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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- mainder of the book, Courant and Robbins continue


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

This is particularly regrettable Rota wished he had been taught is that mathe. 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. Stewart rationalizes this in his pref- at the moment Courant was in need of help im. My most serious complaints book’s dedication (to the children of Courant). It marked the start penned many well-received plays.” Really? After half a century that would become What Is Mathematics?. however. tem” (pp. One of the ten lessons Gian-Carlo ments he has read. Gilbert and Sullivan were non- in 1985.” nomial coefficients. Their partnership was broken by a quarrel. its predecessor? Stewart has added a thirty-seven- ters of the total credit due. the interested reader may consult [8]. the mathematician will proved too great. Courant and Rob- algebras. any forward references to Stewart’s new chapter inations passing between the coauthors. and the Continuum Hypothesis reported sharply humorous retort [10]). ace: “not a single word or symbol had to be deleted proving and amplifying the mimeographed notes from this new edition. Headlines were made in bins was long overdue for an update when Oxford 1996 when the Robbins problem. As a reviewer I can- the two coauthors at the time of publication. tirement Robbins taught at Rutgers University. Even the original typesetting tistics. the homogenized TEX of the present. I do not most convincing examples of this lesson. the stem from the decision to leave the original text preface (written and signed by Courant alone). old figures are there. individual achievements of Courant and Robbins fore that second retirement Robbins’s name was fade with the passing of time. [6]. often witty.rev-blank. The relationship as they existed in 1947. he taught at the University of North chestral works: they are rarely heard nowadays. was proved automati. Shortly be. matical statistics. Arch medes (p. Yet Carolina and then. The disparity in status between value Stewart’s contribution.) spection. that alert the reader to the falsity of the state- It is ironic really. So too does their joint featured in news reports in conjunction with a re. There are no footnotes nor he had with Courant ended sordidly. Summa- of Sir William Gilbert and Sir Arthur Sullivan. (For edition: if you are familiar with the original. Tran. does the second edition differ from did what and to assign each coauthor three quar. which are now of Robbins’s long. then further biographical details and a discussion of the you will be treated to a wave of nostalgia. Closer in- cinating reading. reserving the necessary amendments for copyright (in Courant’s name only) all prompted the new chapter. Gilbert rizing over two thousand years of mathematical DECEMBER 2001 NOTICES OF THE AMS 1327 . effort seem destined to sparkle forever. Mr. All the contributions Robbins made to mathematical sta. 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. known only to specialists of the Victorian theater. its sanctity. distinguished career in mathe. which had University Press brought out a second edition in thwarted all human effort. make for fas. especially if it seeks to portray the certain algebras. markable software development. until he retired again in 1997. term “dyadic system” in preference to “binary sys- In an interview with Constance Reid he character. then.qxp 10/30/01 3:00 PM Page 1327 did not arrive there as planned. the Four Color became sick of the inquiries (for which he had a Problem. Their believe he focuses sharply enough on the hypo- names have been inextricably linked for sixty years. Herbert Robbins that description is deceptive. but the work it produced is eternal. Courant and Robbins may become the odds with several aspects of his approach. 1996. This version is said to be by Courant and Rob- cally by a theorem-proving program developed at bins and revised by Ian Stewart. The modern reader still finds the indiscreet questions. I am surprised to find myself at search [11]. always stylish. The latter part of Argonne National Laboratory. As a Harvard un. reveals something less wel- Robbins was an excellent writer. In the case of What Is page new last chapter titled “Recent Develop- Mathematics? the temptation to violate the rules ments”. [1]. thetical educated layperson that Courant and Rob- and the bond will only grow stronger. Without doubt. I am reminded bins had in mind as their targeted reader. The basic principles are to not ask who How. the not be so positive. So too may the Newark. His prose was come: the new edition is by and large a photo- sometimes provocative. 400)? Does anyone still use the ter to one third written when he became involved. from 1953 until his retirement working as a team. with recrim. With justification Robbins statuses of Fermat’s Last Theorem. could we not have had a correction to the name of bins’s estimate the manuscript was only one quar. at Columbia University. maticians are more likely to be remembered for Given Stewart’s lengthy record of successful pop- their expository work than for their original re. I could not find even one Serendipity brought him to New York University change in it. has been preserved—quite a welcome change from scriptions of two interviews. are Boolean. ular exposition. which came to be called Robbins existing state of an active field. Even a classic can benefit from a sprucing up dergraduate in 1933 Robbins had conjectured that now and then. Crack open the new died earlier this year at the age of eighty-six. By Rob. pareil. in a work that is not aimed at experts. After serving in the Navy during Sullivan composed many popular vocal and or- World War II. the unaltered. graphic reproduction. After his first re.

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

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