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The Mathematical Intelligencer encourages comments about the material in this issue. Letters to the editor should be sent to the editor-in-chief, Sheldon Axler.

Russophobia This letter has been prompted by an interview with Igor R. Shafarevich published in The Mathematical Intelligencer (Spring 1989). In that interview, conducted by Smilka Zdravkovska, the interviewer stated that Shafarevich's article Russophobia "is widely discussed among mathematicians, and occasionally it provokes sharp disagreements. Some consider it unfair, and even accuse you of anti-Semitism." In his response Shafarevich tried to soften the issue. He suggested that "the Russians and Jews must learn to listen and discuss each other's opinions," which sounds like a reasonable proposition. We wrote a review of Russophobia that was rejected by the editor of this journal as being insufficiently mathematical in content, but the editor allowed us to write a short letter stating our disagreement with the above-mentioned part of the interview. In his native country, where Russophobia is used as a modern antiSemitic manifesto by Russian nationalist groups to incite violence against the Jewish minority, Shafarevich is known as a rabid anti-Semite. In fact, to say that Russophobia is savagely anti-Semitic would be an understatement. It is a fanatical book crammed with arbitrary statements and "proofs" that are simply examples or quotations. The counter-examples are conveniently omitted, whereas quotations are either taken out of context or subject to different interpretations. The only criterion for including the material in Russophobia is its seeming ability to support Shafarevich's sick idea. This idea is, briefly, as follows. Whatever the ostensible occupations of Jews have been, be they capitalists or communists, poets, scientists, or politicians, wherever and whenever they have lived, their true goal has been to promote their Jewish values, which are repugnant to the values of all other peoples. At this point in history, their main goal is to destroy the Russian people. 4

The Nazi-type tirades of Russophobia triggered a massive chorus of indignation in nonmathematical media. An interested reader can be referred, for example, to the following articles: "The Closing of the Russian Mind" by L. Greenfeld in The New Republic, 5 February 1990; "From Russia with Hate" by W. Laqueur in the same issue of The New Republic; "Ordinary Fascism" by B. Khazanov in Novoye Russkoye Slovo, 11 August 1989 (in Russian). Shafarevich says that if it his last act, he must warn the Russian people of the dangers of the Jewish conspiracy. Shafarevich blames the Russian revolution and even alleged Russian alcoholism on this old familiar scapegoat. This fantasy is especially dangerous at this time because of the political instability in Russia.

Lawrence A. Shepp AT&T Bell Laboratories 600 Mountain Avenue Murray Hill, NJ 07974-2070 USA

Eugene Veklerov Lawrence Berkeley Laboratory University of California Berkeley, CA 94720 USA

Fract/fls. I am taking the liberty of c o m m e n t i n g on Steven Krantz's review of the books Beauty of Fractals and The Science of Fractal Images (Mathematical InteUigencer, vol. 11, no. 4, 12-16). These books are meant for a general reader. This means they should be judged by the quality of the presentation and the effectiveness of their communication of the subject. It is the job of a reviewer to judge how well the books succeed in these aims. Many of us are interested in the problem of explaining mathematics to the public. Carefully written and illustrated books will greatly increase the understanding of mathematics and therefore will draw more people into the profession.

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Instead the review blasts Mandelbrot because a) Douady named the Mandelbrot set after him, and b) a quote of Kadanoff warned of the danger of form over substance in fractal geometry. The truth is that the concept "fractal" has b e c o m e a universal one in science (physics, chemistry, biology . . . . ). However imprecisely in a mathematical sense it may be used, it has struck a very resonant chord. The collection of papers dedicated to Mandelbrot and comprising the recent volume of Physica D [38(1989), 1-383] gives some sense and background for this phenomenon, at least in physics. In particular there is a contribution from Kadanoff concerning multifractals (credited to Mandelbrot) arising in a beautifully simple model of avalanche. In any case, the fractal sets arising in 1-dimensional complex analytic dynamics contain very rich veins of mathematical gold. The pleasure some of us have in seeing them is exceeded only by our satisfaction in understanding what we are seeing. Albert Marden School of Mathematics University of Minnesota Minneapolis, MN 55455 USA

.Mathematics in Poems. Your Winter 1990 issue excerpts an article by Jonathan Holden that calls Wallace Stevens "the most mathematically sophisticated of recent American poets." As evidence the excerpt cites passages in Stevens that remind Holden of mathematical ideas. But a more objective test of such sophistication is the explicit use of mathematical terms; and in reading Stevens for many years I have found only two trivial references to mathematics. Rather, Holden's citations reflect Stevens's unquestioned life-long interest in philosophy, especiaUy esthetics and epistemology. Of twentieth-century American poets with some standing, I have read none who meets my test better than Kenneth Rexroth, from w h o s e w o r k s I recall a b o u t six p o e m s w i t h m a t h e m a t i c a l r e f e r e n c e s . A m o n g recent English p o e t s I nominate William Empson, the footnotes to whose verse show his conscious metaphorical use of mathematical ideas. A memorable, but seemingly isolated, achievement is Hyam Plutzik's An Equation--sort of an ode to a parabola. And Don DeLillo's 1976 novel Ratner's Star contains a marvelous piece of mathematical light verse, supposedly the words to a song at a math department party in an international research institute. If one asks what major poet shows most familiarity with the mathematics of his or her own time, then the answer, almost surely, is Geoffrey Chaucer. The Canterbury Tales uses mathematical jargon to explain how, in the Franklin's Tale, a savant performs a miracle. In-

deed, Chaucer's interest in astronomy (some say his characters are astrological stereotypes) led him to master sufficiently its technicalities so that, for his son, he could write A Treatise on the Astrolabe, a user's manual that includes instructions h o w to solve what we would n o w call trigonometry problems. In breadth of knowledge, the "Father of English Literature" did notably better than any of his descendants. John S. Lew Mathematical Sciences Department International Business Machines Corporation Thomas J. Watson Research Center Box 218 Yorktown Heights, NY 10598 USA

Sarcasm?, It was nice to see the article by W. M. Priestley on Mathematics and Poetry (Mathematical Intelligencer, vol. 12, no. 1, 1990), but I think that his treatment of literary history needs a couple of modifications. First, Priestley expresses surprise that G. H. Hardy thought people in the 1930s could believe in "good poetry'" that lacked significant meaning. He has failed to recognize that Hardy's brief discussion of that topic is a paraphrase from the famous essay On the Name and Nature of Poetry b y the s c h o l a r a n d p o e t A. E. Housman. That essay was first presented as a lecture at Cambridge in 1933 (possibly with Hardy in the audience), so there is nothing surprising in the reference. More important is Pope's celebrated Epitaph Intended for Sir Isaac Newton, Nature, and Nature's Laws lay hid in Night. God said, Let Newton be.t and All was Light. Priestley announces that it was intended to be taken as playfully composed sarcasm. It is of course impossible to be sure what was in Pope's mind, but the evidence is very heavily against any such interpretation. As the lines are so well known, perhaps readers will not mind if I set out some of the evidence.

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1. Newton, a revered old man, died in 1726. Pope's work, whose full title is Epitaph. Intended for Sir Isaac Newton, in Westminster-Abbey, appeared in a newspaper in 1730, w h e n Newton's monument was of current interest (it was finally erected the next year). Playful sarcasm would have been unlikely at that time. 2. When Pope in 1735 wrote a poetic defense of his own character and writings, it was in the form of an Epistle to Dr. Arbuthnot, and the recipient is clearly presumed to be on Pope's side. Arbuthnot was indeed an old friend of Pope's, a collaborator in the "Scriblerus" jokes of the 1710s and the creator of the figure "John Bull"; but he was also a Fellow of the Royal Society who had been an advocate of Newton and his work for almost 40 years. 3. Even relatively hostile criticism, as in Samuel Johnson's article on Pope's epitaphs (1756), did not suggest that anyone found this poem sarcastic. 4. Pope's other references to Newton are very favorable. In the Essay on Man (1733), which Priestley mentions, Pope does assert Newton's inferiority to supernatural beings; but the point is that even such a man as N e w t o n is inferior, and the superior ones themselves (it is said) "admired such wisdom in a human shape." More striking still is the passage in the Dunciad Variorum (1727), where the prophet of Dullness tells his disciples that they might attack " a Newton's Genius, or a Seraph's flame"; struck then by a glimmering of reason, he goes on to warn them not to scorn " t h e source of Newton's Light . . . . your GOD." No one, I think, is likely to see sarcasm in this closely parallel passage. 5. Finally, Priestley printed only the familiar English part of the epitaph. In the original it is preceded by some words in Latin, which in translation say "This marble acknowledges that he was mortal. Nature, Time, and the Heavens bear witness to his immortality." All of this does not mean that Pope totally accepted the near-deification of Newton after his death. (Marjorie Hope Nicolson has a careful discussion of his views in her book Newton Demands the Muse). But there can be very little doubt that the epitaph was meant seriously. William C. Waterhouse Department of Mathematics Pennsylvania State University University Park, PA 16802 USA 9W. M. Priestley RepliesWilliam Waterhouse is right to take me to task about "playfully composed sarcasm." I intended only to paraphrase the words I was about to quote from Magill's Quotations in Context. My phrase was intended to suggest sarcasm that was not mean and biting, but 6

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tempered by playfulness to the point that it could be "half-admiring, half-satiric." Magill does not mention sarcasm, probably because he was wise e n o u g h - - a s I was n o t - - t o foresee that its mention would misleadingly suggest sarcasm directed at Newton. There is no doubt that Pope on occasion expressed great admiration for Newton. What, then, is being satirized? Marjorie Hope Nicolson, in Newton Demands the Muse, writes that Pope replied in the Dunciad of 1728 to the "lavish and unrestrained adulation'" of the Newtonians "who seemed to feel that the Revelation according to Newton was greater than that according to Moses or St. John the Divine." It is Pope's use of the word "all" that suggests he is satirizing the views of those leaning toward the Newtonians. (In playing with the familiar phrase from Genesis--God said, Let there be light: and there was light - - Pope has replaced "there" by "all.") If "all" refers to everything of significance about "Nature and Nature's Laws," who but an ally of the Newtonians would seriously hold the view that all was light after N e w t o n ? N e w t o n ' s f a m o u s "'queries," k n o w n to Pope, show that the great ocean of truth that lay undiscovered by Newton contained secrets of nature not yet brought to light. To take Pope's words at face value, it seems to me, is to place Pope alongside the Newtonians, where he does not belong. I had known Housman's The Name and Nature of Poetry only through the disconnected fragments of its oft-quoted pasages. Having now at last read the whole essay, I see that Waterhouse is right. Housman defines poetry (which "transfuses emotion") in such a way that poetry has little to do with intellect (which "transmits thought"). However, Housman includes a passage indicating he believes t h a t - - t h o u g h the result cannot be said by Housman to be "more poetical"-greatness of poetry does involve the combination of intellect and emotion: "When Shakespeare fills such poetry with thought, and thought which is worthy of it . . . . those songs, the v e r y summits of lyrical achievement, are indeed greater and more moving poems . . . . " Yet Hardy appears to suggest that the conventional wisdom of the 1930s might ignore coherence and depth in judging the greatness of poetry. Except for Hardy's coyness on this point, my admiration for the Apology is without qualification. It is one of the most beautiful books I know. I am grateful to Waterhouse for this thoughtful response, and also for his implicit suggestion of a similarity between poetry and mathematics that I had not noted: To speak unambiguously about either requires a surprising, and sometimes agonizing, precision of language. W. M. Priestley Department of Mathematics and Computer Science University of the South Sewanee, TN 37375 USA

Karen V. H. Parshall*

A Century-Old Snapshot of American Mathematics Karen V. H. Parshall In the last quarter of the nineteenth century, American mathematics underwent a series of dramatic changes that propelled it onto the international scene. The beginning of this crucial period saw the opening of the Johns Hopkins University in Baltimore with its explicitly articulated purpose of training students at an advanced, graduate level. As its first mathematician, the University chose the gifted and ebullient, if touchy and eccentric, James Joseph Sylvester (1814-1897). Its decision not only rescued Sylvester from a forced retirement in his native England but also gave him the opportunity to do in America what he had never been able to do in Great Britain: to found a research-level school of mathematics [1]. Although Sylvester supervised the dissertations of eight students during his seven-year reign at Hopkins, his Ph.D.'s went out from the exhilarating atmosphere of their mathematical seminarium into the stale air of the mathematical classroom of American academe. Arriving on the scene in the early 1880s, Sylvester's students struggled to keep their research interests alive as they taught undergraduate mathematics for upwards of twenty hours a week in mathematical isolation [2]. Furthermore, w h e n Sylvester left Hopkins in 1883 to assume Oxford's Savilian Chair of Geometry, America's one real training ground for untried mathematical talent lost its mentor. Over the next decade, Americans looked almost exclusively to the Continent for their mathematical inspiration, and they found it most often in the lecture hall and seminar room of Felix Klein. By 1900, though, the educational situation back home had improved markedly. Traditional, colonial,

* C o l u m n Editor's address: D e p a r t m e n t s of Mathematics and History, University of Virginia, Charlottesville, VA 22903 USA.

liberal-arts colleges like Harvard and Yale had responded to the Hopkins challenge and had transmuted into American variants of the modern German university. State and land-grant institutions such as the Universities of Michigan and Wisconsin, which were mere fledglings in the 1860s, had strengthened their graduate and professional programs. Private phil a n t h r o p y h a d created u n i v e r s i t i e s like Cornell (opened 1868), Clark (opened 1889), and Chicago (opened 1892) de novo. Concurrent with, but by no means independent of, these developments, an American mathematical community came into its own. The New York Mathematical Society, formed as a local organization in 1888, went national by popular d e m a n d in 1894. The socalled "Zero-th International Congress" convened in Chicago in 1893 as part of the World's Fair and Columbian Exposition and brought a respectable number of American mathematicians in contact with one another as well as with the event's keynote speaker, Felix Klein. Finally, the country in which mathematics journals, even of the recreational variety, had had a mortality rate of one-hundred percent prior to 1876, supported no fewer than four journals aimed at a research-level audience in 1900. In short, American mathematics reached its critical mass during the 1890s [3]. In publishing his book, The Teaching and History of Mathematics in the United States, precisely at the turn of that decade in 1890, the mathematical historian Florian Cajori (1859-1930) perhaps unwittingly provided us with a snapshot of an American mathematical community on the verge of major change [4]. Given the rapidfire sequence of events already underway as he wrote, however, the picture he presented looked surprisingly unpromising. The Swiss-born Cajori, himself a product of the American system of higher education of the 1880s

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with a B.S. (1883) and M.S. (1886) from the University of Wisconsin, conducted his study over a period of several years under the auspices of the United States Bureau of Education [5]. In keeping with the reformminded spirit of m a n y educators of his day, Cajori sought to d o c u m e n t the changing t r e n d s in the teaching of his specialty, mathematics, within the broader context of the history of mathematics generally [6]. To this end, he subdivided American mathematics into three developmental stages: the period of British influence, which lasted roughly until 1820 and which saw American mathematics teaching dominated by British or British-inspired texts; the analogous period of French influence, which persisted through the next fifty-odd years of the century; and the then present, which seemed to call for Americans to take control of their own mathematical destiny [7]. Collating information from sources such as college catalogues as well as from hundreds of personal inquiries to professors of mathematics around the country, Cajori patched together fairly complete, if often anecdotal, histories of twenty-two programs from their in-

A large part of the mathematics professor's time in 1890 was spent in teaching. cepfion through the 1888-1889 academic year. In addition, he c o n d u c t e d a m u c h more broadly-based survey of colleges and universities in order to get a sense of the overall level of mathematics education nationwide. While his information-gathering and reporting skills may have fallen woefully short of the mark even by 1890s standards, Cajori nevertheless succeeded in amassing some useful and revealing data in his book. Consider, first, the twenty-two institutions he chose to profile: the United States Military Academy, Harvard*, Yale*, Princeton*, D a r t m o u t h , Bowdoin, Georgetown, Cornell*, the Virginia Military Academy, Tulane, Washington University, Johns Hopkins*, and the Universities of Virginia*, North* and South Carolina*, Alabama, Mississippi, Kentucky, Tennessee, Texas*, Michigan*, and Wisconsin*. Although he provided no explicit justification for his selection of these - - a n d not other--schools, a glance at the list would suggest an effort to achieve geographical balance (at least relative to the region east of the Mississippi), to represent long-established as well as newer institutions, and to cover the full range of mathematical curricula from undergraduate programs in both the liberal arts and applied molds to graduate-level courses of study. Of course, appearances may be deceiving. Cajori may only have had sufficient information for precisely these schools, but regardless of the selection criteria used, his sample reflected a fairly accurate image 8

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James Joseph Sylvester of the opportunities available for graduate training in the United States. Of those schools (marked with an asterisk in the list above) supporting at least some sort of a graduate program in 1888-1889, the case of Cornell typifies the situation at only the very best of the lot [8]. By the standards of the day, Cornell supported a huge mathematics faculty at seven strong, counting all of the ranks from professor to instructor. The leader of this group, James Edward Oliver (1829-1895), had graduated in 1849 from a midcentury Harvard dominated mathematically by Benjamin Peirce (1809-1880) and had returned in the midfifties to pursue Peirce's advanced course in mathematics through the Lawrence Scientific School [9]. Earning his living in Cambridge at the federally supported office of the American Nautical Almanac, Oliver entered the academic ranks in 1871 when he accepted the Assistant Professorship of Mathematics at Cornell. Only two years later, he assumed the mathematical chair and directed Cornell mathematics from that vantage point until his death in 1895. A modest and unassuming man, Oliver pursued mathematics more for his own pleasure than for the reputation publication might have brought, yet he was fully attuned to the growing importance and desirability of publication within the emergent mathematical community. Thus, in responding to Cajori's queries about his program, Oliver rather wistfully explained that [w]e are not unmindful of the fact that by publishing more, we could help to strengthen the university, and that we ought to do so if it were possible. Indeed, every one of us five [in 1886-1887] is now preparing work for publication or expects to be doing so this summer, but

Again, by the American standards of the time, these students had a fair range of courses in both pure and applied mathematics from which to choose. On the applied side, Oliver and Wait taught mathematical The duty that sapped their energies most completely optics, the mathematical theory of electricity and magnetism, celestial mechanics, and rational dynamics, was teaching. During the 1886-1887 academic year, Oliver, to- while McMahon offered a course in the mathematical gether with his four colleagues, Associate Professor theory of sound, Hathaway gave molecular dynamics, Lucien Augustus Wait (1846-1913), Assistant Pro- and Studley taught descriptive and physical asfessor George William Jones (1837-1911), and in- tronomy. As for the pure side, Oliver lectured on the structors James McMahon (1856-1922) and Arthur theory of functions, finite differences, vector analysis, Safford Hathaway (1855-1934), taught an average of hyper-geometry, matrices and multiple algebra, and seventeen to twenty hours each week. The next year the general theory of algebraic curves and surfaces, two more instructors, Duane Studley (d. 1947?) and while Jones covered probability, lines and surfaces of George Egbert Fisher (1863-1920), joined the staff, but the first and second orders, and modern synthetic gethis 40% increase in the teaching faculty hardly less- ometry. The instructors r o u n d e d out the offerings ened the burden. According to Oliver, " . . . our de- with McMahon doing quantics with applications to gepartment's whole teaching force, composed of only ometry, Hathaway covering differential equations and about one-eleventh of all active resident professors, number theory, and Fisher handling advanced differhas to do about one-ninth of all the teaching in the ential and integral calculus [14]. By and large, these courses hit at the level of the most sophisticated university" [11]. In spite of this load, which Oliver clearly viewed as British texts available in the various areas, books like inequitable, he and his colleagues managed to make a George Salmon's (1819-1904) Higher Algebra, Thomas fair showing publication-wise in 1887-1888. Oliver Muir's (1844-1934) Determinants, A. R. Forsyth's (1858-1942) Differential Equations, and the Theory of seemed quite proud to report that Equations by William Burnside (1852-1927) and Arthur Panton (d. 1906). In a few cases, though, the Cornell Even had they h a d more time for research, faculty exposed its students to Continental matheA m e r i c a n m a t h e m a t i c i a n s in higher educa- matics through texts such as the Trait~ des fonctions eltion w o u l d have lacked w h a t w a s quickly be- liptiques by George Halphen (1844-1889) and Richard (1831-1916) edition of Peter Lejeune-Dicoming one of the trade's m o s t fundamental Dedekind's richlet's (1805-1859) Zahlentheorie [15]. Given that the tools--the journal. faculty reserved the right to cancel any course with an enrollment of three or fewer, however, most of this Professor Oliver has sent two or three short articles to mathematics did not reach the students in any given the [Annals], and has read, at the National Academy [of year. Still, at least the possibility of pursuing work at Science]'s meeting in Washington, a preliminary paper on this depth and of this extent existed in the Cornell the Sun's Rotation, which will appear in the Astronomical graduate program, and the situation was roughly analJournal. Professor Jones and Mr. Hathaway have litho- ogous at Harvard, Yale, Johns Hopkins, and the Unigraphed a little Treatise on Projective Geometry. Mr. McMahon has sent to the [Annals] a note on the circular versities of Texas and Wisconsin. Programs like that of points at infinity, and has also sent to the Educational the University of South Carolina, however, reflected Times, London, solutions (with extensions) of various the other end of the graduate spectrum in the United problems. Other work by members of the department is States at this time [16]. likely to appear during the summer, including a new ediSouth Carolina's Ellery William Davis (1857-1918), tion of the Treatiseon Trigonometry[12]. like Cornell's Hathaway, had studied under Sylvester The latter work comprised part of the popular series of at Johns Hopkins and had actually earned the Ph.D. textbooks by Oliver, Wait, and Jones designed pri- there in 1884. After spending four years on the faculty marily for use in the college classroom [13]. Thus, the at the Florida Agricultural College, he accepted the Cornell faculty, although perhaps more active in text- mathematical chair at South Carolina in 1888. As his book writing than in original research, was nonethe- university's Mathematics Department, Davis not only less alive mathematically. In Oliver's view, only a suf- taught undergraduate courses for thirteen hours a ficiently high level of vitality would successfully attract week but also instituted a graduate course of studies that increasingly desirable entity--the graduate stu- immediately upon his arrival in Columbia [17]. Ald e n t - t o the department. Apparently, though, he though he apparently had no clientele, he was preand his colleagues attained the necessary level, for pared to offer "algebra (theory of equations, theory of their program drew in eleven graduate students in the determinants, etc.), geometry (projective geometry, higher plane curves, etc.), calculus (differential equa1887-1888 academic year. such work progresses very slowly because the more immediate duties of each day leave us so little of that freshness without which good theoretical work can not be done [101.

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tions and finite differences), elliptic functions, astronomy, and quaternions," in short, the same curriculum he himself had followed at Hopkins [18]. With no mathematical companionship and only the American Journal to keep his research spark alive, Davis labored under a great mathematical handicap at South Carolina. If, as Oliver believed, a strong record of publication attracted graduate students, Davis's efforts seemed doomed to failure [19]. Yet, compared to the vast majority of American colleges around 1890, mathematical life looked positively rosy at South Carolina and simply too good to be true at Cornell. In his more widely ranging survey, Cajori gathered information from representatives of 168 col10

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leges and universities as to the state of mathematics in their respective ins.titutions [20]. The picture that e m e r g e d - - a l t h o u g h undoubtedly skewed by the absence of replies from some of the more mathematically progressive schools like Harvard, Yale, Cornell, the newly founded Clark, and the Universities of Michigan and Wisconsin--was probably fairly accurate nonetheless. A large part of the mathematics professor's time in 1890 was spent in teaching. Of 117 respondents, 12% taught more than twenty hours a week, 35% lectured between sixteen and twenty hours weekly, 27% spent from eleven to fifteen hours in the classroom, and only 6% taught six or fewer hours [21]. These hours were not necessarily devoted solely to mathematics teaching either. Although 39% of the 111 answering the query "What other subjects do you teach?" only taught mathematics, 30% gave instruction in at least one other subject and another 30% covered two or more additional areas [22]. While all of these figures would probably have compared favorably to midcentury statistics, they still suggested that time for research was at a premium in 1890 and course diversity mitigated against specialization. Even had they had more time for research, American mathematicians in higher education would have lacked what was quickly becoming one of the trade's most f u n d a m e n t a l t o o l s - - t h e journal. Of the 135 schools replying to Cajori's question "What mathematics journals are taken?," 62% subscribed to no journals of any sort whatsoever. Admittedly a grim overall state of affairs, 47 institutions or 35% of all respondents did take at least one research-level journal, and of these, 32% (or 11% of all respondents) got at least one from abroad [23]. Cajori's survey, then, depicted a mathematical community in 1890 which could have gone either way. Teaching loads had come down some, but they were still heavy; specialization, at least relative to teaching, had taken place, but diversity still blurred focus; graduate programs had developed, but they still had a long way to go to compete successfully with the programs in Germany; and research was increasingly recognized as desirable in spite of a lack of real incentives for its pursuit. The events of the decade from 1890 to 1900 determined the tilt of American mathematics to the positive. The New York (and later American) Mathematical Society began issuing its Bulletin in 1891, a publication that helped to unite America's farflung mathematical constituency into a community and to inform it of current developments both at home and abroad. American students, who increasingly went to Europe to study after Sylvester's departure from Hopkins, returned to transplant not only the mathematics but also the research ethos they had obtained. This transplantation was made possible by the everimproving state of American higher education. The

The establishment of new research-oriented universities like Clark and Chicago had served to accelerate further the development of programs at relatively older institutions such as Harvard, Yale, Princeton, and the land-grant universities of the Midwest. e s t a b l i s h m e n t of n e w r e s e a r c h - o r i e n t e d universities like Clark and Chicago h a d served to accelerate further the d e v e l o p m e n t of programs at relatively older instit u t i o n s such as H a r v a r d , Yale, P r i n c e t o n , a n d the land-grant universities of the Midwest. Thus, unbek n o w n s t to him, the changes that Cajori implicitly advocated in his s u r v e y were already taking place. H a d h e conducted a follow-up survey in 1910, Cajori would have taken a m a r k e d l y different snapshot, one that w o u l d have incorporated as given the complicated educational, social, a n d professional forces at work in the formation of a research-level mathematical comm u n i t y in America [24].

References 1. For more on these developments at Hopkins, see Karen Hunger Parshall, "America's First School of Mathematical Research: James Joseph Sylvester at The Johns Hopkins University 1876-1883/' Archive for History of Exact Sciences 38 (1988), 153-196. 2. Florian Cajori, The Teaching and History of Mathematics in the United States (Washington: Government Printing Ofrice, 1890), 345-349. The figures for weekly teaching loads were culled from the information Cajori gave on these pages. 3. David Rowe and I are presently in the final stages of work on a book, entitled The Emergence of an American Mathematical Research Community: J. J. Sylvester, Felix Klein, and E. H. Moore (tentatively to be published in the American Mathematical Society's series in the History of Mathematics), which traces the developments just outlined. We have sketched our views in our article "American Mathematics Comes of Age: 1875-1900," pp. 3-28, in Peter Duren, et al., ed., A Century of Mathematics in America--Part III, Providence: American Mathematical Society (1989). 4. Judith V. Grabiner also pointed out this historical use of Cajori's book in her article, "Mathematics in America: The First Hundred Years," pp. 9-24, in Dalton Tarwater, ed., The Bicentennial Tribute to American Mathematics 1776-1976, The Mathematical Association of America (1977). 5. On Cajori's life and work, see David Eugene Smith, "Florian Cajori," Bulletin of the American Mathematical Society 36 (1930), 777-780; and Raymond Clare Archibald, "Florian Cajori," Isis 17 (1932), 384-407. 6. Uta C. Merzbach made this point in her article "'The Study of the History of Mathematics in America: A Centennial Sketch," pp. 639-666, in Peter Duren, et al., ed., A Century of Mathematics in America--Part III, Providence: American Mathematical Society (1989). 7. Recent scholarship has shown that while there is an ele-

merit of truth in Cajori's analysis, it is much too simplistic to be historically useful. See, in particular, Helena M. Pycior, "British Synthetic vs. French Analytic Styles of Algebra in the Early American Republic," in David E. Rowe and John McCleary, ed., The History of Modern Mathematics, 2 vols., Boston: Academic Press (1989), 1, 125-154. 8. For the discussion of Cornell and its faculty, see Cajori, pp. 176-187. 9. For an idea of the advanced, state-of-the-art nature of Peirce's curriculum, see Cajori, pp. 137-138. 10. Ibid., p. 180. 11. Ibid., p. 186. 12. Ibid., p. 181. 13. James Oliver, Lucien Wait, and George Jones, A Treatise on Trigonometry, 4th ed., Ithaca: G. W. Jones (1890); and A Treatise on Algebra, 2d ed., New York: Dudley F. Finch (1887). 14. Cajori, pp. 183-184. 15. Ibid., pp. 184-185. The textbooks in question are: George Salmon, Lessons Introductory to the Modern Higher Algebra, 4th ed., Dublin: Hodges, Figgis, & Co. (1885); Thomas Muir, The Theory of Determinants With Graduated Sets of Exercises for Use in Colleges and Schools, London: Macmillan & Co. (1882); A. R. Forsyth, A Treatiseon Differential Equations, London: Macmillan & Co. (1885); William Burnside and Arthur Panton, The Theory of Equations: With an Introduction to the Theory of Binary Algebraic Forms, Dublin: Hodges, Figgis, & Co. (1886); Georges Halphen, Trait~ des fonctions elliptiques et de leurs applications, 3 vols., Paris: Gauthier-ViUars, 1886-1891); and Peter Lejeune-Dirichlet, Vorlesungen iiber Zahlentheorie, ed. Richard Dedekind, Braunschweig: F. Vieweg und Sohn (1880). 16. For the sketch on the University of South Carolina and its faculty, see Cajori, pp. 208-214. 17. See note [2] above. 18. Cajori, p. 214. 19. In fact, Davis left South Carolina in 1893, presumably for the greener pastures of the University of Nebraska. He remained in Nebraska for the rest of his career, eventually assuming the Deanship of the College of Arts and Sciences. 20. Cajori, pp. 196-349. Cajori did not present any of the information gathered from his survey in a coherent way, so the figures that follow have been compiled, as best as possible, from his write-up. Much of the information he gave cannot be used to draw any meaningful conclusions due to the imprecision of the questions as posed. For example, in trying to get a sense of the educational backgrounds of those in his sample space, he asked them to "State time of your special preparation for teaching mathematics . . . . " [Cajori, p. 145]. Some respondents interpreted this as asking for: 1) the number of years teaching experience they had had, 2) the number of years of college they had had, 3) the number of years of graduate training they had received, and 4) the number of years of special instruction in mathematical pedagogy they had had. Thus, his data cannot be used to get an educational profile of his sample space. This probably also accounts for the discrepancy between the figures that follow and those given by Grabiner. 21. Cajori, pp. 345-349. 22. Ibid. 23. Ibid., p. 302. 24. See Della Dumbaugh and Karen Hunger Parshall, "A Profile of the American Mathematical Research Community: 1891-1906," forthcoming. THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 3, 1990

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The Opinion column offers mathematicians the opportunity to write about any issue of interest to the international mathematical community. Disagreements and controversy are welcome. An Opinion should be submitted to the editor-in-chief, Sheldon Axler.

