rev-blank.qxp 10/30/01 3:00 PM Page 1325 Book Review What Is Mathematics? An Elementary Approach to Ideas and Methods Reviewed by Brian E. Blank What Is Mathematics? An Elementary Approach The book I pulled to Ideas and Methods off the shelf that day Richard Courant and Herbert Robbins did more than open Second edition, revised by Ian Stewart my eyes; it changed Oxford University Press, 1996 my life. I graduated ISBN 0-19-510519-2 from high school later that year, $19.95, 566 pages armed with a twenty- Thirty-two years ago while browsing in my high dollar book certifi- school library, I happened upon an intriguingly ti- cate. It might have tled expository book. I knew astonishingly little of been prudent to save the practice of mathematics at the time. I was my award for the aware that geometry derived from Thales, Pythago- college textbooks ras, and Euclid. Whether or not any geometric the- that I would soon re- orems were discovered after Euclid I could not quire. However, with no further access to say. That algebra and trigonometry were the results my school library, I of conscious development would no more have oc- had a more pressing need. For less than nine dol- curred to me than the idea that the English lan- lars What Is Mathematics? by Richard Courant and guage was the creation of dedicated professional Herbert Robbins became the first volume in my col- linguists. I knew the names and work of many sci- lection of mathematics books. entists—Copernicus, Kepler, and Galileo in as- tronomy; Darwin, Mendel, and Pasteur in biology; Only a sketchy summary of this well-known text Boyle, Lavoisier, and Curie in chemistry; need be given here. The first chapter is largely de- Archimedes, Newton, and Einstein in physics; Jen- voted to number theory. A chapter on real and ner, Harvey, and Koch in medicine; and many oth- complex numbers follows. The discussion includes ers, none of whom were mathematicians. Although Dedekind cuts, Cantor’s theory of cardinal numbers, my recreational reading of Hall and Knight had ex- and Liouville’s construction of transcendental num- posed me to an odd assortment of surnames, such bers. The third chapter is concerned with field ex- as Venn and Horner, I knew of no first-rate scien- tensions and geometric constructions. It includes tist in the field of mathematics. Indeed, I did not a thorough investigation of the impossible straight- really know that there was such a field. edge-and-compass constructions of classical Greek geometry (stopping short of a proof of Lindemann’s Brian E. Blank is professor of mathematics at Washing- Theorem). Chapters on projective geometry and ton University, St. Louis. His e-mail address is topology come next. In preparation for the re- [email protected]. mainder of the book, Courant and Robbins continue DECEMBER 2001 NOTICES OF THE AMS 1325 rev-blank.qxp 10/30/01 3:00 PM Page 1326 with a chapter that introduces the reader to the rig- intellectual maturity” that is mentioned vaguely in orous definition of limit, reinforced by genuine the preface is likely to be a steep hurdle for the proofs of the Intermediate and Extreme Value The- hypothetical layperson Courant hoped to capture orems. Optimization and calculus constitute the as his reader. Twenty-nine pages into the book final two chapters in a rather unusual ordering: the the Prime Number Theorem is stated. It is not calculus of variations is introduced before the cal- proved, of course, but the statement alone is a culus, the definite integral before the derivative. tough swallow for the reader who has not yet seen As is often the case with skilled exposition of the natural logarithm. (At this point Courant and deep subject matter, What Is Mathematics? can be Robbins introduce the logarithm as area under a read at different levels. The professional mathe- curve, a concept that is not made precise until the matician who has not yet encountered the book is last chapter.) The reader who persists soon comes likely to find new things in the chapters on pro- to the Law of Quadratic Reciprocity, a theorem jective geometry and optimization. Also reward- that is exceptionally beautiful to those who have ing are the numerous small details that illuminate an ample supply of that maturity thing, but is material that is more old hat. For example, an ex- thoroughly bewildering to those who do not. cellent “proof” that all positive integers are equal Did Courant and Robbins miscalculate? Not at warns the reader against improper use of mathe- all! As the preface says, a genuine comprehension matical induction. Later on, a quite different “proof” of mathematics cannot be acquired through “pain- of the same absurdity illustrates the need for es- less entertainment.” There