JOHN EWING
Opinion
Predicting the Future of Scho ary Pub ishing I believe that the motion picture is destined to revolutionize our educational system and that in a few years it will supplant largely, if not entirely, the use of textbooks. - Thomas Edison, 1 922 It is probable that television drama of high caliber and produced by first-rate artists will materially raise the level of dramatic taste of the nation. -David Sarnoff, 1 939
hen Orville Wright flew his airplane over a small stretch of rolling grassland in 1903, the managing editor of Scientific American1 predicted that thousands of planes would soon fly over every city, delivering patrons to theaters. On the eve of the First World War, two famous British aviators2 argued that planes would prevent wars in the future (because they brought people together). Scientists, engineers, and futurists have always conjectured the consequences of technology. In the case of planes, the experts were right in recognizing that they would profoundly affect our lives in the coming century . . . but they were cer tainly wrong in foretelling what that effect would be. Once again the experts are predicting the future. The di gerati3 tell us that the Internet has changed everything, that technology will revolutionize the way we do business, and that nothing will again be the same. Maybe. But the experts provide few facts to back their predictions, and they preach a digital future as an act of faith rather than a reasoned conclusion. It's hard to tell hype from reality when some one promotes technology with religious zeal. What about scholarly publishing? Here, a special group of experts is predicting (and promoting) the future. The ex perts foretell the imminent collapse of scholarly journals and some advocate revolutionary replacements-refereed This article
IS
postings, e-prints, and overlays. In many countries, gov ernment agencies have embraced these predictions, pro viding support for alternatives-PubMed Central, the Pub lic Library of Science, the arXiv. And experts offer miraculous solutions to previously intractable problems, describing a revolution in scholarly publishing that will pro vide universal free access to scholarship, at no cost to any one. The "free" alternatives seem to be enticing solutions to our present, very real problems. How can we predict the future rather than merely wish for it? Good predictions are difficult without facts-facts about the past and about the present. This sounds obvious, but, amazingly, experts and enthusiasts often dismiss past experience. They argue that because everything will soon change, experience is not relevant. This kind of sophism is especially prevalent in discussions about the Internet, where experts tell us the old rules no longer apply. But they are wrong: Making predictions without facts is mysticism, not science.
based on a talk given at the Conference on Electronic Information and Communication, Tsinghua University, China, August 29-31, 2002. A version of
this paper will appear in the proceedings of that conference.
© 2003 SPRINGER-VERU\G NEW YORK, VOLUME 25, NUMBER 2 , 2003
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Facts
Here are some facts about the current environment in scholarly publishing. Alternatives to journals have been widely publicized, and some of these are remarkably successful. The best known in mathematics are the arXiv (http://www.arxiv.org) and MPRESS (http://mathnet.preprints.org). The former is a repository of papers, and the latter is a distributed sys tem with links to repositories, including the arXiv itself. •
•
•
As of mid-2002, the mathematics section of the arXiv holds approximately 20,000 papers, with about 1 5,000 of those contributed by individuals and the remainder mi grated from previously existing preprint servers. 4 MPRESS has links to about 25,000 papers (including those in the arXiv). Since 1998, mathematicians have contributed 1 2,618 pa pers to the arXiv (through mid-2002). During this same time, Math Reviews indexed more than 280,000 journal articles.
Alternatives to journals are more popular in some fields than in others, but they have played a prominent role in discussions about electronic publishing. In 2001 the Asso ciation of Learned and Professional Society Publishers con ducted a survey of scholars in many disciplines. 5 One part of the survey considered preprint servers. •
•
When asked whether preprint servers were important in their work, about one-third (32%) said yes. (Among physi cists, 55% answered yes.) When asked whether they used preprint servers, 12% of the respondents said they did. (Among physicists, 32% did.)
Most scholars don't understand the scholarly litera ture-not its content but rather its extent and complexity. When mathematicians think about "journals, " they think about the best known and most visible-the ones they scan on the new-journals-shelf in the library. But the mathe matical literature is far more complex and diverse. MR di vides all journals into two classes: those from which every article is either indexed or reviewed (the "cover-to-cover" journals) and those from which articles are selected for in clusion (the "others"). • • • • •
• •
In 2001 MR indexed or reviewed 5 1,72 1 journal articles. 6 Those articles came from 1 , 1 72 distinct journals. In 200 1 , 59 1 (50%) of the journals were "cover-to-cover." That left 581 (50%) classified as "other." And 30,924 (60%) of the articles were in "cover-to-cover" journals. Leaving 20,797 (40%) articles in the "other" journals. This means that 40% of the journal literature is outside the "mainstream" mathematics journals!
Almost all discussions about scholarly communication focus on electronic publishing. There is a recognition that the transition from paper to electronic is proceeding more slowly than first imagined, but almost no one understands how slowly.
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THE MATHEMATICAL INTELLIGENCER
•
•
•
In 2001 only 46 (4%) of the journals covered by MR were primarily-electronic. 7 Only 1,272 (2.5%) of the articles were in primarily elec tronic journals. On the other hand, approximately 34,000 (67%) of all ar ticles had links, meaning that at least that many are avail able in electronic form.
Mathematicians have always know that past literature is important. Because MR recently added reference lists for articles for some journals, it is now possible to make the dependence more precise. The reference lists currently cover journals from 1998 to the present and include 336,201 citations to journal articles. •
•
Of all references, 53% were to articles published prior to 1990. More than 28% were to articles published prior to 1980.
This is especially striking because the number of journal articles increased over time. Examining the number of pa pers covered annually by MR from 1950-1990, the per centage of MR items cited in recent papers varies between one and two percent for almost every year during the en tire period (Fig. 1). Many scholars have commented about the high cost of commercial journals, but few have noted their number and size. They are gradually dominating the scholarly literature. • • •
In 2001 only 349 (30%) of the journals were commercial. Yet they published 25,008 (48%) of the articles! Moreover, looking back ten years, we see a clear trend. In 1991 only 24% of the journals were commercial, pub lishing 38% of the articles.
What drives the expansion of commercial journals? While most scholars concentrate on the costs ofjournals, revenues are the crucial figures in understanding journal economics. •
•
A rough estimate8 suggests that the revenue from each article in commercial journals is about $4,000 (which may be off by a factor of 2.) Therefore, the 25,000 math ematics articles in commercial journals in 2001 gener ated about $ 100 million in revenue for commercial pub lishers. An even rougher estimate suggests that for the noncom mercial journals, each article generates about half as
Citations
Percent of MR items
o.oo% -Mmmmmmmm-m-m-m-m-m-m-mm-mmMI! 1990 1980 1970 1960 1940 2000 1950
i@tijil;iiM
•
much revenue. Even for these, therefore, the total rev enue was about $50 million in 2001. (Again, this may vary by a factor of 2.) And it's important to remember that mathematics is only a small fraction of all scholarly publishing. There are about 25,000 journals in science, technology, and medi cine alone. 9 Just one commercial publisher, Elsevier, de rives more than a billion dollars in revenue from its sci ence journals.
These are the facts: Many scholars (although not most) promote alternatives to journals, but many fewer actually use them. Journals continue to dominate the scholarly lit erature in mathematics. Almost all journals are in both pa per and electronic format, and almost none are electronic only. The journal literature is highly dispersed, contained in many journals, including those that cover disciplines out side mathematics. The older literature is extremely impor tant for current research. And finally, commercial journals are taking over an ever-larger fraction of the literature, with enormous financial incentives driving the trend. What should we conclude from these facts? Here are two alternative predictions. Prediction 1: The alternative models expand and pressure journals. The independents, with only scant operating mar gins, diminish further. The commercial journals, with deep pockets, continue to expand and add features. Commercial publishers consolidate and eventu ally dominate the schol arly literature.
norance? Of course we should. If we fail to recognize that 40% of mathematical scholarship is published in multidis ciplinary journals, we will design alternatives that ignore almost half the literature. If we believe the e-only journals are growing in number, when the number is shrinking, we may invest in the wrong trend. And if we ignore the fact that commercial journals take up not just more dollars but more shelf space as well, we risk sitting by like Nero while scholarly publishing is destroyed. Ignorance about the past and present of scholarly publishing is more than careless exuberance about the future; it means we can neither pre dict that future nor understand how to shape it. Remem ber-while some ecological disasters are caused by greed or malevolence, most catastrophes occur because well-in tentioned people did not foresee the consequences of new technology. The ecology of scholarly publishing is embedded in the far larger ecology of publishing, which currently has many forces driving change, and few of those forces have any thing to do with scholarship or the academy. Ecological disaster for scholarly publishing would be swift and (largely) unnoticed by anyone outside academic life. Many of the alternatives to journals may temporarily solve the problem of costs and speed of publication. For those who believe scholarly journals are merely a way for publishers to sell re search back to the scholars who cre ated it, this may seem like a fine so lution. But people with publishing ex perience do not subscribe to this reductionist view. Jour nals are not just a way to distribute words on pieces of pa per or screens; they are complicated institutions, involving authors, editors, libraries, researchers, publishers, profes sional societies, and administrators. Each has a role to play, and each has interests represented by the institution. The institution of journals exists because scholarly pub lishing is not meant only for today's scholars but for future scholars as well-for our children and our children's chil dren. Scholarly communication is more than sending pa pers to one's colleagues. Validation? Archiving? Financial incentives? These are all about sustaining scholarship for the future, not about exchanging papers in the present. Who will watch over collections when enthusiastic volunteers move on? Who will pay the costs of ever-changing servers and software to keep papers accessible? Who will provide the huge sums for archiving-not only saving the bits but updating the format of millions of papers? Surely we should not rely on government agencies, which have an increas ingly short-term view in all their activities. Many of the experts on electronic publishing assure us that these questions have easy answers. But we need to re member the lessons of the past: Predicting the conse quences of technology is an uncertain business. Can we solve the problem of archiving in the future? To wave our hands with the assurance that technology will find solu-
Scholarly publishing is not
meant only for today's scholars but for future scholars as well.
Prediction 2:
The alternative models expand and pressure journals, driving out the independent journals. The alter native models solve all their problems-financing, cover ing the dispersed literature, archiving, etc. The commercial publishers close down their journals and walk away with their enormous profits. Which of these predictions is correct? Many scholars hope for the second; only the first is supported by the facts. Ecology
What's bad about promoting technology rather than pre dicting its consequences? We discovered the answer re cently when we examined what a century of technological progress had wrought. The answer is ecology. We normally think of ecology in terms of our natural en vironment, but ecology can refer to any system and its re lationship to the surrounding environment. The ecology of scholarly publishing includes many things-a system of ref ereeing and reviewing, the use of publications in hiring and promotion, the way in which scholars view their legacy of research. Most experts on electronic publishing dismiss these things as unimportant; it's why so many get predic tions wrong. Should we worry about wrong predictions based on ig-
VOLUME 25, NUMBER 2 , 2003
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tions is like waving our hands for nuclear waste or carbon dioxide or fluorocarbons. We need to worry about the fu ture because no one else will worry about something as fragile as scholarship.
AUTHOR
Conclusions
What should we conclude? Should we steadfastly maintain the status quo? Do we avoid technology altogether? Of course not. We should experiment; we should try out new things; we should tinker with technology and find better ways to communicate. But in carrying out our experiments, we need to be cau tious and we need to be humble. We should remember that in the past smart people were unable to predict the effects of technology. There is no reason to believe that today's smart people are any better at predicting than yesterday's. Trying out anything that comes to mind without under standing the effect on the entire system of scholarly com munication may be exciting, but surely it is not wise. We also need to be forward-looking. The essence of scholarship is what we leave for future generations, not what we produce for today's. Scholarly communication is not about us-it's about the future of our discipline. Many enthusiasts who promote new projects ignore this princi ple. Make changes now, they argue, and worry about whether they are sustainable later. But if scholars them selves don't worry about their future, who will? What about the experts? Treat them with skepticism. More information is better? Maybe. But nearly everyone is experiencing information overload today; perhaps the qual ity of information is more important than the quantity. Faster is always better? Maybe. But the bottleneck on the Internet is the person receiving the information, who often is not able to process what is already provided. The Inter net will solve the problems of scholarly communication? Maybe. But scholarship and the Internet are different in an essential way: The nature of scholarship is long-term; that of the Internet is transitory. Finally, be especially skeptical of the experts who demand that you be either with them or against them. Subscribe to their vision of the future or be branded a Luddite. This is a false dichotomy-resist it. Responsible caution is not the same as mindless obsti nacy. It is possible to promote electronic publishing with out promoting the dissolution of institutions that have served us well. It is possible to cultivate and shape those institutions without ripping out their roots. It is possible to have a revolution without renaming the months. If we have learned anything in the past century, it is that even the most useful technology can destroy those things we value most.
USA
by counting submissions for each year at http://arxiv.org/archive/ math. The total number of papers, including those migrated from other preprint servers, can be determined using the total number of papers given at http://front. math.ucdavis.edu/. 5. The ALPSP research study on authors ' and readers' views of elec tronic research communication,
Alma Swan & Sheridan Brown, Key
Perspectives Ltd, ISBN 090734123-3. The survey was sent to ap proximately 14,000 authors of scientific papers across many fields. The response rate was about 9%. 6 . For present purposes, books, proceedings, and all items other than journal articles are not counted. 7. The term "primarily electronic" is not precise, but indicates journals that are either electronic only or that have a subsidiary paper copy added to the electronic version, which is viewed as primary. 8. Competition and cooperation: Libraries and publishers in the transi tion to electronic scholarly journals (see §2), A. M. Odlyzko. Journal of Electronic Publishing
4(4) (June 1999), www. press.umich.edu/
jep/, in the online collection The Transition from Paper: Where Are
NOTES 1. Waldemar Kaempfert, 1 9 1 3
We Going and How Will We Get There?, R.
S. Berry and A. S. Mof
fatt, eds., American Academy of Arts & Sciences, www.amacad.org/
2 . Claude Graham-White and Harry Harper, 191 4
publications/trans.htm, and in J. Scholarly Publishing 30(4) (July
3. A term used i n Digital Mythologies, Thomas Valovic, Rutgers Uni
1999), pp. 163-185.
versity Press, New Brunswick, 2000. 4. The number of papers contributed by individuals can be determined
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THE MATHEMATICAL INTEULIGENCER
9. This figure is often quoted by the Association for Research Libraries, although it is hard to determine its precise source.
EISSO J. ATZEMA
Into the Woods· Norbert Wiener in Maine mong mathematicians, Norbert Wiener (1894-1964) counts as a major contributor to such fields as harmonic analysis, stochastic processes, and quantum mechanics. To a more general audience, Wiener is mostly known as a former child prodigy, the father of the field of cybernetics, and the author of a number of more popular works, including a well-written, two-volume autobiography. For just about anybody who has ever heard of him, he is remembered as a long-time fix ture at Massachusetts Institute of Technology with a repu tation for distracted-professor behavior. Indeed, Wiener's association with MIT went as far back as 1919, but he set tled on the north shore of the Charles River basin only af ter several forays into other careers, including a stay at the University of Maine.1 All of these forays are discussed in greater or lesser de tail in the first volume of Wiener's autobiography. Arguably, Wiener's short spell as a college instructor at the Univer sity of Maine in Orono comes closest to Wiener's ultimate line of work In this article, I will complement Wiener's story of his stay in Orono by using archival sources at the University of Maine and Wiener's letters to his relatives as preserved at MIT. This yields both a snapshot of a transient phase in his life and a portrait of a state university outside the main currents of academic activity. Looking for a Job
In the Spring of 1916 young Norbert Wiener was at a cross roads. At the age of 2 1 , he was no longer a child prodigy.
The time had come to prove himself as an academic and, more mundanely, to earn a living. Most acutely of all, it was time to free himself from the sway that his domineering fa ther still held over him, despite his accomplishments. In the previous two years, while studying in Cambridge (UK), Gottingen, and New York, he had been able to hobnob with the likes of Russell, Hardy, Landau, and Dewey. In the fall of 1915, he had obtained a one-year assist antship in the Department of Philosophy at Harvard, where both semesters he taught two sections of an elementary philosophy course, a course in logic, and a Freshmen tu torial in philosophy. In addition, availing himself of the pre rogative of every Harvard Ph.D. to offer a free series of Do cent Lectures, he taught a course in constructive logic. By all appearances, Wiener was heading toward a career in logic. Prospects of finding a more permanent position in his chosen field, however, looked rather bleak The chair of the Department had bluntly told him that he was not worthy of much recommendation, and in general Harvard seemed to have very little interest in helping him to secure a position to his liking. Giving in to severe pressure from his father, Wiener decided to look for a teaching position in mathematics. 2
1There is no authoritative book-length biography of Wiener (the closest is John von Neumann and Norbert Wiener by Stephen J. Heims). The two volumes of his au
tobiography [1 2], [1 3] remain the best source on his life. More detailed information on Wiener's mathematics can be found in [4]. 2See [1 2 , pp. 238-239] .
© 2003 SPRINGER-VERLAG NEW YORK, VOLUME 25, NUMBER 2 , 2003
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At that time, practically the only way for anyone with out the right connections or a well-established reputation to obtain a teaching position on any level was to register with a so-called teachers' agency. Functioning very much like a modem head-hunter bureau, the agency would serve as an intermediary between the job-seeker and any insti tutions that wanted to fill a teaching position. To Wiener, the whole idea of a recruiting agency was utterly humiliat ing, but he did register with the Boston-based Fisk Teach ers' Agency as a Ph.D. seeking to teach mathematics. Prob ably much to his relief, he did not have to wait long before he was notified that several institutions had expressed an interest in him. Most of these were far away from Wiener's beloved New England. The list, however, also included the University of Maine, and before the semester was over Wiener found himself corresponding with James Norris Hart, the chair of the Department of Mathematics & As tronomy and Dean at large, about salary and courses to teach. The University
Founded in 1865 at Orono (ME) as a land-grant agricultural institution known as Maine State College of Agriculture and the Mechanical Arts, the University of Maine had a re spectable tradition of programs in direct support of Maine's mostly rural economy. Like most land-grant institutions, practically from its inception the school had sought to serve a broader public as well. Already in the early 1870s a course (or "major") in Literature and Science had been established to accommodate students who did not desire a career in agriculture or in engineering-chiefly women, who by state law had been allowed to attend the school since 1872. In 1899, Maine State College was renamed the University of Maine in recognition of its widened mission. By 1 9 16, the school had developed into a full-scale university with an enrollment of over 1200 students3 and course offerings that ranged from state-of-the-art engineering courses to much of the curriculum traditionally covered by Bates, Bowdoin, and Colby, the three liberal arts colleges in the state of Maine. Indeed, much to the frustration of those three col leges, not only did the University award BS degrees , it also awarded the BA, and the traditional classical languages re quirements for these were upheld much more strictly than they were at the three liberal arts colleges. While the Col leges of Engineering and Agriculture definitely dominated the school, the University of Maine could also boast of a College of Arts and Sciences, a College of Pharmacy, and a law school (located in Bangor rather than on Campus). 4 As for the teaching of mathematics, initially there were few students and even fewer required courses, most of which were taught by Bowdoin graduate (class of 1861) and
second president of the university, Merritt Caldwell Fernald (1838-19 16), who also taught most of the physics and phi losophy courses. In 1880, Professor of Civil Engineering George Herbert Hamlin was Professor of Mathematics and Drawing for a year, after which nobody seems to have been specifically in charge of the teaching of mathematics until 1887. Most likely, during that period the Professors of Mil itary Science and Tactics-all West Point graduates, start ing with 2nd Lieutenant Edgar Howe (class of 1878)5-did most of the teaching. In 1887, Maine State College graduate (class of 1885) James Norris Hart (1861-1958), at that time principal of the Machias grade and high school, was hired as the ninth full-time faculty member to teach mathematics and mathematical drawing as well as the occasional fresh man and sophomore rhetoric class. After the creation of the Department of Mathematics and Astronomy in 1890, Hart was promoted to Professor of Mathematics and Astronomy and made head of the Department. He would officially serve in that position until 1929, although his successor Willard (see below) began doing much of the work in 1924. An ef fective administrator, Hart was made the first Dean of the university in 1903 and for three months in 1910 he also served as acting president. The Department thrived under Hart's tenure, and by 1916 it counted three professors other than Hart and two instructor positions. Most of the faculty in the Department had better formal qualifications than ei ther Fernald or Hart. 6 Hart's Regime
A civil engineer by undergraduate training, after his ap pointment Hart quickly obtained the CE degree from Maine State College ( 1890), and starting in 1894 he was on leave for further mathematical study at the University of Chicago. In 1897, he returned with an MS degree in mathematics. By that time, he was still essentially the only mathematics in structor and the school had to scramble to offer all the usual courses in mathematics during his absence. 7 Ten years later, the situation had improved considerably with the establishment of one assistant (tutor) position and the appointment of Harley Richard Willard (1875-1946), a Dart mouth graduate (class of 1899) with strong New England ties, as assistant professor. Like Hart before him, Willard continued to improve his academic credentials after hav ing been appointed: from 1909 through 1912, Willard was on a three-year leave of absence to obtain a PhD in math ematical astronomy (Yale, 1912). Upon his return, Willard was promoted to associate professor, and University of Maine graduate (class of 1907) Lowell Jacob Reed (1886-1966) was promoted from instructor to assistant pro fessor. Like Willard, Reed soon thereafter obtained a PhD on a leave of absence, in his case in statistics (Penn State,
3For 1916, the official enrollment was 1276 students (UMSR). 40n the University of Maine, see [11]. 50n Howe, see [11, p. 80]. Later he had a distinguished military career in the West and the Philippines (see USMAFS). 60n the teaching of mathematics at the University of Maine, see [11] . The principal primary sources are the Annual Reports and Course Catalogs of the university as well as the Hart Papers (�UMEH), consisting of Dean Hart's professional correspondence between 1903 and 1929. 7See the State College's campus newspaper The Cadet 9:4 (Oct. 1894), p. 128. During his absence, Hart's courses were taught by President Harris (a former math ematics teacher) and the professor of military science, Mark Hersey (West Point, 1887), on top of their regular assignments.
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Norbert Wiener at the Aberdeen Proving Ground (Maryland) in 1917.
1915). Unlike Hart and Willard, Reed did not stay at the University of Maine for a very long time. For much of the First World War, both Willard and Reed were on leave in Washington, Willard as a statistician to the Hoover Food Administration and Reed as the director of the Bureau of Tabulation and Statistics. While Willard returned to Orono in 1918, Reed elected not to do so and accepted a position at Johns Hopkins University under Raymond Pearl (1879-1940), the former head of the Experimental Station of the University of Maine. In close collaboration, they would, among other things, rediscover the importance of the logistic curve in biology and be among the first to warn about the long-term dangers of tobacco use. At the end of his career, Reed even served as President of Johns Hop kins University for a three-year period. 8 Reed's tenure at the University of Maine seems to have been a very typical one for the Department around the turn of the century. Whereas Hart, Willard, and Reed remained the only professors until 1 9 1 7, several instructors were ap pointed during the period, and most of them were working on advanced degrees. Once the degree was obtained, al most without exception the instructor would leave for a more promising or better-paying position or to pursue an academic degree elsewhere. Typically, these instructors were recent graduates with degrees from non-Ivy League institutions, often located in the mid-West or having a strong religious affiliation. For instance, University of Illi nois graduate (class of 1898) and later mathematics edu cator Arthur Robert Crathome (1873-1946) was a mathe matics instructor at the university between 1898 and 1900 before obtaining a PhD in Germany (Gottingen, 1907). Like-
wise, shortly before obtaining an MA in mathematics from the University of Maine, Iowa native and St. Olaf College graduate (class of 1903) Martin Andrew Nordgaard (18731952) was appointed as an instructor, which position he held until 1916. After obtaining his MA from the university, Nordgaard spent his summers at Teachers' College in New York to work on an MA in mathematics education. In 1916, he left for an instructor position at his alma mater and later wrote a PhD thesis in history of mathematics un der David Eugene Smith.9 In 1914, Georgetown College (KY) graduate (class of 1914) Roscoe Woods (1889-1863) was ap pointed as instructor. Having done graduate work in Chicago during the summer before he arrived in Maine, Woods ob tained an MA from the University of Maine in 1916 and left a year later to pursue a PhD at the University of Illinois.10 Crathome ended his career as a full Professor at the Uni versity of Illinois, while Woods became a fixture at the Uni versity of Iowa, and Nordgaard was at the now-defunct Up sala College (NJ) for much of his career. Given the rather transient nature of the instructor posi tions in the department, proven effectiveness of the in structor was rather important, and Hart rarely seems to have erred in his assessments of his appointees. Certainly, problems with instructors sometimes did arise, but these were mostly on the personal level (notably alcoholism). Many of the instructors, particularly the local ones, were quite popular among the students. Appointment
In the Spring of 1916, business must have seemed as usual for Hart, as he began planning for the next academic year.
80n Reed and Pearl, see [8, pp. 71 -76]. Their work on the logistic curve is extensively discussed in [1 0]. 90n Nordgaard, see [6]. Nordgaard 's PhD was also published as a book. See [5]. 1 0See UIFC. His PhD of 1 920 (under Arthur B. Coble} was also published as an article. See [1 4].
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The Great War had begun to cast its shadow over the uni versity and the department, but no Americans had yet been called to serve. Both Willard and Reed would be able to teach their courses themselves, as would the second as sistant professor, old hand Truman Hamlin (MA University of Missouri, 1902). Of the three instructors of that year, Roscoe Woods had become a popular teacher and was will ing to stay on for at least another year. John Roberts, who had come on board in March after another old hand (Wal ter Wilbur) had abruptly left to become a high school teacher, was also sure to come back. Nordgaard, however, had resigned to accept a position at his alma mater. An other instructor would be needed, and by April, Hart had put in a request with a number of teachers' agencies (in cluding Fisk's). It is not clear from the archives exactly how many candidates were considered for the position and which were notified by the teaching agencies. Hart does not seem to have approached any local candidates or in quired among his contacts. Only two letters of rejection and one application remain. On April 13, Norbert Wiener wrote a long letter of ap plication to the President of the University of Maine indi cating that he applied at the suggestion of the Fisk Agency.11 The letter was forwarded to Hart and it must have piqued his interest. A letter of April 14 to Wiener's first ref erence, the Employment Agency at Harvard, only resulted in a terse reply by William Fogg Osgood (1864-- 1943), chair of the Department of Mathematics, to whom the letter was forwarded. According to Osgood, Wiener had not done enough work in the department for Osgood to be able to assess his attainments. 12 Hart was more lucky writing di rectly to the Harvard mathematician and logician Edward Huntington (1874-1952), whose name Wiener had given as his other reference. As Hart was careful to point out, he found Wiener's research credentials very impressive, but was skeptical about Wiener's ability to teach a class of av erage college freshmen. Huntington's reply of just one day later, while clearly designed as a letter of recommendation, also is rather revealing about the impression Wiener was most likely to make on people. Writes Huntington, I am not surprised that you are rather skeptical as to Dr. Wiener's fitness for your work, and I think that this skep ticism is in a measure justified, on account of the fact that Dr. Wiener has had, as yet, no experience of this kind of teaching. On the other hand, he is not at all the type of over-brilliant, unpractical scholar which one might nat urally suppose him to be. We have had other ''precocious" students here, whom I would not for a moment recom mend for your position. He is not of that type. His bril-
liancy consists simply in having a mind which works more smoothly and rapidly than is the case with most of us-the kind of mind that is likely to do well anything that he undertakes to do. He has a good sense and good humor, and infinite patience, and has been successful in private tutoring and in handling conference sections. Wh·ile his appointment would certainly be in the nature of an experiment, I believe that the experiment would be worth making. I have talked with him a good deal about methods of teaching mathematics, and have been greatly impressed with the soundness of his · i deas, which are pro gressive and at the same time thoroughly sensible. [UMEW: Huntington to Hart, 411 51191 6] Note that Huntington, though probably knowing nothing about Hart's view on mathematics teaching, did not hesi tate to present the progressive movement in teaching as a positive development while at the same time contrasting progressivism with sensibility. Ambivalence toward pro gressive education was widespread in academia and Hunt ington may have expressed only the expected. Be that as it may, Huntington's last sentence almost certainly would have resonated with Hart. Although keen on many issues in the progressive camp, Hart loathed any kind of radical ism, including the extremes within the progressive move ment.13 On April 20 he writes Huntington again, this time with questions about Wiener's personal habits. In particular, he would like to have more information on Wiener's stance with regard to tobacco and alcohol. Not that he wants any one to totally abstain from either, but as he put it, the school "has had one or two men whose habits were objectionable and we wish to be careful not to repeat the experience." [UMEW: Hart to Huntington, 4/20/1916] Inevitably perhaps, he has a second concern. "Other things being equal, we would rather not take a man of the Jewish faith, but we can see that some Jews might be more acceptable than some protestants." Obviously, in the case of Hart's first concern, he need not have worried. As Huntington is quick to point out, the Wiener family was raised under a strictly vegetarian regime which also included abstinence from tobacco and alcohol. Regarding Wiener's faith, Huntington notes that of course Wiener is of Jewish extraction, but that his father has sent him to Sunday school. He would expect Wiener to be tol erant toward any genuine form of religious belief. 14 In the meantime, Hart had also written to Wiener him self and asked him about his physical appearance and his teaching philosophy. Wiener's prompt reply (with a photo graph enclosed), together with Huntington's response,
11UMEW: Wiener to Aley, 4/10/1916. 12UMEW: Osgood to Hart, 4/18/1916 . Osgood's reply is somewhat puzzling. It is true that Wiener had not had any regular duties at Harvard. On the other hand, the Wiener family and the Osgoods were friendly, and Osgood would definitely have known Wiener fairly well. It should be noted that there are various passages in Wiener's autobiography that seem to indicate that Osgood and Wiener did not get along very well. See for instance [12, p. 231] and [12, p. 271]. 13See Hart's critical commentary on progressive reforms in education in his article [2]. Surely Huntington was referring to the ideas of John Dewey and other rad1cal educators. About the extremes of the progressive movement in education and its reception in academia, see for instance [9], especially p. 126. 14UMEW: Huntington to Hart, 4/20/1916.