Mathematics and Ethics* Reuben Hersh I want to start off by correcting any possible false impression that I'm going to tell you what is ethical, or that I've solved any big problem regarding mathematics and ethics, because I certainly haven't and make no such claim. Of course, the next question you ask is, w h y am I standing up here anyhow? It's only because I have thought about the question, and in the process of thinking about it I have had some ideas that I'd like to offer you. The observation that got me started on this was that in many professional fields there has been for a while a well-established concern with ethics. What that means varies from field to field. But the idea that a professional association of engineers or statisticians might concern itself with ethical behavior in that field is not radical at all. It's a very standard thing. Often it's done officially by the establishment. Often there are active concerns on the part of special organizations, editorials in journals, and so on. One of the first organizations of this type that I had contact with, long before I was a mathematician, was the Society for Social Responsibility in Science. I'm not sure it still exists. In its day, the 1950s and the 1960s, it was primarily concerned with nuclear arms, nuclear warfare, nuclear destruction of the h u m a n race. It consisted largely of physicists, many of them Quakers or Quaker sympathizers. They took the position that there was a question of social responsibility, for the physicist particularly, about whether to work on nu-

clear weapons. Some people refused to work on nuclear w e a p o n s or quit military jobs. Whether y o u agree with that or not, this was a legitimate issue in the physics community [7], [8], [13], [15]. Another example arose with the environmentalist m o v e m e n t . R a l p h N a d e r w a s an o u t s t a n d i n g spokesman. This movement involved biologists and also chemists, because chemists do a lot of polluting. Not chemists themselves, but the things that chemists create. There again was a question of social responsibility, which is one aspect of ethics.

* This p a p e r is b a s e d o n a talk that w a s given first to t h e N e w Eng l a n d Section of t h e M a t h e m a t i c a l Association of A m e r i c a in Nov e m b e r 1987 in W a l t h a m , M A , as t h e D a n Christie Memorial Lecture, a n d again in N o v e m b e r 1988 to t h e S o u t h e r n California Section of t h e M A A m e e t i n g in C l a r e m o n t . 12 THE MATHEMATICALINTELLIGENCERVOL. 12, NO. 3 9 1990Springer-VerlagNew York

I have in my hand two actual codes of ethics. One was adopted by the statisticians' society, and the other by the professional engineers" society. These are not so political. They have more to do with proper behavior toward one's client--ethical issues of that sort. No doubt you could find other examples. For a mathematician, it's natural to ask w h y we don't seem to be concerned about ethical issues or discuss them? It is true, as many of you know, that recently there was a referendum in the American Mathematical Society (AMS). There was a long, drawn-out political hassle, and in the end five motions were passed by the membership. The one that is probably most controversial says that the AMS should not involve itself in helping the Star Wars (SDI) activity to recruit among AMS members. That issue certainly has ethical implications. But it was a one-time, ad hoc thing, not an indication of continuing concern or involvement with ethical issues by mathematicians. In my opinion, the reason it became a big issue in the AMS was that there had already developed strong opposition to the SDI among physicists and computer scientists, both in individual departments and in national organizations. I think that was w h y some mathematicians felt we should also get involved. In the end, after a lot of back-and-forth haggling, the membership approved the anfi-SDI motion. So there is an example of an ethical issue that did come before and actually passed the American Mathematical Society. That's not the main thing I want to talk about. I just mention it because some of you might have it on your mind and might remember it. The thing that is striking, you see, is that in all the other examples I've g i v e n - - t h e biologists' involvement in environmental issues, and the chemists as well, and the physicists in nuclear war, and the statisticians requiring that if you are a good statistician you won't give away your client's d a t a - - t h e s e are all different, but they have one thing in common. They are all in some way intrinsic to the actual practice of the particular profession. The physicists are the ones who make the bombs, the chemists are the ones who pollute, and so on. When I thought about the situation of mathematicians, I found I was oscillating between two different viewpoints. On the one hand, a mathematid a n is somebody who solves a problem or proves a theorem and, of course, publishes it. And it's hard to see significant ethical content in improving the value of a constant in some formula or calculating something n e w - - s a y , the cohomology of some group. You might say it's beautiful or you might say it's difficult, but it's hard to see any good or evil there in the w a y physicists and biologists, a n d so on, do have ethical problems. On the other hand, if you step back from that particular way of looking at the role of mathematicians and just think about your own activity or mine, think of what we actually do daily and yearly, there

are constant decisions and conflicts involving right and wrong. The ethical demands of all the scientific groups seem to fall into three categories: What you owe the client, what you owe your profession, and what you owe the public. Now, if you are a mathematics professor, the word "client" may be unfamiliar. Who is the client, anyhow? But there is always a client in the sense of the one who's paying your salary. The ethics of the statisticians and engineers prominently feature duties to the client. And then there is the profession. What do y o u owe your partner, your colleague, or your fellow professional? In some ethical codes that's up at the top. I think that's the w a y it is with lawyers. Doing something unethical means treating some other lawyer unfairly. Duty to the public is an afterthought. N o w to the mathematicians. I can list five different categories of people to w h o m we have duties: staff, students, colleagues, administrators, and ourselves. First are the staff, the people who do the work that we don't want to do. It would be interesting to think about the situation or treatment of the non-faculty employees of your department. Do you regard it as equitable? If you don't, does anybody ever try to do anything about it? Then there are the students. For instance, there is the problem of mathematical illiteracy. ! don't mean to suggest that we owe students mathematical illiteracy. Rather, the existence of mathematical illiteracy poses an ethical issue. Is the prevalence of mathematical illiteracy among students in part a responsibility of us, their teachers? If so, what can we do about it? This issue needs to be mentioned because so many of us deny our responsibility and blame the high schools. Next example: grading. Again, we don't usually think of this as an ethical issue. We try to make it a mechanical matter, a rule, and let a machine do it. But despite our machinery, there are always hassles and disagreements about grades. I think that the grade I finally give, whether it's a number or a letter, is not just an objective application of some rule, but also to some extent an ethical choice. What do I think is more important, more valuable than something else? I would say grading should be included in the ethical life of the mathematician. I've had a student from some place in the Near East tell me that if I didn't change his grade, he'd have to go home and go into the army and get killed, and it would be my fault. For all I knew it might have been true, except I didn't change his grade and he was still there a year later. That's an extreme example of an ethical issue: m u r d e r associated with grades. Finally, gender and ethnicity. This has been subject of a good deal of talk in recent years. There are, to some extent, special programs to help women and to help Blacks, Hispanics, and Indians attain a higher level in mathematics. Not many people here, I would THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 3, 1990 1 ~

guess, are involved in that activity. And there are certainly differences of opinion about it. But it's a clear case of an ethical issue [2], [6], [10], [11], [12], [18], [191. Colleagues. This, I think, is the big one, the one that most of us are most involved in. Hiring, tenure, prom o t i o n - - t h e s e are the i s s u e s t h a t d e p a r t m e n t meetings hassle about. Sometimes, I suppose, decisions are made entirely on an objective basis of what's best for the department. And then again, sometimes people help their friends. But before you get down to who gets hired or w h o gets tenure, there have to be assumptions about what's important, what's legitimate, what you want to do. It's usually supposed that this is already given. Everybody should already know

If our research work is almost devoid of ethical content, then it becomes all the more essential to heed our general ethical obligation as citizens, teachers, and colleagues, lest the temptation of the ivory tower rob us of our h u m a n nature. what the department needs to do to improve itself. But actually, that's not tenable. The standards for hiring, promotions, and so on are subject to differences of opinion, depending on what you believe in and what you think is the right thing for the department to be doing. In other words, your ethical stance. Here is a story about an ethical problem in relations between colleagues. It's a little out of date, but interesting. You probably k n o w that back in the 1930s many mathematicians were leaving Germany in order not to be killed. Emil Artin was one of the great algebraists of his time. Artin wasn't Jewish, but his wife Natasha was half-Jewish, and they had two kids. Artin was approached by Helmut Hasse, w h o was another outstanding algebraist. Hasse was almost a pure Aryan, though he did have a Jewish great-grandfather. He had become the head of the Institute at G6ttingen after Courant and Weyl and Neugebauer had been kicked out. Artin was planning to leave because of his wife's being half-Jewish, their children quarterJewish. Hasse said he could give Artin a deal. The kids could be made Aryan [14], [16], [17], [20]! Do you see any ethical questions there? Hasse was a great mathematician. After the war he was quoted as being annoyed that some of the de-Nazification programs instituted by the American army were too severe. And he wasn't the worst. There were people like Teichmuller, and Bieberbach, brilliant mathematicians w h o were whole-hearted, all-out Nazis. Their ideology affected their professional work too, driving people like Landau off the lecture platform. Probably 14

THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 3, 1990

it could never happen here. But racism is a problem everywhere. It's not only a political problem, it's an ethical problem. We tend, many of us, to throw it under the rug, to think it's of no relevance to us. But maybe learning a little history will enlighten us about that. So much for ethical p r o b l e m s b e t w e e n colleagues. Finally, what do we owe to the Dean, the Provost, the Chancellor? What they usually expect is that you should get grants, visibility, and things like that. That d e m a n d from administrators is b a s e d on certain values. It's based on a particular idea of what the university is and what the department should become. If those values are accepted, then our present situation follows absolutely. The m a t h e m a t i c s d e p a r t m e n t should get out there and bring it in! But this value system is also arguable. There are some of us w h o think otherwise. And recognizing that there is an ethical conflict here can only help to clarify our possibilities and our alternatives. Now, to yourself! Does anybody here remember Polonius, in Hamlet? Eventually Polonius gets Hamlet's sword through his gut, but he leaves us this memorable line: "And this above all, to thine own self be true.'" So far I've carefully avoided giving you my o w n values. So there's no w a y anyone can disagree with me. I've just listed points of value judgment in our profession. I'm sure there are others that I have forgotten. But you're undoubtedly about to point out that all this really has nothing to do with mathematics. It has to do with academic life. A French professor or a mechanical engineering professor would be involved in the same issues. I've been talking as if we're all academics. Of course, this isn't true. Some people here must be working in industry or other things. But being an instructor or professor involves you in all these interactions with people: students, faculty, staff, administration. And these all have an ethical component. However, this does not really deal with the issue I started with, which was what about mathematicians as mathematicians? Just because we're mathematicians, are there issues we have to face in the same w a y engineers have ethical issues they have to face? Here I think we are forced to recognize the irritatingly vague line between pure and applied mathematics. To the extent that it is really involved in the so-called "real world," applied mathematics brings in the same ethical issues as engineering or any other applied science. For instance, nowadays people are using big computers to figure out secondary oil recovery. The people who do this are both geophysicists and applied mathematicians. The ethical issues for applied mathematicians are the same as for geophysicists. What are the consequences of this activity for the environment, for the economy? To the extent that applied mathematicians get in-

volved with a real world activity like geology or engineering, they have to deal with the ethical issues of that field, not because they are mathematicians but because they are involved in that application. Therefore, let me acknowledge the separation and ask: W h a t about pure mathematics a n d mathematicians w h o merely prove theorems? Is there a n y ethical c o m p o n e n t c o m p a r a b l e to w h a t y o u find in other fields of science? Of course, depending on w h a t y o u include as ethics, you can say yes or no. "It's unethical to prove an ugly t h e o r e m . " "It's unethical to republish u n d e r a different tire a trivial paper that y o u have already published." As expressions of the taste or the standards of the field, these statements are correct. But still, one laughs at the word "ethical" here. It just doesn't make sense to use the same language for such issues of taste in pure mathematics as for air pollution or nuclear war. There are "ethical" issues in pure mathematical research. But they cannot w i t h s t a n d comparison with the major issues of h u m a n survival arising in "real w o r l d " science. In pure mathematics, w h e n restricted just to research and not considering the rest of our professional life, the ethical c o m p o n e n t is very small. Not zero, but so small it's hard to take very seriously. In fact this m a y be a characteristic, a defining characteristic of pure mathematics. I can't think of a n y other field of w h i c h y o u could say that. That's w h y p e o p l e say mathematicians live in an ivory tower. One answer to this could be, "Well, this is fine! There's no need for mathematicians to have a code of ethics, because what we do matters so little that we can do whatever we like." A n d I might agree with that. I'm not going to start advocating a code of ethics in mathematics at this point. But w h e n I think about this attitude, I find it scary. Because it m e a n s that if we become totally immersed in research on pure mathematics, we can enter a mental state that is rather inhuman, totally cut off frem humanity. That's a thing we could worry about a little bit. Therefore, I come to a conclusion for most of us, those w h o are not doing pure research a h u n d r e d percent of the time or w h o are not in the institutes for advanced studies, but have students and colleagues a n d staff and administrators. We mathematicians, I think, have a special n e e d to take all these other re-

sponsibilities very seriously. Because unlike people in other fields, our research work does not automatically involve h u m a n concern. My conclusion: If our research w o r k is almost devoid of ethical content, then it becomes all the more essential to heed our general ethical obligation as citizens, teachers, and colleagues, lest the temptation of the ivory tower rob us of our h u m a n nature.

Bibliography

1. S. Axler, Pseudo-mathematics vs. Mathematics, unpubfished manuscript. 2. L. Blum, Women in mathematics: an international perspective, eight years later, The Mathematical Intelligencer 9, no. 2 (1987) 28-32. 3. C. Davis, The purge, pp. 413-428 of A Century of Mathematics in America, Part I, Providence: Amer. Math. Soc. (1988). 4. , A Hippocratic oath for mathematicians, Science for Peace Seminar (15 Nov. 1988). 5. P. Davis and R. Hersh, Descartes' dream, Boston: Harcourt Brace Jovanovich (1986). 6. G. A. Freiman, It seems I am a Jew: a Samizdat essay. Translated, edited, and with an introduction by Melvyn B. Nathanson, Carbondale: Southern Illinois University Press (1980). 7. Loren R. Graham, Between science and values, New York: Columbia University Press (1981). 8. S. J. Heims, John von Neumann and Norbert Wiener, Cambridge: M.I.T. Press (1980). 9. N. Koblitz, Mathematics as propaganda, Mathematics Tomorrow (Lynn Arthur Steen, ed.) New York: SpringerVerlag (1981) 111-120. 10. R. B. Landis, The case for minority engineering programs, Engineering Education 78, no. 8 (1988), 756-761. 11. E. H. Luchins, Sex differences in mathematics: how not to deal with them, American Mathematical Monthly 86 (1979) 161-168. 12. Vivienne Mayes, Lee Lorch at Fisk: A tribute, American Mathematical Monthly 83 (1976) 708-711. 13. H. J. Morgenthau, Science: servant or master? New York: New American Library (1972). 14. C. Reid, Courant in G6ttingen and New York, New York: Springer-Veflag, (1976) 203. 15. R. W. Reid, Tongues of conscience, London: Constable and Co. (1969). 16. S. L. Segal, Helmut Hasse in 1934, Historia Mathematica 7 (1980) 46-56. 17. , Mathematics and German politics: the national socialist experience, Historia Mathematics 13 (1986) 118-135. 18. L. W. Sells, Mathematics--a critical filter, The Science Teacher 45 (February 1978), 28-29. 19. - - - , Leverage for equal opportunity through mastery of mathematics, Women and Minorities in Science, (Sheila M. Humphreys, ed.) Boulder: Westview Press, American Association for the Advancement of Science (1982) 7-26. 20. C. L. Siegel, On the history of the Frankfurt mathematics seminar, The Mathematical Intelligencer 1, no. 4 (1979) 223-230. Department of Mathematics and Statistics University of New Mexico Albuquerque, NM 87131 USA THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 3, 1990 1 5

Recollections of Mathematics in a Country Under Siege Neal Koblitz

An interview with Professor Ho~ng Tu.y, Director of the Hanoi Mathematical Institute. Koblitz: Professor Tuy, please start by telling us a little about your family background and early years. T~y: I was born in 1927 in the village of Xuan Oai (now called Di~n Quang) about 20 kilometers south of Oa Nafig. My father, who died when I was 4 years old, was a low-level mandarin u n d e r the earlier system, having passed the old-style examinations on the Chinese dassics. In the 1920s the modern French system was just in the process of being introduced in Vietnam. Despite my father's official rank, my family was poor, and several of my brothers had to go to work at a very young age. In my family there was a tradition of noncollaboration with the colonial regime. One of m y ancestors,

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Hoang Di.~u, was governor of Hanoi in the 1880s. He heroically defended Hanoi against the French, but the city fell. Believing himself to be responsible, he took his own life rather than allow himself to be captured by the enemy. His resistance and suicide are considered to have been acts of great patriotism. In 1945, when the revolution took Hanoi back from the French, for a time Hanoi was renamed Hoang Di.4u City. In my generation we tried to continue this tradition of resistance. But most of the time we could do nothing. One of my brothers, w h o was training to become an artist, was expelled from painting school because of his participation in a student strike. My eldest brother, w h o was a secondary school teacher--this was a high intellectual position in our society at that t i m e - - w a s fired for anticolonialist activity. So there was in my family a tradition of culture and a sense of patriotism that was handed d o w n from generation to generation.

THE MATHEMATICAL INTELL1GENCER VOL. 12, NO~ 3 9 1990 Springer-Verlag New York

Koblitz: How did you become interested in mathematics? T~y: When I went to school as a young boy in the village, I was very good in two subjects: literature and mathematics. Then I went to Hfi~, where m y brother was a high school teacher, and entered the lyc6e there, one of the three best in Vietnam at that time. But, unfortunately, I had very bad teachers of literature; on the other hand, the math instructors were quite good. Koblitz: These were the years of World War II. Were there interruptions in your schooling?

already heard of L~ Van Thi~m, later to become the founder of Vietnam's mathematical institutions, and I wanted to study under his guidance. It was rumored that he would return from Europe that year to become dean at Hanoi University, but as it turned out he did not return until 1949. The university followed the French system, and I entered the mathematics program of the faculty of general science. But two months later, in December 1946, war broke out. The French invaded, Hanoi fell, and the university closed. Koblitz: What did you do?

T~y: Well, my marks were not very good, and I was ill. At age 15, I had to miss a year of school because of respiratory problems and partial paralysis. I had paralysis for three months, and I was becoming fearful that it was incurable. But then I was suddenly cured by a skillful acupuncturist. This was 1942. We were occupied by the Japanese as well as the French. There were frequent bombings by American airplanes, even in my home village, because it was situated between two rivers, near two bridges on the trans-Indochinese railway. Nearly every day we had to run for cover in the trenches. After missing a whole year, I transferred from the lyc4e to a private school, where I was able to skip two grades and thereby graduate a year early, in 1946. But there were problems, because in 1945 the revolution had begun. Koblitz: How did that affect your plans? T~y: These were very revolutionary times. After receiving a level-I baccalaur6at, I returned to my village to take part. I was 18, and I knew that there was a danger that I would not be able to finish school with a level-II diploma: To receive a baccalaur6at II, one had to pass two parts of a difficult examination that was given twice a year, in May and September. After returning to Hfi~ in February 1946, I had only three months of study before the May session, and so expected only to take the first part. But my performance was the top in the class, and so I decided to take the second part right away, even though I had not prepared for it. To my surprise I was the highest scoring candidate on the whole exam. So I received my diploma, and had time to rest a little and then work to earn money for the trip to Hanoi.

T~y: I took w h a t r e m a i n e d of my m o n e y , a n d bought mathematics books to study later. Then I returned to m y village south of ~)a Nafig. In early 1947, the situation in Vietnam was complicated. After World War II, by agreement of the Allies, the Chinese armies of Chiang Kai-shek had occupied the north of Vietnam (extending to the south past ~)~ Nafig) and the British had occupied the south. Their purpose had been to disarm the Japanese. But then the French arranged with the British and the Chinese to replace their armies. Thus, the French army was already present in many cities, including -E)a Nafig, when we organized to defend our independence. Our military situation was very unfavorable. We resisted the French for two or three months, but then the Vietnamese army withdrew. I must say that we w i t h d r e w in considerable disorder, because we were surprised by the scale of the French attack. As the army withdrew from the towns, large numbers of civilians evacuated with them, leaving the plains for the highlands. It was terrible. We burned everything, so that the enemy could not use our facilities. Koblitz: Where did you go? Did you live with your family? T~y: For about two months I lived with my mother and brothers in the mountains of the western part of our province. Then they moved to a place about 100 kilometers to the south, and I went to teach secondary school in the province of Qu~ng Ngai. At that time Qu~ng Ngai had the best high school in our free region (which was called the Fifth Liberated Zone), and I taught mathematics there from 1947 to 1951. Koblitz: Did education proceed normally during this period?

Koblitz: Did you go directly on to the university? T~y: In the summer of 1946 1 earned money giving private lessons, so that I could afford to go. Then in late September, I took the train from Hfi~ to Hanoi in order to attend the university there. At that time I had

Tgy: To some extent, yes. Our free region was relatively stable, with a high level of economic and political organization and cultural life. Koblitz: It was at this time that you wrote a textbook? THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 3, 1990

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these youngsters: a book of poetry by the well-known contemporary Vietnamese poet T6 Hfi'u, and my geometry book. Koblitz: Were you able to study math in Qu~ng Ngdi? T~y: Yes, while teaching I also studied mathematics from the books I had bought in Hanoi. Koblitz: Could you visit your family during this period? Tuy: Yes, in our region we were able to maintain rail service. On the weekend I would take the train south to see them. The train went by night, without lights, so as to avoid detection by the French. For the first two years it had a steam-powered locomotive. But in 1949 the locomotive was destroyed in an air attack. Then at first we tried to power the "train" (which consisted of only a single car) using an automobile motor. But it broke, and we had no spare parts. After that the train car was pushed uphill by four people. It w o u l d travel at 7 or 8 km/hr, and of course faster downhill, so that we could leave Quang Ngal at nightfall and arrive in the south before daybreak. I think this was a unique form of transportation: a humanpowered car on rails. One could also travel by bicycle. However, bicycles were expensive, and I could not afford to buy one. So I always went by train. Although life proceeded normally for the most part, this was a hard time. In 1948 the secondary school where I taught was completely destroyed in a half hour of bombing by French Spitfire planes. Seventeen high school students died, and one woman teacher, a friend of mine. Koblitz: Why did the French bomb a school?

A page from Ho/Ing T.uy's 1949 geometry textbook. T~y: Yes, it was printed in 1949 by our anti-French resistance press. It was only an elementary geometry book for high schools, but perhaps it was the first mathematics book published by a guerrilla movement. I was amused to see a reference to m y geometry book in a recent popular novel. You know, in 1954 our country was partitioned at the 17th parallel. At that time some parts of the south had been liberated, but because of the Geneva Accords, the soldiers and many teachers from the liberated zones went north. Unfortunately, many of the schoolchildren could not leave with their teachers, and were left to their own devices to continue their education. According to the popular novel, there were two books that were most prized by 18

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T~y: You see, at this time the French army was terrible. They b o m b e d anything. Even an individual walking in the road would be strafed. We had thought that since it was very well known that we were a high school, there was no need for camouflage. After the b o m b i n g , we became m o r e p r u d e n t , and m o v e d classes to houses in another village and used camouflage. Koblitz: When did you decide to leave the south? T~y: In 1949, with Hanoi occupied and the university closed, some classes in university mathematics were established in the liberated zones in the mountains 200-300 kilometers north of Hanoi, near the Chinese border. In addition, two other rudimentary universities had already been set up in the free regions: one under Professor Ngu~,~n Tht~c H/lo in the Fourth Liberated Zone, and one under Professor N g u ~ n X i ~ in the northwest.

Intellectuals of the Viet Minh [anti-French resistance movement] organized an examination, which was administered by the Ministry of National Education. I was one of two candidates from my district in the south who took the exam. You must understand that the examination process was long and complicated, because the exam questions and our answers had to be carried over the mountain trail (later to be called the H6" Chf Minh Trail) by guerrilla courier. Normally it took about three months to get a letter from the north. But I must say that it was completely reliable, nothing was ever lost or misdelivered; in fact, the Viet Minh post worked much more efficiently than the Vietnam postal service does today. After I sent off my answers, I had to wait for eight months to hear the results. The exam tested general first-year university mathematics, mainly calculus and mechanics. Despite our primitive conditions, the exam was a rigorous one, and it was administered under strict conditions. The committee to administer the exam in our region was appointed directly by the Viet Minh governing council of the Fifth Liberated Zone. So the exam had a high prestige, and people were very impressed when the good news came of my success on the examination. Koblitz: Did you go north as soon as you heard? T~y: No, this was late 1949. I taught for two more years. In 1951 I learned definitively that L~ Van Thi6m had returned to Vietnam and was working in the liberated zones of the north. I then asked for permission to go north, and it was granted. Koblitz: How did you travel north? T.uy: There was only one w a y - - o n foot through the mountains. At this time the H6" Chi Minh Trail was still a trail in the proper sense, a narrow footpath. But it was very well organized. Every 30 kilometers there was a station where one could spend the night and a guide to take us to the next station. But, of course, there were many dangers. Koblitz: What were the main dangers? T.uy: There were t h r e e - - t h e French, malaria, and tigers. Koblitz: How long did your trek north take? T.uy: The actual walk took three months. The beginning, in the region near O~ Nafig, was relatively easy, since we could walk in the plains, taking the road by night. The dties and towns were abandoned, because of the French bombing. But they were not occupied. Further north, however, the French occupied all the

Vietnam.

lowlands, and we had to keep to the mountains. That was the hardest part of the w a l k - - i n the mountains of what is now Binh Tri. Thi~n province. Of course, we did everything possible to lighten our load. We carried only rice and salt for food. Before I left, I had taken my math books, removed the covers, and cut out the margins on every page so that they would be lighter for the journey north. Then, when we entered the Fourth Liberated Zone near H~ T/nh, just south of the city of Vinh, it was again safe to walk in the plains, and so we made better time. But I paused for two months in the Fourth Zone, giving private lessons to earn money to continue. It is THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 3, 1990

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interesting to recall that even during wartime there were schools functioning throughout the liberated regions; it was not hard for a qualified person to find a teaching job, and private tutors were in demand. I gave lessons during the summer h o l i d a y s - - J u l y to September of 1951--when high school students especially wanted extracurricular lessons. These small private classes were organized in houses having a room with a blackboard, and each pupil would bring his or her own small chair. After the interruption to give lessons in the Fourth Zone, in one month's time I was able to complete the distance, detouring around the occupied Hanoi region, and finally reaching the liberated university in the far north.

T.uy: No, on three o c c a s i o n s I h a d to a t t e n d meetings of the education ministry to discuss improvement of secondary education in the liberated zones. Since I had a reputation as a successful teacher, I wrote many reports on this subject. These meetings were a considerable distance away, either in southern Ha Tuy~n province or southern Ba4 Thai province. They were held near the Viet Minh government headquarters. But we would never know the exact location. We were simply told a place we had to reach, then a guide would take us along a complicated route to the location of the ministry, which changed frequently. We traveled by bicycles, which we had stripped d o w n so that we could more easily push them uphill in the

Koblitz: When you reached the north, did you immediately enroll in classes?

We did everything possible to lighten the load. We carried only rice and salt for food. Before I left, I had taken my math books, removed the covers, and cut out the margins on every page so that they would be lighter for the journey north.

T.uy: No. After all the hardships of the trip, when I arrived I learned that only the lower-level university courses had opened. Since I had already finished that level, I instead started teaching secondary school, and continued studying on my own. At that time I also met L~ V~n Thi~m. Koblitz: He was one of the main reasons why you had gone north? T~y: Yes, he was like an idol among young people at the t i m e - - t h e first Vietnamese to return with a French doctor of mathematics degree. He had returned first to the far south of Vietnam in 1949. Then a few months later he made the long trek north in order to help start the university. Of course, he was older, and a very eminent person. So the Viet Minh government provided him with escorts for protection and porterage. Koblitz: Was all the university-level instruction in the far north? T~y: No, as I mentioned before, during this time NguS,~n Th~c Hao, a former high school teacher of mine, was giving advanced classes in a place about half-way between Hanoi and H ~ , in the Fourth Liber.^? ated Zone. And Nguy~n Xlen was doing the same in a place to the west of Hanoi until 1950, when he agreed to join with L~ V~n Thi~m. Then from 1950-54 the main center for university-level instruction was in the far north, only a few kilometers from the Chinese border, where e n e m y planes would be hesitant to come, because of the French fear of involving the Chinese. Koblitz: So you remained in the border area the whole time? 20

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forest with their load. We even took off the brakes. Going downhill, we w o u l d stick branches in the spokes to slow the wheels. In this w a y we w o u l d travel the 100-200 kilometers to the meetings in two or three days. Koblitz: Were there any advantages for you in studying mathematics in the north rather than in your home province in the south? T.uy: Oh, yes. In addition to the presence of L~ Van Thi~m, there were also better books. Until then I had been able to get only old French books--parts of Vessiot-Montel, Papellier, Goursat, etc. N o w I could b u y newer Russian books in bookstores of the liberated region. Koblitz: Did you know Russian?

T.uy: No, I had to learn it. I found a very old book called Russian in Three Months, intended mainly for businesspeople, from which I learned the grammar rules and a few words. Then I immediately started reading I. P. Natanson's Theory of Functions of a Real Variable. For the first page or two I needed a dictionary for almost every word, then less and less, until I could read it fluently. So Natanson was the first Russian book I read. And I must say, it is an excellent book. I was fascinated by measure theory, Lebesgue integration, and so on. That text greatly influenced my early mathematical interests.

Koblitz: How did you organize your studies?

TOy:From 1951-55 1 closely followed the Soviet university program in mathematics--studying book by book on my own. In 1955, with Hanoi liberated, the university there reopened. L~ Van Thi~m became the rector. In September of that year I started teaching at Hanoi University. At that time I was well known as one of the best high school teachers in the liberated zones. In addition, I had read a lot about secondary education in other countries, especially the Soviet Union. So in 1955 the new government appointed me to chair a committee on reform of the secondary school system, even though I was only 27 and others on the committee were much older. Two years later, in S e p t e m b e r 1957, I w e n t to Moscow for further study.

TOy: After one year I had written a Candidate's dissertation in real analysis under Menshov's supervision. Instead of returning to Vietnam after the year, I was allowed to remain for a few months to complete the formalities--first switching from the upgrade-ofqualifications to the Candidate's Degree program, then ensuring publication of m y results, then the thesis defense. I received my degree in April of 1959. Koblitz: What were your reactions to Moscow? Did you have trouble adjusting to a world so different from Hanoi?

TOy: Yes, it was my first trip abroad. It took two weeks by train, through China.

