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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-
brian@math.wustl.edu. mainder of the book, Courant and Robbins continue

DECEMBER 2001 NOTICES OF THE AMS 1325

Although he “had not the faintest acquaintance Though the prerequisites for What Is Mathe. cellent “proof” that all positive integers are equal Did Courant and Robbins miscalculate? Not at warns the reader against improper use of mathe. It is not calculus of variations is introduced before the cal. If tion. more recent figure. a concept that is not made precise until the matician who has not yet encountered the book is last chapter. In the matter of proofs I foresaw that their book could serve as the text for cannot think of a single misjudgment. In fact. effect a tantalizing title can have on marketabil- several distinguished scholars have had the op. an ex. The title notwithstanding. There is no bet- a wide readership that would comprise a broad ter way to begin the acquisition of intellectual spectrum of educated laypersons. that is exceptionally beautiful to those who have ing are the numerous small details that illuminate an ample supply of that maturity thing.” Hermann Weyl thought it “a copies up until 1976 is at all accurate. balance of intuition and rigor is just right for a pop- jors and have titles such as Introduction to Math. In 1938 at the age of twenty-three.qxp 10/30/01 3:00 PM Page 1326 with a chapter that introduces the reader to the rig. Bell described the work as “inspirational Constance Reid’s ballpark estimate of over 100. heeded Thomas Mann. E. a theorem jective geometry and optimization.” In those days—the first edition appeared in Courant had ambitious expectations for a work he 1941.” There are plenty of popular tablishing the existence of an extremum before it books that run away from every mathematical dif- is determined.) The reader who persists soon comes likely to find new things in the chapters on pro. of course. he considered the title “a little bit dishonest. Courant and Robbins has ten for the professional mathematician.” matics? are minimal. to the Law of Quadratic Reciprocity.000 collateral reading. sity Press did not respond to my request for a viewer with the slightest humility would be ex. Robbins. curve. but the statement alone is a culus. intellectual maturity” that is mentioned vaguely in orous definition of limit. maturity. Optimization and calculus constitute the as his reader. ficulty. What Is only be superfluous. They are often in. Twenty-nine pages into the book final two chapters in a rather unusual ordering: the the Prime Number Theorem is stated. I should point out that I brought some. Reviewing the first edi. Courant hoped for one who perseveres can master. (Oxford Univer- chimed in with high praise. (At this point Courant and deep subject matter. reinforced by genuine the preface is likely to be a steep hurdle for the proofs of the Intermediate and Extreme Value The. less entertainment. followed by three revisions at two-year in. as the book is often called. a genuine comprehension matical induction. hypothetical layperson Courant hoped to capture orems. 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. ity.” I often struggled. part. thoroughly bewildering to those who do not. the “certain degree of he was assigned such a course when William Feller 1326 NOTICES OF THE AMS VOLUME 48. What Is Mathematics? “presupposes only knowl. If the world was the way we wished. To counter the perception of Mathematics? turned into a bitter blow for Herbert immodesty. Accordingly. T. Every proof “college courses of an unconventional type on the that is included. who had experienced the bins. The professional mathe. all! As the preface says. Any subsequent re.” For Marston Morse it purchases by individuals could never have was simply “a work of art. NUMBER 11 . Also reward. proved. then What edge that a good high school course could im. What Is Mathematics? can be Robbins introduce the logarithm as area under a read at different levels. To my mind the tended for sophomore or junior mathematics ma. with or interest in either probability or statistics. the reader who seeks such a treatment has Of course. 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.) cused for feeling that his further say-so would A disappointment to its senior author.” Although school curriculum.” he In the sixty years during which Courant and Rob.rev-blank. text “easily understandable. then annual work of high perfection. the definite integral before the derivative. ular book that intends to lay open the real sub- ematical Reasoning. Later on. ideas and methods behind them. tough swallow for the reader who has not yet seen As is often the case with skilled exposition of the natural logarithm. always been well served. For example. What Is Mathematics? was not writ. is there because it contains an idea that any- ceiving What Is Mathematics?. a quite different “proof” of mathematics cannot be acquired through “pain- of the same absurdity illustrates the need for es. no matter how difficult for the be- fundamental concepts of mathematics. ginner. 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. sales of What Is portunity to sing its praises.” Even Albert Einstein amounted to more than a trickle. has been in print. Whereas Einstein found the York University as an instructor one year later. Is Mathematics? would have sold like nickel beer.” In con. He came to New profound ignorance. Mathematics? did not reach Courant’s hopes. but is material that is more old hat. Courant and Robbins stance of mathematics.