10
THE MATHEMATICAL I NTELLIGENCER
must have satisfied Hart, and on April 25, he writes to Wiener to offer him the job. 15 The salary is set at $800, for which Wiener will have to teach three lower-level courses. In addition to his teaching duties, Wiener is supposed to keep office hours and proctor exams as need be. Also, Hart mentions the student-run Mathematics Club, then just founded, for which Wiener may have to give a talk as well. In a short letter later that week, Wiener gladly accepts the position. 16 A New Career
Wiener seemed to be on track to become a college teacher. For the next semester, he was assigned to teach one sec tion of Trigonometry & Algebra, one of Sophomore Calcu lus, and one of Solid Geometry. As Hart had made clear in Wiener's letter of appointment, these classes were to be conducted "entirely by the text-book and recitation plan," [UMEW: Hart to Wiener, 4/25/1916], and Wiener professed to have no problems with this somewhat conventional ap proach. Yet it must have been clear to everybody who knew him, including Hart, that Wiener's heart lay elsewhere. Indeed, it must have been obvious to Hart that Wiener was entirely serious when in his letter of acceptance, he in quired about his options to teach a course in logic on top of his regular duties, as this was his "Fach" and he would like to "keep his hands in it." [UMEW: Wiener to Hart, 4/27/1916] Much to his credit, Hart did talk with Wallace Craig, the Professor of Philosophy, and Wiener ended up teaching a course in elementary logic during his first se mester at the university. Also, in addition to Wiener's reg ular load, Hart offered to have him teach the history of mathematics course. As Hart explained, given Wiener's ob vious language skills and his interest in philosophy, he might enjoy teaching the course. 17 Wiener seems not to have appreciated that Hart was go ing out of his way to accommodate his young instructor. Worse, in his focus on making good he would often forget himself and lapse into his former cocky prodigy mode. A case in point is his reply to Hart's suggestion that he teach the history of mathematics course. As Hart wrote, this course could be taught either by textbook or by lectures, and the last time it was taught, Cajori's History ofMathematics was used. Completely ignoring the intent of Hart's suggestion, Wiener's reply starts out by trashing Cajori's book As Wiener writes, he has had a look at Cajori's book and, I find it much too dry and schematic to appeal either to me, or for the matter of that, to my students, so I think that perhaps the best plan will be to retain it as a book of reference, in which reading will be required, but to base the lectures on Moritz Cantor's Vorlesungen tiber die Geschichte der Mathematik, and to use the sources them-
selves or such other works as may be useful for particu lar periods. [ UMEW: Wiener to Hart, 61811 91 6} In the remainder of his letter, he wonders among other things whether the library at the University of Maine may have the complete mathematical writings of Descartes, Leibniz, Pascal, and Newton available for student use, and sets forth several ideas about how to teach the history of modem mathematics, including the idea of an abstract group. To his credit, after getting carried away by what wonderful things he might teach, Wiener does note that all his suggestions are only tentative, and that what he will ac tually teach will (of course) depend on the capabilities of the students. He neglects to write that he is actually will ing to teach the course. One wonders what Hart thought. In his response, he wisely ignores most of Wiener's questions and suggestions and only gives a list of the relatively few books the library does hold, concluding that these books "should suffice to conduct a class in history of mathematics." [UMEW: Hart to Wiener, 6/16/1917] From later correspondence, it is clear that Wiener did end up using the Cajori book 18 The Faculty
In the spirit of the times, Wiener spent most of the summer before coming to Maine at Officers' Training Camp in Plattsburg (NY). In a somewhat Rooseveltian fashion, Wiener's feelings of physical inaptness probably provided an additional motive to try to join the army, as did his vague desire to liberate himself from the influence of his father. In his case, however, his poor eyesight and general clum siness prevented him from receiving a commission, and at the end of the summer Wiener made his way to Maine with, as he wrote in his autobiography, "no particular feeling of accomplishment." Probably he could have added that he did not have high expectations of the year to come either. Wiener's first letter home to his mother is quite positive about his new colleagues. A day later, in a letter to his father, his tone is similar, but he seems to have changed his mind a little bit. "There is nobody here who knows a continental about my sort of work," he writes, but "all seem properly im pressed with the amount of paper consumed by my articles, and treat me as equal." [MITW: Wiener to Father, 9/16/1916] Wiener has a harder time reciprocating. He describes Willard "a simple farmer sort of fellow," and immediately goes on, "I have not met a man on the faculty yet, who did not seem to have several pounds of hayseed concealed on his person." In a post script, he only makes things worse: "I do not mean to look down on the faculty here. Anyway, I can't. They are mostly six foot Yankees. They are quite nice." Obviously, Wiener is getting a little carried away by the flow of his writing (something characteristic of the later
15UMEW: Hart to Wiener, 4/25/1 9 1 6. 16UMEW: Wiener to Hart, 4/27/1 91 6. 17UMEW: Hart to Wiener, 6/2/19 1 6. 181n one of his letters home, he starts out his usual list of complaints with "Cajori's History of Mathematics, which is our textbook, is vile" [MI1W: Wiener to Father, 1 1/29/1 9 1 6] .
12
THE MATHEMATICAL INTELLIGENCER
Wiener as well). Nevertheless, after no more than a week, Wiener seems to have made up his mind about the Uni versity of Maine. His first letter to his sister Constance says, It 'is quite possible to live at this university, but it is more or less difficult to do much more. There is practically no intellectual life here, and what is there is confined to very few persons. The men here tell me candidly, without any prompting, that the students come here merely to increase their earning capacity, without the slightest thought of acquiring any ideas or other intellectual impedimenta in the process. [MITW: Wiener to Constance (Conta), 91241 1916] This time, the administration takes a beating as well. "The blasted blights that run this Gehenna," he notes, "have not had enough sawdust off the back of their faces to dream of the comfort of the faculty in their petty scheme of things. The faculty are supposed to appear at lectures to release the wisdom learned in their infancy and to sleep the re mainder of the day and night in somewhat out of the way lodging houses." In the course of the year, many such well-crafted dia tribes were to follow. A letter to his sister from October 18 is a highlight: We receive a lot of under-educated, uncivilized farmer boys, put them under an uneducated, uncultivated fac ulty-! am speaking of the average, not of many note worthy exceptions-and expect them to be educated men. I sit at the same table with a professor of Apples, who possesses a great facial resemblance to a hornless animal that cleaveth the hoof and a greater intellectual Tesem blance to another lowly beast that cleaveth not the hoof Near to me sits a lady professor of dishwashing, while another female professor unfavored of countenance in vests the chair of basting and darning with a strange and wondrous dignity. Philosophy is too refined an intel lectual pabulum for Orono, thistles is more what they want. . . . The only cultivation in Omno is to be found in the Department of Agriculture. [MITW: Wiener to Con stance, 1 0/ 1 811 916] Forty years later, Wiener would pen a somewhat more bal anced but rather similar characterization in his autobiog raphy: The oldeT, peTmanent professors WeTefor the most part bro ken men, who had long given up any hope of intellectual accomplishment or of advancement in their careeT. A few of them still showed ragged traces of cultural ambitions, but the greateT part WeTe resigned to their failure. The
younger men WeTe almost all transients like myself, who had been bought wholesale at teacheTs ' agencies afteT the cream of the crop had been skimmed off through the spe cial efforts of their professors. The men who remained, the traveling guests of the univeTsity, had no inteTest what ever in the place, and had as their sole ambition to leave it as soon as possible, before their employment theTe should fix on them too definitely the stigma of unemployableness [sic] a t more desirable positions. Few and far between are the individuals who do not perish of intellectual atrophy at such places [ 12, pp. 239-240]. For all their exaggeration and verbal glitter, these obser vations hold some veracity. The University of Maine was typical of rural universities at the time, and even to this day one could claim that the school lacks the allure and intel lectual aura of the more urban universities in New England. Still, the faculty had its share of illuminati, as Wiener had to admit. Near the end of his stay, for instance, he had be come part of the social circle around Raymond Pearl, the energetic director of Maine's Experiment Station, and he had also made contact with the zoologist Alice Middleton Boring (1883-1955).19 Both were well-established on cam pus, and would also have a successful career in academia. It seems Wiener took offence more at the pretentiousness of some of the faculty than at their actual lack of "culture." In the same letter to his sister quoted above, for instance, Wiener notes that Hart is no scholar either, but that he has enough common sense not to pretend to be one. All in all, one can also easily imagine that Wiener's atti tude would not exactly endear him to his colleagues or, for that matter, to his students. The Students
By the time Wiener arrived in Maine, the student popula tion at the university had acquired a reputation of stubborn resistance toward teachers and situations it did not like. In the early 1880s, the students had forced the unpopular Edgar Howe to resign20; more recently, in 1909, the uni versity had been the venue of one of the nation's first stu dent strikes, protesting the administration's no-hazing pol icy.21 In between, the university's crackdown on cheating had also been the cause of considerable tension between the students and the faculty.22 No wonder, therefore, that Hart explicitly asked Huntington about Wiener's ability to control his classes. As for Wiener himself, he probably was hiding his apprehensions behind his usual bravado. From the start, as with the faculty, the students did not quite meet with Wiener's approval. Reporting on his proc toring an early exam, he writes that "viler work than those students did I never saw," to continue in seemingly genuine amazement, "one man got half of an answer right out of a
190n Boring, see [7]. Boring was at the University of Maine from 1909 to 1 9 1 8, after which she left for China. When Wiener was at Cornell, he boarded with a younger brother of Bonng. He was to meet her again in China when he was on a sabbatical there. In his autobiography, he rather curiously refers to Boring as Elisabeth. 2osee [1 1, p. 80]. 21See [1 1 , pp. 61 -63]. 22See [1 1 , p. 60].
VOLUME 25, NUMBER 2, 2003
13
dozen." [MITW: Wiener to Father, 9/16/1916] Barely a week later, when writing his parents about his course assign ments, his tone is rather vengeful:
Professor Segall of the French department tells me that he shoved off several rots and spots that are looking for snap courses into my history of mathematics. They will have another think coming. There will be a high death-rate among such of my students as are suffering from infan tile paralysis of the intellect. (MITW: Wiener to Constance, 911 811916] In spite of this ominous beginning, in the first weeks every thing seems to have gone smoothly. Four weeks into the year, however, Wiener shares some concerns with his mother:
I am dog-tired today and pretty blue. I gave my students in trig an exam, and they did vilely. I am no teacher, I guess, I felt awfully discouraged & I told Dean Hart, which perhaps I shouldn 't have done. I hate those pesky little exams. [MITW: Wiener to Mother, 1 0/61191 6] A week later, everything is "hunky-dory" (Wiener's expres sion) again. Hart has visited his classes and was content. Something must have happened, though: Wiener goes on to describe how now he gives his students a lot of individ ual attention outside of class, which seems to be appreci ated and makes most of them "come around." Still, ac cording to Wiener, "there are about eight or nine students who will neither pay attention, nor pem1it anyone else to do so. I am going to make an example of some of them if this keeps up-pour encourager les autres." [MITW: Wiener to Mother, 10/1211916] For a while after this, Wiener is quiet about his classes until mid-November when he complains to his father that he is having discipline problems in one of his trigonometry classes. As Wiener writes, "one of them behaves all right, but the other contains two or three first-class muckers who do nothing but make a disturbance by throwing chalk, eras ing one another's work at the board, chatting in class etc." [MITW: Wiener to Father, 1 110 111916] A week later, he has the solution:
I have started to send from the room any boy who makes the least disturbance in class and they don't like that. I think this has caused a very marked improvement in class discipline. I have been too clement in the past, but from now on I am going to nail the lid down. I have found that a laissez-jaire policy is no good. [MITW: Wiener to Fa ther, 1 110111916] After this letter, the only remark about his teaching is in a letter from February in which he mentions that he is teach ing next to Hart's classroom and that Hart has complained that he (Wiener) pitches his voice too high and that he should not address his students by Mr. So-and-so. Appar ently Hart also remarked to Wiener that his classes "were
14
THE MATHEMATICAL INTELLIGENCER
J. N. Hart, Professor of Mathematics and Astronomy and Department Head (1 890-1929), the University of Maine.
inclined to play horse with him" [MITW: Wiener to Mother, 2/20/1917). Clearly, Wiener had a discipline problem in his classes, as is also confirmed in his autobiography:
The students in those days were largely a strapping lot of youngfarmers and lumberjacks, who managed to be quite as idle and collegiate as the students of universities of the Ivy League, but at one-third the expense. Their sole interests were infootball and nagging the lives out of their professors. As I was young, nervous, and responsive, I was their chosen victim. Most of my courses were dull routine to them, and many is the penny which I heard dropped in class to annoy me. [ 12, p. 240] By the time Wiener wrote this, he was mature enough to acknowledge that his poor eyesight interfered with his teaching. In 1916, he clearly was not. In his letter home, he never even mentions his myopia. To be sure, his near-sight edness was by no means the only cause of his discipline problems. Equally important must have been Wiener's at titude toward the students' collegiate behavior and their relative lack of interest in matters academic. At this stage in Wiener's career, it was incomprehensible to him how any university student could not display intellectual curiosity, in or outside the classroom. A case in point is his scathing judgment of the student body after a lecture on the fourth dimension he gave for the Mathematics Club on December
6, 1916. Just the year before, Einstein's theory of general relativity, which heavily relies on four-dimensional geome try, had been published, and there was a considerable pop ular interest in the topic. 23 If we are to believe Wiener, how ever, this vogue must have completely bypassed Maine. To quote his report to his parents,
Upon my word, I felt as I were giving a lecture on Musi cal appreciation before a Deaf-mute asylum. . . . We abound in clubs here: Mathematics clubs, Information clubs etc., but nothing seems to be discussed in the Mathematics club that Constance couldn 't understand just as well without a lecture, and the Information Club is under the painful ne cessity ofsugar-coating dry papers on utterly arid subjects with sensational titles, such as "The Fourth Dimension, " in order to allure a reasonable number of students to pass a half hour in a state of somnolence, in which by some ut terly incomprehensible miracle, they may manage to pick up some scrap of "culchm.ceah" [MITW: Wim1m· to Father, 1 01271191 6].24 Clearly, just as with the faculty, there was little the stu dents could do right in Wiener's eyes, and he did not feel it incumbent on him to try to help them improve. In his big ger trig classes (about 20 students each), the students did not let him get away with this attitude. In the smaller and somewhat more advanced classes that Wiener taught, it probably did not cause major problems. Indeed, one of the reasons why there is almost nothing about his teaching in his letters home during the spring semester may have been that he had much smaller classes (about ten students each). More likely, by spring, Wiener had convinced himself that being a college instructor was no longer a challenge to him and that he had to move on. He may just no longer have cared about teaching. Moving On
For one thing, the war was looming. Wiener had hardly been successful in his attempts to prove that he could be a worthy soldier, but he had not given up hope. If he was not going to be independent of his parents through his mathematics, he might through the military. Much of his correspondence home is taken up by his somewhat com pulsive interest in making good again, this time by service to his country. The situation at the University of Maine cer tainly made it easy to become involved in the war effort. In the first few months after he had arrived, the so-called preparedness movement had obtained a strong hold on the university, and the military had become very visible on cam pus. The hours of classes had been changed to facilitate
drill, and there was strong pressure on faculty and student alike not to "shirk responsibility. " 25 Since there was a great shortage of drill officers, Wiener volunteered as a drill master. As before, he failed miserably, and soon his ser vices as a drillmaster were no longer requested. By then, it was April 1917, and the United States had en tered the war, or as Wiener wrote in capitals at the bottom of one of his last letters home of that year, the "big shindy" had arrived. Around the same time, in view of the expected large drop in enrollment because of enlistment, the ad ministration terminated two faculty positions effective im mediately, discontinued all non-essential telephone ser vices, and announced the release of all instructors for the next year. New instructors could be appointed only to re place faculty called upon to serve the nation or when en rollment would unexpectedly go up. 26 Wiener's stay at the university was not to last. He was notified by Hart about the likelihood of this decision even before the nation de clared war on Germany, and it would seem that at first he tried to find a position elsewhere. 27 In the Hart papers there are a few requests for information from teachers' agencies, and Wiener also registered with the Harvard Job Service. In early May, however, as soon as the semester was over, Wiener resigned from his instructor position before being officially released, in an attempt to receive a commission through the newly formed R.O.T.C. program at Harvard. As so many times before, he went through the program, but did not receive a commission-this time on account of his poor eyesight and high blood pressure. All of Wiener's attempts to get away from his domi neering father had failed, and once again it was his father who found him employment. Wiener himself had made con tacts with a shipyard in Lynn (MA) and, much to his satis faction, had begun to do some engineering work, but his father put an end to this. Deeming his son too clumsy for engineering work, he this time arranged for a job as a hack editor with the Encyclopedia Americana in New York, a job Wiener did not dare to tum down. Only almost a year later did he briefly return to mathematics by joining as a civilian the team of mostly enlisted scientists that Oswald Veblen was collecting at Aberdeen Proving Grounds. Not satisfied by this contribution to the war effort, Wiener en listed as a private in October 1917, and it was not until af ter demobilization in 1919 that he would fully establish him self with the mathematical community and land a position at MIT. It was only then that Wiener's father finally relented and allowed his son to stand on his own feet. As for the University of Maine, the Department of Math ematics and Astronomy lost four of its teachers. Along with Wiener, the instructors Woods and Roberts were let go as
23Publ1c interest in higher-dimensional geometry already existed before general relativity. See for Instance the essay competition on the top1c sponsored by Scientific American in 1 909, leading to the publication of [3]. 241n the flies of the Math Club, th1s talk is (erroneously?) listed as given on 1 2/6. Other faculty talks that semester were "Curve Fitting" (by John Miner of the Experi
ment Stat1on) and "Methods and Symbols Used in Mathematics Before the 1 6th Century" (by Truman Hamlin). 25See [ 1 1 , p. 1 03] . 26See [ 1 1 , p. 1 05]. In the event, enrollment was 91 3 students for 1 9 1 7 and 21 1 for 1 9 1 8. 27UMEW: Hart to W1ener, 3/16/ 1 9 1 7.
VOLUME 25, NUMBER 2 , 2003
15
well, while Hamlin accepted a position at Union College
true debut i n academia. At best, his stint in Orono was a
(NY). By September, however, Willard and Reed had gone
trial run, as Wiener himself appositely calls the chapter on
to work in Washington, and University of Maine graduate
it in his autobiography.
Maynard Jordan (class of1916) and Quentin Weaver Stauf fer (PhD Muehlenberg,
Conversely, one might wonder about the impact Wiener's
1913) were hired as instructors to
stay may have had on the University of Maine. Although the
replace them. In addition, Hart was allowed to appoint an
faculty at the time counted several PhDs, there were few that
in the Mathe
assistant professor. After a short search, this position went
had been hired with a PhD in hand, and none
to another PhD in mathematics (Columbia,
1909), Myron
matics Department. But after Wiener left, the department be
(1874-??). Hart's correspondence contains lit
gan to hire faculty who held a PhD. Indeed, in a letter to Percy
Owen Tripp
tle about Stauffer, but his concerns about Tripp initially
F. Smith (Yale) about the opening of the assistant professor
were similar to his concerns about Wiener. In the end, Tripp
ship that was to go to Tripp, Hart explicitly mentioned that
does not seem to have had much difficulty establishing him
the candidate needed to have a doctor's degree.31 It does not
self. It may have helped that Tripp was quite a bit older
seem unreasonable to assume that Hart had become con
than Wiener, had attended Teachers College after receiv
vinced that the department needed an influx of highly trained
ing his PhD in group theory, and had several years of high school teaching under his belt.
As
Hart put it in a letter to
mathematicians to improve the level
(if not
necessarily the
quality) of teaching. Wiener's appointment may very well
his colleague Townsend in Illinois, Tripp probably would
have been Hart's first attempt to act on this conviction.
never be popular with the students, but would at least com mand their respect. zs
not intervened, more than likely Wiener and the University
If it was an "experiment, " it was a failure. Had the war
During the final stages of the war, Hart and Tripp were
of Maine still would have parted ways. Wiener's world was
left mostly on their own to teach the greatly increased num
and would remain incompatible with the rural institution
ber of mathematics courses required for the Student Army
that the University of Maine was at that time. One crucial
Training Program that was brought on campus to train sol
aspect of this divide was aptly summarized by Hart in one
diers for the war. Stauffer had enlisted, and his replace
of his letters of recommendation for Wiener:
ment, the New Hampshire State College graduate (class of
1916) John Dane Lary, fell victim to the infamous influenza
university, nor had he left a great many hard feelings.
Dr. Wiener is a fine mathematician, as, of course, you will know, and an enthusiastic and hard working teacher. I am obliged to confess, however, that he has not succeeded in understanding well our students, and has not been suc cessful with those indifferent or of only ordinary ability. His nearsightedness is quite a handicap, and his failure to see the students ' point of view another. If he could be placed where his work would be mainly with graduate students, I think he would do well [ UMEH: Hart to Mary L. Wheeler, 6/2511 9 1 7].
Conclusion
At least equally important, however, was the cultural divide.
(Spanish Flu) pandemic that hit the campus in the Fall of
1918.29 After the war was over, Tripp stayed on for a few years, but in 1921 he accepted a position at Wittenberg Col lege (now Wittenberg University, OH), where he spent the rest of his career. By then, Willard had returned from Wash ington, and the department was almost back to the way it was before Wiener was hired. It was easy to forget the Wiener intermezzo. He had not made a real mark on the
Wiener would not be remembered on campus until
1953,
It is not that the University of Maine was such a backward
in
when his less than flattering description of the University
place. Although located in New England, as a land-grant
of Maine in his autobiography caused quite a stir among its
stitution, the university looked to the mid-West for inspira
alumni. One alumna was sufficiently upset to send a tran
tion rather than to the traditional strongholds of academic
scription of the most damning passages to then-President
learning in the East. Many of the faculty of the Mathematics
Hauck, wondering if anything could be done against this
Department had received their training
"immature and biased judgment. " In his response, Hauck
deed, even the President of the University at the time, Robert
in the mid-West. In
said he had not read the autobiography, but he did know
Judson Aley, was a transplant from Indiana. The "culture"
that there were still a few people on campus who remem
that Wiener so badly missed in Maine was that of the New
bered Wiener and were quite amused by what he had writ
England cultural elite, a tradition that the university never as
ten. Indeed, a chuckle and a shrug was probably the best
pired to emulate (and one that Wiener hated on other occa
reaction. 30 With hindsight, it is obvious that Wiener would
sions). The one important aspect of the New England cul
never have made a good college teacher at a rural univer
tural tradition that the university did share was its affinity to
sity and that his stay in Maine was a stage in his own edu
Protestantism, evidenced by the number of faculty members
cation more than anything else. Wiener still had many is
who attended colleges with strong religious affiliations (e.g.,
sues to deal with before he would be ready to make his
Nordgaard, Woods). This was no help to Wiener, as a Jew.
28UMEH: Hart to Townsend, 2/8/1 9 1 8. 290n the pandemic, see [ 1 ] . 30UMEW: Mrs. Frank Lamb to Arthur Hauck, 8/3 1 /1 953 and Hauck's reply of 9/1 1 / 1 953. 31UMEH: Hart to Smith, 9/1 3/1 9 1 7 .
16
THE MATHEMATICAL INTELLIGENCER
A U T HOR
Abbreviations
MITW = UIFC =
MIT, Archives: Wiener Papers University of Iowa, Special Collections: Faculty Charts UMEW = University of Maine, Special Collections: Wiener File University of Maine, Special Collections: Hart UMEH = Papers University of Maine, Student Records: Student UMSR Records USMAFS = United States Military Academy, Archives: Records of Former Students =
EISSO J. ATZEMA
REFERENCES [1 ] Richard Collier, The Plague of the Spanish Lady . . (American edi .
tion: 1 996).
USA
[2] J . N . Hart, What mathematical knowledge and ability may reason ably be expected of the student entering college, Mathematics Teacher
6(1 9 1 3/1 4), pp. 1 58-1 65.
[3] Henry P. Manntng (ed .) The Fourth Dimension Simply Explained (Munn & Company, 1 91 0).
[4] Pesi Masani, Norbert Wiener, 1 894-1 964 (Birkhauser, 1 990). [5] Martin A. Nordgaard A historical survey of algebraic methods of approximating the roots of numerical higher equations up to the year 1 8 1 9
(Teachers' College, Columbia University: 1 922) (= Con
tributions to Education , Vol. 1 23). [6] Olaf M. Norlie, School Calendar, 1 824- 1 924. A Who 's Who among Teachers in the Norweg1an Lutheran Synods in America
(Augsburg
[1 0] Bonnie Shulman, "Using Original Sources to Teach the Logistic
[7] Marilyn Ogilvie & Clifford Choquette A Dame Full of Vim and Vigor:
[1 1 ] David C. Smith, The First Century. A History of the University of
Publishing House, 1 924).
Equation, " The UMAP Journal 1 8 4(1 997), pp. 375-400.
A Biography of Alice Middleton Boring, Biologist in China
(Harwood
Press, 1 999). [8] Michael Olinick, An Introduction to Mathematical Models in the So Cial and Life Sciences
(Addison-Wesley, 1 978).
Maine, 1 865- 1 965
(University of Maine at Orono Press, 1 979).
[1 2] Norbert Wiener, Ex-Prodigy (Simon & Schuster, 1 953).
[1 3] Norbert Wiener, I am a Mathematician (Simon & Schuster, 1 956). [1 4] Roscoe Woods, The elliptic modular function associated with the
[9] Diane Ravitch, Left Back. A Century of Failed School Reforms (Si mon & Schuster, 2000).
elliptic norm curve E 7, Transactions of the American Mathemati cal Society
23:2 (1 922), pp. 1 79-1 97.
Erratum
As a result of a production error, the wrong chess position ap peared in Diagram 1 1 of "The Mathematical Knight" (vol. 25, no. l, page 27). The correct diagram appears below. Even with White to move, the knight on b2 cannot stop the pawn's pro-
motion: it would take three moves for the lrnight to reach ad jacent square a2, and four to reach al due to the edge of the board, in each case one too many. In fact the knight helps Black, by blocking White's own king from coming to the rescue. Diagram 1 1
5
4
5
4
5
4
5
6
4
3
4
3
4
5
4
5
3
4
3
4
3
4
5
4
2
3
2
3
4
3
4
5
3
2
3
2
3
4
3
4
2
1
4
3
2
3
4
5
3
4*
1
2
3
4
3
4
tLl
3
2
3
2
3
4
5 White loses
VOLUME 25. NUMBER 2. 2003
17
Mathematic a l l y Bent
C o l i n Ad a m s ,
Ed itor
I
Wiling Away the Hours Colin Adams The proof is in the pudding.
Opening a copy of The Mathematical Intelligencer you may ask yourself
uneasily, "What is this anyway-a mathematical journal, or what?" Or you may ask, "Where am !?" Or even "Who am !?" This sense of disorienta tion is at its most acute when you
Princeton, NJ, 1987 "Andrew, I am not going to ask you again. Go up there and clean out the attic." "Come on honey, relax. I just want to drink beer and watch the game. Is that too much to ask?" "Andrew, you are going up there right now. And you are staying up there every Sunday until you finish cleaning that attic."
open to Colin Adams's column. Rela:c. Breathe regularly. It's mathematical, it's a humor column, and it may even be harmless.
Andrew's diary: Year 1 :
Discovered trunk filled with old clothes. Have had a lot of fun play ing pirate. Base of trunk measures 3 ft X 4 ft. Strange choice of measure ments. Why not 3 X 3 or 3 X 5? What's special about 3 X 4? Tried to sneak the TV up, but Francine caught me. Have managed to catch the eye of the twelve-year-old boy next door through the attic win dow. He likes my antics with the eye patch. Year 2: Found an old harmonica. Have taught myself how to play the blues. Can do a convincing locomotive whis tle. Am trying to teach Howard Tani yama (the boy next door, now thirteen) Morse code using a flashlight. He's a slow learner. Found my old coin col lection. Had 25 nickels and 144 pen nies. So I have 169 coins. Go figure. Year 3:
Column editor's address: Colin Adams, Department of Mathematics, Bronfman Science Center, Williams College, Williamstown, MA 0 1 267 USA e-mail:
[email protected]
18
Found my old box of Hardy Boys mysteries. They are surprisingly good, even on the second or third read. Howard (now fourteen) has picked up Morse code. Have come to realize that fourteen-year-olds don't have a lot in teresting to say. Been teaching him cal culus j ust to while away the time.
THE MATHEMATICAL INTELLIGENCER © 2003 SPRINGER-VERLAG NEW YORK
Was pacing out the attic when I re alized that it was 9 yards across and 12 yards deep. That means the diagonal is 1 5 yards long. Remarkable how nicely those numbers work out. Year 4: Have become
quite good at bal ancing an old Barbie Doll on my nose. Wonder if the students would enjoy it before class some day. Have be friended a spider, whom I am calling Charlotte. I help to catch her flies and she's happy to hang out with me. I cre ate unusual structures upon which she spins her webs. Some of the resulting webs remind me of elliptic curves. Howard (now 15) is grappling with pu berty. As I said, he's a little behind. I am giving him what advice I can about women, but he's having trouble ac cepting wisdom from a man flashing it to him using Morse code on a flashlight through the attic window. His mother seems to have noticed me as well. She doesn't look thrilled. Year
5: Found Charlotte gone and Charlotte's web destroyed. Believe it was a rogue frog that somehow got into the attic. Have heard him sometimes. "Ribet, ribet." Quite a blow. I was shook up. Wanted to talk to Howard about it, but he hasn't been around lately. I think he may be flashing Morse code with the girl who lives on the other side of his house, Karen Shimura. What monster have I created? Been thinking about the dimensions of the attic again. 92 + 122 = 152 . So 9, 12, and 15 are a Pythagorean triple, a set of three integers x, y, and z that sat isfy x2 + y2 = z2. Seem to be lots of them. Found my old McGovern campaign stuff. Bunch of it was in a cubical box with edge length 3 ft. Rest was in a cu bical box with edge length 4 ft. Both boxes are beginning to fray. Don't know why I didn't find one cubical box to put it all in. Perhaps I was planning to glue to gether a bunch of boxes, one for the
buttons, one for the bumper stickers, and one for the literature. It would be a storage system of modular form. Year 6:
Realized that if I kept my old campaign literature in two 4-dimen sional boxes of edge lengths 3 and 4, I couldn't replace them with a single 4-di mensional box with edge length any in teger, as there is no integer z such that z-1 .x4 + ?f. Or at least that appears to be the case. But maybe there is some other choice of integers for the first two boxes so the third box has integer edge length. Haven't had a lot of luck finding such a triple of integers yet. I feel like I'm going a little crazy up =
here. Six years of Sundays is a lot of time. I'm only semi-stable. Maybe I should buckle down and clean the place. I'm very concerned about Howard and this whole Taniyama-Shimura re lationship. Seems to be progressing too fast, but that is only conjecture. I'm not faulting Howard, but I am concerned. Year 7:
Started thinking about situa tion where I keep the campaign litera ture in 5th or 6th dimension. Maybe just go to n dimensions. Could it be that an equation of the form xn + yn Z11 has no integer triples (.x, y, z) that satisfy it for any integer n > 2? Prob lem sounds vaguely familiar. Wish I
Mathematical Word Processing
=
•
Typesetting
•
had some paper to write on. Using the margins of the Hardy Boy books in the meantime. Flashed the question to Howie and he is intrigued. I was trying to explain to him my approach through modular forms and elliptic curves, but the flashlight batteries died. Hallelujah. Turns out I have solved a 350-year-old conjecture! Francine is letting me out of cleaning the attic. What a break Howie's working on the Hodge Conjecture. Francine is starting to make noises about the basement. But at least for the time being, I get to watch football on Sundays.
