THE NEWSLETTER OF THE MATHEMATICAL ASSOCIATION OF AMERICA Mathematician Awarded Nobel Prize Volume 14, Number 6 Keith Devlin The awarding of the Nobel Prize in econom­ It was the application ics to the American John Nash on October of Nash's work in eco­ II th meant that for the firsttime in the 93-year nomic theory that led to history of the Nobel Prizes, the prize was his recent Nobel Prize, In this Issue awarded for work in pure mathematics. which he shares with fellow American John When the Swedish chemist, engineer, and phi­ Harsanyi and German 3 MAA Secretary's lanthropistAlfred Bernhard Nobel established Reinhard Selten. Report the awards in 1901, he stipulated chemistry, Nash's contribution to physics, physiology and medicine, and litera­ the combined work ture, but did not create a prize for mathematics. 4 Joint Mathematics which won the award It has been rumored that a particularly bad was in game theory. Meetings Update experience in mathematics at high school led to this exclusion of the "queen of sciences", or Nash's key idea-known nowadays as Nash 6 Search Committee it may simply be that Nobel felt that math­ equilibrium-was developed in his Ph.D. the­ Diary ematics was not, in itself, of sufficient sis submitted to the Princeton University relevance to human development to warrant Mathematics Department in 1950, when Nash its own award. Whateverthe reason, the math­ was just 22 years old. The thesis had taken him 10 Networks in ematicians have had to make do with their a mere two years to complete. He had received FOCUS own special prize, the Fields Medal, though both his B.S. and M.S. degrees in an equally this differs significantly from the Nobel Prize rapid three-year period starting in 1945 and fin- by being restricted to mathematicians who are 13 More on the MAA less than 40 years of age. See Nobel Prize on page 5 Gopher Growing Optimism that Fermat's Last Theorem 14 Project NExT is Launched has been Solved at Last old problem. On June 23, 1993, speaking to a Keith Devlin packed audience at a mathematics conference 17 News from the Hopes rose dramatically in late October that at Cambridge University'S Newton Institute, Sections Fermat's Last Theorem might have been Wiles had outlined a proofof a technical result solved at last, when, on October 25th, about elliptic curves (a special case of the 25 Personal Opinion Princeton mathematician Andrew Wiles re­ Shimura-Taniyama Conjecture), and claimed leased two manuscripts claiming to prove the that this result implied Fermat's Last Theorem result. The first of these papers, a long one as a consequence. The connection between the 33 Employment titled "Modular elliptic curves and Fermat's full version of the Shimura-Taniyama Conjec­ Opportunities Last Theorem", contains the bulk of Wiles' ture and Fermat's Last Theorem had been argument. The second paper,titled "Ring theo­ established by Kenneth Ribet of UC Berkeley retic properties of certain Heeke algebras", in 1986. was written jointly by Wiles and a colleague, Though the bulk of Wiles' enormously com­ Richard Taylor, and provides a key step Wiles plex proof was agreed to have been correct, The Mathematical uses in his proof. within a few weeks a significant mistake was Association of America The new announcement came justover a year discovered in the part of the argument that led 1529 Eighteenth Street, NW to the Last Theorem itself. Washington, DC 20036 afterWilesmade hisinitialdramatic announce­ ment that he had solved Fermat's 350-year See Fermat on page 3 You've been good this year. Give yourself a treat! See page 30 FOCUS December 1994 FOCUS Editorial FOCUS is published by The Mathematical Association of America, 1529 Eighteenth The Calculus Ultrafilter Street Northwest, Washington, DC 20036­ 1385, sixtimes a year: February,April, June, In September, William Dunham and I were the guests on National Public Radio's Talk August, October, and December. ofthe Nation, an hour-long talk show with listener phone-ins. The event that occasioned Editorand Chairofthe MAA Newsletter NPR to devote an entire show to mathematics was the publication in the late summer of Editorial Committee: Keith J. Devlin, our two most recent books, Dunham's The Mathematical Universe (Wiley) and my own Saint Mary's College of California Mathematics: The Science ofPatterns (W. H. Freeman, Scientific American Library). Associate Editor: Donald J. Albers, One of the phone-in comments came from a listener in California-Jim, I believe his MAA Associate Executive Director and name was-who suggested that high schools spend far too much time teaching students Director of Publications and Programs calculus, and that the time might be better spent providing instruction in other subjects. Managing Editor: Harry Waldman, MAA Radio being the medium it is, the