the mathematician will proved too great. Crack open the new died earlier this year at the age of eighty-six. could we not have had a correction to the name of bins’s estimate the manuscript was only one quar. ular exposition. There are no footnotes nor he had with Courant ended sordidly. [6]. reveals something less wel- Robbins was an excellent writer. Herbert Robbins that description is deceptive. from 1953 until his retirement working as a team. its sanctity.rev-blank. 1996. until he retired again in 1997. term “dyadic system” in preference to “binary sys- In an interview with Constance Reid he character. the stem from the decision to leave the original text preface (written and signed by Courant alone). the Four Color became sick of the inquiries (for which he had a Problem. however.qxp 10/30/01 3:00 PM Page 1327 did not arrive there as planned. Without doubt. at Columbia University. the homogenized TEX of the present.” Really? After half a century that would become What Is Mathematics?.) spection. Closer in- cinating reading. graphic reproduction. 8–9)? I suppose that there is nothing ized his collaboration with Courant as “pretty wrong with retaining the notation Cin for the bi- close. reserving the necessary amendments for copyright (in Courant’s name only) all prompted the new chapter. Tran. was proved automati. and the Continuum Hypothesis reported sharply humorous retort [10]). So too may the Newark. I am surprised to find myself at search [11]. Courant and Rob- algebras. Summa- of Sir William Gilbert and Sir Arthur Sullivan. My most serious complaints book’s dedication (to the children of Courant). The modern reader still finds the indiscreet questions. After serving in the Navy during Sullivan composed many popular vocal and or- World War II. Mr. has been preserved—quite a welcome change from scriptions of two interviews. often witty. The relationship as they existed in 1947. I am reminded bins had in mind as their targeted reader. I do not most convincing examples of this lesson. Gilbert and Sullivan were non- in 1985. then further biographical details and a discussion of the you will be treated to a wave of nostalgia. distinguished career in mathe. Stewart rationalizes this in his pref- at the moment Courant was in need of help im. pareil. tirement Robbins taught at Rutgers University. Even a classic can benefit from a sprucing up dergraduate in 1933 Robbins had conjectured that now and then. which are now of Robbins’s long. thetical educated layperson that Courant and Rob- and the bond will only grow stronger. make for fas. with recrim. Shortly be. After his first re. Gilbert rizing over two thousand years of mathematical DECEMBER 2001 NOTICES OF THE AMS 1327 . but the work it produced is eternal. in a work that is not aimed at experts. old figures are there. the unaltered. With justification Robbins statuses of Fermat’s Last Theorem. As a Harvard un. ace: “not a single word or symbol had to be deleted proving and amplifying the mimeographed notes from this new edition. its predecessor? Stewart has added a thirty-seven- ters of the total credit due. Their partnership was broken by a quarrel. then. All the contributions Robbins made to mathematical sta. (For edition: if you are familiar with the original. matical statistics. which came to be called Robbins existing state of an active field. I could not find even one Serendipity brought him to New York University change in it.” nomial coefficients. The basic principles are to not ask who How. especially if it seeks to portray the certain algebras. effort seem destined to sparkle forever. 400)? Does anyone still use the ter to one third written when he became involved. Arch medes (p. tem” (pp. any forward references to Stewart’s new chapter inations passing between the coauthors. This is particularly regrettable Rota wished he had been taught is that mathe. are Boolean. known only to specialists of the Victorian theater. markable software development. always stylish. but it does seem dated now Mathematical collaboration is such a wonderful that mathematical notation has standardized on process that conventions have evolved to protect something else. The disparity in status between value Stewart’s contribution. maticians are more likely to be remembered for Given Stewart’s lengthy record of successful pop- their expository work than for their original re. The latter part of Argonne National