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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


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

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

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

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