TOy: For the first few weeks in Moscow I was very enthusiastic. I had heard so much about the Soviet Union, about such great names of mathematics as Kolmogorov, Aleksandrov, Pontryagin. I remember being impressed by the majestic central building of Moscow State University on Lenin Hills. But after a while I started to really miss Vietnamese food, for example. Moreover, w h e n I left Vietnam m y wife was pregnant with our first child, and a month after my arrival in Moscow I received a telegram informing me of the birth of a son. That made me really homesick.

Koblitz: Were there many Vietnamese studying in Moscow at that time?

Koblitz: Was that your first long period away from your wife?

TOy: No, only about a hundred students. I was one of 9 or 10 at the advanced (graduate) level.

TOy: No, we met when I was in the south. She was a student, preparing to become a teacher, and I was a teacher. In 1951 we had just become engaged. My enthusiasm for L~ V~n Thi~m was so great that I decided to leave m y fiancee to go to the north. At that time correspondence was very difficult--a full year to send a letter and receive the reply from the south. Then in 1957 my enthusiasm for Soviet mathematics was so great that I had to leave her again for 20 months.

Koblitz: How did you travel to Moscow? Was that the first time you left Vietnam?

Koblitz: Did you enter a regular program for the Candidate's Degree [= U.S. Ph.D.]? Tuy: No, initially I was supposed to come only for a year for a program of "upgrading qualifications" in the Mechanico-Mathematics Faculty of Moscow State University. I chose to study real analysis, and was assigned two s u p e r v i s o r s - - D . E. Menshov and G. E. Shilov. I think that when I met them they were skeptical about my qualifications. They asked me a series of questions, some of which I could answer right away but some of which were extremely hard. I remember that one of the difficult questions was the following: Given a set A C [0, 1] of positive Lebesgue measure, prove that the set {x + y [ x , y ~ A} contains an interval. Shilov gave me a week to find the solution, which I fortunately managed to do. I later learned that this had been p r o v e d as a proposition in a recent paper of his. My proof was different from Shilov's. He apparently was favorably impressed and after that had confidence in me. That year he invited me to his home for N e w Year's Eve, which is the biggest family holiday in the Soviet Union. Koblitz: How did your studies go?

Koblitz: After receiving your Candidate's Degree in Moscow, did you return to Hanoi? TOy: Yes, I became chair of the mathematics department at Hanoi University. Since 1959 I have lived always in Hanoi, except for trips abroad. Koblitz: Did your mathematical interests change after returning to Vietnam?

TOy: I began my career in real analysis, and published 5 papers in that field in Soviet journals. But then I realized that, as a research field, real analysis is not so useful for my country. It is a very beautiful theory, of course, but it is a bit too theoretical, a bit far from applications (at least this was true at that t i m e - now it seems that this is changing). In 1961 I became interested in operations research. In 1962 I sent my first paper on mathematical proTHE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 3, 1990

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gramming to Kantorovich, and I visited him in Novosibirsk later that year for a few weeks. Koblitz: That was when operations research was first introduced in Vietnam? T.uy: Yes, in 1961 I had heard that Chinese mathematicians had started working in this field. Even the famous n u m b e r theorist, algebraist, and function theorist Hua Lo-keng was actively promoting the field. So when T.a Quang Bu'~, the minister of higher education and also a mathematician, visited China that year, I asked him to get information about applications of o p e r a t i o n s r e s e a r c h . After his r e t u r n , I started working in that direction in earnest. Koblitz: Did you also visit China yourself? T.uy: My first visit to China (not counting transit by train w h e n going to Moscow) was in 1963, when I spent a month lecturing at the Mathematics Institute of the Chinese Academy of Sciences, the Mathematics Department of Beijing University, and several other universities. I visited again in 1964 for three weeks. In those visits I met with such distinguished mathematicians as Hua Lo-keng, Wu Tsin Muo, Cheng Minde, Gu Chaohao, H u Guoding, and a y o u n g w o m a n named Gui Xiang Yun who later became a prominent figure in operations research in China. (I had first met Gu C h a o h a o and H u G u o d i n g w h e n w e were in Moscow.) But I lost contact with them all at the time of the Cultural Revolution in China. Koblitz: At this time, in 1964, the American war in Vietnam was intensifying. How did this affect mathematical life in Hanoi? T.uy: Because of the bombing, in May 1965 the university was evacuated to a forest area 170 kilometers to the northwest of Hanoi, not far from the city of ThAi N g u ~ n . At that time I was dean of the Faculty of Mathematics and Physics. We had about 250 students in the mathematical sciences. Koblitz: How was life in evacuation? Did you have problems of hunger, malaria, other tropical diseases, and so on? T.uy: There was no malaria. Malaria had been eradicated in North Vietnam as a result of the antimalarial campaigns of 1955-59. Koblitz: This is surprising, since now Vietnam is certainly included among the countries with endemic malaria. T.uy: Yes, unfortunately, there has been a resurgence. In the case of Vietnam this was brought on by 22

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the unification with the south in 1975. The movement of people between north and south spread the disease. We knew that malaria had not been eliminated in the south. In 1974, the health minister of North Vietnam made a trip to the liberated zones of the south in order to conduct a systematic study of malaria there. He himself then caught the disease, and died. Koblitz: How about other tropical diseases--schistosomiasis, leishmaniasis, giardiasis, etc.--in the forest near Thdi Ngugfn? T.uy: Those diseases are generally not as life-threatening as malaria. In any case, despite our primitive conditions in evacuation, disease was not common. Of course, we took precautions, such as cooking food well. But there was very little absenteeism from class due to disease. Nor was food a problem where we were. However, life was very hard. Families were separated. My wife had to go with the high school where she taught, which was evacuated to a place 30 kilometers to the southeast of Hanoi, the opposite direction from the university. All three of our children went with her, because they could go to school in that location. The main danger was the bombing. American aircraft could come at any time. We had classrooms made of bamboo in scattered locations hidden under the trees. Right next to the desks below our legs we had dug trenches, so that we could dive into them in an instant. Koblitz: Did any bombs hit the university in the forest? T.uy: No, but some bombs fell quite close. One time an American pilot was captured right near us. Koblitz: And you were able to do mathematics during all this? T~y: Yes, in fact our general morale was so high that we continued our seminars on a regular basis during this whole period. The Mathematical Society, which was founded in 1965 by L6 Van Thi~m (I was general secretary), organized joint seminars in optimization, probability, functional analysis, algebra, numerical analysis. People from Hanoi University, the Pedagogical Institute, and the Polytechnic Institute participated (the Hanoi MathematicaI Institute was not formed until 1970). Since the three institutions had been evacuated in different directions from Hanoi, we held the seminars in Hanoi. They met twice a month. I must say, people were very diligent about attending. Many of us would take advantage of the opportunity to visit our families. Since my wife and children were on the opposite side of Hanoi from the university, it was much more convenient for me to visit them after the seminars held in Hanoi.

Ho/mg Tuy (left)with Cheng Minde at the Great Wall, 1963.

Left to right: Hua Lo-keng, Hoang Tuy, Wu Tsin Muo in Beijing, 1964.

Koblitz: That was the time of the visit by the famous

T~y: In mid-1968 I was invited to head the newly created mathematics division of the State Committee for Science and Technology.

French mathematician Alexandre Grothendieck? T.uy: Yes, he visited in November 1967. The first few days we organized his lectures in Hanoi. But one day a missile exploded only 100-200 meters from the lecture hall. As a result the higher education minister Ta Quang Bu'~ ordered us to be evacuated. I remember that Grothendieck was delighted with the news that we were being evacuated, and approached the unusual situation in a spirit of adventure. Koblitz: And Grothendieck continued his lectures in the forest of Thdi Ngug~n? T~y: Oh, yes. He gave a "short course" in category theory, homological algebra, and algebraic geometry, with Oo~n Quj~nh as the main translator from French into Vietnamese. Grothendieck would lecture for four hours in the morning, and then hold consultations all afternoon. Even so, he always complained that he was underemployed. He was a strict vegetarian and observed a fast day every Monday. Koblitz: How long did the university remain in the

forest? T~y: The university was in evacuation for four years; it reopened in Hanoi in September 1969. Then again in 1972-73 there was another evacuation, this time to a place closer to Hanoi. By then I was no longer affiliated with Hanoi University. Koblitz: When did you leave the university?

Koblitz: Is that when the Mathematics Institute was

formed? T~y: Yes, soon after. The decision to establish the institute was made in 1969, and it opened in 1970, headed by L~ Van Thi~m. At that time it had only 20 members and was located in the building of the State Committee. Koblitz: Was the Mathematics Institute also evacuated from Hanoi during the bombing? T~y: Yes, for a year or so starting in mid-1972, we set up the institute in a place 50 or 60 kilometers from Hanoi. But in that period I spent a total of only one day there, since I wanted to live in Hanoi. I was especially concerned about my many books. At that time there was a termite problem in m y fiat in Hanoi, and I felt I had to care for and watch over my books. Koblitz: So you stayed in Hanoi even during the "'Christmas bombings" of December 1972?

T~y: Until the fifth day. In the first days of the bombing only the outskirts of Hanoi were targeted. Then on day 4, it reached the center: the railway star-ion and B.ach Mai hospital were hit. After that an order came that all non-military personnel must leave the city. So I left to go to my wife, who was with her school in evacuation. THE MATHEMATICALINTELLIGENCER VOL. 12, NO. 3, 1990 2 3

TOy: No, but in the aftermath of the invasion there were discussions of evacuation, because we feared that China would invade again and perhaps penetrate farther. Not only the Mathematics Institute but many other institutions made evacuation plans. We would have moved our institute to H6" Chi Minh City, where a small branch of the institute (now the Center for Applied Mathematics of H6" Chi Minh City) had recently been established. But I was opposed to an early evacuation largely because of the costs involved. I suggested that in the event of another invasion we should remain in Hanoi until the last possible moment. Of course, as it turned out the feared second invasion never took place.

Ho/~ng T.uy in front of the Hanoi Mathematical Institute, 1989. While traveling on the road from Hanoi, by good fortune I saw my wife coming in the opposite direction. She h a d b e e n very worried because of the bombing and was returning to Hanoi to look for me, despite the evacuation order. So it was very fortunate that we saw each other on the way. Soon after, we heard on the radio about a cessation of bombing for Christmas. So we r e t u r n e d to our home and spent the night of December 25 in our house. Early on the 26th we left Hanoi. The same day Kham Thi6n Street was bombed. Many people who had not left early enough were killed. We learned that the restaurant where we had eaten ph(~' [traditional Vietnamese soup, eaten for breakfast] that very morning had been destroyed only hours later. Koblitz: Did everyone return to Hanoi right after the peace agreement was signed on January 27, 19737

TOy: No. I returned within a few days after the peace agreement, but I went back alone. I needed time to arrange for the rest of the family to return. Conditions in Hanoi were very bad, and it took time. We needed a period during which people were returning gradually. The Mathematics Institute returned to Hanoi two or three months later. Koblitz: When did the institute move to its present location, at the National Center for Scientific Research?

TOy: In 1982. Before that, from 1975-82, it was located in cramped quarters on :D.6i Carl Street. Koblitz: Was there any disruption of activities at the Mathematics Institute at the time of the Chinese invasion in February 1979? Did you evacuate Hanoi? 24

THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 3, 1990

Koblitz: You were saying that the place on ,DO.i Cd~ Street was inadequate. So by 1982 you needed a larger facility for the institute? TOy: Our quarters on D.6i Carl were very primitive; Prime Minister Ph.am V~n :D6"ng himself arranged for a building at the National Center to be constructed especially for mathematics. Koblitz: What is the history of PMm VSn ~D6hg's personal interest in the Mathematics Institute? When did you first meet him?

TOy: I knew him slightly in the 1940s, when he visited our school in Quang Ngai several times. He was then the representative of the central Vietnamese government in the Fifth and Sixth Liberated Zones. After 1960 I got to know him better. Soon after the war ended in 1975, he made a trip to Moscow, where he visited Kantorovich's institute. (Kantorovich had by then moved to Moscow from Novosibirsk.) Kantorovich gave Ph.am V~n D6"ng a copy of his book on optimal m e t h o d s in economics. Upon r e t u r n i n g to Hanoi, Pham V~n D6"ng asked me to read the book and report to him on it. The timing was lucky for me. Some French colleagues, who had noticed that I was on the program committee for a conference in Budapest to be held in August 1976, had taken the opportunity to invite me to visit France. However, I received the invitation only one month in advance, and at that time we had many b u r e a u c r a t i c obstacles to o v e r c o m e in t r a v e l i n g abroad, especially to the West. Koblitz: Why was that? TOy: Well, of course, there were financial obstacles to travel to the West. But even in cases when our hosts would pay for all travel and expenses, it was still difficult. During the war and the immediate post-war years the West was considered very remote, even somewhat dangerous. That attitude has changed, for-

tunately, and now we have no difficulty in obtaining government permission for graduate students and mathematicians to travel to the West, provided that they have international support for their travel and expenses. Koblitz: So how did you manage to go to France in 1976?

T.uy: It was at that time that I went to Ph.am Van O6"ng to report on Kantorovich's book. After making my report, I told him about the invitation to France, and asked if I could go. He said, "Sure, no problem," and with the prime minister's support I was able to complete the formalities in one week, which was record time. That trip to France in 1976 was my first visit to a Western country. (Later I visited Canada for the 1979 Montreal conference on mathematical programming, and in 1981 I made my first visit to the United States.) Koblitz: And Pha.m V~n ~DOhg has been a supporter of the Mathematics Institute since then? T~y: Yes, I would say that in the government he has been the most influential and consistent supporter of mathematical development. In 1980 he visited us on O.6i C~fi Street. Seeing how poor our conditions were --conditions in all of the institutes were bad, but ours were even worse--Ph.am Van O6"ng promised to do something about it. We later learned that he had asked the minister of construction to build us a center for the mathematical sciences as soon as possible. It was built very quickly for V i e t n a m - - i n a y e a r - - a n d it n o w houses the institutes of mathematics, mechanics, and computer science. Koblitz: Did you ever meet H0" Chi Minh? T.uy: Twice. In 1956 he visited Hanoi University and observed some classes. After watching me teach, he s h o o k m y h a n d and asked a few questions. The second time was in August 1969, just a month before his death. At that time there was a big problem with queues to buy beer. Actually, there were long lines to get served in the shops generally--for rice rations, clothes, and m a n y p r o d u c t s - - b e c a u s e production lagged far behind the demand. But the lines for beer were a special cause for concern because of frequent disputes, even fights, which disrupted public order. H6" Chl Minh suspected that there must be a scientific approach to reducing the length of the queues. So he asked the State Committee for Science and Technology to look into the matter. Since there was no possibility of increasing beer production, the problem had to be treated as a purely organizational o n e - - a s a problem in operations control. So I headed the group that studied the situation.

Our first meeting with the government leadership was delayed because of the ill health of H6" Chi Minh. Then a few days later I received a call to go to Ph.am Van D6"ng's office. I didn't know w h y I had been summoned and thought that he probably wanted to discuss the Mathematics Institute, which was just then in the process of being set up. I remember that my car was late, and when I arrived at Ph.am Van O6"ng's ofrice several high-ranking members of government had already arrived. Ph.am Van O6"ng said to me, "Oh, I see you're late. If that's the way you do operations research, this project will come to naught!" I apologized and was given a seat next to him. Only then I suddenly noticed H6" Chi Minh among the people seated around the table. Koblitz: HO" Chf Minh was then 79. How involved was he in the discussions? T.uy: When I saw him, he was frail, but mentally in full command. He asked many questions and was quite disturbed about the beer lines. I remember the first words he asked me: "Can't you find a simpler word than v~.n trf~ [for operations research]? The President himself has never before encountered this word in Vietnamese." Koblitz: Had you invented the word? T~y: I had taken it from Chinese. You see, operations research was a new science--it was introduced in Vietnam in 1961. For a long time I could not find the right w o r d for it in Vietnamese. Then I decided to adopt the term that my Chinese colleagues were using. This was quite natural, since Chinese plays a role for Vietnamese s o m e w h a t like Latin does for French and English.

Prime Minister Pham V~n O6"ng (left) and Ho~ng Tuy. THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 3, 1990 2 5

Koblitz: But H6"Chf Minh thought the word was too obscure? T.uy: Well, he was interested in w h y I had chosen it. Later, w h e n I approached him to say good-bye, he asked me if I knew the etymology of the term. I did not, and he proceeded to explain the origin of the word in classical Chinese literature. It was used in a famous novel to denote a certain intricate form of art which is perhaps analogous to operations research. H6" Chi Minh knew Chinese literature w e l l - - h e had even written poems in Chinese--and so he could tell me the etymology of the term vd.n trf~, which I had brought into Vietnamese. Incidentally, in the early 1970s the term v~.n trf~ entered into popular usage; it became fashionable as a way to refer to finding an optimal solution to anything.

Reunion of the leading mathematicians of Vietnam, May. 1986.

Koblitz: Were you often responsible for inventing mathematical terms in Vietnamese? T~y: I participated in this: from 1959-61, as a m e m b e r of our Commission on Scientific Terminology, I helped produce the first Vietnamese-English-Russian dictionary of scientific terms, which included the terminology of contemporary mathematical research. But the person most responsible for developing scientific terms was the mathematician Hoang Xu~n Han, who emigrated from Hanoi to Paris after the French occupied Hanoi and had been a secondary school teacher of L~ Van Thi~m. Besides being a mathematician, Hoang Xuan Han had a great erudition in both Vietnamese and Chinese literature, and in France he changed his interests from mathematics to literary studies. He had started developing scientific terminology in the early 1940s, along with N g u ~ n X i ~ and N g u ~ n Th~c H~o. Koblitz: Returning to the problem of using operations research to reduce beer lines, was your committee able to do anything? T~y: Yes, two or three months later we were able to report back to Ph.am Van O6"ng (H6" Chi Minh had died in September 1969) with specific recommendations. Our suggestions were actually implemented, and they did improve the situation for a couple of years. But then the bombing of Hanoi resumed, there were tremendous organizational difficulties, and everything was back to where we started. Koblitz: When you speak of your ties with government leaders and the importance attached to mathematics by the Vietnamese government ever since the days of the Viet Minh, one gets the impression that mathematics occupies a special place in Vietnamese science. Who were the mathema26

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T.a Quang Bu'~ (center) with colleagues.

ticians responsible for such a prominent place of mathematics in Vietnam? Are they still alive? T.uy: In May 1986, I organized a meeting in Hanoi of mathematicians of various generations. We knew that T.a Quang Bu'~ was in ill health (and, in fact, three months later he died), so it was to honor him and to bring together for one last time all the mathematical generations of Vietnam from youngest to oldest. Although T.a Quang Bu'~ did not make any major contributions to mathematical research, he had a deep appreciation and an understanding at the near-research level of m a n y fields of mathematics and physics. He was the first General Secretary of the State Committee for Science and Technology, and he was an excellent minister of higher education. He had a close relation with Ph.am Van D6"ng, and his rapport with Laurent Schwartz was partly responsible for our early mathematical ties with France. Several other important figures in the development of mathematics in Vietnam were present: L~ Van Thi~m, whose role had been decisive and who had

been an idol of my generation when we were young; Ngu~,~n Th~c Hao, my former high school teacher, who in the 1940s and 50s directed university classes in the Fourth Liberated Zone, and others. Koblitz: How would you summarize the strengths and weaknesses of Vietnamese mathematics at the present time, in particular in your institute? T.uy: First of all, we have a strong tradition in analysis, including classical analysis, functional analysis, p.d.e., convex and nonlinear analysis, and of course complex analysis, where L~ V~n Thi~m did his work. A former student of Nevanlinna, L~ Vfin Thi~m became known for a pioneering solution to an inverse problem in meromorphic function theory and went on to develop a group of people in Vietnam working in complex analysis. In optimization we have one of the strongest groups in the institute, with good contacts in other countries and many publications in international journals. Our work is closely related to nonlinear analysis. Our institute also has a number of researchers in algebra and algebraic geometry, including algebraic topology and the theory of singularities. In probability theory we have some good specialists, but they are isolated and do not really work as a strong group. Our most serious weakness is in applied mathematics, especially those aspects which depend upon availability of equipment, a modern infrastructure, a high level of i n d u s t r y - - n o n e of which we have in Vietnam. Koblitz: But you switched from real analysis to operations research in the belief that it could be applied in Vietnam. Have you been disappointed in the extent to which you have been able to find applications of your work in the conditions of Vietnam?

cally satisfying to me because of its completeness and elegance. But I was young and enthusiastic and so was able to switch to a new direction. After my visit with Kantorovich in Novosibirsk in 1962, where I reported on my work on the nonlinear transportation problem, I became more convinced than ever of the need to change fully to the new field. Starting in 1962, I no longer worked in real analysis. Koblitz: How rapidly did your work in optimization progress?

T.uy: In 1964, when I returned to Kantorovich's institute, I could already report on much more serious results. This was my research on concave minimization, which was my first work to have a significant influence on the field and bring me international recognition. Koblitz: What is concave minimization? Can you explain its importance?

T.uy: Yes. Before, people had extensively studied the problem of minimizing a convex function over a convex set. There it suffices to use local information, i.e., the usual techniques of analysis can be employed. But my work on the transportation problem had led me to see the importance of the concave analog, which was more difficult. Let me give a simple illustration. Using the earlier methods in the transportation problem, one w o u l d have to assume that cost is a convex function of distance, i.e., that the cost per kilometer increases with the distance traveled. This is what is mathematically convenient, but it does not correspond to reality. In practice, there is a fixed cost and then a marginal cost which decreases as a function of distance. Cost versus Distance

T.uy: From the very beginning we have made many efforts to use mathematics to solve practical problems. In 1961-62 1 myself worked on a problem in transportation-reorganizing the logistics of trucking so as to reduce the distance that trucks travel empty. I later learned that we had ventured into this applied problem earlier than had Soviet mathematicians. However, once the Soviet research started in about 1963, they were able to carry it through much better than we could. When the war started in Vietnam, all progress in practical applications of operations research came to a halt. I should say that w h e n I started studying operations research, at first I was not very pleased with the type of mathematics that was used. The first text on linear programming available to me was not a good book, and the field struck me as boring in comparison to the beauty of measure theory, which was more aestheti-

mathematically easier

more realistic

Koblitz: But concave minimization is more intractable? T.uy: Yes, nowadays we would call it an "NP-hard" problem. We didn't have such precise terminology in the early 1960s, but it was recognized by Dantzig and others to be an intrinsically difficult problem. THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 3, 1990

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Koblitz: How did you tackle this problem? T.uy: I proposed a new kind of cutting plane. Cutring planes were introduced in integer programming by Gomory in the 1950s, and then they were used in convex programming, too. In 1964 I suggested a new type of cut, which would enable one to carry out a concave minimization algorithm. Koblitz: But you say you were unable to follow through with the practical side of this work? T~y: Yes, unfortunately. After I introduced this kind of cut and elaborated a method of solution for the concave minimization problem, the next step should have been to test the algorithms on computers and see how one could improve them. But we had no such possibility in Vietnam. Soon we were in evacuation, and I had to p u t out of my mind any thought of working on implementation. So of necessity I worked on more abstract aspects, i.e., the general theory. Thus, my next results concerned convex inequalities and the Hahn-Banach theorem. One of these results is sometimes known as "Tuy's inconsistency condition."

P r i m e M i n i s t e r P h r j m V a n O ~ ' n g h i m s e l f arr a n g e d f o r a b u i l d i n g a t the N a t i o n a l C e n t e r to be c o n s t r u c t e d e s p e c i a l l y f o r m a t h e m a t i c s .

Koblitz: How long did it take for your work to be recognized in the West? T~y: In those years I had almost no contact with the West. It was only in 1972, when I first met Vic Klee in Warsaw, that I learned from him that several people in the U.S. were very interested in my 1964 paper. For several years, my cutting plane had been known as the "Tu1 cut." The reason for the wrong spelling of my name was that my work first appeared in Russian with my name as Tyf~ in Cyrillic letters; this was rend e r e d into English by the A.M.S. transliteration system, as if I were a Soviet mathematician. The person w h o corrected this was Egon Balas, a Romanian emigrant to the U.S., now at Carnegie-Mellon University in Pittsburgh. In a 1971 article he referred to me as a "North Vietnamese mathematician" and wrote my name correctly for the first time. Koblitz: You mentioned a visit to Poland. How often were you able to travel abroad during the war years? T.uy: In 1966 I attended the International Congress of Mathematicians in Moscow, and the following year I went to the Soviet Union for a short visit as part of a 28

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delegation from the State Committee for Science and Technology. But my next opportunity to travel abroad came only in 1972, when J. Los invited me to spend three months in Poland. He was organizing a semester on mathematical economics similar to the programs in various branches of mathematics that were later held at the Banach Center. That visit to Poland was m y first experience lecturing in English. At the time I had only partial success with that language. I remember that after my first talk in English, one of my listeners came up to me and said, "From your talk I can tell that you speak French very well!" Koblitz: How have ties with Western mathematicians developed? Have many visited Vietnam? T.uy: During the war, we had only a few visits. Besides Grothendieck, there was Chandler Davis from C a n a d a and Laurent Schwartz, Martineau, Malgrange, and Chenciner from France. Then in the 1970s and early 1980s many mathematicians came from France: Tatar, Puel, Bardos, Dacunha-Castelle, Y. Amice, and some French-Vietnamese such as Frederic Pham, L~ Dfing Tr~ing, and Bfli Tr.ong Li~fi. The statistician Klaus Krickeberg has a regular association with our institute: he has even learned our language and lectures in Vietnamese. Pierre Cartier has visited us several times. We have also had guests from other countries, such as Bj6rk from Sweden and Saito from Japan. But traditionally among the Western countries our closest ties have been with the French. In recent years, however, it seems there has been a slight decrease in our level of contact with France. Koblitz: Why is that? T.uy: Perhaps partly for financial reasons. In addition, some people in both France and Vietnam feel that, n o w that mathematics is in a healthy state, more attention should be paid to other areas, particularly in applied science. Also, I have the impression that the French government has shifted its focus more to African countries. But in the last few years our ties with some other countries, such as West Germany and Japan, have increased markedly. We have also established closer relations with mathematicians in the U.S., Sweden, Great Britain, Italy. We now have a regular exchange of journals with the Italian Mathematical Society. Koblitz: It seems that many more young Vietnamese mathematicians are traveling to the West and Japan than ever before, isn't that true?

T~y: Yes, we are proud that many of our young researchers have received fellowships to study abroad.

For instance, during the last two years about 8 or 10 have received Humboldt fellowships to study in the German Federal Republic; and the Japanese Society for the Promotion of Science has awarded us several fellowships. At the Mathematics Institute we try to take advantage of opportunities provided by such foundations. Here there is no special arrangement, no bilateral agreement with Vietnam. The fellowships are very competitive, and our researchers apply along with everyone else. Thus, this is true cooperation, rather than aid going in only one direction: we feel that we are contributing to the international mathematical community as well as receiving. Of course, aid as such will continue to be necessary for many years, and we always welcome the special arrangements with different governments and institutions in the developed countries, in particular the Soviet Union and the countries of Eastern Europe. But our increasing use of the normal channels of exchange with the West and J a p a n - - g r a d u a t e fellowships, post-docs, visiting professorships--is a positive and encouraging development. I am very pleased that many of my y o u n g colleagues are n o w established mathematicians with normal international ties. This has also helped alleviate some of the institute's most pressing material problems. Koblitz: How is that? T~y: Well, when a member goes abroad, he or she receives a salary from which a fair amount can be saved. This can be used to help meet family needs after returning, so that the mathematician may be able to get along without working at a second job. Moreover, a certain percent of the salary earned abroad is normally donated to the institute. Koblitz: This is a kind of tax? T~y: No, it is not like the government's tax. It is not official, but rather purely voluntary. I am glad to say that in our institute most of our colleagues are willing to do this. We all understand that the financial support from the government is not sufficient even to maintain the normal functioning of the institute, let alone to meet our increasing needs. For example, we recently b o u g h t t w o n e w m i c r o c o m p u t e r s and a second photocopier in Thailand (our first copier was donated by the late Ed Cooperman of the U.S. Committee for Scientific Cooperation with Vietnam), and we continually need to buy toner, accessories, spare parts. This would not be possible without the donations of institute members returning from fellowships and visiting professorships abroad. Koblitz: In the ten years that our U.S. Committee for Scientific Cooperation with Vietnam has been arranging

Ho/mg Tuy.

visits of Vietnamese scientists to the U.S.--there have been almost 200 such visits--we have never had a case of a scholar deciding not to return. When I mention this to people in other developing countries, they are astonished, especially in view of the gigantic difference in material conditions of scientists in the U.S. and Vietnam. Why have there been no defections among the visitors? T.uy: Most Vietnamese scientists think that visits to foreign countries are necessary for their research. But what we can do best is always in our own country. Of course, we are pleased to have opportunities to spend extended periods abroad. But on the other hand, we are really happy in human terms only in Vietnam. Koblitz: On the question of international ties, how extensive are contacts with other Southeast Asian countries and with India? T~y: We are now thinking about how to develop relations with India and with the neighboring countries. Here there is a contradictory situation. On the one hand, it would be natural to develop cooperation first with neighboring countries. However, most of those THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 3, 1990

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countries are very poor, and it is difficult to find funding for joint activities. We have some ties with Singapore. In 1978, when I met Lee Peng Yee, we agreed that we should develop cooperation. A few years ago a young mathematician from our institute visited Singapore. In 1979 there was a conference in Singapore of mathematicians from Southeast Asian countries. It was sponsored by the Southeast Asian Mathematical Society and the French government, and Laurent Schwartz arranged for the French e m b a s s y to s u p p o r t the p a r t i c i p a t i o n of Vietnam. In general, the only time we see mathematicians from the region is at international conferences. And I must say, I'm not very optimistic about much imp r o v e m e n t in the next few years. Here I am not thinking of the political aspect but rather the financial aspect: neither the Vietnamese government nor the governments of neighboring countries are prepared to give money for this cooperation. Much more favorable conditions exist in the case of the developed countries: when they invite us, they pay for transportation and all expenses; and in certain cases--particularly West Germany, France, and Japan--the governments also support the travel of their mathematicians to Vietnam. Even in the case of India we have difficulties. There is some cooperation between the two countries in applied science--agriculture and m e d i c i n e - - b u t less in theoretical areas. There have been only two visits of our mathematicians to India, and none of Indian mathematicians here. Koblitz: Are the practical difficulties the only ones? Could it be that mathematicians in developing countries-such as Vietnam and India--are not aware of or interested in each other's work and view ties with the developed countries as more prestigious and valuable? Tgy: It is true that a mathematician in a developing country usually has in mind cooperation with a developed country. For example, if his institute can get the money to invite someone, he would prefer to invite someone from the West, since such a visit may lead to expanded cooperation with a d e v e l o p e d country. Thus, to develop our ties with countries such as India, a special effort must be made. Unless there is a source of financial support, the process is likely to go very, very slowly. Koblitz: What can Vietnam offer to countries of the region in mathematics? Tgy: In several areas of mathematics, for example in optimization, we have a group in Hanoi which is, I think, among the strongest in Asia. We could train graduate students from the region in these fields. A relatively small a m o u n t of m o n e y - - s a y , $200 per 30

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m o n t h - - w o u l d fully s u p p o r t a foreign student in Vietnam. Meanwhile, Vietnam could send students to the nearby countries for training in applied areas, such as statistics or computer programming. Perhaps from time to time we could also exchange visiting professors with countries in the region. However, I think that an exchange of students will be more realistic to implement on a larger scale. Koblitz: Earlier you mentioned the increasing needs of the Hanoi Mathematical Institute. Is that because it has increased in size? Tgy: No, there has been only a modest increase in number in recent years: ten years ago we had about 60 mathematicians, and now we have 78. But our level of activity has increased much faster, resulting in a great demand for books, computers, etc. You can imagine - - j u s t to type a paper is a problem in our country. There is no money from the government to buy typewriters--for that we rely upon the donations of our members. Koblitz: So the institute members are increasingly publishing in international journals? Tgy: Yes, b u t there are some special difficulties which are frequently encountered by mathematicians in developing countries. After we complete a paper, we often have long delays and other problems before it is published. We must contend with the slowness and unreliability of the mail, with the slowness of the reviewing process. In the U.S., if you don't hear anything after five or six months, you can call the editor and usually speed up the process. We have no w a y of doing that. Of course, a letter takes longer, especially from Vietnam, and it is not as effective as a phone call. In my o w n case, even though I am known in my field, some of my articles have taken a year and a half to be refereed. Then we have to sign and return the copyright agreement, which may take another two or three months. Some journal editors are not very sensitive to the situation in developing countries. They want three copies of the paper in the correct format. Then they ask us to make minor editorial corrections that they could easily do themselves. All of this adds considerably to the publication delays. Koblitz: When young mathematicians return from study abroad, do they have trouble adjusting to the conditions in Vietnam? For example, someone who just received a Candidate's degree in Moscow will find that in Vietnam there are far fewer people to talk with in his or her field. T~y: Yes, there is the problem of an atmosphere of isolation. A number of mathematicians trained in the

application to the economy of Vietnam. Do the members play a role in the educational system or in industry? Out of the 78, how many teach at the university or work on practical problems? T.uy: I would say that about 15 are working on industrial applications--mainly people in optimization, statistics, or p . d . e . - - a n d another 15 are teaching. We send lecturers to other cifies--Hfi~, Oh Lat, H6" Chi Minh City, Vinh--as well as to Hanoi University and the Hanoi Polytechnic Institute. Koblitz: That still leaves about fifty who are receiving salaries for pure research alone. How do you justify that in the impoverished conditions of Vietnam? Do people outside the institute try to get you to change your focus? Or do you find universal agreement about the importance of theoretical research ? Ho~ng Tuy (left) with L. V. Kantorovich in Moscow, 1981.