Computer Algebra
The Gold Standard for Mathematical Publishing
VOLUME 25. NUMBER 2. 2003
19
1$ffii•i§ir@ih$11@%§#fii.Jrrl§i'd
What This Country Needs Is an 1 8e P iece* J effrey Shallit
This column is a place for those bits of contagious mathematics that travel from person to person in the community, because they are so elegant, suprising, or appealing that one has an urge to pass them on. Contributions are most welcome.
M i c h ael Kleber a n d Ravi Vaki l ,
M
ost businesses in the United States currently make change using four 1 different types of coins: 1¢ (cent), 5¢ (nickel), 10¢ (dime), and 25¢ (quarter). For people who make change on a daily basis, it is desirable to make change in as efficient a manner as possible. One criterion for efficiency is to use the smallest number of coins. For example, to make change for 30¢, one could, at least in principle, give a customer 30 1cent coins, but most would probably prefer receiving a quarter and a nickel. Formally, we can define the optimal representation problem as follows: given a set of D integer denominations < en and an integer N :=:: e1 < e2 < 0, we wish to express N as a non-neg ative integer linear combination N = l l "' i"'n a;e; such that the number of coins S = "l1"'i"'D ai is minimized. In order that every number actually have a representation, we demand that e1 = 1. If (a1, a2, . . . , av) is the D-tuple that minimizes S, then we say it is an opti mal representation, and we define ·
·
·
opt (N; e1, e2,
•
•
•
, en)
:=
S.
The optimal denomination prob lem is to find denominations that min imize the average cost of making change. We assume that every amount of change between 0¢ and 99¢ is equally likely. 2 We then ask, what choice of D denominations minimizes the average number of coins needed to make change? More formally, solving the optimal denomination problem for D denominations up to the limit L means determining the denominations , en which minimize e1, e2, ·
•
•
cost(L; e1, e2, 1 L
·
L
·
Q -5, f
· , en)
:=
opt(i; e1, e2,
.
•
•
, en).
E d itors
For the current system, where (e1 , = ( 1 , 5 , 1 0 , 25), a simple com putation determines that cost(IOO; 1, 5, 10, 25) = 4. 7. In other words, on aver age a change-maker must return 4. 7 coins with every transaction. Can we do better? Indeed we can. There are exactly two sets of four de nominations that minimize cost(IOO; e1, e2, e3, e4); namely, (1, 5, 18, 25) and (1, 5, 18, 29). Both have an average cost of 3.89. We would therefore gain about 1 7% efficiency in change-making by switching to either of these four-coin systems. The first system, (1, 5, 18, 25), possesses the notable advantage that we only need make one small alter ation in the current system: replace the current 1 0¢ coin with a new 18¢ coin. This explains the title of this article. Table 1 gives the optimal denomi nations of size D for 1 -s D -s 7, and their associated costs. Although the system ( 1 , 5, 18, 25) would be superior to the current (1, 5, 10, 25) for change-making, it may be difficult to convince people to accept the removal of the popular dime. Thus it may be worthwhile to consider a dif ferent question: what single denomina tion could we add to ( 1 , 5, 10, 25) to achieve the maximum improvement in cost? The unique answer is 32¢; this im proves cost(100; 1 , 5, 10, 25) = 4. 7 to cost(100; 1 , 5, 10, 25, 32) = 3.46. If we also allow the infrequently used 50¢ piece as a legitimate denomination, then the maximum improvement comes from adding an 18¢ piece; this improves cost(IOO; 1 , 5, 10, 25, 50) 4.2 to cost(I OO; 1, 5, 10, 18, 25, 50) 3. 18. Yet another reason to add an 18¢ piece to US coinage! Other countries provide different problems. In Canada, the coin denom-
e2, e;3, e4)
=
=
Please send all submissions to the Mathematical Entertainments Editor, Ravi Vakil, Stanford University,
Department of Mathematics, Bldg. 380,
*"What this country needs is a really good five-cent cigar." T. R. Marshall (US Vice-President), New York Tri
bune, January 4, 1 920. ' Informally, a 1 -cent co1n is usually called a penny, but this usage is frowned upon by numismatists.
Stanford, CA 94305-2 1 25 , USA
2This assumption is probably inaccurate for several reasons, not least being the fact that many items have prices that end in the digit 9. Also, Benford 's law on first digits of random numbers may play a role; see
e-mail:
[email protected]
Raimi [7].
20
THE MATHEMATICAL INTELLIGENCER © 2003 SPRINGER-VERLAG NEW YORK
Table 1 . Optimal Denominations for Change-Making for 1
D
7
Denominations
D
(e1 ,
·
·
·
,
cost(1 00; e1,
eo)
(1) 2
·
·
·
,
e0)
49.5
( 1 , 1 0)
9
(1 , 1 1 ) 3
(1 , 1 2 , 1 9)
5. 1 5
( 1 , 5, 1 8, 25)
3 . 89
5
(1 , 5, 1 6, 23, 33)
3.29
6
( 1 , 4, 6, 2 1 , 30, 37)
2.92
4
( 1 , 5 , 1 8, 29)
( 1 , 5, 8, 20, 3 1 , 33) 7
( 1 , 4, 9, 1 1 , 26, 38, 44)
inations currently in wide circulation are 1¢, 5¢ , 10¢, 25¢, 100¢ (called a "loonie" for the loon on the reverse), and 200¢ (called a "toonie"). The small est denomination of paper money in wide circulation is a $5 bill. Assuming each amount of change between 0¢ and 499¢ is equally likely, the average cost of making change in Canada is cost(500; 1, 5, 10, 25, 1 00, 200) 5.9. This can be best improved by adding an 83¢ coin; we have cost(500; 1, 5, 10, 25, 83, 100, 200) = 4.578. On the other hand, the new Euro system introduced by the European Union provides coins of denomination .01, .02, .05, . 1 , .2, .5, 1 , and 2 Euro. For this system we have cost(500; 1 , 2, 5, 10, 20, 50, 100, 200) 4.6. This can be best improved (to av erage cost 3.92) by adding a coin of de nomination 1.33 or 1.37 Euro. =
=
2 . 65
with denominations ( 1 , 7, 10) the greedy algorithm gives the representa tion 14 = 10 + 1 + 1 + 1 + 1, whereas 7 + 7 uses fewer coins. Unfortunately, none of the optimal sets of denominations in Table 1 for D 2 3 gives optimal representations when used greedily. For example, when we try to greedily make change for 24¢ using the system ( 1 , 12, 19), we get 19 + 1 + 1 + 1 + 1 + 1 , a far cry from the optimal representation 12 + 12. This suggests considering a varia tion on the optimal denomination prob lem, where cost is replaced by the anal ogous function gcost, and we count
D
Computational Questions
So far I have focused on systems par ticular to the US, Canada, and Europe, but a good mathematician will want more general results. Let us examine the computational complexity of the problems we have studied, and some related ones.
TABLE 2. Optimal Denominations for Greedy Change-Making
(e• •
.
, eo)
(1 )* 2
only the cost of greedy representa tions. For the current system we still have gcost(1000; 1 , 5, 10, 25) = 4.7. Table 2 displays the results for optimal sets of denominations. An asterisk de notes an optimal set for which the greedy representation is always an op timal representation. We also might consider what single denomination could be added to the current US system ( 1 , 5, 10, 25) to best improve the greedy cost. It turns out that adding either a 2¢-piece or a 3¢ piece improves gcost(lOO;-) from 4.7 to 3.9, and this is the best possible 1coin improvement. It is interesting to note that the US actually had a 2¢-piece from 1864 to 1873, and has had two dif ferent 3¢-pieces: one made in silver from 1851 to 1873, and one made in nickel from 1865 to 1889.
(1 , 1 0)*
gcost(1 00; e1,
, eo)
49.5 9
(1 , 1 1 )*
Greedy Methods for
3
5.26
(1 , 5, 23)'
Change-Making
One nice feature of the current set of US denominations ( 1 , 5, 10, 25) is that the greedy algorithm determines the representation with the minimum num ber of coins. By the greedy algorithm, I mean the following procedure: given a number N to be represented as a non negative integer linear combination of denominations e1 < e2 < < en, take as many copies an of the largest denomination en as possible, so that anen :S N. Then set N : = N - anen and continue the procedure with the remaining smaller denominations. Use of the greedy algorithm provides a sim ple, easily-remembered method for making change. Not all sets of denom inations have the property that the greedy method always determines the optimal representation. For example,
(1 , 5, 22)'
4
(1 , 3, 1 1 , 37)*
4.1
(1 , 3, 1 1 , 38)* 5
(1 , 3, 7, 1 6, 40)
3.46
(1 , 3, 7, 1 6, 4 1 ) (1 , 3, 7, 1 8, 44)* (1 , 3, 7, 1 8, 45) (1 , 3, 8, 20, 44)* (1 , 3, 8, 20, 45) 6
(1 , 2, 5, 1 1 , 25, 62)
3.13
(1 , 2, 5, 1 1 , 25, 63) (1 , 2, 5, 1 3, 29, 64) (1 , 2, 5, 1 3, 29, 65) 7
(1 , 2, 5, 8, 1 7 , 27, 63)
2.86
(27 other sets omitted) (1 , 2, 5, 8, 1 9, 30, 63)* (1 , 2, 5, 8, 1 9, 30, 64) (1 , 2, 5, 8, 1 9, 30, 66)* (1 , 2, 5, 8, 1 9, 30, 67)
VOLUME 25, NUMBER 2 . 2003
21
1. Suppose we are given an amount of change to make, say N, and a system of denominations, say 1 = e1 < ez < · · · < en. How easy is it to compute opt(N; e1, e2, · · · , en) or find an optimal repre sentation N = "2:.1-s;-sna;e;, i.e., one which minimizes "2:.1-s; -sna;? The answer depends on how N and the e; are written down. If they are writ ten in ordinary decimal notation, or in binary, then there is no fast algorithm known to solve this problem. In fact, it follows easily from results of Lueker [5] that this problem is NP-hard; roughly speaking, this means it is at least as hard as many famous combinatorial prob lems, such as the traveling salesman problem, for which no polynomial-time algorithm is currently known. If, on the other hand, N and the e; are represented in unary, then a simple dynamic programing algorithm such as the one given in [ 10] solves the optimal representation problem in polynomial time. 2. Suppose we are given N and a system of denominations. How easy is it to determine if the greedy represen tation for N is actually optimal? Kozen and Zaks [ 4] have shown that this problem is co-NP-complete if the data are provided in ordinary decimal, or binary. This strongly suggests there is no efficient algorithm for this problem. 3. Suppose we are given a system of denominations. How easy is it to de cide whether the greedy algorithm al ways produces an optimal representa tion, for all values of N? It turns out that this problem can be solved efficiently; this surprising result is due to Pearson [6] . Since Pearson's result appeared only in an obscure technical report, I give a few details. Suppose the greedy algorithm for the system of denominations 1 = e1 < ez < · · · < en is not always optimal. Pearson showed there exist integers i, j with 1 :so:: j :so:: i < D such that the min imal representation of the minimal counterexample is of the form O · e1 + 0 · e2 + · · · + O · ej - 1 + (aj + 1)ej + aj+ 1ej+1 + · · · + anen, where the greedy representation of ei + 1 - 1 is
22
THE MATHEMATICAL INTELLIGENCER
This gives the following algorithm for finding the smallest number such that the greedy algorithm fails to be op timal (or x if no such number exists):
(e1 , e2, · · · , en) := x for j : = 1 to D - 1 do for i : j to D - 1 do Let "2:. �s;sna;e; be the greedy representation of e; + 1 - 1 aJ : = aj + 1 for k : = 1 to j - 1 do ak : = 0 T : = "2:. 1 si-sna;e; if r < m then Let "2:.�s; -sn b;e; be the greedy representation of T if "2:. 1 s isnb; > "2:.�s ; -sn a; then m := r return(m) PEARSON TEST ?n
=
(Here the scope of the loops is denoted by indentation.) It is easy to see that this algorithm performs O(n:'l) arithmetic operations on numbers of size O(en). 4. Suppose we are given N and a system of denominations. How easy is it to compute cost(N; e1, e2, · · · , en)? Since opt(N; e1, ez, · · · , en) = (N + 1)cost(N + 1; e1, e2, · · · , en) - N cost(N; e1, ez, · · · , en), any algorithm to compute cost would also provide an algorithm to compute opt. It follows that computing cost is NP-hard under Turing reductions. 5. Suppose we are given L and D and want to find an optimal set of de nominations that minimizes the aver age cost of making change for all amounts from 0 to L - 1? I don't know the computational complexity of this problem, but it seems quite hard. The data presented in Table 1 were com puted using a brute-force enumeration of possibilities, but with some tricks to speed up the computation. 6. A related problem is the Fmbe nius pr-oblem. Here we are given a set of D denominations e1 < ez < · · · < en with gcd(e1, e2, · en) = 1, and we •
•
want to find the largest integer N which cannot be expressed in the form "2:.1-s ; -sna;e; with the a; non-negative in tegers. There is a huge literature on this problem (see, for example, Guy [2, pp. 1 13-1 14 ]), but only recently have re searchers considered its computational complexity. Kannan [3] gave an algo rithm for the Frobenius problem that runs in polynomial time if the dimension D is fixed. On the other hand, Ramirez Alfonsin [8] has shown that the general Frobenius problem is NP-hard. 7. Another related problem is the postage stamp pr-oblem. There are two flavors. The "local" problem asks, given a set of D denominations 1 = e1 < e2 < · · · < en and a bound h, what is the smallest integer N which cannot be represented in the form N = "2:. 1 -s ; -sna;e; where the a; are non-nega tive integers and "2:.1sisna; :so:: h? In the "global" version, we are given D and h and want to find the set of denomina tions that maximizes N. There is a larger literature on these two problems (see Guy [2, pp. 123-127]), with much effort devoted to finding efficient algo rithms for small D. However, I recently showed [9] that the local postage stamp problem is NP hard under Turing reductions, and that there is a polynomial-time algorithm for every fixed D. Asymptotic Results
Now we turn to some asymptotic esti mates. Let optcost(L, D) denote the mini mum value of cost(L; e1 , e2, , eD) over all suitable values of e1, · · · , ev. Can we find good upper and lower bounds on optcost(L, D)? One way to find an upper bound is as follows: let k IL 1ml, and define 1 e; = ki - for 1 :so:: i :so:: D. In this case, the greedy algorithm always finds the op timal representation for any N, and it turns out to be the base-k expansion of N. Letting sk(N) denote the sum of the digits in the base-k expansion of N, we find •
•
•
=
cost(L; e1, e2, · · · , en) = gcost(L; e 1 , e2, · · · , ev) 1 L
I
O<S i SL - 1
sk(i).
Hence (1)
where Sk(N) : = 'io-5i
� N log N + N F.k 2log k
(
)
log N log k '
(2)
where Fk is a continuous, nonpositive, nowhere-differentiable function of pe riod 1; see, for example, [ 1 ] . Combin ing (1) and (2), we obtain the upper bound
Sk(kN + a) = kSk(N)
+
2
+ ask(N) +
a(a
·
problem.
Theoret.
(1 994), 377-388.
[5] G. S. Lueker. Two N P-complete problems in
nonnegative
integer
programming.
Technical Report TR- 1 78, Computer Sci ence Laboratory, Department of Electrical Engineering, Princeton University, March 1 975.
[6] D. Pearson. A polynomial-time algorithm for the change-making problem . Technical puter Science, Cornell U niversity, June 1 994. Available from http://citeseer. nj.nec. com/pearson94polynomialtime.html. [7] R. A. Raimi. The first digit problem. Amer. Math. Monthly 83
Frobenius
problem.
the local postage stamp problem. SIGA C T (2002) , 90-94.
[1 0] J. W. Wright. The change-making prob lem. J. Assoc. Comput. Mach. 22 (1 975), 1 25-1 28.
(1 975), 3 1 -47. Department of Computer Science University of Waterloo
Springer-Verlag, 2nd edition, 1 994.
[3] R. Kannan. Lattice translates of a polytope
·
16
[9] J . Shallit. The computational complexity of
[2] R. K. Guy. Unsolved Problems in Number Theory
Comb1natorica
(1 996), 1 43-1 47.
Ia fonction "somme des chiffres." Enseign. Math. 21
(1 976), 5 2 1 -538.
[8] J . L. Ramirez-Aifonsin. Complexity of the
REFERENCES [1] H. Delange. Sur Ia fonction sommatoire de
2
one can compute Sk (N) in time poly nomial in the number of digits in k and N. This provides a fast way to compute the upper bound (1). For a lower bound, one may reason as follows: fix a set of D denominations e1, e2, , eD, and consider the num·
Comput. Sci. 1 23
News 33 (1 )
1)
-
the change-making
I obtained the results in Tables 1 and 2 in October 1999. Erik Demaine kindly pointed out that similar results were posted to the Usenet newsgroup sci.math by Jeffry Johnston and Bill Kinnersley in April 2000. I thank Troy Vasiga, Ming-wei Wang, and Erik Demaine for their helpful comments.
Furthermore, using the identity
1)N
·
Acknowledgments
D liD optcost(L, D) ::; L . 2
-
·
Report TR 94-1 433, Department of Com
-
k(k
[4] D. Kazen and S. Zaks. Optimal bounds for
, ber of different D-tuples (a l > a2, aD) such that 'i1-5i-5D a i :S k. A simple combinatorial argument shows that D / this number is ( J ) . Now if (DJ/) ::; L/2, it follows that for at least L/2 choices of N, 1 :S N :S L, any represen tation for N must use at least k + 1 coins, and hence optcost(L, D) 2: l.(k + 1). Now (D + k) :S (k+D)n.' if D is D D' 2 fixed and L � oc, this gives the lower bound of optcost(L, D) = D(L lln:>. ·
1 optcost(L, D) ::; 8 11-'ml (L) L
Waterloo, ON N2L 3G1
and the Frobenius problem. Combinator
Canada
ica 1 2
e-mail: shallit@graceland. uwaterloo.ca
(1 992), 1 6 1 -1 77.
The Cosmological Complaint 17u·
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VOLUME 25, NUMBER 2. 2003
23
MICHAEL S. LONGUET -HIGGINS
N ested Tri aco ntah ed ral S h e s Or H ow to G row a Q u as i - crystal
he principal motivation for this article is illustrated by Figure 1, reproduced from Figure 2. 6 of Janot [ 1]. This is a micrograph of a crystal grain of the alloy Al Cu Li; the grain is about 1!2 mm in diameter. (For further details see Lang et al. [2]. ) On examination it will be seen to have the form of an almost perfect triacontahedronthat is, a polyhedron with 30 rhombic faces, first described mathematically by Kepler [3] and named after him by Kowalewski [4] . Remarkably, the symmetry group of the triacontahedron is not one of the classical crystallographic groups, for it includes fivefold rotations. Such crystals, ap parently defying the laws of crystallography, have come to be known as "quasi-crystals" [ 1 ] . Since their discovery b y X-ray diffraction analysis in 1984 [5] , quasi-crystals have been intensively studied, yet some puzzles remain. MacKay [6] proposed a crystal pat tern related to the Ammann (3D Penrose) tiling, where the basic cells are all golden rhombohedra (see below). Each cell or vertex of the pattern is decorated in a systematic way by atoms of the metal alloy. The pattern of cells by itself is sometimes called a "toy model" for the crystal. But the Penrose model has the drawback that its growth requires hypothetical long-range physical forces, or else complicated matching rules [7] unlikely to be realised in practice. Another feature requiring explanation is the re markable smoothness of some of the crystal faces shown
Figure 1. Triacontahedral crystals of AI Li Cu (courtesy of M. Audier).
© 2003 SPRINGER-VERLAG NEW YORK, VOLUME 25, NUMBER 2 . 2003
25
Figure 2. A sharp rhombohedron S, and five S 's surrounding an edge.
in Figure 1; see also Janot [ 1 ] , Fig. 4. 1 . Hence a different toy model seems to be required. Crystals normally grow by the gradual accretion of atoms at their surfaces along certain "growth-lines" [8]. Hence we require an explanation of how an assembly of golden rhombohedra in the form of a perfect triacontahe dron can grow, step by step, into a slightly larger triacon tahedron by the addition of only a thin surface layer of golden rhombohedra. Such an explanation will be provided below. It should be of interest to geometers, and may have
Figure 3. A flat rhombohedron F, and ten F's surrounding an edge.
26
THE MATHEMATICAL INTELLIGENCER
something to offer to physicists, who have now undertaken the study of how quasi-crystals grow in nature. First recall some known geometrical results. The two "golden rhombohedra" are illustrated in Figures 2 and 3. Sometimes [ 1 ] , [9] called "acute" and "obtuse," they will here be given the more homely names "sharp" and "flat." Thus a sharp rhombohedron may be derived, in imagina tion, by taking two opposite vertices of a cube and draw ing them apart until the six rhombic faces have diagonals in the ratio T : 1 , where T is the golden ratio (V5 + 1)/2 =
Figure 4. Two views of the rhombic dodecahedron D.
1 . 6 18 . . . . Then it may be shown that the dihedral angle at each edge leading to a sharp vertex is equal to 2 1T/5; hence precisely 5 of the acute rhombohedra can be fitted round each sharp edge; see Figure 2. The six remaining edges have dihedral angles equal to 31T/5. Aflat rhombohedron can be obtained by taking two opposite vertices of a cube and pushing them inwards, until the diagonals of the rhombic faces are in the ratio l : r. The dihedral angles at the edges are then equal to 1r/6 or 41T/5. Hence precisely 10 F 's can be fitted round an edge; see Figure 3. The sharp and flat rhombohedra will be denoted by S and F, respectively. From the above description it will be clear that the faces of F are all congruent to the faces of S, although turned, so to speak, through goa about an axis normal to the plane, provided both F and S have the same edge-length.
A basic assumption made in the present article is that each of the golden rhombohedra can be decorated with atoms of the metal alloy in some way, as was suggested by MacKay [6). The discussion and description of the assemblies of golden rhombohedra will be assisted by the use of a cer tain set of colored blocks [ 10] held together magnetically. In the illustrations the sharp rhombohedra are colored red and the flat rhombohedra are yellow. The simplest such assembly consists of two sharp and two flat rhombohedra which combine to form Bilinski's [ 1 1 ] rhombic dodecahedron D , whose edges are each parallel to one ofjour different vectors. Figure 4 shows two views. Note that, apart from the 14 vertices on the surface, the model has one additional vertex in the interior.
Figure 5. Two views of the rhombic icosahedron /.
VOLUME 25, NUMBER 2, 2003
27
Next in the series of rhombohedra is the rhombic icosa hedron I whose edges are each parallel to one offive dif ferent vectors; see Figure 5. The model consists of 5 sharp and 5 flat golden rhombohedra. There are 4 vertices in the interior. It will be noted that I has two enantiomorphic forms. Each may be obtained from the other by reversing a constituent D. Next is the Kepler Ball K (see Fig. 6) whose edges are each parallel to one of six vectors. (For this reason, D, I, and K can be considered as the projections onto three-di mensional space of parts of hypercubic lattices in 4, 5, and 6 dimensions, respectively.) As noted by Kowalewski [4) and MacKay [6), it can be assembled from ten sharp and ten flat rhombohedra. The fivefold assembly in Figure 2 is first inverted, then five F 's are placed around the outside to form a hollow. A rhombic icosahedron I is then placed in the hollow to form a K. The surface of the model has 32 vertices; there are also 10 in the interior. Note that the arrangement of the internal components has no overall ro tational or reflexive symmetry. Interestingly, this is not the only possible way to build a triacontahedron. As shown in Figure 7, we may build up a Kepler Ball in an essentially different way [ 10], starting with a flat rhombohedron, placing on it three sharp rhom bohedra in a left-handed symmetric way and building up the rest of the ball maintaining always a three-fold axis of rotational symmetry. (We could also start with right handed symmetry, producing the mirror image.) This al ternative composition of a Kepler Ball will be denoted by K*. It also consists of ten sharp and ten flat rhombohedra. That the number (ten) of sharp rhombohedra in K* is the same as in K follows from the fact that the volumes of F and S are incommensurable, being in the ratio T, an irra tional number. For a proof see Appendix A From Figure 7 it can be seen that K* is indeed distinct from K, for K* has no vertex surrounded by five S 's.
Figure 6. Assembly of a Kepler Ball K.
28
T H E MATHEMATICAL INTELLIGENCER
Clearly one can build up either of the triacontahedra K or K* by adding 5 further rhombohedra of each kind onto one side of an I; and by adding 5 further sharp and 5 flat rhombohedra on the other side of the I we get a configura tion E of two K's having an I in common: see Figure 8. Ei ther or both of the K 's may of course be replaced by a K*. Similarly we may construct a useful combination of two K's having in common a single flat rhombohedron; see Fig ure 9. This has been called "Siamese Twins" [ 10], and will be denoted by T. Again, there is a corresponding "Siamese Twins" in which each K is replaced by a K*. A Kepler Ball K or K* constructed from golden rhom bohedra of unit edge-length evidently has the same unit edge-length. It may be denoted specifically by K(1). I shall now show how it is possible to construct a Kepler Ball K(2), with twice the edge-length, from golden rhombohedra of unit edge-length, and then in general how to derive a Kepler Ball K(n + 1) of side n + 1 from a K(n). For simplic ity and ease of description I shall pay particular attention to arrangements with some degree of rotational symmetry. Construction of a K(2) This, then, is our first question: is it possible to construct a K(2) symmetrically by surrounding a K(l) completely by layers of sharp and flat rhombohedra? Consider first the thickness of any such layers. Let hs denote the altitude of a sharp rhombohedron S, that is, the perpendicular distance between opposite plane faces, and let hp be the altitude of a flat rhombohedron F. It may eas ily be shown [9) that hs Thp, where T is the golden ratio. The altitude of a Kepler Ball equals 2hs + 2hp. (As proof, we may define a worm as a line drawn from the center of one face w1 of a Kepler Ball to the center of the opposite face w2 of the corresponding golden rhombohedron; then from Wz to the opposite face wa of the adjacent rhombo hedron, and so on, ending at the face wn of the Kepler Ball =
4.
5 Figure 7. Assembly of K*.
VOLUME 25, NUMBER 2 , 2003
29
Figure 8. An E.
opposite to w1 . Thus a Kepler Ball may be thought of as a can of 15 worms, with 3 worms passing through the centre of each rhombohedron. The two ends of the worm lie on two opposite faces of the Ball. At least one of the worms (and hence all of them) will be found to pass through two F 's and two S's. The altitude of the Kepler Ball (the dis tance between opposite faces) is just the sum of the alti tudes of all the golden rhombohedra through which the worm passes, that is to say, 2hF + 2h8.)
Figure 9. "Siamese Twins" T.
30
THE MATHEMATICAL INTELLIGENCER
Now the K(1) and K(2) cannot be concentric, for then their corresponding faces, having edges of length 1 and 2, respectively, would have centres on the same radius, which is clearly impossible. But we may assume that the centres of K(l) and K(2) lie on the same trigonal axis of symme try. We can then say that the thickness of the layer of rhom bohedra covering one face of K( 1) , when added to the thick ness of the layer on the opposite face, must be equal to the difference in altitudes of K(1) and K(2). Since K(2) has
5
6
Figure 1 0. Type A construction of K(2) from K(1). (Continued on next page.) VOLUME 25, NUMBER 2, 2003
31
11 Figure 1 0 continued. Type A construction of K(2) from K(1 ).
32
T H E MATHEMATICAL INTELLIGENCER
12
twice the dimensions of K(1), this equals the altitude of K(1), namely 2hs + 2hp. Consider three rhombic faces of a K(1) having a com mon vertex on the axis of symmetry. Call these "a-faces." The next 6 adjacent faces we may call "b-faces," the next 3 neighbors "c-faces," the next 6 "d-faces," and so on, end ing with 3 "g-faces"; in all, 30. Starting with a K(1), let us leave the a-faces bare but lay an F on each of the b-faces, as in Figure 10. 1. Then add three F's to complete the a(2)-face, and place an S on each of the c-faces (Figure 10.2). Turning the model over, add an S to each d-face and an F cheirally to each e-face (Figure 10.3). Then add an S and a second F to each e-face (Figure 10.4). Next add one S to eachj-face and 3 further S's cheirally (Fig ure 10.5). Now we can add an F to each g-face and fill up the center with F's to form a cap (Figure 10.6). The central F will be the beginning of a second K*(1) coaxial with the first. Thus we cover the centre of Figure 10.6 with S's cheirally (Figure 10. 7), adding F 's around the rim so as to complete the b(2) faces. In Figure 10.8 we fill the cracks with F's and S's, in a zigzag fashion, then build up the centre with a layer of S's (Figure 10.9). Add F's so as to complete the d(2) and e(2) faces (Figure 10. 10), and finally add F's to complete the K(2) (Figure 10. 1 1). A side view is shown in Figure 10.12. The interior configuration of this K(2) is trigonally sym metric throughout. On the surface just 12 S's are visible: the minimum number. For, at each of the 12 pentagonal vertices of the K(2) there must be at least one S, and no two vertices of K(2) can share the same S. Of particular interest is the sequence of thicknesses of the layers of rhombohedra covering the faces of the K(1). From the a-faces to the g-faces, the thicknesses are, in order,
Note that the sum of the thicknesses on pairs of opposite faces (a, g), (b,j), etc., sum to 2hp + 2hs. Any solution yield ing the above sequence I shall call "Type A."
In any Type A solution the pattern of rhombohedra need not be coaxial with the original K(1) or K*( 1). However, there is a second, altogether different, type of solution which I shall call Type B. Beginning with the initial K(l), lay on the a-faces a cap of 7 F 's, as in Figure 1 1 . 1 . Next place an S on each b-face (Figure 1 1.2), place an F cheirally on each c-face, adding 3 further S's (Figure 1 1.3); place an S on each d-face and a second F cheirally on the c-faces. Add 3 further S 's (Figure 1 1 .4). Turning the model over, complete the b(2)-faces with F 's and fill in with S 's to ob tain Figure 1 1.5. Add an S to each e-face and an F to each !-face, and fill in with F 's (Figure 1 1 .6). Now the exterior surface of 1 1.6 is identical in form with that of 10.6 (though the interior differs). We can therefore build up from 1 1.6 in the same way as from 10.6 to get Figure 1 1. 7, which is the same as Figure 10. 1 1. However, the side view, Figure 1 1.8, shows that the pattern of S's on the outer shell is dif ferent from the Type A covering (Figure 10. 12), being es sentially cheiral. Note that in building up from stage 1 1.6 we have constructed a second K*(1) coaxial with the first, so as to form a Siamese Twin T. The sequence of layer thicknesses is now as follows: hp, hs, 2hp, hp + hs, 2hs, 2hp + hs
and
hp + 2hs.
