THE NEWSLETTER OF THE MA THEMATICAL ASSOCIATION OF AMERICA AParade ofErrors c1usion that what Wiles had done was prove that Keith Devlin the Last Theorem held in a universe governed by Throughoutits long history,Fermat'sLast Theo­ hyperbolic geometry. Observing that in a hyper­ rem seems to have generated errors. The long bolic universe it is possible to square the circle, In this Issue list of those that were led astray by this most vos Savant concluded her reasoning in this way: tantalizing ofproblems almost certainly started 2 Editorial • Image "Has Fermat's last theorem been proved, or has it with Fermat himself, who surely did not have a not? I would say it has not; if we reject a hyper­ Problem Ahead? proof, and continuedwith such illustrious names bolic method of squaring the circle, we should as Euler and Lame. also reject a hyperbolic proofofFennat's last theo­ 3 The Hiccup in Wiles' Time alone will tell if Andrew Wiles is to join rem. This is not a matter of merely changing the Proof this list. Even if he does, his recent work on rules (for example, using a ruler as a measuring elliptic curves will stand on its own merits as a device instead ofa straightedge). It is much more 4 Secretary's Report phenomenal piece ofresearch. The same cannot significant than that.Instead, itisa matter ofchang­ be said for the unfortunate article that appeared ing whole definitions. And, regardless, it is logically inconsistent to rejecta hyperbolic method 5 1994 NSF Calculus in Parade magazine on November 21 last year. Written by columnist Marilyn vos Savant, the of squaring the circle and accept a hyperbolic Reform Workshops article was titled "The World's Most Famous method of proving EL.T!" Math Problem Has Finally Been Solved...Or 7 The New Job Diary For the record, the source ofvosSavant'ssubstan­ Has It?" tial confusion seems to be this. Wiles' proofuses "Have you ever tried to prove it, and if so, did objects called modular curves. These are regular 10 Edyth May SlitTe you succeed?" a reader had asked, with refer­ algebraic objects that Fermat himselfwould have Award Winners ence to the Last Theorem. no trouble recognizing were he to come back to life today. Seen from the viewpoint of complex "No, I never tried," replied vos Savant, "and I 12 Personal Opinion analysis, modular curves may be presented interms don't think I would have succeeded even if I of the upper half plane. The upper half plane can had." be used to construct a model of hyperbolic geom­ Letters to the Editor 13 Up to this point, Iam sure everyone would agree. etry. And there, at last, you see mention of It was what came next that caused mathemati­ hyperbolic geometry. Wiles' proof can be pre­ 17 Washington cians across the country tochoke on their Sunday sented with no reference to hyperbolic geometry Outlook morning muffins. "Moreover, I don't think the whatsoever. current work succeeds in proving 'Fermat's last In response to the vos Savant article, MAA Presi­ 18 MAA Strategic Plan theorem' either---even if no mathematical er­ dent-Elect Kenneth Ross wrote to her. He also rors are discovered in it," the article continued. asked FOCUS ifitwould publish his letter.Along 21 Addresses from Vos Savant then goes on to explain the reason­ with many other mathematicians, he was con­ cerned that the record be set straight. Monthly Centennial ing behind this extraordinary claim. To make a Mathematicians are well used to finding errors in Celebration long story short, it seems that vos Savant had picked up on a mention ofhyperbolic geometry proofs, but theyare hardly so stupidthat they would that appeared in accounts of recent work con­ fall into the kind of trap vos Savant described. Employment 46 nected to Fermat's Last Theorem, including the Ross eventually received a short, ungracious re- Opportunities work ofWiles. Putting 2 and 2 togetherto make 163, the Parade columnist jumped to the con- Please see Parade on page 3 The Mathematical Association of America MINNEAPOLIS MATHFEST 1529 Eighteenth Street, NW Washington, DC 20036 See page 28 FOCUS April 1994 FOCUS Editorial FOCUS is published by The Mathematical Association ofAmerica, 1529 Eighteenth Street Northwest, Washington, DC 20036­ Image Problems Ahead? 