Laboratory. that alert the reader to the falsity of the state- It is ironic really. Even the original typesetting tistics. His prose was come: the new edition is by and large a photo- sometimes provocative. the not be so positive. In the case of What Is page new last chapter titled “Recent Develop- Mathematics? the temptation to violate the rules ments”. By Rob. the interested reader may consult [8]. he taught at the University of North chestral works: they are rarely heard nowadays. This version is said to be by Courant and Rob- cally by a theorem-proving program developed at bins and revised by Ian Stewart. Yet Carolina and then. As a reviewer I can- the two coauthors at the time of publication. which had University Press brought out a second edition in thwarted all human effort. Their believe he focuses sharply enough on the hypo- names have been inextricably linked for sixty years. does the second edition differ from did what and to assign each coauthor three quar. Headlines were made in bins was long overdue for an update when Oxford 1996 when the Robbins problem. So too does their joint featured in news reports in conjunction with a re. individual achievements of Courant and Robbins fore that second retirement Robbins’s name was fade with the passing of time. [1]. Courant and Robbins may become the odds with several aspects of his approach. It marked the start penned many well-received plays. One of the ten lessons Gian-Carlo ments he has read.

It is hardly shocking to see lateral reading. The high school in which I discovered What lier they hint at analytic continuation when they Is Mathematics? was boarded up many years ago. however. Why? The original authors certainly is time to transfer responsibility for teaching geom- knew about the concept and chose not to include etry to the history department. given way to the caffè latte trade. Surely this not. Stewart relates Robbins in a time when education is so often con- a challenge to the basic continuity assumption fused with painless entertainment? I think that that underlies the solution given by Courant and these questions are already being answered. when given a gratuitous. you can the attention spans of their intended readers. In recent years. Ironically. rected to Courant and Robbins twenty years ago. it a real bargain. Do not the standard Courant and Robbins set. However. As Hardy they forbear mentioning the Riemann Hypothesis. At a minimum. put it.” he admits that an inspired mathematical text can serve a to bending his rules on occasion. super. the treat. fractals can be wrote the senior manager of a supercomputer amusing and even the source of serious mathe. Though ficial treatment. Archimedes its statement and. Courant and Robbins deftly picked tion. That there can be no question that Stewart’s contribu.rev-blank. Tempus fugit. Körner’s It would be pointless and misleading to continue book can fairly be described as “inspirational col- with further quibbles. as Gillman has dation for the sort of student I would have di- pointed out in his review [5]. it dimension. fitting clid has increasingly been charged with irrele- for its place in the chapter on topology. more Yet.qxp 10/30/01 3:00 PM Page 1328 developments. considering the longevity of Robbins limit their treatment of the zeta function this classic. Granted. review urged the placement of The Pleasures of tion enhances What Is Mathematics?. His is not an isolated voice. That danger is Turning the yellowed pages of my first edition. Given that the Counting in high school libraries where it can in- original text is included verbatim in the new edi. In the matter good high school course that Courant spoke of is of Whitney’s problem of an inverted pendulum on in danger of vanishing? How many students will a train. still buy What Is Mathematics? for less than twenty trast. without further preparation. pp. because languages die and mathematical ideas do mann Hypothesis ever be proved true. are they Courant-Robbins wor. Stewart has been much less selective in telling dollars (although it is no longer clothbound. At the time of this writing. I fear the beginner will find the de. “In an age when computing power is abun- Stewart as a pretext for a digression on Hausdorff dant these maths are obsolete. fluence the talented student who might consider 1328 NOTICES OF THE AMS VOLUME 48. “Greek mathematics is permanent. I do not think the average reader will. not stopping there. For example. (real z ). 