Soviet Union stopped doing research when they returned to Vietnam. In my case I was lucky, because in '62 and '64 I could visit Kantorovich. My trips to the Soviet Union, China, and Poland during the 1960s and early 1970s were very valuable to my research. Also, since I was largely self-educated in mathematics, I was accustomed to working in conditions of isolation. In general, h o w well someone adjusts to work in Vietnam depends on the particular individual and to some extent also upon the field. Koblitz: It must be difficult adjusting to the material deprivations in Vietnam. What is the salary of a mathematician at the institute? Is is possible to live on that?

T~y: Most people recognize that fundamental research, even though it has no immediate economic impact, is important for the future. From time to time we hear warnings about taking a theoretical orientation. But such views are not so influential compared to the general support in society for fundamental research. It is important to note that our institute itself performs a teaching function at the graduate level. We now have over two dozen graduate students working for the Ph.D. It is certainly more cost-effective to train a number of scientists in the country rather than send everyone abroad for graduate education. Moreover, a good level of graduate education in mathematics is necessary for the country to attain a high level in science, university teaching, and engineering training.

T.uy: My salary is 25,000(tongs per month; someone at the associate professor rank receives about 20,000. At present this is equivalent to four or five U.S. dollars per month. One can survive on this, because many basic items are highly subsidized--rice, rent, and so on. One would be able to eat rice, salt, and a few vegetables. However, to live more normally one needs another source of income. For mathematicians the most common source is private tutoring, usually preparing high school students for the university entrance examinations. Our system is difficult for a foreigner to understand. There are some peculiarities in the salary system. A worker can earn up to 150,000 dongs/month. I have a nephew who works in a factory and earns three times what I do. Of course, nobody thinks that this is a normal situation, and we expect that some changes will occur in the near future.

Tgy: Classes will start officially in September 1989. Before that, from February to September, we are giving intensive language instruction in English and Russian. At present we have seventy students.

Koblitz: The Mathematics Institute seems to be oriented largely toward basic research, in most cases in fields with no

Koblitz: Is it true that some of Vietnam's leaders in the mathematical sciences will be teaching there?

Koblitz: I hear that you are part of a group that is starting a private university here in Hanoi. Is that true? Is it really possible to do this in a socialist country?

Tgy: Yes, in fact, we have already obtained permission to grant degrees which will be recognized by the higher education ministry. But it cannot really be called a university. It consists only of a mathematics and computer science faculty. Later we plan to add programs in mechanics and economic management. Koblitz: Has it opened yet?

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T.uy: So far the main people involved are the algebraist Hohng Xuan Sinh (she will be rector of the school), the algebraic topologist H u ~ n h Mfli, Phan Oinh Di.~u in computer science, Ngu~r~n Oinh Tri in analysis, B~i Tr.ong L.u'u in mechanics, and myself. Koblitz: There are rumors that you left Hanoi University in 1968 because of a policy dispute, and that ever since then you have been dissatisfied with the quality of the program at the university. Do your efforts in starting a new institution of higher education stem from a dissatisfaction with mathematical education at Hanoi University? T~y: Well, many people have seen the need for improvement of our university system, not only in mathematical education but in all fields. Of course, economic difficulties are partly to blame for the problems. But I think that there are other reasons. Right now in mathematics H a n o i University has fewer students than in the 1960s, and three times as many faculty. We are hopeful that a private college can initiate the type of rapid innovations that are necessary to upgrade higher education in Vietnam. Such initiatives are much more difficult in the state system. If we are successful, we may eventually apply to become part of the government's university system. But at this stage we cannot yet be sure of the results. The project is an experiment for Vietnam. Koblitz: You have been deeply involved in education in Vietnam not only at the postgraduate and undergraduate levels but also at the secondary school level. It is there that Vietnam has acquired an especially high reputation because of its teams' outstanding performance in the International Mathematical Olympiads. For example, last year in Australia, the team placed fifth, ahead of the sixth-place U.S. team. Could you give the history of Vietnamese participation and evaluate its significance? Does it reflect a high general level of secondary education in mathematics in the country? T~y: In 1973 1 happened to be in Moscow during the Olympiad, which was held there that year. One day, my friend V. A. Skvortsov (who was one of the main organizers of the Olympiads) invited me to the presidium for the closing ceremonies. Later we discussed the possibility of Vietnamese participation the following year, w h e n the Olympiad was scheduled to take place in East Berlin. The main difficulty was that the hosts would have to cover not only the usual local expenses b u t also travel costs. I talked with my German friends, w h o were very supportive, and they agreed. Next we had to get permission from the Vietnamese government. I approached Ph.am V~n O6ng personally. He gave his agreement, saying: "The only thing I ask of you is to try not to be in last place." 32

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In fact, we were not expecting much. I remember that at the Moscow Olympiad the Cuban team w o n an honorable mention for the first time, and they were very pleased. So we expected at most such a result. But the Vietnamese team in 1974 was very strong, winning several prizes. It was a surprise to everyone, including us. As far as its significance is concerned, of course it is encouraging for us, because it proves the potential of our youth, and perhaps something about the quality of education. But sometimes the interest in the Olympiad in Vietnam is excessive. The winners become more popular than the best mathematicians in the country! And in recent times we have become aware of some difficulties. Koblitz: What type of difficulties? T.uy: N o w the Olympiad includes questions related to computer science. Our youngsters are not used to this kind of mathematics. There is a great gap between our needs and our possibilities not only in computer science but also in more elementary mathematics. For instance, we have few pocket calculators, and so we continue to use log tables. We have failed to keep up with modern developments both in equipment and in teacher training. To speak frankly, in recent years we have had a certain deterioration in the educational system. Scientists have become anxious about the situation. I was recently appointed chair of a commission to look into secondary education in mathematics. There is some disorganization in mathematics education. You see, we were influenced by the French reforms in the style of Bourbaki, also by the Soviet reforms under Kolmogorov. Both largely failed: I agree with Pontryagin's criticism of the excessive abstraction and formalism. In France, there is a joke: A youngster is asked, What do you get w h e n the sets {blue cars} and {red cars} intersect? He answers: catastrophe! Perhaps we were fortunate that we had not been able to keep up fully with the Bourbaki and Kolmogorov reforms. However, books had been rewritten, a n d t h e situation has c a u s e d c o n f u s i o n a m o n g teachers. For example, before the " n e w math" movement took hold, I had written some high school textbooks as part of our efforts in the mid- and late 1950s to upgrade education. Then changes were written into later editions of my book in accordance with the new trend of greater abstraction. My name still appeared as author, however. I did not agree with the changes, and insisted that my name be removed from subsequent editions. Koblitz: Have attitudes of school and university students changed in recent years?

T uy: It seems to me that the general mood is that young people want to do something more concrete, applied. There are many contradictions. What has caused us concern is that university students are not so diligent as ten or fifteen years ago. Part of the reason is our economic difficulties: people are under increasing pressure to earn money, working at different jobs. But also part of the problem is stagnancy in the university administration. That is w h y a few of us took the initiative to establish a private college of mathematical sciences. Whatever happens, there will always be a number of young people dedicated to science. But the fact that students are not so diligent as in the past has been demoralizing for the teachers. Koblitz: How many children do you have? Are any of them in mathematics? T.uy: I have three sons and a daughter. My eldest son is n o w at the Asian Institute of Technology in B a n g k o k s t u d y i n g i n d u s t r i a l e n g i n e e r i n g . My daughter works at the institute of Computer Science here in Hanoi. My middle son is the mathematician--

he is n o w studying in Odessa. My youngest son is at the Hanoi Polytechnic Institute. I can't say what his main interest is. I think football. Koblitz: Specifically in reference to math and science, earlier we discussed the remarkable fact that there have been virtually no "'defections" or "'brain drain" in any of the exchanges with Western countries. As contact with and influence of the West increases, is this likely to change? How would you feel about a young Vietnamese going for graduate studies or a post-doc in the West and then deciding not to return? T~y: As Vietnam opens up more to the outside world, I think it is inevitable that the number of young scientists who choose to remain in the wealthier countries will increase. In the present that represents a loss, but in the future one cannot say. It depends on the particular individual. Some Vietnamese immigrants and second generation immigrants in Western countries have reestablished ties with their home country, and in some cases they have supported the universities and institutes in Vietnam materially and scientifically. THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 3, 1990

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Koblitz: Officially, the Vietnamese government has a policy of reconciliation with the refugees who fled to the U.S., including those who served the Americans and the South Vietnamese regime before 1975. But in psychological terms, is this possible so soon after the war? How has reconciliation been progressing on a personal level? T~y: The consequences of a long war are not fully u n d e r s t o o d by m a n y people. Other nations have simply not experienced such a long war of such intensity. After the partition of Vietnam in 1954, most of my brothers were in the north. But m y sister and a

younger brother were in the south. At first my brother could not find much he could do; finally, he joined the a r m y of the Thi.~u regime. He was then killed. I learned about this in 1975. His wife committed suicide, leaving four young children. When we learned of this, my family took responsibility for the education of their children. Many, many families were divided. Sometimes even the husband was in the north, the wife in the south. You can imagine what kind of problems arose from that situation. The separation was not for one or two years, but for more than twenty years. Then the two social systems were completely different. After the war we have become much more tolerant, but it takes time to normalize all these things. In many cases, who was on which side depended on fate, on circumstance. In some cases the individual bears full responsibility, but in other cases he had no choice. I am happy that in recent years most people from both sides have become more tolerant. For example, the refugees who left after 1975 are now allowed to return to visit relatives. In most cases there is no problem. In 1985, I visited Boston to participate in a symposium on mathematical programming, When I came to register, I was told that several cousins from California had inquired about my arrival. Someone had seen my name in the announcement (I was on the program committee), and had told my cousins, who flew across the country to see me. I was very touched by this. W h e n we were y o u n g we had been very close. T h r o u g h all the years we h a d preserved the best feelings. When we met in Boston we felt very close again. Of course, we both understood that our conversations should avoid some delicate points. In Vietnam we have the tradition that feelings of friendship and family ties are very lasting. The words in Vietnamese are c6 tinh, c6 nghid. They mean: warm feelings of friendship, loyalty, remembering good times and the kindness of friends. Perhaps this is connected with the Confucian tradition. I don't know to what extent this has changed from the war. The years of war created a very complicated situation for observing this tradition. I am a man who from youth has nearly always lived in wartime: the Japanese troops in the 1940s, the French war, the American war. At the same time I am a mathematician, with the same concerns as mathematicians everywhere. Perhaps some would view the years of war as a horrible aberration, a part of history that should be put out of our minds. But the experiences of those years were central to my formation, and they remain a part of me.

Hoang Tu.y Institute of Mathematics Hanoi Socialist Republic of Vietnam 34

THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 3, 1990

Neal Koblitz Department of Mathematics University of Washington Seattle, WA 98195 USA

Are These the Most Beautiful? David Wells

In the Fall 1988 Mathematical Intelligencer (vol. 10, no. 4) readers were asked to evaluate 24 theorems, on a scale from 0 to 10, for beauty. I received 76 completed questionnaires, including 11 from a preliminary version (plus 10 extra, noted below.) One person assigned each theorem a score of 0, with the comment, "Maths is a tool. Art has beauty"; that response was excluded from the averages listed below, as was another that awarded very many zeros, four who left many blanks, and two w h o awarded numerous 10s. The 24 theorems are listed below, ordered by their average score from the remaining 68 responses 9

(11)

The order of a subgroup divides the order of the group.

5.3

(12)

Any square matrix satisfies its characteristic equation.

5.2

(13)

A regular icosahedron inscribed in a regular octahedron divides the edges in the Golden Ratio. 1 1 2x3x4 4x5x6 1 + 6x7x8 ,rr-3

5.0

(14)

9 "

Rank

Theorem

Average

(1) (2)

d ~' = - 1

7.7

Euler's formula for a polyhedron:

7.5

(15)

4.8

4

If the points of the plane are each coloured red, yellow, or blue,

4.7

V+F=E+2

(3) (4) (5) (6)

The number of primes is infinite.

7.5

There are 5 regular polyhedra. 1 1 1 1 + ~ + ~ + ~ + . . . = "rr2/6.

7.0

A continuous mapping of the closed unit disk into itself has a fixed point.

6.8

(7)

There is no rational number whose square is 2.

6.7

(8) (9)

~r is transcendental.

6.5

Every plane map can be coloured with 4 colours.

6.2

Every prime number of the form 4n + 1 is the sum of two integral squares in exactly one way.

6.0

(10)

7.0

THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 3 9 1990 Springer-Verlag New York 3 7

there is a pair of points of the same colour of mutual distance unity. (16)

The number of partitions of an integer into odd integers is equal to the number of partitions into distinct integers.

4.7

(17)

Every number greater than 77 is the sum of integers, the sum of whose reciprocals is 1.

4.7

(18)

The number of representations of an odd number as the sum of 4 squares is 8 times the sum of its divisors; of an even number, 24 times the sum of its odd divisors.

4.7

(19)

There is no equilateral triangle whose vertices are plane lattice points.

4.7

(20)

At any party, there is a pair of people w h o have the same number of friends present.

4.7

(21)

Write d o w n the multiples of root 2, ignoring fractional parts, and underneath write the numbers missing from the first sequence. 12 4 5 7 8 91112 3 6 10 13 17 20 23 27 30 The difference is 2n in the nth place.

4.2

(22)

The word problem for groups is unsolvable.

4.1

(23)

The maximum area of a quadrilateral with sides a , b , c , d is [(s - a)(s - b)(s - c)(s - d)] w, where s is half the perimeter.

3.9

[(1 - x)(1 - x2)(1 - x3)(1 - x4)... 16

3.9

= p(4) + p(9)x + p(14)xa + .... where p ( n ) is the number of partitions of n. The following comments are divided into themes. Unattributed quotes are from respondents. T h e m e 1: Are T h e o r e m s Beautiful? Tony Gardiner argued that "Theorems aren't usually 'beautiful'. It's the ideas and proofs that appeal," and remarked of the theorems he had not scored, "The rest are hard to s c o r e - - e i t h e r because they aren't really beautiful, however important, or because the formulation given gets in the way . . . . " Several re38

THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 3, 1990

T h e m e 2: Social Factors Might some votes have gone to (1), (3), (5), (7), and (8) because they are 'known' to be beautiful? I am suspicious that (1) received so many scores in the 7 - 1 0 range. This would surprise me, because I suspect that mathematicians are more i n d e p e n d e n t than most people [13] of others' opinions. (The ten extra forms referred to above came from Eliot Jacobson's students in his number theory course that emphasises the role of beauty. I noted that they gave no zeros at all.) T h e m e 3: C h a n g e s in A p p r e c i a t i o n over T i m e

5 [ ( 1 - - X5)(1 -- x l O ) ( I -- X 1 3 . . 9 15

(24)

s p o n d e n t s disliked judging theorems. (How many readers did not reply for such reasons?) Benno Artmann wrote "for me it is impossible to judge a 'pure fact' "; this is consistent with his interest in Bourbaki and the axiomatic development of structures. Thomas Drucker: "One does not have to be a Russellian to feel that much of mathematics has to do with deriving consequences from assumptions. As a result, any 'theorem' cannot be isolated from the assumptions under which it is derived." Gerhard Domanski: "Sometimes I find a problem more beautiful than its solution. I find also beauty in mathematical ideas or constructions, such as the Turing machine, fractals, twistors, and so on . . . . The ordering of a whole field, like the work of Bourbaki 9 . . is of great beauty to me." R. P. Lewis writes, ' ( 1 ) . . . I award 10 points not so much for the equation itself as for Complex Analysis as a whole.' To what extent was the good score for (4) a vote for the beauty of the Platonic solids themselves?

There was a notable number of low scores for the high rank theorems 9 Le Lionnais has one explanation [7]: "Euler's formula ei~' = - 1 establishes what appeared in its time to be a fantastic connection between the most important numbers in mathematics . . . It was generally considered 'the most beautiful formula of mathematics' . . . Today the intrinsic reason for this compatibility has become so obvious that the same formula n o w seems, if not insipid, at least entirely natural." Le Lionnais, unfortunately, does not qualify " n o w seems" by asking, "'to whom?" H o w does judgment change with time? Burnside [1], referring to % group which is . . . abstractly e q u i v a l e n t to that of the p e r m u t a t i o n s of f o u r symbols," wrote, "in the latter form the problem presented would to many minds be almost repulsive in its naked f o r m a l i t y . . . " Earlier [2], perspective projection was, "'a process occasionally resorted to by geometers of our o w n country, but generally e s t e e m e d . . , to be a species of

'geometrical trickery', by which, 'our notions of elegance or geometrical purity may be violated . . . . ' " I am sympathetic to Tito Tonietti: "'Beauty, even in mathematics, d e p e n d s upon historical and cultural contexts, and therefore tends to elude numerical interpretation." Compare the psychological concept of habituation. Can and do mathematicians deliberately undo such effects by placing themselves empathically in the position of the original discoverers? Gerhard Domanski wrote out the entire questionnaire by hand, explaining, "As I wrote d o w n the theorem I tried to remember the feelings I had when I first heard of it. In this way I gave the scores."

Theme 4: Simplicity and Brevity No criteria are more often associated with beauty than simplicity and brevity. M. Gunzler wished (6) had a simpler proof. David Halprin wrote "'the beauty that I find in mathematics 9 . . is more to be found in the clever and/or succinct w a y it is proven." David Singmaster m a r k e d (10) down somewhat, because it does not have a simple proof. I feel that this indicates its depth and mark it up accordingly. Are there no symphonies or epics in the world of beautiful proofs? Some chess players prefer the elegant simplicity of the endgame, others appreciate the complexity of the middle game. Either way, pleasure is derived from the reduction of complexity to simplicity, but the preferred level of complexity differs from player to player. Are mathematicians similarly varied? Roger Penrose [10] asked whether an u n a d o r n e d square grid was beautiful, or was it too simple? He concluded that he preferred his non-periodic tessellations. But the question is a good one. How simple can a beautiful entity be? Are easy theorems less beautiful? One respondent marked down (11) and (20) for being "too easy," and (22) for being "'too difficult." David Gurarie marked down (11) and (1) for being too simple, and another r e s p o n d e n t referred to theorems that are true by virtue of the definition of their terms, which could have been a dig at (1). Theorem (20) is extraordinarily simple but more than a quarter of the respondents scored it 7 +.

Theme 5: Surprise Yannis Haralambous wrote: "a beautiful t h e o r e m must be surprising and deep. It must provide you with a new vision o f . . . mathematics," and mentioned the prime number theorem (which was by far the most

popular suggestion for theorems that ought to have been included in the quiz). R. P. Lewis: "(24) is top of m y list, because it is surprising, not readily generalizable, and difficult to prove. It is also important." (12 + in the margin!) Jonathan Watson criticised a lack of novelty, in this sense: "(24), (23), (17) . . . seem to tell us little that is new about the concepts that appear in them." Penrose [11] qualifies Atiyah's suggestion "that elegance is more or less synonymous with simplicity" by daiming that "one should say that it has to do with unexpected simplicity." Surprise and novelty are expected to provoke emotion, often pleasant, but also often negative. N e w styles in popular and high culture have a novelty value, albeit temporary. As usual there is a psychological connection. Human beings do not respond to just any stimulus: they do tend to respond to novelty, surprisingness, incongruity, and complexity. But what happens w h e n the novelty wears off? Surprise is also associated with mystery. Einstein asserted, "The most beautiful thing we can experience is the mysterious. It is the source of all true art and science." But what happens w h e n the mystery is resolved? Is the beauty transformed into another beauty, or may it evaporate? I included (21) and (17) because they initially mystified and surprised me. At second sight, (17) remains so, and scores quite highly, but (21) is at most pretty. (How do mathematicians tend to distinguish between beautiful and pretty?)

Theme 6: Depth Look at theorem (24). Oh, come on now, Ladies and Gentlemen! Please! Isn't this difficult, deep, surprising, and simple relative to its subject matter?! What more do you want? It is quoted by Littlewood [8] in his review of Ramanujan's collected works as of "'supreme beauty." I wondered what readers would think of it: but I never supposed that it would rank last, with (19), (20), and (21). R. P. Lewis illustrated the variety of responses when he suggested that among theorems not included I could have chosen "Most of Ramanujan's work,'" adding, "'(21) is pretty, but easy to prove, and not so deep." Depth seems not so important to respondents, which makes me feel that m y interpretation of depth may be idiosyncratic. I was surprised that theorem (8), which is surely deep, ranks below (5), to which Le Lionnais's a r g u m e n t might apply, but (8) has no simple proof9 Is simplicity that important? (18) also scored poorly. Is it no longer deep or difficult? Alan Laverty and Alfredo Octavio suggested that it would be harder and more beautiful if it answered THE MATHEMATICAL INTELL1GENCER VOL. 12, NO. 3, 1990 ~ 9

the same problem for non-zero squares. Daniel Shanks once asked whether the quadratic reciprocity law is deep, and concluded that it is not, any longer. Can loss of depth have destroyed the beauty of (24)?

Theme 7: Fields of Interest Robert Anderssen argued that judgements of mathematical beauty "will not be universal, but will depend on the background of the mathematician (algebraist, geometer, analyst, etc.)" S. Liu, writing from P h y s i c s R e v i e w (a handful of respondents identified themselves as non-pure-mathematicians), admitted "'my answers reflect a preference for the algebraic and number-theoretical over the geometrical, topological, and analytical theorems,' and continued: "I love classical Euclidean g e o m e t r y - - a subject which originally attracted me to mathematics. However, within the context of your questionnaire, the purely geometrical theorems pale by comparison." Should readers have been asked to respond only to those theorems with which they were extremely familiar? (22) is the only item that should not have been included, because so many left it blank. Was it outside the main field of interest of most respondents, and rated down for that reason?

Theme 8: Differences in Form Two r e s p o n d e n t s suggested that e i" + 1 = 0 was (much) superior, combining "the five most important constants." Can a small and "inessential" change in a theorem change its aesthetic value? How would i i = e -~'t2 have scored? Two noted that (19) is equivalent to the irrationality of V 3 and one suggested that (7) and (19) are equivalent. Equivalent or related? When inversion is applied to a theorem in Euclidean geometry are the new and original theorems automatically perceived as equally beautiful? I feel not, and naturally not if surprise is an aesthetic variable. Are a t h e o r e m and its dual equally beautiful? Douglas H o f s t a d t e r s u g g e s t e d t h a t D e s a r g u e s ' s theorem (its own dual) might have been included, and w o u l d have given a v e r y high score to Morley's theorem on the trisectors of the angles of a triangle. Now, Morley's theorem follows from the trigonometrical identity, 1/4 sin 30 = [sin 0] [sin (~/3 - 0)] [sin ('rr/3 + 0)]. How come one particular transformation of this identity into triangle terms is thought so beautiful? Is it partly a surprise factor, which the pedestrian identity lacks? 40

THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 3, 1990

Theme 9: General versus Specific Hardly touched on by respondents, the question of general vs. specific seems important to me so I shall quote Paul Halmos [5]: "'Stein's (harmonic analysis) and Shelah's (set theory) . . . represent what seem to be two diametrically opposite psychological attitudes to m a t h e m a t i c s . . . The contrast between them can be described (inaccurately, but perhaps suggestively) by the words special and general . . . . Stein talked about singular integrals . . . [Shelah] said, early on: 'I love mathematics because I love generality,' and he was off and running, classifying structures whose elements were structures of structures of structures." Freeman Dyson [4] has discussed what he calls "accidental beauty" and associated it with unfashionable mathematics. Roger Sollie, a physicist, admitted, "I tend to favour 'formulas' involving ~r," and scored (14) almost as high as (5) and (8). Is "rr, and anything to do with it, coloured by the feeling that -a" is unique, that there is no other number like it?

Theme 10: Idiosyncratic Responses Several readers illustrated the breadth of individual responses. Mood was relevant to Alan Laverty: "The scores I gave to [several] would fluctuate according to m o o d and circumstance. Extreme example: at one point I was considering giving (13) a 10, but I finally decided it just didn't thrill me very much." He gave i t a 2. Shirley Ulrich "'could not assign comparative scores to the . . . items considered as one group," so split them into geometric items and numeric items, and scored each group separately. R. S. D. Thomas wrote: "I feel that negativity [(7), (8), (19) and (22)] makes beauty hard to achieve.'" Philosophical orientation came out in the response of Jonathan Watson (software designer, philosophy major, reads M a t h e m a t i c a l Intelligencer for foundational interest): "I am a constructivist.., and so lowered the score for (3), although you can also express that theorem constructively." He adds, " . . . the questionnaire indirectly raises f o u n d a t i o n a l i s s u e s - - o n e theorem is as true as another, but beauty is a human criterion. And beauty is tied to usefulness."

Conclusion From a small survey, crude in construction, no positive conclusion is safe. However, I will draw the negative conclusion that the idea that mathematicians largely agree in their aesthetic judgements is at best grossly oversimplified. Sylvester described mathematics as the study of difference in similarity and similarity in difference. He was not characterising only

mathematics. Aesthetics has the same complexity, and both perspectives require investigation. I will comment on some possibilities for further research. Hardy asserted that a beautiful piece of mathematics should display generality, unexpectedness, depth, inevitability, and economy. "Inevitability" is perhaps Hardy's o w n idiosyncracy: it is not in other analyses I have come across. Should it be? Such lists, not linked to actual examples, perhaps represent the maximum possible level of agreement, precisely because they are so unspecific. At the level of this questionnaire, the variety of responses suggests that individuals' interpretations of those generalities are quite varied. Are they? How? Why? Halmos's generality-specificity dimension may be compared to this comment by Saunders Mac Lane [9]: "I adopted a standard position--you must specify the subject of interest, set up the needed axioms, and define the terms of reference. Atiyah much preferred the style of the theoretical physicists. For them, when a n e w idea comes up, one does not pause to define it, because to do so would be a damaging constraint. Instead they talk around about the idea, develop its various connections, and finally come up with a much more supple and richer notion . . . . However I persisted in the position that as mathematicians we must know whereof we speak . . . . This instance may serve to illustrate the point that there is now no agreement as to h o w to do mathematics . . . . " Apart from asking--Was there ever?---such differences in approach will almost certainly affect aesthetic judgements; many other broad differences between mathematicians may have the same effect. Changes over time seem to be central for the individual and explain h o w one criterion can contradict another. Surprise and mystery will be strongest at the start. An initial solution may introduce a degree of generality, depth, and simplicity, to be followed by further questions and further solutions, since the richest problems do not reach a final state in their first incarnation. A n e w point of view raises surprise anew, muddies the apparently clear waters, and suggests greater depth or broader generality. H o w do aesthetic j u d g e m e n t s change and develop, in quantity and quality, during this temporal roller coaster? Poincar~ and von Neumann, among others, have emphasised the role of aesthetic judgement as a heuristic aid in the process of mathematics, though liable to mislead on occasion, like all such assistance. H o w do individuals' judgements aid them in their work, at every level from preference for geometry over analysis, or whatever, to the most microscopic levels of mathematical thinking? Mathematical aesthetics shares much with the aesthetics of other subjects and not just the hard sciences. There is no space to discuss a variety of examples, though I will mention the related concepts of isomor-

phism and metaphor. Here is one view of surprise [6]: "Fine writing, according to Addison, consists of sentiments which are natural, without being obvious . . . . On the other hand, productions which are merely surprising, without being natural, can never give any lasting entertainment to the mind." H o w might "natural" be interpreted in mathematical terms? Le Lionnais used the same word. Is it truth that is both natural and beautiful? H o w about Hardy's "inevitable?" Is not group theory an historically inevitable development, and also natural, in the sense that group structures were there to be detected, sooner or later? Is not the naturalness and beauty of such structures related to depth and the role of abstraction, which provides a ground, as it were, against which the individuality of other less general mathematical entities is highlighted? Mathematics, I am sure, can only be most deeply understood in the context of all human life. In particular, beauty in mathematics must be incorporated into any adequate epistemology of mathematics. Philosophies of mathematics that ignore beauty will be inherently defective and incapable of effectively interpreting the activities of mathematicians [12].

