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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