32–35]. very long time. What effect will shifts There are some other places where old friends of interest have on Courant and Robbins? How of What Is Mathematics? may wish that Stewart had will What Is Mathematics? hold up now that the used his author’s license less freely. ther solved or taken over by other methods. tail overwhelming in several places. is used by vance. The problems for it. ap- thy? preciation of them can wane. tells us what will be remembered when Aeschylus is forgotten. (As long as I am men. I would be re. Stewart’s chapter may be regarded as a wel- out the highlights and kept the discussion within come bonus. has “made no attempt to introduce new topics Certainly Euclid’s Elements has demonstrated that have recently come to prominence. that makes than sixty years. and us about the progress that has been made in less you don’t get as much change back). A few pages ear. For Robbins. company in a highly lauded glimpse of our digital matics. By con.” Hardy penned that thought as Courant and is too much! Robbins prepared for publication. even Eu- ment of dimension by Courant and Robbins. NUMBER 11 . I am prompted to reflect on many ζ(s) to (real) s ≥ 1 (pp. For a long time Although Stewart asserts in his preface that he it seemed like a safe bet. On balance take my word for it—see the Notices review [4]. formally substitute z = ix into the power series for The bookstore where I bought it has long since exp(z) . especially great in the new discussions that receive seeing annotations in my own hand that I no longer inadequate foundation. things. a neat little riddle that attracted the at. Courant and remember making.” So fractals inevitably follow. 480–481). we will know should the (unstated) Generalized Rie. Stewart rattles off permanent even than Greek literature. be prepared to fight their way through Courant and tention of Littlewood [9.) tle resemblance to Courant and Robbins. theorems once proved stay proved forever. that challenge itself relies on many reasons I find that Körner’s The Pleasures of the assumption that the train can move in a way Counting [7] makes a more accessible recommen- that is physically impossible. for character that is rarely found elsewhere. see so far as to extend the domain of thors are dead. and it is excellent. Courant and Rob. tioning the work of other reviewers. We mathematicians the zeta function to complex s . That in itself is not a bad thing: though it bears lit- miss if I did not steer you to [3].” It allows the beginner to peek into that even an accomplished author can fall short of the mathematician’s art. This is a matter of the camel’s nose slipping which geometry entered the schools have been ei- under the tent: once Hausdorff dimension enters. Thus. future [2]. like to think that our subject has a permanent bins must have come to the same conclusion. Now its two au- on this basis.

Cambridge University Press.qxp 10/30/01 3:00 PM Page 1329 a career in mathematics. Math. edited by Béla Bollobás.). Math Soc. 5–24. The Pleasures of Counting. 2 (1991).). Sci. Boston. 276–284. You never know what life it might change when some curious stu- dent pulls it down from the shelf. Birkhäuser. References [1] DONALD ALBERS and GERALD ALEXANDERSON (eds. 1 (1986). What Is Mathematics? should be in every high school library. Revised Edi- tion. Review of The Pleasures of Counting. 75 years of Herbert Robbins: The professional and personal sides. 48 (1942). [9] J. Review of What Is Mathematics?. 1996. Springer- Verlag. Amer. 1996. Monthly 105 (1998). Amer. BELL. The contributions of Herbert Robbins to mathematical statistics. E. Basic Books. Review of What Is Mathematics?. Man- agement Sci. Indiscrete Thoughts (Fabrizio Palombi. [2] JAMES BAILEY. 810–812. Cambridge University Press. [3] E. [5] LEONARD GILLMAN. 1986. [4] BRIAN E. 45 (1998). [8] TZE LEUNG LAI and DAVID SIEGMUND. Math. After Thought. J. [11] G IAN -C ARLO R OTA . Soc. Birkhäuser.rev-blank. Notices Amer. Math- ematical People: Profiles and Interviews. New York. T. ed. Math. BLANK. [10] CONSTANCE REID. Sta- tist. Courant in Göttingen and New York: The Story of an Improbable Mathematician. Littlewood’s Miscellany. 1997. LITTLEWOOD. DECEMBER 2001 NOTICES OF THE AMS 1329 . KÖRNER. 396–400. 485–488. [6] Z. I would urge no less for Courant and Robbins. 1985. GOVINDARAJULU. [7] T. Amer. 1976. W. Bull.