References 1. W. Burnside, Proceedings of the London Mathematical Society (2), 7 (1980), 4. 2. Mr. Davies, Historical notices respecting an ancient problem, The Mathematician 3 (1849), 225. 3. T. Dreyfus and T. Eisenberg, On the aesthetics of mathematical thought, For the Learning of Mathematics 6 (1986). See also the letter in the next issue and the author's reply. 4. Freeman J. Dyson, Unfashionable pursuits, The Mathematical Intelligencer 5, no. 3 (1983), 47. 5. P. R. Halmos, Why is a congress? The Mathematical Intelligencer 9, no. 2 (1987), 20. 6. David Hume, On simplicity and refinement in writing, Selected English Essays, W. Peacock, (ed.) Oxford: Oxford University Press (1911), 152. 7. F. Le Lionnais, Beauty in mathematics, Great Currents of Mathematical Thought, (F. Le Lionnais, ed.), Pinter and Kline, trans. New York: Dover, n.d. 128. 8. J. E. Littlewood, A Mathematician's Miscellany, New York: Methuen (1963), 85. 9. Saunders Mac Lane, The health of mathematics, The Mathematical Intelligencer 5, no. 4 (1983), 53. 10. Roger Penrose, The role of aesthetics in pure and applied mathematical research, Bulletin of the Institute of Mathematics and its Applications 10 (1974), 268. 11. Ibid., 267. 12. David Wells, Beauty, mathematics, and Philip Kitcher, Studies of Meaning, Language and Change 21 (1988). 13. David Wells, Mathematicians and dissidence, Studies of Meaning, Language and Change 17 (1986).

19 Menelik Road London NW2 3RJ England THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 3, 1990

41

The Infidel Is Innocent Adrian P. Simpson

1.

The

Quasi-Religious

Sect

" . . . They move in dark, old places of the world: Like mariners, once healthy and clear-eyed, Who, when their ship was holed, could not admit Ruin." William Ashbless (Tim Powers), The Twelve Hours of the Night

Every year it's the same: You can always spot them. They have a glazed look. Their lips move rhythmically; mindless, they m u r m u r their near-silent litany. They move, drugged by their opiate religion, roaming the corridors like the stunned spectres of their remaining youth. As you near one of these dazed, shocked novices you can hear their ritualistic chant: "Given an epsilon 9 . . there is a delta . . . . " They are (pause for suspense to build) the First-Year Analysts. Dragged unwilling from their mothers" arms (or kicked out willingly by their fathers' feet) they find scant refuge in the evil clutches of the Standardites. What, then, is the cause of all this misery and suffering? What despotic deeds are being perpetrated on the youth of our proud nation? The cause of their hurt is analysis; or rather, the huge leap between the intuitive calculus of school and the quasi-religious formality of "real" mathematics. We have all been through it; some more recently than others, a few more painfully than most. The root of the problem is the concept of the infinitesimal. At school the little lambs 1 are gently herded towards

the fold of calculus through the idea of a tangent to a curve. The argument goes: If we wish to find the tangent to the graph of f at (a,f(a)), we draw a line from (a,f(a)) to (a + to,f(a + to)) and find its slope. We then choose tl with Itll < It01 and repeat the process. We continue choosing smaller increments until we find the slope of the line from (a,f(a)) to (a + t,f(a + t)) for a very, very tiny2 t indeed. So the basic argument that is presented to the unwilling child is that

f'(a) = f(a + h) - fla) h

1 Given the state of today's schoolchildren, "little lambs" may be an inappropriate metaphor. 2 The teacher, having been through the transition to university analysis, tries to save his/her flock by avoiding the word "infinitesimal." THEMATHEMATICALINTELLIGENCERVOL.12, NO. 3 9 1990Springer-VerlagNew York 43

with a spectacularly heavy splash in this particular o i n t m e n t w h e n one of the most brilliant pieces of vitriolic invective in the history of mathematics poured from the pen of one Bishop Berkeley. It appears that a friend of Newton, one E d m u n d Halley (yes--that Halley), had persuaded a friend of the good bishop of the "inconceivability of the doctrines of Christianity." Since this took place in the days w h e n clergymen were supposed to view God more as our omnipotent creator than as a cool dude who enjoys nothing more than an afternoon chat with a high-ranking British policeman, Berkeley took offence and set out to show that this new-fangled calculus thing had no clearer basis than religion did. In his famous treatise The Analyst, with the beautiful subtitle of A Discourse Addressed to an Infidel Mathematician, he pointed out that if the increment h is a nonzero number, no matter how small, then it cannot be ignored if the integrity of mathematics is not to be plunged into "much emptiness, darkness and confusion"; if the n u m b e r is zero, t h e n we cannot go around dividing by it without "'shocking good sense." These were logically valid (if scathingly phrased) arguments, and mathematics w a s - - e v e n t u a l l y - - f o r c e d to change its ideas. Admittedly, even though a rigorous new formulation did not appear for many years, mathematicians continued to use the calculus--and people say that moral standards have declined! In the fulness of time, it fell to the great Weierstrass to light the mantle of honour, and mathematics was presented with the famous

f'(a)

lim f(a + h) - f ( a ) :

h--*O

h

"

or: "f is differentiable at a, with derivative f'(a), if, given an e > 0, there exists a 8 > 0, with f defined on (a - 8, a + 8), such that An

analysis lecturer at work.

0 < Ih] < 8 ~

for the merest smidgen of an h. So, if f(x) = x2 then (a + h ) 2 -

f'(a) =

a2

h a2 +

2ah

+

h2 -

a2

=2a+h but, since h is the just the m e r e s t s m i d g e n of a number, we can ignore it! Nope! Sorry, you can't do that! The argument above is very much the same one that Newton used w h e n inventing the calculus (give or take a few magical, mystical words like "fluxion" and "nascent increment"). Unfortunately, the fly landed 44 THE MATHEMATICALINTELLIGENCERVOL. 12, NO. 3, 1990

f(a + h) h

-

f(a)

-f'(a)

<e."

The eyes began to glaze and the minds began to soften. Just recently, in mathematical terms, the arguments of Bishop Berkeley have been re-examined. In 1966, Abraham Robinson (who can be called father, mother, and midwife of this reappraisal 3) realized the dream of Leibniz. This co-founder of the calculus had anticipated most of Berkeley's arguments and had said that he wished for a n u m b e r system that contained both infinitely small and infinitely large elements as well as the real

3 Q u i t e a n achievement! (He h i m s e l f refers to t h e subject as his stepchild.)

numbers. These extra numbers were to have all the "usual" properties of arithmetic. In his b o o k , Non-Standard Analysis, R o b i n s o n brought Leibniz's dream to life not only by producing, rigorously, just such a number system, b u t also by creating a method for applying the idea of infinitesimals to many branches of mathematics, opening up a new and fertile approach to numerous problems. This article is intended as an informal introduction to the beauty of the system's construction, showing how proofs in analysis can be presented without the ritual of the Quasi-religious sect, noting progress that the n e w method has provided in various research areas, a n d - - a s a w h o l e - - p r o v i n g that Berkeley's Infidel is innocent.

2. Motivation "Let's start at the very beginning, A very good place to start, When you write you begin with A,B,C, When you differentiate you begin with "given an e there is a 8" "" Julie Andrews (???), The Sound of Music Refuting the a r g u m e n t s of dead b i s h o p s is fine sport, but what use is this new tool to those of us who have gone through a period of glazed eyes, a confused hatred for the calligraphic dexterity of the Greeks, 4 and a large bar bill? Is one method of viewing analysis better than another? Must we hide our carefully written lecture notes from the eyes of fellow mathematicians, lest we are mistaken for Philistines who have not yet heard of Robinson's revolution? Is the deft handling of Greek letters to become a lost art? Yes, no, and probably not are the respective ans w e r s - a n d I'm not so sure about the "yes." Infinitesimal arguments are no more right or wrong than e-8 ones. What they do is make ideas clearer, texts shorter and, most importantly, proofs intuitive. It is much easier to see that a function is continuous at c if f(x) is the merest smidgen away from f(c) whenever x is infinitely close to c, than it is to digest the "given an e, there is a 8 . . . argument. To drive the point home, this section will lead to the definition of a differential with much less fuss than is required using standard concepts. Using infinitesimals, integration too loses its mysterious cloak of partitions, lim sups, lim infs, and so on. They are replaced by the "schoolchild" notion of an 4 O.K. H a n d s u p all those w h o have s p e n t 10 m i n u t e s trying to d r a w a recognizable lower case zeta whilst the lecturer continues, oblivious to your increasing frustration.

infinite s u m on infinitely thin slivers that has been used since long before there were any Greek letters. Unfortunately, trees are only a partially renewable resource and, to prevent a drop in the oxygen levels that would result from the amount of extra paper I would need to adequately cover these topics, I will rely exclusively on differentiation as an inspirational device. With this new tool, Leibniz's notation -~ becomes (almost) a fraction, and can be treated as if it is one. The new method makes continuity cushy, differentiation a doddle, integration intelligible, and analysis accessible. Before this article degenerates into some appalling reject advertising copy for non-standard analysis ("new biodegradable, 16-bit x 4-fold oversampling, whiter-than-white non-standard analysis--reaches the parts other analyses cannot reach!"), I should point out that all those clich6s you hate hearing about free lunches and swings and roundabouts apply as much here as elsewhere in life. I really should have said that the ultimate goal of Robinson's work is to make proofs more easily understood. For the honest mathematician, that ultimate goal can take nearly as much effort to achieve as passing the e-8 barrier. Constructing the n u m b e r system that Leibniz longed for is an arduous task. H o w e v e r , because few people in the world are honest mathematicians, once the construction has been done by someone, most of the world can go about using infinitesimals in the safe knowledge that they are free from attack b y mathematicians and bishops s alike. What, then, is an infinitesimal? Here we have:

Definition 2.1 In an ordered field extension R* D R, an element e ( ~* is said to be an infinitesimal if lel < r for all r (R,r>0. There is a similar notion for negative infinitesimals. It is at this juncture that some antagonists crow like malevolent roosters, heralding the dawn of a n e w Hell. "What is r = ~e?" they cry. The point is that e ( {0} (otherwise r = ~ would bring a contradiction): e ( R*\R or e = 0. The theme of the next section is actually to construct such "numbers." For now let us just assume that they exist and form a proper, ordered extension of the reals, and we will see just how useful they can be. To allow access to a very powerful theorem, consider an equivalence relation on this extension:

Definition 2.2 x,y E ~* are said to be infinitely close if Ix - Yl is infinitesimal. We write "x ~ y" to denote this.

s But n o t n e c e s s a r i l y b y m a t h e m a t i c i a n s w h o are t h e m s e l v e s Bishops? THE MATHEMATICALINTELLIGENCERVOL. 12, NO. 3, 1990 45

that h could not be ignored. If, however, we define the differential as the standard part of the "schoolchild's" idea, for h infinitesimal, we get f'(a) = st(2a + h) =

2a.

Thus we not only get the correct answer, we get it rigorously because st(x) is well-defined on a rigorously constructed field. N o w comes the juicy stuff. D e f i n i t i o n 2.3 For a real-valued function f at a real point a,

define

f'(a) = st The author persuades a friend to use infinitesimals.

f(a

+ ax) ~xx

f(a +

THEOREM 2.1 Every finite x ~ ~* is infinitely close to a unique real u E ff~.

for Ax -~ O, Ax ~a 0,

J

if st

It says m u c h of the subject that one of its most useful theorems can now be proved.

q

f(a) /

ax) -f(a)] Ax

is the same for all infinitesimal nonzero ax. Leave f'(a) undefined otherwise. This, after only three definitions and one theorem, is equivalent to the definition given after 25 lectures of background work in an introductory analysis course. And not even an ~ in sight!

Proof: Suppose x is finite (that is, IxI < t for some t

R). Let S = {s E R I s < x}. Then S is non-empty and has an upper bound. Because R is complete, S has a least upper b o u n d u ( R. So for every r > O, r E R x~u+r. Thus x-u~r.

Also, for any positive real r, U

-

-

r<~x,

SO

-(x

- u) ~< r.

That is, for all real, positive r, Ix - uJ <<-r. So x - u is infinitesimal, and x is infinitely close to u. Uniqueness is easy. 9 This u is called the standard part of x, u = st(x). If x is infinite (i.e., Ix[ > r Vr ~ R) then st(x) is undefined. It is through the use of the function st(x) that the transfer from the larger field (called the hyperreals) to the reals is performed. When in Section 1 we had fix) = x2 and obtained

3. D a r k R o o m s a n d L a r g e W h i s k i e s

It is no use pretending that this "free lunch" of analysis w h i c h the h y p e r r e a l s allows us to c o n s u m e escapes the clich6. This section is an introduction to the bill that must be paid. It is a hefty invoice, including large tips to the head waiter of logic, but, as we have seen, once paid, the amount of food is limitless, tasty, and digestible. 6 To pretend, however, that this article can itemise the whole "bill" that is the basis of non-standard analysis w o u l d be to stretch the b o u n d s of plausibility. This section sets out to rigorously construct the hyperreals, to introduce the connection between this n e w field and the reals via first order predicate calculus, and finally to explain, quite informally, the larger concept of the non-standard model. This section is fairly technical and is best followed by a long period in a dark room with a large whisky. It is through the more general concept of the nonstandard model that the path to the hyperreals normally passes. We, however, will take the alternative route (avoiding low bridges) provided for us by the ultrafilter construction.

f'(a) = 2a + h , we were persuaded by Bishop Berkeley, quite rightly, 4 6 THE MATHEMATICALINTELLIGENCERVOL. 12, NO. 3, 1990

6 M y t h a n k s to R e n t - a - S i m i l e @ for t h e i r F l o g - a - M e t a p h o r - t o - D e a t h | service.

The direction taken is reminiscent of the construction of the reals via C a u c h y s e q u e n c e s , a n d this analogy will crop up from time to time. Whereas in building the reals we consider equivalence classes of sequences with the same limit, here we consider an equivalence class of sequences that has the property of being equal for certain subsets of their indexing set. For example, we might have the sequences (1, 1, 1, 1. . . . ) and (1, 0, 1, 0 . . . . ) in the same equivalence class if the set {1, 3, 5 . . . . } C N is o n e of these subsets. This set of subsets is called an ultrafilter.

Definition 3.1 A free ultrafilter on ~ is a set 1I of subsets of with the properties (i) (ii) (iii) (iv) (v)

0 f 1,t if A, B E ll then A N B E lI if A E lt with N D B D A then B ~ lI if A C N then either A ~ lI or (N\A) ~ it if A C ~ is finite then A ~ lI.

(It follows from the axiom of choice that there is a free ultrafilter on a n y infinite set. If y o u can't accept the axiom of choice, y o u have my permission to view this article as h e r e s y - - I believe the pages make very good firelighters!) We can thus define the relation discussed above:

Definition 3.5 Let r = [(ri)], s = [(si)]. Then define (i) (ii) (iii) (iv)

r + * s = [(ri + si)] r "* s = [(ri" si)] r < * s c : ~ { i E • l r i < si} ~ lI r =* s r [(ri) ] = [(si)].

With these definitions we obtain R*, the hyperreals, as a field extension of R. The m a p * : ~ --~ R* is injective a n d preserves the field operations a n d order. N o w I must show that there is something non-standard about some of these numbers, because if there isn't I could be in a lot of trouble from those of y o u w h o f o u g h t through all that h e a v y construction stuff. Consider [(1, 2, 3, 4 . . . . )] = 00. Note that (o cannot equal a standard number r = [(r, r, r, r . . . . )], as the set on which (r, r, r . . . . ) and (1, 2, 3 . . . . ) coincide can consist of at most one element, so, by definition 3.1(v), the two sequences cannot be equivalent. In fact (o >* r for all r ~ (R)*, that is, (o is infinite. Similarly e = (o-1 = [(1, 1/2, 1/3, 1/4. . . . )] is an infinitesimal. The b e a u t y of this t h e o r y is rooted in one easily stated concept: A n y statement in first-order logic that is true for ~ is true for R* if the statement is liberally sprinkled with stars. To p u t some bones of formality into the skin of this idea, we need to know a little about first-order logic a n d star-sprinkling.

Definition 3.2 If (ri)i~~, (si)i~N are sequences of reals, define -r~ by (r~) - u (s~) ~ {i ~ N I ri = si} ~ 1I. This is an equivalence relation. It e q u a t e s two sequences if the set on which they coincide belongs to the ultrafilter. So, only one of the two sequences (0, 1, 0, 1 . . . . ) and (1, 0, 1, 0 . . . . ) is equivalent to (1, 1, 1, 1. . . . ) and the other to (0, 0, 0, 0 . . . . ). Which depends on the ultrafilter c h o s e n - - t h e r e are many. If we denote the equivalence class of (ri)i~ N as [(ri)], we can define the hyperreals.

The new method makes continuity cushy, differentiation a doddle, integration intelligible, and analysis accessible.

Definition 3.3 Let ~* denote the set of equivalence classes of sequences of R induced by the relation - u. R* is called the set of hyperreals. Definition 3.4 Ifr E R, then r* = [(r, r, r . . . . )] E R* and we define (~)* = {r* [ r ~ ~}. This is called the set of standard reals.

nOW

tO reaa

~ec[lon

a,

THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 3, 1990

47

Definition 3.6 For a set S, an n-ary relation P on S is a subset of Srt. If (a1, a2. . . . . an) E P we write P(a1, a2, 9

9

9

,

art).

The *-transform P* of P is the set of all n-tuples (r 1, r 2, r 3, . . . . rrt) in (R*)rt such that if r ~ = [(~, r2k, ~, . . .)], then {i E N I P(r x, q . . . . . r 0}ELI. That is, an n-ary relation on the reals can be extended to an n-ary relation on the hyperreals by the rule that the P holds on an n-tuple in (R*)rt if the set of places where P holds on the representative real ntuple sequence is in the ultrafilter. For example, consider the relation < on R. We can define <* as the set of ordered pairs (a,b), a = [(ai)] E R* and b = [(bi)] E R* such that {i E ~ [a i < bi} E 1[. This is equivalent to definition 3.5 (iii). Definition 3.6 allows us to define the *-transform of a real-valued function by considering it as an (n + 1)ary relation. This is not sufficient for our purposes, however. There w o u l d be no point in defining the *-transform of the sine function only to discover that sin*(x) # sin* (x + 2"rr) for real x embedded in the new field, let alone for hyperreal values. We need some concept of the preservation of rules across the *-transform. To cut down on the need for paper, dark rooms, and large whiskies, I will make life easier by pointing you in the direction of the bibliography for the construction of a first-order predicate calculus language and interpretation sufficient for my purposes (cf. Hurd and Loeb, etc.) and assume that job done. N o w we construct the *-transform of a sentence in that language by "star-ing" each relation, function, and term in the sentence. For example, consider the sentence

which could be interpreted as "for all hyperreal x, the hyperreal sine of x equals the hyperreal sine of x + 2~r." N o w the crux of this theory can be stated in one beautiful theorem.

THEOREM 3.1 If ~ is a simple sentence which is true for R, then 4" (its *-transform) is true for R*. This single theorem lets us transfer nearly all of the work done in standard analysis to the non-standard form. Nearly all of the rules for standard functions transfer to the non-standard extension with ease. Life nearly becomes a bed of roses! " O . K . , " I hear the skeptics cry," w h e r e ' s the catch?" I would like to be able to reply that there isn't one, but I'm much too honest for my own good (he lied!). Glance back a paragraph and you will notice the cunning wording to the effect that nearly all of standard analysis can be transferred. Everything that I have said so far is v a l i d - - b u t only for first-order logic, that is, logic that allows quanfitiers to range over individuals, but not over properties, functions, or relations. So any statement in standard analysis concerning, say, "all non-empty sets of reals," "the existence of a continuous function," and so on, cannot be expressed in the language we created, and we cannot apply Theorem 3.1.

A proof should never read like a detective story, with subplots littered throughout the text and all the revelations left to the last paragraph.

(Vx)[R(x) ~ E(S(x),S(A(x,P(2, ~r))))] where the relations are

R(x) iff x is real E(x,y) iff x and y are equal and the functions are

P(x,y) = x ' y A(x,y)

= x + y

S(x) = sin(x). So the sentence could be interpreted as "for all real x, sin(x) = sin(x + 2"rr)." The *-transform of it is

(Vx)[R*(x) ~ E*(S*(x),S*(A*(x,P*(2, ~r))))] 48

THE MATHEMATICALINTELLIGENCERVOL. 12, NO. 3, 1990

We cannot, however, merely skip through the last few pages and replace the adjective "first-order" with "'second-order, ''7 as there are severe philosophical problems with quantifying over predicates which make such a solution unacceptable. There is a solution, though, and it is important to glance at it, as this more intricate method generates not only the non-standard analysis that we have already created, but also leads to applications as diverse as modelling the spin of sub-atomic particles, the verification of computer programs, and stochastic integration in hyperfinite dimensional linear spaces. (O.K. where's the whisky bottle?) The general method of producing a non-standard m o d e l is to create a huge "pile" of the standard theory. So, for non-standard analysis, we w o u l d 7 Second-orderlogicallowsquantification over properties, functions and so on.

create a massive edifice with all the mathematical objects that we wish to consider therein, such as sets of real-valued functions, n o n - e m p t y sets of reals, and SO o n .

We do this b y d e f i n i n g a superstructure Un=o,...,| X~ w h e r e X0 = X, Xn+l =

as 3r =

P(uk=0,...,n Xd,

P(S) indicates the p o w e r set of S and X is the ground set of the whole theory (R in our case, but it could be a topological space, a Hilbert space, etc.). This superstructure 3~ contains everything that we w a n t - - a l l the relations and functions of our original construction, a set containing all n o n - e m p t y sets of reals, a set of continuous functions, and so forth. The advantage of such a vast m o n u m e n t of mathematics is that it not only produces an effect similar to that experienced after a heavy night's drinking, but, since everything is a member of the superstructure (rather than, say, a subset of, or function on, the reals) we can quantify over it and use first-order logic. With numerous changes to the details, we can run through the ultrafilter construction in the first part of this section a n d e v e n t u a l l y p r o v e a m o r e p o w e r f u l version of Theorem 3.1 for the *-transform of anything. N o w , where did I put that bottle of L a p h r o a i g . . . !

4. The M o r n i n g After One of the good things about constructions is that, after t h e y have b e e n p e r f o r m e d , they can be completely ignored. We n e e d only drag out the blueprints of the hyperreals in order to prove that our infinitesimal m e t h o d s are valid and impervious to the Miltonian imagery of Bishop Berkeley. This section is meant as an Alka-Seltzer | for the unsettling effect of the last few pages. I intend to s h o w the efficiency and comprehensibility inherent in the non-standard m e t h o d of simple analysis by proving a theorem (the chain rule) using both the old and n e w methods. I should point out that w e need not actually prove the result t w i c e - - t h e non-standard theorem, restricted to the reals, is precisely the same as the real theorem, and the standard statement, once proved, can have Theorem 3.1 applied to it, to give the nonstandard one. (Isn't life wonderful?) N o w , on with the c o n t e s t . . . T H E O R E M 4.1 If g is differentiable at a and f is differentiable at g(a), then fog is differentiable at a with

0~o~)'(a) = f'(g(a)) .g'(a). Seconds out, round o n e . . .

Standard Proof: Define qffh) =

' f(g(a + h)) - f i g ( a ) ) if g(a + h) - g(a) # O g(a + h) - g(a) ' .f'(g(a))

, if g(a + h) - g(a) = O.

As f is differenfiable at g(a), for e > 0 there is a 8' > 0 so that 0 < Ikl < 8 ' ~

f(g(a + k))k - fig(a)) _ f'(g(a)) < e. (1)

Also, g is differentiable at a, so it is certainly continu o u s at a. That is, there is a 8 > 0 so that Ihl < 8 ~ Ig(a + h) - g(a)l < 8'.

(2)

So, if ]hi < 8, we can take k = g(a + h) - g(a). If 0 < ]kI < 8', from (1) w e have fig(a) + k)k - f(g(a)) _ f'(g(a)) < whence Iq0(h) - f'(g(a)) I < ~. N o w if k = 0, g(a + h) - g(a) = 0, so we certainly have The after-effects of Section 3.

I~(h) - f'(g(a))l < THE MATHEMATICALINTELLIGENCERVOL. 12, NO. 3, 1990 4 9

and we have shown that ~ is continuous at 0. Consider (fog)'(a)

=

limf(g(a + h)) - f(g(a)) h

_ limq~(h)[g(a + h) - g(a)] -

h--,o

h

= f'(g(a)) "g'(a)

9

Phew! Exactly what that proof is intuitively saying becomes lost in the mire between "'Define" and " 9 By contrast, let's look at the non-standard proof. Seconds out, round t w o . . . N o n - S t a n d a r d Proof: Let x = g(a), Ax = g(a + aa) g(a) for any nonzero infinitesimal Aa. I claim that Ax is infinitesimal. This follows from Definition 2.3 that

Ax

-aa

~

g'(a)

i.e., a x = (g'(a) +

)aa,

where (g'(a) + e) is finite. Let y = f(z), a y = f(z + Az) - f(z) for any nonzero infinitesimal M,. By Definition 2.3 ~y --

az

~ f'(z),

i.e., 5. T h e U l t i m a t e M u l t i - F u n c t i o n

a y = (f'(z) + a ) a z

for some infinitesimal ~. N o w this holds for all z for which f' is defined, so take z = x, and for all infinitesimal ~Lz, so take az = ax. ay = r

+ a)ax

SO

Ay

Ax (f'(x) +

Taking standard parts (fog)'(a) = f'(x)g'(a) = f'(g(a))g'(a).

9

A proof should never read like a detective story, with subplots littered throughout the text and all the revelations left to the last paragraph. Infinitesimal methods make avoiding this "Poirot-style" trap considerably easier. 50

THE MATHEMATICAL INTELLIGENCERVOL. 12, NO. 3, 1990

Tool

Though Section 4 introduced the idea of using nonstandard analysis for teaching calculus, that is not really the major impact of this work. You will remember that the end of Section 3 introduced the idea of a non-standard model for many branches of mathematics (assuming you were still conscious at that point). I can n o w sketch the advantage that the more complicated construction gives. One of the first areas where the non-standard model was applied was perturbation t h e o r y - - t h e study of the change in b e h a v i o u r of equations with small changes in their form. (Let's face it--if it says "small changes" on the label, then it's ripe for Robinson!) An example of this idea is the discovery of a particular cycle shape of the equation ~x" + (x2 - 1)x' + x = a for small ~. If you draw a graph of solutions for a in the vicinity of 1, you get closed cycle shapes that change size suddenly. The cycle shape was said to look like a duck (probably to those whose only experience of a real duck is marinated in red wine and

served in an orange and garlic sauce). The first such was discovered in 1980 after the duck-hunting season was extended by taking e to be infinitesimal. These ducks are therefore called canards to this day [Ref: Lutz and Goze, Ch IV.8; Diener]. The uses of infinitesimal methods range far from their original home of calculus. A non-standard approach has been applied to the problem of verifying the correctness of computer programs. Whereas, in a standard world, we could only consider a program executing a loop a finite number of times, with nonstandard models, we can consider a loop as a potentially infinite one, without getting bogged d o w n in the mathematics of limits and partial functions. [Ref: Hurd, Nonstandard Analysis--Recent Developments, Paper 8]. There are many other applications: see the bibliography.

6. R e d e m p t i o n "'Malignant possibilities stand rockfirm as facts.'" Thomas Hardy, Tess of the D'Urbervilles

The infidel is innocent. The sins that Berkeley saw him swirling in have been absolved and he can be led back into the church of mathematics and may take a seat in the highest of pews. Not only is the construction of the hyperreals firm enough to be a platform for future analysts to stand on from their earliest days, it is also high enough for the present priests of the Quasi-religious sect to launch new and more powerful theorems from. Having stolen most of his thunder, I will give the last word to Bishop Berkeley: He who can digest a second or third fluxion, a second or third difference, need not, methinks, be squeamish about any point in divinity.

Meanwhile, if you want me, I'll be writing the next preface to Crockford's.

A Quasi-religious sect novice after redemption.

Acknowledgments: The author gratefully acknowledges the help of Ian Stewart and Darren Rogan for various forms of advice, proofreading and editing; CPMSDOSUGUK journal and Punch for the illustrations; Warwick University Mathematics Institute and Christ Church College, Canterbury for their facilities; the distillers of Laphroaig Islay malt whisky for their invaluable aid with the last parts of Section 3.

A n n o t a t e d Bibliography Bishop Berkeley, The analyst: A discourse addressed to an infidel mathematician, The World of Mathematics, vol. 1 (J. R. Newman, ed.), London: Allen and Unwin (1956), 288-293. A thin succulent layer of mathematics with a strong sprinkling of invective, steeped in a vitriolic sauce of sarcasm. Brilliant!! M. Diener, The canard unchained or how fast/slow dynamical systems bifurcate, Mathematical Intelligencer 6, no. 3 (1984), 38-49. A canny introduction to duck-hunting. S. Haack, Philosophy of Logics, Cambridge: Cambridge University Press (1978). Discusses some of the problems with second-order logic and gives the basisfor the language used in Section 3. A. E. Hurd (ed.), Nonstandard Analysis---Recent Developments, New York: Springer-Verlag (1983). A series of papers (all quite technical) on recent uses of Robinson"s work. Includes a paper by Richter and Szabo on program verification (cf. Section 5). A. E. Hurd and P. A. Loeb, An Introduction to Nonstandard Real Analysis, London: Academic Press (1985). Chapter 1 contains the construction on which Section 3 was based. H.J. Keisler, Foundations of Infinitesimal Calculus, Boston: Prindle, Weber and Schmidt (1976). The definitive book on the teaching and understanding of nonstandard analysis. R. Lutz and M. Goze, Nonstandard Analysis: A PracticalGuide with Applications, New York: Springer-Verlag (1981). A chirpy, if technical, exploration of non-standard analysis. Good humour let down by poor English! Chapter IV.8 discusses canards (cf. Section 5). A. Robinson, Non-Standard Analysis, Amsterdam: NorthHolland (1966). The original text upon which all of the theory is built. M. Spivak, Calculus, New York: Benjamin (1967). The introductory text to university-level analysis. The New English bible of the Quasi-religious Sect! I. Stewart, The Problems of Mathematics, Oxford: Oxford University Press (1987). Chapter 7 introduces the subject for the beginner. K. D. Stroyan and W. A. J. Luxemburg, An Introduction to the Theory of Infinitesimals, New York: Academic Press (1976). Pitched highly, but contains a good ultrafilter construction in chapter 1. D. O. Tall, Infinitesimals Constructed Algebraically and Interpreted Geometrically, Coventry: University of Warwick, preprint (1979). Gives a construction of another extension containing infinitesimals with a neat way of visualising them. Tudor Court, Wattisfield Road Walsham le Willows Bury St. Edmunds Suffolk IP31 3BD, United Kingdom THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 3, 1990

51

Ian Stewart* The catapult that Archimedes built, the gambling-houses that Descartes frequented in his dissolute youth, the field where Galois fought his duel, the bridge where Hamilton carved quaternions-not all of these monuments to mathematical history survive today, but the mathematician on vacation can still find many reminders of our subject's glorious and inglorious past: statues, plaques, graves, the cafd where the famous conjecture was made, the desk where the

famous initials are scratched, birthplaces, houses, memorials. Does your hometown have a mathematical tourist attraction? Have you encountered a mathematical sight on your travels? If so, we invite you to submit to this column a picture, a description of its mathematical significance, and either a map or directions so that others may follow in your tracks. Please send all submissions to the European Editor, Ian Stewart.