are plenty of popular tablishing the existence of an extremum before it books that run away from every mathematical dif- is determined. ficulty; the reader who seeks such a treatment has Of course, What Is Mathematics? was not writ- always been well served. Courant and Robbins has ten for the professional mathematician. Many of earned what ought to be a permanent place in the the topics that are covered in the first six chapters mathematical literature by conveying not only a bring to mind the “transition” books that have treasure-trove of mathematical facts but also the sprouted up in the last decade. They are often in- ideas and methods behind them. To my mind the tended for sophomore or junior mathematics ma- balance of intuition and rigor is just right for a pop- jors and have titles such as Introduction to Math- ular book that intends to lay open the real sub- ematical Reasoning. In fact, Courant and Robbins stance of mathematics. In the matter of proofs I foresaw that their book could serve as the text for cannot think of a single misjudgment. Every proof “college courses of an unconventional type on the that is included, no matter how difficult for the be- fundamental concepts of mathematics.” In con- ginner, is there because it contains an idea that any- ceiving What Is Mathematics?, Courant hoped for one who perseveres can master. There is no bet- a wide readership that would comprise a broad ter way to begin the acquisition of intellectual spectrum of educated laypersons. Accordingly, maturity. What Is Mathematics? “presupposes only knowl- If the world was the way we wished, then What edge that a good high school course could im- Is Mathematics? would have sold like nickel beer. part.” In those days—the first edition appeared in Courant had ambitious expectations for a work he 1941, followed by three revisions at two-year in- said “expresses my own personal views and aims tervals—calculus was rarely if ever part of the high more than any other of my publications.” Although school curriculum. he considered the title “a little bit dishonest,” he In the sixty years during which Courant and Rob- heeded Thomas Mann, who had experienced the bins, as the book is often called, has been in print, effect a tantalizing title can have on marketabil- several distinguished scholars have had the op- ity. The title notwithstanding, sales of What Is portunity to sing its praises. Reviewing the first edi- Mathematics? did not reach Courant’s hopes. If tion, E. T. Bell described the work as “inspirational Constance Reid’s ballpark estimate of over 100,000 collateral reading.” Hermann Weyl thought it “a copies up until 1976 is at all accurate, then annual work of high perfection.” For Marston Morse it purchases by individuals could never have was simply “a work of art.” Even Albert Einstein amounted to more than a trickle. (Oxford Univer- chimed in with high praise. Any subsequent re- sity Press did not respond to my request for a viewer with the slightest humility would be ex- more recent figure.) cused for feeling that his further say-so would A disappointment to its senior author, What Is only be superfluous. To counter the perception of Mathematics? turned into a bitter blow for Herbert immodesty, I should point out that I brought some- Robbins. In 1938 at the age of twenty-three, Rob- thing to my first reading of Courant and Robbins bins completed his dissertation in topology under that none of these learned scientists could boast: the direction of Hassler Whitney. He came to New profound ignorance. Whereas Einstein found the York University as an instructor one year later. text “easily understandable,” I often struggled. Although he “had not the faintest acquaintance Though the prerequisites for What Is Mathe- with or interest in either probability or statistics,” matics? are minimal, the “certain degree of he was assigned such a course when William Feller 1326 NOTICES OF THE AMS VOLUME 48, NUMBER 11 rev-blank.qxp 10/30/01 3:00 PM Page 1327 did not arrive there as planned. It marked the start penned many well-received plays, which are now of Robbins’s long, distinguished career in mathe- known only to specialists of the Victorian theater. matical statistics. After serving in the Navy during Sullivan composed many popular vocal and or- World War II, he taught at the University of North chestral works: they are rarely heard nowadays. Yet Carolina and then, from 1953 until his retirement working as a team, Gilbert and Sullivan were non- in 1985, at Columbia University. After his first re- pareil. Their partnership was broken by a quarrel, tirement Robbins taught at Rutgers University, but the work it produced is eternal. So too may the Newark, until he retired again in 1997.