(B)
Any solution with the above sequence I shall say is of Type B. Types A and B are the only possible thickness sequences in a trigonally symmetric covering of K(l). For, consider segments of the golden rhombohedra along the axis of sym metry (Figure 12). These must sum to the diameter dK of a K*(l). But by Figure 7, dK is the sum of dp and ds, where dp and ds denote the segments of F and S along their tri gonal axes of symmetry. These two segments are either on the same side of the original K*(1 ), as in Figure 12. 1 , or on opposite sides, as in Figures 12.2 and 12.3. These two pos sibilities yield solutions of Type A and Type B, respectively. There are no others.
2 Figure 1 1 . Type B construction of K(2) from K(1 ). (Continued on next page.)
VOLUME 25, NUMBER 2, 2003
33
8
7
Figure 1 1 continued. Type B construction of K(2) from K(1 ).
34
THE MATHEMATICAL INTELLIGENCER
Some terminology will be useful. Defme a carpet of rhombohedra as an (n X n X 1) array of golden rhombo hedra (of the same kind), covering an n X n rhombic face such as b(n), for example. All the rhombohedra are ori ented identically. Afringe is an (n X 1 X 1) array, oriented similarly, adjoining the "edge" of two different arrays, and a tassel is a single cell, i.e., a (1 X 1 X 1) array at the join or extension of two or more fringes. In a Type A construction, take the following steps, cor responding to Figures 13. 1 to 13. 1 1:
ds
ds
ds
Figure 1 2. Segments of the trigonal axis of K(2).
Construction of a
K(3}
Just as K(2) was constructed from K(I), so one can also construct a K(3) by starting from a K(2) and surrounding it by a relatively thin layer of golden rhombohedra. It will save space, however, if I describe at once how to derive a K(n + 1) from a K(n) for general values ofn, using the case n 2 as an illustration (Figs. 13 and 14). =
(1) Leave the a(n)-faces bare, and cover each of the b(n) faces with a carpet of F's. (2) Complete the a(n + 1)'s with three fringes of F's and lay a carpet of S's on each of the c(n)-faces. (3) Turn the model over. Lay a carpet of S's on each of the d(n)-faces. (4) Lay a carpet ofF's cheirally on each e(n)-face and a car pet of S's on eachj(n)-face, with a cheiral fringe of S 's. (5) Lay a second carpet of F's, cheirally, on each of the carpets covering the e(n)-faces. (6) Lay a carpet of F's on each of the f(n)-faces, and fill in with fringes of F's and a tassel in the centre. The latter will be the start of a coaxial F"( 1 ). (7) Cover the upper surface cheirally with a layer of F 's, leaving three zigzag canyons meeting at the centre. (8) Fill in the canyons with F 's and S's. (9) Cover the F 's with a layer of S's. (10) Complete d(3)-face with a carpet of F 's. (This also completes the e(3)-faces.) ( 1 1 ) Add F's to complete the K(n + 1). Figures 13. 1 1 and 13. 12 are overhead and side views, re spectively, of a completed K(3) (Type A). The Type B con struction, illustrated in Figures 14. 1 to 14.8, is as follows:
2.
Figure 13. Type A construction of K(3) from K(2). (Continued on next page.) VOLUME 25, NUMBER 2, 2003
35
7 Figure 13 continued. Type A construction of K(3) from K(2).
36
THE MATHEMATICAL INTELLIGENCER
8
11.
12.
Figure 13 continued. Type A construction of K(3) from K(2).
(1) Lay a carpet of F 's on each of the a(n)-faces and com plete the a(n + 1)-faces by adding fringes of F's and an F tassel at the top. (2) Lay a carpet of S's on each of the b(n)-faces. (3) Lay a carpet of F's, cheirally, on each of the c(n)-faces, and add fringes of S's. (4) Lay a second carpet of F 's on each of the first, and add a carpet of S 's to each d(n)-face. (5) Tum the model over and add a second carpet of S's to each carpet on a d(n)-face. (6) Lay a carpet of S's on each e(n)-face and a carpet of F's on each f(n)-face, adding fringes of F's. Figure 14.6 will be seen to be very similar to Figure 13.6, and so K(n + 1) may be completed in the same way, as in Figure 14.7. The side view (Fig. 14.8) shows that the outer shell of the complete model is cheiral, unlike the Type A shell (Fig. 13. 12).
The proof that Types A and B are the only ones possi ble follows from a generalisation of Figure 12. We simply have to replace the segment dK by the diameter of a K(n), which is n times as long. The segments dF and ds remain unchanged. Thus the whole K(n + 1) is covered by a layer of rhombohedra no more than four deep.
Discussion
The method of obtaining a K(n + 1) by covering the faces of a K(n) with thin layers of F's and S's, which we have demonstrated in the cases n = 1 and 2, is evidently quite general. It seems possible that this or the pentagonal con struction described below is a method by which large tria contahedral quasi-crystals are actually formed. This does not of course exclude the possibility that larger assemblies of golden rhombohedra are involved, particularly at the ini tial stages of crystallisation.
VOLUME 25, NUMBER 2 , 2003
37
5.
6.
Figure 1 4. Type B construction of K(3) from K(2). 38
THE MATHEMATlCAL INTELLIGENCER
7.
8.
Figure 14 continued. Type B construction of K(3) from K(2).
For example, to construct a K(3) we can start with two Siamese Twins and a Siamese Triplet (Fig. 15. 1). When brought together they appear as in Figure 15.2. Building on these, the completed K(3) has digonal symmetry about a vertical axis. To construct a K(2) whose outer shell has pentagonal sym metry, let us label the thirty faces of a K(1) as follows: call the five faces surrounding a given pentagonal vertex a's; the five adjoining faces {3's; the next ten adjoining faces (which are all parallel to the pentagonal axis) y's; the next five O's, and the last five E's. Taking the K(1), leave the a-faces bare and lay one F on each �face. Next lay an S on each of the y-faces. Proceeding cheirally in a way very similar to the trig onal Type A construction, we may arrive finally at a K(2) whose outer shell is shown in Figure 16. The view from the opposite end is similar. In fact there is a second K(1), coax ial with the first, along the pentagonal axis. Since the interior of a K(1) does not have five-fold rotational symmetry, nei ther does the resulting K(2). But the K(2) may be constructed so as to have reflexive symmetry in the central point 0, say. By reversing each K(1) individually, we get a Kepler Star sur rounding 0, that is, a configuration consisting of 20 S's meet ing at the central point; see [ 12] and [ 10]. The sequence of thicknesses of the layers of rhombo hedra surrounding the original K(1) will be seen to be 0, hF, hF + hs, hF + 2hs, 2hF + 2hs
(C)
for the faces a, {3, -y, 8, and E, respectively. This sequence corresponds to Type A in a trigonal construction. There is also a Type D construction with the sequence (D)
which corresponds to the Type B trigonal construction. This starts by placing an S on each a-face. The final shell is not pentagonally symmetric.
Note that a complete K(n), whatever its internal struc ture, must have equal numbers of component S 's and F's. For, it would be possible in theory to construct a K(n) from ten S(n)'s and ten F(n)'s. Each S(n) would be formed of an n X n X n lattice of S(1)'s and similarly each F(n) could be formed of an n X n X n lattice of F(l)'s. In this instance, therefore, the K(n) has 10n3 S(1)'s, and similarly 10n3 F(1)'s. By Appendix A, these numbers of S's and F's are indepen dent of their particular internal arrangement in the K(n). I have purposely not discussed the possible decoration of the rhombohedra by the different metal species in the alloy. Although the X-ray diffraction of the corresponding crystal pattern will indicate the predominance of icosahedral sym metry, it should be noted that in many respects the particular arrangements described here are not unique. For example, in places where a triacontahedron occurs locally, a I0' may be replaced by a K or vice versa This will tend to blur the peak densities in the observed diffraction pattern, although the po sitions of the chief maxima will remain invariant. It will be noted that the method of assembly proposed above does not require the existence of such long-range forces as would be needed to assemble an Ammann tiling, considering the absence of any simple set of local match ing rules in three dimensions. For other types of quasi-crystal, including those with icosahedral symmetry but not producing grains in the form of triacontahedra, the reader is referred to Janot's textbook [ 1 ] , especially Table 2.1. The existence of systems with 2dimensional, decagonal phases may be noted. An analogous model for the growth of such quasi-crystals is suggested in Appendix B. Appendix A
Consider the volume Vs of a sharp rhombohedron S. As seen from Figure 2, Vs equals the area R of a rhombic base times
VOLUME 25, NUMBER 2, 2003
39
2. Figure 1 5. A diagonal basis for K(3).
the height of S, that is, the perpendicular distance between two opposite faces. Similarly VF equals R X HF (Figure 3). Hence
Vs VF
=
Hs HF
(A. l )
But from Figure 4 we can see that 2Hs L8, where Ls is the length of the long diagonal of a rhombic face; and sim=
40
THE MATHEMATICAL INTELLIGENCER
ilarly from Figure 4 we see that 2HF LF, where LF is the length of the short diagonal. Therefore =
Hs HF
=
Ls = T, LF
(A.2)
the golden ratio. Thus the volumes V8 and VF are in the same ratio T.
Figure 16. A pentagonally constructed K(2).
Now K* and K have the same total volume. If K* con sists of m1 S's and n 1 F's while K consists of m2 S's and n2 F's then we must have
hence (A.4)
There are two solutions: a Type A solution is shown for n = 1 in Figure 18. The inner decagon Ll(1) and the outer decagon Ll(2) have two sides in common, so that the thick ness of the covering layer on those two sides is zero. On the next two sides the thickness is h<J>. On the next it is (hct> + hi), and so on. The complete sequence is 0, h¢, hct> + hi, hct> + 2hi
and
2h¢ + 2hi.
(A')
The Type B solution is shown in Figure 19. The sequence of layer thicknesses is now
Thus, if m1 * mz, we have Vs nz - n1 = m 1 - mz Vp
(A.5)
a rational number. But we saw earlier that VsfVF equals r, m2 and n1 = n2. an irrational number. Hence m1 The above proof clearly applies to any other two differ ent assemblies of S 's and F 's having the same exterior sur faces. =
Appendix B. Crystal Growth In Two Dimensions
Corresponding to the golden rhombohedra F and S, con sider the two elementary rhombuses (or tiles) having acute angles of 7T/5 and 2 7T/5. Denote these by
h¢, h'£., hct> + h'£. , 2hct> + h'I.
and
hct> + 2hi.
(B ' )
The sequences (A') and (B' ) are analogous to (A) and (B), respectively. The sum of the thicknesses on opposite sides of the decagon Ll(1) is always 2h¢ + 2h'i., which is the alti tude of a Ll(1). In the case n = 2, the Type A and Type B solutions are shown in Figures 20 and 21, respectively. The situation for general values of n is most easily understood from Figures 18 and 20. In Figure 20 the arrangement of tiles between any two parallel sides of the inner and outer decagons is repeated. If one unit of each of the repeated sections were removed, we should retrieve Figure 18, the solution for n = 1; and if one more were removed, we should obtain simply Figure 17.1. Re versing the process and inserting an extra section near the middle of each edge in Figure 20, we obtain the solution for n = 3, and so on. By induction, it is clear that we have a Type A solution for any positive integer n. Similarly for Type B.
VOLUME 25, NUMBER 2 , 2003
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Figure 1 7. Three tilings of a decagon.
Figure 18. Type A construction of ll(2) from 11(1).
Figure 1 9. Type B construction of Ll(2) from ll(1).
Figure 20. Type A construction of 11(3) from J-(2).
Figure 21. Type B construction of ll(3) from ll(2).
42
THE MATHEMATICAL INTELLIGENCER
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REFERENCES
1 . Janot. C . , Quasicrystals: A primer (2nd ed .). Oxford, Clarendon Press, 1 994. 2 . Lang, J . M . , Audier, M., Dubost, B., and Sainfort, P., Growth mor
phologies of the AI-Li-Cu icosahedral phase. J. Crystal Growth 83
(1 987), 456-465 .
MATHEMATICS:
•
A lgebra
•
Combinatoric and Graph Theory
•
Computational Mathematic and Scientific Computing
3. Keppler, I. (sic) , Harmonices Mundi, Liber V. Vienna, Lincii Austriae. I. Plancus, 1 61 9.
•
Dynamical Sy tern
4. Kowalewski, G . , Der Keplersche K6rper und Andere Bauspiele . Leipzig, K.F. Koehlers Antiquarium, 1 938. 5. Shechtman, D . , Blech, 1 . , Gratias, D., and Cahn, J. W., Metallic
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Fluid and Mechanic
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Functional Analysis and
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General Mathematic
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Geometry and Topology
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Mathematical B iology
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Mathematical Method
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MathematicaJ Phy ic
Operator Theory
phase with long-range orientational order and no translational sym metry. Phys. Rev. Lett. 53 (1 984), 1 95 1 -1 953. 6. MacKay, A.L. , Crystallography and the Penrose pattern. Physica 1 1 4A (1 982), 609-61 3 .
7. Katz, A . , Theory o f matching rules for the 3-dimensional Penrose tilings. Commun. Math. Phys. 1 1 8 (1 988), 263-288. 8. Burton, W. K . , Cabrera, N. and Frank, C. F. , The growth of crystals and the equilibrium structure of their surfaces. Phil. Trans. R. Soc. Land.
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(1 960), 251 -262.
1 2. Unkelbach, H . , Die kantensymmetrisch gleichkantigen Polyeder. Deutsche Mathematik 5
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lj¥1(9·\· (.j
Dav i d E .
From KOnigsberg to GOttingen: A Sketch of H i l bert's Early Career David E. Rowe
Send submissions to David E. Rowe,
Rowe ,
E d itor
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I
avid Hilbert's remarkable career falls into two clearly distinct peri ods: the quiet Konigsberg phase, from his birth on 23 January 1862 to his full maturity as one of Germany's leading mathematicians, followed by the tu multuous Gottingen years. The latter began with his appointment in Gottin gen in 1895 and ended with his death on 14 February 1943 when Nazi Ger many had already entered its death throes. It would be difficult to exag gerate the contrast between these two phases, just as it remains difficult to picture life in Germany before the two world wars. 1 The East Prussian city of Hilbert's youth stood for solidity, in tegrity, and a harsh, no-frills lifestyle rooted in the Protestant work ethic. Hilbert identified with Konigsberg's so cial ambience all his life. In Gottingen, his students and friendly admirers loved to mimic his distinctive East Prussian accent with its unfamiliar twang that set him apart from those who cultivated the high German of good Hanoverians. Neither did Hilbert make any particular effort to hide his own preference for Konigsberg over Gottingen; he treasured no honor higher than the one the city of his birth bestowed on him in 1930 when he was made an honorary citizen [Reid 1970] . For many years Konigsberg had been the most important Prussian out post in eastern Europe. Situated on the Pregel, only a short distance inland from the Baltic Sea, Konigsberg lay at the heart of a vibrant trade network that linked it with St. Petersburg to the east and Hamburg to the west. But by the end of the nineteenth century the German Empire had become an indus trial power whose economic strength stemmed from the steel and chemical firms of the Rhineland. The plains east of Berlin, mainly lands owned by Junker aristocrats-men like Otto von Bismarck and others who dominated
the officer corps of the Prussian army-became increasingly marginal for the economy of the Second Reich. Konigsberg retained its importance as a regional center, but the surrounding terrain to which it belonged was part of a semi-feudal world with an uncer tain future. Academic life in Germany had a strong local coloring through most of the nineteenth century, and the rela tionship between town and gown was quite different at Konigsberg than in the quaint atmosphere of Gottingen. Still, both cities were proud of their dis tinguished universities which devel oped strong traditions in mathematics and the natural sciences. The "Al bertina" in Konigsberg was founded by Duke Albrecht of Brandenburg in 1544 as a Protestant bulwark against the Counterreformation. Its later fame be gan in the eighteenth century when the great philosopher Immanuel Kant graced its faculty. The much younger Georgia Augusta in Gottingen, as the product of a more enlightened, Kantian age, managed to attract a number of leading scholars right from the start. Founded in 1 737 by the Elector of Hanover George II, who happened also to be King of England, Gottingen became the prototype for the modem universi ties of the nineteenth century by ele vating the status of its philosophical faculty, whose members no longer had to answer to their colleagues in theol ogy. Gottingen's reputation in mathe matics, astronomy, and physics rose sharply at the beginning of the nine teenth century after it acquired the ser vices of Carl Friedrich Gauss, who taught there from 1807 to 1855. Al though distant and aloof toward the younger generation of mathematicians in Germany who looked up to him, Gauss exerted a deep influence on such figures as Dirichlet, Jacobi, Eisen stein, and Kummer, who pursued
Fachbereich 1 7 - Mathematik, Johannes Gutenberg University,
' Hermann Weyl broke Hilbert's career into five phases based on his evolving mathematical interests in [Weyl
055099 Mainz, Germany.
1 944], but this breakdown obscures some of the most striking features of his bipartite career.
44
THE MATHEMATICAL INTELLIGENCER © 2003 SPRINGER-VERLAG NEW YORK
higher mathematics, and particularly number theory, with an extraordinary passion. Gauss's fame and influence, mediated by the impressive achieve ments of these and other leading num ber-theorists, made itself felt on Hilbert as well, and in a truly profound way. Hilbert's professional mathematical career began officially in Konigsberg in 1886 when he joined the faculty as a Privatdozent, an unsalaried position that left a young man with meager re sources in a precarious situation. In Hilbert's case, it took another six years before he became a Professor Extra ordinarius in 1892, and then, only a year afterward, he was elevated to Or dinarius at the age of thirty-one. It was a long, hard climb for Hilbert, but along the way he acquired an enormous breadth of mathematical knowledge that served him well during the second phase of his career, which began in 1895 when he was called to Gottingen as the successor of his former teacher Heinrich Weber. This geographic shift meant leaving behind that familiar en vironment he had enjoyed throughout his student years and beyond-an at mosphere of quiet solitude conducive to deep mathematical contemplation to enter another kind of academic world, one only just taking form and to which Hilbert was called upon to con tribute in an extraordinary way. During the next twenty years Hilbert emerged as the era's most influential mathemati cian, attracting scores of talented disci ples from around the world. Whereas in Konigsberg he had not even one doc toral student, in Gottingen he super vised the dissertations of no fewer than sixty aspiring mathematicians between the time of his arrival and the outbreak of World War I. That averages out to better than three a year, an impressive figure even today; in those days, it was simply unheard of. A number of these Hilbert students went on to ef\ioy dis tinguished careers, and a few left a deep imprint on twentieth-century mathe matics, among them Hermann Weyl, Richard Courant, Erhard Schmidt, and Erich Heeke. The Gottingen years were thus the period of Hilbert's real fame and glory, and it is only natural that they have re ceived far more attention than the
Konigsberg period. These more distant events in time and space have been di minished all the more because the later Hilbert, the deeply dedicated teacher whose self-confidence and sarcastic wit set the tone for Gottingen's whole mathematical community, so quickly became a legend. The young Hilbert, who worked in virtual isolation, pub lishing only a small fraction of the mathematical ideas that occupied his attention, this Hilbert was never well known and was therefore easily for gotten after he stepped up to the Paris podium in 1900. Yet the Konigsberg pe riod not only had a deep significance for him personally but also laid the foundations for the dramatic success
The first to exert significant influence on H ilbert was Heinrich Weber . he later enjoyed in Gottingen. Hilbert's triumph there was a matter of reaping the fruits of the fifteen years he had spent as a student and aspiring acade mic in the city of his birth. Hilbert in Konigsberg
No university played a greater part in Germany's sudden ascendancy in the world of mathematics than did Konigs berg's Albertina. Its fame in the natural sciences was launched during the 1830s by the physicist Franz Neumann and the brilliant analyst Carl Gustav Ja cob Jacobi. Following Jacobi's depar ture for Berlin in 1844, his successor and leading disciple, Friedrich Riche lot, continued to promote the ideals of the master. Largely through his efforts, the dynamic Konigsberg tradition in mathematical physics, analysis, and analytical geometry made its influence felt in nearly early university in Ger many. Hilbert's initial exposure to higher mathematics came through two representatives of the Jacobian tradi tion in analysis. In the winter semester of 1880-8 1, Hilbert studied analyti cal geometry with Georg Rosenhain
and differential calculus with Louis Saalschlitz. Both were natives of Konigsberg who had gone on to take their doctorates under Jacobi and Rich elot. Rosenhain had helped pioneer the theory of so-called untraelliptic func tions, which generalized the elliptic functions first introduced by Niels Hen rik Abel and Jacobi. This field of re search, soon extended by Karl Weier strass and Bernhard Riemann, would come to dominate complex analysis throughout the nineteenth century. After a brief exposure to classical analysis, Hilbert spent his second se mester in Heidelberg, where he took courses from Lazarus Fuchs on inte gral calculus and the theory of invari ants. He then returned to Konigsberg, where he stayed for the next four years until the completion of his studies. Hilbert thus had plenty of opportuni ties to learn mathematics from world class researchers. Still, it seems the first mathematician to exert a signifi cant influence on him was the multi talented Heinrich Weber, who held the chair in mathematics at Konigsberg from 1875 to 1883. Weber was born and raised in Heidelberg, where he earned his doctorate in 1863. During that pe riod, however, his teachers included the physicist Gustav Kirchhoff and the geometer Otto Hesse, former students of Neumann and Jacobi who trans planted the Konigsberg tradition to the southern German soil of Baden. Fol lowing the advice of his Heidelberg teachers, Weber spent the next three years as a postdoctoral student in Konigsberg, where he and a number of others worked under the supervision of Neumann and Richelot. Nine years later, he succeeded Richelot as profes sor of mathematics at Konigsberg, a logical enough choice given Weber's strong ties with the Konigsberg math ematical tradition. At the time Hilbert first met him, Weber had just completed a paper with Richard Dedekind titled "Theorie der algebraischen Funktionen einer Veran derlichen," which was published some what later in Grelle's Journal in 1882. This work has come to be regarded as a landmark in the history of mathe matics, as it presents all the tools nec essary to give purely arithmetic proofs
VOLUME 25, NUMBER 2, 2003
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Protocol for a talk on algebraic curves presented by Hilbert in May 1884 in Lindemann's sem inar.
of Riemann's main theorems in the the ory of algebraic functions. The late Jean Dieudonne wrote of this paper of Dedekind and Weber that its "remark able originality . . . is only barely sur passed by that of Riemann" in the en tire history of algebraic geometry. Strong praise indeed! A new insight gained from this approach-and even more apparent in the closely related work of Kronecker-was a deep, but clear analogy between algebraic num ber fields and the fields of meromor phic functions associated with Rie mann surfaces. Hilbert's later work in number theory, as well as that of his students Erich Heeke, Otto Blumen thal, and Rudolf Fueter, was strongly motivated by this analogy. A clear sign of this can be seen from one of the most important of Hilbert's 23 prob lems, the twelfth, which had to do with
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THE MATHEMATICAL INTELLIGENCER
Abelian extensions of algebraic num ber fields. Its aim was to explore the full implications of the analogy first studied by Kronecker and Hilbert's teacher, Heinrich Weber. Hilbert called this prospect "one of the deepest and most profound problems of number theory and function theory." Very few details are known about Hilbert's personal contacts with Hein rich Weber, but we do know that the young Konigsberger attended Weber's courses on number theory and on the theory of elliptic functions. He also participated in a seminar Weber di rected on invariant theory. The latter field soon became Hilbert's primary area of expertise, although his interest in algebraic number theory and alge braic functions continued. When Weber left Konigsberg in 1883, he was succeeded by Ferdinand
Lindemann, a geometer who had stud ied under Alfred Clebsch, but also with the latter's youngest disciple, Felix Klein. These names conjure up a host of new connections, personal as well as intellectual, many of which proved important for Hilbert's subsequent ca reer. Alfred Clebsch had studied in Konigsberg under Franz Neumann and Jacobi's pupil, Otto Hesse; he also be friended Neumann's son Carl, who be came a distinguished mathematical physicist, first in Ttibingen and then Leipzig. A major episode that helped solidify Konigsberg's mathematical legacy took place in 1869 when Clebsch and Carl Neumann co-founded Mathe matische Annalen, which later, under the editorship of Felix Klein and then Hilbert, came to be regarded as the pre mier mathematics journal in the world. Clebsch had a leading role in propagat ing Konigsberg's distinguished mathe matical tradition. After his sudden death at age thirty-nine, his student Lindemann prepared the publication of Clebsch's lectures on projective geom etry, a project supervised by Klein. Ferdinand Lindemann, arriving in 1883, provided a vital link between Konigsberg and Felix Klein, who was in Leipzig from 1880 to 1886 and after ward in Gottingen. Although Linde mann's influence on Hilbert was by no means deep, he did act as a catalyst for what became a surprisingly lively, though small, mathematical commu nity. Far more decisive for Hilbert's development were his contacts with two other mathematicians: Hermann Minkowski, a fellow student, and Lin demann's colleague, Adolf Hurwitz, who was only three years older than Hilbert. These three brilliant upstarts would become the stars of their gen eration in Germany. It was through Minkowski and Hurwitz that Hilbert learned how to "talk mathematics," the art of conversing about ideas and prob lems. Each of the three had his own distinctive style and personality. Unlike Hilbert, whose mathematical gifts took long to ripen, Minkowski was a child prodigy who completed the cur riculum at the Altstatdisches Gymna sium in Konigsberg by the age of fif teen. Two years later he submitted a paper to the Academie des Sciences in
Paris in response to a prize announce ment for the problem of expressing a positive integer as the sum of five squares. Although it was written in German, contrary to the stipulations of the Academy, Minkowski's paper was awarded the Grand Prix des Sciences Mathematiques, creating a minor sen sation. A year later, Minkowski began his studies at the Albertina alongside Hilbert. His principal teachers were Heinrich Weber and the physicist Waldemar Voigt. After five semesters, however, he left Konigsberg to study in Berlin, where he attended the lectures of such luminaries as Weierstrass, Kummer, Helmholtz, Kirchhoff, and above all Leopold Kronecker. He then returned to Konigsberg, passed his doctoral exams, and submitted a dissertation on the theory of quadratic forms; it was published soon thereafter in volume 7 of Acta Mathematica. Minkowski was a sharp-witted, brash young man, and Hilbert clearly enjoyed his sarcastic sense of humor. Hilbert's relationship with Adolf Hur witz, who came to Konigsberg in the spring of 1884 to assume a newly cre ated Extraordinariat, was far more formal, befitting that of professor and student, despite their closeness in age. Nevertheless, it was the quiet Hurwitz who exerted the most decisive influ ence on Hilbert's mathematical sensi bilities, and in particular his longing for universal breadth of knowledge. Adolf Hurwitz grew up in Hildesheim, where his father was a merchant. Like Minkowski, he came from a Jewish family, and, like him, too, his talent for mathematics was apparent from an early age. His Gymnasium teacher was Hermann Schubert, creator of the Schubert calculus (the topic of Hilbert's 1 5th problem), and one of the leading experts in the difficult field of enu merative geometry (see [Kleiman 1976]). Hurwitz absorbed his teacher's work like a sponge, and by the time he was only seventeen they had published a paper together in the Gottinger Nachrichten that dealt with Chasles's theory of characteristics. It was Schu bert who advised Hurwitz to study un der Klein, who was then teaching at the Technische Hochschule in Munich. Hurwitz's family had strong reser-
H ilbert in a contemplative mood.
vations about the wisdom of pursuing an academic career, which was hard enough for a family with limited means but particularly difficult for Jews, as many Germany universities were re luctant to appoint Jews to their facul ties. Hurwitz also had health problems, and this worried his parents as well. Felix Klein, already a powerful figure in German mathematics by 1880, did his best to persuade the family to let his star pupil go on. He wrote Hurwitz's father: "Above all I want to stress that among the totality of young people with whom I have up until now worked there was not one who in specifically mathematical talent could measure up to your son. From now on your son will enjoy a brilliant scientific career, which is all the more certain because his gifts are combined with endearing personality traits" [Rowe 1986, p. 432]. Hurwitz was, in fact, a charming, mod est, and very warm individual. He was also a talented pianist, who loved to gather with friends and family for mu sic-making. Years later in Zurich, Al bert Einstein was a regular participant in such festivities and became a close friend of the Hurwitz clan. Hurwitz went on to become Klein's star pupil, but he also studied in Berlin before habilitating in Gottingen. Dur ing the 1860s and 70s, Berlin held far more attraction for aspiring young mathematicians than any other univer sity in Germany. Led by Ernst Eduard
Kummer, Karl Weierstrass, and Leopold Kronecker, Berlin also virtually mo nopolized university appointments in Prussia. In Hurwitz's case, this experi ence afforded him the opportunity to broaden his knowledge in complex analysis. He had already mastered the theory of Riemann surfaces, a staple of Klein's school, so he now immersed himself in the competing approach, based on the analytic extension of power series representations, that had been developed by Weierstrass and his students. Still, even more interesting for Hurwitz than Weierstrass's lectures
Felix Klein as portrayed in a painting by Max Liebermann in 1 9 1 2.