The Left-Handed Pythagoras Catriona Byrne The dedicated mathematical tourist in northern France will have few problems locating this relief of Pythagoras, dating from the twelfth century, among the nearly 1800 sculpted figures of the cathedral of Notre Dame de Chartres. The town is dominated by its cathedral, rising between the narrow streets of the old quarter, close to the river Eure. Pythagoras sits at the foot of the arch above the right-hand door of the magnificent West (Main) Portal. This arch depicts the Greek philosophers and their attributes. For his discovery that intervals in music may be expressed mathematically, Pythagoras is associated with music as well as geometry. Chartres is a masterpiece of the ingenious system of proportions underlying medieval architecture. For medieval Europeans, God was the Great Geometer; this concept inspired the architect. Chartres was a reputed center of philosophical study, devoted particularly to Plato's book Timaeus, which held that the whole universe expresses "measurable harmony." Thus the geometric properties of the cathedral's architecture were probably well understood and planned. Appropriately, the town lies in the middle of a "plane," and you can see the spires from literally miles and miles a w a y - - i t really gives you a gut feeling for what it means to say that a manifold is locally fiat. There is a famous pilgrimage from Paris to Chartres every year at Easter: hundreds of people walk the 96 kilometers, and of course long before they get there,

* Column Editor's address: Mathematics Institute, University of Warwick, CoventryCV4 7AL England. 52

they have the illusion that they have arrived, because they can see the cathedral. It must be maddening. Note that Pythagoras is depicted as left-handed. Apparently, the proportion of left-handed mathematicians is distinctly higher than the proportion of lefthanded people in a random population. Tiergartenstrafle 17 D-6900 Heidelberg 1 Federal Republic of Germany

Arrow indicates location of Pythagoras on the right portal.

THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 3 9 1990 Springer-Verlag New York

Cathedral of Notre Dame de Chartres.

The left-handed Pythagoras. THE MATHEMATICALINTELLIGENCER VOL. 12, NO. 3, 1990 ~ 3

A Centennial: Wilhelm Killing and the Exceptional Groups Sigurdur Helgason

In his article [3] in the Mathematical Intelligencer (vol. 11, no. 3) A. John Coleman gives a colorful biography of the mathematician Wilhelm Killing and offers an admiring appraisal of his paper [8]. The subject of this paper, the classification of the simple Lie algebras over C, has indeed turned out to be a milestone in the history of mathematics. At the conclusion of his article Coleman lists six reasons why he considers [8] to be such an epoch-making paper, the first one being that it furnished the impetus towards the problem of classifying finite simple groups. Here one could add that the answer to that problem was also motivated by and was partly provided by the classification of simple Lie algebras through Claude Chevalley's paper [2b]. In two sections of his article, entitled "Killing Intervenes" and "The Still Point of the Turning World," Coleman discusses the work of Killing and Elie Cartan on the classification of simple Lie algebras over C. I would like to add a few comments to his discussion (see also [7b]). While Sophus Lie and some of his associates in Leipzig attempted the problem of classifying all local transformation groups of R n, Killing set himself the problem of finding all possible Zusammensetzungen of r-parameter groups. In other words, he was interested in all possible ways in which a vector space could be turned into a Lie algebra. While Lie was motivated by applications to differential equations, Killing was led to his problem from his work in geometry. Let ~ be a simple Lie algebra over C. Then g is isomorphic to the Lie algebra of linear transformations adX of g given by adX(Y) = IX, Y], with X running through g and [,] denoting the bracket product in the Lie algebra. To study this family it is natural to try to diagonalize the operators ad(X) as effectively as possible. This is the motivation for the definition of a Cartan subalgebra as a subalgebra ~ which is 54

(i) A maximal abelian subalgebra of g. (ii) For each H E ~, adH is a diagonalizable linear transformation of 0. For the classical simple Lie algebras the existence of such an ~ is a simple matter; for the general case Killing gave only an incomplete proof. The gap was filled in Cartan's thesis. Even today the general existence proof for a Cartan subalgebra is not easy. The usual proof proceeds using theorems of Engel and of Lie on nilpotent and solvable Lie algebras, respectively; an entirely different proof, realizing a possibility suggested by Cartan [le], p. 23, was devised by Roger Richardson [11]. In [8] Killing introduced many of the fundamental concepts in the modern theory of simple Lie algebras, for example the following:

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(i) The roots of ~, which are by his definition the roots of the characteristic equation det (hi - ad X) = 0. Twice the second coefficient in this equation, which equals Tr(ad X)2, is now customarily called the Killing form. However, Cartan made much more use of this form. (ii) A basis r 1. . . . . oq of the roots; all other roots are integral linear combinations. He then introduced the matrix (aq) where

- aq = the largest integer q such that oLi + qoLj is a root. The matrix (aii) is now called the Cartan matrix. Killing had discovered that the bracket relations in ~, in particular the Jacobi identity, are reflected in certain additive properties of the roots. This motivated his introduction of (aq) described above. Killing's method of classification is a grandiose conception and consists of two main steps: Step I. Find certain necessary conditions on the maor Cartan [la], w 5). Then classify all equivalence classes of matrices (aq) satisfying these conditions.

Wilhelm Killing

trix (aij) (cf. Killing [8], w

Step II. Show that for each equivalence class of matrices (aq) there exists exactly one simple Lie algebra over C having it as a Cartan matrix. Step II is explicitly stated in Cartan [la], beginning o f w 8. Roughly speaking, this method is one that is still used today (among others). Instead of Step I, one carries out the equivalent classification of root systems by means of the Coxeter-Dynkin diagrams. As stated by Cartan [la] p. 410, Step I above was completely carried out by Killing. While Killing recognized that A3 = D3 (the local isomorphism between SU(4) and SO(6)) he did not notice that E4 = F4, although as Cartan remarked this is immediate from his root tables in [8], pp. 30-31. While Trace (ad X)2 is nowadays called the Killing form and the matrix (aq) called the Cartan matrix, in view of the above it would have been reasonable on historical grounds to interchange the names. Step II above, the existence and uniqueness of a simple Lie algebra with a given Cartan matrix, is a more difficult problem. This is where Killing's work is most defective, although the existence result at the end of [8] is correct. For the matrices (aij) for the classical Lie algebras A l, Bl, Ct, D 1the uniqueness is stated in Killing ([8], p. 42) and is verified in detail in Cartan [lb], Ch. V, p. 72-87. The exceptional simple Lie algebras are the subject

of the final w in Killing's paper. This is certainly his most remarkable discovery, although these algebras appeared to him at first as a kind of a nuisance, which he tried hard to eliminate. Even Lie, who was generally critical of Killing's work, expressed in letters to Felix Klein his admiration of the results [6a]. While these exceptional Lie algebras may at first have been rather unwelcome, they have subsequently played important roles in Lie theory, for example by forcing one to find a priori conceptual proofs rather than case-bycase verifications. Extrapolation from the classical Lie algebras has to be done with care: for example in [7c] a result appears about invariants, which holds for all the classical Lie algebras (real or complex) but fails for exactly four of the seventeen real exceptional Lie algebras. It m a y at first a p p e a r strange that after working so hard at obtaining the classification, mathematicians should work equally hard avoiding the use thereof. The aim is of course thereby to gain better understanding. The role of the exceptional Lie groups in the construction of the sporadic simple finite groups and the use of the exceptional groups in modern string theory are among the many unexpected bonuses. For the actual existence of the exceptional Lie algebras Killing indicates in w how to determine the structural constants on the basis of the matrix (aq); the Jacobi identity then has to be verified in each case. Killing does this for G2; for the others this would necessitate a huge computation and from the indications in Killing ([8], p. 48), it is hard to say how far he had progressed. Killing had also tried to represent G2 as a THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 3, 1990 5 5

genuine local transformation group in R n. He found that n would necessarily have to be larger than 4; w h e n he communicated this to Friedrich Engel, he (Engel [4a]) and independently Cartan ([lb], p. 281) showed that G 2 could be realized as the stability group of the system

dx 3 + x~ dx2 - x2 dx~ = 0 dx4 + x3 dx~ - x~ dx3 = 0 dx 5 + x2 dx3 - x3 dx2 = 0 i n R s. In his papers [la], [lb] Cartan devotes much work to Step II (i.e., existence and uniqueness) for the excep-

It may at first appear strange that after w o r k i n g so h a r d a t o b t a i n i n g the c l a s s i f i c a t i o n , mathematicians should w o r k e q u a l l y h a r d a v o i d i n g the use thereof. tional groups. In [la] he states without proof an explicit representation for F4 as a stability group of a Pfaffian system in R TM (in analogy with G2 above) and indicates representations for E6, E7, and Es as groups of contact transformations in R 16, R27, and R 29, respectively. In his thesis Cartan determines the structural constants on the basis of the matrix (aq), remarking explicitly how this implies the uniqueness (cf. [lb, p. 93]). For the existence, the Jacobi identity would have to be verified; since the structural constants given by Cartan have a very simple and symmetric form it is possible that he indeed did verify the Jacobi identity, but he is silent on this point. This verification would be unnecessary if the above models that Cartan gives in [la] could be shown to have Lie algebras with the structural constants mentioned. In his thesis he also gives a second set of models for the exceptional groups, but again with somewhat sketchy proofs. Nevertheless, one is completely convinced that Cartan proved for himself both the existence and uniqueness of the exceptional groups and that the full details were left out only because of their complexity. Subsequently, general a priori proofs have been given for both parts of Step II. The uniqueness was proved by H e r m a n n Weyl [12]; for the existence, a proof was given by Ernst Witt [13] and by Chevalley [2a] (with full details in Harish-Chandra [5]). Coming back to Coleman's article, I agree with him that in the past Killing's work has been overshadowed by the work of Cartan; however, in recent years I believe that Killing's work has been better recognized and Coleman's article is also a valuable contribution in this regard. Coleman also asks: Why was Killing's work neglected? I think that in modern terminology it is fair to 56

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say that Cartan's thesis represented a friendly takeover. In addition to many other new results, Cartan's thesis gave a complete solution of the classification problem, so at least for this problem it was no longer necessary to struggle with Killing's unrigorous papers. In addition to giving a clear exposition, Cartan's thesis represented first-rate scholarship; in particular it gives precise references to and a fair and detailed evaluation of Killing's work, warts and all. Cartan's thesis is actually much easier reading than his later papers in differential geometry, which reflected his accumulated experience with Lie groups combined with his unusual geometric intuition. A more challenging question seems to be: Why did it take so long for Cartan's thesis to get assimilated in spite of the recognized importance of its results and the clarity of its exposition? Although the results seem specific and concrete to us, perhaps Cartan's contemporaries saw them as relatively abstract, in that application of specific transformation groups to differential equations was no longer the dominant theme. At any rate, it seems that Cartan had the field of Lie algebra theory pretty much to himself for about a quarter century. (A notable exception: the Levi decomposition in paper [9] from 1905). Here it should perhaps be noted that Cartan's Lie algebra papers subsequent to his thesis, for example his paper [ld] on the classification of simple Lie algebras over R and his paper [lc] on representation theory, were considerably more difficult than his thesis. For the structure theory and classification of simple Lie algebras ~ over R, the basic tool is the concept of a Cartan subspace a defined by the properties: (i) For each H ~ a, adH is a (real) diagonalizable linear transformation of ~. (ii) a is maximal relative to (i).

In m o d e m t e r m i n o l o g y i t is f a i r to s a y t h a t C a r t a n ' s t h e s i s represented a f r i e n d l y t a k e over. The corresponding root theory is more complicated in that twice a root can now be a root and, in addition, the multiplicity of a root may now be larger than one. From the classification one can deduce that the root pattern, together with the multiplicities, determines g up to isomorphism ([7a] p. 535); however, a simple direct proof of this fact does not seem to be available. On page 30 of [3] Coleman speculates that perhaps Cartan might not have become a research mathematician if he had been unaware of Killing's work (for example, if it had never been published for lack of rigor). With m y admiration of Cartan's genius, I consider this possibility extremely remote. Cartan was at the top of

his class at t~cole Normale [6b]; he and fellow students h a d been very m u c h immersed in the work of Lie as well as in differential geometry through the teaching of Gaston Darboux. If Killing h a d not come along, Cartan might have w o r k e d on infinite Lie groups earlier t h a n he did (in 1902) as well as on differential systems (1899) and differential geometry (1910). If not earlier, it seems quite possible that he would have disc o v e r e d the classification of simple Lie a l g e b r a s t h r o u g h his differential geometric work on symmetric

W h y d i d i t t a k e s o l o n g f o r C a r t a n ' s t h e s i s to g e t a s s i m i l a t e d in s p i t e o f the r e c o g n i z e d i m p o r t a n c e o f its r e s u l t s a n d the c l a r i t y o f its exposition? spaces (1926), because their classification is equivalent to that of the simple Lie algebras over R. This is of course pure speculation. It s e e m s ironic t h a t w h i l e Lie [10, vol. 3, pp. 768-771] subjected Killing's work to a severe criticism, one can argue that Killing's paper [8] was the first spark t h a t led to the f o r g i n g of the t h e o r y of Lie groups and Lie algebras into a mathematical force in its o w n right, i n d e p e n d e n t of differential equations, exerting ever-increasing influence in mathematics and mathematical physics. Therefore it seems appropriate for all m a t h e m a t i c i a n s to c o m m e m o r a t e W i l h e l m Killing on this centenary of his epoch-making paper.

Sophus Lie

5. 6a.

References

6b. la. I~. Cartan, Uber die einfachen Transformationsgruppen, Leipzig Ber., 1893, pp. 395-420; reprint, Oeuvres completes, Vol. I, no. 1, Paris: Gauthier-Villars (1952), 107-132. lb. - - , Sur la structure des groupes de transformations finis et continus, Th~se, Paris Nony, 1894; reprint, Oeuvres completes, vol. I, no. 1, Paris: Gauthier Villars (1952), 137-287. lc. - - , Les groupes projectifs qui ne laissent invariants aucune multiplicit6 plane, Bull. Soc. Sci. Math. 41 (1913), 53-96. ld. - - , Les groupes r6els simples finis et continus, Ann. Sci. EcoleNorm. Sup. 31 (1914), 263-355. le. , Groupes simples closet ouverts et g4om6trie riemannienne, J. Math. Pures Appl. (9) 8 (1929), 1-33. 2a. C. Chevalley, Sur la classification des algebras de Lie simples et de leurs repr6sentations, C.R. Acad. Sci. Paris 227 (1948), 1136-1138. 2b. , Sur certains gToupes simples, T6hoku Math. J. 7 (1955), 14-66. 3. A.J. Coleman, The greatest mathematical paper of all time, The Mathematical Intelligencer 11, no. 3 (1989), 29-38. 4a. F. Engel, Sur un groupe simple ~ quatorze param~tres, C.R. Acad. Sci. Paris 116 (1893), 786-788. 4b. , Wilhelm Killing (obituary), Jber. Deutsch. Math. Verein. 39 (1930), 140-154.

7a. 7b. 7c. 8. 9. 10. 11. 12. 13.

Harish-Chandra, On some applications of the universal enveloping algebra of a semisimple Lie algebra, Trans. Amer. Math. Soc. 70 (1951), 28-96. T. Hawkins, Wilhelm Killing and the structure of Lie algebras, Archivefor Hist. Exact. Sci. 26 (1982), 126-192. , Elie Cartan and the prehistory of the representation theory of Lie algebras, preprint 1984. S. Helgason, Differential Geometry, Lie groups and Symmetric Spaces, New York: Academic Press (1978). , Invariant differential equations on homogeneous manifolds, Bull. Amer. Math. Soc. 83 (1977), 751-774. , Some results in invariant theory, Bull. Amer. Math. Soc. 68 (1962), 367-371. W. Killing, Die Zusammensetzung der stetigen endlichen Transformationsgruppen II, Math. Ann. 33 (1889), 1-48. E.E. Levi, Sulla struttura dei gruppi continui. Atti Accad. Sci. Torino 60 (1905), 551-565. S. Lie and F. Engel, Theorieder Transformationsgruppen, 3 vols. Leipzig: Teubner (1888-1893). R. Richardson, Compact real forms of a semisimple Lie algebra. J. Differential Geometry 2 (1968), 411-420. H. Weyl, The structure and representations of continuous groups, New Jersey: Inst. Adv. Study Princeton, Notes. (1935). E. Witt, Spiegelungsgruppen und Aufz/ihlung halbeinfacher Liescher Ringe, Abh. Math. Sem. Univ. Hamburg 14 (1941), 289-322.

Department of Mathematics M.I.T. Cambridge, MA 02139 USA THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 3, 1990 5 7

So Far, So Good: My Life Up to Now Edwin Hewitt

I was born in Everett, Washington (USA) on 20 January 1920. I am the youngest of three children born to Irenaeus Prime Hewitt and Margaret Guthrie Hewitt. I had two siblings: William Guthrie Hewitt (1916-1983) and Helen Hewitt Arthur (1918-1953). Our parents were old American stock. On my father's side we (and myriad other Hewitts) are descended from Thomas Hewitt, who flourished, as the saying goes, in North Stonington, Connecticut, in 1639. On my mother's side there are a lot of Pennsylvania Dutch ancestors, some with a wild streak that keeps cropping up unto the third and fourth generation. My father was trained as a lawyer, but in my lifetime never practiced law. My mother had a bachelor's degree in English from the University of Nebraska. During my first decade she was a housewife, but also taught English as a freelance and for two years studied English in the graduate school of the University of Washington. My early memories are for the most part happy, though I carry to this day recollections of stern discipline meted out for this or that childish crime. I suppose I was precocious, though by no means a prodigy. An early memory is of being taken to a meeting of the Everett Rotary Club and being stood up on a table to recite the Gettysburg Address. My mother began to teach all three of her children French, using a little book called French without Tears. I can still quote passages from that little book, and recite the Gettysburg address. I was an insufferable child. I h a d - - a n d still suffer f r o m - - a terrible habit of mouthing off. Recognizing this weakness has at least preserved me from the folly of becoming a department chairman or dean: I knew that at some inopportune moment I would lose control of my tongue and then lose my job. 58

M y m o t h e r was d e v o t e d to literature. D u r i n g summer vacations in the 1920s, she and her children read aloud to each other Pilgrim's Progress, David Copperfield, Dombey and Son, Oliver Twist, and finally Vanity Fair. I recall my pride at being given the honor of reading the last installment of Vanity Fair to my mother and siblings. In 1931, my mother took all three of her children to St. Louis, Missouri, where she enrolled us in the Principia Academy, a Christian Science school. I have no idea where she got the money. I know now that she was determined to get out of Everett, Washington, and a w a y from my father. I spent three academic years at Principia and two at a Christian Science country boarding school called The Leelanau School, located 30 miles west of Traverse City, Michigan. I got a good education at both schools, though I had unh a p p y times at both. Part of the process of growing up, no doubt. I learned mathematics and French with little effort. In grammar school I had a hard time with written arithmetic, t h o u g h m e n t a l arithmetic and w o r d problems were simple. At ten years of age I heard about logarithms, at thirteen about trigonometric functions. I had a marvelous mathematics teacher at Principia, Paul C. Dietz. He gave me my head and the run of his exiguous mathematical library. I read trigonometry on my own, ditto Euclid (I loved ruler-andcompass constructions), built models of the regular solids, wrote out on tape an estimate of 1200!, found in Thornton C. Frye's probability text, and enjoyed wielding a Keuffel & Esser log log duplex sliderule, which I still have. Of course, it is only an antique now. I had a fine teacher of French at Principia, Monsieur Robbins. He gave me a good start in French, and

THE MATHEMATICAL INTELLIGENCER VOL. 12, NO, 3 9 1990 Springer-Verlag New York

turned me loose to read Les Mis&ables on my own: this was my first experience of reading a serious book in a foreign tongue. The Latin teacher at Principia was a Mrs. Semple, a lady of the old school. We learned our Latin from her a n d m u c h more a b o u t w h a t the Germans called Umgang mit Menschen. I entered Harvard in 1936 on a Samuel Crocker Lawrence scholarship in the a m o u n t of $500 per annum. A half-century ago Harvard charged $400 per a n n u m tuition and $100 per a n n u m for dormitory rooms [at least for the needy among us]. With waiting on tables for my meals and savings from summer jobs, I had no money problems. My six years at Harvard were happy. I melted into the crowd, no longer an underage, undersized misfit. The mathematics faculty was awe-inspiring. George David Birkhoff, an Olympian figure, was also dean of the Faculty of Arts and Sciences. William Caspar Graustein was a dignified man who always wore black three-piece suits. Edward Vermilye Huntington was an elderly scholar, plainly not part of the power structure. Joseph Leonard Walsh was greatly respected for his flawless lectures, though not for the Walsh-Rademacher functions: we had not heard of them. The y o u n g e r faculty i n c l u d e d David Vernon Widder, Saunders Mac Lane, Garrett Birkhoff, Willard van Orman Quine [I think he was in the Philosophy Department: he gave brilliant courses on mathematical logic], and Marshall Harvey Stone. In the autumn of 1937, Stone accepted me as an advisee. Thirty-four years of age and an associate professor [shortly thereafter promoted], he was at the meridian height of his mathematical powers. He had the s t r o n g e s t mind I h a d ever e n c o u n t e r e d . He had written a huge and, to me, incomprehensible book; he was a son of the Chief Justice of the United States; his lectures were known for their clarity and vigor. I took but one course from Stone, the theory of functions of a real variable, in 1939-1940. This course put my feet on the path to becoming a mathematician. The concepts set forth in this course have been with me for fortynine years; some of them appear in the book that Karl Stromberg and I wrote. From Stone and his fellow mathematicians at Harvard, I learned vital lessons about our wonderful subject: Rule #1. Respect the profession. Rule #2. In case of doubt, see Rule #1. My studies under Stone's policy of benign indifference led me into set-theoretic topology, to the neglect of much else. In 1941 1 proved my first theorem. If X is a metric space without isolated points, then X contains complementary dense subsets. Early in 1942, I lectured on this and related matters to the Harvard Colloquium. G. D. Birkhoff did me the h o n o r of attending. At the end he asked, "Mr. Hewitt, h o w much of this would you have without the axiom of choice?" I

Edwin Hewitt

replied, "Nothing at all, Sir." With that, G.D. said something like "hump" and turned about to register his scorn at a whippersnapper w h o couldn't construct the things he was talking about. I got m y Ph.D. in June 1942 for a thesis on set-theoretic topology written under Professor Stone. Saunders Mac Lane read the manuscript and half way through found an unfortunate blunder. Luckily the lacuna could be filled. George Mackey got his Ph.D. the same day I got mine, also from Stone. Professor Stone has been a decisive influence in my life for over half a century. My debt to him is second only to that I owe m y dear parents. It is a marvel that Stone took me as his prot6g6, and over the years increasingly as his friend. Military service loomed for all young men in my position at that time. Nonetheless, I accepted an instructorship at Harvard in June 1942 and taught there happily for six months. This was a carefree time. I was the assistant senior tutor at a Harvard house and so had free room and board. I had my first serious girlfriend, an e i g h t e e n - y e a r old Radcliffe s o p h o m o r e . (She d r o p p e d me early in 1943.) I found mathematics of consuming interest and rejoiced in learning mathematics and in discovering new facts. In January 1943 John M. Harlan paid a visit to Harvard. He was a prominent N e w York lawyer who had been recruited by W. Barton Leach to organize and run an Operations Research Section at the 8th Bomber THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 3, 1990 5 9

Command in England. Someone, probably Oswald Veblen, had given Harlan my and Frank Stewart's names. Harlan offered us positions with his group. We thought for as long as thirty seconds before saying "yes," and in April 1943 found ourselves in High Wycombe, 30 miles outside of London. Harlan was a colonel, and there were perhaps two other commissioned officers in the group. The rest of us, some 40 in number, were civilian scientists. I was put onto the problem of defending B-17 and B-24 bombers against German fighter planes. Astonishingly, here was the B-17 with machines guns all over it and no doctrine whatever for gunners to aim by. I attended an RAF gunnery school, where a sensible and simple system was taught. I recomputed the RAF data for our ammunition and airspeeds, and added n e w rules to deal with head-on attacks, which the night-flying RAF did not have to cope with. My first job was to convince Col. Harlan that I had the right answers. He gave me a lawyeflike grilling. Once satisfied that I was right, he turned me loose. I got great assistance. Major George R. Weinbrenner was a great help until he was shot down. (He coldcocked a g u a r d on a train in G e r m a n y , w a l k e d through France, and lived the life of Riley on the Riviera until US forces liberated him in 1944.) I had able assistance from Sergeant John S. JiUson, whom I had known at Harvard. There was a superb draftsman named Bosch. Porter Henry, a N e w York journalist, joined us and performed prodigies. I went all over the 8th Bomber Command, lecturing to bomber crews and passing out skillfully written training literature from Porter Henry. I felt it necessary to fly and w o u n d up with seven missions to Germany and France as a bombardier-gunner. In the late summer of 1944 1 was rotated back to the USA. I spent some weeks with Saunders Mac Lane's Applied Mathematics Panel at Columbia University and about 1 January 1945 joined an Operations Research Section at the headquarters of the 20th Air Force in the Pentagon. I wanted nothing so much as to go to the Marianas and help bomb Japan, but I never got my orders. I learned 20 years later that my mother had telephoned a deputy secretary of the Navy and asked him not to let me be sent out a second time. Altogther, m y two and a half years with the Air Force were the best times of my life up to now. I've been a fortunate fellow. I've climbed some nontrivial mountains. I've skied across Lapland. I've driven 165 miles per hour in a motor car. I've known some wonderful women, two of whom I married. But nothing else even comes close to hanging around the Air Force and getting rides in their beautiful machines. I was discharged in September 1945. Powerful forces were guiding my d e s f i n y - - I suspect Veblen or Stone. At any rate, H e n r y Allen Moe of the Guggenheim Foundation awarded me out of the blue a $2500 fel60

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lowship for reconversion to civilian life. Though I was of two minds about taking up mathematics again and thought of having a whirl at law school, I took the money and with my beautiful young wife, Carol Blanchard Hewitt, went to Princeton for the year 19451946. There was a galaxy of eminences at the Institute for Advanced Study: Albert Einstein, Carl Ludwig Siegel, John yon Neumann (who was around only part of the time) Hermann Weyl, Kurt G6del, James W. A l e x a n d e r , M a r s t o n M o r s e . Richard A r e n s w a s M o r s e ' s assistant; Ernst Strauss, Einstein's. At Princeton University there were Wedderbum (a man of small stature, no longer young, who wore well-cut tweeds), H. P. Robertson, L. P. Eisenhart, Salomon Bochner, Ralph Fox, Emil Artin, Solomon Lefschetz, Claude Chevalley. I was very much an Unterspieler but enjoyed watching the big shots, attending their seminars, and observing their foibles. I was horrified o n e day to watch Chevalley bait Weyl like a terrier nipping at a water buffalo, while Weyl was trying to give a lecture on Lie groups. My o w n research paid off quickly, though of course with the usual struggles. I solved two old problems posed by Paul Urysohn in the 1920s and discovered what are n o w called realcompact or Hewitt spaces. This came about through my efforts to understand the ring of all real-valued continuous [not necessarily bounded] functions on a completely regular T0-space. I was guided in part by a casual remark made by Gel'fand and Kolmogorov (Doklady Akad. Nauk SSSR 22 [1939], 11-15). Along the w a y I found a novel class of real-closed fields that superficially resemble the real n u m b e r field and have since become the building blocks for nonstandard analysis. I had no luck in talking to Artin about these hyperreal fields, though he had done interesting work on real-closed fields in the 1920s. (My published "proof" that hyperreal fields are real-closed is false: John Isbell earned my gratitude by giving a correct proof some years later.) In fact, what I got from Artin was a quick brush-off. My ultrafilters also struck no responsive chords. Only Irving Kaplansky seemed to think my ideas had merit. My first paper on the subject was published only in 1948. It got a lukewarm review from Dieudonn~. Later Leonard Gillman, Meyer Jerison, and Melvin Henriksen put me under a great debt by taking the matter up afresh and doing a whole lot with rings of continuous functions. A list of citations published a few years back by Joseph Schatz lists my 1948 paper as one of the most cited papers of the past fifty years. Three Soviet mathematicians came to the hyperreal field problem many years later: the hugely talented brothers Gregory and David Chudnovsky and that great fellow Misha Antonovskij. The four of us published a joint paper in 1983 on rings of continuous functions. Nothing like coming back to a problem after 35 years!

The author as a Harvard freshman in 1936; between George Mackey and Frank Stewart at Harvard around 1940; and conquering the Alps in 1952.

