objects’ and the rules governing operations with them.” It doesn’t matter what mathematical things are: it’s what they do that counts. Thus mathematics hovers uneasily between the real and the not-real; its meaning does not reside in formal abstractions, but neither is it tangible. This may cause problems for philosophers who like tidy categories, but it is the great strength of mathematics—what I have elsewhere called its “unreal reality.” Mathematics links the abstract world of mental concepts to the real world of physical things without being located completely in either.

I first encountered What Is Mathematics? in 1963. I was about to take up a place at Cambridge University, and the book was recommended reading for prospective mathematics students. Even today, anyone who wants an advance look at university mathematics could profitably skim through its pages. However, you do not have to be a budding mathematician to get a great deal of pleasure and insight out of Courant and Robbins’s masterpiece. You do need a modest attention span, an interest in mathematics for its own sake, and enough background not to feel out of your depth. High-school algebra, basic calculus, and trigonometric functions are enough, although a bit of Euclidean geometry helps.

One might expect a book whose most recent edition was prepared nearly fifty years ago to seem old-fashioned, its terminology dated, its viewpoint out of line with current fashions. In fact, What Is Mathematics? has worn amazingly well. Its emphasis on problem-solving is up to date, and its choice of material has lasted so well that not a single word or symbol had to be deleted from this new edition.

In case you imagine this is because nothing ever changes in mathematics, I direct your attention to the new chapter, “Recent Developments,” which will show you just how rapid the changes have been. No, the book has worn well because although mathematics is still growing, it is the sort of subject in which old discoveries seldom become obsolete. You cannot “unprove” a theorem. True, you might occasionally find that a long-accepted proof is wrong—it has happened. But then it was never proved in the first place. However, new viewpoints can often render old proofs obsolete, or old facts no longer interesting. What Is Mathematics? has worn well because Richard Courant and Herbert Robbins displayed impeccable taste in their choice of material.

Formal mathematics is like spelling and grammar—a matter of the correct application of local rules. Meaningful mathematics is like journalism—it tells an interesting story. Unlike some journalism, the story has to be true. The best mathematics is like literature—it brings a story to life before your eyes and involves you in it, intellectually and emotionally. Mathematically speaking, What Is Mathematics? is a very literate work. The main purpose of the new chapter is to bring Courant and Robbins’s stories up to date—for example, to describe proofs of the Four Color Theorem and Fermat’s Last Theorem. These were major open problems when Courant and Robbins wrote their masterpiece, but they have since been solved. I do have one genuine mathematical quibble (see §9 of “Recent Developments”). I think that the particular issue involved is very much a case where the viewpoint has changed. Courant and Robbins’s argument is correct, within their stated assumptions, but those assumptions no longer seem as reasonable as they did.

I have made no attempt to introduce new topics that have recently come to prominence, such as chaos, broken symmetry, or the many other intriguing mathematical inventions and discoveries of the late twentieth century. You can find those in many sources, in particular my book From Here to Infinity, which can be seen as a kind of companion piece to this new edition of What Is Mathematics?. My rule has been to add only material that brings the original up to date—although I have bent it on a few occasions and have been tempted to break it on others.

What Is Mathematics?

Unique.

Ian Stewart

Coventry
June 1995

PREFACE TO THE REVISED EDITIONS

During the last years the force of events has led to an increased demand for mathematical information and training. Now more than ever there exists the danger of frustration and disillusionment unless students and teachers try to look beyond mathematical formalism and manipulation and to grasp the real essence of mathematics. This book was written for such students and teachers, and the response to the first edition encourages the authors in the hope that it will be helpful.

Criticism by many readers has led to numerous corrections and improvements. For generous help with the preparation of the third revised edition cordial thanks are due to Mrs. Natascha Artin.

R. Courant

New Rochelle, N. Y.
March 18, 1943
October 10, 1945
October 28, 1947

PREFACE TO THE FIRST EDITION

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 modern 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.

Grateful acknowledgement is due to the help of many friends. Discussions with Niels Bohr, Kurt Friedrichs, and Otto Neugebauer have influenced the philosophical and historical attitude; Edna Kramer has given much constructive criticism from the standpoint of the teacher; David Gilbarg prepared the first lecture notes from which the book originated; Ernest Courant, Norman Davids, Charles de Prima, Alfred Horn, Herbert Mintzer, Wolfgang Wasow, and others helped in the endless task of writing and rewriting the manuscript, and contributed much in improving details; Donald Flanders made many valuable suggestions and scrutinized the manuscript for the printer; John Knudsen, Hertha von Gumppenberg, Irving Ritter, and Otto Neugebauer prepared the drawings; H. Whitney contributed to the collection of exercises in the appendix. The General Education Board of the Rockefeller Foundation generously supported the development of courses and notes which then became the basis of the book. Thanks are also due to the Waverly Press, and in particular Mr. Grover C. Orth, for their extremely competent work; and to the Oxford University Press, in particular Mr. Philip Vaudrin and Mr. W. Oman, for their encouraging initiative and coöperation.

R. Courant

New Rochelle, N. Y.
August 22, 1941

HOW TO USE THE BOOK

The book is written in a systematic order, but it is by no means necessary for the reader to plow through it page by page and chapter by chapter. For example, the historical and philosophical introduction might best be postponed until the rest of the book has been read. The different chapters are largely independent of one another. Often the beginning of a section will be easy to understand. The path then leads gradually upward, becoming steeper toward the end of a chapter and in the supplements. Thus the reader who wants general information rather than specific knowledge may be content with a selection of material that can be made by avoiding the more detailed discussions.

The student with slight mathematical background will have to make a choice. Asterisks or small print indicate parts that may be omitted at a first reading without seriously impairing the understanding of subsequent parts. Moreover, no harm will be done if the study of the book is confined to those sections or chapters in which the reader is most interested. Most of the exercises are not of a routine nature; the more difficult ones are marked with an asterisk. The reader should not be alarmed if he cannot solve many of these.

High school teachers may find helpful material for clubs or selected groups of students in the chapters on geometrical constructions and on maxima and minima.

It is hoped that the book will serve both college students from freshman to graduate level and professional men who are genuinely interested in science. Moreover, it may serve as a basis for college courses of an unconventional type on the fundamental concepts of mathematics. Chapters III, IV, and V could be used for a course in geometry, while Chapters VI and VIII together form a self-contained presentation of the calculus with emphasis on understanding rather than routine. They could be used as an introductory text by a teacher who is willing to make active contributions in supplementing the material according to specific needs and especially in providing further numerical examples. Numerous exercises scattered throughout the text and an additional collection at the end should facilitate the use of the book in the class room.

It is even hoped that the scholar will find something of interest in details and in certain elementary discussions that contain the germ of a broader development.