VOLUME 25, NUMBER 2, 2003
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were those of Kronecker. As he related in a letter to his friend Luigi Bianchi, he came to Berlin mainly to attend Kro necker's lectures on number theory (Hurwitz to Bianchi, 20 March 1882, in [Bianchi 1959], p. 80). Minkowski had also studied under Kronecker during the early 1880s and, like Hurwitz, he too came away deeply impressed by the elderly master's achievements. At the same time, both Minkowski and Hurwitz found Kronecker's personality much less appealing than his mathe matics. Thus, Hurwitz in his letter to Bianchi, called him "that great, but very vain mathematician," and Minkowski's letters to Hilbert from the late 1880s and early 1890s are strewn with simi lar remarks (see [Minkowski 1973, pp. 64, 80, 107]). How soon Hilbert came to share this opinion is difficult to say. When he arrived in Konigsberg, Hur witz was at the height of his powers, and he opened up whole new mathe matical vistas to Hilbert, who looked up to him with admiration mixed with a tinge of envy. He later said about him that "Minkowski and I were totally overwhelmed by his knowledge and we never thought we would ever come that far" [Blumenthal 1935, p. 390]. Hurwitz's appointment had been engi neered by Ferdinand Lindemann, Hilbert's future Doktorvater. Linde mann had been called to Konigsberg shortly after winning international ac claim in 1882 for his proof that 7T is a transcendental number, thereby fmally establishing the impossibility of squar ing the circle. This odd tale has some interesting ties with Hilbert's life story as well. The young Hilbert was rather fond of Lindemann, but he also clearly appreciated how his teacher's success in solving this famous problem had ef fectively launched his career. In fact, the truly difficult work had already been accomplished by Hermite some ten years earlier in proving the tran scendence of e. Hilbert would later show, in 1892, that the transcendence of 7T follows very easily once Hermite's fundamental methods are called into play. Still, many claims to fame in the history of mathematics have been more a matter of luck than genius, and in this sense Lindemann's case was in
48
THE MATHEMATICAL INTELUGENCER
Adolf Hurwitz (1 859-1919).
no way exceptional. His subsequent career was marked by a series of pur ported solutions of Fermat's famous last conjecture. Both Hurwitz and Minkowski expressed something close to shock at the elementary blunders Lindemann committed along the way; Hilbert, on the other hand, seems to have done his best to avert his eyes from these embarrassments. Lindemann's memoirs contain many anecdotes about mathematical life in Konigsberg during the ten years he taught there. Lindemann had been re luctant to leave his professorship in the comfortable environs of Freiburg im Breisgau, but he overcame these doubts after a colleague told him that he simply could not tum down the chair that had once been held by Ja cobi. Konigsberg's isolated location certainly contributed to the univer sity's stagnation during the 1 860s and 70s. But judging from Lindemann's rec ollections, it would seem that many professors felt inclined to follow in the footsteps of their predecessors, with the effect of stifling new innovative ideas. When Lindemann came to Konigsberg in 1883 he found the facil ities woefully inadequate. The mathe matics seminar library, which con sisted of little more than the twenty or so volumes of the Mathematische An nalen that had appeared since 1869,
was housed in the university's deten tion quarters (the Karzer), along with some of Konigsberg's rowdier stu dents. After several years of futile negotiations, Lindemann managed to obtain a separate room for the mathe matics seminar, although the mathe maticians were forced to share this with a professor of medicine who used it to store pharmaceutical prepara tions. Only with Hilbert's appointment as Lindemann's successor did the mathematics seminar and library re ceive its own quarters, taking over a room on the top floor of the university building that had formerly been occu pied by one Secretary Lorkowsky. In 1883 Konigsberg had no special lecture halls for mathematics. Jacobi and his successors had delivered their lectures at a podium next to which stood two blackboards on wooden easels of the same type used in the schools of that day. Lindemann put forth a motion that the mathematics lectures be held in a special room in which a large slate blackboard was to be installed. But the faculty voted this proposal down, pointing out that math ematics had been taught successfully in Konigsberg for some fifty years with out any such modem devices. The fac ulty also pointed out to Lindemann that classrooms could not be reserved for
Hermann Minkowski (1 864-1 909).
particular subjects as they were allo cated according to the seniority of the professors who requested them. Soon after this, he went directly to the uni versity's Kurator, and not long after ward a modern blackboard was in stalled in one of the classrooms. Lindemann recalled how impressed he was by most of the older students who had studied under Heinrich We ber. He even admitted that some of them had already surpassed him in their knowledge of mathematics. Both of Lindemann's teachers, Clebsch and Klein, had excelled in attracting small groups of dedicated disciples, and Lin demann did his best to follow suit. Once a week, he held a colloquium in his home, which he organized together with Hurwitz. The two most active stu dents at these meetings were Hilbert and Emil Wiechert, who later became an outstanding geophysicist and a colleague of Hilbert's in Gottingen. Minkowski also took part in these col loquia. After the meetings, which were sometimes attended by old-timers like Rosenhain, the group would gather in a restaurant for dinner, talking mathe matics the whole time. On other occasions, Hilbert would join his cohorts on a mathematical walk along the Paradeplatz before the university building. One day he ran into Lindemann there and proceeded to tell him about some results he had ob tained on continued fractions. Hilbert wondered if this work might prove ac ceptable for his dissertation. Immedi ately thereafter, Lindemann went home only to discover that Hilbert's main findings had already been estab lished by Jacobi. The result of this en counter was that Lindemann suggested another dissertation topic to him, one concerned with the invariant-theoretic properties of spherical harmonics. He knew that Hilbert had studied invariant theory in a seminar with Weber, but he probably did not anticipate that his pupil would strike out in a direction different from the methods he himself knew best. Lindemann soon came to realize that the complicated symbolic calculus of Siegfried Aronhold and Al fred Clebsch had little appeal for Hilbert, just as it was quite foreign to
Heinrich Weber. Lindemann recalled how he later met Weber, and when he told him about Hilbert's dissertation re search the latter responded in surprise: he would have saved such a beautiful theme for himself rather than assign ing it to a student ([Lindemann, 1971, p. 91]). As it turned out, the topic pointed Hilbert in a direction that would dominate his interests for the next eight years, during which time he made his name as one of the world's authorities on invariant theory. Entering Klein's Circle
After taking his doctorate in 1885, Hilbert spent one last summer in Konigsberg rounding out his educa tion. Like many other aspiring mathe maticians in Germany, he took and passed the Staatsexamen, which en sured that he could apply for a teach ing position at a Gymnasium in the event that his plans to pursue a uni versity career failed to fructify. During this final semester, he attended Linde mann's geometry course, which dealt with Plucker's line geometry and Lie's sphere geometry, but he also followed Hurwitz's lectures on modular func tions. Then he departed for Leipzig, where he was welcomed by Hurwitz's former mentor, Felix Klein. Hilbert spent the winter semester of 1885-86 working under Klein, who led an active group of young researchers. One of these, Eduard Study, shared Hilbert's interests in algebraic geometry and in variant theory. Study, however, fa vored the symbolic methods of Aron hold and Clebsch, whereas Hilbert was experimenting with other formalisms. Whether for this reason or simply be cause he found Study a bit overbearing in his opinions about mathematics, a rather tense relationship developed be tween them. For his part, Study may well have sensed that Klein was more than a little interested in this new comer passing through from Konigs berg. His own future dealings with Klein were often unpleasant, and he was not alone in feeling that Klein treated him unfairly. Recognizing Hilbert's talent, Klein quickly took him under his wing, con fiding to him the latest events of inter-
est in Leipzig. First among these was Klein's own imminent departure for Gottingen and his efforts to appoint So phus Lie as his successor. After con siderable squabbling, Klein was able to inform Hilbert at year's end that Lie had received an official offer from the Saxon Ministry and had accepted it. This unofficial news was conveyed by Hilbert to Hurwitz in a letter that con tinued: "New Year's Eve I was invited to Prof. Klein's and found myself in very small but exclusive company con sisting of Prof. Klein, his wife, the Prague Privatdozent Dr. Georg Pick, and me. We drank a New Year's punch with great pleasure and chattered about all conceivable and inconceiv able things, amusing ourselves won derfully. Prof. Klein tried hard to con vince me to spend the next semester studying in Paris. He described Paris as a beehive of activity, especially among the young mathematicians, and thought that in view of this a period of study there would be most stimulating and profitable for me" (Hilbert to Hurwitz, 2 January 1886, Mathematiker-Archiv, Niedersachsische Staats- und Univer sitatsbibliothek Gottingen). Klein even joked that he should try to befriend Henri Poincare and join him in drinking a toast to "mathematical brotherhood." For Felix Klein, memories remained fresh of the Parisian mathematical scene he and Lie had encountered in the spring of 1870. Even fresher, to be sure, was the painful experience of watching Poincare sail by him during the early 1880s when both were working inten sively on the theory of automorphic functions (see [Gray 1984]). But if Klein had special reasons for counseling Hilbert to visit Paris, such advice would hardly have been rare. Indeed, Linde mann had also studied in Paris during the mid-1870s, and this had given him the opportunity to learn directly from the source about Charles Hermite's proof of the transcendence of e, a result the French mathematician regarded as among his most important [Lindemann, 1971, p. 70]. Lindemann clearly thought so, too, since it led him to pursue the closely connected problem of proving the transcendence of 7T, the solution to which made him famous.
VOLUME 25, NUMBER 2 , 2003
49
Hilbert's journey to Paris turned
son why others should not live by the schedule.
they spanned practically every area of
out to be considerably less eventful
same insane
Even before
higher mathematics of the day: from in
than those by Klein and Lindemann
Hilbert had departed from Paris, Klein
variant theory, number theory, and an alytic, projective, algebraic, and differ
had been. Joining his rival invariant
was scolding him for not taking full ad
theorist, Study, whom Klein had also
vantage of what the city had to offer:
ential
persuaded to make the trip, Hilbert got
"hold before your eyes, " he wrote his
potential theory, differential equations,
geometry,
to
Galois
theory,
to meet most of the mathematicians he
young protege, "that the opportunity
function theory, and even hydrody
had hoped to see, including Poincare,
you have now will never come again. "
namics. During his entire nine years on
Emile Picard, and Paul Appell. Still,
That proved to b e true only for the time
the Konigsberg faculty he never lec
difficulties
being.
second
tured on any subject more than once,
vented Hilbert from breaking the ice
chance in 1900, but in the meantime a
with the exception of a one-hour course
with many, including Poincare. Politi
subdued rivalry developed between
on determinants. Surely Hilbert read a
cal tensions between France and Ger
him and Poincare, the era's leading
great deal, but more important to him
many
mathematician. Hilbert returned to the
still were the opportunities to talk
had
with
not
the
language
lessened
pre
noticeably
during these years, but Hermite, the leader of a burgeoning school in analy sis, took a strong internationalist stance
He
would
have
his
H ilbert 's courses were often
in an effort to cultivate better relations among mathematicians. Hermite had
ination played a central role in his cess in Gottingen simply would not have been possible.
only two or
centenary of the birth of Gauss, and he never tired of extolling the achieve
the start, the power of the spoken word that stimulates the mathematical imag work Without it, his phenomenal suc
attended by
attended the ceremonies in 1877 that were held in Gottingen honoring the
about mathematics. Thus, right from
REFERENCES
three auditors.
[Bianchi 1 959] Luigi Bianchi, Opere, vol. XI,
warmly urged him to pledge himself to
quiet environs of Konigsberg, submit
[Blumenthal 1 935] Otto Blumenthal, "Lebens
the noble cause of promoting the in
ted
the
geschichte," in [Hilbert 1 935, pp. 388-429] .
ternational
summer of 1886, and began his six-year
[Browder 1 976], Felix Browder, ed. , Mathe
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of
German
mathematicians.
Lindemann recalled how Hermite had
brotherhood
of
mathe
maticians, and Klein may well have
his
period as
Habilitationsschrift
in
Rome, Edizioni Cremonese, 1 959.
Privatdozent.
matical Developments Arising from Hilbert's
had this incident in mind when he
The following spring, Minkowski de
urged Hilbert to do the same when he
parted for Bonn where he, too, began teaching as a Privatdozent. Thereafter,
met Poincare. within
France
for
Symposia in Pure Mathematics,
vol. 28. Providence.
[Frei 1 985] Gunther Frei, Der Briefwechsel
Hurwitz was Hilbert's primary source
David Hilbert- Felix Klein (1 886-1 9 1 8) ,
German
of intellectual stimulation. Probably the
beiten aus der N iedersachsischen Staats
Hermite had long been the primary conduit
Problems ,
Ar
work in pure mathematics, and despite
most memorable moments for Hilbert,
und Universitatsbibliothek Gottingen, Bd. 1 9.
his advanced age-he was then 64-it
when he thought back on the years
Gottingen: Vandenhoeck & Ruprecht, 1 985.
was he who took the strongest interest
from 1886 to 1892, were the almost
[Gray 1 984] Jeremy Gray, Linear Differential
in the two young visitors. Hilbert found
daily walks with Hurwitz. He recalled
Equations and Group Theory from Riemann
him absolutely fascinating and delight
how together they wandered through
ful. He reported to Klein how Hermite
nearly every comer of mathematics,
to Poincare.
Basel: Birkauser, 1 984.
[Hilbert 1 935] David Hilbert, Gesammelte Ab hand/ungen, vol. 3.
had told both him and Study about his
with his friend and former teacher act
reciprocity theorem for binary invari
ing as guide. No doubt they often dis
[Kleiman 1 976] Steven Kleiman, "Rigorous Foundation of Schubert's Enumerative Cal
ants, encouraging them to try to extend
cussed ideas that Hilbert was working
it to ternary forms. He also told them
on in the context of his lecture courses.
about
correspondence
Konigsberg was just about the last
with Sylvester, who was struggling to
place one could find mathematics stu
his
ongoing
culus," in [Browder 1 976], pp. 445-482. [Lindemann
1 97 1 ]
Ferdinand
Lebenserinnerungen.
find an independent proof of Gordan's
dents during the late 1880s and early
theorem (David Hilbert to Felix Klein,
1890s, and Hilbert's courses were often
an David Hilbert,
2 1 April 1886 [Frei 1985, p. 9]). Klein
attended by no more than two or three
Zassenhaus.
kept urging Hilbert to seek out as many
auditors, sometimes even fewer! He
contacts as possible,
complained about these circumstances
but time was
Berlin: Springer, 1 935.
Lindemann,
Munchen, 1 97 1 .
[Minkowski 1 973] Hermann Minkowski, Briefe Hg. L. ROdenberg und H .
New York: Springer-Verlag,
1 973.
[Reid 1 970] Constance Reid, Hilbert. New York:
short and he had to worry about his
occasionally, but never seemed to be
first priority, namely completing his
really bothered. His goal was to become
Habilitationsschrijt
a truly universal mathematician. In
matics' at Gottingen in the Era of Felix Klein. "
deed,
isis 77:422-449.
so that he could
qualify for a position as a Privatdozent
his
lecture
courses,
supple
in Konigsberg. Klein, who could roll
mented by nearly daily
more into one day than most people
with Hurwitz, served as his primary ve
his Mathematical Work," Bulletin of the Amer
would attempt in a week, saw no rea-
hicle for attaining that purpose, and
ican Mathematical Society 50:61 2-654.
50
THE MATHEMATICAL INTELLIGENCER
discussions
Springer-Verlag, 1 970. [Rowe 1 986] David E. Rowe, " 'Jewish Mathe
[Weyl 1 944] Hermann Weyl, "David Hilbert and
JEAN-MARC LEVY-LEBLOND
Co u m e I a' s Fo rm u a This paper is dedicated to the memory of Pierre Souffrin, who initiated me to Columella's formula and should have co-authored this paper.
ythagoras, Thales, Eudoxus, Euclid, Archimedes, Hero-these names, still asso ciated with deep geometrical or arithmetical results, may lead us to believe that the mathematical science of Antiquity was but a forerunner of our own. It may be sobering as well as entertaining to dig up some strange statements, hardly remembered today, which impel us to recognize some ele ments of strangeness in ancient knowledge. Let me show you a minor but amusing geometrical result of this sort. Columella was a Roman author of the 1st century, con sidered the founder of agronomy. In his magnum opus De re rustica, he describes in Chapter 2 of Book V the ways to cal culate the areas of fields with various geometrical patterns. He considers in particular the rather odd case of a field with the shape of a segment of a circle. I will not dwell here on the empirical import of this problem; it requires some stretch of the imagination to believe that such fields were common and that the computation of their area was a question of practical interest. Disregarding agronomy and economics, let us concentrate on geometry. Here is Columella's recipe: "If [the field] is less than a circle, it is measured as fol lows. Consider an arc the base of which is 16 feet, and the height 4. The base and the height together make 20 feet. Multiplied by 4 [the height] , we obtain 80, the half of which is 40. The base being 16 feet, its half is 8. Mul tiplied by itself, it gives 64, the fourteenth part of which
is a little over 4. Added to 40, this gives 44. I say that this is how many square feet there are in the arc [the segment)."
As usual then, and for many centuries yet to come, the for mula was not expressed in general terms, but given through a specific example. At about the same period, this algo rithm was reported by the great mathematician Hero of Alexandria in his Geometria as having been known for quite some time. Although little quoted in medi1Eval times, it reappeared (with the same numerical values as with Col umella) in the Ludi matematici ( 1450 circa) of the Re naissance polymath Leon Battista Alberti, poet, artist, ar 1 chitect, and engineer. Columella's formula is better discussed once transcribed in our modem literal formalism. Calling C the chord of the arc (the "base" of the segment, in Columella's terms), and H its height (Fig. 1 ), the area of the segment takes the value: (1)
1 For a study of the geometrical work of Alberti, see Pierre Souffrin, "La Geometria practica dans les Ludi rerum matematicarum", Albertiana 1 (1 998), 87-1 04. This pa per contains original references and erudite comments on Columella's formula (pp. 1 01 - 1 03).
© 2003 SPRINGER-VERLAG NEW YORK, VOLUME 25, NUMBER 2, 2003
51
•
'
'
'
'
'
The role of the last term in (1), with the strange factor 1114, is by no means negligible, as shown on Figure 2 and espe cially Figure 3 by the curves labeled "quasi-Columella," showing the result obtained from the formula (2). Besides the "1114 puzzle," the weirdness of Columella's formula (1) fully appears when it is expressed by characterizing the seg ment of the circle, according to present-day uses, by a more natural (to us!) parameter, to wit, the angle e correspond ing to the arc. Taking the radius R of the circle as unity for simplicity, we obtain the formula
'
R. � l ,
Ac
1 0
=
(
_!_ ! 1 - cos 2 2
)(
)
_!_ ! ! ! + 1 - cos + 2 sin sin2 2 2 14 2'
(3)
which looks awkward enough when compared to the ex act expression A =
ljlijil;iiM The first term on the r.h.s. may readily be understood as the area of a triangle with height H and base H + C (the thin triangle in Figure 1), corresponding to the truncated Columella's formula AquasiCot =
-
n
--
1 2H(H + C).
(2)
t
ee - sine).
(4)
Yet the graphical comparison (Fig. 2) reveals that Col umella's formula indeed is quite good, at least for segments with angles up to e = 57T/4. The fact is all the more sur prising since the power series development in e of Col umella's formula (3) does not coincide at low orders with the exact formula (4); the latter starts at third order, while the former has a non-vanishing second-order term. More generally, the function (4) is odd, which is not the case for (3). This means that Columella's formula (like the other ones to be discussed below) gives a rather bad result for low values of the angle e, that is, for small segments of cir cles, as Figure 3 clearly shows; but in these circumstances,
Area A
0,7Sn
O,Sn
0,2Sn
0,2Sn
+§lriil;i#W 52
THE MATHEMATICAL INTELLIGENCER
o,sn
0.7Sn
n
1 ,25n
l ,Sn
1 , 7Sn
2n
i
\
Area approxJ Area exact 1,1
Columella
Angle 8 1 ,S n
1 ,7Sn
2n
0,9
l@'ijii;i¥+ if one believes in the practical value of these formulas for measuring field areas, they would not be of much interest anyway. It is possible to give an interpretation of Columella's for mula, and perhaps to form an idea of how it was arrived at. Of course, the following does not purport to be a gen uine historical reconstruction, if only because it is couched in modem algebraic terms; still, these comments may help elucidate the reason for the precise form of formula (1) and its efficiency. Suppose then that we want to evaluate the area of a seg ment of a circle in the spirit of Antiquity. In keeping with the supposedly concrete nature of the problem as one of field surveying, it is natural to define the dimensions of the circle segment by the lengths C and H of its chord and height, readily accessible to measurement, unlike the radius and angle (think, for instance, of the difficulty in locating on the ground the center of the circle). Now, back in those times of antique geometry, an area necessarily had to be computed by multiplying two lengths. Dimensionless func tions of numbers, such as the angle defining an arc, or, more generally, arbitrary functions of the ratio H!C, cannot enter the calculation. With the two available lengths C and H, we may form three areas only, namely, C2, CH, and H2. The ob vious method, then, is to look for the area of the segment as some linear combination of these three quantities: A
=
aH2 + {3HC + yC2.
(5)
Now, suppose we know the value (exact or approximate) of the area of a full circle, that is, 7T (where for simplicity we take the radius of the circle to be unity). We try to find the
a,
unknown coefficients {3, y by fitting the formula (5) to the simple cases of the half-circle and quarter-circle. For the half-circle, with C 2 and H 1, we obtain
27T = a
1
=
=
+ 2{3 + 4y.
=
And for the quarter-circle, with C we get
..!_7T - ..!_ 4
2
=
(l- Y2)a 2
V2 and H
(6) =
+ cV2 - 1){3 + 2y
= %a - f3
+
2y + V2({3
-
1
-
a).
v;,
(7)
Now, there appears in (7) that monster of irrationality, V2, a scandal which we, as good Ancients, definitely want to avoid (note that we have not yet any reason, at that time and for a long time after-to think of 7T as irrational). Noth ing is easier: it suffices to choose
{
a = f3
(8)
to get rid of this radical horror. Rewriting (6) and (7) in the light of (8), we are led to the equations
= 3a t7T � = a 1
-7r
+
1
with the immediate solution
a = f3 =
1 2' -
+ 4y
'V I
4y (9)
7T - 3 = --. 8
VOLUME 25. NUMBER 2 . 2003
53
It only remains to choose a numerical value for 'TT. The simplest (biblical) choice 7T = 3 leads to the quasi-Col umella formula obtained by deleting the last term in (1). The better (Egyptian) choice 7T = 22/7 leads to the full Col umella formula (1). It is now plain where the "strange fac tor" 1/14 comes from. The next best rational approxima tion to 'TT, that is 355/1 13, would allow for an improved value 8/1 13 (to be compared with 1/14 = 8/1 12); but the differ ence in the result is hardly worth noticing. Of course, if the above "derivation" explains the value of the coefficients in ( 1), it still is somewhat of a mystery why a (rational) fit at the two points e = 7T/2, 7T suffices to obtain such a good agreement. It turns out that the de rivatives of the functions (3) and (4), which obviously both vanish at e = 0, remain quite close up to e = 'TT, which gives the inflexion point for the graph of the ex act function and is very near that of the Columella ap proximation. Let us note again that Columella's formula works well only for circle segments not much greater than a half-cir cle. For larger segments, an approximate value may be obtained by subtracting the area of the (smaller) com plementary segment calculated via Columella's formula from the area of the full circle. However, to remain true to the spirit of the formula, this calculation requires the evaluation of the height H' of the complementary seg ment in terms of the given height H of the large segment (the value C of the chord being common to both seg ments). Although simple, the relation to be used, that is, HH' = (C/2)2, spoils the simplicity of the initial formula, in that the area to be subtracted, when expressed in terms of H and C, is no longer an elementary quadratic function. Other related half-empirical formulas for the area of a segment of circle were known and used in the Middle Ages. A practical treatise of the XVth century, quoted by Souf frin,2 proposes the following formula:
A U T H O R
J EAN - MARC livv- LEBLOND c �
France e-
: �.fr
Jeanc L.Avy·Leblond was known as a ma hemal physicist He IS now a leas as wei known for hiS wn connect.ng mathema cs. soence. era ure and the s. He founder and edl or of he arty Ali8ge (QJture Science Technique). which h ex1raord'naty ambl oo exlends • s CXN· erage to
of the above.
11 3 Axv = - H[C - -(C - 2H)] 14 14 6 X 11 11 X 11 fi2 HC. = + 14 X 14 14 X 14
With only two terms, it gives a very good result indeed (Fig. 3), much better than the truncated Columella's formula, be tween e = 7T/2 and e = 'TT, for which last value it is "exact" with 7T = 22/7, so that the reason for the " 14" coefficient clearly is the same as above. I must admit that I did not find a plausible derivation or explanation for (1 1).
2Pierre Souffrin, ibid., p. 1 03
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THE MATHEMATICAL INTELLIGENCER
(1 1)
li·iM•effli·i§rr€1hfiii.Jj!Q?il
Around the Graves of Petrovskii and Pontryagin Pieter Maritz
Does your hometown have any mathematical tourist attractions such as statues, plaques, graves, the caje where thefamous conjecture was made, the desk where the famous initials are scratched, birthplaces, houses, or memorials? Have you encountered a mathematical sight on your travels? If so, we invite you to submit to this column a picture, a description of its mathematical significance, and either a map or directions so that others may follow in your tracks.
Please send all submissions to Mathematical Tourist Editor, Dirk Huylebrouck, Aartshertogstraat 42, 8400
Oostende, Belgium
e-mail: dirk.huylebrouck@ping. be
G
D i rk H uy l e b ro u c k ,
Editor
rand Prince Vasili'i III of Moscow ruled from 1505 to 1533. He fought Lithuania, staging three campaigns aimed at Smolensk before that town was finally captured in 1514; the treaty of 1522 confirmed Russian gains. The Muscovite ruler founded the Convent of the Smolensk Icon of Our La dy in 1524 to commemorate the return of Smolensk to Russia. This convent, nowadays the No vodevichii Convent, was built as a fortress in a bend of the Moscow River, and became an important component of the capital's southern defense belt which consisted of several important monasteries. The fortress-convent played a part of some consequence in the struggle against the Tartar and Pol ish invaders. In 1591 the Crimean Tar tars, on one of their annual raids into the heart of Muscovy, tried to cross the Moscow River and capture Moscow, but the convent's cannon compelled the enemy to retreat. In early Septem ber 1612, during the "Time of Trou bles," the Second Volunteer Corps led by Prince Dmitril Pozharski'i reached Moscow and besieged the Poles. In early November 1612 the Russians stormed Moscow and Pozharskil 's troops de feated the Polish detachments at the convent and also captured Polish posi tions in the heart of the city, in partic ular in the Kremlin. In the 16th and 17th centuries the convent enjoyed a privileged position because it was here that women from the royal family and top-ranking boyar families took the veil. With the death of Tsar Feodor I (1584-1598) the dy nasty of the ruling branch of Rurik was extinct. Feodor's wife, Irina (the sister of Boris Godunov), refused the crown and entered the Novodevichi'i Convent. Patriarch Job summoned a national as sembly to elect a tsar. On the nomina tion of the patriarch, Boris Godunov was elected. Initially, he was reluctant to accept, but yielded to the mass of people who thronged his cell in the Novodevichii Convent. Boris ruled
I
from 1598 to 1605. Princess Sophia Alexeevna ( 1 657-1704), half-sister of Peter the Great, was confined in the Novodevichii Convent for 15 years (from 1689 to 1704) following an abortive attempt to deprive Peter of the throne. Peter the Great (1682-1725), who was not particularly fond of con vents, tried to use the Novodevichi'i Convent for practical purposes. On his orders, an orphanage for foundlings was set up at the convent, and later on distinguished veteran soldiers were given food and shelter there. In 1812, shortly before the retreat of the French from Moscow, Napoleon ordered the convent to be blown up. Trenches were dug and filled with powder kegs. The convent was saved by the valiant Sis ter Sara, the convent's treasurer, who managed to extinguish the fuses con nected to the kegs. In more recent years, a Theological Institute for the training of Church dignitaries and a Theological Course as a seminary for priests were established on June 14, 1944, at the Novodevichi'i Convent [ 1 , p . 686]. The convent is a unique 16th-17th century architectural ensemble. It is dominated by the huge five-domed Cathedral of the Smolensk Icon of Our Lady (1524-1525), which was modeled on the Cathedral of the Assumption in the Kremlin. However, the proportions are slightly different: the full-blown cupolas are closer together and their general appearance is more slender than those of the Cathedral of the Assump tion. Beginning in 1474, Ivan III (the Great) sent an agent to Venice and re peatedly invited Italian architects and other masters to come to work for him in Moscow. The volunteers included the famous Bolognese architect and engi neer Alberto Fioravanti, together with such prominent builders as Marco Ruffo, Pietro Antonio Solaria, and An tonio Gilardi. Fioravanti, who lived in Russia from 1475 to 1479, erected the Cathedral of the Assumption in the Kremlin on the Vladimir model. The bell
© 2003 SPRINGER-VERLAG NEW YORK , VOLUME 25, NUMBER 2 , 2003
55
Figure 1 . Bell tower and cathedral of the Novodevichii Convent.
tower of the Novodevichil Convent (erected in 1689-1690) rivals the famous Bell Tower of Ivan the Great (erected in 1 505--1508) in the Kremlin (Fig. 1). As soon as the convent was founded, a cemetery was opened on its grounds, which subsequently became a tradi tional burial place for the church dig nitaries and feudal lords of Moscow and, in the 19th century, of the intelli gentsia and merchants. The graves of quite a number of public and cultural figures famous in Russian history have been preserved in the burial grounds. In 1898, the so-called New Cemetery was established behind the south wall of the convent. Surrounded by a wall (built in 1898--1904), it became the most venerated cemetery in Moscow. Here lie the bodies of outstanding writ ers and poets such as N. Gogol and A. Cheknov, the artists V. Serov and I. Levitan, politicians such as V. M. Molo tov, N. S. Khrushchev, A. Gromyko, as well as famous scientists, such as the mathematicians I. G. Petrovski1 and L. S. Pontryagin, among others.