In 1946-1947, I taught at Bryn Mawr College. I was not housebroken enough for Bryn Mawr. Although John Oxtoby and Anna Pell Wheeler were wonderful to me and expressed regret at my departure after one year, they must have felt relieved to have this loose cannon out of their pleasant demesne. I made friends among students at Bryn Mawr whom, 42 years later, I still cherish as my close friends. I went to the University of Chicago in 1947 to take a half-time position with a military research project called CHORE [Chicago Ordnance Research] and a half-time position with the Department of Mathematics at the University of Chicago. This move was a disaster. I lost the military job by running afoul of the regular army colonel, Frank Fenton Reed, w h o was the Army liaison officer. I then found that Marshall Stone had no interest whatever in hiring me full time in his department. The man in charge of CHORE, Dean Walter Bartky, was decent enough about my plight but could do nothing to mitigate it. So I served out my one-year appointment in wretched circumstances. In the spring of 1948, I generated job offers from two large midwestern universities and from the University of Washington. All salaries offered were $4250 for the academic year. One of the midwest schools offered me twelve hours per week of teaching: four sec-

tions of beginning calculus. The other midwest school is in an agricultural state. I had enough Harvard snobbery to refuse to go where [ thought I would be in the middle of a cornfield. The University of Washington offered a nine-hour teaching load, a course in the theory of functions of a complex variable, and the opportunity of living in Seattle, which I r e m e m b e r e d pleasantly from m y childhood. So in August 1948 m y young wife and I, with most of our worldly goods in a station wagon and accompanied by my recently widowed father, set out for the Pacific Northwest. I was taken with Seattle. We were welcomed by old f r i e n d s f r o m the 1920s, n o w r a t h e r gray at the temples. We f o u n d reasonable living accommodations. The department was a comfortable little group, which was for the most part unconcerned with research. Professors Z. W. Birnbaum (a Polish-born statistician), Ross A. Beamont, and Herbert S. Zuckerman rapidly became good friends. In particular, I found in Z u c k e r m a n a wonderful friend and collaborator. I learned most of my classical mathematics from him, a side that I had foolishly neglected in order to follow the sirens of functional analysis and set-theoretic topology. Herbert and I worked together from the autumn of 1948 until the day of his tragic and untimely death in June 1970. I am proud that my name stands THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 3, 1990

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alongside his, attached to now classical theorems in Diophantine approximation, the structure of semigroups, and abstract harmonic analysis. In 1950 I attended the International Congress of Mathematicians at Harvard and met the great Swedish mathematician Arne Beurling. I had studied his enigmatic 1938 paper on the convolution algebra of functions of finite variation on the real line and asked him some questions about his proofs. Within minutes he invited me to spend a year with him in Uppsala. So my wife and I set about learning Swedish. In August 1951 we sailed for Copenhagen on the M.S. Stockholm. We had a splendid year in Sweden, though my hopes of achieving great things in research with Beurling never came to fruition. We made friends in Sweden w h o are today still close friends at the houseguest level. I can carry on a conversation in Swedish and read Dagens Nyheter. In the summer of 1952 my wife was pregnant with our first daughter, Greta. We travelled in Germany, the Netherlands, France, Switzerland, and Britian. We met many interesting people, among them Wilhelm Blaschke, E. Witt (who pointed out that I nearly stole his name), Hel Braun (who was a close friend up to her death), Helmut and Irma Grunsky (also lifelong friends), M. Plancherel (who gave me Kirschwasser in his garden in Z~irich), Jean Dieudonn6 (who entertained me and others lavishly in Nancy), and Henri Cartan. We spent several weeks in Paris as guests of the late Leonard Jimmy Savage and his charming wife and young son Sam. They had an enormous apartment not far from La Sant6. Jimmy and I did much of the research for our paper on product measures while sitting on a bench in the Luxemburg gardens, surrounded by nursemaids with prams. This paper contains the Hewitt-Savage 0-1 law, along with a construction of measures on extreme points of a convex set that is a special case of what later became Choquet theory. Back in Seattle in the autumn of 1952 and settled down to a quiet life, I found myself picking up doctoral students. I was enthusiastic about mathematics; I gave lectures that were attention-getters though not show-stoppers; and over the years, a number of fine young men and w o m e n came to me for guidance as they w o r k e d toward the doctorate. The first was George H. Swift (1954), who now is one of my best friends. Thirty-six others followed George over the years. Some have vanished, to my sorrow. Many hold positions of great influence and trust in academe and industry. One has even been a university president: Albert J. Froderberg. With Karl Stromberg and Kenneth Ross I have written many papers and three books. I am grateful to these fine scientists and good friends. It was seldom easy to work with me. Wis Comfort has maintained close ties of friendship as he has m o u n t e d to the academic empyrean. Leonard 62

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Yap, n o w a senior member of the faculty at the National University of Singapore, has been a stalwart friend. Those whom I don't list will forgive me. They know that I love them all and that I rejoice in their successes and mourn their misfortunes: and there are far more of the former than the latter. In 1967 1 met and instantly took to Mark Aronovich Naimark. In 1969 1 went to Moscow for seven months to work with him under the inter-Academy exchange program. By this time, little Greta had grown into a sixteen-year-old woman, her mother had divorced me, and I had remarried. Greta and I went to Moscow; my second wife joined us later and we had many an adventure. I had begun to study Russian during World War II, partly out of boredom when there was nothing better to do. I had continued my Russian study over the intervening years and was fluent when we arrived in the USSR. Greta, who had had one year of Russian in a Seattle high school, plunged into a Moscow high school and did very well. She made many Russian friends, received and declined a proposal of marriage, saw sights that no adult could have, and came away with a strong feeling for Russia and the Russians. She can speak Russian well enough to fool native speakers and can imitate both Moscow and Kishinev accents. I took the academic year 1972-1973 at the University of Texas in Austin, intending to stay permanently. There too I had many dear friends, but I missed the rain in Seattle, and I missed my Seattle friends. After spending the summer of 1973 in Russia with Greta and her younger sister Lise, I went back to the University of Washington. Lise majored in Russian at Middlebury, has a master's degree in Russian, and is n o w a professional translator of Russian into English. I spent the years 1973-1975 as vice-chairman and chairman of the University of Washington Faculty Senate. This was good sport but of negative value to my career as a mathematician. I took a lot of unpopular positions, presented honestly the faculty's point of view, and once got on the AP wire with a rash statement about not hounding faculty for overdue library books. The fact is, of course, that the faculty have no p o w e r in running the university. We're hired help without a union to help us. The occasional star gets exceptional treatment from the administration, especially if he is a surgeon with a large team and a large budget. But the hard-eyed businessmen who run the university are not really interested in research as such or even in teaching. Naturally they like intercollegiate football. What they are mainly enamored of is garnering government contracts and trafficking in real estate. My great friend Heinz Bauer of the University of Erlangen-Niirnberg put me in for an Alexander von Humboldt-Stiftung prize. Through his efforts I was awarded one of these marvelous prizes. With it I spent the m o n t h s September 1975-April 1976 and Sep-

Hewittfest at the University of Washington, May 1988. From left to right: Shozo Koshi, Leonard Yap, Edwin Hewitt, Marshall Stone, Oliver Chen.

tember 1986-December 1986 at the University of Erlangen and the University of Passau. To live in West Germany was a superb experience. I love the country, a n d I l o v e t h e l a n g u a g e [ w h i c h I l e a r n e d in 1938-1940]; and I a m - - o r was in 1975--fascinated by the Wehrmacht's and the Luftwaffe's performance in World War II. In Erlangen I quickly met Gunter Ritter, a Ph.D. student of Bauer who was assigned the care and feeding of this eccentric American. Ritter and I hit it off at once. Over the past thirteen years we have written four or five substantial joint papers, and the Ritters have spent many months in Seattle. In 1982 I taught for a semester at the University of Alaska in Fairbanks. This was a great adventure with good colleagues and interesting students at a fine university in a burgeoning state. The following year I accepted a permanent appointment at Fairbanks but at the last moment reneged. My friends were justifiably chagrined and angry. When the oil market collapsed a year later, they must have been relieved not to have an additional tenured professor to support. I continue to treasure my Fairbanks friends, in particular Jack Distad and Bill Phillips. I've been abundantly blessed for many years by attracting gifted collaborators. I've already mentioned a number of them. Let me list some others. The great analyst and great gentleman K6saku Yosida spent 1950-1951 at the University of Washington under the auspices of our chairman, Roy M. Winger. Yosida and I wrote a paper on finitely additive measures that is still well regarded. In Princeton in 1956 1 ran into E. P. Wigner at a party given by Kees Gugelot and wrote a paper with him at the party. Shizuo Kakutani and I have written two papers about measures on groups. One of these includes the first construction of what are

now called Kronecker sets, which are remarkable sets in topological groups. I had the privilege of writing a paper with I. I. Hirschman, Jr., on ~p Fourier transforms on groups and a paper with John H. Williamson on Dirichlet series. Both collaborations gave me a lot of pleasure. A casual remark made in Moscow to Dusa McDuff led to a paper with her in Mat. Sbornik that has not received the attention it deserves. It shows that M(G) for an infinite compact non-Abelian group G can have infinite-dimensional simple homomorphic images: a startling result in view of the docile behavior of ~X(G). Robert Edwards, who now leads a secluded life in Canberra, has been a good friend since 1952. We have collaborated frequently, with Robert applying his great powers to whatever problem is at hand. Gavin Brown of the University of N e w South Wales has taught me a lot of classical analysis and neoclassical analysis on groups. Our labors have resulted in three papers. Shozo Koshi of the University of Hokkaido is the hardest worker I've ever met. We have collaborated happily on two papers. My final Ph.D. student, Nakhl6 Asmar, with unrivaled generosity asked me to be his co-author on what is really his work, the last item in my bibliography. Teaching has been a vital part of my career. I love to get in front of a class or a colloquium a u d i e n c e - - a n y audience, in fact. Professor Stone will forgive me for quoting one of his great aphorisms: "Teaching is a form of public entertainment, much like acting, which it closely resembles." In 1984-1986 I taught a two-year honors analysis class at the University of Washington. These splendid young people were the best undergraduates I ever taught. I recall especially Eric Stromberg and Marrena Lindberg, w h o have extraordinary talents in several fields and are blessed with ebullient personalities. NakhM Asmar took his Ph.D. in June 1986. I knew that I w o u l d never again get students like these. I found that I could feather my nest nicely by taking partial retirement in 1986. [Did y o u hear about the chap w h o always gave his parties in the basement? He liked to nether his fest.] Other straws in the wind whispered plainly that it was time to go. And so I retired on 1 July 1986. Since then I've taught part-time at the University of Washington, travelled and lectured in eleven countries, and done a certain amount of research. I am learning Chinese. In May 1988 my friends put on the HEWITTFEST. Here I am, still having fun and grateful for every new experience. As my dear friend Ursula Leonhardtyon Barchewitz says: Jeder Tag ist ein Geschenk.

Department of Mathematics University of Washington Seattle, WA 98195 USA THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 3, 1990

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Steven H. Weintraub* For the general philosophy of this section see Vol. 9, No. 1 (1987). A bullet (e) placed beside a problem indicates a submission without solution; a dagger (t") indicates that it is not new. Contributors to this column who wish an acknowledgement of their contribution should enclose a self-addressed postcard. Problem solutions should

be received by 1 November 1990. This column will have a new editor as of Vol. 13, No. 1 (1991). Problem solutions and other correspondence should be directed to David Gale, Department of Mathematics, University of California, Berkeley, CA 94720 USA.

Problems Accompanying remarks: Ana 90-5 by the Column Editor W e are all familiar w i t h the r e m a r k of F e r m a t accomp a n y i n g the s t a t e m e n t of the Fermat conjecture: I h a v e d i s c o v e r e d a w o n d e r f u l p r o o f of this fact b u t this m a r g i n is too small to contain it. Readers are invited to s u b m i t r e m a r k s a c c o m p a n y i n g the s t a t e m e n t s of other f a m o u s conjectures or t h e o r e m s .

Minimizing a form: Problem 90-6 by Stephen Wynn (Brighton, England) Let p be a p r i m e c o n g r u e n t to 1 m o d u l o 3. For an integer n, let ~ -= n ( m o d p), 0 ~ ~ < p. Let w satisfy w 3 1 (rood p), w ~6 1 ( m o d p), a n d set v = w 2. Let x 0 = m i n {max(-d,h-~,a-b-) I a = 1 . . . . . Yo = XoW, Z0 =

p - 1},

XoV.

L e t f(x,y,z)

=

x2

+ y2 +

Z 2 __ x y

-- y z

- - XZ. S h o w

that

f(xo, Yo,Zo) = p.

*Column editor's address: Department of M a t h e m a t i c s , State University, Baton Rouge, LA 70803-4918 USA. 64

Louisiana

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Untangling DNA De Witt Sumners

Some History Knot theory--a new kind of applicable mathematics! Some might scoff at the idea, believing knot theory to be a prime example of esoteric pure mathematics, or (worse yet) recreational doodling for terminally bored topologists. My aim is to convince you that knot theory is yet another example of what Edward E. David, Jr., [D] eloquently terms "the seemingly inevitable utility of mathematics conceived symbolically without reference to the real world." I will freely admit that knot theory did not begin as pure mathematics. Knot theory began as applied mathematics, and now, in its second century, is enjoying new applications in science that go back to its beginnings. The family tree of modern knot theory has its roots in nineteenth-century physics, beginning with the work of Gauss on computing inductance (linking numbers) in a system of linked circular wires. Gauss's student Listing also thought about knots and was the person who coined the word topology. Things really got started in 1867 with Kelvin's model of the atom [T]. In his model, an atom was a configuration of linked vortex tubes in the ether. Kelvin's idea was based on the work of Helmholtz, who proved that, once created, a vortex tube (an invariant solid toms) in the flow of a perfect fluid would be immortal. Kelvin was spurred on by seeing the smoke ring experiments of Tait, in which the rings underwent elastic collisions, exhibiting interesting modes of vibration. Kelvin wanted to produce a kinetic theory of gases, a theory that would explain multiple lines in the emission spectrum of various elements. A swirling vortex tube would absorb and emit energy at certain fundamental frequencies; linked vortex tubes would explain multiple spectral lines. The main advantage of the Kelvin atom over the Lucretius-Newton atom was that the indivisible bits of the Kelvin atom (the vortex tubes) would be held together by the forces of topology (linking), avoiding the problems inherent in

devising forces to hold together an atom made up of little billiard balls. Tait soon discovered the main disadvantage of the Kelvin model--with no pure mathematics (algebraic topology) to help out, knots and links are impossible to characterize. In true pioneering spirit, Tait set out to build a table of the elements--a knot table--and the rest is history. Knot theory is the study of entanglement and symmetry of elastic graphs in 3-space. It is a study of embedding-pathology and has proven to be fundamental as a laboratory for the development of algebraic-topological invariants and in the understanding of the topology of 3-manifolds. During the last 100 years, topologists have developed the discrete geometric language of knots to a fine mathematical art [BZ,K]. Recently, the unexpected (coming from Von Neumann algebras, which came from quantum mechanics) discovery by Vaughan Jones U] of new polynomial invariants which help with the knot classification problem has brought intense attention to the subject

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[L], spawning that might be called the new "combinatorial" knot theory. Whatever it is, however, knot theory isn't just pure mathematics anymore. It is a prototype of what Lynn A. Steen [St] calls the scienceof patterns: "theory built on relations among patterns and on applications derived from the fit between pattern and observation." The precise descriptive and calculational power of knot theory has recently been put to work in the description and computation of molecular configurations [W, Si]. One of the most interesting new scientific applications of topology is the use of knot theory in the analysis of DNA experiments [S,WC,WM]. This article will focus on the tangle model [S1,ES] for site-specific recombination, which utilizes some state-of-the-art pure mathematics (the Cyclic Surgery Theorem [CG], the Knot Complement Theorem [GL], and Property R [G]) to help in the solution of tangle equations posed by DNA experiments.

these vital life processes, manipulate cellular DNA in topologically interesting and nontrivial ways as shown in Figure 1. These enzyme actions include promoting w r i t h i n g (coiling up) of the DNA molecules a n d passing one DNA molecule through another by an enzyme-bridged transient break in one of the molecules (switching a crossover). Enzymes that promote recombination (recombinases) function by breaking a pair of DNA strands and recombining them to different ends (smoothing a crossover). If one correctly regards DNA as very thin string, these enzyme actions are the very stuff of which the new combinatorial knot theory is made.

The Topological Approach to Enzymology

In order to describe and understand what these enzymes are doing, it is clear that a dose of knot theory ought to help. An interesting development for topology has been the recent (circa 1983) emergence of a new experimental protocol, the topological approach to enzymology [WC], which aims to exploit knot theory directly to unravel the secrets of e n z y m e action. Here's how it works. Focus attention on an enzyme that mediates a local DNA interaction. Because there is at present no direct observational method (either in the cell or in a laboratory) for enzyme action, one must rely on indirect methods. One can deduce facts about enzyme mechanism by detecting a topological enzyme signature, the change the enzyme causes in the topological state (embedding) of the molecule upon which it is acting. In many cases, the natural substrate for the enzyme action is linear DNA. The problem for the molecular detective is that linear DNA cannot trap topological changes caused by an enzyme m there can be no interesting (observable) topology (knots) in an unconstrained linear piece of string. The trick is to get a particular enzyme to act on circular DNA molecules. This can be done by manufacturing (via cloning) artificial circular molecules with which the enzyme will react. When an enzyme acts on circular DNA moleo cules, some of the enzymatic changes can be trapped in the form of DNA knots and links. One performs laboratory (in vitro) experiments, in which a high concentration of purified enzyme is reacted with a large collection of circular molecules (the substrate). In such experiments, it is possible to control the amount of supercoiling (Figure 3), the knot type, and the linking of the family of substrate molecules. Using a new biological technique (rec A coating) to enhance viewing under the electron microscope, one can observe the reaction product, an enzyme-specific family of DNA knots and links. The mathematical problem: given the

DNA molecules are long and thin, and the packing of DNA into the cell nucleus is very complex. If one scales up the cell nucleus to the size of a basketball, the DNA inside scales up to the thickness of thin fishing line, with 200 km of that line inside the nuclear basketball. Most cellular DNA is double-stranded (duplex), consisting of two linear backbones of alternating sugar and phosphorus. Attached to each sugar molecule is one of the four bases: A = Adenine, T = Thymine, C = Cytosine, G = Guanine. A ladder is formed by h y d r o g e n bonding between base pairs, with A bonding with T, and C bonding with G. The base-pair sequence (or code) for a linear segment of duplex DNA is obtained by reading along one of the two backbones and is a word in the letters {A,C,G,T}. In the classical Crick-Watson double helix model for DNA, the ladder is twisted in a right-hand helical fashion with an average and nearly constant pitch of approximately 10.5 base pairs per full helical twist. The local helical pitch of duplex DNA is a function of the base-pair sequence, and if a DNA molecule is under stress or constrained to live on a surface, the helical pitch can change. Duplex DNA can exist in nature in dosed circular form, where the rungs of the ladder lie on a twisted cylinder. Duplex DNA in vivo (in the cell) is usually a linear molecule but constrained by periodic attachment to a protein scaffold. The packing, twisting, and topological constraints all taken together mean that topological entanglement poses problems for the DNA molecules in the cell nucleus. This entanglement would interfere with, and be exacerbated by, the vital life processes of replication, transcription, and recombination. The biological solution to this entanglement problem is the existence of enzymes (topoisomerases) which, in order to mediate 72

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Knot theory began as applied mathematics, and now, in its second century, is enjoying new applications in science that go back to its beginnings.

geometry (supercoiling) and topology (knot type) of both the substrate and the product families, deduce the enzyme mechanism. This is a piece of detective work, with knot theory doing the detecting. By understanding the behavior of an enzyme in a controlled laboratory situation, scientists are one step closer to understanding its behavior in the cell. In many cases, an experiment begins with a large collection of unknotted and unlinked supercoiled circular DNA substrate molecules. The product of a reaction is a family of enzyme-specific DNA knots (and/or links). The first hurdle to be cleared on the way to observing these reaction products is to fractionate the product family according to knot/link type. This is achieved by the process of gel electrophoresis, in which the negatively charged DNA molecules are forced through a resistive medium (the protein gel) under the influence of an electric field. The gel mobility of DNA molecules can discriminate between very subtle differences in DNA molecular weight and/or shape. Molecules (of the same molecular weight) that are geometrically similar have the same velocity through the gel, and at the end of a gel run, these molecules are grouped together in discrete gel bands. In many experiments, conclusions are drawn directly from the band structure at the end of a gel run; the problem is to build a model that explains the gel band structures.

Figure 1. Topoisomerase action showing Topo I (a), Topo II (b), and Recombinase (c). The lines represent DNA backbones, and Crick-Watson helical twist is omitted.

In order to employ the topological method, however, one must observe the DNA knot and link products, so the circular molecules are removed from the gel. After removal, the molecules are prepared for viewing under the electron microscope by coating them with a protein called rec A. This coating increases the DNA diameter from 20 to 100 Angstroms, stiffens the molecule, and lengthens it by r e d u d n g the helical pitch of the core DNA. This fattening and stiffening facilitates the unambiguous determination of crossovers in an electron micrograph of a configuration of DNA cirdes and reduces the number of extraneous crossovers. This new precision in the determination of reaction products opens the door for detailed mathematical analysis, the building of topological models for enzyme action.

The Geometry of DNA The mathematical idealization of it as a flexible embedded atoms are vertices, the bonds dings of the same graph are

of a molecule is to think molecular g r a p h - - t h e are edges. Two embedequivalent if there is an

If one scales up the cell nucleus to the size of a basketball, the DNA inside scales up to the thickness of thin fishing line, with 200 km of that line inside the nuclear basketball. orientation-preserving homeomorphism of R3 which, upon restriction, takes one embedding to the other. Equivalently, there is a motion of 3-space (an ambient isotopy) that moves one embedded graph to a position congruent to the other. A pair of nonequivalent erabeddings of a molecular graph form a pair of topological isomers (topoisomers). In molecular biology, an enzyme that manufactures topoisomers is called a topoisomerase. Type I topoisomerase (TOPO I) acts via the passage of single or double-stranded DNA through an enzyme-bridged transient break in single-stranded DNA (Figure la). Type II topoisomerase (TOPO II) acts via the passage of double-stranded DNA through an enzyme-bridged transient break in double-stranded DNA (Figure lb). Topoisomerase is ubiquitous--the cells of all prokaryotic organisms (such as bacteria) and all eukaryotic organisms (such as humans) manufacture both T O P O I and TOPO II. The elucidation of differences between prokaryotic and eukaryotic topoisomerases, between T O P O I and TOPO II, and between topoisomerase produced by different cell lines has used (and will continue to use) geometry and topology in non-trivial ways [Wa]. The topological approach to enzymology uses knots and links to detect enzyme action on circular DNA. It has its roots in the study of topoisomerase action on unknotted circular DNA, which uses linking numbers THE MATHEMATICAL 1NTELLIGENCER VOL. 12, NO. 3, 1990 7 3

Figure3. Twist and writhe: One twist, no supercoils (a) and no twists, one supercoil Co). James H. White relating these three quantities is the Conservation Law [P]: Lk(~) = Tw(~R) + Wr(~). figure ~. ~lgn convennons: nvvon onenmnon ta) ana crossover signs (b). of the two backbone strands of circular duplex DNA to detect the difference between T O P O I and TOPO II. The differential geometry of smooth ribbons in R 3 plays a fundamental role in the analysis of the geometry of circular duplex DNA [BC]. A circular duplex DNA molecule can be modelled as a smooth ribbon in R 3, that is, a smooth embedding f: S1 x [-1,1] R 3. The axis of 9t is flS 1 x {0}). Orient the axis, and orient the two boundary components of 3~ parallel to the axis of 9t, as shown in Figure 2a. One can devise differential-geometric invariants (not topological invariants) of ~ , the twist (which measures the amount of twisting of the ribbon about its axis), and the writhe (which measures the non-planarity of the axis of ~). With the crossover sign convention of Figure 2b, the signed crossover number of a planar projection of an oriented curve in R 3 is the sum of all of the signed crossover numbers and is independent of the orientation of the curve. The writhe of ~ (denoted Wr(~R)) is the average value of the signed crossover n u m b e r of planar projections of the axis of 9t, averaged over all planar projections. The linking number of ~ (denoted Lk(~)) is the homological linking number of the oriented 2-component link 3~ and is computed from any planar projection of O~ as half of the sum of all signed crossovers b e t w e e n the circular c o m p o n e n t s (selfcrossings are ignored). The twist of 9t (denoted Tw(~R)) is the integral of the incremental twist of ~ about its axis, integrated around the axis. In Figure 3a the axis of ~ is planar, and so Wr(~) = 0, and Lk(~) = Tw(~) = - 1. One can convert by ambient isotopy the local left-hand twist of the ribbon of 3a to a global negative supercoil of the ribbon of Figure 3b, in which Wr(~) = Lk(~) = - 1, and Tw(~) = 0. The result of fundamental importance proved by 74

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For a duplex DNA ribbon ~, Tw(~) is (approximately) a constant, so a change in Lk(~) due to toposiomerase action is converted to a change in Wr(~t) (some supercoils): ~Lk(~) = aWr(~R). Passing one strand of DNA through another is a very "small" move, involving a small region of the molecule, and is not in principle observable. A supercoil, on the other hand, is much larger (about 160 base pairs) and is observable in terms of gel mobility. The locally constant twist of D N A in effect magnifies T O P O I and II action on circular duplex DNA to a level observable in a gel. In a gel, adjacent bands in a Topo I experiment on unknotted circular duplex DNA are interpreted to represent molecules that differ by one in Wr (number of supercoils), or equivalently, molecules that differ by one in Lk. When TOPO I and TOPO II gels are compared, the TOPO II gel bands line up opposite the even TOPO I gel bands and are interpreted to represent molecules that differ by two in Wr (or Lk). In this manner, differential geometry plays a critical role in interpretation of experimental results on circular duplex DNA which differentiate between Topo I and Topo II [Wa].

Site-Specific Recombination Site-specific recombination is one way nature alters the genetic code of an organism, either by moving a block of DNA to another position on the molecule (an action performed by transposase) or by integrating a block of alien DNA into the molecule (an action performed by integrase). A recombination site for a site-specific recombination enzyme (the recombinase) is a short linear piece of duplex DNA whose code is recognized

Figure 4. Recombination synaptic complex, direct repeats. A single line represents duplex DNA; supercoiling is omitted. by the enzyme. A pair of sites (either on the same or on different molecules) is juxtaposed in the presence of the enzyme. The juxtaposition of the pair of sites is the result of enzyme manipulation of the DNA, or random thermal motion, or both. The juxtaposed sites are then bound by the enzyme. This stage of the reaction is called synapsis, and the complex formed by the substrate together with the binding enzyme is called the synaptic complex. In a single recombination event, the enzyme performs a double-stranded break at each site, recombines the ends differently, and then releases the resulting molecule(s). The local schematics of a single site-specific recombination event are shown in Figure lc. We call the unbound DNA molecule(s) before recombination takes place the substrate and the unbound DNA molecule(s) after recombination, the

product. The process of site-specific recombination clearly involves some interesting topological changes in the substrate. In order to trap some of these topological changes, one creates circular duplex DNA substrate molecules with both recombination sites on the same molecule. The linear base-pair sequence induces a local orientation at each site, and the local orientation of each site induces a global orientation on the ambient circle. If the global orientations induced by the two sites agree, this configuration is called direct repeats, and if the orientations disagree, this configuration is called inverted repeats. If the substrate is a single circle with direct repeats, the recombination product is a pair of circles and can form a DNA link (or catenane) (Figure 4). If the substrate is a pair of circles with one site each, the product is a single circle and can form a DNA knot (Figure 4 read in reverse). If the substrate is a single circle with inverted repeats, the product is a single circle and can form a DNA knot (Figure 9).

The Mathematics of Tangles and 4-Plats In his seminal paper in knot theory [C], John H. Conway introduced the concept of a (2-stfing) tangle.

Figure 5. Rational tangles. Consider n o w the unit ball D in R 3 = xyz space. The equator o n S 2 = OD is the intersection of S2 with the xy plane. Thinking of the positive y axis as north, and the positive x axis as east, label the four compass points {NE, NW, SW, SE} on the equatorial circle. A 2-string tangle (or just tangle) will denote any configuration (B,t) of two unoriented arcs (t) embedded in a 3-ball (B), satisfying the following conditions: (i) the arcs meet the boundary of B in endpoints, and (ii) there is a fixed ofientation-preserving homeomorphism from B to D that takes the 4 arc endpoints to the 4 distinguished equatorial points. By means of the homeomorphism from B to D, we can consider any two tangles as lying in D. Two tangles are isomorphic if it is possible to superimpose the arcs of one tangle upon the arcs of the other by means of ambient isotopy in the interior of D, leaving the b o u n d a r y pointwise fixed. Equivalently, tangles (D, tl) and (D, t2) are isomorphic if and only if there exists an orientation-preserving homeomorphism H: (D, tl) --~ (D, t2) which is the identity on OD. Mathematically, there is a well-understood class of tangles that look like DNA micrographs and are created by twisting pairs of strands about each other. These tangles are called rational tangles and were classified up to isomorphism by Conway. The tangles of Figures 5a and 5b are trivial; a tangle is rational if and only if it can be converted to a trivial tangle by an ambient isotopy of D that moves the boundary. A rational tangle can be (non-uniquely) represented by a vector (a 1. . . . . a,) with integer entries, each entry corresponding to a number of vertical or horizontal halftwists, the last half-twist on the fight (coded by a n, whose strings connect to NE and SE) being horizontal. For any rational tangle (other than the four exceptional THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 3, 1990 7 5

Figure 6. Other kinds of tangles: locally knotted (a) and prime (b). tangles {(0), (0,0) and ( ___1)}, there is a canonical vector representative (called the Conway symbol for the tangle) with lall 1> 2, and all nonzero entries of the same sign. This canonical vector representative corresponds geometrically to a minimal alternating projection for the tangle. We will take {(0), (0,0) and ( - 1 ) } as the Conway symbols for the four exceptional tangles. The entries of any vector representative can be used in a continued fraction calculation to compute a rational number [3/o~ ( Q u {~}, which classifies the tangle up to isomorphism: ~/(x = an + 1~(an-1 +

1/(an-2 + . . . ) ) .

In general, tangles can be divided into three classes. We say that a tangle is locally knotted if it can be isotoped so as to isolate a local knot in either of its strands (Figure 6a). A tangle is prime [L1] if it is neither locally knotted nor rational (Figure 6b). Given a pair of tangles (A and B), there are two constructions that can be performed on them. One is tangle addition, in which the NE point of A is joined to the NW point of B, and the SE point of A is joined to the SW point of B, forming the tangle A + B (Figure 7a). In general, the operation of tangle addition is associative but not commutative, and A + B can fail to be a rational tangle, e v e n if both A and B are rational. Another operation is the numerator construction, in which the NW point of A is joined to the NE point of A and the SW point of A is joined to the SE point of A, forming the knot (or 2-component link) N(A) (Figure 7b). If A and B are tangles, then A and B are called summands of N (A + B). Closely related to rational tangles is a large class of knots and links of two unknotted components, the class of 4-plats (2-bridge knots and links). Some 4-plats are shown in Figure 8. If A and B are rational tangles, then N (A + B) is a 4-plat. Like rational tangles, 4-plats have vector representatives [c1. . . . . C2k+1] with integer entries, where the integers code for half-twists between strings. Every 4-plat (except [0]) has a pair of canonical vector representatives (called the Conway symbols for the 4-plat) with ci I> 1 for all i. One of these canonical representatives is the other read in reverse, 76

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Figure 7. Tangle constructions. and each corresponds geometrically to a minimal alternating projection for the 4-plat. We take [0] as the Conway symbol for the unlink of two unknotted components. Both rational tangles and 4-plats are studied and classified by means of derived 3-manifolds, their 2-fold branched cyclic covering spaces. The tangle A is rational if and only if its 2-fold branched cyclic covering space is a solid torus S1 x D 2. The knot (link) K is a 4-plat if and only if its 2-fold branched cyclic cover is a lens space. If [Q. . . . . C2k+1] is a Conway symbol for a 4-plat, then the 2-fold branched cyclic cover is the lens space L(oL,~), where the lens space indices are computed by the continued fraction:

~/c~ = 1/(Q + 1/(C2 + . . . ) ) . Lens spaces are fundamental objects in the study of 3-manifolds, and the fundamental group of L(ot,f3) is the cyclic group of order o~.