responses Bill and I gave led the conversation in the direction of the ubiquity of mathematics, and then the host moved us on to other matters. Production Specialist: Amy S. Fabbri, MAA We never did get a chance to come back to what I think is a particularly important matter. The plain truth is I agree with the listener. I think schools do spend too much time, far Proofreader: Meredith Zimmerman, too much time, teaching calculus. Come to that, I think colleges and universities spend MAA too much time teaching calculus as well. Copy Editor: Nancy Wilson, Saint Mary's College of California In fact, I did begin my response by saying I thought the listener's point had some validity, but I quickly thought better offollowing up on the matter when I realized that I could not Letters to the editor should be addressed to: possibly make my case adequately in the 'sound bite' form of a radio talk show. When Keith Devlin, Saint Mary's College of treading on sacred and hallowed ground, one should tread slowly and warily. And the California, P.O. Box 3517, Moraga, CA 94575; e-mail: [email protected] world of mathematics knows no ground more hallowed than the teaching of calculus. The FOCUS subscription price to individual I guess it is time I defined my terms. By 'teaching calculus' I (and presumably the listener members of the Association is $6.00, from California) mean the kind of activity that, judging from the results it produces, included in the annual dues. (Annual dues concentrates on developing in the student the ability to differentiate and integrate lots of for regular members, exclusive of annual functions. In nearly 25 years of college and university mathematics teaching, I have subscription prices for MAA journals, are encountered scores and scores of students who had become quite expert in performing $68.00. Student and unemployed members this task. Just over halfway through that 25 year period, I started asking those highly receive a 66 percent discount; emeritus members receive a 50 percent discount; new proficient differentiators and integrators to explain to me, in as simple a fashion as they members receive a 40 percent discount for could, just what the processes of differentiation and integration actually are, and what the first two membership years.) exactly are those things called functions that these two processes appear to act upon. Copyright © 1994 by The Mathematical Ifyou have ever tried this, you will surely not be surprised when I say that I almost never Association of America (Incorporated). received an answer that indicated any real understanding on the part of the student. Sure, Educational institutions may reproduce some could give the standard limit definitions, but they clearly did not understand the articles for their own use, but not for sale, definitions-and it would be a remarkable student who did, since it took mathematicians provided that the following citation is used: "Reprinted with permission of FOCUS, the a couple of thousand years to sort out the notion of a limit, and I think most of us who newsletter ofThe Mathematical Association call ourselves professional mathematicians really only understand it when we start to of America (Incorporated)." teach the stuff, either in graduate school or beyond. But I was not looking for a regurgi­ tation of a formal definition. What I wanted was some sense of what the calculus is all Second-class postage paid at Washington, DC and additional mailing offices. about. Why was it invented? What were the basic intuitions that Newton and Leibniz Postmaster: Send address changes to the drew on in order to arrive at the notion of (what we now call) differentiation? Why was Membership and Subscriptions De­ it necessary to develop an elaborate calculus, a battery oftechniques to compute deriva­ partment, The MathematicalAssociation of tives and integrals? What did they think of the fundamental theorem of calculus, and why America, 1529Eighteenth Street Northwest, was that result so important? Washington, DC 20036-1385. Having students who graduate from high school, or college for that matter, without being ISSN: 0731-2040 able to differentiate or integrate does not bother me--especially since computers can do Printed in the United States of America. Printed on recycled paper. the whole job faster and more reliably than people. But having students graduate without a sense of what the calculus is about bothers me greatly. The invention of the calculus was an enormously significant event in the history of humankind, having ramifications that affected most aspects of our lives in one way or another. We fail our students badly if we do not convey to them some sense of that giant leap for mankind. A few will need to know the how of calculus, and that need can be handled when it arises.