56
THE MATHEMATICAL INTELLIGENCER
One of Peter the Great's most am bitious cultural undertakings was the founding of the Imperial Academy of Sciences in St. Petersburg in 1724. The official opening took place some months after his death in 1 725. The Academy initially had three depart ments, namely, Mathematics, Physics, and History, as well as a section for the Arts. Although the Academy operated at first on a small scale and consisted of only seventeen specialists, all of them foreigners and most of them Ger man-speaking, it became before long, as intended, the main directing center of science and scholarship in the Russ ian Empire [2, p. 319] . The first Russ ian academician was not elected until 20 years after the founding of the Acad emy, and a century and a half passed before the ethnic Russians won control of the Academy [3, p. 15]. Egorovshchina
N. V. Bugayev, a member of the De partment of Mathematics of the Fac ulty of Physics and Mathematics of the
Moscow State University (MSU) and one of the founder members of the Moscow Mathematical Society (MMS) in 1864 [4], was an exponent of the philosophical idealism that Marxism condemned. He had a religious per spective and opposed materialism. Bugayev and his colleagues V. Ya. Tsinger and P. A. Nekrasov champi oned the development of a more com prehensive theory of functions that could include discontinuous as well as continuous functions. D. F. Egorov, born on December 10, 1869 (pre-revo lutionary dates are given in the Old Style), in Moscow, enrolled as an un dergraduate in this Department in the fall of 1887, graduating in 1 89 1 . He re ceived the degree of Doctor of Pure Mathematics in 190 1. He spent the aca demic year 1902-1903 in Berlin, Paris, and Gottingen, and was appointed on March 14, 1904, as Ordinary Professor in the Department of Mathematics, MSU. Egorov wrote papers on number theory, partial differential equations, differential geometry, calculus of vari-
ations, theory of functions of a real variable, and integral equations. An in novation in teaching at the beginning of the twentieth century was the cre ation by B. K. Mlodzeevskii and his younger colleague Egorov of scientific seminars at the MSU. Egorov's semi nars were divided into groups accord ing to subjects, and the result of the joint work of each group on its subject was read as a paper by one of the mem bers of the group to a general meeting of the seminar. The Moscow school of the theory of functions of a real vari able was expanding under the influ ence of Egorov and N. N. Luzin (18831950, one of Egorov's most important students), and originated from Egorov's seminar on mathematical analysis. It is from Bugayev that they inherited an interest in the theory of functions. Among Egorov's other students were V. V. Golubev, I. G. Petrovskil, I. I. Priv alov, and V. V. Stepanov [5]. Egorov was a leading figure in the MMS. He was elected Secretary in 1917, Vice-President in 192 1 , and Pres ident in 1923. In 192 1 , under the influ ence of the Society, the Institute for Mechanics and Mathematics was cre ated at MSU to promote research, and in 1923 Egorov became its Director. He had been elected a Corresponding Member of the Russian Academy of Sciences in 1924, and an Honorary Member of the Academy of Sciences of the USSR in 1929. (In 1917 the Imper ial Academy of Sciences was renamed and became the Russian Academy of Sciences; in 1925 the name changed to the Academy of Sciences of the USSR, sometimes referred to as the Soviet Academy of Sciences; in 199 1 , after the disintegration of the Soviet Union, the Academy was transformed into the Russian Academy of Sciences [6] .) The Church experienced a wave of violent repression after the Revolution that culminated in a mass execution of clergy in 1922-1923. The attack was re newed by Stalin in 1928. As Marxists, the Soviet leaders were of the opinion that the advance of science would nec essarily cause the retreat of religion. The founders of the Moscow School of Mathematics, whose members shared a religious outlook, were a targeted group. During the next ten years,
nearly all religious communities were swept away. Egorov publicly defended his leadership in the Russian Orthodox Church. In the mid-1920s a "war" was declared on Egorov in his capacity as Head of both the Society and the Insti tute. In 1929, he was dismissed as Director of the Institute, which was re organized, and "a sharp proletarianisa tion of personnel was conducted" [4, p. 29] . In [7], S. S. Demidov writes that right after the 1917 Revolution, one of the principal goals of the Soviet power became breaking the old university system and establishing a new one that would train scientific personnel de voted to a completely different ideol ogy. Although the structure of students could be changed quite rapidly, the au thorities had to rely on those few pro fessors who had actively supported establishing the new power: 0. Yu. Schmidt, algebra, appointed in 1926, and a member of the Communist Party, took over from Egorov as Director of the Institute; S. A. Yanovskaya, a lead ing Communist Party ideologist among Moscow mathematicians at MSU, ran a seminar on Methodology of Mathemat ics and Natural Sciences; M. Y. Vygod skii, an old-time communist, worked at the Institute of Red Professors. Unlike these three professors, the militant Czech Marxist Ernest (AmoS!:) Kol'man exercised ideological influence from the outside, participating now and then in meetings and publishing articles. Kol' man was born in Prague in a mid dle-class German Jewish family [8] . He received his doctorate at the Karlov University in Prague, was mobilized into the Austro-Hungarian army and sent to the Russian front, where he was captured in 1916. He supported the Bolshevik revolution, joined the Com munist Party, and stayed in the Soviet Union. He was connected for a time with the Institute of Red Professors, and with the Marx-Engels-Lenin Insti tute. He held important posts in the Moscow Committee and the Central Committee of the Communist Party during the 1920s and 1930s, and even headed the Department of Science of the Moscow Party Committee in 1936--1937. Egorov was singled out by Kol'man as one of the "saboteurs" of the new order. Egorov was given a public
rebuke at a meeting of graduate students of MSU on December 2 1 , 1929; the ac cusations against him were "religious political" [4], "criticizing his ossification, inertia, lack of political zeal in reform ing and renovating pedagogical research and scientific activity, and methodol ogy" [7]. Egorov resisted to the end. His role was significant at the Kharkov First All-Union Congress of Mathematicians, which took place in June 1930; true to his convictions, he was an10ng those who rejected the proposal to sign a greeting to the Communist Party Con gress going on at the same time. Soon afterward he was arrested. The first meeting of the MMS after Egorov's arrest (conducted as if noth ing had happened) was devoted to re ports by S. P. Finikov, one of Egorov's students, and by A. G. Kurosh, who had been a student of P. S. Aleksandrov. For this, Kurosh was expelled from the Komsomol (but later readmitted). Such a response the instigators of the cam paign against Egorov could not toler ate, and as a result "an initiative group of young Soviet mathematicians" was set up, and it raised the question of re structuring the MMS. This group, in 1930, published the "Declaration of the Initiative Group for the Reorganization of the Mathematical Society. " The "Declaration" had five signatories: L. A. Lyusternik, L. G. Shnirel'man, A. Gel'fond, L. Pontryagin, and Nekrasov (no initials were given for Nekrasov). The following are excerpts from the opening paragraph of the "Declaration" [4]: "The intensified class war in the USSR has pushed the right wing of the professorate into the camp of counter revolution. . . . Thanks go to the bril liant efforts of the OGPU [predecessor to the KGB], for uncovering the crimes of a whole series of scientific bonzes . . . Professor Egorov was arrested for participation in a counter-revolution ary organization . . . . Egorov is the pre server of academic traditions, against which the proletarian student body had already undertaken struggle. " At the next meeting of the MMS on November 2 1 , 1930, the "Initiative Group" took control. Egorov, Finikov, and Appel' rot were expelled and the Society adapted the "Declaration." Lyustemik and Gel'fond became respectively, Editor-
VOLUME 25, NUMBER 2. 2003
57
in-Chief and Secretary of the journal Matematicheski'i Sbornik, published by the MMS. On the verso of the front cover of volume 38(1)(2) (1931) of this j ournal, they wrote: "Until recently the Mathematical Society had retained its cliquish academic character. Prof Egorov, a reactionary and a church man, headed the Society. He opposed the policies of Soviet power" [8]. Soon after being expelled from the MMS, Egorov lost his position at the MSU. Af ter his arrest, he was intensively inter rogated and exiled to Kazan, where his friend Chebotarev had the courage to attempt to help him. He found Egorov, already seriously ill, on a hunger strike in the prison. On September 10, 193 1 , E gorov died i n Chebotarev's home, in his arms [9, p. 24] . Only Chebotarev and Egorov's wife were present at his funeral. His grave is in the Arskoe Cemetery in Kazan. After the arrest, ex ile, and death of his teacher, Luzin was in line to be the next target. He was a religious believer who did not accept the new regime. Luzitaniya
Luzin enrolled in the Department of Mathematics and Physics at the MSU in 1901. Following trips to France and Germany, Mlodzeevskil, in the aca demic year 1900-190 1 , gave the first course in the theory of functions of a real variable at the MSU. For the first time the words "set," the "power" of a set, "countable sets," etc., were used in the lectures. Egorov, as well as the first-year student Luzin, attended the lectures. At MSU, Luzin made the ac quaintance of V. V. Golubev (a student of Egorov) and P. A Florenskii, a fu ture Russian Orthodox priest. Floren skii's interest was aroused by Bugayev's ideas about the mathemat ics of discontinuous functions, and he chose as his research topic "The idea of discontinuity as an element of worldview." The relationships between theology and science, which were pri mary concerns for Bugayev, became so for Florenskii as well. Upon graduating in Mathematics from the University in 1904, Florenskii enrolled in the Moscow Theological Academy, and af ter his graduation from the Academy in 1908 was appointed to its faculty. The
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THE MATHEMATICAL INTELLIGENCER
unsuccessful revolution of 1905 re sulted in the closing of the MSU, and Egorov advised his student Luzin to go to Paris. Accompanied by his friend Golubev, Luzin spent part of the 1905-1906 academic year in Paris, at tending lectures by Poincare, Borel, Hadamard, and Darboux. Back in Moscow toward the end of 1909, Luzin studied one year with Egorov, and then went to Gottingen and Paris for the years 19 10-1914. One of Egorov's students, V. V. Stepanov, wrote, "We regard the date of birth of the new School of Mathematics in Moscow to be the beginning of 191 1, when Egorov's note 'Sur les suites des fonctions mesurables,' appeared in the Comptes Rendus 152 of the Paris Academy" [ 10, p. 145] . That note con tained the now-well-known "Egorov's Theorem" in Measure Theory. On his return to Moscow, Luzin was awarded the degree of Doctor of Pure Mathe matics on the basis of his famous trea tise The Integral and Trigonometric Series [ 1 1, p. 277] , and was appointed Professor in 1917 [ 1 1, p. 286], [4]. In [ 12 ] , M. A Lavrent'ev writes that this treatise differed sharply from conven tional dissertations, in the sense that each section contained new formu lations, new approaches to classical problems, as well as concrete results, and there were problems with sketched proofs incorporating such phrases as "it seems to me" and "I am convinced." Some members of the Leningrad School did not consider it much of a contribution to science; for example, Academician V. A. Steklov, when he read it, wrote comments in the margin, such as "it seems to him, but it doesn't seem to me" and "Gottingen chatter." Over the next ten years Luzin and Egorov attracted a whole series of students who became first-rate mathematicians, thus creating the Moscow School of Mathematics. These two colleagues were completely different individuals: Luzin was extroverted and theatrical, inspiring real devotion among these students and young colleagues, whereas Egorov was reserved and formal, a closed man [ 1 1 ] , [4] , [9] . Luzin's seminars attracted the at tention and admiration of mathemati cians such as I. I. Privalov, V. V.
Stepanov, M. Ya. Suslin, P. S. Aleksan drov, D. E. Men 'shov, and A Ya. Khinchin. Student papers were also read, and some of these contained important mathematical discoveries. Luzin and his students began an inten sive study of the theory of effective sets (that is, sets which can be constructed without the axiom of choice), a field that Luzin named the "descriptive the ory of sets" [ 1 1 , p. 29 1 ] . In October 1915, P. S. Aleksandrov, then a nine teen-year-old seminar student, gave an affirmative answer to Luzin's question: "Does an uncountable Borel set (B-set) have the power of the continuum?" This result was published in 1916 in Comptes Rend us Acad. Sci. Paris 162. In that paper, an operation (later on called the A-operation) was introduced for the first time. Aleksandrov was able to use this operation to prove that every uncountable Borel set contains a perfect set, and therefore has the power of the continuum. Following Luzin's suggestion, M. Ya. Suslin stud ied one of Lebesgue's papers, found an error in it, and discovered in the process a class of sets different from B-sets, and called them A-sets (some times also called Suslin sets). These sets were produced by means of the operation that Suslin proposed to call the A-operation, in honor of Aleksan drov. Suslin's result was published in 19 1 7, also in Paris; see A P. Yushke vich [ 13, p. 13]. A couple of years later, Luzin introduced a new class of sets which he called projective sets, and in his paper "Memoir on analytic and pro jective spaces" (French), Matemati cheski'i Sbornik 33(3) (1926), Luzin called A-sets analytic sets, an act that filled Aleksandrov with great bitter ness toward Luzin; that was perhaps the major cause of the deterioration of the relationship between student and teacher [ 13 ] . [Editor's addendum: This dispute is discussed in depth by G. G. Lorentz in "Who discovered analytic sets?", The Mathematical Intelligencer 23(4) (200 1), 28-32.] The Civil War which followed the Oc tober Revolution was accompanied by great hardships and suffering through out Russia. In February 1918 (New Style), the sixteen-year-old Nadezhda
(Nadia) Sergeevna Allilueva becarne the second wife of the thirty-eight yea r-old Joseph Stalin [ 14, p. 46}. Na di.a, who in the spirit of the socialist emancipation of wornen retained her birth narne after rnarriage, was known to be an ardent partisan of the revolution. During that same year, Luzin, Khinchin, Men'shov, and Suslin took jobs at the newly opened Polytechnic Institute in Ivanovo-Voznesensk, the present Ivanovo. Golubev and Privalov settled down at the Universities of Sara tov and Gor' kii, respectively [ 12]. Suslin wanted to transfer to Saratov. "One expected Luzin to give him a rec ommendation. But Luzin failed to give it," according to P. S. Aleksandrov [ 13, pp. 13, 14]. Suslin went to his parents in the village of Krasavka in the Sara tov district, and shortly thereafter he contracted typhus and died in 19 19. In later years, P. S. Aleksandrov regarded Luzin as morally responsible for Suslin's death; he in fact thought that Luzin was conscience-stricken for many years, mainly because Suslin's portrait was always on Luzin's desk [ 13]. The circle of devoted students and younger colleagues who surrounded Luzin and who attended his seminars
were jokingly referred to as "Luzi taniya." Luzin's return to Moscow in the early 1920s led to a further expansion of his seminar, and also the solution of all the original problems it had studied, except the most intractable. As a result, some Luzitaniyans began branching out into areas not always approved by Luzin. Among the Luzitaniyans during this period were some of his colleagues and older students ( [ 1 1 , p. 295]): Privalov, Stepanov, P. S. Aleksandrov, Men'shov, and Khinchin. In the second (post-revolutionary) generation of students were P. S. Uryson, V. N. Veniaminov, A. N. Kolmogorov, V. V. Nemytskii, N. K. Bari, S. S. Kovner, V. I. Glivenko, L. A. Lyusternik, M. A. Lavrent'ev, and I. G. Shnirel'man. Then P. S. Novikov, L. V. Keldysh, and E. A. Selivanovskii made up the third and last generation of Luzitaniyans. In 192 1, Uryson began his research in point set topology. P. S. Aleksandrov soon joined him, and they wrote a series of papers in abstract topology. In the spring of 1923, P. S. Aleksandrov and Uryson ar rived at Gottingen. They were the first scientists to be "sent out" of Russia since the revolution. In the summer of 1924 Uryson set off again with Alek sandrov on a trip through Germany, Holland, and France. They visited
L. E. J. Brouwer in Amsterdam (Fig. 2). Uryson, for whom he developed some thing like the attachment to a lost son, particularly impressed Brouwer. After this visit the two Russians traveled to Brittany where they rented a cottage. Uryson died at the age of twenty-six while swimming in the Atlantic with Aleksandrov at Batz-sur-Mer in 1924 in stormy weather. His grave is in the old cemetery in Batz [ 15]. Brouwer was heartbroken, and decided to look after the scientific estate of Uryson [ 16], [ 17, pp. 958, 959]. When Constance Reid vis ited Aleksandrov in Moscow in 197 1 , there was a photograph o f Uryson o n a table in his living room [ 18, p. 107]. Beginning in 1923, P. S. Aleksandrov had returned each year to Gottingen, either alone or accompanied by coun trymen. From 1926 through 1930 Courant arranged for him to give regular lec tures [ 18, p. 106] Between 1926 and 1 930, P. S. Alek sandrov laid the basis for the combi natorial topology of point sets, which included the homology theory of di mension. In spite of the lack of har mony among the Luzitaniyans, the years 1922-1926 can still be regarded as a good period for the MMS. There not only appeared a whole cycle of papers by Luzin, Privalov, Men'shov,
Figure 2. P. S. Aleksandrov, L. E. J. Brouwer, and P. S. Uryson.
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Uryson, P. S. Aleksandrov, and others, but also a number of fundamental works by the next generation of math ematicians-Kolmogorov, Lavrent' ev, Petrovskii, etc. There were diverse ap plications of Egorov's 19 1 1 paper on the metric theory offunct·ions (Soviet writers used this term to denote the area of the theory of functions in which the concepts of measure and integral are central): to boundary-value prob lems (Luzin, Privalov, Golubev), to number theory (Khinchin, Shnirel'man), to probability theory (Khinchin, Kol mogorov), to problems of mathemati cal physics (Gyunter), to integral equa tions (Egorov), to geometry (A. D. Aleksandrov), to the calculus of varia tions (Lavrent' ev, Lyustemik, Bogolyu bov), to the theory of almost periodic functions (Stepanov), to partial-differ ential equations (Sobolev), . . . "And the school of Luzin disintegrated into nu merous brilliant new schools," wrote Lyustemik [ 1 1 ] . Luzin was elected to full membership in the Soviet Academy of Sciences in 1929, and in the same year was put in charge of the division of the theory of functions at the Steklov Mathematical Institute, which operated under the direction of the So viet Academy of Sciences in Leningrad. His famous work Ler;ons sur les en sembles analytiques et leurs applica tions was published by Gauthier-Vil lars, Paris in 1930, as part of a series of monographs edited by E. Borel. The Rockefeller Foundation supported its publication, and H. Lebesgue wrote the foreword. (This foreword was omitted from the Russian translation published
Figure 3. The Luzin Tree.
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THE MATHEMATICAL INTELLIGENCER
in 1953.) A major reason for the final breakup of Luzitaniya may have been Luzin's preoccupation with the prepa ration of the Ler;ons, and the fact that he spent the years 1928-1930 in Paris working on this monograph. The pro Luzin camp at MSU included P. S. Novikov, A. A. Lyapunov, and the two women Lydia V. Keldysh and Nina K. Bari. Novikov gave courses on de scriptive function theory, while A. A. Lyapunov was the last mathematician of that era who actively studied de scriptive set theory at MSU. He intro duced the concepts of R-sets and of de scriptively measurable sets. After his return to the Soviet Union in the au tumn of 1930, Luzin decided to leave the MSU and concentrate on the work of the Academy of Sciences; it appears that he had no more students [ 1 1, p. 300). P. S. Aleksandrov was elected President of the MMS in 1932, and his presidential tenure continued until 1964. Figure 3 shows a photograph, taken in the Department of Mathematics at MSU in 1992, of the so-called Luzin Tree (for detail, see Fig. 4), containing the names and photographs of some of Luzin's students. Stalin 's wife, Nadia, died the night of November 8-9, 1982. That was after a banquet at the apartment of Kli menti Voroshilov, Stalin's war min ister. Stalin addressed his wife rudely in front of others a.t the banquet, and with the words "Don't you 'Hey, you' me, " Nadia left the room, walked around the Krernlin court with her close friend, Polina Zhernchuzhina,
the Jewish wife of the Chai1man (that ·is, Premier) of the Council of the Peo ple's Comm issars of the USSR, V. M. Molotov, and then, seemingly calrned, retired to her apartment. That same night Nadia shot herself with a little Walther pistol given to her by her brother Pa vel Alliluev. The fu n eml i n the Novodev·ichii Cernete1y was pub lic. Nadia was buried in a coffin and in the ground (Fig. 5), in contmst to the usual cremation funeral rite of the Bolsheviks [ 1 9, p. 289}. After the fu ner·al the country was unofficially in formed that she had died of acute ap pendicitis. Only after Stalin 's death in 1 953, did the Russian public dis cover that she had not d·ied of natural causes. But at that time he had so much blood on his hands that the in telligentsia immediately started say ing that he had killed his wife [ 1 9, p. 289}. By 1927, the Academy was the most important unreformed tsarist institu tion: not a single academician was a Party member. This state of affairs did not last long. A typical Soviet proce dure in taking control of pre-Revolu tionary institutions was to impose communist members and simultane ously expel older members, and pre cisely that happened after the Acad emy elections of February 13, 1929, the turning point in the Sovietization of the Academy [3, pp. 1 14, 1 15 ) . During Stalin's reconstruction throughout the 1930s, the Academy strengthened its status and controlled all scientific re search conducted in the country. The V. A. Steklov Mathematical Institute (popularly known as the "Steklovka") was moved, with its first Director I. M. Vinogradov, from Leningrad to Moscow in 1934 (Fig. 6). Vinogradov was fa mous for his work on the Waring and Goldbach problems. He was the per manent Director of the "Steklovka" practically to his death in 1983. Luzin's authority, as new head of the Mathematical Group of the Academy, was increased, yet there was a care fully planned campaign for his dis missal, described by Demidov [7, pp. 47, 48). The assault was led in the me dia by Kol' man, who accused Luzin of representing a school, which had long
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been known for "idealistic" views. (The campaign against P. A. Florenskil was also started at that time; he was ar rested on February 25, 1933. Kol'man
was among his most zealous attackers in the media.) In April 1936, Kol'man wrote, "This reactionary way of think ing has been preserved intact by one of
the 'pillars' of the school whose name is Luzin and who has rendered it a more modem pro-fascist coloring" [7]. To support a trial of Luzin, they staged
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Figure 5. The grave of Nadezhda Sergeevna Allilueva in Novodevichii Cemetery.
an act of sabotage. He was invited to a graduation examination in mathemat ics at a Moscow secondary school and was asked to contribute to the news paper Izvestiya an article giving his im pressions [20, p. 139]. The masterminds knew beforehand that Luzin would overdo his praise; they counted on his tendency to give inflated evaluations of the work of others. They were not wrong: The issue of Izvestiya dated June 2 7, 1936, carried a short article by Luzin titled "A pleasant disappoint ment." He was expecting the usual low level of academic performance, and was "pleasantly disappointed": "I could find no weak students" [8] . Luzin also acknowledged the school Director's personal contribution. On July 2, 1936, Izvestiya published the answer of the school Director, who argued that Luzin was biased, that the students did not have mathematical skills and could not work with textbooks properly. Luzin was severely criticized for praising the school: "Wasn't it your goal to white wash our shortcomings and thus to damage our school?" On July 3, Pravda carried an editorial titled "On enemies hiding behind a Soviet mask." The ar ticle says, among other things, "We well know that N. Luzin is an anti-So viet person." Further articles in Pravda followed on July 9, 10, 12, 13, 14, and 1 5 in the campaign against Luzin. At various research and academic institu tions, meetings took place where Luzin was branded as an enemy of the state.
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At MSU, Yanovskaya made a report de nouncing Luzin, and she was echoed by Lyusternik, Bukhgolz, P. S. Aleksan drov, and Kolmogorov. The main events took place at the session of a special committee set up by the Presidium of the USSR Academy of Sciences on July 7, 1936. The chair man of the committee was the Vice President of the USSR Academy of Sci ences, G. M. Krzhizhanovskii, and the committee was composed of Academi cians A. E. Fersman, S. N. Bernstein, 0. Yu. Schmidt, I. M. Vinogradov, A. N. Bach, N. P. Gorbunov, Associate Mem bers I. G. Shnirel'man, S. L. Sobolev, P. S. Aleksandrov, and A. Ya. Khinchin. Also on the committee were A. N. Krylov, A. Gel'fond, L. A. Lyusternik, and B . J. Segal. The major charges against Luzin boiled down to the fol lowing: (1) he systematically gave un deserved favorable reviews of the works of other mathematicians; (2) he published his best works abroad while only minor papers were published in the USSR; (3) he misappropriated re sults of his own disciples, like those of Suslin, Aleksandrov, Kolmogorov, Lavrent'ev, and P. S. Novikov. The most aggressive mathematicians oppos ing him were Aleksandrov, Schmidt, Sobolev, Gel'fond, and Lyustemik, while only Bernstein and Krylov tried to de fend their colleague [7]. The commit tee's report is given in Pravda, August 6, 1936, p. 3. The well-known physicist and Nobel laureate Pyotr Leonidovich Kapitsa had the courage to protest openly against this onslaught on Luzin. On July 6, 1936, Kapitsa wrote a letter to V. M. Molotov protesting the campaign against Luzin [ 8 ] . A day or two later, this letter was returned to Kapitsa; in the upper left corner of the first page were the words: "Not wanted, return to citizen Kapitsa. V. Molotov." For years Kapitsa had been at Cambridge Uni versity, England, working in the Cavendish Laboratory with Ruther ford. Kapitsa spent his summers at his dacha in the village of Zhukovka in the Soviet Union. When he wanted to re turn to England at the end of the sum mer of 1934, the Soviet authorities re fused permission, and he was given his own institute in Moscow. In 1937,
Kapitsa wrote a letter to protest the ar rest of the physicist Fok, and in 1980 he addressed one to Yurii Andropov (then head of the KGB) to protest the treatment of the physicists Sakharov and Orlov. The campaign against Luzin was abruptly stopped sometime between July 1 1 and July 13, 1936, by somebody high up. He got off lightly for those times, and instead of the expected ex pulsion from the Academy, was given a reprimand and a warning by the Pre sidium of the Academy of Sciences. Yushkevich [ 13, pp. 14-17], explains that a perusal of Luzin's publications shows no trace of appropriation of Suslin's ideas, and that Luzin's refusal in 1919 to recommend Suslin for a pro fessional position at the University of Saratov was due to Suslin's lack of per formance and could hardly be called "Suslin's persecution by Luzin, " as was done by Aleksandrov at Luzin's hear ing. However, at the hearing, Luzin re ferred to Suslin's death in these terms: "And then a catastrophe occurred that has weighed heavily on all my life" [ 13]. In [20] , Demidov and Ford offer the hypothesis that the attack on Luzin was halted by Stalin himself, who was at that stage hopeful of a N azi-Soviet pact. In 1990 the article "The Fate of Greatness" by Vitalii Shentalinskil was
Figure 6. I. M. Vinogradov.
published in the journal Ogonyok. This article, based on documents in the archives of the KGB, is devoted to the circumstances of the arrest and exe cution of P. A Florenskii in 1937. These documents show that the OGPU con structed a counter-revolutionary orga nization, "Party for the Rebirth of Rus sia. " This new affair apparently began with the arrests in January and Febru ary 1933 of Profs. P. V. Gidulyanov and Florenskii, respectively. The first to "confess" was Gidulyanov, and he im plicated S. A Chaplygin (Mathematics, MSU), Luzin, and Florenskii as mem bers of the "Committee for National Organization" (or just, "the national center") within their fictitious organi zation. Hitler became Chancellor of Germany on January 30, 1933, and the OGPU took the opportunity to add the label "fascist" to "national center. " In his testimony given on March 3, 4, and 5, Florenskii named himself as the head of the organization, with Luzin and Gidulyanov as his immediate deputies, and with Luzin in the special role of making connection with the outside world. On June 30 the "investigation" reached its conclusion, and a docu ment of 30 pages was issued: "The or ganization was headed by the leader ship center consisting of Professors Florenskii, Gidulyanov, and Academi cians Chaplygin and Luzin. Connection had been made with the emigre White Guards and a confidential meeting with Hitler had been arranged" [20]. The Special Troika of the Moscow district of the OGPU on July 26, 1933, sentenced the participants of the "cen ter": Gidulyanov was exiled and Flo renskii was convicted and sentenced on November 10, 1933, to 10 years in corrective labor camps. Although Luzin and Chaplygin were not touched at this time, the OGPU preserved this mater ial about their supposed membership in the leadership of the National Fas cist Center up to 1936 [20, pp. 144, 145]. The attack on Luzin may have been started by local initiative with the in volvement of the OGPU. A campaign against the National Fascist C enter, however, would touch on the Soviet re lationship with Germany. During the years 1935--1936 Stalin was seeking good relations with Berlin. He sent his
personal confidant David Kandelaki to Berlin as a Soviet trade representative to pursue secret negotiations for a non aggression pact [20] . As a result of the efforts of Kandelaki, a trade agreement with Gem1any was concluded in May 1936. According to the Demidov-Ford hypothesis, Stalin could have regarded as badly timed a major affair that would serve no other purpose than anti-Nazi propaganda. This, together with preparations for the much more important trial against political oppo nents L. B. Kamenev, G. E. Zinoviev, and fourteen others, to be held in pub lic between August 19 and 24, 1936, could have caused Stalin to abort the campaign against Luzin. Luzin never knew the precipice before which he stood. Academician Chaplygin also never learned of the materials against him contained in the files of the OGPU, but in 1937 he was rearrested and shot. Florensk!1 was a priest and, unlike Luzin, openly identified himself with the Church. The years 1936-1939 saw the liquidation of a number of bishops and clergy, and following the decision of a Special Troika of the Leningrad district of the Soviet Military Intelli gence (NKVD), Florensk!1 was shot on December 8, 1937 [7], [20] . The MSU today acknowledges Florenskl1 as one of the "outstanding thinkers of the tum of the century" [2 1 ] . L. V. Keldysh (wife of P . S . Novikov and sister of M. V. Keldysh) was, ac cording to Sossinsky [22 ] , the only one of Luzin's students to remain faithful to him during the difficult years 19361937 when considerable pressure was applied on her to denounce him pub licly in the framework of the campaign against him. Apart from his work in the theory of functions of a complex vari able, M. V. Keldysh is known for his long tenure as the President of the Academy of Sciences of the USSR, 1961-1975, and also for the institute named after him [23] . So, in a sense, "the young" won. In 1939, Kolmogorov was elected Member of the Academy of Sciences. Gel'fond, Pontryagin, and Khinchin were elected Associate Members; in 1946 Lavrent' ev became Academician and Lyustemik an Associate Member. P. S. Aleksandrov became Academician only in 1953, after
Luzin's death [ 13 ] . Luzin withdrew from most of his mathematical activi ties and died of natural causes in 1950. From 1964 until 1966, and again from 1973 to 1985, A N. Kolmogorov was President of the MMS. The "black angel" of the Moscow mathematical circles, Kol 'man, disap peared from the political arena in 1937 when he was arrested and dismissed from his posts. This was revealed only on November 14, 1989 in the newspa per Vechernyaya moskva [7] . He ceased to play any significant role in the life of the mathematical community of the USSR.
The death of Nadezhda Sergeevna Allilueva was not, in all probability, murder, and it did not lead quickly to a morbid deteriora tion of Stalin 's dealings with his political associates. That change seems to date from De cember 1 , 1 934, and the murder of S. M. Kirov, Party boss of Leningrad and long-term protege of Stalin. Late in 1934, Stalin lashed out to right and left. A grim nationwide search swept up hundreds of thousands of suspects in every walk of life; more than 75% of the Central Committee of the Commu nist Party were killed or imprisoned. The "Great Purge" (Yezhovshchina) of 1934-1 938 sent thousands before the firing squads, imprisoned thousands, and put untold numbers into forced labor camps or in exile in Siberian wastelands. Every Soviet citizen was given to understand that Stalin was the state and that to question his lead ership or policies would be to invite a charge of treason. Following the Munich Pact of Sep tember 1 938, Great Britain and France sought to establish an alliance with the USSR, but Stalin saw an al liance with Nazi Germany as a safer bet. The Soviets believed that joining with France and England would have guaranteed a Nazi attack on the So viet Union. On August 23, 1 939, in Stalin 's presence, Hitler's Foreign Minister, Joach'im von Ribbentrop, and the USSR's new Commissar of Foreign Affairs, V. M. Molotov (still Chairman of the Council of People's Commissars), signed the Russo German Non-aggression Pact in
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Moscow. The previous Commissar of Foreign Affairs, Maxim Litvinov, a Jew w'ith an English wife, could not be the best intermediary between Stalin and Hitler, so he had been re placed on May 3, 1 939. But also Molo tov could hardly qualify under the Reich's racial laws: he was Aryan himself, but he had a Jewish wife. The German diplomats in Moscow, most of them proponents of a Russo-Ger man understanding, withheld this embarrassing fact from the tempera mental Fuhrer. The Pact came as a surprise not only to France and England but also to the Soviet people, who had been sub jected to years of anti-Nazi propa ganda. They accepted the Pact with that acquiescence which years of to talitarian rule might be expected to produce [24, p. 51 8]. Since the be ginning ofhis political life, Hitler had declared the destruction of Commu nism as one of his primary objectives, so, in spite of the agreement, he launched a massive attack against the USSR on June 22, 1 941. This news was announced 'in a radio broadcast by Molotov on that same day; see [25]
Figure 7. I. G. Petrovskii.