The Tangle Model for Site-Specific Recombination One fundamental observation behind this model is that pairs of DNA strands are often seen in electron micrographs winding about each other. Both rational tangles and 4-plats are formed by twisting pairs of strands about each other, and they look like DNA micrographs. Moreover, most of the products of recombination experiments performed on unknotted substrate are 4-plats. The other fundamental observation is that a pair of sites bound by an enzyme looks like a tangle. Having been struck by these observations, I first began thinking about using tangles to model DNA enzyme action in early 1986. In July of 1986 I attended the Artin's Braid Group conference at Santa Cruz. Just prior to going to Santa Cruz, I discussed m y tangle ideas with molecular biologists Nicholas Cozzarelli and Sylvia Spengler in Berkeley. By this time I had figured out that, given that the tangles of interest were rational, and the products were 4-plats, in certain cases I could manipulate the classifying symbols to solve the tangle equations posed by DNA experi-

ments. While describing the model to Raymond Lickorish at Santa Cruz, we discussed methods for detecting that tangles are rational. Raymond then said the magic words to me: "The Cyclic Surgery Theorem!" How right he was. We will use tangles to build a model that will aim to compute the topology of the synaptic complex in a single recombination event, given the topology of the substrate and product [$1, ES]. In site-specific recombination on circular DNA substrate, two kinds of geometric manipulation of the DNA occur. The first is a global ambient isotopy, in which a pair of recombination sites is juxtaposed in space and the enzyme binds to the molecule(s), forming the synaptic complex. Once synapsis is achieved, the next move is local and due entirely to enzyme action. Within the region controlled (bound) by the enzyme, the substrate is broken at each site, and the ends are recombined. We will model this local move. We model the enzyme itself (or the interior of its sphere of influence) as a 3-ball B3. The two recombination sites (and some contiguous DNA) form a tangle of two arcs in the enzyme ball. During the local phase of recombination, we assume that the action takes place entirely in the interior of the enzyme ball and that the substrate configuration out-

We are now in position to use some brandnew pure mathematics, the recent celebrated result by Cameron Gordon and John Luecke that knots are uniquely determined by their complements. side and on the b o u n d a r y of the ball remains fixed while the strands are being broken and recombined inside the enzyme ball. For symmetry of mathematical exposition, we take the point of view that the recombination event is taking place in the 3-sphere S3 (instead of R3), because the boundary S2 of the enzyme ball separates the pair (S3, substrate) into two complementary tangles identified along their common boundary. If two tangles A and B are identified along their common boundary, the r e s u l t i n g k n o t (link) can be t h o u g h t of as N (A + B). In Figure 9, the dotted circle represents an equatorial circle on the enzyme S2. At synapsis, just prior to recombination, the enzyme S2 divides the substrate into two complementary tangles, the substrate tangle S (outside the enzyme ball), and the site tangle T (inside the enzyme ball). At synapsis, just after recombination, the enzyme S2 divides the substrate into two complementary tangles, the same substrate tangle S, and the recombinant tangle R. So the local phase of sitespecific recombination is modelled as tangle surgery, in which the site tangle T is deleted from the synaptic complex and replaced by the recombinant tangle R. As in Figure 9, the knot type of the substrate and that of

the product each yield a tangle equation in the variables S, T, and R: Substrate Equation: N(S + T) = Substrate Product Equation: N(S + R ) = Product We wish to treat each of S, T, and R as unknown recombination variables and solve the substrate a n d product equations for these u n k n o w n s . Because a single recombination event yields only two equations in three unknowns, the best we can hope for, given only this information, is to solve for any two in terms of a third. The analysis is helped along at this point by making the following biologically reasonable assumption:

Biological Assumption: T and R are enzyme-determined constants, independent of the variable geometry of the substrate S. In some experiments, the substrate may be a large number of circular molecules, all the same knot type, but equipped with varying degrees of supercoiling. Recombination can trap some of this "trivial" geometry, producing a distribution of product DNA knot (link) types from a single substrate knot type. In such an experiment, then, the substrate tangle S can vary over a number of configurations, but the tangles T and R are enzyme-determined constants, and appear in a number of pairs of equations, one for each different product. In other experiments, iterated recombination can occur at a single binding encounter between substrate and enzyme. This means that one can often obtain enough information about S, T, and R to prove that they must be rational, and then to solve the recombination equations for them.

Tn3 Resolvase Tn3 Resolvase is a site-specific recombinase that reacts with certain circular duplex DNA substrate with directly repeated recombination sites [WD]. One begins with supercoiled unknotted DNA substrate and treats it with resolvase. The principal product of this reaction is known to be the DNA 4-plat [2] (the Hopf link), pictured as the recombination product in Figure 4. Resolvase is known to act dispersively in this situation-to bind to the circular DNA, to mediate a single recombination event, and then to release the linked product. It is also k n o w n that resolvase and free (unbound) DNA links do not react. However, once in 20 encounters, resolvase acts processively--additional recombinant strand exchanges are promoted prior to the release of the product, with yield decreasing exponentially with increasing number of strand exchanges at a single binding encounter with the enzyme. Two successive rounds of recombination produce the DNA 4plat [2,1,1] (the figure-8 knot); three successive rounds of recombination produce the DNA 4-plat [1,1,1,1,1] THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 3, 1990 7 7

Figure 8. DNA 4-plats (Tn3) (a) shows the Whitehead link [1,1,1,1,1]; (b) the knot (the Whitehead link, shown in Figure 8a); four successive rounds of recombination produce the DNA 4-plat [1,2,1,1,1] (the knot 62, shown in Figure 8b). The discovery of the DNA knot [1,2,1,1,1] substantiated a model for Tn3 resolvase mechanism [WD]. The electron micrograph of the Whitehead link is from [KS], and the electron micrograph of 62 is from [WD]. One can prove [ES] that the equations determined by the first two rounds of iterated resolvase recombination have four solutions for the tangle pair {S,R}. The third round of iterated recombination is then used to discard three of these solutions. This theorem can be viewed as a mathematical proof of resolvase synaptic complex structure: the model proposed in [WD] is the only explanation for the first three observed products of iterated Tn3 recombination, assuming that iterated recombination acts by adding on copies of the recombinant tangle R.

THEOREM 1: Suppose that tangles S, T, and R satisfy the following equations: (i) N(S + T) = [1] (the unknot); (ii) N(S + R) = [2] (the Hopf link); (iii) N(S + R + R) = [2,1,1] (the figure-8 knot). Then {S,R} = {(-3,0),(1)}, {(3,0),(-1)}, { ( - 2 , - 3 , - 1 ) , ( 1 ) } , or {(2,3,1),(-1)}. The first (and mathematically most interesting) step in the proof of this theorem is to argue that solutions {S,R} must be rational tangles. Now S, R, and (S + R) are locally unknotted, because N(S + R) is the Hopf link, which has two unknotted components. Any local knot in a tangle summand would persist in the Hopf link. Likewise, T is locally u n k n o t t e d , b e c a u s e N(S + T) is the unknot. Let A' denote the 2-fold branched cyclic cover of the tangle A; then OA' = S1 x S1. If A is a prime tangle, then the inclusion homomorphism injects ~I(OA') = Z ~ Z into "~I(A') [L1]. If both A and B are prime tangles, and N(A + B)' denotes the 2-fold branched cyclic cover, then ~I(N (A + B)') contains a subgroup isomorphic to Z @ Z. Because no cyclic group contains Z ~ Z, no 4-plat has two prime summands. This means that if A and B are locally unknotted tangles, and N(A + B) is a 4-plat, 78

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[1,2,1,1,1].

then at least one of A and B must be a rational tangle. From equation (ii) above, we conclude that at least one of {S,R} is rational. Suppose that S is rational and that R is prime. Given that N((S + R) + R) is a knot, one can argue ILl} that (S + R) is also a prime tangle. From equation (iii), we then have that the 4-plat [2,1,1] admits two prime summands, which is impossible. We t h e r e f o r e conclude that R m u s t be a rational tangle. The next step is to argue that S is a rational tangle. S14ppose that S is a prime tangle. Then T must be a rational tangle, because N(S + T) is the unknot. Passing to 2-fold branched cyclic covers, we have that N(S + T)' = S3, and T' is homeomorphic to S1 x D2, so S' is a bounded knot complement in S3. We know that R is a rational tangle, and can argue that equation (iii) implies that (R + R) is likewise rational. Again passing to the 2-fold branched cyclic covers of equations (ii) and (iii), we obtain the equations N(S + R)' = L(2,1), and N(S + (R + R))' = L(5,3). Because R' and (R + R)' are each homeomorphic to a solid torus S1 x D 2, this means that there are two attachments of a solid torus to S' along OS' = S1 x S1, yielding the lens spaces L(2,1) and L(5,3). The process of adding on a solid torus along its boundary is called Dehn Surgery, and the Cyclic Surgery Theorem [CG] now applies to this situation to argue that, since the orders of the cyclic fundamental groups of the lens spaces differ by more than one, the only w a y this can happen is for S' to be a Seifert fiber space, hence a torus knot complement. Fortunately, the results of Dehn surgery on torus knot complements are well understood, and one can show that in fact S' must be the complement of the unknot (a solid torus), which means that S is a rational tangle. The proof now amounts to computing the rational solutions to equations (ii) and (iii), exploiting the dassifying schemes for rational tangles and 4-plats. Claus Ernst and I worked out all the details of the above rationality argument and developed what we call a "calculus for rational tangles" to perform such calculations [ES]. O n e can use this calculus of symbols to solve equations (ii) and (iii), obtaining the four solutions

above. Because each of the unoriented 4-plat products in equations (ii) and (iii) is achiral (equivalent to its m i r r o r image, o b t a i n e d by r e v e r s i n g all of the crossings), given any solution set {S,R} to equations (ii) and (iii), its mirror image { - S , - R} must also be a solution. So the mathematical situation, given equations (i)-(iii), is that we have two pairs of mirror image solutions for {S,R}. In order to decide which is the biologically correct solution, we must utilize more experimental evidence. The third round of iterated resolvase recombination determines which of these four solutions is the correct one: THEOREM 2: Suppose that tangles S, T, and R satisfy the following equations: (i) N(S + T) = [1] (the unknot); (ii) N(S + R) = [2] (the Hopf link); (iii) N(S + R + R) = [2,1,1] (the figure-8 knot); (iv) N(S + R + R + R) = [1,1,1,1,1] (the Whitehead link). Then, S = ( - 3 , 0 ) , R = (1), and N(S + R + R + R + R) = [1,2,1,1,1]. Of the four solutions in Theorem 1, only {S,R} = {(-3,0),(1)} is a solution to equation (iv). The correct global topology of the first round of iterated Tn3 recombination on the u n k n o t is shown in Figure 4. Moreover, the first 3 rounds of iterated Tn3 recombination uniquely determine N(S + R + R + R + R), the result of 4 rounds of iterated recombination. It is the 4-plat knot [1,2,1,1,1] (62 in the knot tables), and this DNA knot has been observed (Figure 8b).

Phage k Integrase Bacteriophage lambda is a virus that infects bacteria, inserting its own genetic material into that of the host. The genetic insertion mechanism is site-specific recombination mediated by the recombinase Int. In order for in vivo insertion to take place, one recombination site is on a circular viral DNA molecule, and the other site is on a host DNA molecule. To employ the topological method to study Int recombination in vitro, one synthesizes unknotted duplex circular substrate with both Int sites on the same molecule. The first experiment we discuss [SS] was performed on both re-

Figure 9. Substrate and product equations (inverted repeats).

laxed (no supercoiling) and supercoiled u n k n o t t e d DNA with direct repeats (producing DNA links), and the second experiment [SS] was performed on both relaxed and supercoiled unknotted DNA with inverted repeats (producing DNA knots). Although Int does not perform iterated recombination, one round of recombination produces a remarkable family of reaction products, the family of (2,k) torus knots and links. Specifically, for inverted repeats, the family of 4-plat DNA knots {[-(2k + 1)] [ 0 ~ k ~ 11} was observed. The relaxed substrate produced the unknot [ - 1 ] and the trefoil [ - 3 ] , and the supercoiled substrate produced the knots of higher crossover number. A hypothetical synthesis for the Int knot [ - 5 ] is shown in Figure 9. For direct repeats, the family of 4-plat DNA links {[-2k] [ 0 ~ k ~ 11} was observed, with relaxed substrate producing the unlink [0] and the Hopf link [ -2], and supercoiled substrate producing the links of higher crossover number. The Int knot [ - 13] and the Int link [ - 4 ] from [SS] are shown in Figure 10.

Figure 10. DNA 4-plats (Int). Left photograph shows the knot [ - 13]; right photograph shows the link [ - 4]. THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 3, 1990 7 9

THEOREM 3: Suppose that there exist tangles S k (k = 0,1) which satisfy the following: (i) N(S k + T) = [1] (the unknot), k = 0,1, and (ii) N(Sk + R) = [ - ( 2 k + 1)], k = 0,1. Then T and So are rational tangles. As in the proof of Theorem 1, one argues that T, R, and Sk are locally u n k n o t t e d (k = 0,1). If T is a prime tangle, then Sk is rational, k = 0,1. At the level of 2fold branched cyclic coverings, this means that T' is a knot complement that admits two distinct Dehn surgeries that produce S3. We k n o w that the surgeries are different because of the equations in (ii) of the hypothesis. We are n o w in position to use some brandn e w pure mathematics, the recent celebrated result by C a m e r o n G o r d o n a n d John Luecke [GL, Ci,Li] that knots are uniquely determined by their complements. Gordon and Luecke prove that there is only one way to do Dehn surgery on a nontrivial knot complement to produce S3--the obvious one, to p u t the solid torus back in the same w a y it came out. This means that T' m u s t be a trivial knot complement, a solid torus, and that the tangle T is rational. A similar analysis proves that So is likewise rational. If one than adds another experimental result to the hypotheses of Theorem 3, namely that N(S 2 + R) = [ - 5 ] , one can s h o w (in a m a n n e r similar to the proof of Theorem 1) that R must also be a rational tangle. The analogous theorem holds for the case of direct repeats a n d uses the recent proof of Property R for knots [G] to prove that the analogue of SOis a rational tangle. The remaining mathematical problem is that there does not seem to be e n o u g h experimental information available at the time of this writing to prove that Sk is rational for 1 ~< k ~< 11, and to nail d o w n T, R, a n d Sk uniquely. Molecular biologists believe that Int recombination proceeds as in Figure 9. In a n y event, I hope that you m a y n o w have some r e a s o n to q u e s t i o n the f o l l o w i n g s t a t e m e n t [Ci]: "'Gordon a n d Luecke's solution [GL] to Tietze's 80year-old problem has no immediate practical or even theoretical applications." The author gratefully acknowledges research support from the United States Office of Naval Research a n d the National Science Foundation, and the hospitality of the Mathematical Sciences Research Institute.

[D] [ES] [G] [GL] [J] [K] [KS]

[L] [L1] [Li] [P] [Si] [SS]

[St] [S]

[$1]

[T] [W] [Wa] [WM]

References [BC] W.R. Bauer, F. H. C. Crick, and J. H. White, Supercoiled DNA, Scientific American 243 (1980), 100-113. [BZ] G. Burde and H. Zieschang, Knots, Berlin, de Gruyter (1985). [Ci] B.A. Cipra, To have and have knot: when are knots alike?, Science 241 (1988), 1291-1292. [C] J.H. Conway, On enumeration of knots and links, and some of their algebraic properties, Computational Problems in Abstract Algebra, Proc. Conf. Oxford 1967, Pergamon (1970), 329-358. [CG] M.C. Culler, C. M. Gordon, J. Luecke, and P. B. Shalen, Dehn surgery on knots, Ann. of Math. 125 (1987), 237-300. 80

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[WC] [WD]

E.E. David, Jr., Renewing U.S. mathematics: an agenda to begin the second century, Notices of the A.M.S. 35 (1988), 1119-1123. C. Ernst and D. W. Sumners, A calculus for rational tangles: applications to DNA recombination, preprint, Florida State University (1988). D. Gabai, Foliations and surgery on knots, Bull. A.M.S. 15 (1986), 83-97. C.M. Gordon and J. Luecke, Knots are determined by their complements, preprint, University of Texas (1988). V . F . R . Jones, A polynomial invariant for knots and links via Von Neumann Algebras, Bull. A.M.S. 12 (1985), 103-111. L . H . Kauffman, On knots, Ann. of Math. Studies 115, Princeton Univ. Press (1987). M.A. Krasnow, A. Stasiak, S. J. Spengler, F. Dean, T. KoUer, and N. R. Cozzarelli, Determination of the absolute handedness of knots and catenanes of DNA, Nature 304 (1983), 559-560. W . B . R . Lickorish, Polynomials for links, Bull. L.M.S. 20 (1988), 558-588. W . B . R . Lickorish, Prime knots and tangles, Trans. A.M.S. 267 (1981), 321-332. R. Lipkin, Unraveling more than just knots, Insight 4 (Oct. 24, 1988), 54-55. W . F . Pohl, DNA and differential geometry, The Mathematical Intelligencer 3 (1980), 20-27. J. Simon, Topological chirality of certain molecules, Topology 25 (1986), 229-235. S.J. Spengler, A. Stasiak, and N. R. Cozzarelli, The stereostructure of knots and catenanes produced by phage ~ integrative recombination: implications for mechanism and DNA structure, Cell 42 (1985), 325-334. L . A . Steen, The science of patterns, Science 240 (1988), 611-616. D.W. Sumners, The role of knot theory in DNA research, Geometry and Topology, Manifolds, Varieties and Knots (C. McCrory, T. Schifrin, eds.), New York: Marcel Dekker (1987), 297-318. D.W. Sumners, Knots, macromolecules and chemical dynamics, Graph Theory and Topology in Chemistry (R. B. King and D. H. Rouvray, eds.), Studies in Physical and Theoretical Chemistry 51, New York: Elsevier (1987), 3-22. W. Thompson, On vortex atoms, Philosophical Magazine 34 (July, 1867), 15-24. D.M. Walba, Topological stereochemistry, Tetrahedron 41 (1985), 3161-3212. J.C. Wang, DNA topoisomerases, Scientific American 247 (1982), 94-109. J. H. White, K. C. Millett, and N. R. Cozzarelli, Description of the topological entanglement of DNA catenanes and knots by a powerful method involving strand passage and recombination, J. Mol. Biology 197 (1987), 585-603. S. A. Wasserman and N. R. Cozzarelli, Biochemical topology: applications to DNA recombination and replication, Science 232 (1986), 951-960. S. A. Wasserman, J. M. Dungan, N. R. Cozzarelli, Discovery of a predicted DNA knot substantiates a model for site-specific recombination, Science 229 (1985), 171-174.

Department of Mathematics Florida State University Tallahassee, FL 32306 USA

Chandler Davis*

Discrete Thoughts: Essays on Mathematics, Science and Philosophy by Mark Kac, Gian-Carlo Rota, and Jacob T. Schwartz Boston: Birkh/iuser, 1986, xii + 264 pp.; US $65

Reviewed by Lawrence Zalcman "Mathematicians, like Proust and everyone else, are at their best when writing about their first love." Two of the authors of this book, Rota and Schwartz, are past recipients of the American Mathematical Society's Leroy P. Steele Prize; the third, the late Mark Kac, won the Chauvenet Prize of the Mathematical Association of America twice, the only individual ever to do so. All three are (or were) members of the U.S. National Academy of Sciences. In short, the authors are among the very best we have; and their best is very good indeed. Inevitably, one approaches this book with high hopes. Happily, one is not disappointed. Discrete Thoughts is a collection of essays and other occasional pieces on mathematics, the mathematical sciences, philosophy, and education. Of the twentysix essays in the book, six (four of them by Schwartz) deal with computers and computer science, four with philosophy (reflecting Rota's interest in phenomenology), two (by Kac) with statistics, two more (also by Kac) with u n i v e r s i t y education, and one (by Schwartz) with "economics, mathematical and empirical." The remaining eleven essays, on core mathematics and mathematicians, include survey articles, book reviews, the introductions to at least two books, a eulogy, and a couple of installments of Rota's "discrete thoughts," which lend the book its name. The various items here anthologized were written over a period of almost a quarter century, from 1962 to 1985; fully half of them appeared between 1974 and * Column editor's address: Mathematics Department, University of Toronto, Toronto, Ontario M5S 1A1 Canada

1978. For the most part, they have aged gracefully. In any collection of this sort there is bound to be a certain amount of repetition; here, amazingly, it does not jar. In fact, the repetition may even serve a useful function of reinforcement. In any case, w h e n a mathematician writes well, one can forgive him anything, even repeating himself. How does one summarize the argument of a book of essays? Better to say something about the authors, and then let them speak for themselves. Predictably, the style, intellectual and personal, of the individual

"'There is only one source of corruption in mathematics, and that is motivation by anything except curiosity and desire for understanding and harmony.'" authors comes through very clearly in these pieces: Kac is wise; Schwartz is penetrating; Rota is brilliant. Kac is wise. One cannot fail to be impressed by his good sense, his measured judgment, his sense of proportion. Everything he writes is worth reading, pondering. Naturally, that includes the survey articles (especially the masterful "Mathematics: Trends") here reprinted; it also i n c l u d e s his t w o pieces on the university, "'Academic Responsibility" and "Doing Away with Science." Even when he writes about computers, a subject in which he was not an expert, he manages to educate and entertain. Schwartz is penetrating. His "The Pernicious Influence of Mathematics on Science" should be required reading for all scientists and mathematicians. The introductory section of his essay "Economics, Mathematical and Empirical" contains the best single answer to the question "Why is mathematics useful?" that I have ever encountered. And w h e n Schwartz writes on computers, he writes with complete authority.

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Rota is brilliant. H e is a m a s t e r of the e p i g r a m , a n d his short pieces are s t u d d e d with m e m o r a b l e formulations. Even the articles on p h e n o m e n o l o g y are l u c i d ) Rota's eulogy of U l a m is a masterpiece of the genre, with an astonishing conclusion that s o m e h o w calls to m i n d t h e u n f o r g e t t a b l e final p a r a g r a p h of J a m e s Joyce's short sto W " T h e D e a d . " N o w let the a u t h o r s s p e a k for t h e m s e l v e s . H e r e is Kac: The main purpose of professional education is development of skills; the main purpose of education in subjects like mathematics, physics or philosophy is development of attitudes. (p. 12) There is only one source of corruption in mathematics, and that is motivation by anything except curiosity and desire for understanding and harmony . . . . Not even the lofty terms of serving the needs of society and mankind are free from corrupting influences . . . . It is not corrupting either for the science or for the individual to be engaged in a problem which can be applied to physics, astronomy, or even ballistics; what is corrupting is to become engaged in such a problem solely because it can be applied. (p. 13) iT]here are worse things than being wrong, and being dull and pedantic are surely among them. (p. 16) iT]here is nothing wrong with vague questions; it is the combination of vague questions and vague answers that is bad. Many imprecisely stated questions have a tremendous amount of good science in them. (pp. 34-5) Culture like nature has its ecological aspects, and the price for interfering with established equilibria may be catastrophically high. (p. 103) iT]he only mathematical problems which arose during our century and are not traceable to the nineteenth century are conceptual problems generated by computers. (p. 204) A n d Schwartz: That form of wisdom which is the opposite of singlemindedness, the ability to keep many threads in hand, to draw for an argument from many disparate sources, is quite foreign to mathematics. This inability accounts for much of the difficulty which mathematics experiences in attempting to penetrate the social sciences. (p. 22) To find the simple in the complex, the finite in the infinite - - t h a t is not a bad description of the aim and essence of mathematics. (p. 64)

1 Here one m a y register surprise at a curious omission. According to Rota (p. 175), G6del considered E d m u n d Husserl the greatest philosopher since Leibniz, a n d Weyl had a similar opinion. (As ever,

the truth is not so simple. See, for instance, Hao Wang's fascinating study, Reflections on Kurt GOdel [Cambridge: MATPress, 1987].) That mathematicians with philosophical inclinations should find Husserl's philosophy particularly congenial is hardly surprising in view of Husserl's own training as a mathematician: he was a student of Weierstrass. The real mystery is why Rota gives no hint of this additional, and important, mathematical connection. There have, of course, also been philosophers turned mathematicians: Leibniz is (arguably) the most famous example; L. E. J. Brouwer is another. 82 THE MATHEMATICALINTELLIGENCERVOL. 12, NO- 3, 1990

In th[e] quest for simplification, mathematics stands to computer science as diamond mining to coal mining. (p.

64) [S]ince in this area [computer science] it is so hard to distinguish real technical content from the arbitrary effects of corporate investment, salesmanship, and accidents of priority, it is gratifying that a solid theoretical inheritance has begun to emerge from th[e] raw systems material. (p. 109) Concerning artificial intelligence, I think it well to comment with high hope concerning its long-term future, skeptically concerning its accomplishments to date, and in repudiation of the relentless overselling which has surrounded it, which I believe discourages those sober scientific discernments which are essential to the real progress of the subject. (p. 113) The construction of artificial intelligence would affect the circumstances of human life profoundly. The appearance of intelligent beings other than man would surely create a new economics, a new sociology, and a new history. (p. 184)

"'In thle] quest for simplification, mathematics stands to computer science as diamond mining to coal mining.'" A n d , finally, Rota: It is much easier for a mathematician to read a physics book after the physics becomes obsolete, and that is in fact what usually happens. (p. 1) Gifted expositors of mathematics are rare, indeed rarer than successful researchers. It is unfortunate that they are not rewarded as they deserve, in our present idiotic pecking order. (p. 1) Mathematics, like theology and all free creations of the Mind, obeys the inexorable laws of the imaginary. (p. 154) We often hear that m a t h e m a t i c s consists mainly in " p r o v i n g theorems." Is a writer's job mainly that of "writing sentences"? (p. 154) As mathematical logic becomes ever more central within mathematics, its contributions to the philosophical understanding of foundations wane to the point of irrelevance. (p. 167) The Russells, the Spenglers, the Toynbees, and their third-rate cohorts have lowered the understanding of phi-

losophy to a level unseen since the seventh century. (p. 175)

On page 136, lines - 3 to - 1, the phrase " i n d e e d . . . return" is a doublet and should be deleted. On page 234, line - 10, insert "one of" after "of." On page 256, The Galileo of biology is yet to come, and without him the methods of biology will not appreciably differ from those line 7, read "'and" for " a d " ; and on page 259, line of a stamp collector. (p. 179) -13, read "the" for "it." And these do not exhaust If we are to set the new sciences on firm, autonomous, the list! The editors and publisher would do well to formal foundations, then a drastic overhaul of Aristotelian reflect upon what the wisest of men had to say on the logic is in order. (p. 180) subject of errors (Ecclesiastes 9:18, 10:1). Another complaint is that no indication of any sort "'It is much easier for a mathematician to is given as to where the individual pieces of the book read a physics book after the physics becomes originally appeared. This is especially curious in view obsolete, and that is in fact w h a t usually of the detailed pages of acknowledgment (xi-xii) that precede the text. Nor can one find very much good to happens. "" say about the repellent yellow dust jacket, decorated (if that is the word) with random-looking quadrilat[A]nything that happened before Leibniz is not history, erals and triangles and little black squares; at least it but paleontology. (p. 231) can be discarded. But the harshest criticism must be [T]he private lives of some of the greatest philosophers are in pathetic contrast to their writings. Plato bowed to petty reserved for the greed--there really is no other word tyrants, Leibniz was a con man of sorts, Hegel had illegiti- for i t - - o f the publisher in setting the price of a book of mate children, etc. The personalities of some of the best 264 pages of mostly nontechnical text at a whopping philosophers of the Twentieth Century could make a gal- $65. Wisdom's price may be above rubies, but surely lery of horrors. (p. 245) this is going too far! No one has the faintest idea how the process of scientific Priced b e y o n d pocketbooks of ordinary mortals, induction words, and in calling it a "process" we may be Discrete Thoughts is (if my local bookseller is to be bealready making a dangerous assumption. (p. 263) lieved) already out of print. A pity: in paperback, it might well have become a mathematical best seller. Go And finally, my favorite passage in the book: to your library and read it. Many of us remember Old Europe and its mitteleurop~ische Kultur. The sleek blue express trains would pull out from Milan and Rome, Venice and Florence, bearing the tag Department of Mathematics Mitropa, carrying away haughty ladies with guttural ac- Bar-Ilan University cents, who would get busy on their manuscripts as soon 52100 Ramat-Gan as the train set in motion. Their heavenly destinations, Israel Vienna and Berlin, Prague and Stockholm, were pictured as the meccas of exquisite artistic refinement and unfathomable learning, a vision shared with proud superiority by the natives of Mitropa. To them, Americans were blundering infants, Italians frivolous fakers, Englishmen eccentric gentlemen who cultivated Greek or mathematics as a hobby, everyone else a barbarian. Only the French could stand up to them, a painful thorn in the thick layers of their cultural fat, a feared enemy richly endowed with the deadly weapons of wit and elegance. It is hard to believe that this very same world of stifling Kultur and deadening prejudice (now largely wiped out) produced a majority of the great minds of the West . . . . (p. 232) "If you put your best foot forward, you have to put your worst foot last." Discrete Thoughts is a wonderful book, but the authors have been ill-served by their publisher. For one thing, the volume has been sloppily edited. Misspelled words abound: "emphaisis" (p. 78), "'denumberable" (p. 83), "equillibrium" (p. 140), "infallability" (p. 204), "curiousity" (p. 234), "devestation" (p. 237). On page 101, there are actually three errors: "prophesizes" (line 1), "example" (line 3, should be "examples"), and "is" (line 13, should be "are"). On page 102, line - 7 , reference is made to a nonexistent w On page 134, the formula marked ( + ) is incorrect: each m* should be replaced by m +. THE MATHEMATICALINTELLIGENCER VOL. 12, NO. 3, 1990 8 3

Robin Wilson*

International Congress of Mathematicians

As part of the 400th anniversary celebrations of Columbus's voyage to America, a "World Congress of Mathematicians" took place in Chicago in 1893. Fortyfive mathematicians were present; the opening speech was given by Felix Khan. Since that first congress, twenty ICMs have been held, with several thousand attending. Although all past congresses have been held in Europe or North America, the 1990 congress is scheduled for Kyoto.

Only three ICMs have been commemorated by postage stamps--Moscow (1966), Helsinki (1978), and Warsaw (1982/3). The Warsaw set, featuring Wadaw Sierpi~ski and three other eminent Polish mathematicians, was issued in 1982, although the Congress was eventually postponed to 1983; the Banach stamp from this set was illustrated in the "Stamp Corner" of Vol. 8, No. 3 of the InteUigencer.

* C o l u m n editor's address: Faculty of M a t h e m a t i c s , T h e O p e n University, Milton K e y n e s MK7 6AA E n g l a n d 84

THE MATHEMATICALINTELL|GENCERVOL, 12, NO. 3 9 1990Springer-VerlagNew York