1385,six times a year: February.April.June, August. October. and December. To be or not to be? That is the question when it comes to Andrew Wiles' announced proofof EditorandChairof theMAANewsletter Fermat's Last Theorem, reported in the August 1993 issue of FOCUS, not to mention through­ Editorial Committee: Keith J. Devlin, out the world's press. Saint Mary's College of California Wiles made his claimat the end ofa series ofthree lectures he gave at a small meetingofnumber Associate Editor: Donald J. Albers, theorists at the Isaac Newton Institute in Cambridge, England, last June. The powerful new MAA Associate Executive Director, and techniques he outlined in his proof, together with his own track record as a research mathema­ Director of Publications and Programs tician, were enough to convince the audience that the new proof was probably correct. And, ManagingEditor: HarryWaldman, MAA since that audience included many ofthe world's most highly qualified experts in the area, that Production Specialist: Amy E. was good enough for the rest of us. Such was the complexity of Wiles' argumentthat, even with Stephenson, MAA his 200-page proof in front of us, most of us would in any case have to rely on the judgment of those experts. Not that Wiles was giving out copies of the proof. So we accepted the Proofreader: Meredith Zimmerman, MAA assurances of others that the argument looked sound. CopyEditor: Nancy Wilson. Saint Mary's Several months later, with doubt still hanging over the proof, it seems to me that our attitude College of California was perfectly reasonable. Mathematics is unique to human learning in having an ultimate, decisive notion ofproof. But in the form the discipline is actually practiced, things are never Letters to the editor should be addressed to: Keith Devlin. Saint Mary's College of so cut and dried. We make guesses, we follow hunches, we make value judgments. Mostly, we California. P.O. Box 3517, Moraga. CA are right. But, occasionally, we are wrong. 94575, E-mail: devlin@ strnarys-ca.edu As I recall, everyone involved said at the time that there was, of course, always the possibility The FOCUS subscription price to ofsome subtle errorin the proof. This is normal in mathematics, especially with a proofas long, individual members of the Association is as complex, and as novel, as Wiles'. Wiles himself refused to circulate copies of his paper, $6.00, included in the annual dues. (Annual preferring toawait the outcomeof the refereeing process for publication inthejournallnventiones dues for regular members, exclusive of Mathematicae. It was very much business as usual in the mathematical community. annual subscription prices for MAA journals. are $68.00. Student and What was different in this case was that a proofofFermat's Last Theorem was big news, not unemployed members receive a 66 percent just within mathematics but in the world at large. Within hours ofWiles' announcement, the discount; emeritus members receive a 50 world's press were converging on Cambridge. A "probable proof' is not news; a theorem is. percent discount; new members receive a 40 percent discount for the first two Unusually for mathematicians, an on-the-spot decision was necessary. membership years.) So impressive were Wiles' new techniques, that most people who were able to appreciate fully Copyright © 1994 by The Mathematical what he had done were prepared to stick their necks out and vote for it being correct. Even now, Association of America (Incorporated). when the proof seems to be stuck on a particularly thorny detail, no one has reduced their Educational institutions may reproduce admiration for what is acknowledged to be one ofthe most impressive pieces ofmathematics articles for their own use, but not for sale, produced in this century. ThatWiles' work constitutes a majoradvance in number theory is not providedthat the following citation is used: in doubt. The only question is, do his results imply Fermat's Last Theorem? "Reprinted with permission of FOCUS, the Newsletter of The Mathematical Actually, that is not the only question. Given the immense amount of publicity that has sur­ Association of America (Incorporated)." rounded the whole affair, what do we say if the present proofturns out to be unfixable? Or even Second-class postage paid at Washington, if, contrary to everyone's expectations, Fermat's marginal comment turns out to be false? Do DC and additional mailing offices. we face an "image problem"? We probably do, but I don't think it is because we were wrong Postmaster: Send address changes to the about Wiles. There is plenty of evidence to indicate that part ofthe reason why so many young Membership and Subscriptions De­ people are put off by mathematics at an early age is that it seems so cold, so factual and partment, The Mathematical Association impersonal. Mathematicians portray themselves as a breed apart, perfectly logical beings who of America, 1529 Eighteenth Street Northwest, Washington, DC 20036-1385. always know the right trick, the appropriate substitution, the relevant lemma. We write books that are heavy on fact and logic, but low in passion and personality. Not for us the squabbles ISSN: 0731-2040 ofthe physicists or the cosmologists, to say nothing of the folks in comp. lit. and economics. Printed in the United States ofAmerica. Just take a look at almost any mathematics book. To the uninitiated, it must seem as though we Printed on recycled paper.