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for Molotov's speech. Immediately af ter World War II the Party underwent a bloodless purge to rid it ofpolitically illiterate lackeys who had wormed their way into subordinate posts during the conflict [26, p. 585]. According to rev elations by Khrushchev, Stalin even threatened to kill Molotov, Malenkov, and Mikoyan, in fact all the old mem bers of the Politburo. The Politburo was renamed in 1952 the Presidium [19, p. 530]. Petrovskii
Ivan Georgievich Petrovskii (Fig. 7) was born on January 18, 1901, in Sevsk, Orlov guberniya, Russia, a grandson of a "merchant of the first guild" [27]. Af ter graduating from the technical high school in Sevsk in 1917, he entered the MSU, where he intended to study Biol ogy and Chemistry. When the revolu tion broke out, Petrovskii returned home to Sevsk, and the family moved south to Elizavetburg, where he en tered the local technical school for ma chine construction. There, for the first time, he came across a serious mathe matics book and found it interesting. In 1922, as a proletarian, he returned to the MSU and settled on Mathemat ics. In [27], E. M. Landis, (a student of Petrovskil), describes him "as intellectu ally deliberate": Petrovskil and Men' shov "reasoned slowly" when compared to Kolmogorov, M. V. Keldysh, and A. A. Lyapunov [27, p. 68]. Petrovskii gradu ated in 1927, and remained at the MSU until 1930 as a graduate student of Egorov [28], [29 ] . From 1929 to 1933, Petrovskii was Assistant Professor and Dozent at the MSU. The Faculty of Mathematics was opened in 1933. It in cluded three divisions: Mathematics, Mechanics, and Astronomy. Later the Astronomy Division was transferred to the Department of Physics; the re maining two divisions are commonly known as Mekh-Mat. Petrovskii be came Professor in 1933, and in 1935 Doctor of Physical-Mathematical Sci ences. During World War II he was Dean of Mekh-Mat. From 1951 he was Head of the Department of Differential Equations, and from the same year un til his death on January 15, 1973 in Moscow, he was Rector of the MSU [29]. TI:te Petrovskii stamp displayed in
Figure 8. The Petrovskii stamp.
Figure 8 was issued by the Soviet Union in 1973 marking his death. From 1943, Petrovskii also worked at the V. A. Steklov Mathematical In stitute at the Academy of Sciences, of which he was Vice-Director from 194 7 to 1949. He was also Editor-in-Chief of Mathematicheskii Sbornik. In the fall of 1946 a number of new mathematical academicians were to be appointed to the Soviet Academy of Sciences. One of the candidates was P. S. Aleksan drov, Corresponding Member since 1929, and closest friend of Academi cian Kolmogorov (Aleksandrov once described his friendship with Kol mogorov as one "which in the whole 53 years was always unclouded" [30, p. 34]). During the initial discussions of the candidates, Luzin noted Aleksan drov's great contributions to topology and supported his candidacy. A few hours later, however, just before the vote, Luzin stated that what the coun try needed at that moment was an "ap plied" profile, and that Aleksandrov's candidacy was inappropriate because of his interest in very abstract prob lems. It was at that point that Kol mogorov lost his temper and slapped Luzin very slightly in the face. When S. I. Vavilov, President of the Academy of Sciences, told Stalin about this inci dent, the latter replied, "Well, such things occur with us as well." During the 1946 elections the following "ap plied" mathematicians were appointed
Figure 9. The grave of I. G. Petrovskii in the Novodevichii Convent Cemetery.
to the Academy: M. V. Keldysh, M. A. Lavrent'ev, A. N. Dinnik, and Petrov skii. From 1953 to 1973 Petrovskil was a member of the Presidium of the Academy. He was twice awarded the State Prize of the USSR, and received the title Hero of Socialist Labor. He was also a member of the Soviet Com mittee for the Defence of Peace and Vice-President of the Institute of So viet-American Relations. His first research (still as a student) dealt with the Dirichlet problem for Laplace's equation ( 1928) and the the ory of functions of a real variable ( 1929). In a 1929 paper he solved a problem proposed by Lebesgue, namely, to establish whether a function is uniquely determined by its general ized derivative. In the early 1930s he began research on the topology of al gebraic curves and surfaces, and in 1933 he proved Hilbert's hypothesis that a curve of the sixth order cannot consist of eleven ovals lying outside each other. The results that Petrovskil obtained in 1934 on the solvability of the first boundary-value problem for the heat equation were widely applied in the theory of probability, especially in connection with the Khinchin-Kol mogorov law of the iterated logarithm. During 1937-1938 Petrovskil studied classes of systems of partial-differen tial equations; in 1 93 7 he published his
proof that the Cauchy problem for non linear systems of differential equa tions, called "hyperbolic" by him, is well posed. Also in 1937, he introduced his notion of elliptic systems and showed that when the functions are analytic, all sufficiently smooth solu tions will be analytic, thereby giving a more complete solution of Hilbert's nineteenth problem. Petrovskil 's re sults were the starting point for nu merous investigations on systems of partial-differential equations. In 1954 he examined lines and surfaces of dis continuity of solutions of the wave equation. Petrovskil was gentle and tactful, and talked cautiously: "You see, what very often helped me solve problems was lack of knowledge. Had I known that people had unsuccessfully tried to solve a certain problem in the past, I certainly would not have attempted it" [27]. Quite a number of well-known specialists in the theory of differential equations were his students, and his seminar became a leading center for the study of partial-differential equa tions [29]. Landis [27] writes that dur ing his most creative period, Petrovskil celebrated each year with a new, first rate work. It used to be said at Mekh Mat that Petrovskil chooses a problem in the spring, solves it in the summer, lectures on it in the fall, and publishes
it in the winter. Incidentally, Professor Olga Arsienevna Olelnik, who com pleted her thesis under the guidance of Petrovskil, passed away on October 1 1, 200 1 . She made her mark throughout the world with her (almost three hun dred) articles and monographs. For a long period she was Head of the De partment of Differential Equations at MSU and was the adviser of more than fifty mathematicians, some twenty of whom obtained the degree of Doctor of Science. A. B. Sossinsky [22, p. 227] de scribes the period 1957-1968 at Mekh Mat as a "Golden Era, " when mathe matics and mathematicians flourished in a highly stimulating environment. The person most responsible for this was Rector Petrovskil, who will be re membered for his honesty, personal courage, and outstanding ability as an administrator. Although Petrovskil be came rector under Stalin, he managed to concentrate a great deal of power in his own hands ("although he's not even in the party" [22]) and used it to expand and enrich the university in general. On the other hand, that period also over lapped with the years of partiinost nauki ("political orientation of sci ence"), which manifested itself in ac tions such as the Lysenko theory in biology [27], and denunciations of cy bernetics (a "bourgeois pseudo-sci ence"), of psychoanalysis (another "capitalist fraudulent science"), of mathematical methods in economics (as "inapplicable in principle"), and of "Mach-inspired" Physics. In 1934-1935 Trofim Lysenko promoted a wide cam paign, supported by Kol'man [ 3 1 ] , against genetics (bear i n mind that Gregor Mendel was a monk), and in sisted that characteristics developed through environmental influences could be inherited. Helped by faked experi ments, Lysenko opened a campaign for the domination of Soviet biology. The debate was linked to the economic dif ficulties of the USSR, and Lysenko's theories enabled the government to claim that Soviet agriculture was on the verge of spectacular developments. Leading geneticist Nicholas I. Vavilov refused to subscribe to Lysenko's the ory (which was supported by Stalin) and was packed off in 1940 to Siberia
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as a "British spy." Lysenko replaced Vavilov as Director of the Institute of Genetics of the Academy of Sciences, a position that he occupied until Janu ary 1965 [ 1 , p. 749]. Even Khrushchev had a weakness for the man who promised to work such miracles for So viet agriculture. During a special assembly of the Mekh-Mat Research Council, one mathematician after an other came forward and "confessed his sins." In particular, Kolmogorov re nounced his genetics-related works on probability, and Khinchin published a paper in a slanderous anti-Einstein collection of papers. Of the leading Moscow mathematicians connected with genetics, only A. A. Lyapunov stood firm. This in the long run did great harm to his career [27] . He moved to Siberia in 1961 to become a profes sor in the Department of Mathematics, University of Novosibirsk. After World War II, Kol'man re turned to Prague. In 1948, after the change in government in Czechoslova kia, he was arrested and deported to Moscow, where he spent three and a half years in Lubyanka Prison. After his release, he took up science writing again. In 1953 Kol'man was apparently one of the first in the Soviet Union to defend Cybernetics [8], the science that A. A. Lyapunov helped to intro duce and for which he was severely criticized. V. M. Molotov served as Commissar of Foreign Affairs from 1 939 to 1 949, and again after Stalin's death from 1 953 to 1 956, when he became Min ister of State Control. Khrushchev dis missed him from all his posts in July 1 957. Molotov was expelled from the Party in 1962, was reinstated as a member in 1 984, and died in Moscow on November 8, 1 986 [32], [33]; see also the last section of this article. Among all this, Mekh-Mat, until the end of 1968, was a haven where the ob jective value of one's research work was one's best asset. The people re sponsible for this state of affairs, ac cording to Sossinsky [22 ] , were Petro vski'i, P. S. Aleksandrov (head of the otdelenie matematiki of Mekh-Mat),
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N. V. Efimov (Mekh-Mat Dean, 19591969), and Kolmogorov, because he "symbolized total scientific involve ment." I. R. Shafarevich, widely ac knowledged by his colleagues to be a brilliant mathematician (a Wunderkind [27]), lost his position as Professor at Mekh-Mat in 1 948, but Petrovski'i in vited him back when he became Rec tor of the MSU. Shafarevich became a vocal critic of the Soviet government in the 1970s. In retaliation for his ac tivities as a dissident, he was ousted from the Steklov Institute Scientific Council, but not fired from the Steklov Institute due to a rule that a member of the Russian Academy cannot be dis missed from an institute of the Acad emy. In [9] , Shafarevich mentions the great role played by Petrovskil both at MSU and at Mekh-Mat, and that he, around the time of Stalin's death, pulled MSU from a depression. Sha farevich also recalls his collaborations with Solzhenitsyn, Sakharov, and other dissidents, and that Petrovskil (who, like Vinogradov, never signed a state ment condemning Solzhenitsyn and Sakharov) was able to delay Shafare vich's second dismissal from Mekh Mat. In fact, Petrovskii managed to de lay it to his death. The Soviet government repressed and controlled various forms of intel lectual inquiry during those years. Satirical works by the Soviet writers Yulil Daniel and Andre'i Sinyavski'i were smuggled out of the country and pub lished in the West under pseudonyms. Both men were indicted and tried for the crime of producing "anti-Soviet propaganda, harmful to the Soviet peo ple," [34, p. 142]. In February 1966, Daniel and Sinyavskii were convicted and sentenced to five and seven years of forced labor, respectively. There was also a network of anti-communists consisting of writers, artists, mathe maticians, and other intellectuals, who not only expressed their solidarity for Daniel and Sinyavskii, but also even supported the United States's inter vention in Vietnam. During 1965 Stephen Smale and Laurent Schwartz collaborated to internationalize oppo sition to the Vietnam War. The 1966 Moscow International Congress of
Mathematicians (ICM-66) provided an opportunity to petition mathemati cians from all over the world. The In ternational Appeal Against the War in Vietnam had been conceived by a Japanese mathematician and Schwartz supported the effort. Prior to the Con gress, Schwartz enlisted the help of Chandler Davis. The Appeal's mission was in line with Soviet policy, but there was a reluctance at the Congress to permit political activity outside of rigid government control. Petrovski'i, Presi dent of the Congress, refused a request for a meeting room by a Schwartz Davis-led delegation. Access to dupli cation machinery, as a potential pro paganda tool, was out of the question [34]. Chandler Davis solicited the help of the North Vietnamese government, at whose embassy Davis typed the petition and the North Vietnamese mimeographed it. In October 1964, Leonid I. Brezhnev replaced Khrushchev as First Secre tary of the Communist Party of the So viet Union; his title was changed to "General Secretary" in April 1966. For several years he shared power with A. Kosygin and N. Podgorny'i. In 1977, Brezhnev replaced Podgorny'i as Chair man of the Presidium of the Supreme Soviet, and thus became head of state as well as head of the Communist Party. During his rule, the Soviet Union experienced a period of stability but then of stagnation. Brezhnev's health worsened in the late 1970s, and he rel egated many of his duties to col leagues. Unwilling to bring younger leaders into top positions, the aging So viet leadership was unable to adopt in novative solutions to the country's growing problems. Coupled with the tight controls on public debate and expression of ideas that continued throughout Brezhnev's reign, the im mobility of the Soviet leadership under Brezhnev ensured that his successors would face serious, and eventually in surmountable problems. Upon Brezh nev's death in 1982, Yuri'i Andropov succeeded him. The year 1968 was the year in which Russian tanks crushed the Prague Up rising. But Mekh-Mat remained rela tively free thanks to the efforts of
Petrovskfi ("the last decent rector of MSU" [35]). After the Daniel-Sinyavskil trial, some groups of people wrote letters to official institutions with protests, about violations of rights. Early in 1968, A. S. Esenin-Volpin, a mathematical logician and well-known human-rights activist, and son of the great Russian poet Sergei Esenin, was taken to a psikhuska, a special psychi atric hospital for political deviants. This shocked the Moscow mathemati cal community and, uncharacteristi cally, it reacted by writing a letter of protest signed by 99 of the leading mathematicians of the day. The letter was almost immediately published in the West, against the wishes of its authors and cosigners. The violations were so evident that Esenin-Volpin was transferred to the Academy's hospital where he was out of danger. Some of the cosigners were subjected to humil iating public disavowal procedures, and some eventually lost their jobs. A year later Efimov was replaced as the Dean of Mekh-Mat by Ogibalov, who was one of the heroes of the Yezhovshchina of 1934-1938. Every side of Mekh-Mat's life was terribly af fected: entrance examinations (and the organization of systematic anti-Semitic practices at these examinations [22 ] , [35], [36]), selection o f graduate stu dents and new staff members, etc. Kol'man, who had again returned to Prague, took a critical attitude to the entry of Soviet troops. Eventually, he immigrated to Sweden, where in 1982 he published his memoirs in German and Russian, under the title "We should not have lived that way." During the last years of his life, Petrovskii conducted a joint seminar with the physicist I. M. Lifshits. Petrov skii also took part in archeological digs in Novgorod; he knew a great deal about painting (in spite of being color blind); and he was an avid bibliophile; his widow donated his splendid library of 30,000 volumes to the MSU [ 2 1 ] . "Petrovskii i s one o f the few mathe maticians whose work shapes the face of modem mathematics. However, he regarded being a rector as the most important thing in his life, even more important than his mathematical re-
search," according to P. S. Aleksandrov and 0. A. Oleinik, see [28]. Pontryagin
Lev Semenovich Pontryagin (Fig. 10) was born on September 3, 1908, in Moscow, Russia. His father, Semen Akimovich Pontryagin, was a minor civil servant from the town of Trubchevsk in the province of Orlov. His mother Tat'yana Andreevna Pontryagina, was originally from a peasant family in the province of Yaroslav, but worked as a tailor in Moscow [37]. At the age of 13 Pontryagin suffered an ac cident and an explosion left him blind [38]. His mother, with no mathematical training or knowledge, made by deter mination and extreme efforts a major contribution to mathematics by enabling her son to become a mathe matician against all odds. For many years she read scientific work aloud to him, writing in the formulae in his manuscripts, correcting his work, and so on [9]. Pontryagin entered the MSU in 1925 as a student in the Faculty of Physics and Mathematics. Although he
Figure 1 0.
L. s.
Pontryagin.
could not make notes during lectures, he was able to remember complicated manipulations and to grasp the topics presented. P. S. Aleksandrov influ enced him, and the direction of Alek sandrov's research, namely topology, also determined the area of Pontrya gin's work for many years. In 1927 (the year of his father's death), Pontryagin produced his first independent results, in which he strengthened and general ized Aleksandrov's principle of duality. After graduating from the MSU in 1929, Pontryagin became a member of Mekh Mat [39]. In 1932 he produced the most sig nificant of these duality results when he proved the duality between the ho mology groups of an arbitrary bounded closed set in Euclidean space and the homology groups in the complement. Pontryagin's duality law not only gave rise to a vast new field of topological re search, but, of equal significance, en abled him to construct a general theory of characters for commutative topolog ical groups. This theory, historically, was the first really exceptional achieve ment in algebraic topology. Pontryagin summarized his research in his mono graph Topological Groups, which ap peared in 1938 in Russian, and in 1939 in English.
He worked for 25 years in topology and algebra. In 1934, Pontryagin proved Hilbert's Fifth Problem for Abelian groups using the theory of characters on locally compact Abelian groups, which he had introduced. In that same year, Elie Cartan, the father of Henri Cartan, visited Moscow and lectured in Mekh-Mat on the problem of calculat ing the homology groups of the classi-
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cal compact Lie groups. Pontryagin, in 1935, was able to solve the problem completely using a totally different ap proach from the one suggested by Car tan. In [38, p. 14], Pontryagin writes that finding the solution to Cartan's problem of determining the Betti numbers of compact Lie groups was his greatest achievement. During the years 1935 to the end of the 1940s, Pontryagin did his fundamental research on homotopy the ory and the theory of skew products. In 1935, Pontryagin became affili ated with "Steklovka," where he was Head of the Differential Equations De partment. He was allowed to keep his position at Mekh-Mat. At the time of Luzin's trial in 1936, Pontryagin openly expressed criticism of Aleksandrov at a large meeting. After that, Pontryagin felt that he was a mathematician in his own right, independent of Aleksan drov. In 1939 he was elected Corre sponding Member of the Academy of Sciences of the USSR, and full mem bership came in 1958 [38, p. 2 1 ] . This last election was planned and carried through by I. M. Vinogradov. In 1940 he was one of the first recipients of the Stalin Prize (later called the State Prize) for his monograph Topological Groups [38, p. 13]. Pontryagin was hon ored with the Order of the Red Banner of Labor, the Order of the Badge of Honor, the Golden Star of a Hero of So cialist Labor, more than once with the Order of Lenin, and also with the LobachevskiiPrize for his research. He was also Editor-in-Chief of Matem aticheskii Sbornik for some time. Pontryagin [38, p. 18], writes that long before World War II, he had lived with a sense of a dreadful impending danger. Molotov's speech of June 22, 194 1 , filled him with a sense of cata strophe, and his fears, in the face of an uncertain future, pushed him into a marriage which did not last for long. In 1932, Pontryagin made contact with the physicist A A Andronov, and they met several times a year (right up to the time of Andronov's death in 1952) to discuss important problems in the theory of oscillations and the the ory of automatic control in which Andronov was engaged. In 1932, with Andronov, Pontryagin produced a pub lication on coarse dynamical systems,
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and in 1942 he produced another on the zeros of certain elementary transcen dental functions [37, p. 149]. In 1952, the nature of Pontryagin's work changed drastically: he concentrated wholly on optimal control theory. The main rea sons for this, according to Pontryagin himself [38, p. 18], were threefold: his long-time desire to move to more ap plied problems; the interests of his stu dents E. F. Mishchenko, R. V. Gamkre lidze, and V. G. Boltyanskil ; and pressure from the Steklov Institute (in the person of its acting Director M. V. Keldysh) for him to move to more ap plied problems. Landis [27, p. 66] ex presses his own opinion, which is also supported by some of his friends, why Pontryagin moved away from topol ogy: Some French mathematicians (the students of Cartan, Serre, Thorn . . . ) created a new theory of so-called fibre spaces that made possible simple so lutions of problems that Pontryagin struggled with. Indeed, the "French works" made many beautiful geomet ric results of Pontryagin's era into comparative trivialities, see D. B. Fuchs [42, p. 2 1 7]. In 1952, Pontryagin and his seminar began by studying Andronov's book on the theory of oscillations, and contin ued after that by working exclusively on applied problems, in particular dif ferential equations, control theory, and optimal control processes. They pur sued these fields until Pontryagin's death on May 3, 1988. One of Pontrya gin's greatest achievements in connec tion with the mathematical theory of control was to define the concept of a controllable system and to formulate the problem of optimal control for a system of this kind. He stated the necessary optimal condition-the now well-known Pontryagin Maximum Principle. This principle is the core of the mathematical theory created by Pontryagin's school. Pontryagin's stu dents subsequently obtained an essen tial generalization of the maximum principle, to which were added exis tence and uniqueness theorems of op timal control, sufficient conditions for optimality, and other results. Pontrya gin and his seminar also obtained im portant results in the field of differen tial games.
The activities of Pontryagin and his school did not arouse sympathy or sup port from the older generation of math ematicians. Aleksandrov, for example, regarded it as disloyalty to Pontrya gin's main work in the field of topol ogy; and at Mishchenko's Ph.D. exam ination Kolmogorov was sarcastic about studying the vacuum type oscil lator [38, p. 19]. In 1954, Pontryagin be gan to give a course in ordinary differ ential equations in Mehk-Mat, while Mishchenko and Gamkrelidze were in charge of the exercises on this course. The intrusion of these three into the field of ordinary differential equations was not met with great enthusiasm in the Faculty. When, at some stage, Pon tryagin wanted to have his course notes published by the University Press to be ready before the examina tion, Petrovskil, the Rector of MSU and Head of the Department of Differential Equations, "was not able" to persuade the manager of the University Press to do the job. When after three years Pon tryagin decided to end his relationship with the lecturer, Petrovskil would not allow either Mishchenko or Gamkre lidze to continue with the course. Their activities in that area then came to an end. The lecture notes were published in 1961 , and in 1975 were awarded a State Prize for published textbooks. In 1961, Pontryagin published The Math ematical Theory of Optimal Processes with Boltyanskil, Gamrelidze, and Mishchenko, and they received the Lenin prize for this publication in 1962 [38] . An English translation appeared in 1962. Furthermore, an English trans lation of his book Ordinary Differen tial Equations also appeared in 1962. The Organizing Committee of the ICM-58 in Edinburgh invited Pontrya gin to deliver a plenary lecture on topology, but he suggested another subject: The mathematical theory of optimal processes [38] (in [43, p. 59] , however, the title is given as "Optimal processes of regulation"). The lecture was delivered in Russian. He also de livered a plenary lecture at the ICM-70 in Nice on differential games, this time in English [38, p. 22] . (In [43, p. 36] one reads that English became the lingua franca of the Congress and that: "All the plenary speakers, including the
Russians, gave their talks in that lan guage, with the exception of Pontrya gin, who used French. " In [43] the title of Pontryagin's talk is also given in French: "Les jeux differentiaux lineaires.") Before the ICM in Nice, Pontryagin was elected to be the So viet representative on the Executive Committee of the IMU. Apparently N. N. Bogolyubov had already promised this post to Academician I. N. Vekua, but I. M. Vinogradov, Chairman of the National Committee of Soviet Mathe maticians, and M. V. Keldysh, President of the Academy of Sciences of the USSR, decided otherwise. They rec ommended Pontryagin, and in the As sembly of 1970, he was elected a Vice President of the IMU for the period 1971-1974 [38, p. 22], [44]. At the same occasion Pontryagin gave a one-hour lecture on differential games. He also served on the Executive Committee of the IMU for the period 1975-1978. In [38, p. 23] , Pontryagin says that when the 1971-1974 Executive Committee considered proposals for the composi tion of its 1975-1978 successor, an at tempt was made by the Zionists to take the IMU into their hands, and that he managed to repel that attack Accord ing to Pontryagin, "they" tried to raise Professor N. Jacobson, "a mediocre scientist but an aggressive Zionist," to the Presidency of the IMU. The Coun cil of the London Mathematical Soci ety, the British Library Lending Divi sion, and the Editor of the Russian Mathematical Surveys dissociated themselves from this personal attack on Professor Jacobson, and stated that Pontryagin's accusations were without foundation [38]. Professor Jacobson also responded by stating that he was offered, and reluctantly accepted, a po sition as a Vice-President of the IMU on the Executive Committee to serve out the term of professor A. A. Albert, who died on June 6, 1972. Jacobson also made it clear then that he would not serve beyond 1974, and, in [38, pp. 23, 24] , states that he was unaware of any outside "Zionist" pressure to influ ence the nominations, and that he was in any case the only Jew on the Exec utive Committee. Professor D. Mont gomery was elected the new President of the IMU for the period 1975-1978,
while Pontryagin served as a member of the Executive Committee during that period [44]. In 1979, the Chairman of the Orga nizing Committee of the ICM (Warsaw 1982), Cz. Olech, had to negotiate with the Chairman and Vice-Chairman of the National Committee of Soviet Mathematicians, I. M. Vinogradov and Pontryagin, respectively, about the participation of the Soviet mathemati cians in the ICM-82 in Warsaw [45, p. 2 15]. The negotiations stalled, and in points one and two of a signed six point protocol of the negotiations, the Russians objected to racist propa ganda of the Zionists, and to western mathematicians with Zionist ideology using the ICM for anti-Soviet political activity. A couple of months later it was agreed that the Soviet mathematicians would participate in the ICM-82 in War saw on a large scale. Plainly the con flict had been due largely to Vino gradov and Pontryagin. Their policy met resolute resistance from the Poles, and the leadership of the Soviet Acad emy of Sciences did not endorse it [ 45, p. 218]. The ICM-82 was postponed due to the events in Poland, and took place between August 16 and August 24, 1983, in Warsaw. The fact that Pontryagin in his old age become a "fierce and vicious anti Semite," as reflected in this incident, is also noted by Landis [27, p . 66] . B. A. Rosenfeld [46, p. 82] , makes a remark in the same vein: "In the years before the war Pontryagin was a very pleas ant man, for the anti-Semitism that he became famous for in the last years of his life had not completely surfaced." Sossinsky [22, p. 233] describes how he was recruited by Kolmogorov in 1962 to teach some exercise classes in cal culus in Moscow Special School No. 18, a boarding school for talented and carefully selected out-of-town students interested in Mathematics and Physics, which Kolmogorov had founded with the help of Petrovskil and I. Kikoin, the H-bomb physicist. One of the teachers was V. A. Skvortsov ("another unusual mathematician, for many years the president of the University English Club"), who years later accompanied the author of this article to Novode vichii Cemetery. After his resignation
from Mekh-Mat in 1974, Sossinsky, in 1975, got a job as a Mathematics Edi tor at the popular science magazine Kvant. Kolmogorov and Kikoin had founded this magazine in 1969, with the assistance of Petrovskii, in the frame work of the Academy of Sciences. In 1980, Pontryagin made an unsuccess ful attempt to wrest the magazine away from them. Pontryagin accused Kol mogorov of ruining the secondary school curriculum by introducing ab stract, set-theoretic, and "Semitic" math ematics in place of traditional, applica tions-oriented "Russian mathematics" [22 ] . The takeover bid failed because Pontryagin's followers had not done their homework properly: the Mathe matical Section of the Academy did not have any legal authority to control the magazine. Sossinsky, who was present at these proceedings in the Steklov In stitute, remembers the tragic and odi ous figure of Pontryagin, not as the great mathematician that he once was, but as an old man nervously clicking the beads of his rosary and lashing out at Kikoin and Kolmogorov. These virulent attacks, supported by the anti-Semitic remarks of Vinogradov, were simply ignored by Kikoin [22, p. 240] . I conclude this section, and intro duce the next one, with the words of Shen [47]: "Also the science was in herently connected with the general (terrible) political climate, repressions of Soviet times, etc. Pontryagin's 'anti Semitism' is a small aspect of the gen eral situation and-1 believe-should not be considered in isolation." Anti-Semitism: A few Incidents over the Years 1 93Q-2000
An extensive treatment of the at tempts by the Stalinist regime to suppress Soviet Jewry in public life is given by G. Kostyrchenko [48]. Kostyrchenko uses hundreds ofprevi ously unpublished and recently de classified documents discovered mainly in the archives of the Central Com mittee of the Communist Party of the USSR, of the KGB, and of the archives of the Federal Counter-Intelligence Service. By the end of World War II, Stalin was preparing a crackdown. Solomon
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Figure 1 1 . A. G. Kurosh, A. A. Glagolev, A. N. Kolmogorov, I. G. Petrovskii, L.
Mikhoels, a People's Artist of the USSR, had been Chairman of the Jew ish Anti-Fascist Committee (JAG), set up by the government during World War II. On its behalf he had traveled in the USA, giving speeches and col lecting funds for the devastated Jew ish community in the USSR. All of the JAG's activities, members, contacts, and pronouncements at home and in foreign parts, must have been checked and approved by the appropriate So viet authorities [24, p. 683]. Almost all of the well-known Jews in the So viet Union were on the JAG [19, p. 504}. Stalin appointed Solomon Lo zovskii:, the Head of the Soviet Infor mation Bureau (of Jewish origin, who under Trotsky's leadershipjoined the Bolsheviks in August 191 7, and was former Assistant Foreign Minis ter) , to the committee as its political commissar, and he also found a use for Molotov's wife, Polina Zhem chuzhina, as patroness of the com mittee. In February 1 948 Mikhoels met his death while visiting the city of Minsk; Stalin ordered his liquida tion, but the official version cause of his death was given as "automobile accident" [ 19, p. 530}. The first am bassador of the new state of Israel, Golda Meir, arrived in the USSR on September 3, 1 948. At a reception given by the Ministry of Foreign Af fairs, Meir was approached by Polina Zhemchuzhina, who addressed her in
70
THE MATHEMATICAL INTELLIGENCER
S.
Pontryagin (1 940).
Yiddish. Early in 1949 Stalin launched a mass'ive campaign against "home less cosmopolitans. " The great major ity of the unmasked "cosmopolitans " were Jews [19}. Lozovskii: and the other members of the JAG were ar rested and most of them would be shot in the summer of 1.952. The end of the JAG made possible the arrest ofPolina Zhemchuzhina. She told her life story under interrogation: "My name is Perl Semyonovna. Zhemchuzhina was my Party pseudonym " [19, p. 293}. She was exiled to the distant Kus tanai: oblast, where she was known as "Object Number 12. " In March 1949, the Soviet public was informed of a reshuffle in the highest state posi tions: Molotov surrendered the portfo lio of Foreign Affairs, Bulganin the Ministry of Defense, and Mikoyan the Ministry of Foreign Trade. These changes bespoke Stalin's suspicions of his closest lieutenants. In January 1953 a team of secret police brought "Object Number 12" to a Moscow jail from where she was taken to the Lubyanka for interrogation. She was the intermediary through whom Molo tov had been recruited in Stalin's last concocted thriller "as an enemy agent" [1 9, p. 552]. The story was that Zion ists had infiltrated even the highest levels of the political elite, that the sin ister Jewish organization ''Joint" was bent on destroying the Russian peo ple, that traitorous doctors (most of
them eminent Jewish doctors) were kUling statesmen. The Molotovs, hus band and wife, survived only because of Stalin 's death on March 5, 1953. In [49] , I. I. Piatetski-Shapiro recalls his experiences as a Jewish student of A 0. Gel'fond at Mekh-Mat. He gradu ated from the MSU in 1951 , but was de nied admission to enter the graduate
Figure 12. The grave of L.
S. Pontryagin in the
Novodevichii Convent Cemetery.
Figure 1 3. Grave of the Molotovs in Novode vichii Cemetery.
school of MSU, in spite of the MMS re ward for young mathematicians that he had won and in spite of Gel' fond's rec ommendation. Unfortunately for him, 1951 was a year of great anti-Semitism in Russia. He was accepted as a stu dent by the Moscow Pedagogical Insti tute in 1952 after he had obtained a "C" (satisfactory), the highest grade for a Jew. After he defended his Ph.D. the sis, Piatetski-Shapiro went to Kaluga for three years. He recalls the inspira tion that he obtained from I. R. Sha farevich's seminars, and that the latter was willing to read his paper at the ICM-62 in Stockholm, after he was re fused permission to go [49]. Piatetski Shapiro expressed his feelings later, when Shafarevich wrote the paper "Russophobia. " "It was very unpleasant to see a man I respected so much be come a leader of anti-Semitism." This paper, written in 1982, was circulated informally in Russia prior to its publi-
cation by a German publisher in 1989 and after that by a Russian publisher in 1991. Barbara Spector [50] reports on a letter signed by the President of the National Academy of Sciences of the USA, F. Press, and the Foreign Secre tary of the Academy, J. B. Wyngaarden, to the foreign Associate Member (since 1974) Shafarevich, then Head of the Algebra Section of the "Steklovka" in Moscow. This letter was sparked in part by what Press and Wyngaarden re fer to as "anti-Semitic writings" in a Russophobia", asking Shafarevich "to consider whether it is appropriate for you to maintain your membership," and it states further that "we are in formed that there are few, if any, Jew ish members of the Steklov Institute in Moscow, even though many of the out standing mathematicians of Russia are Jewish." The letter evoked divergent opinions from mathematicians and members of the NAS. Some felt that the move was hasty, but about 350 US mathematicians, including some Soviet emigres, signed an open letter to Sha farevich, decrying "the numerous anti Semitic sentiments" in "Russophobia" [52]. On the other hand, some of Sha farevich's former Jewish students (see [50] for some names) declared that he had helped to advance the careers of Jews in the years when roadblocks were routinely placed in their way by prominent Soviet mathematicians, such as Vinogradov; the latter was "proud that under his leadership the Institute has become free of Jews." NAS foreign associates Jean-Pierre Serre and Henri Cartan were the first to write to Press protesting the Academy's move, and later on Serge Lang also objected to the manner in which Press and Wyngaar den acted and to "the low standards of the NAS letter." On the other hand, Stephen Smale expressed support for the NAS letter: "What Shafarevich is doing is extremely destructive, and a big contribution to anti-Semitism in Russia. He legitimizes it. For NAS to speak out on that is great." In his re sponse to the article by Spector [50], Shafarevich in [ 5 1 ] denies that "Russo phobia" is anti-Semitic; he argues that for a long time he had not the slightest opportunity to influence decisions taken by the Scientific Council of the Steklov
Institute, and that the reactions of some of his former Jewish students are in sharp contrast with Wyngaarden's accusation that he (Shafarevich) used his position "to interfere with the ca reers of young Jewish mathemati cians. " He writes that in his paper he discusses the activities of certain Jew ish radical literary and political groups and trends (during the 1 9 1 7 Revolution as well as in recent times) and that there is a detailed explanation of his view that this does not concern the Jewish nation as a whole. However, Pi atetski-Shapiro [49] reads "Russopho bia" as calling the October Revolution a plot of Jews against the Russian peo ple. Spector [53] writes, based on in formation supplied by a researcher who traveled to Russia with NAS fund ing, about Shafarevich's ultra-national ist activities, his association with "right-wing, anti-Semitic publications" like Den and Nash Sovremennik, and that he was part of the same broad po litical current as Pamyat, the best known Russian extremist nationalist movement with an anti-Semitic ideol ogy. In [51 ] , Shafarevich challenges the authors of these statements to support them by a single fact or to refute them openly. The Council of the AMS at its annual winter meeting, held on Janu ary 12, 1993, passed a resolution [54] in which it expressed its condemnation of the anti-Semitic writings of Shafare vich, stating that he "has used his highly respected position as an emi nent mathematician to give special weight to his words of hatred, which are contrary to the spirit of mathemat ics and science." Semyon Reznik [55] writes that when time came for the So viet dissidents to define what they stood for, it became increasingly obvi ous that for some of them, like Sakharov, democracy was the only al ternative to the communist regime, while for others, like Shafarevich, democracy was an even greater evil than communism. The latter group dreamed about replacing a bad form of dictatorship with a "good" one based on an ideology of Russian chauvinism, which they preferred to define as "pa triotism." In his conversations with Smilka Zdravkovska [9] , Shafarevich defends his article by saying that many
VOLUME 25, NUMBER 2, 2003
71
very sharp judgments have appeared in journals with wide distribution, putting down Russian history, culture, and na tional character, and that if this ideol ogy be accepted without reserve, it will have a destructive influence on Russia. "In my paper I tried to formulate and discuss those objections, keeping at the level of logic and facts. " I n the years o f Stalin's crackdown on "cosmopolitans" almost no Jews were admitted to Mekh-Mat. Then the problem suddenly vanished [42 ] . It worsened after the Esenin-Volpin af fair; in 1970-1988 there would gener ally be only three to five Jews or half Jews among Mekh-Mat's class of 500. Jews and other unwanted applicants were kept out by giving them specially difficult problems at oral examinations [22] , [35 ] , [36]. A. Vershik [36] thinks the objective of holding down the number of tal ented people was at least as important, though never stated; cf. [22 ] . Interest ing data showing blatant discrimination against students with some Jewish par ent or grandparent can be seen in [35 ] . When the character o f the entrance ex aminations became known to school graduates, those suspected of being Jewish began to apply to other places, mostly to faculties of applied mathe matics where there was no discrimina tion, like, for example, the Institute for Petrochemical and Natural Gas Indus try (nickname: "Kerosinka"). In [56], one finds a short account of the expe riences of the sixteen-year-old Edward Frenkel (his father was Jewish, but not his mother) when he presented himself to the examiners of the Department of Mathematics at the MSU in 1984. He failed the entrance examinations be cause of the familiar story: students from Jewish backgrounds were asked questions significantly more difficult than other candidates. So, Frenkel en rolled at Kerosinka, studied mathemat ics-and also followed the unofficial evening classes at the MSU, offered by professors of the MSU for students who were not legally enrolled there. After graduating from Kerosinka he was ad mitted to Harvard, where he received his Ph.D. in 199 1 ; he became a full pro fessor at the University of California at Berkeley at the age of twenty-nine. In
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THE MATHEMATICAL INTELLIGENCER
[36], Vershik writes: "Is there anything surprising about the drain of Russian science, emigration, apathy, and the low prestige of official institutes and academics? All of this was predictable from what was done." Shen [35, p. 10] writes that since the beginning of per estroika one could openly and safely write about anti-Semitism. The discus sion continued inside the University, but died down gradually, because dis crimination in entrance examinations had ceased, and many of those involved had scattered all over the world. The "Union of Councils for Jews in the Former Soviet Union" (UCSJ) is a Jewish human-rights organization with monitoring bureaus in Russia and six countries of the former Soviet Union that has reported on anti-Semitism and other violations of human rights for 30 years [57]. The new report by UCSJ concludes that although President Vladimir Putin has made positive ges tures towards the Jewish community and has strongly condemned anti Semitism, "the situation in Russia must be closely monitored."
[6] The Russian Academy of Sciences ' 2 75th Anniversary. Historic short review.
Avail
able at URL http://www.turpion.com/mc/ ras-info.htm (accessed March 7, 2001 ). [7] S. S. Demidov. The Moscow School of the Theory of Functions in the
1 930s.
In:
Golden years of Moscow Mathematics. Editors: S. Zdravkovska and P. L. Duren. History of Mathematics, Volume 6. AMS, LMS, Providence: Rhode Island, 1 993, 35-53. [8] A. Shields. Egorov and Luzin: Part 2. The Mathematical lntelligencer 1 1 (2)
(1 989),
5-8. [9] S. Zdravkovska. Listening to Igor Ros tislavovich Shafarevich. The Mathematical lntelligencer 1 1 (2)
(1 989), 1 6-28.
[1 0] P. I. Kuznecov. DmitriT Fedorovich Egorov. Russian Math. Surveys
26(5) (1 9 7 1 ) , 1 25-
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Historia
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Acknowledgments
Ivan Vtotov took the photograph of the Novodevichi'i Convent on January 13, 200 1 . The author gratefully acknowl edges his permission to use it in this paper as Figure 1 . The author also expresses his gratitude to Professor Valentin Skvortsov of the Moscow State University for accompanying him to the Cemetery. The author took the photographs in figures 3, 5, 9, 12, and 13 in August 1992.
ematicians .
I n : Golden years of Moscow
Mathematics. Editors: S. Zdravkovska and P. L. Duren. History of Mathematics, Vol ume 6. AMS, LMS, Providence: Rhode Is land, 1 993, 1 -33. [1 4] R. H . McNeal. Stalin: Man and Ruler. New York University Press, New York, 1 990. [ 1 5] F. Palmeira and M. A. Shubin. Pavel Samuilovic Urysohn. The Mathematical ln telligencer 1 2(4)
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[1 6] J. J. O'Connor and E. F. Robertson. Pavel Samuilovich Urysohn.
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http : //www- h i st o r y . m c s . st - an d rews . ac. uk/history/Mathematicians/PetrovskiT.
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VOLUME 25, NUMBER 2, 2003
73
l$@ii•i§!.Shl¥11§.h§4fh,j,i§.id
Capita l ism Overturned The Solution M ichael Kleber
M i c hael K l e b e r and Ravi Vaki l ,
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view every character is right-side-up when encountered. Credit goes to Jessica Polito for IV, and to Claire Ruberman for pointing out that KICRBOXED is a second 9-letter word that reads the same upside-down. Thanks also to Dan Asimov, Thomas Colthurst, David Miller, and Kiran Kedlaya for com ments; all are doubtless annoyed at my ignoring some of their good advice. And r does too look like L upside-down.
apitalism Overturned was published in the Winter issue of The Mathe matical InteUigencer (Vol. 25, No. 1). The puzzle is embedded in a Mobius strip: answers extending past the right edge re-enter on the left after a half twist, affecting both their position and the orientation of their characters. Numbers in the grid indicate the posi tion and orientation of the first letter in each word, and from that point of
This column is a place for those bits of contagious mathematics that travel from person to person in the community, because they are so elegant, suprising, or appealing that one has an urge to pass them on. Contributions are most welcome.
E d itors
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Mathematical Entertainments Editor, Ravi Vakil, Stanford University,
Department of Mathematics, Bldg. 380, Stanford, CA 94305-21 25, USA e-mail:
[email protected]
74
THE MATHEMATICAL INTELLIGENCER © 2003 SPRINGER-VERLAG NEW YORK
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Prime Maze The Solution Dean Hickerson
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his problem was published in the Winter Issue of the The Mathe matical Intelligencer (Vol. 25, No. 1). It was originally posted to rec. puzzles in October 199 1 and is reprinted with the author's permission. The Problem A
number theorist set his car's odome ter to zero and then went for a drive along the roads shown below, starting and ending in his home town. He no ticed that his odometer reading was a prime number each time he entered a town. Where did he live and how jar did he drive? (Notes to avoid trick solutions:
Every town is at an intersection of roads and every such intersection has a town. Distances on the map are in miles and are exact. The odometer measures miles and is accurate. The driver didn't reset his odometer at any time after he started. He never turned around outside of a town, but he did sometimes leave a town by the road on which he arrived. He didn't add ex tra distance by driving within a town or raising his car off the ground and spinning his wheels or getting out of the car while someone else drove it. He didn't subtract distance by driving backward. He stayed on the roads shown here at all times. Etc.)
5
4 13
The Answer
The number theorist lived in city B. He traveled 523 miles, through the cities listed here: BJ S J I K Q L I K P T U P T U WVOVN E N O VN 0 V O G F O VOFENVWUTPUTPKILQKILMRQKPTHC D CADB
© 2003 SPRINGER-VERLAG NEW YORK, VOLUME 25, NUMBER 2 , 2003
75
Solving the Puzzle
It's not hard to write a computer program to solve the puz zle, but it can also be done by hand in an hour or so. Here are some things to consider: After the first move or two, the odd-length roads are un usable, and the graph is broken into 4 pieces: the center, consisting of H, K, P, T, U, and W, and 3 others, attached to the center at H, K, and W. The first or second move carries you from one of the 3 outer pieces to another one along a road of odd length. Somehow you have to get back to the piece you started in. Large gaps between con secutive primes impose con straints. In particular, the gap of 18 between 523 and 541 can't be crossed, so the to tal length is at most 523. Some searching shows that there's no solution of length 1 13 or less, so the path must cross the gap of 14 between 1 1 3 and 127, which can only be done on the road from T to U. Working backward from there shows that the path must start at B. Working forward from 127 gets you to V at 139. Some how you must get back to H. Working mod 3 shows that you can't get there without an intervening trip to K. And looking at the possible paths from V to K and K to H shows that you must cross from V at 281 to K at 349 and from K at 431 to H at 457. Filling in the rest of the path is easy.
length. I listed the possible trips through the center and de cided that the overall structure of the path would be this: Follow an odd-length road from the H-section to the K-sec tion and get to town K at 71. Go through the center, end ing at W at 137. Move within the W-section, returning to W at 283. Go through the center, reaching K at 349. Move within the K-section, returning to K at 431 . Go through the center, reaching H at 457, and move within the H-section to return to the start at 523. Next I worked on the W section. Between 137 and 283, there are 5 gaps of length 10 or 12, so I put in just one road of length 12 and all oth ers of length at most 8. After that, the W-section was pretty much forced. Building the K-section was troublesome. There had to be a path from K to K, from 349 to 431. But with almost any choice for the odd-length roads joining the K-section to the others, it would be possible to reach K at 79. Note that the primes after 79 and after 349 have similar structure for a while:
Several people asked me
how I created the puzzle, so
I wrote down what I could
remember of the process.
How the Puzzle Was Created
Several people asked me how I created the puzzle, so I wrote down what I could remember of the process: I started by listing primes and the gaps between them. I noticed the gaps of 14 from 1 13 to 127, 293 to 307, and 3 1 7 t o 331 , and the gap o f 18 from 523 to 541 . S o I decided to have the total length be 523 (or as close as I could get) and have just one road of length 14. (I later added another, but it's not usable for the gaps listed above.) I soon came up with the "center", consisting of H, K, P, T, U, and W. Then I decided to build 3 more pieces, attached to the center at H, K, and W, and joined to each other by roads of odd
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THE MATHEMATICAL INTELLIGENCER
79 83 89 97 349 353 359 367
101
103 107 109 1 13 127 373 379 383 389 397
The first time there's a prime above 349 without a corre sponding one above 79 is at 389. So any path starting at K at 349 leads to one starting at 79, giving many opportuni ties for a solution shorter than 523. (E.g., in the final map, you can follow the route SWUTPUTPKIL, with several short continuations after that; fortunately none take you back to S.) After quite a bit of experimentation, I came up with a usable K-section. The H-section was easy. Finally, I experimented with different odd-length con necting roads, to make sure that there were no unwanted short solutions. I worked mostly by hand, but used a computer to test various choices for the odd lengths and the lengths IJ, MJ, and RS. The main problem was to prevent solutions of length 1 13 or less.
lil§'h§'.{j
O s m o Pekon e n ,
Ed itor
I
Weighing the Odds: A Course in Probability and Statistics by David Williams CAMBRIDGE. CAMBRIDGE UNIVERSITY PRESS. 2001 560 pp. €46.69 ISBN 052 1 0061 8 X
=
REVIEWED BY MARC YOR
Feel like writing a reviewfor The Mathematical Intelligencer? You are welcome to submit an unsolicited review of a book of your choice; or, if you would welcome being assigned a book to review, please write us, telling us your expertise and your predilections.
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D
(the odds!) is modified as our partial knowledge of the random phenomena under study evolves. For example, as sume the two random variables of in terest to you are N1 and C = N2/N1, with N1 and N2 independent gaussian variables: thus C is a Cauchy variable which is not even integrable! But now assume you know N1 x * 0, that is, the first random phenomenon is "frozen"; then the second variable C be comes a gaussian ( N2/x) which has excellent integrability properties. This book is of the same character as the three earlier ones by Williams, that is: the author addresses under graduate students (roughly!)-indeed anyone who wishes to understand some probabilistic and statistical as pects of our daily lives-and this leads the author to many informal discus sions of =
avid Williams, an author well lrnown in the probabilists' commu nity, has written several books which have had an important influence from the moment they were published. This was true, first of all, of his book about Brownian motion and diffusions (Wi ley, 1979)-the first book on this sub ject of the "post-Ito-McKean" era, whose basic reference, Diffusion Processes and their Sample Paths (Springer, 1965) is fundamental but not easy to comprehend. Then, together with Chris Rogers, Williams wrote a two-volume treatise (Wiley, 1987, now republished by Cam bridge University Press), in which the authors develop the topics of the 1979 book, among many others, with the help of stochastic calculus and the Strasbourg general theory of stochas tic processes. The third book of D. Williams (Cam bridge University Press, 1991) takes the viewpoint that "most problems" in probability can be solved by (con struction and) use of suitable martin gales. In each book, the author engages the reader in dialogue and conveys his own enthusiasm. The emphasis is much more on the importance (and/or the beauty) of the results, their ap plications, and the power of the meth ods that yield them, than on rigorous step-by-step elaboration of definition, lemma, theorem, proof, corollary. Williams's new book, with the well conceived title Weighing the Odds, is an uninterrupted discussion between author and reader on how randomness
• • •
games of chance on television (p. 15); introduction to genetics (pp. 87-95); significance of cancer rates around certain nuclear plants (p. 239), a topic which has aroused some debate in Great Britain and in France. . . .
The book also contains some more formal "openings": go and look, the au thor tells us, at such-and-such a book on percolation (p. 439), stochastic geometry (p. 438), and so on. He makes us pause, with him, between the fre quentist and bayesian philosophies, the scale tilting to one side or the other (p. 3 1 1) depending on the question be ing studied. The author laments the wide gap which still exists between the Kolmogorov-type probability frame work and quantum probability. Of course, Williams's style of expo sition has its drawbacks. Often this text may be preferred by the reader "who lrnows already" (this is not surprising), while the serious beginner will have to look elsewhere for the missing E's and o's. But this is being too harsh. It would be fairer to say that Williams, rather than asking the reader to ponder the finest hypotheses of a theorem, directs attention to examples and counterex-
© 2003 SPRINGER-VERLAG NEW YORK. VOLUME 25. NUMBER 2 . 2003
77
amples (e.g., lack of uniform integra bility, p. 414), or to inconsistencies of notation which reveal some difficulties, as with conditional probabilities (pp. 258--260). Conditional probabilities are indeed the Ariadne thread of the whole book, as I hinted above with the Gauss Cauchy example. Let us now go through the overall structure of the book. The first five chapters, roughly 150 pages, are de voted to the fundamental notions of probability, which are systematically presented in correspondence with those of statistics. There follow three chapters, roughly 200 pages, on statis tics: confidence intervals, bayesian sta tistics, linear models (this topic takes up the whole of Chapter 8, 100 pages in itself). Then the author returns to probability in Chapter 9 on conditional expectations, martingales, and Poisson processes; and Chapter 10 is devoted to quantum probability. The book ends with four appendices, concerning a number of technical points (for example, the axiom of choice!) which were skipped over in the main text; the solutions of the principal exer cises (Appendix B); and an invitation to and overview of the literature (Appen dix D). The book contains a great num ber of exercises, updates (sometimes with Star-trek-type adventures), discus sion of some classical questions: the bus paradox (p. 434), the problem of mo ments, for which B. Simon's 2000 article is exploited, and so on. This book is often reminiscent of the classic Volume 1 by W. Feller; still, the discussion is much more directed at the reader, as I have said. It is re plete with pedagogical tricks, and con stantly is on the lookout for the sim plest solution. Often the author leads the reader to a wrong path, then re turns later to the good road; in Chap ter 10, the reader is initiated to quan tum probability in a series of steps, leading to a good understanding. Many notes of humour are scattered through the text; I enjoyed, for instance, the sobs notation (p. 1 70) supposed to bring sta tisticians to tears. In summary, while progressing through the book, the reader evolves from elementary questions of heads and tails toward a global vision of probabil-
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THE MATHEMATICAL INTELLIGENCER
ity and statistics, with the author vow ing (see, e.g., the first lines of the Pref ace) to reconcile these two domains, which have separated, then divorced. It is a goldmine of information in both do mains, giving access to the most ad vanced knowledge and techniques. I recommend it as a working tool for a specific class of French students just below the graduate level: those who are preparing for the "agn!gation de Math ematiques: option probabilites et sta tistiques" and who in the future will teach candidates for the Ecole Poly technique, Ecoles Normales, etc. (There are plenty of potential readers through out the world, but this reviewer can be
The author leads t h e reade r to a wron g path , then retu rns late r to the good road . most precise in describing a natural readership in France.) Bravo, David, for this new tour de force! Many readers will be looking for ward to your next volume, but this should not distract you from l 'art d'etre grand-pere, especially in the bicenten nial year of Victor Hugo. Laboratoire de Probabilites et Modeles Aleatoires Universite Paris VI 1 75, rue du Chevaleret 7501 3 Paris France e-mail:
[email protected]
From Trotsky to Godel: The Life of Jean van Heijenoort by Anita Burdman Feferman NATICK, MASS. : AK PETERS, 2001 PAPERBACK 4 1 6 pp, US $24.95 ISBN 1 -5688 1 - 1 48-9
REVIEWED BY JOHN W. DAWSON, Jr.
T
his is a paperback reissue of the bi ography of Jean van Heijenoort
originally published in 1993 under the title Politics, Logic, and Love. It is to be hoped that the new title will attract greater attention to this vivid portrait of the life of a remarkable and complex man. Within the mathematical commu nity van Heijenoort is best known as a historian of modem logic, whose From Frege to Godel: A Source Book in Math ematical Logic, 1 8 79-1931 (Harvard University Press, 1967) remains an ex emplar of what a source book should be. Some are also familiar with his edi tion of Jacques Herbrand's Ecrits Logiques (Presses Universitaires de France, Paris, 1968), but relatively few are aware of his later book With Trotsky in Exile: From Prinkipo to Coyoacan (Harvard, 1978), in which he recounted the seven years from 1932 to 1939 during which he lived with Leon Trotsky as his bodyguard, secretary, and translator. After those turbulent years van Hei jenoort entered upon a life of scholar ship. He left Trotsky's circle not long before the latter's assassination, ob tained a doctorate in mathematics in 1949 from New York University (for a dissertation in differential geometry), and stayed on as a professor there un til 1965, when he moved to Brandeis University to teach logic. In the 1950s he became an archival consultant for the Trotsky papers at Harvard, and dur ing the last four years of his life he held a half-time appointment at Stanford, where he served both as an archival assistant to the Hoover Institution (whose Trotsky holdings complement those at Harvard) and as a co-editor of Kurt Godel's Collected Works. Feferman's biography is based on extensive personal interviews she had with van Heijenoort during his years at Stanford. It is a sensitive account that opens with his childhood memory of his father coughing up blood. The year was 19 14: the Germans had occupied the small French town of Creil where young Jean had been born two years before, and in the wake of the German invasion all medical personnel had gone away. Van Heijenoort's father, a Dutch citizen, was not subject to French military service and had re mained in Creil with his wife and son,
where he worked as a decorative painter of metal safes. But he devel oped a gastric ulcer that began to he morrhage, and with no one with med ical knowledge to tum to, he became a casualty of the war. That searing event left van Heijenoort without any male ancestor to serve as a role model and caused a series of fur ther calamities. Without a provider in the family, van Heijenoort's mother Helene was forced to seek employment. Her family had lived in Creil for genera tions, but by marrying a foreigner she had lost her French citizenship and "was required to carry papers designating her as an alien" (p. 9). As a Dutch widow she was further ineligible for "almost every . . . job . . . except private domestic work" and had "to report to the police once a week, . . . [and] refrain from leaving . . . Creil [without) a special permit for each departure" (p. 16). When she eventually found work as a hotel chambermaid, Helene had to send her son to live with an aunt. The resulting separation was painful, and the only way she could escape from her predicament was to marry a French man. When she did so, her citizenship rights were fully restored, but Jean jealously resented the new man in her life and never accepted him as a step father. As Feferman notes, such possessive attachments were characteristic of all of van Heijenoort's relationships with women. As with his mother, he at first put all his lovers on a pedestal; but "when the honeymoon was over and the relationship came down to earth, there was always serious trouble" (p. 2 1). He was "the quintessential roman tic Frenchman," who in his affairs with women "forgot about logic and reason" and consequently suffered "pain, re morse, and eventually . . . [a] violent end" (p. xiii). (He was murdered in 1986 by his last wife-his fourth or fifth, "depending," he said, "on how you count.") For van Heijenoort, "almost every thing was an all or nothing affair" (p. xiii). As a youth he turned abruptly away from an early devotion to religion and "reapplied his zealousness to his studies" (p. 30). He was a brilliant stu dent, first in Creil, then at the college
in Clermont and finally at the lycee Saint-Louis in Paris. But he resented the regimentation of the French school system, and, disillusioned with the world political situation, became ac tive in a secret Marxist society. There his linguistic abilities impressed his companions and he was recruited to join Trotsky as a translator. Accord ingly, with little hesitation, he forsook his scholastic achievements and prospects and became a revolutionary. Van Heijenoort followed Trotsky from Prinkipo, an island in the Sea of Marmora, through France, Norway, and finally to Coyoacan in Mexico. But intrigues within Trotsky's circle, in-
Recru i te d to joi n Trotsk y as a translato r , he forsook h i s scholast i c p rospects and became a revolut iona r y . eluding a clandestine affair between van Heijenoort and Frida Kahlo, the wife of Diego Rivera, eventually led to conflicts. Van Heijenoort left Coyoacan in November of 1939, less than a year before Trotsky's murder (an event van Heijenoort believed he could have thwarted, had he heard the speech pat terns of the Spanish assassin, who claimed to be a Belgian). At the mo ment he learned of the assassination, van Heijenoort felt that "darkness set in." He remained active in the Trot skyite cause for seven more years, but eventually his growing awareness of the evils of Stalinism led him to become disillusioned with Marxist-Leninist ide ology. Part III of Feferman's biography is devoted to van Heijenoort's return to the world of scholarship. As a non mathematician, she focuses more on his personal life and relationships than on his mathematical work. But an appen-
dix at the end of the volume, written by her husband, the distinguished logician Solomon Feferman, provides an over view of van Heijenoort's contributions to mathematics and philosophy. The last two chapters of the book chronicle van Heijenoort's on-again, off-again relationship with Ana Maria Zamora, the daughter of Adolfo Zamora, who had been a comrade of van Hei jenoort's during his days with Trotsky and had served as Trotsky's and Rivera's laWYer. Van, as he was by then known among his mathematical colleagues, became reacquainted with Ana Maria (whom he had first known as a child) in 1958, while assisting the Harvard li brary in its negotiations to acquire Trotsky's papers. The two quickly fell in love, and a tempestuous relationship ensued. They were married, divorced, and eventually remarried, but could not live happily either together or apart. Eventually Ana Maria began to show signs of serious mental distur bance, culminating in suicidal and homicidal depression. Believing he could somehow control her impulses, van Heijenoort returned to be with her in her home in Mexico City; and there, while he slept, she shot first him and then herself. At the time of his death, I was work ing with Van on the Godel Project. He was a vital contributor to that effort, a friendly collaborator, and a meticulous scholar. But as Feferman stresses throughout her book, he was also a very private person whose life was carefully compartmentalized. I knew nothing of his private life and was stunned by his murder. Only posthu mously, through this book, did I be come aware of how complex his per sonality really was. Van Heijenoort is buried in the Zamora family crypt in Mexico City, but there is no outward indication of the fact. Should an epitaph ever be in scribed for him, it ought to be one of Pascal's Pensees that he particularly appreciated: "The heart has its reasons reason knows nothing about. "
Department of Mathematics Pennsylvania State University York, PA 1 7 403-3398 USA e-mail:
[email protected]
VOLUME 25, NUMBER 2 , 2003
79
k1£1,J.IQ•IQ•t41
Robin Wilson
The Phi lamath' s A l phabet-A A
bacus: The abacus has appeared in various forms around the world, originally as a sand tray containing pebbles, and is still in use in the Far East for everyday calculations. The Chinese version consists of a frame with beads, called a suan pan. This stamp was issued by Liberia as part of a millennium set celebrating Chinese science. Abel: Niels Henrik Abel (1802-1826) was the first to prove that no general formula can exist for solving the gen eral polynomial equation of degree five or more. He also contributed to the study of elliptic functions, and several
I
mathematical concepts (Abelian group, Abelian integral, etc.) are named after him. This Abel statue by the sculptor Gustav Vigeland was erected in Oslo in 1908. Al-Khwarizmi: Muhammad ibn Musa al-Khwarizmi (c. 750-850) lived in Baghdad and wrote influential works on arithmetic and algebra. His Arith metic introduced the Hindu-Arabic place-value system to the Islamic world, while his Kitab al-jabr wal muqabala gave us the word algebra: al-jabr involves adding a positive quan tity to each side of an equation to elim inate a negative one. Archimedes: Archimedes (c. 287-2 12 BCE) lived in Syracuse, in Sicily. Among his many achievements he calculated the surface areas and volumes of spheres and cylinders, proved that 7T lies between 310/71 and 31h investi gated the "Archimedean spiral," and listed all thirteen semi-regular polyhe-
3 50
Abel
AJ-Kh
Please send all submissions to the Stamp Corner Editor, Robin Wilson, Faculty of Mathematics,
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Aahoka column
THE MATHEMATICAL INTELLIGENCER © 2003 SPRINGER-VERLAG NEW YORK
rizml
dra. This Greek stamp features the Archimedes principle, which he used to test the purity of a gold crown, and his law of moments. Ashoka column: Around 250 BCE King Ashoka of India became the first Bud dhist monarch. His conversion was cel ebrated by the construction of many pillars, some containing the earliest known appearances of what became our Hindu-Arabic numerals. This Nepalese stamp shows the Ashoka col umn in Lumbini, Buddha's birthplace. Astrolabe: The astrolabe can be traced back to Greek times, but reached its maturity during the Islamic period. It consists of a brass disc suspended by a ring, and has a circular scale on its rim and an attached rotating bar: to measure the altitude of a planet or star, the observer looks along the bar at the object and reads the altitude from the scale. Featured here is al-Zarqali, Cor doba maker of astrolabes.
